Complete Boundary Phase Diagram of the Spin-$\frac{1}{2}$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime

TL;DR

This paper provides an exact phase diagram for the spin-1/2 XXZ chain with boundary fields in the gapped regime, revealing boundary phase transitions and the structure of the Hilbert space.

Contribution

It offers the complete boundary phase diagram of the XXZ chain with boundary fields using Bethe ansatz, identifying boundary phase transitions and the structure of boundary-bound states.

Findings

Identification of multiple boundary phases depending on boundary fields.

Discovery of two types of boundary phase transitions.

Analysis of boundary-bound states and Hilbert space structure.

Abstract

We consider the spin $\frac{1}{2}$ XXZ chain with diagonal boundary fields and solve it exactly using Bethe ansatz in the gapped anti-ferromagnetic regime and obtain the complete phase boundary diagram. Depending on the values of the boundary fields, the system exhibits several phases which can be categorized based on the ground state exhibited by the system and also based on the number of bound states localized at the boundaries. We show that the Hilbert space is comprised of a certain number of towers whose number depends on the number of boundary bound states exhibited by the system. The system undergoes boundary phase transitions when boundary fields are varied across certain critical values. There exist two types of phase transitions. In the first type the ground state of the system undergoes a change. In the second type, named the `Eigenstate phase transition', the number of towers of the Hilbert space changes, which is again associated with the change in the number of boundary bound states exhibited by the system. We use the DMRG and exact diagonalization techniques to probe the signature of the Eigenstate phase transition and the ground state phase transition by analyzing the spin profiles in each eigenstate.