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Factorized finite-size Ising model spin matrix elements from Separation of Variables

G von Gehlen, N. Z. Iorgov, Stanislav Zdislavovich Pakuliak, V. N. Shadura

Published Aug 4, 2013• 14 citations
OpenAlex

Abstract

Using the Sklyanin–Kharchev–Lebedev method of separation of variables adapted to the cyclic Baxter–Bazhanov–Stroganov or the τ(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.

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Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter--Bazhanov--Stroganov or τ(2)τ^{(2)}τ(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.
The Baxter-Bazhanov-Stroganov model (also known as the τ^(2) model) has attracted much interest because it provides a tool for solving the integrable chiral Z_N-Potts model. It can be formulated as a face spin model or via cyclic L-operators. Using the latter formulation and the Sklyanin-Kharchev-Lebedev approach, we give the explicit derivation of the eigenvectors of the component B_n(λ) of the monodromy matrix for the fully inhomogeneous chain of finite length. For the periodic chain we obtain the Baxter T-Q-equations via separation of variables. The functional relations for the transfer matrices of the τ^(2) model guarantee non-trivial solutions to the Baxter equations. For the N=2 case, which is free fermion point of a generalized Ising model, the Baxter equations are solved explicitly.
Using some modification of the standard fermion technique we derive factorized formulas for spin operator matrix elements (form factors) between general eigenstates of the Hamiltonian of the quantum Ising chain in a transverse field of finite length. The derivation is based on the approach recently used to derive factorized formulas for Z N -spin operator matrix elements between ground eigenstates of the Hamiltonian of the Z N -symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulas for the matrix elements of the Ising chain coincide with the corresponding expressions obtained by the separation of variables method.
Using some modification of the standard fermion technique we derive factorized formula for spin operator matrix elements (form-factors) between general eigenstates of the Hamiltonian of quantum Ising chain in a transverse field of finite length. The derivation is based on the approach recently used to derive factorized formula for Z_N-spin operator matrix elements between ground eigenstates of the Hamiltonian of the Z_N-symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulas for the matrix elements of Ising chain coincide with the corresponding expressions obtained by the Separation of Variables Method.