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arXiv:0812.3657 · Physical Review B

SU(N) Heisenberg model on the square lattice: A continuous- N quantum Monte Carlo study

K. S. D. Beach, Fabien Alet, Matthieu Mambrini, Sylvain Capponi

Published Nov 2, 2009• 86 citations
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Abstract

A quantum phase transition is typically induced by tuning an external parameter that appears as a coupling constant in the Hamiltonian. Another route is to vary the global symmetry of the system, generalizing, e.g., SU(2) to $\text{SU}(N)$. In that case, however, the discrete nature of the control parameter prevents one from identifying and characterizing the transition. We show how this limitation can be overcome for the $\text{SU}(N)$ Heisenberg model with the help of a singlet projector algorithm that can treat $N$ continuously. On the square lattice, we find a direct, continuous phase transition between N\'eel-ordered and crystalline bond-ordered phases at ${N}_{\text{c}}=4.57(5)$ with critical exponents $z=1$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.81(3)$.

DOI
10.1103/physrevb.80.184401

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A quantum phase transition is typically induced by tuning an external parameter that appears as a coupling constant in the Hamiltonian. Another route is to vary the global symmetry of the system, generalizing, e.g., SU(2) to SU(N)\text{SU}(N)SU(N). In that case, however, the discrete nature of the control parameter prevents one from identifying and characterizing the transition. We show how this limitation can be overcome for the SU(N)\text{SU}(N)SU(N) Heisenberg model with the help of a singlet projector algorithm that can treat NNN continuously. On the square lattice, we find a direct, continuous phase transition between N'eel-ordered and crystalline bond-ordered phases at Nc=4.57(5){N}_{\text{c}}=4.57(5)Nc​=4.57(5) with critical exponents z=1z=1z=1 and \ensuremathβ/\ensuremathν=0.81(3)\ensuremath{\beta}/\ensuremath{\nu}=0.81(3)\ensuremathβ/\ensuremathν=0.81(3).
To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU(NNN) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU(NNN) N\'eel and valence-bond solid phases is characterized for N=2N=2N=2, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size (L=256L=256L=256) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y\ensuremath{\equiv}1/\ensuremath{\nu}$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field ${\ensuremath{\Psi}}^{2}$ for the SU(3) model.
We study the quantum phase transition between the superfluid and valence bond solid in easy-plane J-Q models on the square lattice. The Hamiltonian we study is a linear combination of two model Hamiltonians: (1) an SU(2) symmetric model, which is the well known J-Q model that does not show any direct signs of a discontinuous transition on the largest lattices and is presumed continuous and (2) an easy-plane version of the J-Q model, which shows clear evidence for a first-order transition even on rather small lattices of size $L\ensuremath{\approx}16$. A parameter $0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ [$\ensuremath{\lambda}=0$ being the easy-plane model and $\ensuremath{\lambda}=1$ being the SU(2) symmetric J-Q model] allows us to smoothly interpolate between these two limiting models. We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) to investigate the nature of this transition as $\ensuremath{\lambda}$ is varied---here we present studies for $\ensuremath{\lambda}=0,0.5,0.75,0.85,0.95$, and 1. While we find that the first-order transition weakens as $\ensuremath{\lambda}$ is increased from 0 to 1, we find no evidence that the transition becomes continuous until the SU(2) symmetric point, $\ensuremath{\lambda}=1$. We thus conclude that the square-lattice superfluid-VBS transition in the two-component easy-plane model is generically first order.
We investigate the ground state of the SU(4)\mathrm{SU}(4)SU(4) Heisenberg model on a square lattice with spatial anisotropy on each plaquette bond using the tensor-network method based on infinite projected entangled pair states. We find that the SU(4)\mathrm{SU}(4)SU(4) singlet ground state appears in the strongly anisotropic limit, whereas N\'eel and valence-bond crystal orders coexist in the nearly isotropic limit. By examining the intermediate parameter region, we identify a phase transition between these phases. The nature of the phase transition is likely to be of first order, and the transition point is estimated to be around ${J}^{\ensuremath{'}}/J\ensuremath{\approx}0.85(5)$, where JJJ and ${J}^{\ensuremath{'}}$ are the interaction strengths of intra- and interplaquette bonds, respectively. We also calculate the anisotropy dependence of singlet correlations on a plaquette bond, which will be useful for future experiments of ultracold atoms in optical lattices.
We investigate thermal phase transitions to a valence-bond solid phase in SU(N)\mathrm{SU}(N)SU(N) Heisenberg models with four- or six-body interactions on a square or honeycomb lattice, respectively. In both cases, a thermal phase transition occurs that is accompanied by rotational symmetry breaking of the lattice. We perform quantum Monte Carlo calculations in order to clarify the critical properties of the models. The estimated critical exponents indicate that the universality classes of the square- and honeycomb-lattice cases are identical to those of the classical XYXYXY model with a Z4{Z}_{4}Z4​ symmetry-breaking field and the three-state Potts model, respectively. In the square-lattice case, the thermal exponent, $\ensuremath{\nu}$, monotonically increases as the system approaches the quantum critical point, while the values of the critical exponents, $\ensuremath{\eta}$ and $\ensuremath{\gamma}/\ensuremath{\nu}$, remain constant. From a finite-size scaling analysis, we find that the system exhibits weak universality, because the Z4{Z}_{4}Z4​ symmetry-breaking field is always marginal. In contrast, $\ensuremath{\nu}$ in the honeycomb-lattice case exhibits a constant value, even in the vicinity of the quantum critical point, because the Z3{Z}_{3}Z3​ field remains relevant in the SU(3) and SU(4) cases.
Using quantum Monte Carlo simulations, we study effects of disorder on the S=1/2S=1/2S=1/2 Heisenberg model with exchange constant JJJ on the square lattice supplemented by multispin interactions QQQ. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance rrr as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random QQQ couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent zzz at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.
We consider the S=1/2S=1/2S=1/2 Heisenberg model with nearest-neighbor interaction JJJ and an additional multispin interaction Q3{Q}_{3}Q3​ on the square lattice. The Q3{Q}_{3}Q3​ term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the N\'eel state to the VBS as a function of Q3/J{Q}_{3}/JQ3​/J. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The N\'eel-VBS transition in this case is strongly first order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging U(1)U(1)U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).