On the Six-Vertex Model’s Free Energy

Duminil-Copin, Hugo; Kozłowski, Karol K.; Krachun, Dmitry; Manolescu, Ioan; Tatiana, Tikhonovskaia,

doi:10.1007/s00220-022-04459-x

Cited by 7 publications

(8 citation statements)

References 44 publications

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“…Moreover, when D < 1/2, the work [59] also established the existence of a full large L asymptotic expansion for the sum appearing above and large class of test functions f . Further, the works [29,30] also established the condensation, by different techniques, when N/L = 1/2 and provided an estimation of the first subdominant correction to the sum in that case. Still, the Bethe Ansatz equations for the XXZ chain are also expected [12,31,46], e.g.…”

Section: The Non-linear Integral Equation Approach To the Bethe Ansat...mentioning

confidence: 97%

Dressed energy of the XXZ chain in the complex plane

2021

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We consider the dressed energy $$\varepsilon $$ ε of the XXZ chain in the massless antiferromagnetic parameter regime at $$0< \Delta < 1$$ 0 < Δ < 1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers, it describes the energy of particle–hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures, the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve $$\mathrm{Re}\, \varepsilon (\lambda ) = 0$$ Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function $$\mathrm{Re}\, \varepsilon $$ Re ε in regions of the complex plane, where it is positive.

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Section: The Non-linear Integral Equation Approach To the Bethe Ansat...mentioning

confidence: 97%

Dressed energy of the XXZ chain in the complex plane

2021

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“…To our knowledge, the statement about p max is still proved rigorously only asymptotically, see e.g. [52], though a commonly believed conjecture is that a stronger statement can be made that holds for finite L. In particular, the non-degenerate largest eigenvalue is expected to be found in the subspace with p = p max = L/2 for even finite L.…”

Section: J Stat Mech (2023) 033103mentioning

confidence: 99%

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

Povolotsky¹

2023

J. Stat. Mech.

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This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the 45∘ rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allows us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder, respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov–Stroganov point, while the effective tilt is introduced via the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin–Stroganov–Zagier’s solution of Baxter’s T–Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

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“…Besides the Potts model, there are a few SMMs for which the Helmholtz free energy is exactly known. Among them, we have the ice-type models or six-vertex models, introduced to model crystals with hydrogen bonds, such as ice crystal (see [ 19 ] and references therein, and [ 20 ] for a very recent account). Those models can be thought of as SMMs with support on a two-dimensional subshift of finite type X with entries in the alphabet

.…”

Section: The Sft–smm Correspondence For the Six-vertex Modelsmentioning

confidence: 99%

“…Two other important submodels are the KDP model and the Rys F model, which are described in detail in [ 19 ] (see [ 20 ] as well). They are defined on X by a one-letter interaction of the kind

with

.…”

Section: The Sft–smm Correspondence For the Six-vertex Modelsmentioning

confidence: 99%

“…At sufficiently low temperature, the system has a unique ergodic equilibrium state associated to periodic configurations alternating the symbols

and

. The Helmholtz free energy for this model (which can be found in [ 20 ]) is

with

and

. The SFT–SMM correspondence gives, for each

, a transitive two-dimensional SFT

, on the alphabet

, for which the topological entropy can be exactly computed.…”

Section: The Sft–smm Correspondence For the Six-vertex Modelsmentioning

confidence: 99%

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On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models

Corona

Salgado-García

Ugalde

2022

Entropy

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Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics—a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

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On the Six-Vertex Model’s Free Energy

Cited by 7 publications

References 44 publications

Dressed energy of the XXZ chain in the complex plane

Dressed energy of the XXZ chain in the complex plane

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models

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