Alexander Seel

Integral representations for correlation functions of the XXZ chain at finite temperature

We derive a novel multiple integral representation for a generating function of the σz–σz correlation functions of the spin- XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic Bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics.

Separation of variables in the open XXX chain

We apply the Sklyanin method of separation of variables to the reflection algebra underlying the open spin-12 XXX chain with non-diagonal boundary fields. The spectral problem can be formulated in terms of a TQ-equation which leads to the known Bethe equations for boundary parameters satisfying a constraint. For generic boundary parameters we study the asymptotic behaviour of the solutions of the TQ-equation.

Integral representation of the density matrix of the XXZ chain at finite temperatures

We present an integral formula for the density matrix of a finite segment of the infinitely long spin- XXZ chain. This formula is valid for any temperature and any longitudinal magnetic field.

Functional Bethe ansatz methods for the open XXX chain

We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe-type equations determining the energy spectrum of the spin chain. In a second approach, the hierarchy of functional equations is mapped to an infinite system of nonlinear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite-size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.

A note on the spin-½ XXZ chain concerning its relation to the Bose gas

By considering the one-particle and two-particle scattering data for the spin-½ Heisenberg chain at T = 0 we derive a continuum limit relating the spin chain to the 1D Bose gas. Applying this limit to the quantum transfer matrix approach to the Heisenberg chain we obtain expressions for the correlation functions of the Bose gas at arbitrary temperatures.

A Note on the Bethe Ansatz Solution of the Supersymmetric t-J Model

The three different sets of Bethe ansatz equations describing the Bethe ansatz solution of the supersymmetric t-J model are known to be equivalent. Here we give a new, simplified proof of this fact which relies on the properties of certain polynomials. We also show that the corresponding transfer matrix eigenvalues agree.

The finite temperature density matrix and two-point correlations in the antiferromagnetic XXZ chain

We derive finite temperature versions of integral formulae for the two-point correlation functions in the antiferromagnetic XXZ chain. The derivation is based on the summation of density matrix elements characterizing a finite chain segment of length m. On this occasion we also supply a proof of the basic integral formula for the density matrix presented in an earlier publication.

Algebraic Bethe ansatz for the gl(1|2) generalized model: II. the three gradings

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9 × 9, rational, gl(1|2) invariant R-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t–J model, the supersymmetric U model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.

Correlation functions of the integrable isotropic spin-1 chain at finite temperature

We represent the density matrix of a finite segment of the integrable isotropic spin-1 chain in the thermodynamic limit as a multiple integral. Our integral formula is valid at finite temperature and also includes a homogeneous magnetic field.

Emptiness formation probability at finite temperature for the isotropic Heisenberg chain

Abstract We present an integral formula for a special correlation function of the isotropic spin- 1 2 antiferromagnetic Heisenberg chain. The correlation function describes the probability for the occurrence of a string of consecutive up-spins as a function of temperature, magnetic field and length of the string.