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Low-temperature spectrum of the quantum transfer matrix of the XXZ chain in the massless regime

Saskia Faulmann, Frank Göhmann, Karol K. Kozlowski

TL;DR

This work provides a rigorous solvability framework for the nonlinear integral equations that describe the spectrum of the quantum transfer matrix of the XXZ chain in the massless regime at low temperature. By recasting Bethe Ansatz equations into NLIEs and introducing a contour-deformation strategy, the authors prove existence and uniqueness of solutions in a controlled function space and derive the low-$T$ expansion of the dominant eigenvalue, establishing a direct link to the $c=1$ conformal field theory spectrum. Under two physics-motivated conjectures (dominance and commutativity of limits), the leading temperature dependence of the free energy and the conformal structure of the spectrum are rigorously connected to CFT data, thus providing a mathematically solid realization of long-standing physics predictions. The methodology hinges on auxiliary linear integral equations, a fixed-point theory for a modified NLIE, and a detailed analysis of particle-hole root quantisation, facilitating explicit characterization of the relevant eigenvalues and their thermal form factors. Overall, the results give a rigorous bridge between integrable QTM analysis, finite-temperature Bethe Ansatz, and conformal field theory predictions in the XXZ model’s massless regime.

Abstract

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 $c=1$ conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.

Low-temperature spectrum of the quantum transfer matrix of the XXZ chain in the massless regime

TL;DR

This work provides a rigorous solvability framework for the nonlinear integral equations that describe the spectrum of the quantum transfer matrix of the XXZ chain in the massless regime at low temperature. By recasting Bethe Ansatz equations into NLIEs and introducing a contour-deformation strategy, the authors prove existence and uniqueness of solutions in a controlled function space and derive the low- expansion of the dominant eigenvalue, establishing a direct link to the conformal field theory spectrum. Under two physics-motivated conjectures (dominance and commutativity of limits), the leading temperature dependence of the free energy and the conformal structure of the spectrum are rigorously connected to CFT data, thus providing a mathematically solid realization of long-standing physics predictions. The methodology hinges on auxiliary linear integral equations, a fixed-point theory for a modified NLIE, and a detailed analysis of particle-hole root quantisation, facilitating explicit characterization of the relevant eigenvalues and their thermal form factors. Overall, the results give a rigorous bridge between integrable QTM analysis, finite-temperature Bethe Ansatz, and conformal field theory predictions in the XXZ model’s massless regime.

Abstract

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 conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.
Paper Structure (48 sections, 46 theorems, 778 equations, 10 figures)

This paper contains 48 sections, 46 theorems, 778 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The integrable XXZ spin-1/2 chain
  3. The conformal field theory predictions
  4. CFT predictions from the perspective of our results
  5. The non-linear integral equation approach to the Bethe Ansatz equations
  6. Main results
  7. Auxiliary linear integral equations
  8. Working hypotheses
  9. The main results
  10. An equivalent non-linear integral equation
  11. An auxiliary contour with better properties
  12. An equivalent non-linear integral equation
  13. The low-$T$ expansion of the solution to the non-linear integral equation
  14. Fine properties of the functional space for the non-linear integral equation
  15. The main building ingredients of the functional space
  16. ...and 33 more sections

Key Result

Theorem 1.1

There exists $\eta>0$, $T_0>0$ and $C_{\mathcal{M}}^{(0)}>0$ such that, for any the quantum transfer matrix admits the Eigenvalue In this expression the integration curve $\mathscr{C}_{\mathrm{ref}}$ is as defined in definition contour Cref, c.f. Fig. contour integration Cref, and the logarithm $\mathscr{L}\mathrm{n}_{ \mathscr{C}_{\mathrm{ref}} }$ is introduced in definition logarithm de 1+a.

Figures (10)

  • Figure 1: The integration contour $\widehat{\mathscr{C}}_{\varepsilon}$.
  • Figure 2: Definition of the contours $\Gamma_{\delta_T}^{(L/R)}$ used in the definition of the reference contour $\mathscr{C}_{\mathrm{ref}}$. Note that one has $\mathfrak{t}_{L}^{(\pm)}>0$ and $\mathfrak{t}_{R}^{(\pm)}<0$.
  • Figure 3: Definition of the reference contour $\mathscr{C}_{\mathrm{ref}}$.
  • Figure 4: The curve $\mathscr{C}_{\varepsilon}$ and the deformed strip $\mathcal{M}$.
  • Figure 5: A sketch of the particle and hole parameters (shaded) that have to be removed from the sets $\widehat{ \boldsymbol{ \texttt{Y} } }'$ and $\widehat{ \boldsymbol{ \texttt{X} } }'$ in order to switch to $\widehat{ \boldsymbol{ \texttt{Y} } }_{\mathrm{ref}}'$ and $\widehat{ \boldsymbol{ \texttt{X} } }_{\mathrm{ref}}'$.
  • ...and 5 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • ...and 41 more