Thermodynamics of the Anisotropic Spin - 1 / 2 Heisenberg Chain and Related Quantum Chains ∗

The free energy and correlation lengths of the spin-1/2 XY Z chain are studied at finite temperature. We use the quantum transfer matrix approach and derive non-linear integral equations for all eigenvalues. Analytic results are presented for the low-temperature asymptotics, in particular for the critical XXZ chain in an external magnetic field. These results are compared to predictions by conformal field theory. The integral equations are solved numerically for the non-critical XXZ chain and the related spin-1 biquadratic chain at arbitrary temperature. One of the simplest models of magnetism is the Heisenberg model describing the exchange interaction of spins. Despite its simplicity, in general only approximate methods are available for its study. An exception to this situation is the one-dimensional spin-1/2 case. Very early Bethe [1] constructed the eigenstates for the isotropic spin-1/2 Heisenberg chain. Much later also the fully anisotropic spin-1/2 XY Z chain was discovered to be integrable [2,3] as it is related to the exactly solvable eight-vertex model [4]. In this way the spectrum of the XY Z chain and the ground state correlation lengths are known, cf. [5,6]. The thermodynamics of this model were studied in [7] by an elaborate version of the method used in [8]. Unfortunately, only the free energy of the XY Z chain could be studied within this approach, the correlation functions at finite temperature remained out of reach. In this paper we apply an alternative method to the thermodynamics of inte-grable quantum chains giving the free energy and correlation lengths. Our treatment will follow somewhat the approach of [9,10] where the Suzuki-Trotter formula was employed leading to a mapping of quantum chains at finite temperature to classical two-dimensional lattice models. The quantum transfer matrix of the Heisenberg model was identified as the diagonal-to-diagonal transfer matrix of the exactly solv-able six-vertex model which is identical to the row-to-row transfer matrix of an inhomogeneous six-vertex model. The eigenvalues are known in terms of a Bethe ansatz [11,12,13], the largest one yielding the free energy and the next-leading ones the correlation lengths. The study of the limit of infinite Trotter number poses a certain problem. In [9] this limit was taken numerically by extrapolation of the eigenvalues, in [13,14] it was done analytically for low temperatures. In [10] the limit of infinite Trotter number was taken analytically at the Bethe ansatz level for any finite temperature. The derived equations are not of the integral …