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arXiv:2012.05182 · Journal of Physics A Mathematical and Theoretical

The two-point correlation function in the six-vertex model

P. Belov, Nicolai Reshetikhin

Published Feb 22, 2022• 14 citations
arXivHTMLPDF

Abstract

Abstract We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm. Particular attention is paid to the free fermionic point (Δ = 0), for which the correlation functions are obtained analytically in the thermodynamic limit. A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered (|Δ| < 1) phase and to monitor the logarithm-like behavior of correlation functions there. For the antiferroelectric (Δ < −1) phase, the exponential decrease of correlation functions is observed.

DOI
10.1088/1751-8121/ac578e

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Abstract We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm. Particular attention is paid to the free fermionic point (Δ = 0), for which the correlation functions are obtained analytically in the thermodynamic limit. A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered (|Δ| < 1) phase and to monitor the logarithm-like behavior of correlation functions there. For the antiferroelectric (Δ < −1) phase, the exponential decrease of correlation functions is observed.
Abstract We study numerically the density profile in the six-vertex model with domain wall boundary conditions. Using a Monte Carlo algorithm originally proposed by Allison and Reshetikhin we numerically evaluate the inhomogeneous density profiles in the disordered and antiferromagnetic regimes where frozen corners appear. At the free fermion point we present an exact finite-size formula for the density on the horizontal edges that relies on the imaginary time transfer matrix approach. In all cases where exact analytic forms for the density and the arctic curves are known the numerical method shows perfect agreement with them. This also suggests the possibility of its use for accurate quantitative purposes.
We present results of the Monte-Carlo simulations for scaling of the free energy in dimers on the hexagonal lattice.The traditional Markov-chain Metropolis algorithm and more novel non-Markov Wang-Landau algorithm are applied.We compare the calculated results with the theoretical prediction for the equilateral hexagon and show that the latter algorithm gives more precise results for the dimer model.For a non-hexagonal domain the theoretical results are not available, so we present the numerical results for a certain geometry of the domain.We also study the two-point correlation function in simulations of dimers and the six-vertex model.The logarithmic dependence of the correlation function on the distance, which is in accordance with the Gaussian free field description of fluctuations, is obtained.
Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional NxN square lattice with the domain wall boundary conditions are considered. The one-point correlation function ("boundary polarization") is expressed via the partition function of the model on a sublattice. The partition function is represented in terms of standard objects in the theory of orthogonal polynomials. This representation is used to study the large N limit: the presence of the boundary affects the macroscopic quantities of the model even in this limit. The logarithmic terms obtained are compared with predictions from conformal field theory.
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.
We review our recent work on the boundary correlation functions of integrable vertex models on an N × N lattice with domain wall boundary conditions. Particularly considered is the six vertex model. The general expression of the boundary correlation functions is obtained for the six vertex model by use of the quantum inverse scattering method. We also comment on the potential application of the boundary correlation functions, and the relation between the boundary correlation functions for the nineteen vertex model.
We perform a numerical study of the F model with domain-wall boundary conditions. Various exact results are known for this particular case of the six-vertex model, including closed expressions for the partition function for any system size as well as its asymptotics and leading finite-size corrections. To complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size, up to L=512. We compare the energy to its exactly known large-L asymptotics. We investigate the model's infinite-order phase transition by means of finite-size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with Duine and Barkema. In addition we analyze local properties of the model. Our data are perfectly consistent with analytical expressions for the arctic curves. We investigate the structure inside the temperate region of the lattice, confirming the oscillations in vertex densities that were first observed by Syljuåsen and Zvonarev and recently studied by Lyberg et al. We point out "(anti)ferroelectric" oscillations close to the corresponding frozen regions as well as "higher-order" oscillations forming an intricate pattern with saddle-point-like features.