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Quantum Dimer Models on the Square Lattice In collaboration with D. Banerjee, M. Bogli, C. Hofmann, P. Wilder, and U.-J. Wiese (PRB 90, 245143 (2014) and arXiv:1511.00881). Some of the figures and content are from Debasish, Philippe, Pascal, and online sources, in particular the preprint arXiv:0809.3051. F.-J. Jiang Physics Department, Nataional Taiwan Normal University, Taipei, Taiwan Nov. 2015 Outlines ◮ Introduction: why quantum dimer models. ◮ The established (conjecture) ground state phase diagrams of QDM. ◮ Connecting QDM on the square lattice with U(1) Quantum link model (QLM). ◮ Results : the phase diagram of QDM on the square lattice. ◮ Conclusions. Introduction ◮ In 1986, high temperature (Tc ) superconductivity was discovered in doped cuprate materials. ◮ Since then a lot of efforts have been devoted to understanding the mechanism behind high Tc superconductivity. It is believed that high temperature cuprate superconductors are obtained by doping antiferromagnets with holes or electrons. ◮ ◮ Notice although it is established experimentally that antiferromagnets have massless excitations, no long-range order is found in high temperature cuprate superconductors. ◮ This experimental observation plays the crucial role in constructing the Quantum dimer models (on the square lattice). Heisenberg and Hubbard models ◮ The relevant model for an antiferromagnet (on the square lattice) is the spin-1/2 Heisenberg model: X H=J S~x · S~y . hxyi ◮ The relevant model for cuprate high temperature superconductor (or doped antiferromagnets) is the Hubbard model: X + ci,σ cj,σ + H.c. H = −t hi,ji,σ +U X i ni↑ ni↓ − µ X i (ni↑ + ni↓ ) Heisenberg and Hubbard Models ◮ ◮ At half filling (µ = U/2) and U → ∞, the Heisenberg model is recovered from Hubbard model (identifying 4t 2 /U with J and ~ = c † ~σ c). S 2 Quantum Dimer Models (QDMs) ◮ ◮ Cuprate turns into a high temperature superconductor when doping density x exceeds a critical value (x ≤ 0.2) → most of lattice sites are still occupied by a SU(2) spin. Since no long-range order is observed in high temperature cuprate superconductors, one natural candidate degree of freedom for these high Tc materials is the SU(2)-singlet. ◮ Resonating valence bond theory: first introduced by P. W. Anderson in 1987 in order to understand high temperature (cuprate) superconductor. ◮ Electrons from neightboring copper atoms interact to form a valence bond (singlet, RVB). In particular, the valence bond lucks the electrons inside it. ◮ These electrons can act as mobile Cooper pairs and can superconduct when the cuprate is doped. Quantum Dimer Models (QDMs) ◮ Quantum dimer models (QDMs) were introduce by Rokhsar and Kivelson to model the physics of RVB in lattice spin models. ◮ In addition to (might) be relevant to high temperature cuprate superconductors, QDMs are interests from the point of view of condensed matter physics as well. ◮ Quantum deconfined criticality. ◮ Spin liquid. ◮ Topological order. ◮ Fractionalized spinors. ◮ These are the reasons why QDMs are interesting and have attracted a lot of investigation during the last two decades. Quantum Dimer Models (QDMs) ◮ Degree of freedom: SU(2) singlet valence bonds (or dimers): = √1 [| 2 ↑↓i − | ↓↑i] = ◮ Hamiltonian: X H = [−J (|k ih= | + | =ihk |) + λ (| =ih= | + |kihk|)] = ◮ HJ + Hλ . Constraint: Every site touches exactly one dimer, no long-range dimers and no polymers. Quantum Dimer Models (QDMs) ◮ One example of configuration in quantum dimer model on the square lattice: Quantum Dimer Models (QDMs) H = X = HJ + Hλ . [−J (|k ih= | + | =ihk |) + λ (| =ih= | + |kihk|)] HJ HJ Hλ ◮ ◮ Hλ The kinetic term flips plaquettes (namely, turns a horizontal pair of dimers into a vertical one and vice-versa) and the potential term counts flippable plaquettes. The phase diagrams of QDMs ◮ Intuitively, the system will try to maximize the number of flippable plaquettes when λ/J → −∞. ◮ On the square lattice, such state can be obtained by arranging the dimers in a columnar pattern (four-fold degenerate): The phase diagrams of QDMs ◮ On the contrary, the system will try to minimize the number of flippable plaquettes when λ/J → +∞. ◮ On the square lattice, such state can be obtained by arranging the dimers in a staggered pattern (many such states: superposition of close-pack arrangement of dimers): The phase diagrams of QDMs ◮ Plaquette arrangement of dimers (in the mean field sense, four-fold degenerate, only found on 2D bipartite lattices): The phase diagrams of QDMs ◮ Mixed states: |statesi = c1 |ki + c2 | =i. The phase diagrams of QDMs columnar ??? plaquette staggered λ RK ◮ 2d bipartite lattices. The ??? is a first order transition for honeycomb lattice. The phase diagrams of QDMs columnar ??? Z2 RVB liquid staggered λ RK ◮ 2d (and higher d) non-bipartite The ??? is a unknown √ √ lattices. phase (could be so-called 12 × 12 phase) for triangular lattice. ◮ Z2 RVB phase: a phase with Z2 topological order, i.e. there are four-fold degenerate gapped ground states for 2d lattice with periodic boundary condition. In particular, it has nontrivial excitations. It is a liquid phase because all dimer correlations decay exponentially. The phase diagrams of QDMs columnar ??? U(1) RVB liquid staggered Coulomb λ RK ◮ higher d bipartite lattices. The ??? is unknown for cubic lattice. ◮ U(1) RVB phase: gapless with algebraic dimer correlations. It is called Coulomb phase as well because it has a continuous description that resembles the usual Maxwell action of the free electromagnetic field. U(1) QLM ◮ Motivated by the (lattice QCD) Wilson action (Brower, Chandrasekharan and Wiese, 1997, 1999). ◮ The same symmetry properties. ◮ The same algerbra relations. ◮ Finite-dimensional Hilbert space. U(1) QLM ◮ The Hamiltonian of U(1) QLM: i Xh † † 2 J(U + U ) − λ(U + U ) = H1 + H2 . H=− ◮ ◮ † † U = Uwx Ux,y Uzy Uwz is the typical plaquette operator formed by + + quantum links Uxy = Sxy , where Sxy is a raising operator of 3 electric flux Exy = Sxy which is built from a quantum spin-1/2 of the link xy. Notice the charges Qx at each site x can be determined from it neighboring electric flux by Qx = Ex,x+1̂ + Ex,x+2̂ − Ex−1̂,x − Ex−2̂,x . ◮ For QLM, the Gauss law restrictsP the system to gauge invariant states → Gx |ψi = 0. Here Gx = i (Ex,x+î − Ex−î,x ) is the infinitesimal U(1) gauge transformations and Gx commutes with the Hamiltonian. U(1) QLM ◮ Represent the two possible states of electric flux for a link by the LHS figure. Then in the flux space, a typical configuration of U(1) QLM is given by the RHS figure. + 21 flux − 21 flux Symmetries of U(1) QLM ◮ Symmetris of QLM: ◮ U(1) symmetry. ◮ CTi . ◮ 90 degrees rotation. ◮ CO ′ . ◮ U(1)2 center symmetry. Connecting QDM with U(1) QLM ◮ The Hamiltonian of U(1) QLM: i Xh † 2 † ) = H1 + H2 . ) − λ(U + U J(U + U H=− H1 = H2 = H1 = H2 = ◮ ◮ Notice the first term and second term in the U(1) QLM flips a plaquette 2 and counts the number of flippable plaquettes (notice U = 0), respectively. ◮ In other words, the QLM and QDM have the same Hamiltonians, but realize the corresponding Gauss law differently. Connecting QDM with U(1) QLM ◮ QDM on the square lattice is characterized by the variables Dxy ∈ {0, 1} which indicates the presence or absence of a dimer on the link connecting neighboring sites x and y. ◮ The electric flux variables Exy are related to the dimer variables Dxy by Exy = (−1)x1 +x2 (Dxy − 12 ). Connecting QDM with U(1) QLM ◮ ◮ GLM: Gx |ψi = 0. P Let Gx = i (Ex,x+î − Ex−î,x ) be the infinitesimal U(1) gauge transformations (Gx commutes with the Hamiltonian). Then for dimer covering constraint, namely each site is touched exactly by one dimer, one has X (Dx,x+î + Dx−î,x ) = (−1)x1 +x2 . Gx = (−1)x1 +x2 i ◮ This implies the dimer covering constraint is equivalent to a staggered background electric charge ±1. ◮ Hence the physical states |Φi in the QDM should satisfy Gx |Φi = (−1)x1 +x2 (Gauss law for QDM). Controversial results of quantum dimer model on the square lattice plaquette columnar 2. 3. staggered columnar 1. columnar staggered staggered mixed 0.0 0.6 1.0 λ Numerical (Monte Carlo) Results ◮ We work with the height representation hX̃ of QDM living on the (four) dual lattices x̃: X̃ = (X1 + ◮ 1 1 , X2 + ). 2 2 A B A B A B C D C D C D A B A B A B C D C D C D A B A B A B C D C D C D Numerical Monte Carlo Results ◮ The quantities hA,B,C,D , residing at the sites of a dual sublattice, are related to the electric flux variables on the links by ◮ Ex,x+1̂ = Ex,x+2̂ = ′ 1 [hexX − hexX−2̂ ] mod2 = ± , 2 ′ 1 (−1)x1 +x2 [hexX − hexX−1̂ ] mod2 = ± , 2 X , X ′ ∈ {A, B, C, D}. A B A B 0 − 12 1 + 21 C D C D + 21 0 − 12 1 A B A B 0 − 12 1 + 21 C D C D + 12 0 − 12 1 Numerical Monte Carlo Results ◮ ◮ P We introduce novel order parameters: MX = ex ∈X sexX hexX , with 1 1 sexA = sexC = (−1)(ex1 + 2 )/2 (xe1 + 21 even), sexB = sexD = (−1)(ex1 − 2 )/2 (xe1 + 12 odd) With the order parameters M11 = MA − MB − MC + MD = M1 cos ϕ1 , M22 = MA + MB − MC − MD = M1 sin ϕ1 , M12 = MA − MB − MC − MD = M2 cos ϕ2 , M21 = −MA + MB − MC − MD = M2 sin ϕ2 , ◮ one defines ϕ = 12 (ϕ1 + ϕ2 + π4 ). Columnar: ϕ = 0mod π4 ; plaquette: ϕ = π π 8 mod 4 . Histograms of the Order parameters ❈ ✁✂✄☎❆✆ ✓✕✕ ✸ ✹ ✷ ✓✔✔ ✶ ✶ ✷ ✹ ✸ ✓✕✔ ✷ ✶ ✸ ✹ ✹ ✸ ✓✔✕ ✷ ✶ P✁❆✝✂✞✟✟✞ ✓✕✕ ❈ ❇ ❆ ✓✔✔ ❉ ❆ ❉ ❇ ❈ ✄▼✠✞❉ ✓✕✕ ✌✍ ✌☞ ☛☞ ☛✒ ✎✍ ✑✒ ✓✔✔ ✑✏ ✎✏ ✑✏ ✎✏ ✎✍ ✑✒ ☛✒ ☛☞ ✌☞ ✌✍ ✓✕✔ ❉ ❈ ✓✔✕ ❆ ❇ ❇ ❆ ❈ ❉ ✓✕✔ ✎✏ ✎✍ ✌✍ ✌☞ ✓ ✑✏ ☛☞ ✔✕ ☛✒ ✑✒ ☛✒ ✑✒ ☛☞ ✑✏ ✌☞ ✌✍ ✎✍ ✎✏ ❙✟❆✡✡✞✆✞❉ ✓✕✕ ❙ ✓✔✔ ✓✕✔ ❙ ✓✔✕ Histograms of the Order parameters Figure : Left: λ = 0.8. Right: λ = 0.9. Numerical Results (QMC) ✎ ✮ ❥ ✍✌ ☞☛ ✡ ❁ ✂ ✁✠ ✁✟ ✁✞ ✁✝ ✁✆ ✁☎ ✁✄ ✁✥ ✁✂ ✲ ✁✂ ▲✑✂✥✒❜❏ ✑ ☎ ▲✑✥☎✒❜❏ ✑ ✟ ▲✑✄✝✒❜❏ ✑ ✂✥ ▲✑☎✟✒❜❏ ✑ ✟ ▲✑✝ ✒❜❏ ✑ ✂✥ ❞✏ ✁✥ ✁✄ ✁☎ ✁✆ ✁✝ ❧ ✁✞ ✁✟ ✁✠ ✂ Numerical Results (QMC) ✁ ☎ ✲ ✁ ☎ ✴✟❡ ✲ ✁✝ ✲ ✁✝☎ ✲ ✁✆ ✲ ✁✆☎ ✲ ✁✄ ✲ ✁✄☎ ✲ ✁✂ ✲ ✁✆ ✁✆ ✁✂ ❧ ✁✥ ✁✞ ✝ Numerical Results (Exact Diagonalization) ◮ Based on the angle ϕ as well as how it transforms under various symmetries of the underlying model (QDM on the square lattice), one can construct the corresponding low-energy effective theory. ◮ ρ 1 ∂ ϕ∂ ϕ + ∂ ϕ∂ ϕ + κ(∂i ∂i ϕ)2 + δ cos2 (4ϕ), t t i i 2 c2 which is identical with the effective Lagrangian of the (2 + 1)-dimensional RP(1) model. L= ◮ The rotor spectrum indicates that the columnar phase persists till one reaches the RK point. More Numerical Results (Exact Diagonalization) c2 /ρ Ja2 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 −0.4−0.2 0 ◮ 0.2 0.4 0.6 0.8 λ 1 The phase diagram of QDM on the square lattice ◮ The columnar phase extends all the way to the RK point. No mixed phase and plaquette phase are found. More Numerical Results (QMC) ◮ Two external static charges ±2 relative to the staggered background at two points separated by odd lattices spacings − 21 1 + 12 0 − 12 0 + 21 0 − 21 0 + 21 1 + 12 0 + 12 1 + 12 0 − 21 1 More Numerical Results (QMC) ◮ † The energy density −J(U + U ) for λ = −2 : 140 0 -0.1 120 -0.2 100 -0.3 80 -0.4 60 -0.5 40 -0.6 20 -0.7 0 -0.8 0 20 40 60 80 100 120 140 More Numerical Results (QMC) 1 4 ◮ The total flux 2 fractionalizes into 8 strands (each carries ◮ The interior of the strands displays plaquette order and represent interfaces separating distinct columnar phases whose columns are oriented in orthogonal directions. 0 140 -0.1 120 -0.2 100 -0.3 80 -0.4 60 -0.5 40 -0.6 20 -0.7 0 -0.8 0 20 40 60 80 100 120 140 flux). More Numerical Results (QMC) The energy of the string plays the role of a confining charge-anti-charge potential V (r ) = σr 22 λ=-0.5 λ=0.5 λ=0.8 20 18 V(r) ◮ 16 14 12 10 8 10 15 20 25 30 r 35 40 45 Conclusions ◮ The quantum dimer model(s) (QDMs) are introduced and the possible candidate ground states for the model in the parameter space λ/J are described. ◮ The U(1) quantum link model is introduced as well. ◮ Connection between QDM and U(1) QLM. ◮ We study the phase diagram of QDM on the square lattice by simulating the U(1) QLM in the height variables space. ◮ No mixed and plaquette phases for λ ≤ 1. ◮ Fractionalized flux.