Tensor-Network Analysis of Root Patterns in the XXX Model with Open Boundaries

TL;DR

This paper introduces a tensor-network method to analyze root patterns in the Bethe ansatz equations of the XXX model with open boundaries, revealing complex structured root distributions even without U(1) symmetry.

Contribution

The authors develop a tensor-network approach to determine Bethe root patterns in non-symmetric quantum spin chains, providing explicit structures for large system sizes.

Findings

Bethe and zero roots form structured patterns including strings and lines.

Four types of Bethe roots identified: regular, line, arc, and paired-line.

Root structures are more complex than previously understood, with boundary effects influencing configurations.

Abstract

The string hypothesis for Bethe roots represents a cornerstone in the study of quantum integrable systems, providing access to physical quantities such as the ground-state energy and the finite-temperature free energy. While the $t-W$ scheme and the inhomogeneous $T-Q$ relation have enabled significant methodological advances for systems with broken $U(1)$ symmetry, the underlying physics induced by symmetry breaking remains largely unexplored, due to the previously unknown distributions of the transfer-matrix roots. In this paper, we propose a new approach to determining the patterns of zero roots and Bethe roots for the $\Lambda-\theta$ and inhomogeneous Bethe ansatz equations using tensor-network algorithms. As an explicit example, we consider the isotropic Heisenberg spin chain with non-diagonal boundary conditions. The exact structures of both zero roots and Bethe roots are obtained in the ground state for large system sizes, up to ($N\simeq 60$ and $100$). We find that even in the absence of $U(1)$ symmetry, the Bethe and zero roots still exhibit a highly structured pattern. The zero roots organize into bulk strings, boundary strings, and additional roots, forming two dominant lines with boundary-string attachments. Correspondingly, the Bethe roots can be classified into four distinct types: regular roots, line roots, arc roots, and paired-line roots. These structures are associated with a real-axis line, a vertical line, characteristic arcs in the complex plane, and boundary-induced conjugate pairs. Comparative analysis reveals that the $t-W$ scheme generates significantly simpler root topologies than those obtained via off-diagonal Bethe Ansatz.