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Physical review. B, Condensed matter

Monte Carlo investigation of the resonating-valence-bond ground state and a lattice statistical model

Bill Sutherland

Published Oct 1, 1988• 18 citations
Publisher

Abstract

We have suggested that the properties of the resonating-valence-bond (RVB) ground state can usefully be studied first by a correspondence with a lattice statistics model, and second by embedding this RVB point into a larger family of lattice statistical models---the loop gas. The nature of this loop gas---correlations, phase transitions, and order parameters---is studied in this paper by means of Monte Carlo techniques, on lattices of up to 4608 sites or 9216 bonds. An ordered phase is identified with divergent susceptibility and an order parameter which has the topology of a circle. Since the model is two dimensional, the phase transition is expected to be of Kosterlitz-Thouless (KT) type. The RVB point is within this ordered KT phase.

DOI
10.1103/physrevb.38.6855

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We have suggested that the properties of the resonating-valence-bond (RVB) ground state can usefully be studied first by a correspondence with a lattice statistics model, and second by embedding this RVB point into a larger family of lattice statistical models---the loop gas. The nature of this loop gas---correlations, phase transitions, and order parameters---is studied in this paper by means of Monte Carlo techniques, on lattices of up to 4608 sites or 9216 bonds. An ordered phase is identified with divergent susceptibility and an order parameter which has the topology of a circle. Since the model is two dimensional, the phase transition is expected to be of Kosterlitz-Thouless (KT) type. The RVB point is within this ordered KT phase.
We study a short-range resonating valence bond (RVB) wave function with diagonal links on the square lattice that permits sign-problem free wave function Monte Carlo studies. Special attention is given to entanglement properties, in particular, the study of minimum entropy states (MES) according to the method of Zhang et al. [Phys. Rev. B 85, 235151 (2012)]. We provide evidence that the MES associated with the RVB wave functions can be lifted from an associated quantum dimer picture of these wave functions, where MES states are certain linear combinations of eigenstates of a 't Hooft ``magnetic loop''-type operator. From this identification, we calculate a value consistent with ln(2)ln(2)ln(2) for the topological entanglement entropy directly for the RVB states via wave function Monte Carlo. This corroborates the Z2{\mathbb{Z}}_{2}Z2​ nature of the RVB states. We furthermore define and elaborate on the concept of a ``pre-Kasteleyn'' orientation that may be useful for the study of lattices with nonplanar topology in general.
A one-parameter family of long-range resonating-valence-bond (RVB) state on the square lattice was previously proposed to describe a critical spin liquid (SL) phase of the spin-1/2 frustrated Heisenberg model. We provide evidence that this RVB state in fact also realizes a topological (long-range entangled) Z2{\mathbb{Z}}_{2}Z2​ SL, limited by two transitions to critical SL phases. The topological phase is naturally connected to the Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry of the local tensor. This Rapid Communication shows that, on one hand, spin-1/2 topological SL with C4v{C}_{4v}C4v​ point-group symmetry and SU(2)SU(2)SU(2) spin rotation symmetry exists on the square lattice and, on the other hand, criticality and nonbipartiteness are compatible. We also point out that strong similarities between our phase diagram and the ones of classical interacting dimer models suggest both can be described by similar Kosterlitz-Thouless transitions. This scenario is further supported by the analysis of the one-dimensional boundary state. Forms of parent Hamiltonians hosting the Z2{\mathbb{Z}}_{2}Z2​ SL are suggested.
We study the S=1/2S=1/2S=1/2 Heisenberg (JJJ) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions (QQQ) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large Q/JQ/JQ/J and diverging when Q/JQ/JQ/J approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge c=1c=1c=1 and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of U(1)U(1)U(1) symmetry of the order parameter at T=TcT={T}_{c}T=Tc​ when ${T}_{c}\ensuremath{\rightarrow}0$.
We use Monte Carlo simulations to study properties of Anderson's resonating-valence-bond (RVB) spin-liquid state on the square lattice (i.e., the equal superposition of all pairing of spins into nearest-neighbor singlet pairs) and compare with the classical dimer model (CDM). The latter system also corresponds to the ground state of the Rokhsar-Kivelson quantum dimer model at its critical point. We find that, although spin-spin correlations decay exponentially in the RVB, four-spin valence-bond-solid correlations are critical, qualitatively like the well-known dimer-dimer correlations of the CDM, but decaying more slowly (as $1/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}1.20$, compared with $\ensuremath{\alpha}=2$ for the CDM). We also compute the distribution of monomer (defect) pair separations, which decay by a larger exponent in the RVB than in the CDM. We further study both models in their different winding-number sectors and evaluate the relative weights of different sectors. Like the CDM, all the observed RVB behaviors can be understood in the framework of a mapping to a ``height'' model characterized by a gradient-squared stiffness constant KKK. Four independent measurements consistently show a value ${K}_{\mathrm{RVB}}\ensuremath{\approx}1.6{K}_{\mathrm{CDM}}$, with the same kinds of numerical evaluations of KCDM{K}_{\mathrm{CDM}}KCDM​ giving results in agreement with the rigorously known value ${K}_{\mathrm{CDM}}=\ensuremath{\pi}/16$. The background of a nonzero winding-number gradient W/LW/LW/L introduces spatial anisotropies and an increase in the effective KKK, both of which can be understood as a consequence of anharmonic terms in the height-model free energy, which are of relevance to the recently proposed scenario of ``Cantor deconfinement'' in extended quantum dimer models. In addition to the standard case of short bonds only, we also studied ensembles in which fourth-neighbor (bipartite) bonds are allowed at a density controlled by a tunable fugacity, resulting (as expected) in a smooth reduction of KKK.
The concept of the resonating valence bond (RVB) state was originally proposed by Anderson (1987) for the ground state of the positive-U Hubbard model. However, the on-site pairing long-range order is expected to exist in the ground state of the negative-U Hubbard model and to be incompatible with the RVB order. In this article, we shall rigorously prove that, in fact, these two long-range orderings either coexist or are suppressed simultaneously in the global ground states of the doped Hubbard models.