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arXiv:1011.2603 · Journal of Statistical Mechanics Theory and Experiment

Spin operator matrix elements in the quantum Ising chain: fermion approach

N Iorgov, V Shadura, Yu Tykhyy

Published Feb 18, 2011• 22 citations
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Abstract

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.

DOI
10.1088/1742-5468/2011/02/p02028

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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.
Explicit factorized formulas for the matrix elements (form factors) of the spin operators σ x and σ y between the eigenvectors of the Hamiltonian of the finite quantum periodic XY -chain in a transverse field were derived. The derivation is based on the relations between three models: the model of quantum XY -chain, Ising model on 2D lattice and N = 2 Baxter–Bazhanov–Stroganov τ (2) -model. Due to these relations we transfer the formulas for the form factors of the latter model recently obtained by the use of separation of variables method to the model of quantum XY -chain. Hopefully, the formulas for the form factors will help in analysis of multipoint dynamic correlation functions at a finite temperature. As an example, we re-derive the asymptotics of the two-point correlation function in the disordered phase without the use of the Toeplitz determinants and the Wiener–Hopf factorization method.
We consider the general -symmetric free-fermion model on the finite periodic lattice, which includes as special cases the Ising model on the square and triangular lattices and the -symmetric BBS τ (2) -model with n = 2. Translating Kaufman's fermionic approach to diagonalization of Ising-like transfer matrices into the language of Grassmann integrals, we determine the transfer matrix eigenvectors and observe that they coincide with the eigenvectors of a square lattice Ising transfer matrix. This allows us to find exact finite-lattice form factors of spin operators for the statistical model and the associated finite-length quantum chains, of which the most general is equivalent to the XY chain in a transverse field.
In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length N∈2Z≥2N\in 2Z_{\ge 2}N∈2Z≥2​. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to 2\sqrt{2}2​. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient G(12)G(32)G(\frac{1}{2})G(\frac{3}{2})G(21​)G(23​) appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.
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.