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Dressed energy of the XXZ chain in the complex plane

Saskia Faulmann, Frank Göhmann, Karol K. Kozlowski

Abstract

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

Dressed energy of the XXZ chain in the complex plane

Abstract

We consider the dressed energy of the XXZ chain in the massless antiferromagnetic parameter regime at 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 . We describe this curve and give lower bounds to the function in regions of the complex plane, where it is positive.

Paper Structure

This paper contains 9 sections, 7 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the kernel function
  4. The solvable case $Q = \infty$
  5. The general case of finite $Q$
  6. Proofs
  7. Proof of Theorem \ref{['th:fpoints']}
  8. Proof of Theorem \ref{['th:main']}
  9. Conclusions

Key Result

Theorem \oldthetheorem

Existence and uniqueness of Fermi points DGK14b. Let $\gamma \in (0, \pi/2)$ and

Figures (2)

  • Figure 1: The curves (\ref{['reepszero']}) for $J = 1$, $\gamma = 1.3$ and various values of the magnetic field in units of $h_c = 5.07$. Loosely speaking, Theorem \ref{['th:main']} says that the figure describes the generic situation: ${\rm Re\,} \varepsilon (\lambda) = 0$ is a simple closed curve, situated entirely inside the strip $|{\rm Im\,} \lambda| < \frac{\gamma}{2}$, symmetric with respect to the real and imaginary axis, such that its positive part, ${\rm Re\,} \lambda > 0$, is the graph $\{x(y) + {\rm i} y| y \in (- \gamma/2,\gamma/2)\}$ of a smooth function $x(y)$. At $h = h_c$ this graph develops a cusp which signals the transition to the fully polarized massive regime.
  • Figure 2: We deform the original integration contour, which is a straight line from $-Q_F$ to $Q_F$ to the sketched contour and move the left and the right parts to minus and plus infinity.

Theorems & Definitions (13)

  • Theorem \oldthetheorem
  • Remark
  • Theorem \oldthetheorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 3 more