Thermodynamic limit of the XXZ central spin model with an arbitrary central magnetic field

Abstract

The $U(1)$ symmetry of XXZ central spin model with an arbitrary central magnetic field $\vec{B}$ is broken, since its total spin along $z$-direction is not conserved. We obtain the exact solutions of the system by using the Off-diagonal Bethe Ansatz method. The thermodynamic limit is investigated based on the solutions. We find that the contribution of the inhomogeneous term in the associated $T-Q$ relation to the ground state energy satisfies the $N^{-1}$ scaling law, where $N$ is the total number of spins. This result made it possible to investigate the properties of the system in the thermodynamic limit. By assuming the structural form of the Bethe roots in the thermodynamic limit, we obtain the contribution of the direction of $\vec{B}$ to ground state energy. It is shown that the contribution of the direction of the central magnetic field is a finite value in the thermodynamic limit. This is the phenomenon caused by the $U(1)$ symmetry breaking of the system.

Supersymmetry and integrability for a class of XY central spin models

“…While this approach has proved very successful for understanding specific physical aspects around thermalisation, it is not general. Not only is the approach not applicable to the class of central spin models [2,8,14,17,18,32,44,45,[48][49][50]52] cited in the discussions above, it also does not accommodate the recognised class of integrable BCS models, e.g. [4,10,11,13,29,37,46], and other closely related systems [22,33].…”

“…As mentioned, much of the theoretical interest in central spin models stems from the existence of exact solutions, see e.g. [2,14,15,17,18,32,48,49], and there are ongoing efforts to extend the body of known results. In recent times it was shown that a central spin-1/2 particle interacting with arbitrary bath spins is integrable for XX interactions [44].…”