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Geometry, Frustration, and Exotic Order in Magnetic Systems Kumar S. Raman A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics November, 2005 c Copyright 2005 by Kumar S. Raman. All rights reserved. Abstract This thesis considers two topics in magnetism, the ﬁrst involving classical spins and the second quantum spins. A theme running through this work is how geometric constraints and frustration can substantially inﬂuence the qualitative physics. The ﬁrst topic[1] is the magnetization process of spin ice. Spin ice in a magnetic ﬁeld in the [111] crystallographic direction displays two magnetization plateaux, one at saturation and an intermediate one with ﬁnite entropy. We study the crossovers between the diﬀerent regimes from the viewpoint of (entropically) interacting defects. We develop an analytical theory for the nearest-neighbor spin ice model, which covers most of the magnetization curve. We ﬁnd that the entropy is non-monotonic, exhibiting a giant spike between the two plateaux. The intermediate plateau and crossover region are described by a two-dimensional monomer-dimer model with tunable fugacities. At low ﬁelds, we develop mean-ﬁeld and renormalization group treatments for the extended “string” defects which restore threedimensionality. The second topic[2] is the construction of a family of rotationally invariant, local, S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Speciﬁcally, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of a reference lattice. This strategy leads to Hamiltonians that exhibit i) Z2 RVB phases in two dimensions, ii) U (1) RVB phases with a gapless “photon” in three dimensions, and iii) a Cantor deconﬁned iii region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2 RVB phase and the other a U (1) RVB phase. This construction provides a proof of principle that topological phases can be realized in a local, SU(2)-invariant spin model. iv Acknowledgements This thesis was written under the guidance of Prof. Shivaji L. Sondhi. I began working with Shivaji late in my third year after switching to condensed matter theory from a diﬀerent ﬁeld. That I am still graduating on time at the end of my ﬁfth year and heading to a nice postdoc (Urbana) testiﬁes to his great skill as an advisor. The distinction between “letting one starve”, “catching one a ﬁsh” and “teaching one how to ﬁsh” is sometimes rather subtle. Shivaji kept me on track with plenty of insight and help but also gave me the freedom to work so I was never deprived of the conﬁdence which comes from being able to solve a problem “by myself”. I acknowledge him for this and also thank him for sharing his broad vision of condensed matter theory with me. Prof. Roderich Moessner (ENS, Paris) was a co-advisor on both of the topics presented here. Nearly every aspect of this work has beneﬁted from Roderich’s careful analysis of technical details which range from improving the noise of Monte Carlo simulations to understanding the workings of an RG calculation. The work on spin ice was done in collaboration with Dr. Sergei Isakov (Toronto). While this thesis emphasizes my contribution to that eﬀort, some of Sergei’s results are also presented and acknowledged in the text. The work on RVB phases includes a discussion of a model on the pyrochlore lattice invented by Prof. Steve Kivelson. I would also like to acknowledge Dr. Matt Hastings (LANL) for a highly stimulating discussion which eventually led us to invent the decoration procedure. I thank Prof. David Huse for reading the thesis and for his support during my time at Princeton. I have beneﬁted from interacting with members of the condensed matter physics group. Especially helpful were the excellent courses taught by Shivaji, and Profs. Boris Altshuler, v Ravin Bhatt, Paul Chaikin, Duncan Haldane, and Elliott Lieb. I have enjoyed interacting with Dr. Vadim Oganesyan (Yale) and am currently working with him on a possible extension of the spin ice RG (discussed in the text) to the problem of layered superconductors. During my stay, I did an advanced project in the mathematical physics group with Prof. Lieb on the problem of bosonic jellium. I thank him and also Prof. Robert Seiringer for useful conversations on this topic. I conducted an experimental project in the group of Prof. William Happer on the depolarization of polarized xenon gas. I would like to thank him and members (some former) of his group: Warren Griﬃth, Yuan-yu Jau, Peter Ouyang, Brian Patton, Dan Walter, and especially Nick Kuzma. I am grateful to the physics department for providing me with a teaching assistantship for each semester of my stay and I have beneﬁted from interactions with many students, faculty, and staﬀ. In this regard, I would especially like to acknowledge Profs. David Huse, Peter Meyers, Lyman Page, and Stew Smith, and Dr. Steve Smith. I am also grateful for ﬁnancial support which I received during my ﬁnal year from the McGraw Center for Teaching through its AI liaison program. I would like to thank Pat Barwick, Martin Kicinski, and Laurel Lerner, for helping me with various administrative tasks through the years. I am fortunate that during the past ﬁve years, I have had the support of many colleagues who are also friends. In this regard, I would like to acknowledge Touﬁc Suidan, Sasha Baitine, Chris Beasley, Latham Boyle, Shoibal Chakravarty, Pedro Goldbaum, Karol Gregor, Kevin Huﬀenburger Subroto Mukerjee, David Olson, Vassilios Papathanakos, Srinivas Raghu, and Emil Yuzbashan. I would like to also collectively acknowledge a large number of friends outside of the Princeton physics department. Finally, I turn to my family. Padma, Josh, Ravi, Jaya, and many other relatives have helped me manage the emotional aspects of the graduate school process. The opportunity for me to pursue a career in physics may never have arisen were it not for personal sacriﬁces made by the older generation of my family long before I was born, particularly my uncles G. Natrajan and G. Balachandran on my father’s side and my grandfather, P. V. Chandra, vi on my mother’s side. However, above all I acknowledge my parents G. S. Raman and Gita S. Raman. Their contribution to this work is the kind of debt which can not be quantiﬁed let alone repaid. I close by dedicating this thesis to the two of them. vii Contents Abstract iii Acknowledgements v Contents viii 1 Introduction 1 2 The magnetization process of spin ice in a [111] magnetic field. 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model and notation 2.3 The two [111] magnetization plateaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11 . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Low ﬁeld termination: string defects . . . . . . . . . . . . . . . . . . 14 2.3.2 High ﬁeld termination: monomer defects . . . . . . . . . . . . . . . . 16 2.3.3 Interaction of defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The low ﬁeld regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Mean ﬁeld calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 RG calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.3 Comparison with simulation . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 The high ﬁeld regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Crossing points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Relation to experiment, other theories, and applications . . . . . . . . . . . 27 2.4 viii 2.7.1 2.8 Cooling by adiabatic (de)magnetization . . . . . . . . . . . . . . . . 28 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 SU(2) Invariant spin 1/2 Hamiltonians with RVB and other valence bond phases. 38 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Quantum dimer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Honeycomb lattice: Bipartite physics in d = 2 . . . . . . . . . . . . . . . . . 44 3.3.1 Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.3 Decoration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.4 Square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Other Valence Bond Phases in d = 2 and d = 3 . . . . . . . . . . . . . . . . 54 3.4.1 Non-bipartite lattices in d = 2 . . . . . . . . . . . . . . . . . . . . . 54 3.4.2 Non-bipartite lattices in d = 3 . . . . . . . . . . . . . . . . . . . . . 56 3.4.3 Bipartite lattices in d = 3 . . . . . . . . . . . . . . . . . . . . . . . . 56 Dynamical selection of gauge structures: pyrochlore lattice . . . . . . . . . . 57 3.5.1 The Klein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.2 The Kivelson-Klein model . . . . . . . . . . . . . . . . . . . . . . . . 58 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 3.5 3.6 A An overview of height representation theory 70 A.1 The height representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.2 Application to spin ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 A.2.1 Classical dimers on the honeycomb lattice . . . . . . . . . . . . . . . 71 A.2.2 Interaction between defects in spin ice . . . . . . . . . . . . . . . . . 75 A.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.3 Application to quantum dimer models . . . . . . . . . . . . . . . . . . . . . 77 A.3.1 Quantum dimers on bipartite lattices . . . . . . . . . . . . . . . . . . 77 ix A.3.2 Interaction of defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.3.3 Stability of the RK point . . . . . . . . . . . . . . . . . . . . . . . . 78 B Mean field theory for string defects 82 B.1 Mean ﬁeld calculation of the system response . . . . . . . . . . . . . . . . . 82 B.2 Correlation lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 C Renormalization group treatment of string defects 87 D Sign conventions in the overlap matrix 91 D.1 Overlaps in the fermionic convention . . . . . . . . . . . . . . . . . . . . . . 92 D.2 Honeycomb and diamond lattices . . . . . . . . . . . . . . . . . . . . . . . . 92 D.3 Other bipartite lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 D.4 Kivelson-Klein model on pyrochlore lattice . . . . . . . . . . . . . . . . . . . 93 E Spinon gap for the decorated honeycomb lattice 95 F Classical dimers on the pentagonal lattice 104 References 107 x Chapter 1 Introduction In conventional textbook examples of interacting many body systems, the qualitative physics, such as the phases the system can exhibit, may be obtained from very general features about the system geometry (for example, whether it is periodic), interaction (for example, whether it is short-ranged or long-ranged) and symmetries. In contrast, the central theme of this thesis is the inﬂuence of frustration, arising from microscopic details of the interplay of interactions and geometric constraints, on the macroscopic physics. The two topics considered in this thesis are model spin systems, one involving classical spins and other quantum spins, where frustration gives rise to exotic phase diagrams not easily described in the usual framework of local order parameters and symmetries. A canonical example of frustration and its consequences is the large ground state degeneracy of the classical Ising antiferromagnet on the triangular lattice. The Ising interaction prefers neighboring spins to be oppositely aligned. If we consider a triangular plaquette and anti-align two of the spins, then the third spin is frustrated in that whichever way it points, it is unable to simultaneously satisfy all of its interactions. In contrast, the same interaction on a square lattice can be fully satisﬁed at every site via the Neel conﬁguration. This comparison is shown in Fig. 1.1. In the triangular case, any spin conﬁguration where every triangle has at least one up spin and one down spin is a ground state. The ground state manifold is highly degenerate, the number of states increasing exponentially with system 1 2 size, while the same interaction on the square lattice has only two ground states. ? Figure 1.1: Classical Ising spins with nearest-neighbor antiferromagnetic interaction. On the square lattice, the interaction is optimally satisﬁed by the Neel conﬁguration drawn above. In contrast, spins on the triangular lattice are frustrated in that the third spin is unable to simultaneously satisfy its interaction with its up and down neighbors. This scenario does not arise on the square lattice because in that geometry two neighboring spins do not have a common neighbor. The system can move in this highly degenerate manifold by local spin ﬂips and at low, but nonzero, temperatures, the system will be described by ensemble averages over this manifold. In contrast, a macroscopic perturbation is required to move between the two ground states in the square lattice case. Related to this is the fact that the state of the square lattice Ising antiferromagnet can be described by a local order parameter, for example the magnetization at a given site. Such an order parameter will be zero, upon ensemble averaging, in the triangular case but the ground state is not “disordered” in the sense of a paramagnet, though it shares the same macroscopic symmetries. In the paramagnetic case, interactions are negligible compared to thermal ﬂuctuations and each spin is essentially independent of the others. In the ground state of the triangular antiferromagnet, interactions are strong and ﬂipping a spin will generally require ﬂipping neighboring spins in order to maintain the “one up and one down per triangle” constraint. Recent studies [26, 25, 50, 22], building on the work of Blote et. al. [24], have made important progress in characterizing the order within such “disordered” systems using height representation theory. One feature of the height representation is that excitations of the system appear as vortices in a height ﬁeld which is convenient for analytical treatments. 3 The topic of Chapter 2 is spin ice, where a geometrically constrained ferromagnetic interaction gives rise to frustration. As will be discussed, experimental signatures of the frustration include the retention of entropy at very low temperatures (when naively we expect it should tend to zero); the failure to develop long range magnetic order despite a ferromagnetic Curie constant; and the appearance of two plateaux in the magnetization when a ﬁeld is applied along a particular crystallographic direction. The height representation will be used to characterize the lower plateau and to analyize the string-like excitations which cause its low-ﬁeld termination. An exponentially degenerate ground state implies a ﬁnite entropy (per spin) at zero temperature. Assuming the third law of thermodynamics is correct, behavior such as that described below can not literally occur in a physical system. However, frustration can give rise to a large number of low lying states very close in energy. When viewed at energy scales (i.e. temperatures) much larger than the characteristic level spacing, the behavior is eﬀectively an ensemble average over all of these states. In spin ice, the bandwidth of these states is believed to be much smaller than experimentally relevant temperatures so that while the physical system probably has a true ground state, it is dynamically irrelevant. In the triangular antiferromagnet example and also spin ice, the apparent lack of an order parameter is due to the ensemble averaging which occurs at temperatures of interest. However, frustration can also inﬂuence the zero temperature characteristics as in the topic presented in Chapter 3 of the thesis. There we construct SU(2) invariant spin systems that realize the phase diagrams of quantum dimer models. The construction involves perturbing a class of models called Klein models. These models are antiferromagnetic in nature but also include additional terms which frustrate the system into forming singlets between neighboring spins. The phase diagrams of these models diﬀer substantially if the lattice is bipartite or non-bipartite. In the case of the non-bipartite triangular lattice, the ground state phase diagram features an RVB (resonating valence bond) spin liquid phase. A valence bond state is a wavefunction where each spin forms a singlet pair with one of its nearest neighbors. An RVB state is a superposition over all singlet conﬁgurations connected by 4 local resonance moves. Spin liquids are characterized by rapidly decaying correlations, translational and rotational invariance, and the lack of a local order parameter. However, as in the classical case discussed above, the state is not “disordered” in the paramagnetic sense either. It turns out that while spin liquids do not have a local order parameter, they do possess a global type of order called topological order. Fig. 1.2 explains this notion in more detail. A central feature of topological phases is that they admit fractionalized excitations. In the RVB example, a natural excitation is the spinon, which is a spin 1/2 excitation formed by breaking a singlet (valence bond). The name “fractionalized” arises because when a ground state is described by spin 0 objects, the naively expected spin excitations will have integer spins but in this example, valence bonds (spin 0 objects) admit excitations with half integer spins (spinons). Currently, there is no deﬁnitive experimental realization of spin liquid physics though, as will be discussed, these ideas form the basis of theoretical descriptions of a variety of phenomena in correlated electron systems including high Tc superconductivity. [41] The notions discussed in this brief overview will be made more precise in the respective chapters of this work. The purpose of this introduction was to highlight the common thread connecting the two rather diﬀerent topics discussed in this thesis, namely how frustration can give rise to exotic behavior not easily categorized by conventional paradigms. 5 C1 C1 Figure 1.2: A valence bond state is a state where each spin forms a singlet pair with one of its neighbors. The RVB spin liquid is a quantum superposition of all valence bond states connected by local resonance moves. The above example shows a valence bond covering for a part of a triangular lattice and the dotted lines depict the simplest local move. C1 is a line extending through the system and we see that initially there are three bonds crossing the line and after the ﬂip, there is one. By inspection, we see that if the number of bonds crossing the line is odd (or even) then this property will not be aﬀected by local resonance moves. We could also have drawn a horizontal line and the torus depicts the fact that there are four distinct topological sectors (the number of bonds crossing a horizontal/vertical line may be odd/odd, odd/even, even/odd, or odd/odd). The RVB spin liquid state is a superposition of all valence bond coverings in a given topological sector so the state may be labelled by its winding number. This global property is called topological order. Chapter 2 The magnetization process of spin ice in a [111] magnetic field. 2.1 Introduction The name “spin ice” refers to a class of magnetic compounds that may be described by spins on a lattice obeying a local “ice-rule” constraint. Speciﬁc examples of spin ice compounds we will be interested in are Ho2 Ti2 O7 and Dy2 Ti2 O7 . For a review on spin ice, see Ref. [3]. The dynamical objects in models of these compounds are the large spins of the rare-earth ions (e.g. JHo = 8, JDy = 15/2) which reside on the sites of a pyrochlore lattice, shown in Fig. 2.1. As the ﬁgure shows, each pyrochlore site is a corner shared by two tetrahedra. Fig. 2.2 shows a single tetrahedron inscribed in a cube with some important crystallographic directions labelled. An important eﬀect of the neighboring Ti and O atoms is to cause a crystal ﬁeld anisotropy which strongly favors maximizing the component of the Ho/Dy spin pointing along its “easy-axis” which is the local [111] direction. As Figs. 2.1 and 2.2 show, this axis is the line joining the centers of the two tetrahedra sharing the corner where the spin resides. In this work, we take the anisotropy energy to be inﬁnite so that with respect to one of its tetrahedra, the spin either points “in” towards the center of the tetrahedron or 6 7 AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAA AA AAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAA AAAA AA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA 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The conﬁguration space becomes more constrained when the interaction between spins is considered. As discussed below, the net interaction may be modelled by a ferromagnetic coupling between nearest neighbor spins. With reference to a single tetrahedron, this implies that if a particular spin points “in”, then its interaction is optimized if the other spins of the tetrahedron point “out”. Of course, this is also true for the other spins so that the ferromagnetic nearest-neighbor coupling, combined with the pyrochlore geometry and easy-axis constraint, gives rise to frustration. An optimal conﬁguration for a given tetrahedron is one where two of its spins point inwards and two outwards; for each tetrahedron, there are six such conﬁgurations. An optimal conﬁguration for the whole system is one where the spins obey what is called an “ice-rule”: on every tetrahedron, two spins point in and two out. As is commonly the case in frustrated systems, the number of such optimal conﬁgurations grows exponentially with system size. 8 Figure 2.2: A single tetrahedron inscribed in a cube. The edges of the cube denote the [100] direction and the edges of the tetrahedra are the [110] direction. The [111] directions of the pyrochlore lattice are the body diagonals of the cube, denoted by d̂κ , and indicated by the dashed lines. The idea of geometric frustration being the source of a large “ground state” degeneracy was ﬁrst proposed by Pauling in the context of cubic water ice Ih .[5] The mystery of water ice was that integrating low temperature measurements of its speciﬁc heat suggests that the entropy approaches a (non-zero) constant at zero temperature, in apparent violation of the third law of thermodynamics. An important step in the resolution of this mystery was the formulation by Bernal and Fowler[4] of the ice rules, based on measurements of the crystallographic structure and bond lengths. Referring to the pyrochlore structure discussed above, in water ice the oxygen atoms lie at the centers of the tetrahedra and are surrounded by four hydrogens which lie on the corners of the tetrahedra, each H being “shared” by two O’s. Bernal and Fowler proposed that a hydrogen would not be exactly on a corner but closer to one of the two oxygens sharing it and the optimal conﬁgurations would be where each central oxygen has two hydrogens close to it and two farther away. The resulting degenerate manifold is exactly the ice-rule manifold discussed above for spin ice, where the 4 spins on a tetrahedron play the role of the 4 O-H bond lengths. Pauling[5] showed that the degeneracy gives rise to a macroscopic ground state entropy which agreed well with 9 experiments. Pauling proposed that while the true ground state of ice may be unique, thus satisfying the third law, the frustration gives rise to a large number of closely spaced low lying states. Typical experimental temperatures are much larger than the bandwidth of these states so that the system is eﬀectively described by ensemble averaging over these states. Thus, the true ground state is dynamically irrelevant and the system displays a “zero-point entropy”. Anderson [6] suggested that Pauling’s proposal may have a magnetic analogy and theoretically predicted spin ice roughly forty years before its ﬁrst experimental realization. The discovery[9] was made when it was observed that the pyrochlore compound Dy2 Ti2 O7 did not magnetically order at low temperatures even though measurements of its dc susceptibility suggested a ferromagnetic Curie-Weiss constant. It was suggested that the lack of local order was due to the system ﬂuctuating between degenerate ice-rule states and the hypothesis was conﬁrmed when its measured zero-point entropy was in agreement with Pauling’s prediction for water ice. Spin ice is easier to work with (experimentally) than water ice and may be used to explore various properties of (cubic) water ice. Spin ice is susceptible to a magnetic ﬁeld which allows for the exploration of features without previously observed water ice analogues[7]. The highly anisotropic magnetic response of a single spin ice crystal to magnetic ﬁelds along diﬀerent crystallographic directions was ﬁrst investigated numerically[14] and as unicrystalline samples became available, experimentally[10, 11, 12, 13]. However, vestiges of some of these anisotropic features, such as a sharp spike in the speciﬁc heat for ﬁelds in the [100] direction, were possibly seen earlier in orientationally averaged powder samples.[9] The main subject of the present work is the magnetization process of a unicrystalline sample of spin ice that is placed in an external magnetic ﬁeld in the [111] direction. A sketch of the exotic thermodynamic properties of spin ice in a [111] ﬁeld is given in Fig. 2.3. The striking features are the two magnetization plateaus and the entropy peak which occurs in the vicinity of their crossover. The higher plateau corresponds to saturation: the magnetization fraction of 0.5 is the highest value consistent with the easy axis constraint. The lower 10 plateau, ﬁrst predicted theoretically[14] and explored in Monte Carlo simulations,[14, 16] has been particularly remarkable as it was found to retain a fraction of the zero-ﬁeld spin ice entropy.[11, 21, 22] Vestiges of a [111] entropy peak were seen in experiment prior to the present study. [11] Magnetocaloric measurements of the entropy[15], published after the present work[1] show a clear entropy peak, though somewhat smaller in magnitude than our prediction. The spin-ice results presented in this thesis builds on work initiated in Ref. [22] where the thermodynamics and correlations of the [111] plateau were systematically explored. Diﬀerent regimes of the magnetization curves were identiﬁed, including the mechanisms which terminate the lower plateau at its high and low ﬁeld ends. On the plateau itself, the system is well described by a two-dimensional antiferromagnetic Ising model on a kagome lattice in a longitudinal ﬁeld, which is in turn equivalent to a hexagonal lattice dimer model.[21, 22, 23]. At the high-ﬁeld end, the crossover between the plateaus occurs via the proliferation of monomer defects in the underlying dimer model. At low ﬁelds, a more exotic extended string defect restores three dimensionality. The asymptotic densities of both kinds of defects were estimated in Ref. [22]. The various regimes are indicated in Fig. 2.3. In Section 2.2, we develop a theoretical model for spin-ice in a [111] ﬁeld. In Section 2.3, we discuss the physics of the [111] plateau and introduce the mechanisms which terminate it at low and high ﬁelds. In Section 2.4, we consider the low ﬁeld end of the plateau in detail. We develop mean ﬁeld and renormalization group treatments for the extended string defects, which we use to analyze the in-plane and out-of-plane correlations. We compare these with Monte Carlo simulations by Isakov[34], who used an eﬃcient cluster algorithm to obtain accurate data from the zero ﬁeld to the beginning of the [111] plateau. We ﬁnd that the mean ﬁeld treatment is accurate at the lowest ﬁelds, where the string density would be relatively high. The renormalization group treatment compares well with simulation in the dilute string limit. At even higher ﬁelds, the plateau is approached and the suppression of the entropic activation of strings becomes apparent as a ﬁnite-size eﬀect. 11 In Section 2.5, we study the high ﬁeld end of the plateau. We present the results of Monte Carlo simulations of a two-dimensional Ising antiferromagnet on a kagome lattice, which we compare with an analytical treatment by Isakov[34]. We observe a giant peak in the entropy, which even exceeds the zero ﬁeld Pauling value, despite the fact that a quarter of all spins are pinned. We show that the entropy peak is due to the crossing of an extensive number of energy levels which have macroscopic entropies. We conclude by comparing with experiment and discussing some potential practical implications of the work. 2.2 Model and notation A starting point for a theoretical description of spin ice is the following Hamiltonian, where the spins are treated classically: H = Jij Si · Sj + D (i,j) − E d̂κ(i) · Si i 2 3(Si · rij )(Sj · rij ) − Si · Sj (i,j) +µ 5 rij B · Si , (2.1) i The sums on (i, j) are over all pairs of spins; the spins {Si } are unit-length. The ﬁrst term is an exchange interaction where the constants Jij are expected to decay rapidly with the distance between sites. The second term is a dipolar interaction of strength D, where rij is the vector distance between sites and rij = |rij |. The constant D = µ0 2 4π µ , where µ is the magnetic moment (for Dy, µ ≈ 10µB ). The third term is the easy-axis anisotropy which favors the spin pointing along the local [111] direction, denoted by d̂κ(i) . The energy scale of the anisotropy is given by the constant E > 0,|E| ∼ 50K, which is much larger than typical exchange and dipolar interaction strengths. We take E to be inﬁnite, which turns the anisotropy into a strict constraint. The fourth term is the inﬂuence of a magnetic ﬁeld. Note that the only free parameters in this theory are the exchange constants Jij . Siddharthan et. al. [16] performed Monte Carlo simulations of Eq. 2.1 assuming a purely dipolar interaction, truncated at the sixth neighbor piece, and found good agreement with experiment when the antiferromagnetic exchange interaction was accounted for by reducing 12 the value of D by 25% from the above value. The nearest neighbor piece of the dipolar interaction favors alignment of neighboring spins which, when combined with the easy-axis constraint, gives rise to the ice rule behavior discussed above and seen in experiments. The ice rule energy scale is somewhat reduced by the exchange interaction. It is not obvious why a long range dipolar interaction, which is what was simulated in [16], should give behavior consistent with keeping only the nearest-neighbor piece. That the local Ising axis varies from site to site appears to cause a subtle cancellation of the longer ranged dipolar terms. Siddharthan et. al. [16] truncated the dipolar interaction at the sixth neighbor piece, after observing that the remaining terms were small corrections. Gingras et. al. [17, 18] handled the dipolar lattice sum using an Ewald summation method which permits one to treat the interaction to inﬁnite range. They found the surprising feature that the inﬁnite distance result is closer to the nearest neighbor result than an interaction truncated at, for example, 10 nearest neighbors. The analytical explanation for this remarkable cancellation was provided recently by Isakov et. al. [19, 20], who demonstrated that the long-range dipolar interaction could be replaced with a “model” interaction that has the same ground state manifold as the nearest-neighbor piece but diﬀers from the physical interaction by small terms which decay faster than the 1/r 3 dipolar decay. This recent work provides a theoretical justiﬁcation for our starting point of modelling the eﬀective spin-spin interaction as a ferromagnetic coupling between nearest-neighbors. A subtlety of making this approximation is that the correspondence of the (model) dipolar ground state manifold with the nearest-neighbor ground state (ice rule) manifold does not extend to the excited states of the two Hamiltonians. In particular, if an external ﬁeld is strong enough to take the physical system out of its zero-ﬁeld ground state manifold, then the new state need not correspond to what is obtained by similarly perturbing the nearest-neighbor Hamiltonian. We will return to this point in Section 2.7 but for now, we invoke the nearest-neighbor approximation which, it turns out, can explain many of the experimental features. Because we take the easy-axis anisotropy as inﬁnite, it is convenient to describe the 13 system by the Ising pseudospins σi , where Si = σi d̂κ(i) . The pseudospin σi =+1(-1) if the physical spin points into (out of) its associated up-pointing tetrahedron (“up-pointing” and “down-pointing” are with respect to the [111] direction indicated in Fig. 2.4). We may write an eﬀective Hamiltonian for the pseudospins: H = Jeﬀ <ij> σi σj − gµB J B · d̂κ(i) σi , (2.2) i where Jeﬀ ∼ 1K[11]. Note that the ferromagnetic spin-spin interaction gives rise to an antiferromagnetic interaction between pseudospins. The ice rule constraint says that the pseudospins on each tetrahedron must sum to zero, | κ σκ | = 0. At zero temperature and zero magnetic ﬁeld, the ice rule states span the ground state manifold of our model. 2.3 The two [111] magnetization plateaus At low magnetic ﬁelds (and low temperatures), the system will continue to obey the ice rule, though the magnetic ﬁeld will favor certain states among those in the zero-ﬁeld ground state manifold. For the calculations we will describe, it is useful to visualize the pyrochlore lattice along one of its [111] directions as shown in Fig. 2.4. The lattice is seen to be a stack of alternating kagome and triangular planes, the [111] direction being the direction in which the planes are stacked. In this picture, each spin lies on a corner shared by an up-pointing and down-pointing tetrahedron. If a magnetic ﬁeld is applied along one of the [111] directions (see Fig. 2.4), then ice rule states where spins are aligned with the ﬁeld (as much as possible, given the easy-axis constraint) will be favored. If the [111] ﬁeld is large enough, the spins in the triangular planes, having easy-axes permitting full alignment, will completely polarize and the kagome planes will decouple. The triangular planes are formed by the apical spins of the uptetrahedra. With respect to their up-tetrahedra, these spins are pointing out so the ice rule requires two of the remaining spins to point in and one to point out. The result is that the system is well-described by a two-dimensional model with its own ice-rule: pseudospins on a kagome lattice where on each triangle, two pseudospins point up and the remaining one 14 points down. This describes the basic physics of the lower [111] plateau and the degeneracy of the “kagome ice” manifold accounts for the ﬁnite entropy on the plateau. The higher [111] plateau is a zero entropy state where the spins are completely aligned with the ﬁeld (as much as possible, given the easy-axis constraint); note that this state is outside of the ice rule manifold. The crossover between the plateaus is a competition between satisfying the ice rule and optimizing the ﬁeld energy. We now turn to a brief description of the mechanisms which terminate the lower [111] plateau at low and high ﬁelds. 2.3.1 Low field termination: string defects At ﬁelds slightly lower than the plateau, excitations labelled string defects[22] restore threedimensionality and are responsible for the low ﬁeld termination of the plateau. To describe these defects, we consider the entropic beneﬁt of relaxing the condition that the triangular planes are polarized. Suppose we ﬂip a spin in some triangular layer. This, by itself, would violate the ice rule for the up and down tetrahedron sharing that spin. To restore the ice rule on the two tetrahedra, we need to also ﬂip spins in the kagome layers of each tetrahedron. Flipping these kagome spins requires ﬂipping spins in each of the two neighboring triangular layers, which requires ﬂipping spins in the two next-nearest kagome layers and so on. The resulting “string defect”, illustrated in Fig. 2.5, is an excitation that extends through the system. That such a defect must be inﬁnite in length follows more directly from an argument given in Ref. [22]. Each spin in the kagome layer is a member of an up-triangle which is the base of an up tetrahedron, whose apex is part of the above triangular layer. The ice rule implies that sum of the pseudospins on each tetrahedron is zero. These two facts imply that the pseudospin magnetization of a [111] kagome plane is the equal and opposite of the triangular plane above it. The triangular spins are also members of down tetrahedra corresponding to the down triangles in the above kagome layer. Therefore, we ﬁnd that the local ice rule implies the global property that all of the [111] kagome planes have the 15 same (pseudospin) magnetization, which is equal and opposite to that of the [111] triangular planes. On the plateau, the triangular magnetization is saturated. The preceding global property implies that to reduce this magnetization by ﬂipping a spin in one layer necessarily requires simultaneously ﬂipping spins in all of the other layers. Despite its name, which is historical, a string defect is not a “string”, i. e. a single extended object. Creating a string defect actually involves creating a pair of defects in each kagome layer. A “positive” defect connects the kagome plane to the kagome plane directly above it via a ﬂipped spin in the intermediate triangular plane. Similarly, a “negative” defect connects the kagome plane to the kagome plane directly below it. Two defects in a given plane may be separated by ﬂipping pairs of spins pointing in diﬀerent directions on neighboring triangles of the kagome plane. Every “negative” defect must be directly above a “positive” defect in the layer below. Therefore, creating a string defect actually means creating a defect in each kagome-triangle bilayer. In particular, a dilute gas of N string defects merely means having N defects in each kagome-triangle bilayer (or equivalently, N positive and N negative defects in each kagome layer with appropriate interlayer coupling) and in no sense are particular defects associated with a particular “string”. We may estimate the condition for having a string defect. The energy cost of a string defect may be found by noting that in each kagome-triangle bilayer, we need to ﬂip one triangular and one kagome spin that were previously aligned with the ﬁeld. The cost per bilayer is Es = 8gµB JB/3 so the total energy of a string E ∼ BL, where L is the system size. The thermodynamic state containing a single string defect has an entropy due to our choice of which spin to ﬂip in each triangular layer. This suggests that the entropy of the system S ∼ L ln L. The free energy balance implies that for nonzero temperatures and a given magnetic ﬁeld, string defects are favored in a suﬃciently large system. For a given system size, strings are favored at suﬃciently low magnetic ﬁelds. 16 2.3.2 High field termination: monomer defects On the plateau, the magnetization of the triangular sublattice is saturated and we may consider each kagome plane separately. Thus, the 3-dimensional model may be mapped onto a 2-dimensional one. Each triangle on the kagome plane contains two up pseudospins (σ = 1) and one down pseudospin (σ = −1). The states satisfying this constraint are in one-to-one correspondence with dimer coverings of the dual honeycomb lattice as sketched in Fig. 2.6.[21, 22, 23] The entropy on the plateau may be obtained from the exact solution of the honeycomb dimer problem. [21, 22] In this manner, it is found that the plateau retains an extensive ground state entropy, S/kB = 0.080765 (per spin). If we ﬂip a down (pseudo)spin it violates the ice rule on two triangles, one being uppointing and the other down-pointing as may be readily seen by inspecting Fig. 2.6. Analogous to the discussion of strings, these triangles correspond to positive and negative defects which can move on the lattice. In terms of the dimer description, creating a pair of defects involves replacing a dimer with two monomers (in the dimer description of a string, the pair of defects created in each kagome plane have two dimers/site). The energy cost for creating two monomers is 2E = 4Jeﬀ − 2gµB JB/3. This is balanced against the entropic beneﬁt in that we can place the monomers anywhere on the lattice. The energy cost vanishes at a critical ﬁeld Bc = 6Jeﬀ /(gµB J). At higher ﬁelds, the monomers proliferate leading to complete saturation and an ordered state with zero entropy. 2.3.3 Interaction of defects The picture of string and monomer defects presented above is a simpliﬁcation in that the defects interact. The interaction is entropic in origin and stems from the fact that if the defects are held ﬁxed at various positions, there are a number of ways in which the remaining sites may be covered by hard-core (one dimer/site) dimers. Defect conﬁgurations permitting many dimer coverings would appear more often in a partition function sum than more rigid conﬁgurations. In this sense, we may assign an entropy to each conﬁguration of defects and if these defects are dilute, we may represent the entropy of a conﬁguration as the sum of 17 pairwise interaction terms. In Appendix A, it is shown that at long wavelengths the entropy of two defects in the same plane as a function of defect separation is: S(|r1 − r2 |) = −κp1 p2 ln |r − r | 1 2 τ (2.3) where κ = 12 ; (ri , pi ) are the position and charge (±1) of the ith defect; and τ is a short distance cutoﬀ of order the lattice spacing. Of course, the overall entropy can not actually be negative so more precisely, this function is the entropy relative to that of the reference state of having no defects (which is essentially the case where the two defects are inﬁnitely far apart). This amounts to an overall constant, which we have dropped, of order ln(L/τ ), L being the system size, which arises because the number of dimer coverings increases exponentially with the system size. This coarse-grained description is expected to work at long wavelengths but the details of the kagome lattice enter through the value of κ, as discussed in Appendix A. Eq. 2.3 is expected to capture the leading in-plane interaction at long wavelengths for both the monomer and string defects (in the latter case, we are referring speciﬁcally to the local defects created in the planes). This is because both types of defects (in fact, any kind of local defect) appear as vortices in an underlying height ﬁeld, as discussed in Appendix A. In order to maximize its in-plane entropy, the system will prefer defects of the same (opposite) sign to be far apart (close together). Since the form of this interaction is that of 2d electrostatics, we will refer to the defects as “charges” but this should not conjure up images of wires or vortex lines. Eq. 2.3 only gives the interaction between two defects in the same plane. In the monomer case, this is the only interaction since the individual planes are decoupled. In the case of strings, we have the additional constraint that a positive charge in layer k is rigidly connected to a negative charge in layer k + 1. This negative charge can interact with the other charges in the k + 1 plane so the result is an interlayer coupling between defects in nearest neighbor planes. The details of this coupling are developed in the calculations described in the next section. 18 2.4 The low field regime We now consider the regime of the magnetization curve (Fig. 2.3) near the low ﬁeld termination of the plateau. String defects will be present by the arguments discussed above. In the next two subsections, we investigate the in-plane and out-of-plane correlations for these defects when the density of these defects is in the medium (mean-ﬁeld) and low (renormalization group)density regimes. We obtain the dependence of the defect density on temperature and ﬁeld in both of these limits and compare the analytic results with numerical simulation of the magnetization. 2.4.1 Mean field calculation If the number of string defects is fairly large, we may expect the interaction to be suﬃciently screened to justify the use of variational mean ﬁeld theory[32]. We consider a layered system of two-dimensional planes (indexed by the label k which ranges from −K to K) where each plane contains N positive and N negative charges that interact logarithmically. The string constraint requires that each positive charge in layer k is rigidly connected to a negative charge in the layer k + 1. We impose a periodic boundary condition to connect the positive charges in the Kth layer to the negative charges in the −Kth layer. We formally impose the constraint by writing the “Hamiltonian” in terms of positive charges alone. The planes are stacked in the z-direction. Let xki be the in-plane position of the ith positive charge in the kth layer. In absence of external ﬁelds, the entropy of a particular conﬁguration of N defects is given by: SN = K N ( k=−K i=j V (|xki − xkj |) − N V (|xki − xk+1 |)) j (2.4) i,j Here V (R) = −κ ln(R/τ ), where τ is a hard-core radius deﬁning the minimum separation between two charges and κ = 1/2. The ﬁrst term corresponds to the repulsion of positive charges within the same layer. The absence of a factor of 1 2 in front of this term is due to the string constraint: bringing two positive charges in the same plane close together also involves bringing together their negative partners in the plane above. In terms of our 19 positive charge formulation, this means the repulsion is twice as large. The second term is the interlayer interaction. Physically, a positive charge in layer k has a negative partner in the layer k + 1 which attracts the positive charges in layer k + 1. In terms of our positive charge formulation, like charges in the same plane repel one another but like charges in neighboring planes attract. We assume a variational mean ﬁeld density of the form: k K ρ(x−K 1 , ..., xi , ..., xN ) = N K ρk (xki ) k=−K i=1 (2.5) N which asserts that all particles in a given layer k have the same probability density ρk (x)/N , but the density may vary from layer to layer. We also need the normalizing condition: d2 xρk (x) = N (2.6) A This assures that each layer contains N particles: particles can not jump from layer to layer in this formulation. This trial function implies a variational entropy functional: Sρ,N K 1 − = k=−K − d2 xd2 x (ρk (x) − ρk+1 (x))(ρk (x ) − ρk+1 (x ))V (|x − x |) 2 d2 xρk (x) ln( ρk (x) ) N (2.7) where we have ignored terms of order 1/N . The ﬁrst term is optimized if the density is the same function in every layer: ρk (x) = ρk+1 (x). This leaves the second term, which is maximized (given the normalization constraint) when the density is uniform ρk (x) = N A which gives Sρ,N = (2K + 1)N ln A (A is given in units of τ ). To investigate the linear response of the system, we may apply a perturbing potential to the objects in the k = 0 plane. In particular, we consider the eﬀect of placing a positive charge at the origin of the plane on this uniform density solution. The details of the calculation are given in Appendix B but we may quote the result: 1 x δρ( , k) = 2 2 ξ 4π ξ −1/2 is·( x ) ξ d2 s s2 (s2 + 2) e 1 + s2 + 2 s2 (s2 + 2) 1 + s2 + s2 (s2 + 2) k−1 (2.8) 20 where the in-plane length scale is given by ξ = A 4πκN 1/2 . We note ﬁrst that this expression diverges at small x for k = 0, which is not surprising because the assumption of a linear response would be not be valid so close to the perturbing charge. The expression would be valid at larger k and an interesting feature is that when x = ξ , the decay in the z-direction does not depend on any physical parameters, i.e. there is no length scale in the z direction. We will return to this point in the next section. The correlation length has a temperature and ﬁeld dependence contained in the factor (N/A). We may estimate the number of defects, given a temperature and ﬁeld, from the partition function: Z = e−βA = y (2K+1)N N (N !)2K+1 eSN (2.9) where SN is the entropy of having N defects and y = e−Es /kB T is the fugacity of a positive defect (y 2K+1 is the fugacity of a “string”) where Es = 8gµB JB/3 is the energy per kagometriangle bilayer of a string defect. At mean ﬁeld level, we may replace SN by Sρ,N = (2K + 1)N ln A. In Appendix B, it is shown that N ∼ yA so that we obtain the ﬁnal result: 2 ξ,M F ∼ exp(8gµB JB/3kB T ) 2.4.2 (2.10) RG calculation When the gas of defects is fairly dilute, we may expect that the screening is not eﬀective enough to justify a mean ﬁeld treatment. In this section, we account for ﬂuctuations by making a real space renormalization group calculation using methods similar to the Kosterlitz treatment of the 2d Coulomb gas[31, 33]. The dynamical objects described by Hamiltonian 2.4 are “dipoles of length 1”, i.e. objects consisting of dipoles where the negative charge is rigidly connected to a positive partner one (kagome) plane below. We need to generalize this model in order to do an RG calculation. The generalization that we consider is allowing for dipoles of arbitrary length. An “l-dipole” is an object where the negative charge lies directly l planes above its positive partner to which it is rigidly connected. While the original problem involved 21 just the coupling of nearest neighbor planes, our generalized model involves all possible couplings. Associated with each l-dipole is a fugacity yl /2π (the 2π is for convenience). Fig. 2.7 is a cartoon of our generalized model. The grand partition function for the system may be written as: Z= (yl /2π)N,l {Nk,l } k,l (Nk,l )! Z[{Nk,l }] (2.11) where Nk,l denotes the number of l-dipoles in layer k; N,l is the number of l-dipoles in the system; and Nk is the number of dipoles (of any length) that have their positive ends in layer k. The sum is over all particle number conﬁgurations {Nk,l } that satisfy the charge neutrality constraint in each plane: Nk = l Nk−l,l . The canonical partition function corresponding to a given dipole distribution {Nk,l } is: Z[{Nk,l }] = (1) (2) (1) (2) d2 xk,i d2 xk,i xk,i − xk,i Ωτ k,i τ2 τ2 δ τ exp −H({Nk,l }) (2.12) H({Nk,l }) is the Hamiltonian (actually an entropy) corresponding to the dipole distribution (1) {Nk,l }. The coordinate xk,i is the planar coordinate of the ith positive charge of layer k (2) and xk,i is the planar coordinate of its negative partner which lives in layer k + l(i), l(i) being the length of the dipole being described. The string constraint is imposed by the delta function, where we use the normalization d2 x x R2 τ 2 δ( τ ) = 1. The product is over all positive charges in all layers. The integration is over the space Ωτ . This is deﬁned to be the set of all possible spatial conﬁgurations of the dipole distribution {Nk,l } that respect the hard-core constraint: no two charges in a given plane may be closer than distance τ . Our procedure is an extension of the treatment in Refs. [31],[33]. The ﬁrst part of an RG procedure normally involves integrating over the high momentum modes of the system. In our problem, these correspond to conﬁgurations where in some plane, we have a pair of charges separated by a distance between τ and τ + dτ . We assume a dilute system so only oppositely charged pairs are considered and also the distance between the members of a pair is taken to be much smaller than the distance from the pair to another charge. These approximations were also used in [33] and are based on the Boltzmann factor for conﬁgurations containing same charge pairs being much smaller than that for oppositely 22 charged pairs. Strictly, we would need to keep all of these other states unless they are explicitly shown to be irrelevant in the RG sense but this demonstration is for another day. The basic coarse-graining step in our RG transformation is illustrated in the lower part of Fig. 2.7. Suppose a particular state involves pairing the negative end of an l1 -dipole in layer k with the positive end of an l2 -dipole in layer k + l1 . Viewed at long length scales, we eﬀectively have an (l1 + l2 )-dipole in layer k. We will ﬁnd that integrating over all possible pairings gives a leading term (which just involves replacing Ωτ with Ωτ +dτ ) and a number of correction terms of order dτ where two short dipoles were destroyed and replaced by a longer dipole. Since the procedure respects the charge neutrality constraint, these correction terms will combine with other terms in the grand partition sum. The second step involves rescaling lengths so that the high momentum cutoﬀ, in the new variable, is the same as before. The aim is to see how the fugacities and couplings change as we run this procedure. Details of the calculation are given in Appendix C. Here we give the resulting ﬂow equations: dy1 dt dyl dt dκ dt = (2 − κ)y1 (2.13) = (2 − κ)yl + l−1 ym yl−m (2.14) m=1 = 0 (2.15) where t = ln τ . One notable feature is that the coupling does not change with the ﬂow, even at second order in the fugacity, in contrast with the 2d Coulomb gas[33] where it does vary. This indicates that our defects are stiﬀer objects than usual charges. Another observation is that for the initial conditions of our physical problem, namely that y1 (0) = y0 = 2πe−Es /kb T and yl (0) = 0 for l > 1, the ﬂow equations have an exact solution: yl = y0 τ 2−κ y 0 2−κ l−1 (τ 2−κ − 1) (2.16) Our RG is valid as long as the corrections to the fugacities are small, meaning that the derivatives dyl /dt should be bounded. If we look at the above result, Eq. 2.16, we see that when the term in brackets is greater than 1, yl diverges with l. Therefore, a critical length, 23 which we interpret as an in-plane correlation length, is deﬁned by when the term in brackets equals 1: y0 (ξ 2−κ − 1) = 1 2 − κ ,RG (2.17) Substituting earlier expressions and noting that for our system, κ = 1/2, we ﬁnd that: 2 = ln ξ,RG 32gµB JB ln(e−Es /kB T + 2 − κ) 1+ 9kB T Es /kB T 2 ξ,RG ∼ exp(32gµB JB/9kB T ) (2.18) for the ﬁelds and temperatures of interest. This value is the same as that predicted in Ref. [22] using a free energy argument. For τ < ξ,RG , yl decreases with l which means that states with long dipoles are less probable than states with short dipoles. If τ > ξ,RG , yl diverges with l which suggests that longer dipoles are favored, but, as mentioned above, the RG procedure is no longer valid in this regime. We note that when τ = ξ,RG , yl is independent of l so that, as in the mean ﬁeld calculation discussed above, there is no discernible length scale in the z direction. If τ < ξ,RG , then we may consider an out-of-plane length scale, which we deﬁne nominally as the value of l = lτ for which yl /y1 = 1/e. 1 lτ = 1 + ln ξ 3/2 −1 ,RG (2.19) τ 3/2 −1 We may interpret lτ as the typical length of a string segment that is captured by a tube of diameter τ (where a tube need not be straight). 2.4.3 Comparison with simulation In Fig. 2.8, we show the magnetization as a function of the magnetic ﬁeld strength on a log-log scale, computed by Isakov[34] using an eﬃcient cluster algorithm. The magnetization should scale with the average density of defects, which in turn should scale like the inverse square of the in-plane correlation length. As shown in this ﬁgure, the data at low ﬁelds are well ﬁt by the exponent 8/3 obtained in the mean ﬁeld calculation discussed earlier. At somewhat higher ﬁelds, the data are well ﬁt by the exponent 32/9, obtained by the RG 24 calculation discussed earlier and also in Ref. [22] by looking at the entropic contribution to the free energy. At high ﬁelds, the exponent of 8L/3 (=16 for L=6 (sites), as was the case in the simulations) characterizes a regime where ﬁnite-size eﬀects are important, as discussed below. The low ﬁeld crossover makes qualitative sense in that at low ﬁelds, there will be many defects which screen one another which suggests that a mean ﬁeld treatment may be reasonably accurate. At higher ﬁelds, the gas of defects is more dilute so an RG treatment would be required. The high ﬁeld crossover is a ﬁnite-size eﬀect since the position of a crossover between exponents is system size dependent and the corresponding exponent is also system size dependent, getting steeper with increasing system size. The ﬁnite-size behavior, alluded to in Section 2.3, may be explained as follows. At high magnetic ﬁelds, there are a small number of string defects in the system. The magnetization and the energy of one string defect in a system of size L are −4LgµB J/3 and 4LgµB JB/3 respectively. The energy cost grows linearly with system size and, as mentioned above, the defects are favored solely due to their entropic contribution to the free energy. At suﬃciently high magnetic ﬁelds, a given system will be too small to provide the entropy to balance the energy cost of a string. This will occur when the magnetization per spin reaches the magnetization of a system with one string defect: m = = 1/3 − 2(4L/3)/(16L3 ) gµB J 1/3 − 1/(6L2 ) gµB J. (2.20) In this case, the statistical weight of a single string defect will be a Boltzmann factor exp(−8LgµB JB/3kB T ) and the magnetization will equal: m = [1/3 − C exp(−8LgµB JB/3kB T )]gµB J (2.21) where C is some constant. The crossover between diﬀerent regimes occurs when the magnetization reaches (2.20). We have good agreement with the 8L/3 behavior for a variety of system sizes, including L = 6 which is shown in Fig. 2.8. 25 2.5 The high field regime The physics near the transition may be approximately described by the following Hamiltonian which acts on the kagome lattice: H = K(si , sj )si sj − h si , T i ij (2.22) where the sum is over all nearest neighbors; si are classical Ising spins taking values +1 and −1; h is the strength of a ﬁctitious magnetic ﬁeld; and K(1, 1) = 0, K(1, −1) = K(−1, 1) = K = [gµB JB/6 − Jeﬀ ] /T , and K(−1, −1) = ∞. The coupling constants imply that each triangle of the kagome lattice contains at most one down pseudospin and that down spins cost energy (positive or negative dependent on the magnetic ﬁeld strength). In terms of the dimer representation, this approximation involves considering only states where each site contains one dimer or zero dimers (i. e. a monomer). States involving sites with two or three dimers involve an inﬁnite energy cost so are ignored. Isakov[34] calculated the magnetization and entropy within this simpliﬁed monomerdimer model using the Bethe approximation[28]: 1 1 2 1 + x2 1 2z 3 3xz ln z + ln 2 , S = − 2 + 6xz 4 x (3z − x) m = where x = 2z/(1 + √ (2.23) (2.24) 1 + 8z 2 ) and z = exp(−2K). In Fig. 2.9, we compare these expressions with a Monte Carlo simulation of Hamiltonian 2.2. In this simulation, sites containing two or three dimers are given the appropriate Boltzmann weight instead of being explicitly ignored. The simulation is of a kagome lattice with 16x16 up-triangles (768 total spins). The standard single spin-ﬂip Metropolis algorithm was used, which may explain the inaccuracy in the simulated entropy at low ﬁelds, where a more clever scheme may be needed to sample the degenerate manifold. The entropy was computed, for a given ﬁeld, by integrating from high temperatures (where S/kB = (3/4) ln 2 per atom) to low temperatures. The agreement of the simulation with Isakov’s analytic expression validates the monomer-dimer picture of the transition. 26 There is a giant peak in the entropy at the transition point, S/kB = 1/4 ln(16/5) ≈ 0.291, which exceeds even the zero ﬁeld entropy. The peak is due to the crossing of an extensive number of energy levels which have macroscopic entropies. For B = Bc , the energies of states corresponding to diﬀerent numbers of monomer defects are equal since the monomer and dimer weights are, by deﬁnition, equal at the critical ﬁeld. There are an extensive number of states corresponding to a given number of monomers (below saturation). So while the [111] magnetic ﬁeld selects a subset of the zero ﬁeld spin ice ground state manifold, near the critical ﬁeld, the ground state space also includes a large number of states not in the ice manifold. Our model predicts that these extra states more than compensate causing a peak which exceeds even the zero ﬁeld entropy. 2.6 Crossing points The theory described in the previous section implies that the curves of magnetisation versus ﬁeld, plotted for diﬀerent temperatures, will display a crossing point. This arises simply because the partition function depends on magnetic ﬁeld and temperature eﬀectively only through the combination (B − Bc )/T . Thus, when plotted as a function of B − Bc , the curves coincide only at the point B = Bc . At this point, the Bethe approximation gives a value for the magnetisation of m = 0.4gµB J. In addition, we expect a crossing point at low ﬁelds, due the interplay of string and monomer defects. Indeed, where the plateau is well-formed, the string density is ns ∼ exp(−32gµB JB/9kb T ) and the monomer density is nm ∼ exp(−8Em /7kB T ), where E = gµB J(Bc − B)/3 is the energy of creating one monomer. The crossing point occurs when ns = nb . With logarithmic accuracy, we can write 8gµB J(B − Bc ) 32gµB JB = . 9kb T 21kB T Thus the crossing point lies at B = 3Bc /31. (2.25) 27 2.7 Relation to experiment, other theories, and applications Our model gives a description of the high ﬁeld transition that is qualitatively consistent with experiment for a range of temperatures [11]. The predicted value for the entropy of the lower [111] plateau, based on the exact calculation for classical dimers on the honeycomb lattice[21, 23], has been conﬁrmed in experiment[11]. The magnetization curves cross at a ﬁeld close to the predicted critical ﬁeld[11] and in particular, a peak in the entropy has been recently observed[15]. The height of this experimental peak is smaller than the zero ﬁeld entropy, while our simple model predicts a larger peak. However, a more serious discrepancy is a recent experiment[35] on the spin ice compound Dy2 Ti2 O7 which strongly suggests that at low temperatures, the high ﬁeld transition becomes ﬁrst order. In Ref. [35], the onset of ﬁrst order behavior, indicated by discontinuous magnetization curves and hysteresis, was found to occur for temperatures lower than a critical temperature of Tc ∼ 0.36K (∼ 0.327Jef f,Dy /kB ). Fig. 2.9 shows that our predicted curves remain continuous even at temperatures below this observed Tc . Our initial hypothesis was that this discrepancy is due the long range nature of the dipolar interaction, which we approximated as a nearest neighbor Ising model. In Section 2.2, we discussed recent work[17, 18, 19, 20] that demonstrated that this nearest-neighbor model has the same ground state manifold as a model dipolar interaction. However, this does not imply a correspondence between the excited states of these models. In the nearestneighbor case, the natural excitations at high magnetic ﬁelds are monomer defects and the proliferation of these defects appears to describe the regime where the magnetization curves are continuous. In the regime where the magnetization is discontinuous, the excited states of the full dipolar interaction may diﬀer signiﬁcantly from the excited states of the nearestneighbor model. The simplest way to account for the long-range dipolar interaction is to model the further neighbor interaction terms as giving rise to a magnetic ﬁeld proportional to the magnetization. By assuming the magnetization M , as a function of the eﬀective ﬁeld B + αM , has the same functional form as given in ﬁgure 2.9, we may self-consistently 28 determine M for a given B. Using α as a free parameter, we ﬁnd that this simple model predicts the onset of ﬁrst order behavior, at the experimentally observed critical ﬁeld Bc , only for temperatures in the millikelvin range. To obtain a higher numerical Tc requires a larger α, which causes a lower numerical Bc . To get the numerical Tc to match experiment requires an α so large that our numerical Bc is “negative” (in the sense of artiﬁcially extending the M = 1/3 line of ﬁgure 2.9 for the purpose of a spline ﬁt). Therefore, it appears that the role, if any, of long-range interactions in explaining this feature is something more subtle. Another possibility which we have not considered is the impact of the slowdown of the dynamics which is observed at low temperature.[36] The hysteresis in the magnetization[35] at low temperatures, which was interpreted as indicating a ﬁrst order transition, may be a nonequilibrium eﬀect. As for the crossing points mentioned above, the high-ﬁeld one does indeed appear to be present in the experimental data[10, 35] in the appropriate temperature range. The experimental value of the magnetization at the crossing point is about m = 0.38gµB J, reasonably close to the theoretical value m = 0.4gµB J. By contrast, a crossing point at small ﬁelds is harder to make out, and an approximate estimate of its location gives B = 0.35Bc , in disagreement with the theoretical B = 3Bc /31. 2.7.1 Cooling by adiabatic (de)magnetization At low temperatures, near the degeneracy point, the partition function depends on magnetic ﬁeld and temperature eﬀectively only through the combination (B − Bc )/T . One may thus argue that the spike may be used to eﬀect cooling by adiabatic demagnetization[37] in exactly the same way one may use paramagnets – analogous constraints limit the application in either case. There are two features which may be worth pointing out at this point. Both follow from the fact that – unlike in the case of a paramagnet – Bc = 0. Firstly, maximal cooling occurs at a ﬁnite ﬁeld, namely around Bc . This phenomenon may therefore be useful to eﬀect 29 cooling for a magnet in a ﬁeld, with the restriction that Bc , for a given spin ice compound, is not tunable. Secondly, if B approaches Bc from below, one can in fact obtain “cooling by adiabatic magnetization”, as entropy and magnetization grow together in this regime. 2.8 Conclusions In this paper, we have analyzed in detail the magnetization curve of nearest-neighbor spin ice in a [111] magnetic ﬁeld. The basic ingredient which makes this system particularly interesting is that a uniform ﬁeld can be used to couple to the Ising pseudospins as a staggered ﬁeld[38, 39]. This amounts to the possibility of applying ﬁelds which would have appeared to be rather unnatural in the formulation of a simple Ising model (without the detour via spin ice) on the pyrochlore lattice. As a result, one observes an attractively rich behavior. Perhaps the most salient is the dimensional reduction from pyrochlore to kagome under the application of an external ﬁeld. The restoration of three-dimensionality upon weakening the ﬁeld goes along with the string defects. We hope that the extension developed here of Kosterlitz’s RG treatment to such extended defects might be of more general use. A particularly attractive feature of the monomer-dimer model we have obtained here lies in the fact that the relative monomer and dimer fugacities in the low-temperature (T Jeﬀ ) regime are given by simple Boltzman weights of Zeeman energies. They are thus straightforwardly tunable by changing the strength of the applied ﬁeld. In particular, anisotropic fugacities for the dimers can be obtained by tilting the ﬁeld, and they therefore do not require an actual manipulation (such as an application of anisotropic stress) of the two-dimensional layer. As discussed previously in Ref. [22] the price for our ability to analyze the model in such detail has been the omission of the long-range nature of the dipolar interaction. A truncation of the interaction at only the nearest-neighbor distance would seem a rather drastic step; an expectation of quantitative agreement between experiment and the nearest-neighbor model will in general likely be misplaced. Recent work has shown that this approximation is not 30 entirely unreasonable[19, 20] and this observation might lie at the basis of the fact that the measured dipolar ice entropy agrees so well with Pauling’s estimate. Our ‘prediction’ of the entropy peak between the intermediate and saturated plateaux bears witness to the promise of our approach to unearth at least some qualitative features of interest. Magnetization fraction 31 monomers 1/2 dimers 1/3 dimer-monomer strings 1/6 linear response 0.25 0.50 0.75 1.0 1.5 1.0 1.5 Magnetic field (T) S/k (per spin) 0.3 0.2 0.1 0.25 0.50 0.75 Magnetic field (T) Figure 2.3: Properties of spin-ice as the [111] magnetic ﬁeld is varied. These curves are for illustration and do not show actual numerical or experimental data. We have indicated the regions where various analytic approaches apply. The linear response regime is not directly discussed in this thesis; for a detailed treatment of that regime, the reader should consult Refs. [1, 34]. 32 H [111] Figure 2.4: The ﬁgure depicts the pyrochlore lattice in proﬁle with the [111] direction indicated. The lattice sites are indicated by the black dots and the structure of interwoven kagome and triangular planes is clearly seen. Each triangular spin is a member of an up and down pointing tetrahedron and will point along or against the ﬁeld. 33 H [111] Figure 2.5: A single string defect. The ﬁgure depicts the pyrochlore lattice in proﬁle with the [111] direction indicated. The large black dots indicate that the corresponding triangular spin is oriented against the ﬁeld. This causes the 2-up, 1-down “kagome ice” rule to be violated on the triangles immediately above and below the ﬂipped spin. The resulting defects are indicated by the circles. As discussed in the main text, a pair of defects is created in each layer. A defect which is the base of an up (down) pointing tetrahedron is called “positive” (“negative”). In each plane, the two defects may move independently but the string constraint requires each positive defect to be rigidly connected to a negative defect in the layer above. This condition is denoted by the thick bars. 34 + + + - ++ + + + + - + + + + - + + + ++ + + + - - + - + + + - + + - + + + + + - + + Figure 2.6: Kagome ice. The constraint on the [111] plateau is that each triangle of the kagome lattice must have two up spins and one down spin. The centers of these triangles form a honeycomb lattice as indicated by the dotted lines above. Draw a dimer (indicated by thick lines) on every link containing a down spin. The two up, one down constraint ensures this procedure will place one and only one dimer on every site. Hence the kagome ice manifold maps onto dimer coverings of the honeycomb lattice. 35 + - - - - + + - + - + + - + - - + + + Figure 2.7: Renormalization group analysis of string defects. In the physical model, the dynamical objects are oppositely charged pairs, where the negative charge lies in the plane directly above its positive partner, to which it is rigidly connected. Our generalized Hamiltonian considers objects of arbitrary length. The basic coarse-graining step in our RG transformation is shown in the lower part of the ﬁgure: smaller objects grow into larger objects at the expense of renormalizing the fugacities and coupling. 36 1/3-m 10 0 10 -2 10 -4 10-6 10 10 MC L=6 0.3539 exp(-8/3 x) 0.6278 exp(-32/9 x) 7 6.16 10 exp(-16 x) -8 -10 0 0.5 1 1.5 2 2.5 3 0.4 0.5 0.6 1/3-m gµBJB/kBT 10-1 MC L=6 0.3539 exp(-8/3 x) 0 0.1 0.2 0.3 gµBJB/kBT 1/3-m 10-1 10-2 MC L=6 0.6278 exp(-32/9 x) 10-3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 gµBJB/kBT Figure 2.8: The crossover between exponents. The magnetic ﬁeld on the plateau is too weak to create a string defect. As the ﬁeld is lowered, strings nucleate and the 32/9 line describes the dilute string limit quite accurately. The RG assumes a dilute gas of strings and, expectedly, breaks down at lower ﬁelds where strings proliferate. In this regime, a mean ﬁeld treatment works fairly well as indicated by the 8/3 line. 37 0.52 0.48 m/µ 0.44 0.4 kT/Jeff = 0.1 kT/Jeff = 0.3 kT/Jeff = 0.5 Bethe approximation 0.36 0.32 -4 -2 0 2 4 6 8 10 gµBJ(B-Bc)/3kBT 0.3 kT/Jeff = 0.15 kT/Jeff = 0.3 kT/Jeff = 0.5 Bethe appr. Series exp. Exact result Pauling result 0.25 S/kB 0.2 0.15 0.1 0.05 0 -10 -5 0 5 10 15 gµBJ(B-Bc)/3kBT Figure 2.9: The magnetization (top) and the entropy (bottom) around the transition between the plateaux. The Bethe approximation and higher order series expansion calculations of Isakov[34] are compared to the Monte Carlo results. The exact result for the entropy at zero monomer density and Pauling’s estimate for the entropy at zero magnetic ﬁeld are shown for reference. Chapter 3 SU(2) Invariant spin 1/2 Hamiltonians with RVB and other valence bond phases. 3.1 Introduction Just over thirty years ago Anderson[40] introduced the resonating valence bond (RVB) state as an alternative to Néel ordering in antiferromagnets with strong quantum ﬂuctuations. In essence, he proposed that on a suﬃciently frustrated lattice an S = 1/2 system would exhibit a disordered state at T = 0, which would be captured by a wavefunction of the form |ψ = Ac |c (3.1) c where |c is a conﬁguration of singlet pairings of spins or valence bonds (Fig 3.1). For suﬃciently short-ranged valence bonds this describes, in contrast to the Néel state, a state with short-ranged spin correlations. The discovery of the cuprates and the suggestion that their superconductivity could be traced to RVB physics[41] greatly energized the elucidation of the RVB idea and by now a rather complete understanding of its internal logic has emerged. In modern parlance, 38 39 Figure 3.1: Sample valence bond conﬁgurations. The thick lines represent singlet pairings. The original formulation allowed for bonds of arbitrary length (left). Considerable progress has been made by restricting to conﬁgurations with only nearest-neighbor valence bonds (right). an RVB phase is a topological phase, characterized by excitations with fractional quantum numbers and a low energy gauge structure which mediates topological interactions among the excitations[42]. The excitations include spinons, the S = 1/2 excitations produced by breaking a valence bond, as well as collective excitations within the valence bond manifold (see ﬁgure 3.2). The sr-RVB[43] (short ranged RVB) with short-ranged bonds and gapped spinons will be our concern in this paper. A post-cuprate version with longer-ranged bonds and gapless spinons [44] has also been the subject of recent progress.[45] Readers familiar with one dimensional lore will note that the short-ranged and long-ranged RVBs generalize the physics, respectively, of the Majumdar-Ghosh chain[46] and the Bethe chain [47] to higher dimensions. The important progress that we have described has been kinematical. It has not directly answered the question of realizing RVB phases for actual Hamiltonians. The original proposal was made for the nearest-neighbor Heisenberg model on the triangular lattice but that is now generally believed to exhibit weak Néel order[48]. To make progress on the dynamical front, Rokhsar and Kivelson[49] introduced the quantum dimer model (QDM) which assumes that the low energy dynamics is dominated by valence bond conﬁgurations 40 x Figure 3.2: Typical excitations of an RVB liquid. The left ﬁgure depicts a pair of spinons, fractionalized excitations with S = 1/2, formed by breaking a valence bond. Spinons are unpaired spins which move in the RVB liquid background; the interaction between spinons depends on the lattice geometry. The right ﬁgure depicts a vison, which is an excitation within the valence bond subspace. If we consider an RVB liquid which is an equal amplitude superposition of nearest-neighbor valence bond states, then the vison is deﬁned by the wavefunction |ψv = c (−1)Nc |c where Nc is the number of bonds which cross the dashed line shown in the ﬁgure. Clearly this state is orthogonal to the RVB state, |ψ = c |c. The interaction between spinons and visons is discussed in detail in Ref. [42]. of short range which are taken to be nearest-neighbor in the versions studied to date. Such conﬁgurations are labelled by dimer coverings of the lattice at issue and the quantum dimer Hamiltonian acts in a Hilbert space spanned by such coverings. The program of studying the simplest dimer models has been rather fruitful. It is now clear that Z2 RVB phases may arise on non-bipartite lattices in d ≥ 2[51, 52] while bipartite lattices in d > 2 give rise to U (1) RVB phases that exhibit a gapless “photon”[53, 54]. In addition a variety of crystalline phases have been identiﬁed, most notably a Cantor deconﬁned region[55] of interleaved commensurate and incommensurate valence bond crystals on bipartite lattices in d = 2. The next order of business then, is to ﬁnd rotationally invariant, local, spin Hamiltonians that are accurately described by these well understood dimer models. This is the problem that we solve in this paper thus completing a program initiated by Chayes, Chayes and Kivelson[56]. The strategy we follow is that of constructing Klein Hamiltonians[57] with 41 large energy scales which select nearest neighbor valence bond states as their ground states, separated by a gap from excited states. We then lift this degeneracy by the inclusion of perturbations that precisely mimic the terms in the quantum dimer models of interest. To control this procedure, with its diﬃculties stemming from the non-orthogonality of the valence bond basis, we introduce a parameter which is the extent of decoration of the reference lattice. By suitably tuning this parameter we are able to make our dimer model realizations arbitrarily accurate. An elegant feature of this limit is that it enables us to establish the existence of a spin gap about the nearest-neighbor valence bond manifold and to discuss the states above this gap in terms of “microscopic” spinons whose meaning will become clear below. We note that our tuning procedure occurs within the space of SU(2) invariant Hamiltonians. This is in contrast to approaches involving enlarging the symmetry group to Sp(N ) or SU(N ) and studying the large N limit[58]; in these cases, the applicability of results to Sp(1) ≡ SU(2) is not obvious. This is also a good place to note that there is a considerable body of work on variational[59] and ﬁnite-size studies of some of the two-dimensional phases discussed in this paper, e.g. the early ﬁnite-size study of a multiple-spin Hamiltonian on the triangular lattice that adduced evidence for a topologically ordered phase[60]. A comprehensive review of such work is given in Ref. [61]. This work is complementary to ours as it deals with somewhat simpler Hamiltonians but is unable to access the thermodynamic limit in a controlled fashion. We also note that, there is a large and growing literature on more general models with topological phases which we skip in our focus on S = 1/2 spin systems; this work also ﬁnds inspiration from the proposal that a quantum computer may be robustly created from a topological phase[62]. Finally, we note that there are encouraging reports of spin liquids in experimental systems[63, 64]. In the rest of this chapter we give details of our constructions. We begin with a quick review of quantum dimer models and the known results on their phase diagrams in Section 3.2. In Section 3.3 we explain our strategy with the honeycomb lattice serving as an example; 42 this realizes the physics of bipartite dimer models in d = 2. In Section 3.4 we show how the physics of non-bipartite dimer models in d = 2 and bipartite and non-bipartite dimer models in d > 2 may be obtained from spin models. In Section 3.5 we discuss two spin models on the non-bipartite pyrochlore lattice, one exhibiting a Z2 RVB phase and the other a U (1) RVB phase. We conclude with a summary (Section 3.6) and a set of appendices that contain some technical material. 3.2 Quantum dimer models For a system of spins si on a lattice Λ, a (nearest-neighbor) valence bond state is a product wavefunction of the form Ψ = ij ψij where ψij = √1 (ψ ↑ ψ ↓ i j 2 − ψi↓ ψj↑ ) and the product is over nearest-neighbor pairs (i, j). The product is deﬁned so each spin forms a singlet with exactly one of its neighbors. Each valence bond state corresponds to a hard-core dimer covering of Λ where a dimer connecting two sites corresponds to a singlet bond between the respective spins. Valence bond states are not orthogonal but the overlap between two arbitrary states is exponentially small in the length of closed loops obtained by superposing them. This suggests that they are linearly independent on suﬃciently open lattices and indeed there are proofs for the square and honeycomb lattices[56] and numerical evidence that this is so even on the triangular lattice[65]. The identiﬁcation with dimer coverings suggests that any low energy dynamics restricted to the valence bond manifold can be represented by a quantum dimer Hamiltonian acting on orthogonal dimer states. More precisely, the set of hard core dimer coverings of the underlying lattice may be taken as an orthonormal basis spanning a dimer Hilbert space; the orthonormality of the dimer coverings deﬁnes the inner product on this space. We may introduce dynamics on this dimer Hilbert space and the simplest such Hamiltonian on the square lattice, written down by Rokhsar and Kivelson [49], has the pictorial form H = ∑-t( 〉〈 QDM +h.c.)+v( 〉〈 + 〉〈 ) (3.2) where t and v are positive constants and the sum is over all possible square plaquettes. 43 Evidently, this can be supplemented by kinetic energy terms which act on longer loops and potential energy terms which count more complicated dimer motifs.[66] Fig. 3.3 outlines the appropriate generalizations of these basic operators to the honeycomb and triangular lattices. The passage from valence bonds to dimers, however, has to contend with two complications. One is that one needs to choose a phase convention for the valence bonds, which is subject to restrictions on what signs one can obtain for various couplings in the dimer Hilbert space. The other, already alluded to, is the lack of orthogonality of the valence bond states which makes the transcription from a spin model to a dimer model (and the relationship between the respective Hilbert spaces) non-trivial. We will deal with both problems later in the paper; here we merely wish to alert the reader to their existence. An important property of the quantum dimer Hamiltonian is the existence of the “Rokhsar-Kivelson point” (RK point) t = v, where any equal amplitude superposition of all dimer coverings connected by the operation of the kinetic energy is a ground state. To see this, note that for every ﬂippable plaquette, the second term gives a penalty v while the ﬁrst term gives at most a beneﬁt of −t. Nonﬂippable plaquettes have zero energy according to HQDM . This gives a lower bound for the ground state energy: E0 ≥ min{0, NP (v − t)}, where NP is the number of plaquettes in the lattice. The equal amplitude state has energy nf l (v − t), where nf l is the average number of ﬂippable plaquettes in the state. At v = t, this saturates the lower bound and, since the equal amplitude state is an eigenstate of HQDM (at v = t), we may conclude that it is a ground state when v = t. Thus the ground state correlations at the RK point reduce to those of solvable classical dimer models. Additionally, the inﬁnite temperature static correlations of QDMs also reduce to those of the same classical models. These features, along with the additional one that Hamiltonians of the form (3.2) can be simulated by Monte-Carlo without any sign problems, have been crucial to making progress in determining the phase diagrams of the quantum models. As a consequence of this progress we now know that: 44 i) QDMs on bipartite lattices in d = 2 do not exhibit an RVB phase. The equal amplitude state present at the RK point, v/t = 1, does not extend into a phase. As v/t increases, the system generically passes through a sequence of interleaved commensurate and incommensurate crystalline phases before reaching the staggered valence bond solid (VBS) phase. These incommensurate ground states, whose measure approaches unity near the RK point, turn out to have deconﬁned monomers, a phenomenon coined Cantor deconﬁnement[55]. Some more details about Cantor deconﬁnement are given in Appendix A. As v/t decreases from unity, the system passes through a plaquette phase[67] before undergoing a ﬁrst-order transition to a columnar VBS. The phase diagram for these systems is given in Fig. 3.4. ii) QDMs on non-bipartite lattices in d = 2 may exhibit RVB phases. These are Z2 RVB phases captured by a purely topological BF theory[42]. For the triangular lattice, it has been shown[51] that for v/t > 1, the system is in a staggered VBS; for v/t ≤ 1, there is a deconﬁned RVB liquid phase. As v/t is further reduced, there are probably a small number of VBS phases culminating in the columnar state. The phase diagram for these systems is given in Fig. 3.5. iii) QDMs on non-bipartite lattices in d = 3 and higher also exhibit a Z2 RVB phase[52]. iv) QDMs on bipartite lattices in d = 3 and higher exhibit a U (1) RVB phase with a gapless, linearly dispersing transverse mode, the “photon”[53]. We now turn to the task of constructing spin models whose low energy dynamics is precisely captured by these dimer models. We begin, for pedagogical speciﬁcity, with the honeycomb lattice. 3.3 Honeycomb lattice: Bipartite physics in d = 2 Our strategy for realizing dimer models proceeds in three steps. First, we construct, following Klein, a local spin Hamiltonian that has valence bond states as its ground states. Next we perturb it to obtain a QDM. Finally, we decorate the lattice to simplify the QDM to the well studied form (3.2). In the high decoration limit, we show the existence of a gap and give a description of the spectrum in terms of spinons. 45 3.3.1 Klein model The basic idea of the Klein model consists of considering a cluster of z sites (typically, a spin and its z − 1 neighbors) and deterring, via an energy penalty, this cluster from having maximal total spin Stot = z/2. If two of the spins in the cluster form a singlet bond, this condition is satisﬁed. This is why Klein Hamiltonians naturally lead to valence bond ground states. In particular, for a system of spins si on a lattice Λ, a Klein Hamiltonian is a sum of projection operators P̂N (i) deﬁned as follows. For each site i, consider the neighborhood of spins N (i) consisting of the spin at site i and its (zi − 1) nearest neighbors. Let P̂N (i) be the projector onto the highest total spin state of the cluster. The Klein Hamiltonian is formally given by the expression: HK = P̂N (i) , (3.3) i∈Λ with total spin of cluster N (i) given by: N (i) = S sj (3.4) j∈N (i) We may write P̂N (i) in terms of this operator. For example, if zi is even, then: zi /2−1 P̂N (i) = Ci 2 SN (i) − L(L + 1) . (3.5) L=0 The total spin of this cluster will take values from 0, 1,. . . ,(zi /2) − 1,zi /2. The factors of this product are operators which sequentially annihilate all but the highest spin sector. The form of the operator implies that larger clusters involve higher-order spin interactions– but they always remain local. If the constants Ci in Eq. 3.5 are chosen to be positive, then HK will have non-negative eigenvalues. By construction, valence bond coverings are zero-energy ground states of HK . This is readily seen by noting that the highest spin sector of a cluster is necessarily symmetric under interchange of spins; if two of the spins in a cluster are paired in a singlet, then the state is antisymmetic under their interchange so hence has zero projection in the highest 46 spin sector. We note that HK may also have zero-energy ground states outside of the valence bond manifold. For the honeycomb lattice, there are proofs[56], for a restricted set of boundary conditions, that the ground state manifold of HK is spanned by the set of valence bond coverings. For other lattices, we can explicitly construct many non-valence bond ground states (for example, Fig. 3.11). We will return to this question later but for now, we assume that valence bond states span the ground state manifold of our lattice. For the honeycomb lattice, Chayes et al.[56] have already written down the explicit form of the Klein Hamiltonian in terms of spin operators. Their expression, up to unimportant overall constants, is: H = si · sj + i,j + 1 si · sj 2 i,j 2 ( si · sj )( sk · sr ) 5 ijkr (3.6) The ﬁrst and second terms are over nearest and next nearest neighbors respectively. The third term is over quartets ijkr where i and j are nearest neighbors; k is a neighbor of i diﬀerent from j; and r is a neighbor of j diﬀerent from i. A striking feature of this Hamiltonian is that the leading term is the usual Heisenberg antiferromagnet. 3.3.2 Perturbations We will now perturb the Klein Hamiltonian to obtain a QDM with dynamics. In doing so we will use the overlap expansion invented by Rokhsar and Kivelson[49], which is predicated on the linear independence of the valence bond states. That valence bond states on the honeycomb lattice are linearly independent was proved in Ref. [56]. For the purpose of obtaining the dimer kinetic energy, it is suﬃcient to consider including just an additional nearest neighbor interaction[68], δH = si · sj (3.7) i,j To ﬁrst order in degenerate perturbation theory, we may write this as an eﬀective operator on the valence bond states. First, we deﬁne an orthonormal basis set {|α} in terms of the 47 linearly independent valence bond states {|i}: |α = (S −1/2 )α,i |i (3.8) i Here Sij = i|j is the overlap matrix element between valence bond states |i and |j. The magnitude of the overlap of two valence bond states may be determined by overlaying the two conﬁgurations forming what is called the transition graph[49]. The construction is described in Fig. 3.6. As shown in Fig. 3.6, the transition graph consists of double bonds, where the two states have a bond in common, and closed loops of varying (even) lengths. The magnitude of the overlap Sij is given by 2Nl i xLi where Nl is the number of loops; the product is over all such loops; Li is the length of loop i; and x = √1 . 2 The sign of Sij depends on how we choose to orient the bonds. By orientation, we refer to the fact that a bond between sites 1 and 2 may be interpreted as the singlet bond ψ12 = √1 (ψ ↑ ψ ↓ − ψ ↓ ψ ↑ ) 1 2 1 2 2 or as ψ21 = −ψ12 . The key idea of the overlap expansion is to treat x as a small expansion parameter. We may specify the matrix elements of our eﬀective operator in terms of the {|α} basis: Hαβ = (S −1/2 δHS −1/2 )αβ = (S −1/2 )αi i|δH|j(S −1/2 )jβ (3.9) (3.10) ij If either state |i or |j contains the bond (12), then i| s1 · s2 |j = − 34 i|j. If neither |i nor |j contains the bond (12), then we have a non-zero matrix element only if spins 1 and 2 are members of the same loop in the transition graph. If that is the case, then one may show that i| s1 · s2 |j = (−1)n/2 (∓ 34 )i|j where n is the length of the loop for the case where spins 1 and 2 are separated by an even (odd) number of sites. We now specialize to the honeycomb lattice. As sketched in the Appendix, we may orient the bonds on the honeycomb lattice so that for any two states diﬀering by a (minimal) length 6 loop, the overlap i|j has a positive sign. On the honeycomb lattice, the sign is a matter of convention and we could have chosen the negative sign. Given the positive sign convention, we conclude that our matrix element is given by: i|δH|j = − 3 3nd δij − (2x6 )ij + O(x10 ) 4 4 (3.11) 48 where nd is the number of bonds (half the number of sites) and ij is a matrix that is 1 if states |i and |j diﬀer by a length 6 loop and zero otherwise. We may also expand the overlap matrix: Sij (S −1/2 )ij = δij + 2x6 ij + O(x10 ) (3.12) = δij − x6 ij + O(x8 ) (3.13) Comparing the previous line with Eq. 3.8, we see that within the overlap expansion, each |α has a largest component corresponding to a unique valence bond state. Therefore, we will refer to the orthogonal set {|α} as the set of dimer states corresponding to the valence bond coverings. In writing our eﬀective operator, we absorb the leading term involving the number of dimers nd times the unit operator into our deﬁnition of zero energy. Hαβ = [δαi − x6 αi + O(x8 )] ij 3 × [− (2x6 )ij + O(x10 )] 4 × [δjβ − x6 jβ + O(x8 )] (3.14) 3 ≈ − (2x6 )αβ + O(x10 ) 4 (3.15) We conclude that the leading term in the overlap expansion is an operator with nonzero matrix elements only between dimer states diﬀering by a minimal length 6 loop. All of these nonzero elements have the same value −t = −(3/4)(2x6 ). Thus we have obtained the kinetic energy operator in the quantum dimer model (QDM). Note that we can conclude this only because we were able to deﬁne a bond orientation convention such that all minimal overlaps come with the same sign. Otherwise some oﬀ-diagonal terms would have energy t and the energy bound arguments which we gave previously to conclude that there is an RVB state at the RK point would no longer hold. To obtain the potential energy term as the leading order eﬀect, we need a more complicated interaction which we take to be: δH = J ij si · sj + 49 v ( s1 · s2 )( s3 · s4 )( s5 · s6 ) +( s2 · s3 )( s4 · s5 )( s6 · s1 ) (3.16) where the ﬁrst sum is over nearest neighbors and the second sum is over elementary plaquettes (see Fig. 3.7). The ﬁrst term of Eq. 3.16 gives the QDM kinetic energy. Similarly, consider the operator s12 s34 s56 = ( s1 · s2 )( s3 · s4 )( s5 · s6 ). If valence bond states |i and |j contain the bonds (12), (34), and (56), then i|s12 s34 s56 |j = (− 34 )3 i|j. If state |i contains all three bonds, then the diagonal matrix element i|s12 s34 s56 |i is (− 34 )3 . If state |i is missing one or more bonds, then the diagonal matrix element is zero unless |i contains the three complementary bonds (23), (45), and (61). In this case, the expectation of the operator equals (− 34 )3 x6 , which is higher order in the overlap expansion. It may be shown that oﬀ-diagonal matrix elements evaluate to a term proportional to the overlap of the states, the proportionality constant being of order unity. These results imply that the matrix element of the ring interaction between two valence bond states is given by: 3 3 − 4 vnf l,i δij + O(vx6 ) (3.17) where nf l,i is the number of ﬂippable hexagonal plaquettes in conﬁguration |i. We may write this in terms of dimer states, as discussed above. Absorbing numerical factors into the constants J and v, we arrive at our eﬀective dimer Hamiltonian: Hαβ = −Jx6 αβ + vnf l,α δαβ + O(vx6 + Jx10 ) = −tαβ + vnf l,α δαβ + O(vx6 + tx4 ) (3.18) where t = Jx6 and nf l,α is the number of ﬂippable plaquettes in the valence bond state corresponding to dimer state α. If t and v are of order unity, then the higher order terms will be small compared to the ﬁrst two terms, which act on our dimer states (which are really spin states) in a manner analogous to the QDM kinetic and potential energy operators on usual dimer states. 50 √ For the actual problem at hand, x = (1/ 2) is less than 1 but is by no means tiny. Hence the neglect of other terms induced by our perturbations is not obviously justiﬁed. While we do not need them to be zero, we do need them to be weak enough perturbations so the analysis of Ref. [49] is justiﬁed. What we do learn from the overlap expansion result (3.18) is that the non-orthogonality is a much smaller problem on more open lattices which involve large loops. While the honeycomb is a good candidate on this score, to put the issue beyond doubt we now consider a decorated version of the lattice. 3.3.3 Decoration scheme In this section, we propose a modiﬁcation to our earlier arguments which makes the overlap expansion essentially exact. Consider the decorated honeycomb lattice shown in Fig. 3.8 where we insert N (an even integer) sites between neighboring sites of the usual honeycomb lattice. The dimer structure of this lattice, including the number of dimer states, is exactly the same as before except that having a dimer between sites 1 and 2 corresponds to a chain of (N + 2)/2 dimers beginning at site 1 and ending at site 2. Not having a dimer between sites 1 and 2 corresponds to having a chain of N/2 dimers beginning at site b1 and ending at site a2 . The Klein Hamiltonian is correspondingly modiﬁed by including Klein projectors for the added sites. Majumdar and Ghosh [46] showed that the valence bond state is the only ground state of the Klein Hamiltonian for a one-dimensional spin chain with an even number of spins. Therefore, the conclusions regarding the Klein model on the honeycomb lattice (linear independence of valence bond states, valence bond states span the ground state manifold, etc) carry over directly to the decorated honeycomb lattice. While decorating does not introduce any new technical problems, there is a signiﬁcant technical advantage with respect to the overlap expansion. The smallest two loops on the hexagonal lattice are length 6 and 10 from which we obtained that the relative orders of the leading and error terms in the overlap expansion were x6 and x10 . The smallest loops 51 on the decorated hexagonal lattice have lengths 6(N + 1) and 10(N + 1). Repeating the previous analysis, we will ﬁnd that the leading and error terms in the overlap expansion are x6(N +1) and x10(N +1) . The ratio of error term to leading term has improved from x4 to x4(N +1) . In the large N limit, the error term is “rigorously” negligible but we propose that even fairly small values of N may suﬃce to capture the qualitative features of the large N limit. While we have added complexity to the lattice, we do not have to increase the order of the spin interaction. Consider the following as a perturbation to the decorated honeycomb lattice Klein model: δH = J si · sj + ij v ( s1 · sb1 )( s3 · sb3 )( s5 · sb5 ) +( s1 · sa1 )( s3 · sa3 )( s5 · sa5 ) (3.19) The ﬁrst term is a nearest neighbor interaction over all spins while the second term is over all elementary plaquettes, such as the one in Fig. 3.8. A 6 spin interaction is suﬃcient, even though we have many more spins in the loop, because having a (1b1 ) bond automatically implies the other bonds in the chain connecting 1 and 2. Our previous analysis carries over to the present case and we conclude: Hαβ = −Jx6(N +1) αβ + vnf l,α δαβ + O(vx6(N +1) + Jx10(N +1) ) = −tαβ + vnf l,α δαβ + O(vx6(N +1) + tx4N ) (3.20) where t = Jx6(N +1) and otherwise the notation is the same. Clearly, by decorating enough we can make the matrix elements beyond the dimer model arbitarily small and thus realize the physics, including Cantor deconﬁnement, present in generic, weak perturbations of the honeycomb QDM. 52 Spinons In the highly decorated limit, one may show that nearest-neighbor valence bond states are separated by a ﬁnite gap from the excited states of the Klein model. In this limit, we are connecting a set of Majumdar-Ghosh[46] (MG) chains into a two-dimensional network. We may describe the low energy excited states of our system in terms of the well studied spinon defects of the MG chains, which are widely believed to be gapped[69, 70, 71]. Here we give an outline of our argument and relegate technical details to Appendix E. In Appendix E, we consider what happens when we put these chains together for diﬀerent values of a tunable parameter in our model: the ratio of Klein scales (the coeﬃcient Ci in Eq. 3.5) of the Klein projectors of the decorated and reference sites. For large values of this ratio, the excited states are represented by “microscopic” spinons localized on the reference sites. For small values of the Klein ratio, the excited states are extended and may be interpreted as MG spinons scattering at the vertices. There is a ﬁrst order transition between these limits. In both limits, there is a gap between the VB manifold and the spinon states, as depicted in Fig. 3.9. Spinons are the natural excited states (outside of the VB manifold) in the high decoration limit. For an unperturbed Klein model, the VB manifold is degenerate so these excitations are mobile. The next question is what happens when the degeneracy of the ground state manifold is lifted. We argue that this has a small but vital eﬀect on the spinon dynamics. At the RK point and in liquid phases, we expect the spinons to be deconﬁned. In the crystalline phases, we may consider a pair of test spinons, holding one member ﬁxed and considering the quantum mechanics of the other. If the wavefunction of the non-ﬁxed spinon has spatial extent L, this would have an energy cost of order c L2 where c is the energy cost per unit area of scrambling the crystalline background. The L-dependence of the kinetic energy varies as 1/L2 . The implication is that while c is a much smaller scale than the spinon gap, spinons moving in a crystalline background are conﬁned at suﬃciently long length scales. 53 3.3.4 Square lattice The square lattice is another two-dimensional bipartite lattice for which (nearest-neighbor) valence bond states are linearly independent[56]. We may orient the bonds on the square lattice so that two states diﬀering by a (minimal) length 4 loop, have positive overlap. The problem with applying our approach to the bare square lattice is that the Klein model has ground states outside of the valence bond manifold (Fig. 3.10). These extra states do not arise when we consider the decorated square lattice. Consider perturbing a decorated square lattice Klein model with: δH = J si · sj + ij v ( s1 · sb1 )( s3 · sb3 ) + ( s1 · sa1 )( s3 · sa3 ) (3.21) where the ﬁrst term is a nearest neighbor interaction and the second sum is over all elementary plaquettes, the spins 1234 labelling the 4 sites of a square plaquette in clockwise order and, as before (see Fig. 3.8), the labels ai and bi denoting the ﬁrst counterclockwise and ﬁrst clockwise neighbor of spin i. By arguments similar to the honeycomb case, we obtain an eﬀective Hamiltonian: Hαβ = −Jx4(N +1) αβ + vnf l,α δαβ + O(vx4(N +1) + Jx6(N +1) ) = −tαβ + vnf l,α δαβ + O(vx4(N +1) + tx2N ) (3.22) Here ij is a matrix that is 1 if states |i and |j diﬀer by the (minimal) length 4 loop and zero otherwise. Therefore, we realize the physics of the square lattice QDM. Note that without the decoration, the error would be order x2 = (1/2), as opposed to the bare honeycomb case where the error is order x4 = (1/4). We also emphasize that these “small” terms multiply matrices so in the undecorated case, the overlap expansion is not obviously justiﬁed. 54 3.4 Other Valence Bond Phases in d = 2 and d = 3 We now sketch the application of our strategy to obtain the rest of dimer model physics, including RVB phases. The points to be made concern the choice of lattices and phase conventions. 3.4.1 Non-bipartite lattices in d = 2 The simplest d = 2 non-bipartite lattice is the triangular lattice. Numerical evidence suggests that (nearest-neighbor) valence bond states are linearly independent[65]. As with the square and honeycomb lattices, we may orient bonds so that states diﬀering by a (minimal) length 4 loop, have positive overlap.[72] As with the square lattice, the Klein model admits non-valence bond ground states, though the problem is more serious with the triangular lattice (see Fig. 3.11). Decoration eliminates these possibilities by removing the triangular nearest-neighbor structures. Applying our strategy to the decorated triangular lattice allows us to reproduce the physics of the triangular lattice QDM, including its RVB phase[51]. By calculations similar to Appendix E, one may show that spinon excitations are gapped. For the triangular lattice, it is known that collective excitations within the valence bond manifold are also gapped[51, 73], a conclusion which will remain valid for the decorated case. Therefore, we have constructed a model that shows a stable, SU(2)-invariant RVB liquid phase. Another non-bipartite lattice is the pentagonal lattice[74] shown in Fig. 3.12. There are currently no formal proofs for the pentagonal lattice regarding the issues of linear independence of (nearest-neighbor) valence bond states and whether the set of these states spans the ground state space of the corresponding Klein model. However, it was explained in Ref. [56] that the most important ingredients of their proofs for the honeycomb lattice are its relatively low coordination number (3); relatively large minimum loop size (6); and the absence of triangular structures in the lattice. The pentagonal lattice, has sites of coordination 3 and 4, minimum loop size 8, and no triangular structures, suggesting that the arguments may be adapted to this lattice. As before, it is possible to orient bonds so 55 that the overlap of states diﬀering by a (minimal) length 8 loop always has the same sign. While in the square, triangle, and honeycomb cases, the sign of the overlap is a matter of convention (which we chose as positive), for the pentagonal lattice, only the negative sign is possible. In fact, in Appendix D, it is shown that using the fermionic convention, one may always obtain the negative sign independent of lattice details. Therefore, to generate the QDM kinetic energy, we must perturb the Klein model with a ferromagnetic interaction. From a perturbation of the form: δH = −J si · sj + ij v ( s1 · sb1 )( s3 · sb3 )( s5 · sb5 )( s7 · sb7 ) +( s1 · sa1 )( s3 · sa3 )( s5 · sa5 )( s7 · sa7 ) (3.23) where the ﬁrst term is over all nearest-neighbor spins and the second term is over elementary plaquettes (ai and bi , once again, denoting the ﬁrst counterclockwise and ﬁrst clockwise neighbors of spin i), we may obtain the quantum dimer Hamiltonian for the decorated pentagonal lattice: Hαβ = −tαβ + vnf l,α δαβ + O(vx8(N +1) + tx2N ) (3.24) Here ij is a matrix that is 1 if states |i and |j diﬀer by the (minimal) length 8 loop and zero otherwise. Therefore, we realize the physics of the pentagonal lattice QDM. We have checked that in the classical limit, the dimer-dimer correlations decay exponentially and in Appendix F, we present numerical evidence that monomers are deconﬁned. As both features also hold at the RK point, we may repeat the arguments described in Ref. [51] for the triangular lattice to conclude that the pentagonal lattice QDM also shows an RVB liquid phase, a result which may be transcribed into spin language as discussed above. 56 3.4.2 Non-bipartite lattices in d = 3 The face-centered cubic (FCC) is a three-dimensional non-bipartite Bravais lattice with each site having 12 nearest-neighbors. The undecorated lattice has triangular structures involving two neighboring facial sites and the two corners which are their common neighbors. This will lead to non-valence bond ground states in the FCC Klein model. Decoration eliminates the triangular structure and hence this type of pathology. The shortest resonance loops are length 4. In the fermion sign convention, these loops come with negative sign. A perturbation consisting of a ferromagnetic exchange and 4-spin resonance interaction will reproduce the QDM results for the decorated lattice. We expect the resulting model to show a Z2 RVB phase near its RK point[52]. 3.4.3 Bipartite lattices in d = 3 For the diamond lattice we pursue the same strategy as above. The properties of the diamond lattice we require are the following. The diamond lattice is bipartite, has coordination four, and the shortest resonance loops are of length six. The Klein Hamiltonian again has nearest-neighbour dimer coverings as ground states, although, as for the case of the pentagonal lattice, no theorem exists excluding other ground states. It is likely that extra states may be excluded by decorating the lattice. The number of dimer ground states, ngs , is exponentially large in the number of sites, N , but it is not known exactly. Deﬁning the ground state entropy per site as S = (1/N ) ln ngs , an accurate series expansion by Nagle[29] yields S ≈ 0.265. We now try to mimic an RK quantum dimer model for the diamond lattice. As for the case of the honeycomb lattice, we do this by adding a nearest-neighbour exchange term to induce a kinetic term and in addition, a ring term to generate a potential term. We then expect the resulting model to exhibit, near the eﬀective RK point, a U(1) RVB liquid phase with algebraically decaying correlations as well as gapless photonic gauge excitations, as discussed in detail in Ref. [53]. This liquid phase will give way, upon making the potential term more attractive, to 57 a columnar-type solid. For an increasingly repulsive potential, the scenario of Cantor deconﬁnement predicted for the two-dimensional case is simpliﬁed. Technically, there are no relevant lock-in terms in three dimensions so that the deconﬁned region simply acquires an increasing amount of U(1) ﬂux as v/t is increased through the RK point; ﬁnally, a staggered solid, with the maximal amount of U(1) ﬂux allowed by microscopic constraints,[52, 53, 55] is reached. We cannot say whether this will happen continuously or via a ﬁrst order transition. 3.5 Dynamical selection of gauge structures: pyrochlore lattice We construct a Klein model with Z2 order and a Kivelson-Klein model with U (1) order which takes advantage of the bipartiteness of the dual lattice. This nicely illustrates the dynamical selection of the low-energy gauge structure present in topological phases. 3.5.1 The Klein model The undecorated pyrochlore lattice (Fig. 3.13) does not lend itself straightforwardly as a starting point for dimer models obtained via the Klein route because its basic building block, the tetrahedron, supports more dimer coverings than linearly independent singlet states. By suﬃciently decorating the lattice, the orthogonality problem is solved. The shortest resonance loops are length 6 and the fermionic convention may be used to make the minimal overlaps come with negative sign. Perturbing the decorated pyrochlore Klein model with a ferromagnetic nearest-neighbor interaction and 6-spin ring interaction, we obtain an eﬀective Hamiltonian mimicking the pyrochlore lattice QDM, which includes a Z2 RVB phase. 58 3.5.2 The Kivelson-Klein model A modiﬁed version of the Klein Hamiltonian, which we will refer to as the Kivelson-Klein Hamiltonian, may be used to produce a model displaying a U (1) RVB phase. [75] Its Hamiltonian is of the same form as Eq. 3.3 but the deﬁnition of N (i) is changed. The projection now acts not on a site and its nearest neighbours but instead on the four sites of a tetrahedron: HKK = P̂tet . (3.25) tet We note that the simple decoration trick described above cannot as usefully be applied to the Kivelson-Klein model, as here increasing the number of sites in the tetrahedron does not lead to an increase in the number of dimers required. Ground states of the Kivelson-Klein model Evidently, each state in which each tetrahedron contains at least one singlet bond is a ground state of the above Hamiltonian, Eq. 3.25. How can this be related to dimer coverings of the pyrochlore lattice? First, note that (i) the number of tetrahedra equals twice the number of sites, Nt = 2N , and that (ii) the number of hardcore dimers, Nd ≤ N/2, as a dimer involves two sites. From this it follows that Nd ≤ Nt , the equality sign holding for hardcore dimer coverings. However, the requirement of having at least one dimer in each tetrahedron gives (iii) Nd ≥ Nt . We therefore see that (ii) and (iii) imply that those hardcore dimer coverings of the pyrochlore lattice for which each tetrahedron contains exactly one dimer are ground states of the Kivelson-Klein Hamiltonian. The ensemble of these states maps onto the ground states of the six-vertex model on the diamond lattice, or equivalently onto the ground states of the pyrochlore Ising antiferromagnet. This can be seen as follows. First, note that the lattice of tetrahedra deﬁned by the pyrochlore lattice is the bipartite diamond lattice. One diamond sublattice sits at the centre of the ‘up’ tetrahedra, the other one at the centres of the ‘down’ tetrahedra. Now let us deﬁne the Ising spins on the pyrochlore lattice as follows. For an up (down) tetrahedra, 59 the pair of spins at the two ends of a dimer point up (down), and the other pair points down (up). This deﬁnes a one-to-one mapping of dimer to Ising states; crucially, on each tetrahedron, two spins point up and two point down, thus putting the tetrahedron into an Ising ground state. (The mapping to the six-vertex model on the diamond lattice proceeds by calling an up (down) spin an arrow pointing from the centre of an up (down) tetrahedron to a down (up) tetrahedron). Ground state correlations The total number of ground states (assuming that there are none in addition to the abovementioned dimer states) gives rise to an extensive ground state entropy well-approximated by the Pauling entropy SP auling = (1/2) ln(3/2). Using the mapping to an Ising magnet, it is straightforward to calculate the correlator between singlet bonds averaged over the ground state manifold. To do this, note that each of the six dimer positions on a bond of a given tetrahedron corresponds to an Ising ground state of that tetrahedron. In turn, the corresponding vertex of the six-vertex model describes a net ﬂux, the direction of which is given as follows. Consider a cube circumscribing the tetrahedron in question, so that the bonds of the tetrahedron are face diagonals of the circumscribing cube. The direction of the ﬂux (i.e., the average direction of the four arrows of the given vertex) now points from the centre of the cube through the midpoint of the face of which the bond occupied by the dimer resides. Using the theory developed in Ref. [52], one can read oﬀ that the dimer correlations are simply dipolar. Brieﬂy, this follows from the observation that, upon coarse graining, the smaller the coarse-grained ﬂux, the more microstates (prior to coarse-graining) correspond it. Modelling this by an eﬀective quadratic weight on the ﬂux conﬁgurations leads to simple magnetostatics. For example, the connected correlator between a pair of dimers located on the top of an up tetrahedra, separated by a vector r which makes angle θ with the z axis, is proportional to the dipolar form (3 cos θ 2 − 1)/r 3 . 60 Finally, this model can again in principle be “Rokhsar-Kivelsonized”, i.e. by adding appropriate perturbations to Eq. 3.25, we may generate an eﬀective Hamiltonian which acts on the Kivelson-Klein ground state manifold in a manner similar to Eq. 3.2 on the space of dimer coverings. We do not do this here for a number of reasons. Firstly, the expected phase diagram has the same topology as that discussed for the diamond lattice in the previous section, so no new phases are obtained. Secondly, the dimer dynamics is rather messy. The shortest resonance loop now involves six dimers straddling a hexagonal loop of the pyrochlore lattice – and there are several symmetry-inequivalent loops of this type. (This is reminiscent of the – exactly soluble – kagome dimer model proposed in Ref. [76].) In addition, for a simple perturbing nearest-neighbour exchange, the resonance term vanishes in the leading order of the overlap expansion, so that a more complex perturbation is needed. 3.6 Discussion and outlook We have presented spin-1/2 Heisenberg Hamiltonians that realize a large class of valence bond phases. In particular they realize Z2 RVB phases in d = 2 and d = 3, the U (1) RVB phase in d = 3 and the Cantor deconﬁned region in d = 2. These phases have previously been shown to exist in quantum dimer models with dimers standing in for valence bonds. In this paper we have constructed Klein models that exhibit ground state manifolds spanned by nearest neighbor valence bond states and then perturbed them to to realize quantum dimer models within these manifolds. This perturbation is done within the framework provided by the overlap expansion, made arbitrarily accurate by a decoration procedure that we have introduced. The decoration has the eﬀect of expanding the length scale on which the Hamiltonian acts directly. However in order to stabilize the phases in quantum dimer models, we do not need to go to inﬁnite decoration – it would be enough to suppress subleading terms suﬃciently. In this fashion we obtain spin models with interactions of ﬁnite range[82]. While large decorations are needed to realize the simplest dimer models under analytic control, there is little reason to doubt that the extent of decoration could be reduced drastically, 61 and even eliminated altogether on some of the lattices without sacriﬁcing the various phases of interest. The subleading interactions will not necessarily uniformly tend to subvert such phases and it is also possible to add other terms that would stabilize them. Showing how this can work is an obvious task for the future, as is the construction of mathematically rigorous proofs for various statements in this paper that are made by appealing to a small parameter. One promising approach is that of Ref. [77], which appeared at the same time as our work, where the phases of the d = 2 bipartite quantum dimer model are realized by perturbing a Klein model with ring exchange terms. We note that the RVB phase on the triangular lattice is generically stable to small perturbations, including (perhaps) the error term resulting from a ﬁnite decoration, because it is gapped. We emphasize that our central result has been the demonstration that spin liquid phases can be realized in SU(2)-invariant models. However, the actual models we have given involve rather complicated geometries and Hamiltonians without direct experimental relevance. While it may be possible to engineer highly decorated lattices, a more important task perhaps, now that the question of principle is settled, is to refocus on studying much simpler Hamiltonians. Our insistence on a speciﬁed form of the wavefunction (containing nearest neighbor valence bonds alone) has led to fairly complicated Hamiltonians but simpler Hamiltonians can exhibit the same phases with more elaborate ground state wavefunctions. Indeed, the situation for the simplest lattice under consideration in this paper, the (undecorated) honeycomb lattice, looks quite promising. Previous exact diagonalisations of a J1 − J2 − J3 Heisenberg model on this lattice[78] have clearly demonstrated the existence of the staggered VBS. The current data appears not inconsistent with a scenario in which the magnet leaves the staggered phase but never reaches the ﬂuxless plaquette phase. We are optimistic that it will be possible to realize the physics discussed in this chapter in such simpler models. 62 Figure 3.3: The kinetic (potential) energy operator of the simplest quantum dimer model on a given lattice will ﬂip (count) the dimers on the simplest ﬂippable plaquette. The basic resonance move for the honeycomb lattice and the three basic moves on the triangular lattice are shown. The construction may be obviously generalized to arbitrary lattices and dimensions. 63 Figure 3.4: Phase diagram for a bipartite QDM in d=2 from Fradkin et. al. [55]. i i -i i i i RVB columnar 12x 12 columnar 0 columnar/plaquette staggered 1 staggered v/t Figure 3.5: Phase diagram for a non-bipartite QDM in d=2 from Moessner and Sondhi[51]. 64 Figure 3.6: Transition graph construction for two valence bond coverings of the square lattice (the construction for other lattices is similar). The dots are the lattice sites and the thick lines denote that the two sites form a singlet bond. The singlet orientation may be speciﬁed, for example, by having the bonds point from the gray sites to the white sites. The transition graph is formed by overlaying the two conﬁgurations resulting in a graph (lower) containing double bonds and closed loops of varying (even) lengths. In the above example, there are two double bonds and three loops of lengths 4, 6, and 16. The magnitude the two valence bond coverings is then given by √ of the overlap between 1 . Thus, while the overlap between two arbitrary |S| = 23 (1/ 2)4+6+16 = ( 12 )10 = 1024 valence bond coverings is never zero, it is usually a small number. This is the basis for the overlap expansion discussed in the text. 65 1 6 2 5 3 4 Figure 3.7: The elementary plaquette of the honeycomb lattice. 66 a1 1 b6 a6 b5 b1 a2 6 2 5 3 a5 a3 b3 4 b4 b2 a4 Figure 3.8: The decorated honeycomb lattice where N (an even integer) two-fold sites are inserted between the old sites. This drawing shows N = 4. The labels a1 and b1 designate the ﬁrst counterclockwise and ﬁrst clockwise neighbor of spin 1 where clockwise is with respect to the loop 123456. For the undecorated case, a1 and b1 are just sites 6 and 2. 67 E MG spinon continuum MG spinon continuum Micro-spinon band VB excitations QDM ground state VB excitations QDM ground state Figure 3.9: A cartoon of the spectra for the limiting cases where the Klein ratio is large (left) and small (right). In the small Klein ratio case (right), the lowest spin excitations are described by extended spinon wavefunctions. These scattering states are present even in the large ratio case (left) but here the lowest excitations, which we call “microscopic” spinons, are described by a wavefunction having peaks at the reference sites and decaying on the chains. The decay rate can be made arbitrarily fast by tuning the Klein ratio. The bandwidth of these localized states depends on the decoration and is zero in the inﬁnite decoration limit. Whether or not the valence bond excitations are gapped depends on the details of the speciﬁc QDM. For example, the topological phase on the triangular lattice is gapped while the RK point on bipartite lattices have gapless excitations. Also, the valence bond excitations will not, in general, have a ﬁnite bandwidth; the ﬁgure is drawn this way to emphasize the spin gap. Details are given in Appendix E. 68 Figure 3.10: The Klein model for the square lattice, with periodic boundary conditions, permits ground states which are not in manifold spanned by nearest-neighbor valence bond states such as this one. Here the thin lines form the lattice and the thick lines denote singlet pairings. Note that the Klein condition is satisﬁed at every lattice site. Figure 3.11: The Klein model for the triangular lattice admits many nontrivial non-dimer ground states, such as this one. The thin lines show the lattice and the thick lines denote singlet pairings. The dots represent free spins. Note that the Klein condition is satisﬁed at every lattice site. 69 Figure 3.12: The pentagonal (Sutherland-Shastry) lattice. It is the dual of the ShastrySutherland lattice, which is indicated by the dashed lines. Figure 3.13: The pyrochlore lattice, a network of corner-sharing tetrahedra Appendix A An overview of height representation theory A.1 The height representation The height representation provides a natural way to construct ﬁeld theories of dimer models on bipartite lattices. While our primary concern is the two-dimensional honeycomb lattice, the height mapping applies to any bipartite lattice and may be generalized to arbitrary dimension[52]. The strategy is to ﬁnd a free energy functional which satisﬁes the basic symmetries of the microscopic system and reproduces the important long distance characteristics, such as the decay of dimer-dimer correlations. We then use this functional to consider questions not easily addressed using the microscopic model, such as the nature of the interaction of certain excitations or the stability of the model to generic perturbations. A dimer covering of the honeycomb lattice may be represented by assigning a height variable hi to each site i of the triangular lattice dual to the hexagonal lattice on which the dimers lie. This dual lattice is formed by the centers of the hexagons. The heights are assigned as follows. Assign a value of zero height to some arbitrary site on the triangular lattice. Moving from this site to a nearest neighbor site by moving clockwise around an up(down-) triangle, increase (decrease) the height by +2 (-2) if a dimer is crossed. If a dimer 70 71 is not crossed, then decrease (increase) the height by -1 (+1). According to these rules, traversing a closed loop counter-clockwise in the dual lattice will result in a height diﬀerence of 0 if the constraint of one dimer per site is satisﬁed everywhere inside. Deviations from this constraint (monomers or multiple dimers on a site) manifest as vortices in the height variable. For example, the height diﬀerence is +3 (-3) if a single positive (negative) monomer is enclosed by the loop. Examples of dimer coverings and the associated height mappings are given in Fig. A.1. In a coarse-grained description, the hi are replaced by a real, continuum ﬁeld h(x). We now consider two examples of how this representation may be used to construct ﬁeld theories of dimer models and some applications relevant to this thesis. A.2 A.2.1 Application to spin ice Classical dimers on the honeycomb lattice The tilt of a conﬁguration is the average value of the gradient of its height ﬁeld in a given direction, for example: 1 L |∇x h|dx. In the absence of potentials, the states with small (or zero) tilt will be favored in the free energy for entropy reasons (i.e. there are more such states). This motivates the following guess for the free energy (which is entropic): F {h(x}) = −S{h(x)} = T K dx 2 |∇h|2 + V (h(x)) (A.1) The gradient term penalizes tilted states and the constant K will be determined below. The form of the potential energy may be determined by noting that the physics should not depend on which site we choose as our zero height reference point. Choosing a diﬀerent origin is equivalent to uniformly shifting the heights: h(x) → h(x) + c where c is some integer. In our model, the minimum such shift is one unit so the potential V must be a periodic function of h with period one. A common simpliﬁcation is to approximate V by its lowest frequency component: V (h((x)) ≈ λ cos(2πh) (A.2) 72 If λ < 0, this term favors conﬁgurations where the height function takes integer values everywhere and when combined with the ﬁrst term, would favor the system locking into smooth, ﬂat conﬁgurations. As discussed in Ref. [32], the actual outcome depends on the value of K and it turns out that this phenomenological model also exhibits a rough phase where the potential term is irrelevant (in the sense of the renormalization group). The spin ice [111] magnetization plateau is described by this rough phase so we will neglect the potential term in the following. The next step is to choose the constant K so that our eﬀective theory reproduces the long distance behavior of the microscopic classical dimer problem. For the honeycomb lattice, this problem has an exact solution and the long distance behavior of the dimer-dimer correlation function where both dimers are oriented vertically (labelled the “1” direction below; see Fig. A.1) is: c11 (x) = n1 (x)n1 (0) = 1 1 4πx ) − cos(2θ) cos( 2π 2 r 2 3 (A.3) where r and θ are the usual polar coordinates and n1 (x) is 1 if there is a vertical dimer on the link to the right of position x (which labels a site on the dual height lattice) and 0 otherwise. In terms of heights, the density of dimers having orientation κ is given by nκ = 1−(êκ ·∇lat )h /3 where ∇lat is the lattice gradient and êκ is a unit vector perpendicular to the dimer orientation κ. From this, we may guess the corresponding form in the continuum case: nκ (x) = 1 − (êκ · ∇)h + fκ (h(x), x) 3 (A.4) where fκ (h(x), x) is a function whose form we will now determine. While naively, we may expect just the ﬁrst term to be the correct continuum expression, it turns out (as will be shown below) that this term is insuﬃcient in reproducing the behavior given in Eq. A.3. Because nκ is a local quantity, we expect fκ (h(x), x) will depend only on the local arrangement of dimers which translates into a local distribution of height diﬀerences. Suppose a particular dimer pattern occurs in some part of the lattice and the same dimer pattern occurs elsewhere. Then the value of nκ should be the same in both regions. While by 73 construction, both regions will have the same distributions of height diﬀerences, the values of the heights themselves will in general be uniformly shifted from region to region. An example of this is given in Fig. A.2 which indicates that for the honeycomb (square) lattice, this uniform shift is 3 (4). This implies that for the honeycomb lattice, the function fκ (h(x), x) should be a periodic function of h(x) with period 3. The simplest form of such a function is: fκ (h(x), x) = gκ (x)e 2πih(x) 3 + c.c. (A.5) More generally, we may include terms for the higher wavevectors in height space that also 2π have period 3 ( 4π 3 , 2π, etc.) but the lowest one ( 3 ) will be the most relevant in terms of the long distance behavior. The function gκ (x) is chosen in order to match Eq. A.3. It turns out that gκ (x) = λκ e4πix/3 where λκ is a constant and x refers to the horizontal coordinate (not |x|). This is related to the fact that the macroscopically ﬂat states will have a microscopic height ﬂuctuation with a characteristic wavevector of 1 12 lattice constants (see Fig.A.1). With this conjectured form for the dimer density, we may compute the dimer-dimer correlation function for vertical (κ = 1) dimers. We may proceed using the familiar path integral approach. Our continuum theory implies a partition function given by: Z[J] = D[h]e−S[h;J] (A.6) where the integral is over all possible height conﬁgurations; J(x) is a source term; and the action is given by: S[h; J] = 1 2 1 1 d2 q 2 Kq h h + J h + J h q −q q q −q −q (2π)2 Kq 2 Kq 2 (A.7) where hq denotes the (2d) Fourier transform of h(x) and so on. Performing the Gaussian integral, we obtain: Z[J] = Z[0] exp 1 2 d2 q Jq J−q (2π)2 Kq 2 (A.8) from which we may conclude: hq1 h−q2 = δ2 Z 1 = (2π)2 δ(q1 − q2 ) J=0 δJq1 δJ−q2 Kq12 (A.9) 74 and: 1 (h(x) − h(0))2 = (h(0)2 − h(x)h(0)) 2 d2 q1 d2 q2 (1 − eiq1 ·x )hq1 h−q2 = (2π)2 (2π)2 d2 q (1 − cos q · x) = (2π)2 Kq 2 r 1 ln( ) ≈ 2πK a (A.10) where a << r is a length scale (of order the lattice spacing) associated with regularizing the potential at short distances. This information may be used to obtain the dimer-dimer correlator: n1 (x)n1 (0) = 2πi 1 1 + ∂x h(x)∂x h(0) + λ2 e4πix/3 e 3 (h(x)−h(0) + c.c. (A.11) 9 9 Each of the two averages may be computed using the above results. We obtain for the ﬁrst term: ∂x h(x)∂x h(0) = = = = = This expression, choosing K = π 9, d2 q1 d2 q2 iq1 ·x 2 e qx1 hq1 h−q2 (2π)2 (2π)2 d2 q qx2 eiq·x (2π)2 Kq 2 d2 d2 q 1 − cos q · x dx2 (2π)2 Kq 2 r d2 1 ln( ) 2 dx 2πK a 1 y 2 − x2 2πK r 4 (A.12) reproduces the second term of Eq. A.3. We obtain the other term by using Eq. A.8 with Jq = (eiq·r − 1). e 2πi (h(x)−h(0)) 3 With K = π/9 and λ2 = 1 4π 2 a2 , 2πi )2 d2 q 1 − cos q · r 3 (2π)2 Kq 2 4π 2 1 r ln( ) = exp − 9 2πK a r − 2π2 9K = a = exp ( we obtain Eq. A.3. (A.13) 75 A.2.2 Interaction between defects in spin ice As mentioned in the main text, spin ice on the [111] plateau is a layered system, each layer being described by a classical dimer model. A string defect involves creating a pair of defects in each layer with an added interlayer constraint. The defects are sites of the honeycomb lattice having two dimers; the two defects are on diﬀerent sublattices so may be labelled as “positive” and “negative”. In the continuum theory, we consider a defect density function σ(x) and the multivaluedness of the height ﬁeld is expressed in the equation: C =3 ∇h · dr d2 rσ( r) (A.14) S where S is the area enclosed by the loop C. We may proceed by analogy with the 2d XY model and divide h into “dimer” (spin-wave) and “defect” (vortex) contributions. We follow the presentation of Ref.[31]. h(x) = hsw (x) + hv (x) (A.15) ∇ · ∇hv = 0 (A.16) ∇ × ∇hsw = 0 (A.17) where ∂hv ∂ h̃v ∂ h̃v 2 v It is useful to deﬁne the ﬁeld h̃v such that ( ∂h ∂x , − ∂y ) = ( ∂y , − ∂x ). Notice that |∇hv | = |∇h̃v |2 and Eq. A.14 may be rewritten (using Stokes’s theorem) as: ∇2 h̃v (x) = 3σ(x) (A.18) The equation may be solved using Green’s functions: h̃v (x) = 3 2π dyσ(y) ln( |x − y| ) a Using the above information, we may rewrite the Hamiltonian for the system as: F {h(x}) T K dx|∇h|2 2 K = dx(|∇hsw |2 + |∇hv |2 ) 2 F {h(x}) K + dx|∇h̃v |2 = sw T 2 = (A.19) 76 F {h(x}) K − dxh̃v ∇2 h̃v = sw T 2 F {h(x}) 9K |x − y| = + dxdyσ(x)σ(y) − ln sw T 4π a F {h(x}) 1 |x − y| = + dxdyσ(x)σ(y) −κ ln sw T 2 a (A.20) where κ = 1/2. This shows that the interaction between two defects separated by distance r is given by p1 p2 V (|r 1 − r 2 |) where pi is +1 (-1) for a positive (negative) defect and V (R) = −κ ln(R/a). A.2.3 Summary To summarize, we provided arguments for why the free energy for classical dimers on the honeycomb lattice is given by: K F {h(x}) = T 2 where K = π 9 dx|∇h|2 (A.21) for the honeycomb lattice. If we were working on a lattice where we did not know the exact solution, then this constant K would need to be determined numerically by matching the continuum dimer-dimer correlator with numerical simulation of the correlator of the microscopic model. We then used this to determine the coupling constant of the interaction between defects which was required in the analysis of string defects in spin ice discussed in the main text. We emphasize that this interaction between defects is entropic in origin. It is instructive to phrase this in terms of the microscopic model. If two defects are ﬁxed at positions r1 and r2 and everywhere else, the one dimer per site rule is obeyed, we may ask how many dimer coverings are consistent with the constraint and how this number changes as the defects are moved. If moving the defects increases the number of dimer coverings of the remaining sites, then the defects will move in order to maximize the dimer entropy. What the above analysis yielded is the nontrivial result that the number of dimer coverings scales with defect separation r as r −1/2 at long distances. 77 A.3 A.3.1 Application to quantum dimer models Quantum dimers on bipartite lattices The height representation provides a route to constructing a ﬁeld theory for quantum dimers moving under a quantum dimer Hamiltonian. The action conjectured by Henley[25] for this problem is sometimes called the quantum Lifshitz model: S[h] = 1 d2 x 1 1 (∂τ h)2 + ρ2 (∇h)2 + ρ4 (∇2 h)2 + λ cos(2πh) 2 2 2 (A.22) The coeﬃcient −ρ2 = (v/t)−1 where t and v are the coeﬃcients of the kinetic and potential energy terms in the RK quantum dimer Hamiltonian. The coeﬃcient ρ4 depends on the lattice (for the honeycomb, ρ4 = (π/18)2 ) and, similar to the classical case discussed above, is chosen to correctly recover the dimer-dimer correlations. A special property of the RK point (where v = t) is that the correlations are the same as in the classical model so the procedure is literally the same as what we used above to calculate K, except the action is now described by Eq. A.22. Eq. A.22 is the simplest action which has a number of features which are required to discuss the quantum dimer model but is not easily deduced by deductive argument. Moessner et.al. [50] motivated this form by noting that the gradient-squared term (with ρ2 > 0) and the cosine term (with λ < 0) favor the ﬂat states (columnar and plaquette phases). If ρ2 < 0, then the quadratic term favors the tilted state (staggered phase). At the RK point itself, quadratic term disappears, the cosine term is irrelevant, and the remaining Laplacian squared term does not distinguish between the ﬂat and tilted states. This qualitatively describes the bipartite QDM phase diagram. A.3.2 Interaction of defects As in the classical case, monomer defects may be described as vortices in the height ﬁeld and Eqs. A.14 and A.18 will still apply. We may use this to rewrite the action at the RK point for the case of time independent correlations: S[h] = 1 d2 x ρ4 (∇2 h)2 2 78 = = 9 d2 x ρ4 σ(x)2 2 9ρ4 dxdyσ(x)σ(y)δ(x − y) 2 (A.23) where we have dropped the irrelevant cosine term. In contrast to Eq. A.20, we ﬁnd that monomer defects at the RK point do not interact with one another beyond having hard cores. Therefore, we obtain that monomer defects are deconﬁned at the RK point. A.3.3 Stability of the RK point Fradkin et. al. [55] considered the inﬂuence of generic perturbations on the action (A.22) at the RK point. We will summarize the results of their work of particular importance to this thesis. The ﬁrst observation was that: √ √ 1 1 3 3 ∂y h)( ∂x h + ∂y h) S3 = g3 (∂x h)( ∂x h − 2 2 2 2 (A.24) is a relevant perturbation that will, when g3 is nonzero and negative, drive the system into a tilted phase pointing along one of the three dimer orientations of the honeycomb lattice. Even if g3 were strictly tuned to zero, they proposed higher order terms which could also break the degeneracy of tilts which is a feature of the RK point. Assuming the system was in a tilted phase to the right of the RK point, they then considered the inﬂuence of the most general perturbation allowed by the microscopic model: V (h(x), x) = VG eiGh +Gx ·x (A.25) G where G = (Gh , Gx ) where Gx is a reciprocal vector of the honeycomb lattice and Gh is a reciprocal vector of the height lattice. They found that if the tilts were commensurate with the lattice, then the term in the above sum with the wavevector of the commensuration was a relevant perturbation. In this case, the system would lock into a conﬁning crystalline phase with gapped excitations. If the tilt was incommensurate with the lattice, then the system would be in a crystalline phase with gapless excitations that were logarithmically conﬁned. The magnitude of the commensurate wavevector is inversely proportional to the magnitude of the tilt so as the tilt is made very small, by tuning g3 for example, then the commensurate 79 states would occur less and less often. In addition, the gap in the commensurate phase was estimated to be exponentially small at small tilts. The result is that tuning a generic perturbation will drive the system into a sequence of incommensurate and commensurate phases and in a very tiny region near the RK point, the commensurate phases occupy a set of inﬁnitesimal measure. The sequence of incommensurate crystalline phases form something like a gapless, logarithmically conﬁning “phase”. This phenomenon, called Cantor deconﬁnement[55], is the default scenario unless the microscopic system is highly ﬁne tuned (for example, to the RK QDM Hamiltonian). The construction of the bipartite QDM phase diagram presented in the main text will realize this Cantor deconﬁned scenario because of the small but nonzero error terms. 80 y x a b c 2 0 1 0 2 1 2 0 1 0 2 0 1 2 0 -1 0 1 -1 0 0 1 -1 0 -2 2 0 -2 -1 -3 -1 -3 -5 -7 -9 0 -4 -2 -1 -3 -6 -5 -8 -7 -9 Figure A.1: This ﬁgure gives three representative dimer coverings and the associated height mapping. In each case, the ﬁrst hexagon of the middle row was assigned height 0 and the remaining values were obtained by the rules given in the text. In the columnar state (a), the height is uniform in y for a given x and periodic in x with a period of 1 12 lattice constants. The height ﬁeld obtained by coarse graining will be uniform apart from this microscopic ﬂuctuation so is called a “ﬂat state”. In contrast, the staggered state (c), is the conﬁguration having maximal tilt. (b) shows an arbitrary conﬁguration. 81 2 2 1 0 1 0 2 2 1 0 0 0 -3 -1 -1 1 2 -2 -2 -3 -2 -1 0 3 0 3 0 -1 -4 -1 1 2 1 2 1 -2 -3 -2 0 3 0 3 0 -1 -4 -1 -3 Figure A.2: Periodicity of heights. The top ﬁgure shows a dimer pattern on the honeycomb lattice where on the left and right side, the dimers are arranged in the same columnar pattern. These regions are separated by a staggered region indicated by the circle. The value of any local operator will depend only on the local dimer arrangement and thus should be the same on the left and right of the circle. However, if we look at corresponding plaquettes, we ﬁnd that the values of the height variables on the right have been shifted down by 3 relative to the left side. The precise value of the shift depends on the details of the dimer pattern in the region separating the two columnar regions but the shift will always be some integer multiple of 3. Therefore, the operator should be a periodic function of the height variable with period 3. The bottom ﬁgure shows the similar construction for the square lattice. In this case, the height construction involves gaining +3 if a dimer is crossed when going clockwise around the sites of one of the sublattices and -1 otherwise. Comparing corresponding plaquettes, it is seen that the periodicity is 4. Appendix B Mean field theory for string defects B.1 Mean field calculation of the system response Here we present some details of the mean ﬁeld calculation outlined in section 2.4. In the main text, it was shown that the system prefers the uniform density state when left to itself. We now impose in each layer k a small external potential φkext (x), and study how the system responds. We assume that this potential only acts on the positive charges, which are the dynamical objects in our formulation. This requires adding a term to the Hamiltonian: K N k=1 k k i=1 φext (xi ) and modifying the free energy functional (omitting terms that vanish in a large N limit): Fρ,N T = K 1 k=−K + 2 d2 xd2 x (ρk (x) − ρk+1 (x))(ρk (x ) − ρk+1 (x ))V (|x − x |) d2 xρk (x) ln( ρk (x) )+ N d2 xρk (x)φkext (x) We minimize the functional subject to the normalization constraint, (B.1) 2 k d xρ (x) = N , to obtain the self consistent equation: N exp(−(φcext (x) + d2 x (2ρc (x ) − ρc+1 (x ) − ρc−1 (x ))V (|x − x |)) ρ (x) = 2 d x(exp(−(φcext (x) + d2 x (2ρc (x ) − ρc+1 (x ) − ρc−1 (x ))V (|x − x |)) c (B.2) We note that in the absence of a potential, we recover the uniform solution. If we start with the uniform solution, then we may ask what happens when we turn on a small potential. 82 83 Speciﬁcally, we would like to calculate the susceptibility: χx,x c,c ≡ − δρc (x) δφcext (x ) (B.3) φcext (x)=0 The minus sign follows the convention of [32]. Diﬀerentiating Eq. B.2: χx,x c,c = N exp() δc,c δ(x − x ) d2 x exp() δρc−1 (x ) δρc+1 (x ) δρc (x ) − − )V (|x − x |) δφcext (x) δφcext (x) δφcext (x) N exp() − 2 d2 y exp() ( d x exp())2 δρc−1 (x ) δρc+1 (x ) δρc (x ) − − ) × δc,c δ(y − x ) + d2 x (2 c δφext (x) δφcext (x) δφcext (x) + d2 x (2 × V (|y − x |) φcext (x)=0 = ρ0 δ(x − x )δc,c − − (ρ0 )2 δc,c − NT ,x x ,x d2 x (2χxc,c,x − χxc−1,c − χc+1,c )V (|x − x |) ,x x ,x d2 yd2 x (2χxc,c,x − χxc−1,c − χc+1,c )V (|y − x |) 1 x −x = ρ0 (δ(x − x ) − )δc,c − d2 x (2χxc−c−x − χxc−c−x −1 − χc−c +1 )V (|x − x |) A 1 x −x x −x d2 yd2 x (2χxc−c−x − χ − χ )V (|y − x |) (B.4) + c−c −1 c−c +1 A where in the last line, we have assumed translational invariance. We also assumed that the layers are suﬃciently large (and that the susceptibilities are suﬃciently “well-behaved”) that the limits of integration may be taken to be the entire plane R2 . The following is seen by integrating Eq. B.4: d2 xχx−x c−c = 0 (B.5) This expresses the physical fact that, on average, the change in density due to the perturbing potential is zero, which is another way of saying that charges can not leave the plane. Eq. B.5 allows a simpliﬁcation of Eq. B.4: x −x d2 yd2 x (2χxc−c−x − χxc−c−x −1 − χc−c +1 )V (|y − x |) = x −x −y x −x −y x −x −y d2 yd2 x (2χc−c − χc−c − χc−c −1 +1 )V (|x |) = d2 x V (|x |) x −x −y x −x −y x −x −y d2 y(2χc−c − χc−c − χc−c −1 +1 ) = 0 84 Therefore, we have an expression for the susceptibility (where we have chosen x = 0 and c = 0) given by: χxc = ρ0 (δ(x) − 1 )δc,0 − A d2 y(2χyc − χyc−1 − χyc+1 )V (|x − y|) (B.6) Suppose we apply a small external ﬁeld in one of the layers. We would like to know how the density responds in that plane and in the z-direction. Far away from the charge, the density should be essentially the uniform equilibrium value i.e, ρk (x) = ρ0 . We are interested in length scales associated with the decay of the perturbation in both the plane and z-direction. We assume the perturbation induces a linear response in the density: c (δρ (x))[φext ] = c = − d2 x c δρc (x) φcext (x ) c φ =0 δφext (x ) ext c d2 x χx−x c−c φext (x ) Taking the Fourier transform gives: δρ(q, qz ) = −χ(q, qz )φext (q, qz ) (B.7) where the Fourier transform of Eq. B.6 χqqz = (1 − δq,0 )q 2 4πκ 1 q2 4πκρ0 + 1 − cos qz (B.8) A potential acting in the c = 0 plane would have a functional form φcext (x) = f (|x|)δc,0 or in momentum space,φext (q, qz ) = f (q). The corresponding density perturbation will be: δρ(q, qz ) = − where aq = q2 4πκρ0 1 (1 − δq,0 )q 2 f (q) 4πκ aq − cos qz (B.9) + 1. We invert this in the z-direction: π dqz δρ(q, qz )eiqz n 2π (1 − δq,0 )q 2 f (q) π eiqz n = − dq z 4π 2 κ 2aq − eiqz − e−iqz −π δρ(q, n) = −π (B.10) We make a variable change z = eiqz , dz = izdqz , and the integral becomes a contour integral about the unit circle. (This is for n ≥ 0; for negative n, we need to use the variable change 85 z = e−iqz . Note that the expression is symmetric in n so we will get the same result either way.) δρ(q, n) = where αq = aq + (1 − δq,0 )q 2 f (q) 4π 2 κi zn (z − αq )(z − α−1 q ) dz |z|=1 (B.11) 2 (aq − 1). Notice that if q = 0, aq > 1, αq > 1, and only the pole at z = α−1 q lies inside the contour. Evaluating the residue, we ﬁnd if n ≥ 0: δρ(q, n) = − = − (1 − δq,0 )q 2 f (q) αq 1 2πκ α2q − 1 αnq (1 − δq,0 )q 2 f (q) αq e−(ln αq )n 2πκ α2q + 1 (B.12) As suggested, we see that associated with each in-plane wavenumber q, we have an out of plane length scale given by: ξz (q) = 1 = ln αq ln 1 + 1 q2 4πκρ0 + q2 q2 4πκρ0 ( 4πκρ0 (B.13) + 2) To say something about an in-plane length scale, we assume the perturbation is a point charge (that acts only on the positive charges in the plane), i.e. f (q) = 2πκ . q2 We note that close to our perturbing charge, the assumption of the perturbation being small will break down. We can invert this, at least formally, to get the density perturbation in real space: 1 x δρ( , n) = 2 2 ξ 4π ξ B.2 is·( ξx ) e 2 d s n−1 s2 (s2 + 2) 1 + s2 + 2 s2 (s2 + 2) 1 + s2 + s2 (s2 + 2) (B.14) Correlation lengths As suggested, a natural in-plane length scale comes out of the calculation: ξ = 1 1 )2 2Dκρ0 (B.15) where κ = 1/2 and ρ0 is the density of defects in a layer. This scale depends on the temperature and magnetic ﬁeld through ρ0 . We may estimate ρ0 by considering the grand partition function: Z = e−βA = y (2K+1)N N (N !)2K+1 eSN (B.16) 86 where SN is the entropy of having N defects and y = e−Es /kB T is the fugacity of a positive defect (y 2K+1 is the fugacity of a “string”) where Es = 8gµB JB/3 is the energy per kagometriangle bilayer of a string. At mean ﬁeld level, we may replace SN by Sρ,N = (2K+1)N ln A. Making this substitution, we obtain: ZM F = N (yA)N 2K+1 N! N where N0 is the value for which (yA)N N! ≈ (yA)N0 2K+1 N0 ! (B.17) is a maximum; the latter approximation holds if K is large. Using Stirling’s approximation, we may estimate that N0 ≈ yA. We may then estimate the average number of defects per layer: N = 1 ZM F N N (yA)N 2K+1 N! ≈ N0 ≈ yA (B.18) from which we conclude that: 2 ξ,M F ∼ exp(8gµB JB/3kB T ) (B.19) We can also say something about the out-of-plane correlation by looking at the density perturbation (Eq. B.14) when x = ξ x̂. The result is: 1 δρ(x̂, n) = 2 2 4π ξ eis·x̂ d2 s n−1 s2 (s2 + 2) 1 + s2 + 2 s2 (s2 + 2) 1 + s2 + s2 (s2 + 2) (B.20) The most striking thing about this result is that the rate of decay in the z-direction has no dependence on temperature, although the coeﬃcient of the integral does have a temperature dependence. The implication is that when we place a charge at the origin; move away by an in-plane correlation length, and then look upwards, the decay will be the same regardless of temperature. Thus, there is no temperature-dependent length scale in the zdirection. This same conclusion comes out of the RG calculation, although with a diﬀerent temperature/ﬁeld dependence. Appendix C Renormalization group treatment of string defects In this appendix, we provide details of the renormalization group calculation outlined in section 2.4. Let us begin by considering the canonical partition function Z({Nk,l }), corresponding to the dipole distribution {Nk,l }. We would like to calculate, to order dτ , the result of integrating out those modes where in some plane, an oppositely charged pair is separated by a distance between τ and τ + dτ . We introduce the abbreviation: 2 (1) 2 (2) (1) (2) d xk,i d xk,i xk,i − xk,i dΩτ = τ2 k,i∈Ik τ2 δ τ (C.1) in terms of which the canonical partition function for a given dipolar distribution {Nk,l } may be written: Z({Nk,l }, τ ) = Ωτ dΩτ exp(−H). Our RG calculation has two steps. The ﬁrst step is integrating over short length scales, i.e. those states where at least one pair of charges is separated by a distance between τ and τ + dτ . The second step is to rescale variables to restore the short distance cutoﬀ. When we carry out the ﬁrst step, the result is a zeroth order term and a correction of order dτ : Z({Nk,l }, τ ) = Ωτ +dτ dΩτ exp(−H) + k,l,m,i,j 87 Iklmij (C.2) 88 where Iklmij is the contribution of the conﬁguration that has the negative end of the ith m-dipole of layer k paired with the positive end of the jth (l − m)-dipole of layer k + m. The sum over k is over all planes; the sum over l is over all dipole lengths up to the number of planes; and the sum over m is from 1 to l − 1. The form of this term is given by: Iklmij = Ωτ +dτ −H dΩτ e A (1) (1) (2) (2) (1) (2) d2 xj xj − xj −H(x(2) ,x(1) ) d2 xi xi − xi i j δ δ e (2) τ2 τ τ2 τ d(xi ,τ ) (C.3) (1) The region of integration of the positive charge xj is an annulus of radius τ and thickness (2) (2) dτ centered on the negative charge xi . This region is denoted by d(xi , τ ). The position of this negative charge (and hence the pair) is integrated over the entire area A. Strictly (2) speaking, xi would have to avoid the hard cores of all of the other charges but this introduces an error of order (dτ )2 . Ωτ +dτ is the space of conﬁgurations of the rest of the charges in which the charges are separated from each other by a distance of at least τ + dτ . (2) (1) (2) H(xi , xj ) refers to the piece of the Hamiltonian which involves charges xi (1) and xj and the rest of the Hamiltonian is denoted by H . (1) (1) (2) (1) The xj integration amounts to making the substitution xj = xi + τ ; d2 xj = τ dτ dθ; and integrating over angles. If we denote the latter two of integrals of equation C.3 by I, then: I = dτ τ A (2) d2 xi τ2 2π 0 (2) dθe−H(xi (2) , xi + τ) δ (2) (2) x(1) − x(2) xi − xj + τ i i τ δ τ (C.4) We assume that our gas of defects is suﬃciently dilute that the following distances are much greater than the pair separation τ : (1) the distance of a particle in plane k + m from (1) our pair, (2) the distance of a particle in plane k from the positive charge xi , and (3) the (2) distance of a particle in plane k + l from the negative charge xj . In this dilute limit, we may make the approximation: δ (2) (2) (1) (2) xi − xi xi − xj + τ τ δ τ (1) (2) τ 2 xi − xj ≈ δ A τ (C.5) 89 (2) (1) We also have that H(xi , xj ) is small in this limit, which allows us to expand the exponential and to leading order, the integral may be done exactly [31]. The result is: (1) (2) dτ xi − xj (πκτ 2 )2 rab δ 2π − ea eb ln τ τ A τ a=b I = (1) (2) − xj τ dτ xi ≈ 2π δ τ (C.6) In the penultimate line, the sum refers to a sum over all charges, positive and negative, residing in the plane k + m. This sum term may be neglected in the large A limit, which is why, in contrast to the Kosterlitz calculation[33], the coupling strength does not vary during our RG ﬂow (see equation 2.13). The delta function implies that the m-dipole and (l − m)-dipole have been combined into a larger l-dipole. Returning to our correction term: Iklmij dτ ≈ 2π τ Ωk,l,m τ +dτ dΩk,l,m exp(−H) τ (C.7) where the space Ωk,l,m τ +dτ is analogous to Ωτ +dτ , except that there is one less m-dipole in layer k; one less (l − m)-dipole in layer k + m; and one more l-dipole in layer k. What we are actually interested in is the grand partition function (equation 2.11). Because our RG procedure is consistent with the charge neutrality constraint, the various {Iklmij } may be combined with diﬀerent terms in the grand partition function. When we substitute into Eq. 2.11 and arrange terms, we ﬁnd that: Z = {Nκ,λ } × 1 κ,λ (Nκ,λ )! yλ N,λ κ,λ 2π + Ωτ +dτ dΩτ exp(−H) yλ N,λ k,l,m κ,λ y Nk,l −1 dτ ym yl−m l 2π N k,l τ (2π)2 2π 2π (C.8) Nk,l −1 The prime on the second product means that yl has been taken outside the product. If the fugacities are small, then we may write this in a more convenient way: Z = N,l (yl + dτ l−1 m=1 ym yl−m ) τ {Nk,l } k,l (2π)Nk,l (Nk,l )! Ωτ +dτ dΩτ exp(−H) (C.9) 90 Finally, we rescale lengths, x → x(1 + dτ /τ )−1 , and ﬁnd (dropping primes): Z= {Nk,l } k,l yl N,l 2π (Nk,l )! Ωτ dΩτ exp(−H) (C.10) where yl = (yl + l−1 dτ dτ dτ ym yl−m )(1 + 2 )(1 − κ ) τ m=1 τ τ The ﬂow equations (2.13) follow from this. (C.11) Appendix D Sign conventions in the overlap matrix In the Rokhsar-Kivelson derivation of the dimer model, the overlap matrix between the diﬀerent dimer coverings plays a crucial role. Whether or not the RK point corresponds to an equal-amplitude superposition of all dimer coverings depends on the question of whether the dimer Hamiltonian can be turned into a form where all oﬀ-diagonal matrix elements corresponding to the shortest resonance loops are negative. The leading eﬀect of a perturbing nearest-neighbour exchange ﬁnally gives rise to a constant (for each type of loop) multiplying the overlap matrix (restricted to that type of loop). If the overlap matrix can be written as a term proportional to matrix with entries only 0 or 1, the ground state is indeed given by an equal-amplitude superposition at the RK point. In the following, we demonstrate that a fermionic sign convention may be used to generate the negative sign, independent of lattice details, for models involving valence bond coverings of the lattice. We show that for the honeycomb and diamond lattices, we may obtain the positive sign as well. We also present a convention for the Kivelson-Klein model which gives the negative sign. 91 92 D.1 Overlaps in the fermionic convention A general convention for overlaps can be obtained by employing the so-called fermionic convention, where a valence bond between sites a and b is generated via operators, such as the one placing a fermion with spin up on site a: c†a↑ . The singlet bond is then deﬁned as: 1 |[ab] ≡ d†ab |0 = √ [c†a↑ c†b↓ + c†b↑ c†a↓ ]|0 . 2 (D.1) Here |0 is the vacuum state with no fermions present. Note that d†ab = d†ba , and that these operators, being bilinear in fermions, commute unless they have exactly one site in common. This means that for constructing a valence-bond covering, the order in which the bonds are generated is inconsequential. A loop in the transition graph involving sites h, i, j, and k will lead to the following type of expression in the overlap matrix element calculation: 0|dab · · · dij d†jk d†hi · · · d†ab |0 1 = − 0|dab · · · d†hk · · · d†ab |0 2 (D.2) By induction, a loop in the transition graph involving L dimers in each conﬁguration leads to a factor of (−1/2)L−1 , independently of any further details of the lattice. D.2 Honeycomb and diamond lattices The following approach works for both honeycomb and diamond lattice, both of which have a shortest resonance loop of length six, as in the original benzene picture. As these lattices are bipartite, we can orient each bond to point from one sublattice (A) to the other (B), so that a singlet between sites a and b of sublattices A and B, respectively, has the wavefunction √ |(ab) ≡ [| ↑a ↓b − | ↓a ↑b ] / 2 . (D.3) 93 The two singlet coverings of the benzene loop now have wavefunctions |1 ≡ |(ab)(cd)(ef ) ; |2 ≡ |(bc)(de)(f a) from which it follows that 1|2 = +1/4 (D.4) for any hexagonal plaquette. It is in fact also possible to choose 1|2 = −1/4 (D.5) for the honeycomb lattice. This can, for example, be achieved by choosing any fixed hardcore dimer covering of the triangular lattice which is dual to the honeycomb lattice under consideration. One then multiplies each valence bond state of the honeycomb lattice by (−1)n× , where n× is the number of valence bonds which cross dimers of the triangular dimer covering. This generates the desired eﬀect. D.3 Other bipartite lattices The above construction for generating uniform overlap matrix elements can be generalised to any bipartite lattice. By orienting the bonds from one sublattice to the other, one always obtains an overlap which is positive; its size is the simple product over the individual loops involved in the transition graph. k|l > 0 ; (D.6) indeed, the positive overlap holds true for any value of the loop length, and therefore for an arbitrary pair of valence bond coverings |k, |l. D.4 Kivelson-Klein model on pyrochlore lattice Here we ﬁrst need to establish the possible resonance loops. These involve six dimers on a cluster of twelve sites arranged as follows. 94 Six of the sites sit on a hexagonal ring on the pyrochlore lattice; each link of this hexagonal ring belongs to a diﬀerent tetrahedron. Each of these six tetrahedra contains one dimer linking a site on the hexagonal ring with a site oﬀ the hexagonal ring. As there is a choice of 2 such oﬀ-sites per tetrahedron, the total number of shortest resonance loops corresponding to each hexagonal ring equals 26 = 64. Not all of these loops are symmetry equivalent. These loops all involve moving six dimers. Hence, their overlap in the Fermion convention is given by −1/32. Appendix E Spinon gap for the decorated honeycomb lattice A stable RVB liquid phase requires not only certain properties of the ground state wavefunction but also that the excitation spectrum has a positive lower bound. In this section, we argue that the nearest neighbor valence bond ground states are separated by a ﬁnite gap from the excited states for the case of the decorated honeycomb lattice Klein model. To this end observe that in the highly decorated limit we are connecting a set of Majumdar-Ghosh[46] (MG) chains into a two-dimensional network. The excitations of the chains themselves are well studied: these are spinon defects between the two diﬀerent dimerizations and there is considerable analytical[69, 70] and numerical[71] evidence that they are gapped. In putting the chains together we need to ask if the intersections lead to the emergence of states below the one-spinon continuum on the chains that can ﬁll in the gap. In the inﬁnite decoration limit, it is suﬃcient to consider a single intersection: thus we look for bound states localized near a site of the original honeycomb lattice where three MG chains would cross (see Fig. E.1). With reference to Fig. E.1, we consider the Klein Hamiltonian: HK = P̂N (i) . i 95 (E.1) 96 4a 3a 2a 1a 0 2c 4c 1c 2b 1b 3b 3c 4b Figure E.1: This ﬁgure depicts a spinon at the crossing of the edges of a decorated honeycomb lattice. An up spinon is at position 0 and the other spins are dimerized into singlet pairs. As before, i is the site index; N (i) is the “neighborhood” of i consisting of the site and its nearest neighbors; and P̂N (i) projects the set of spins N (i) onto its highest spin state. Because the highest spin state is symmetric under interchange of spins, if a wavefunction involves spin i forming a singlet with one of its neighbors, this wavefunction will be destroyed by the projector P̂N (i) . A projection operator has only non-negative eigenvalues so nearestneighbor valence bond states are ground states of the Hamiltonian (3.3). We consider a wavefunction describing a single spin at the crossing of three semi-inﬁnite MG chains. |ψ0 = |+0 |00ee . (E.2) Here |+0 denotes an up spin at location 0 and |00ee denotes that “everything else” is dimerized into nearest-neighbor singlet (00) pairs. We consider the action of the Klein Hamiltonian (3.3) on this state. Observe that P̂N (0) , which involves only site 0 and its three neighbors, is the one projector that does not destroy our trial function. We may write this projector in terms of the spins involved: P̂N (0) = Ccr (S 2 − 2)S 2 . (E.3) 1a + S 1b + S 1c and Ccr is a positive constant which sets the energy scale for =S 0 + S Here S 97 violating the Klein condition at the crossing. Then: S2 = Si2 + i · S j 2S i<j i=0,1a,1b,1c 1 3 i · S j · ·4+ 2S 2 2 i<j = = 3+ i · S j . 2S (E.4) i<j Therefore: P̂N (0) = Ccr [3 + i · S j ][1 + 2S i<j = 3+4 i · S j + ( 2S i<j i · S j ] 2S i<j i · S j )( 2S i<j i · S j ) 2S (E.5) i<j 0 · S 1a (3 such terms), The operator S 2 is made of operators involving the central site, 2S 1b (3 such terms). We 1a · S and also interactions between neighbors of the central site, 2S compute the actions of these terms on our trial state. 1a |ψ0 = |00ee 2S 0 · S 1a |+0 |001a,2a 0 · S 2S = = = = = 1 − + z √ |00ee [S0+ S1a + S0− S1a + 2S0z S1a ][0↑ (1a)↑ (2a)↓ − (1a)↓ (2a)↑ ] 2 1 1 1 √ |00ee [−0↓ (1a)↑ (2a)↑ + 0↑ (1a)↑ (2a)↓ + 0↑ (1a)↓ (2a)↑ ] 2 2 2 1 1 √ |00ee [ 0↑ (1a)↑ (2a)↓ − (1a)↓ (2a)↑ + 0↑ (1a)↓ (2a)↑ − 0↓ (1a)↑ (2a)↑ ] 2 2 1 |00ee [ |+0 |001a,2a + |+2a |000,1a ] 2 1 |ψ0 + |ψ2a (E.6) 2 Here |ψ2a means a state having an up spin at site 2a with “everything else” dimerized into singlet pairs. 1b |ψ0 = |00ee |+0 2S 1a · S 1b |001a,2a |001b,2b 1a · S 2S = 1 + S z z − − + |00ee |+0 [S 1a 1b + S1a S1b + 2S1a S1b ][(1a)↑ (2a)↓ − (1a)↓ (2a)↑ ] 2 × [(1b)↑ (2b)↓ − (1b)↓ (2b)↑ ] 98 = 1 + S z z − − + |00ee |+0 [S 1a 1b + S1a S1b + 2S1a S1b ][(1a)↑ (1b)↑ (2a)↓ (2b)↓ 2 − (1a)↑ (1b)↓ (2a)↓ (2b)↑ − (1a)↓ (1b)↑ (2a)↑ (2b)↓ + (1a)↓ (1b)↓ (2a)↑ (2b)↑ ] = + + = 1 |00ee |+0 [−(1a)↑ (1b)↓ (2a)↑ (2b)↓ − (1a)↓ (1b)↑ (2a)↓ (2b)↑ 2 1 1 (1a)↑ (1b)↑ (2a)↓ (2b)↓ + (1a)↑ (1b)↓ (2a)↓ (2b)↑ 2 2 1 1 (1a)↓ (1b)↑ (2a)↑ (2b)↓ + (1a)↓ (1b)↓ (2a)↑ (2b)↑ ] 2 2 1 1 |00ee |+0 [ (1a)↑ (2a)↓ + (1a)↓ (2a)↑ (1b)↑ (2b)↓ + (1b)↓ (2b)↑ 2 2 − (1a)↑ (2a)↑ (1b)↓ (2b)↓ − (1a)↓ (2a)↓ (1b)↑ (2b)↑ ] = 1 |00ee |+0 |101a,2a |101b,2b − |111a,2a |1, −11b,2b 2 − |1, −11a,2a |111b,2b (E.7) In the previous line, the notation |10i,j means that spins i and j are paired in their m = 0 triplet state and so forth. The action of S 2 on |ψ0 generates the two types of terms described above. Note that (E.7) is orthogonal to |ψ0 . The eﬀect of acting with S 2 a second time may be found by examining its eﬀect on the terms in equations (E.6) and (E.7). Many of the generated terms will be orthogonal to |ψ0 but some will not be including, for example: 1a |ψ2a = |+2a |00ee 2S 0 · S 1a [|000,1a ] 0 · S 2S 3 = − |ψ2a 2 (E.8) and 1b 1a · S 2S 1 1 + S z z − − + |00ee |+0 |101a,2a |101b,2b = |00ee |+0 [S 1a 1b + S1a S1b + 2S1a S1b ] 2 4 (1a)↑ (2a)↓ + (1a)↓ (2a)↑ = + = = (1b)↑ (2b)↓ + (1b)↓ (2b)↑ 1 |00ee |+0 (1a)↑ (2a)↑ (1b)↓ (2b)↓ + (1a)↓ (2a)↓ (1b)↑ (2b)↑ 4 1 (1a)↑ (2a)↓ − (1a)↓ (2a)↑ (1b)↑ (2b)↓ − (1b)↓ (2b)↑ 2 1 |00ee |+0 |001a,2a |001b,2b + |111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b 4 1 1 |ψ0 + |00ee |+0 |111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b (E.9) 4 4 99 and similarly, 1b 1a · S 2S = − = − = 1 |00ee |+0 (|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b ) 2 1 |00ee |+0 (1a)↑ (2a)↓ (1b)↓ (2b)↑ + (1a)↓ (2a)↑ (1b)↑ (2b)↓ 2 1 (|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b ) 2 1 |00ee |+0 |101a,2a |101b,2b − |001a,2a |001b,2b 2 1 (|111a,2a |1, −11b,2b + |1, −11a,2a |111b,2b ) 2 1 1 − |ψ0 + |00ee |+0 2|101a,2a |101b,2b − |111a,2a |1, −11b,2b 2 4 − |1, −11a,2a |111b,2b (E.10) Therefore, we ﬁnd that: P̂N (0) |ψ0 = Ccr 54 4 |ψ0 + 4 |ψ2a + |ψ2b + |ψ2c + |α (E.11) where |ψi = |+i |00ee describes an up spin at location i with everything else paired into singlets; Ccr sets the energy scale for violating the Klein condition at the cross; and |α is a piece orthogonal to the spinon states, {|ψi }. We may also compute the eﬀect of operator (3.3) on the state |ψi , where i is a location along a chain; in this case, P̂N (i) is the only operator that does not destroy the state: P̂N (i) |ψi = Cch 5 2 |ψi + |ψi−2 + |ψi+2 + |β (E.12) Cch is the energy scale for violating the Klein condition on the chain and |β is a piece orthogonal to the spinon states. We see that the Klein Hamiltonian acting on a spinon state produces the original state; spinon states where the spin has hopped two sites; and terms orthogonal to all spinon states. This motivates the bound state trial function: |ψ = |ψ0 + y n |ψ2n,i (E.13) n>0;i=a,b,c where y is a variational parameter less than unity. We are interested in the expectation value of the energy for this state (E.13): E= ψ|HK |ψ ψ|ψ (E.14) 100 In calculating this, we need to contend with the non-orthogonality of the spinon states: i ψ|ψj 1 |i±j|/2 = − (E.15) 2 where the minus (plus) is for sites i and j on the same (diﬀerent) chain(s). First, we compute the normalization: ψ|ψ = 1 + 2 n>0;i=a,b,c = 1 + (2 · 3) y 1 n n n>0 = 1+6 ψ|HK |ψ Ccr − 2 y n+m (2m,i ψ|ψ2n,j ) n,m>0;i,j=a,b,c +6 1 m+n y n+m − m,n>0 2 +3 1 |m−n| y n+m − m,n>0 −y/2 2 −y/2 3y 2 (1 − y/2) +6 + 1 + y/2 1 + y/2 (1 + y/2)(1 − y 2 ) 54 = ψ| 4 |ψ0 + 4 |ψ2a + |ψ2b + |ψ2c + A yn n>0,i=a,b,c 54 = ψ| ( + A = y n (0 ψ|ψ2n,i ) + 4 5 2 2 (E.16) |ψ2n,i + |ψ2n−2,i + |ψ2n+2,i + 3Ay)|ψ0 + (4 − A) |ψ2a + |ψ2b + |ψ2c 5 ( y n + y n−1 + y n+1 )|ψ2n,i 2 n>0,i=a,b,c 5 y 3 1 54 54 + 3Ay + (4 − A)(− ) + A[ + + y] + + 3Ay · 3 (− )n 4 2 2 y 4 2 n>0 + (4 − A) 1 1 y n [(− )n−1 · 3 + (− )n+1 · 6] 2 2 n>0 5 1 y n+m (2m,i ψ|ψ2n,j ) + A( + + y) 2 y n,m>0;i,j=a,b,c = + + = + 54 − 6(4 − A) + 3Ay 4 54 + 10A −y/2 A + + 4Ay − 6(4 − A) − 3(4 − A) 3 4 y 1 + y/2 5 2 1 3y (1 − y/2) −y/2 2 A + + y [6 + 2 y 1 + y/2 (1 − y 2 )(1 + y/2) 18 + 66A −y/2 3A 30 + 6A + 3Ay + + + 12Ay 4 4 y 1 + y/2 5A 2 A 3y (1 − y/2) −y/2 2 + + Ay 6 + 2 y 1 + y/2 (1 − y 2 )(1 + y/2) (E.17) 101 where A = Cch /Ccr . Fig. E.2 shows a graph of the expectation value of the energy for several values of A. For small values of A, the best variational wavefunction is an extended scattering state while for large values, the best wavefunction is localized at the cross. 30 a=0.5 a=1 a=2 a=5 a=20 25 E 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 y Figure E.2: This plot gives the energy expectation as a function of the variational parameter y for diﬀerent values of A = Cch /Ccr , which is ratio of Klein scales; the energy is measured in units of Ccr . For large values of this ratio, the minimum occurs for y = 0 which corresponds to spinons localized at the crossing. For small values, the minimum occurs at y = 1, which corresponds to an extended spinon state. At a value slightly greater than A = 1, there is a ﬁrst order “phase transition” between these limits. The important feature is that for any nonzero A, there is an energy gap between spinon states and the valence bond states, which have zero energy. Our analysis has been for the inﬁnite decoration limit. For large, but ﬁnite, decoration, the extended spinon states obtained for small A may be interpreted as MG spinons scattering at the vertices. In this limit, the natural extension of the localized states obtained for large A will involve the wavefunction having peaks at the reference sites and decaying on the chains. There will be a band of such localized states below the scattering states. We may estimate the bandwidth by considering a variational wavefunction where the spinon resides 102 only on the reference sites: |Ψ = eik·n | n (E.18) n where | n denotes a wavefunction of the form (E.2) for the spinon at lattice site n and k is a wavenumber. The variational energy of this trial state may be calculated to leading order in a large N expansion, where N is the number of sites inserted between reference sites in the decoration. Ψ|HK |Ψ Ccr Ψ|Ψ = = ≈ &× Ψ| 54 na + Ψ| nb + Ψ| nc +4 eik·n 4 Ψ|Ψ n m| 54 na + m| nb + m| nc +4 eik·(n−m) 4 Ψ|Ψ n,m √ √ kx 54 kx 3ky 3ky 2N + 16 cos kx + cos( + ) + cos( − ) (x + 2x2N +2 ) 4 2 2 2 2 n Ψ|Ψ ≈ 54 kx + 16x2N cos kx + cos( + 4 2 √ kx 3ky ) + cos( − 2 2 √ 3ky ) 2 Here | na is a state with an up spinon on the a-chain at site n, etc. and x = (E.19) √1 . 2 We see that the band becomes more narrow as the decoration is increased. In our analysis so far, we have considered states where the defects are always an even number of sites away from the reference sites. There is another family of spinon states corresponding to the defects being located on the odd sites. Referring to Fig. E.3, the Klein operator permits the spin at 1a to hop only to the site 1b, which is connected by a dimer to the origin. Therefore, in the large decoration limit, this is equivalent to the MG chain, which we know is gapped. 103 4a 3a 2a 1a 0 2c 4c 3c 1c 2b 1b 3b 4b Figure E.3: This shows a representative of the family of states where a spinon is an odd distance from the origin. In this conﬁguration, the Klein operator may hop the spinon only onto the b-chain. The conﬁguration where the origin forms a singlet with 1c is essentially orthogonal to the given conﬁguration for large decoration. Appendix F Classical dimers on the pentagonal lattice At inﬁnite temperatures, where “inﬁnite” means a temperature that is small compared with the excitation gap of the Klein Hamiltonian but much larger than the energy scales of the quantum dimer Hamiltonian, the dimers are described classically, i.e. thermodynamic quantities are computed as equal-weight averages over all dimer states. The number of dimer states grows exponentially with lattice size. This number may be computed using the method of Kasteleyn [79, 80]. The results are shown in Table F.1. The method also yields the entropy per site in the thermodynamic limit. S = 0.168608 . . . (F.1) The striking feature of table F.1 is that even fairly small systems have an enormous number of dimer coverings so numerical studies of large systems require Monte-Carlo simulations. Fig. F.1 is a Monte-Carlo calculation of monomer-monomer correlation functions C(r) for the pentagonal lattice using the algorithm of Sandvik[81]. C(r) is deﬁned as the number nd (r) of dimer coverings given a pair of test monomers separated by distance r divided by nd (1). The simulation shows monomer deconﬁnement at inﬁnite temperatures, as opposed to the square and honeycomb lattices which show logarithmic conﬁnement. This 104 105 1.1 along (1,0) direction along (1,1) direction C(r) 1.05 1 0.95 0.9 1 10 r Figure F.1: The monomer-monomer correlation function for a 100x100 pentagonal lattice with periodic boundary conditions. The two curves are cuts along the x̂ and x̂+ ŷ directions. The distance r refers to the distance between unit cells. In computing these correlation functions, we take the two test monomers to be on the same sublattice (the pentagonal lattice is a cubic lattice with a six point basis). Each data point is an average over N√= 106 data points and the noise seen in the plot is of the order of Monte Carlo noise 1/ N ∼ 10−3 ∼ 0.1%. indicates a liquid phase at high temperatures. 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