Tensor-network analysis of root patterns in the XXX model with open boundaries

Abstract

The string hypothesis of Bethe roots is a cornerstone in the thermodynamic analysis of quantum integrable systems, since it connects root configurations with physical quantities such as the ground-state energy, surface energy and excitation spectra. For integrable models with U(1) U ( 1 ) symmetry, this connection is well established. When the U(1) U ( 1 ) symmetry is broken by generic non-diagonal boundary fields, however, the off-diagonal Bethe Ansatz leads to an inhomogeneous T\text{--}Q T – Q relation whose Bethe roots have highly nontrivial distributions. This raises two fundamental questions: whether the zero roots and the ODBA Bethe roots still possess regular and classifiable structures in the large-size limit, and whether such structures can be used to extract physical quantities. In this work, we address these two questions for the isotropic Heisenberg spin chain with non-diagonal open boundaries. By combining tensor-network algorithms with Bethe-Ansatz techniques, we determine the zero-root and Bethe-root configurations associated with the \Lambda\text{--}\theta Λ – θ relation and the inhomogeneous Bethe Ansatz equations for large system sizes, up to N\simeq 60 N ≃ 60 and 100 100 . We find that, despite the absence of U(1) U ( 1 ) symmetry, the roots exhibit well-organized patterns. The zero roots form bulk strings, boundary strings, additional roots, and a helical half-spinon, while the ODBA Bethe roots split into four geometric classes: regular roots, line roots, arc roots and paired-line roots. We further show that these root structures are not merely numerical patterns, but contain direct thermodynamic information. In particular, the geometric structure of the ODBA Bethe roots allows us to derive analytic expressions for the surface energy and elementary excitation energies. This demonstrates that the full ODBA Bethe-root configuration can serve as a thermodynamically meaningful object in a U(1) U ( 1 ) -symmetry-broken open spin chain. A comparison with the t\text{--}W t – W scheme shows that the zero-root description has a simpler topology, while the ODBA Bethe roots provide a direct route to physical quantities. Our results therefore provide a framework for studying root structures and thermodynamic properties in quantum integrable models without U(1) U ( 1 ) symmetry.