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arXiv:1003.4805 · Journal of Physics A Mathematical and Theoretical

Spontaneous magnetization of the integrable chiral Potts model

Helen Au-Yang, Jacques H. H. Perk

Published Oct 14, 2011• 7 citations
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Abstract

We show how Z -invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the N asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.

DOI
10.1088/1751-8113/44/44/445005

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We show how Z -invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the N asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.
For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szegő's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.
We adapt our previous results for the “partition function ” of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH, where H is the associated hamiltonian. The spontaneous magnetization Mr can be expressed in terms of particular matrix elements of e −αH S r 1 e−βH, where S1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model. KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 1
In 1993, Baxter gave eigenvalues of the transfer matrix of the N -state superintegrable chiral Potts model with the spin-translation quantum number Q , where m Q = ⌊( NL − L − Q )/ N ⌋. In our previous paper we studied the Q = 0 ground-state sector, when the size L of the transfer matrix is chosen to be a multiple of N . It was shown that the corresponding τ 2 matrix has a degenerate eigenspace generated by the generators of r = m 0 simple algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators E ± m and H m for m = 1, ..., r . Moreover, the corresponding 2 r eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of H m .
We prove an exact finite-volume symmetry formula for two-point functions in the periodic NNN-state superintegrable chiral Potts spin chain. We show that, for every chain length LLL and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy ⟨Z0rZR†r⟩∗=⟨Z0rZL−R†r⟩\langle Z_0^r Z_R^{\dagger r}\rangle^*=\langle Z_0^r Z_{L-R}^{\dagger r}\rangle⟨Z0r​ZR†r​⟩∗=⟨Z0r​ZL−R†r​⟩ for 1⩽r⩽N−11\leqslant r\leqslant N-11⩽r⩽N−1. Hence, whenever LLL is even, the midpoint correlation ⟨Z0rZL/2†r⟩\langle Z_0^r Z_{L/2}^{\dagger r}\rangle⟨Z0r​ZL/2†r​⟩ is real. Then we generalise the three-state chain case to arbitrary NNN and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.
In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic τ 2 ( t q ) model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of τ 2 ( t q ) in the Q = 0 sector is 2 r , with r = ( N − 1) L / N when the size of the transfer matrix L is a multiple of N . We introduce chiral Potts model operators, different from the more commonly used generators of quantum group . From these we can form the generators of a loop algebra . For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra .