arXiv:1908.05773 · Journal of Statistical Mechanics Theory and Experiment

Arctic curve of the free-fermion six-vertex model with reflecting end boundary condition

I R Passos, G A P Ribeiro

Published Nov 1, 2019• 2 citations

Abstract

Abstract We consider the six-vertex model with reflecting end boundary condition. We study the asymptotic behavior of the boundary correlations. This asymptotic behavior is used as an input into the tangent method in order to derive analytically the arctic curve at the free fermion point. The obtained curve is a semicircle, which is in agreement with previous Monte Carlo simulations.

DOI: 10.1088/1742-5468/ab4bbf

Related papers

The density profile of the six vertex model with domain wall boundary conditions

I. Lyberg, V. E. Korepin, Jacopo Viti

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.

arXiv:1612.06758Journal of Statistical Mechanics Theory and ExperimentMay 1, 201717 citations

Boundary correlations for the six-vertex model with reflecting end boundary condition

I R Passos, G A P Ribeiro

Abstract We consider the six-vertex model with reflecting end boundary condition. We compute analytically boundary correlation functions, such as the boundary polarization and the emptiness formation probability. In order to do that, we use the Sklyanin’s reflection algebra to derive recursion relations for the partition function of the model as well as for the boundary correlations in terms of the partition function. Thanks to the Tsuchiya determinant formula, these recursion relations allow the boundary correlations to be also efficiently written in determinant form.

arXiv:1904.06383Journal of Statistical Mechanics Theory and ExperimentAug 1, 20192 citations

Inhomogeneous Gaussian free field inside the interacting arctic curve

Etienne Granet, Louise Budzynski, Jérôme Dubail +1 more

Abstract The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e. models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e. not free, not determinantal) six-vertex model, and provide evidence for the following two claims: (i) the critical fluctuations are given by a Gaussian free field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous , meaning that its coupling constant becomes position-dependent, . The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green’s function of the Laplacian inside the critical domain, but instead, of the Green’s function of a generalized Laplacian parametrized by the function . Surprisingly, we also find that there is a change of regime when the anisotropy parameter Δ is smaller than , with becoming singular at one point.

arXiv:1807.07927Journal of Statistical Mechanics Theory and ExperimentJan 1, 201926 citations

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

P. Belov, Nicolai Reshetikhin

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.

arXiv:2012.05182Journal of Physics A Mathematical and TheoreticalFeb 22, 202214 citations

The arctic curve of the domain wall six-vertex model in its antiferroelectric regime

F Colomo, A G Pronko, Paul Zinn-Justin

An explicit expression for the spatial curve separating the region of ferroelectric order ('frozen' zone) from the disordered one ('temperate' zone) in the six-vertex model with domain wall boundary conditions in its antiferroelectric regime is obtained.

arXiv:1001.2189Journal of Statistical Mechanics Theory and ExperimentMar 4, 201038 citations

Determinant formula for the six-vertex model with reflecting end

Osamu Tsuchiya

Using the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six vertex model with reflecting end.

Jan 1, 199865 citations