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The Roaming Bethe Roots: An Effective Bethe Ansatz Beyond Integrability

Wenlong Zhao, Yunfeng Jiang, Rui-Dong Zhu

Abstract

We propose an effective Bethe ansatz for solving quantum many-body systems near an integrable point. Our approach retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions. These effective roots are determined by minimizing physically motivated cost functions. The resulting off-shell Bethe states serve as approximate eigenstates of the non-integrable models. We assess the quality of the approximation using various physical observables, including the energy eigenvalue, state fidelity, and bipartite entanglement entropy. Our tests show that for models with weak integrability-breaking, the effective Bethe ansatz provides a high-quality approximation to the exact eigenstates over a wide range of deformation parameters. In contrast, for models with strong integrability-breaking interactions, the efficacy of the effective Bethe ansatz degrades relatively quickly as the deformation parameter increases. The efficacy of the method thus offers a useful probe for characterizing the strength of integrability breaking. Within its regime of accuracy, it also provides a new representation of the eigenstates of nearly integrable models, enabling one to exploit the algebraic structure inherited from integrability.

The Roaming Bethe Roots: An Effective Bethe Ansatz Beyond Integrability

Abstract

We propose an effective Bethe ansatz for solving quantum many-body systems near an integrable point. Our approach retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions. These effective roots are determined by minimizing physically motivated cost functions. The resulting off-shell Bethe states serve as approximate eigenstates of the non-integrable models. We assess the quality of the approximation using various physical observables, including the energy eigenvalue, state fidelity, and bipartite entanglement entropy. Our tests show that for models with weak integrability-breaking, the effective Bethe ansatz provides a high-quality approximation to the exact eigenstates over a wide range of deformation parameters. In contrast, for models with strong integrability-breaking interactions, the efficacy of the effective Bethe ansatz degrades relatively quickly as the deformation parameter increases. The efficacy of the method thus offers a useful probe for characterizing the strength of integrability breaking. Within its regime of accuracy, it also provides a new representation of the eigenstates of nearly integrable models, enabling one to exploit the algebraic structure inherited from integrability.

Paper Structure

This paper contains 2 sections, 18 equations, 4 figures, 2 tables.

Table of Contents

  1. Algebraic Bethe ansatz
  2. Numerical Optimization of Effective Bethe Roots

Figures (4)

  • Figure 1: Fidelity and Entanglement Entropy computed in two models.
  • Figure 2: Evolution of the effective Bethe roots of the ground state of two models with $L=8$. The black circles mark the positions of the Bethe roots corresponding to the undeformed ground state. The arrows indicate the directions that the corresponding Bethe roots move as the deformation parameter increases. For the weak integrability-breaking model, the effective Bethe roots changed abruptly at the MG point $\lambda=0.5$, indicated by the green dots in the figure.
  • Figure 3: Fidelity and Entanglement Entropy computed in XXZ model.
  • Figure 4: Fidelity computed in two models based on an effective Bethe ansatz generated from the 8-vertex R-matrix.