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ARTICLE DE FOND QUANTUM SPIN LIQUIDS: QUEST PARTICLE BY FOR THE ODD YONG-BAEK KIM AND ARUN PARAMEKANTI T he world around us is full of materials which exhibit interesting properties, from “familiar stuff” such as fridge magnets, to more “complex stuff” such as superconductors which display zero resistance to current flow when cooled to low temperature. Even a simple crystalline solid has the astonishing property that it emerges as a perfectly periodic pattern of atoms starting from a random soup of atoms in a liquid, simply by cooling to the right temperature. Such states of matter are called “Broken Symmetry” states. While the fluctuating high temperature state looks the same when viewed from any point within it, the low temperature “Broken Symmetry” state (magnet or crystal) picks a particular direction for magnetization, or a specific atomic crystal pattern. Peering inside such a solid we see that not all points in space or not all directions are equivalent. This breaking of symmetries is accompanied by new excitations. A periodic crystal has sharply defined modes of collective atomic oscillations which can extend over the entire crystal, which simply do not exist in the liquid. Similarly a magnet has sharp modes associated with collective periodic motions of the tiny atomic magnetic moments, which again cease to exist in the random high temperature state. A general lesson to take away is that many materials exhibit broken symmetries, and states with such broken symmetries possess new types of excitations. In quantum mechanics, such “wave like” collective excitations can acquire a discrete particulate character, so that physicists talk of “phonons” in a vibrating crystal, and “magnons” in an excited magnet. It may appear that such ideas are applicable only to materials, the study of which is part of Condensed Matter Physics. However, physicists, informed by these SUMMARY Strongly correlated quantum condensed matter phases often have unusual excitations which bear no resemblance to the constituent electrons which form such states of matter. We review recent developments which show that Quantum Spin Liquids, made of insulating electrons in solids, may support emergent particle-like excitations with quantum numbers very different from the electron. examples, use the same language to understand such complex questions as the origin of the Universe and the origin of the slew of subatomic particles found in nature. The premise is that our Universe has a “Broken Symmetry” of some sort, and the properties of various subatomic particles derive from such a Symmetry Breaking principle. The search for the Higgs Boson C the “God Particle” C at the Large Hadron Collider is a quest for evidence of such symmetry breaking in the Universe. In recent years, condensed matter physicists have started to study other states of matter where quantum particles with “strange properties” arise as sharp excitations at low temperature. One example is the Fractional Quantum Hall liquid of strongly interacting electrons confined to two dimensions and moving in a large magnetic field C such a quantum Hall state of ordinary charge-e electrons can exhibit particle-like excitations which carry charge, even though the constituent electron is, by itself, perfectly robust against such splintering. More recent examples include electrically insulating solids which support excitations that slightly resemble electrons, i.e. they behave as mobile spin-1/2 fermions which can transport heat but they carry no charge, even while the constituent electrons are themselves strongly interacting and truly immobile in the insulator. Such insulators, driven by strong electron-electron repulsion are termed Mott Insulators (as opposed to Band Insulators like silicon), and such a weird magnetic state of a Mott Insulator is an example of a Quantum Spin Liquid. Understanding such novel states of matter, the search for such strange quantum excitations C our quest for an “Odd Particle” C and uncovering the underlying order responsible for them, are all topics at the forefront of research in physics. Here, we briefly survey some recent developments in the field of Quantum Spin liquids, highlighting recent Canadian contributions to this field. For technical details and a broader perspective, we refer the reader to reviews by by Wen [1] and by Balents [2]. WHAT IS A QUANTUM SPIN LIQUID Quantum spin liquid (QSL) phases are, broadly speaking, paramagnetic (i.e., non-magnetic) insulating ground states of strongly interacting electrons in a solid, where lattice symmetries are not broken. While magnetically ordered ground states of solids may be easily described by a ‘classical cartoon’ where we stick an arrow on each atom Yong-Baek Kim <ybkim@physics. utoronto.ca> and Arun Paramekanti <arunp@physics. utoronto.ca>, Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 and Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 LA PHYSIQUE AU CANADA / Vol. 68, No. 2 ( avr. à juin 2012 ) C 71 QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI) which shows its magnetic moment magnitude and direction, there is no such simple cartoon picture for a spin liquid. Just as for an ordinary liquid, where drawing a random pattern of atoms yields a mere snapshot which does not capture the thermal motions in the liquid or its ability to flow, cartoon pictures with random orientations of the magnetic moments fail to capture the quantum mechanical fluctuations and superpositions of various magnetic moments. We should therefore view Quantum Spin Liquids as highly quantum entangled states of magnetic moment configurations, with no classical counterpart. FRUSTRATING MAGNETIC ORDER One key notion which seems to be crucial to realizing QSL physics in higher dimensions is “frustration” C we expect QSLs to win over magnetically ordered ground states only if something frustrates magnetic ordering. The simplest example of such frustration arises from lattice geometry. Various lattices shown in Fig.1 (together with examples of materials), such as A particularly well understood example of such a QSL is a one dimensional (1D) Mott insulator, where each site on a periodic chain of atoms seats one electron (with spin-1/2 and charge-e). While the strong electron-electron interaction forbids electrons from moving around freely, leading to something like a ‘traffic jam’, the kinetic motion of the electrons allows them to jiggle slightly, and this quantum jiggling can be shown to lead to spins on neighbouring atoms preferring to anti-align. Such a local tendency to “Néel antiferromagnetism” does not, however, extend to global antiferromagnetic order C such a global staggered pattern of Fig. 1 Materials based on triangular, kagomé, and hyperkagomé lattices which support quantum spin liquid states. moments across the entire lattice in unstable to the tiniest quantum fluctuations. Thus, this 1D Mott insulator is a spin liquid with “quantum the 2D kagomé lattice, the 3D pyrochlore lattice, the 3D mechanically melted” Néel order. What kind of excitations hyperkagomé lattice C lattices made of corner sharing does such a 1D spin liquid support Careful calculations show motifs C turn out to not to favor magnetic orders. Indeed, that such a 1D spin liquid has charge-zero spin-1/2 particle-like many such geometrically frustrated materials tend to order, if excitations which can propagate across the lattice, even while they do, at temperatures much below the Curie-Weiss [3] the electrons themselves stay stuck locally . Such excitations, temperature where one might expect magnetic ordering based dubbed ‘spinons’, may be crudely viewed as mobile domain on mean field theory, suggesting that the frustration induced by walls between states with different orientations of lattice geometry leads to strong fluctuations which suppress the antiferromagnetic order. An alternative rough viewpoint is to transition temperature . Aside from geometry, the exchange view them as arising from splintering a magnon (which carries paths for moments to interact in certain materials may involve spin-1 and charge-zero) into two pieces, each of which carries pairs of spins further apart than nearest neighbors even on a spin-1/2. bipartite lattice C in such cases, even simple bipartite lattices, such as 2D honeycomb [4,5], may frustrate magnetic ordering. Extending this idea to higher dimensions is complicated by In other cases, proximity to an insulator-to-metal transition, several issues. The fact that long range magnetic order is more and the resulting enhancement of local charge fluctuations, is stable in higher dimensions, means that such spin liquid states found to lead to “ring-exchange”, where four or more spins have to be able to compete against and beat out ordinary cooperatively flip as several electrons cooperatively exchange magnetically ordered states by having a lower energy. This positions. Such ring-exchange frustrates magnetic order and depends on the details of the Hamiltonian, and many simple may promote QSL states [6]. model Hamiltonians in 2D or 3D simply end up having magnetically ordered ground states. The second issue is that, GAPPED SPIN LIQUIDS unlike in 1D, there are typically no exact or controlled analytical calculations in higher dimensions, while many The simplest form of a QSL is one with a gap to all excitations, numerical methods available to address such spin Hamiltonians spinful excitations as well as spinless excitations. If the spin also fail for models of direct relevance to materials. However, gap is large, then one may effectively think of neighboring recent detailed studies of numerically tractable toy models, as spins on the lattice as having formed a singlet. However, there well as experiments on a host of frustrated spin-1/2 magnets, may be no unique pattern to which neighbors pair up into appear to be uncovering such novel QSL states in 2D and 3D. singlets. Instead, many many singlet patterns exist at this level, 72 C PHYSICS IN CANADA / VOL. 68, NO. 2 ( Apr.-June 2012 ) QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI)AA so that such ‘dimer configurations’, while they significantly lower the energy and entropy of the system compared to having just unpaired spins, still possess a large degeneracy, with a nonzero entropy per site. The approach to the eventual ground states of such dimers is then via “dimer resonances” within the restricted subspace of dimer configurations. This is the socalled “resonating valence bonds” picture of a QSL. Such fluctuations, where the dimer patterns rearrange, correspond to spinless fluctuations occurring below the spin gap scale. Certain such “quantum dimer models”, for instance on the triangular lattice [7], have been shown to have an eventual QSL ground state where the dimers do not freeze into a regular crystal even down to zero temperature. Such ground states may be viewed as magnetic analogues of Helium-4, where singlet bonds undergo significant zero point motion thus preventing crystallization. Going beyond such a simple model of dimers, one can study microscopic yet numerically tractable spin-1/2 models using quantum Monte Carlo calculations. Such studies show that the triangular lattice antiferromagnetic XXZ model undergoes a transition from a conventionally magnetically ordered state into magnetic states analogous to supersolids [8] instead of forming QSLs, while an extended kagome lattice spin-1/2 XXZ model undergoes a quantum ferromagnet to gapped QSL transition [9]. Recent density-matrix renormalization group calculations [10] on the nearest neighbor Heisenberg model on the kagomé lattice indicate that its ground state may also be such a QSL. Ordinary correlations, such as spin-spin or dimer-dimer correlations, all decay exponentially in such gapped QSLs and they do not possess any broken symmetries. Characterizing such QSLs properly thus requires going beyond simple ideas of “Broken Symmetry” discussed in the Introduction. Such gapped QSLs have been shown to possess, in addition to the odd particles described above, ground state degeneracies which depend on the topology of the manifold on which they reside. Such degeneracies reflect a new form of “topological order” [1] in the ground state. Such gapped QSLs also have a sharp notion of a “topological entanglement entropy” which reflects the topological order [11], and numerical simulations have “measured” this entropy in the extended kagome lattice XXZ model [12]. Such gapped 2D QSLs support two novel particle-like excitations: a gapped bosonic spin-1/2 excitation called a ‘spinon’, and a vortex-like excitation called a ‘vison’. The spinons carry, in addition to their spin quantum number, an emergent Ising ‘electric charge’ (gauge charge), and interact with each other via short range interactions, while the ‘vison’ carries an emergent Ising ‘magnetic flux’ [1,13]. This Ising nature of its excitations leads to the name “ spin liquids” for such QSLs. These excitations, the spinon and the vison, are quite different from the original electrons which make up such a Mott insulator; observing such odd particles in numerics or experiments is a sure signature of a strange quantum state. GAPLESS SPIN LIQUIDS Aside from the Gapped Spin Liquids, discussed above, a less well-understood and perhaps more interesting class is that of Gapless Spin Liquids. Some of these spin liquids are characterized by a gap to spinful excitations but have gapless “dimer resonances” which survive down to the lowest energy. Such gapless dimer fluctuations turn out to be simply describable as “electromagnetic fluctuations” of an emergent gauge field: an emergent “Photon”. The simple pyrochlore spin-1/2 XXZ model is found to have such a novel photon excitation [14,15]. The other, perhaps more experimentally relevant form of a Gapless QSL is characterized by having both gapless spinon excitations (which leads to algebraicallydecaying spin-spin correlations in space and time) as well as gapless gauge fluctuations. Because of the algebraic behavior of the spin-spin correlations, these spin liquid phases are often called “critical spin liquids”, and the spinons and gauge fields in such states are strongly coupled. One may think of such a spin liquid phase as a superposition of valence bonds (or “dimers") between two spins separated in varying distances. As a result, there is no characteristic length scale in the spin correlations, resulting in the algebraic form. The spinons in such spin liquids can obey different types of statistics depending on the microscopic details. When the spinons are fermions, the gapless spinon excitations, for example, may have a spinon Fermi surface [6] or possess a massless Dirac fermion spectrum [16] similar to graphene. On the other hand, if the spinons are bosonic, the corresponding spin liquid can have gapless spinons only when the spin liquid phase is sitting at the critical point. This is because if the bosonic spinons have gapless spectrum, then they would generically be condensed and form a magnetically ordered state. These two possibilities are indeed explored in theoretical literatures to explain powerlaw-in-temperature specific heat observed in many candidate spin liquid materials based on the triangular [17,18,6], kagome [19,20,16], and 3D hyper-kagome lattices [21,22]. Gapless spin liquid phases with fermionic spinons may arise most naturally when the Mott insulator is close a metalinsulator transition [23]. In this case, the fermionic spinons in spin liquid Mott insulator inherit the Fermi surface of electrons in the metallic side. This is thought to occur, for example, in the case of organic salts C (BEDT-TTF)Cu(CN) [17,6] and EtMeSb[Pd(dmit)] [18] based on triangular lattices, and NaIrO [21,22] based on three-dimensional hyper-kagome lattice (a 3D network of corner sharing triangles). Indeed these materials show rather clear power-law specific heat and constant spin susceptibility at low temperatures, consistent with the presence of the Fermi surface. In addition, these Mott insulators become metallic under moderately high external pressure, indicating the proximity to a metal-insulator transition. TOPOLOGICAL MOTT INSULATOR Recently the possibility of novel spin liquid phases with fermionic spinons that have the so-called topological band structure has been proposed [24,25]. The spinons in these phases LA PHYSIQUE AU CANADA / Vol. 68, No. 2 ( avr. à juin 2012 ) C 73 QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI) are gapped in the bulk as they possess the band-insulator-like spectra. However, in analogy to the topological insulators of electrons, the spectrum of such spinon band insulators can be “trivial” or “topological". If the spectrum is “topological”, it means that there is a protected gapless surface state despite a bulk gap, and the surface spectrum in momentum space has an odd number of linearly dispersing Dirac spinons. If the spectrum is “trivial”, gapless surface states are not guaranteed. Such topologically nontrivial spin liquid phases are called topological Mott insulators or fractionalized topological insulators. Currently two different proposals exist to obtain topological Mott insulators [24,25]. (i) One could start from an electronic topological insulator with a strong spin-orbit coupling which is essential for the existence of a “topological” band structure. Increasing the on-site Hubbard interaction may lead to a transition to a spin liquid Mott insulator. Within a slave-particle formulation, where one represents the electron as a composite of the charge-carrying boson and the spin-carrying fermionic spinon, the transition from the spin liquid insulator to the band insulator can be described as Bose condensation of the chargecarrying bosons while the fermionic spinons simply inherit their topological band structure from the electrons [24]. (ii) Alternatively, one may start from spin models with frustrated antiferromagnetic and ferromagnetic exchange couplings. It has been recently shown that the slave-fermion theory of such spin models can support a spin liquid with an emergent spin-orbit coupling, hence the topological band structure of the spinons [25]. In this case, the non-trivial topology of the spinon spectra is emergent, and not inherited from the electrons. In either case, one obtains an insulator with a bulk gap but with strange gapless spin-1/2 and charge-neutral surface states: “surface spinons”. In such topological Mott insulators, there is no electrical conduction. However, gapless surface spinons can conduct heat while the bulk is a thermal insulator. CONCLUSION Condensed matter systems are made of a large number of strongly interacting constituent particles, e.g. electrons, having well-understood properties. 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