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Physical review. B, Condensed matter

Correlation length and free energy of theS=1/2XYZchain

Minoru Takahashi

Published Mar 1, 1991• 69 citations
Publisher

Abstract

A set of equations that allows one to calculate the correlation length and the free energy of the S=1/2 XYZ chain at a given temperature is obtained. It contains infinite unknown numbers and is derived by the largest and the second-largest eigenvalues of the quantum transfer matrix in the limit of infinite Trotter number. The numerical solution of this set of equations gives very accurate values of the free energy and the correlation length at arbitrary temperature. The energy gaps that appear in the correlation length and the free energy at low temperature are discussed.

DOI
10.1103/physrevb.43.5788

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A set of equations that allows one to calculate the correlation length and the free energy of the S=1/2 XYZ chain at a given temperature is obtained. It contains infinite unknown numbers and is derived by the largest and the second-largest eigenvalues of the quantum transfer matrix in the limit of infinite Trotter number. The numerical solution of this set of equations gives very accurate values of the free energy and the correlation length at arbitrary temperature. The energy gaps that appear in the correlation length and the free energy at low temperature are discussed.
The supersymmetric tJ model in one dimension is investigated by means of the quantum transfer matrix (QTM). The free energy of the quantum system is given by just the largest eigenvalue of the QTM whereas the next-largest ones determine the correlation lengths at finite temperature. The first step of the analysis consists of an embedding of the QTM into a family of commuting matrices. In the second step the eigenvalues equations are derived by an algebraic Bethe ansatz. Finally, the eigenvalue equations are transformed into a set of nonlinear integral equations. Some numerical results are presented. The specific heat for instance shows the typical Luttinger liquid behavior at low temperatures.
The quantum mechanical analogue of the classical Perk - Schultz model is considered which comprises the Uimin - Sutherland model and the integrable t - J chain. The quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic Bethe ansatz. Only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature. The Bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions. Furthermore, the eigenvalue of the quantum transfer matrix is expressed in terms of these functions. The integral formulation allows for taking the limit of infinite Trotter - Suzuki number analytically. The low-temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented.
Transfer matrix methods are used to obtain the partition function and correlation length of Ising chains at low temperatures. For the infinite-spin chain, asymptotically exact eigenvalues and eigenvectors are obtained from an iteration scheme, and from a secular equation method. It is also shown how the Ising chain for general spin S (including S to infinity ) may be treated using a 2*2 transfer matrix. For low temperatures (K identical to J/K B T>>1) the inverse correlation length, in units of the spin spacing, is xi -1 =4S e -2K +O(e -4K ) for finite S, and xi -1 =8K e -2K (ln 4K+ gamma +O((lnK)/K)) for infinite spin where gamma is Euler's constant.
The free energy per lattice site of a quantum spin chain in the thermodynamic limit is determined by a single `dominant' Eigenvalue of an associated quantum transfer matrix in the infinite Trotter number limit. For integrable quantum spin chains, related with solutions of the Yang-Baxter equation, an appropriate choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full spectrum allows one to study its universality properties such as the appearance of a conformal spectrum in the low-temperature regime. More generally, accessing the full spectrum is a necessary step for deriving thermal form factor series representations of the correlation functions of local operators for the spin chain under consideration. These are statements that have been established by physicists on a heuristic level and that are calling for a rigorous mathematical justification. In this work we implement certain aspects of this programme with the example of the XXZ quantum spin chain in the antiferromagnetic massless regime and in the low-temperature limit. We rigorously establish the existence, uniqueness and characterise the form of the solutions to the non-linear integral equations that are equivalent to the Bethe Ansatz equations for the quantum transfer matrix of this model. This allows us to describe that part of the quantum transfer matrix spectrum that is related to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter number limit. Within the considered part of the spectrum we rigorously identify the dominant Eigenvalue and show that those correlations lengths that diverge in the low-temperature limit are given, to the leading order, by the spectrum of the free Boson c=1c=1c=1 conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.
The asymptotics of the equal-time one-particle Green's function of the half-filled one-dimensional Hubbard model is studied at finite temperature. We calculate its correlation length by evaluating the largest and the second largest eigenvalues of the Quantum Transfer Matrix (QTM). In order to allow for the genuinely fermionic nature of the one-particle Green's function, we employ the fermionic formulation of the QTM based on the fermionic R -operator of the Hubbard model. The purely imaginary value of the second largest eigenvalue reflects the k F ( = π/2) oscillations of the one-particle Green's function at half-filling. By solving numerically the Bethe ansatz equations with Trotter numbers up to N = 10240, we obtain accurate data for the correlation length at finite temperatures down into the very low-temperature region. The correlation length remains finite even at T = 0 due to the existence of the charge gap. Our numerical data confirm Stafford and Millis' conjecture regarding an analytic expression for the correlation length at T = 0.