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arXiv:1504.07923 · Journal of Physics A Mathematical and Theoretical

Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime

Maxime Dugave, Frank Göhmann, Karol Kajetan Kozlowski, Junji Suzuki

Published Jul 28, 2015• 27 citations
arXivHTMLPDF

Abstract

We consider the spectrum of correlation lengths of the spin− XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy finite sets of algebraic equations, the so-called higher-level Bethe ansatz equations. The behaviour of these equations, if we send the temperature T to zero, is different for zero and non-zero magnetic field h . If h is zero the situation is much like in the case of the usual transfer matrix. Non-trivial higher-level Bethe ansatz equations remain which determine certain complex excitation parameters as functions of hole parameters which are free on a line segment in the complex plane. If h is non-zero, on the other hand, a remarkable restructuring occurs, and all parameters which enter the description of the quantum transfer matrix eigenvalues can be interpreted entirely in terms of particles and holes which are freely located on two curves when T goes to zero.

DOI
10.1088/1751-8113/48/33/334001

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We consider the spectrum of correlation lengths of the spin− XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy finite sets of algebraic equations, the so-called higher-level Bethe ansatz equations. The behaviour of these equations, if we send the temperature T to zero, is different for zero and non-zero magnetic field h . If h is zero the situation is much like in the case of the usual transfer matrix. Non-trivial higher-level Bethe ansatz equations remain which determine certain complex excitation parameters as functions of hole parameters which are free on a line segment in the complex plane. If h is non-zero, on the other hand, a remarkable restructuring occurs, and all parameters which enter the description of the quantum transfer matrix eigenvalues can be interpreted entirely in terms of particles and holes which are freely located on two curves when T goes to zero.
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.
We consider the dressed energy ε\varepsilonε of the XXZ chain in the massless antiferromagnetic parameter regime at 0<Δ<10 < \Delta < 10<Δ<1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers it describes the energy of particle-hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve Re ε(λ)=0{\rm Re}\, \varepsilon (\lambda) = 0Reε(λ)=0. We describe this curve and give lower bounds to the function Re ε{\rm Re}\, \varepsilonReε in regions of the complex plane, where it is positive.
Abstract We consider the dressed energy ε\varepsilon ε ε of the XXZ chain in the massless antiferromagnetic parameter regime at 0<Δ<10< \Delta < 10<Δ<1 0 < Δ < 1 and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers, it describes the energy of particle–hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures, the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve Re ε(λ)=0\mathrm{Re}\, \varepsilon (\lambda ) = 0Reε(λ)=0 Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function Re ε\mathrm{Re}\, \varepsilon Reε Re ε in regions of the complex plane, where it is positive.
Abstract We use the form factors of the quantum transfer matrix in the zero-temperature limit in order to study the two-point ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. We obtain novel form factor series representations of the correlation functions which differ from those derived either from the q -vertex-operator approach or from the algebraic Bethe Ansatz approach to the usual transfer matrix. We advocate that our novel representations are numerically more efficient and allow for a straightforward calculation of the large-distance asymptotic behaviour of the two-point functions. Keeping control over the temperature corrections to the two-point functions we see that these are of order T ∞ in the whole antiferromagnetic massive regime. The isotropic limit of our result yields a novel form factor series representation for the two-point correlation functions of the XXX chain at zero magnetic field.
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.
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.