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arXiv:cond-mat/0607389 · Journal of Statistical Mechanics Theory and Experiment

Free energy of the eight-vertex model with an odd number of lattice sites

Ovidiu I. Pâţu

Published Sep 10, 2007• 2 citations
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Abstract

We calculate the bulk contribution for the doubly degenerate largest eigenvalue of the transfer matrix of the eight-vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show that, as expected, in the thermodynamic limit the result coincides with the one in the N even case.

DOI
10.1088/1742-5468/2007/09/p09007

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We calculate the bulk contribution for the doubly degenerate largest eigenvalue of the transfer matrix of the eight-vertex model with an odd number of lattice sites N in the disordered regime using the generic equation for roots proposed by Fabricius and McCoy. We show that, as expected, in the thermodynamic limit the result coincides with the one in the N even case.
We study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity points η = mK/L with m odd the transfer matrix is known to satisfy the famous “TQ ” equation where Q(v) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(v) matrix is qualitatively different from the case of even N and in particular they satisfy a previously unknown equation which is more general than what is often called “Bethe’s equation. ” For the case of even m where no Q(v) matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the TQ equation. The ground state for the particular case of η = 2K/3 and N odd is investigated in detail.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.
Abstract The square-lattice eight-vertex model with vertex weights a , b , c , d obeying the relation ( a 2 + a b ) ( b 2 + a b ) = ( c 2 + a b ) ( d 2 + a b ) and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2 n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue Θ n = ( a + b ) 2 n + 1 . This proves a conjecture by Stroganov from 2001. The proof uses the supersymmetry of a related XYZ spin-chain Hamiltonian. The eigenstates of the transfer matrix corresponding to Θ n are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights Θ n is the largest eigenvalue of the transfer matrix.
Abstract The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K -matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.
For arbitrary Ising-like models of any dimension and Hamiltonians with a finite support with all possible multispin interactions and boundary conditions with a shift, the exact value of the free energy in the thermodynamic limit is obtained at some parametrically specified set of multispin interaction coefficients. In this case, half of the multispin interaction coefficients and the coordinates of the special eigenvector corresponding to the largest eigenvalue of the elementary transfer matrix are parameters, and the second half of the multispin coefficients is calculated using simple explicit formulas. For models with Hamiltonians invariant under the reversal of signs of all spins, the formulas are simplified. As examples of independent interest, solutions are written for the cases when the support of the Hamiltonian is a simplex, a cube, the support of the ANNNI model in spaces of 2, 3 and arbitrary dimensions.