© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/0401485

Generalized boundary condition, free energies of six-vertex models and fractal dimension

Kazuhiko Minami

Published Jan 25, 2004• 0 citations

Abstract

The structures of the configuration space of the six-vertex models with various boundaries and boundary conditions are investigated, and it is derived that the free energies depend on the boundary conditions, and that they are classified by the fractal structures. The "n-equivalences" of the boundary conditions are defined with a property that the models with n-equivalent boundary conditions result in the identical free energy. The configurations which satisfy the six-vertex restriction are classified, through the n-equivalences, into sets of configurations called islands. It is derived that each island shows a fractal structure when it is mapped to the real axis. Each free energy is expressed by the weighted fractal dimension (multi-fractal dimension) of the island. The fractal dimension of the island has a strict relation with the maximum eigenvalue of the corresponding block element of the transfer matrix. It is also found that the fractal dimensions of islands take the absolute maximum when the boundary is "alternative"; in this case, the free energy equals to the solution obtained by Lieb and Sutherland.

DOI: 10.48550/arxiv.cond-mat/0401485

Related papers

The free energies of six-vertex models and the n-equivalence relation

Kazuhiko Minami

The free energies of six-vertex models on a general domain D with various boundary conditions are investigated with the use of the n-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit (height,width)->(∞,∞). It is derived that the free energies of the model on the domain D are classified through the densities of left/down arrows on the boundary. Specifically, the free energy is identical to that obtained by Lieb [Phys. Rev. Lett. 18, 1046 (1967); 19, 108 (1967); Phys. Rev. 162, 162 (1967)] and Sutherland [Phys. Rev. Lett 19, 103 (1967)] with the cyclic boundary condition when the densities are both equal to 1/2. This fact explains several results already obtained through the transfer matrix calculation. The relation to the domino tiling (or dimer, or matching) problems is also noted.

•Jul 20, 2006•0 citations

An Equivalence Relation of Boundary/Initial Conditions and the Infinite Limit Properties

Kazuhiko Minami

The 'n-equivalences' of boundary conditions of lattice models are introduced and it is derived that the models with n-equivalent boundary conditions result in the identical free energy. It is shown that the free energy of the six-vertex model is classified through the density of left/down arrows on the boundary. The free energy becomes identical to that obtained by Lieb and Sutherland with the periodic boundary condition, if the density of the arrows is equal to 1/2. The relation to the structure of the transfer matrix and a relation to stochastic processes are noted.

arXiv:cond-mat/0503192•Journal of the Physical Society of Japan•May 15, 2005•1 citation

Surface free energy of the critical six-vertex model with free boundaries

Aleksander L. Owczarek, R. J. Baxter

The Bethe ansatz equations are derived for the six-vertex model with general boundary weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit. Finite-size corrections to the free energy (for a restricted class of boundary weights, including those corresponding to a Potts model with free boundaries) are calculated in the critical region. The first-order term gives the surface free energy of the model. The second-order term is found to be - pi okT tan( pi nu /2 mu )c/48N' 2 where c=(1-6 mu 2 /( pi 2 - pi mu )) is the conformal anomaly. This can be compared to - pi kT sin( pi nu / mu )c/6N' 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions. This difference can be explained geometrically using conformal invariance.

Journal of Physics A Mathematical and General•Apr 21, 1989•32 citations

Six-vertex model with domain wall boundary conditions and one-matrix model

Paul Zinn-Justin

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.

arXiv:math-ph/0005008•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 2000•123 citations

Domain-wall boundaries through non-diagonal twists in the six-vertex model

W. Galléas

Abstract In this work we elaborate on a previous result relating the partition function of the six-vertex model with domain-wall boundary conditions to eigenvalues of a transfer matrix. More precisely, we express the aforementioned partition function as a determinant of a matrix with entries that are eigenvalues of the transfer matrix of the anti-periodic six-vertex model.

arXiv:1807.03659•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2019•1 citation

States with v1=lambda, v2=-lambda and reciprocal equations in the six-vertex model

M. J. Rodríguez-Plaza

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,...,vn, the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary. The number obtained is independent of the value of Delta, but depends on N. We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a,b,c such eigenvalues are real.

arXiv:cond-mat/0506613•Jun 23, 2005•1 citation

The structures of the configuration space of the six-vertex models with various boundaries and boundary conditions are investigated, and it is derived that the free energies depend on the boundary conditions, and that they are classified by the fractal structures. The "n-equivalences" of the boundary conditions are defined with a property that the models with n-equivalent boundary conditions result in the identical free energy. The configurations which satisfy the six-vertex restriction are classified, through the n-equivalences, into sets of configurations called islands. It is derived that each island shows a fractal structure when it is mapped to the real axis. Each free energy is expressed by the weighted fractal dimension (multi-fractal dimension) of the island. The fractal dimension of the island has a strict relation with the maximum eigenvalue of the corresponding block element of the transfer matrix. It is also found that the fractal dimensions of islands take the absolute maximum when the boundary is "alternative"; in this case, the free energy equals to the solution obtained by Lieb and Sutherland.

The free energies of six-vertex models on a general domain D with various boundary conditions are investigated with the use of the n-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit (height,width)->(∞,∞). It is derived that the free energies of the model on the domain D are classified through the densities of left/down arrows on the boundary. Specifically, the free energy is identical to that obtained by Lieb [Phys. Rev. Lett. 18, 1046 (1967); 19, 108 (1967); Phys. Rev. 162, 162 (1967)] and Sutherland [Phys. Rev. Lett 19, 103 (1967)] with the cyclic boundary condition when the densities are both equal to 1/2. This fact explains several results already obtained through the transfer matrix calculation. The relation to the domino tiling (or dimer, or matching) problems is also noted.

The 'n-equivalences' of boundary conditions of lattice models are introduced and it is derived that the models with n-equivalent boundary conditions result in the identical free energy. It is shown that the free energy of the six-vertex model is classified through the density of left/down arrows on the boundary. The free energy becomes identical to that obtained by Lieb and Sutherland with the periodic boundary condition, if the density of the arrows is equal to 1/2. The relation to the structure of the transfer matrix and a relation to stochastic processes are noted.

The Bethe ansatz equations are derived for the six-vertex model with general boundary weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit. Finite-size corrections to the free energy (for a restricted class of boundary weights, including those corresponding to a Potts model with free boundaries) are calculated in the critical region. The first-order term gives the surface free energy of the model. The second-order term is found to be - pi okT tan( pi nu /2 mu )c/48N' 2 where c=(1-6 mu 2 /( pi 2 - pi mu )) is the conformal anomaly. This can be compared to - pi kT sin( pi nu / mu )c/6N' 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions. This difference can be explained geometrically using conformal invariance.

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.

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,...,vn, the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary. The number obtained is independent of the value of Delta, but depends on N. We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a,b,c such eigenvalues are real.