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Letters in Mathematical Physics

Dressed energy of the XXZ chain in the complex plane

Saskia Faulmann, Frank Göhmann, Karol Kajetan Kozlowski

Published Oct 1, 2021• 1 citation
Publisher

Abstract

Abstract We consider the dressed energy $$\varepsilon $$ ε of the XXZ chain in the massless antiferromagnetic parameter regime at $$0< \Delta < 1$$ 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 $$\mathrm{Re}\, \varepsilon (\lambda ) = 0$$ Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function $$\mathrm{Re}\, \varepsilon $$ Re ε in regions of the complex plane, where it is positive.

DOI
10.1007/s11005-021-01473-3

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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.
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.
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 review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.