On the exact solvability of the anisotropic central spin model: An operator approach

Abstract

Using an operator approach based on a commutator scheme that has been previously applied to Richardson's reduced BCS model and the inhomogeneous Dicke model, we obtain general exact solvability requirements for an anisotropic central spin model with $XXZ$-type hyperfine coupling between the central spin and the spin bath, without any prior knowledge of integrability of the model. We outline the basic steps of the usage of the operator approach, and pedagogically summarize them into two \emph{Lemmas} and two \emph{Constraints}. Through a step-by-step construction of the eigen-problem, we show that the condition $g_j^{\prime 2}-g_j^2=c$ naturally arises for the model to be exactly solvable, where $c$ is a constant independent of the bath-spin index $j$, and $\{g_j\}$ and $\{g_j^{\prime }\}$ are the longitudinal and transverse hyperfine interactions, respectively. The obtained conditions and the resulting Bethe ansatz equations are consistent with that in previous literature.

5 pages, 0 figure, to appear in Physica A: Statistical Mechanics and its Applications

References in corpus (5)

• Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics
• Exact dynamics in the inhomogeneous central-spin model
• Integrability-based analysis of the hyperfine-interaction -nduced decoherence in quantum dots
• An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models
• Different types of integrability and their relation to decoherence in central spin models

Cited by in corpus (4)

• Exact quantum dynamics of XXZ central spin problems
• Separable and entangled states in the high-spin XX central spin model
• Exact dynamics of the homogeneous two-qubit $XXZ$ central spin model with the spin bath prepared in superpositions of symmetric Dicke states
• Relaxation of antiferromagnetic order and growth of Rényi entropy in a generalized Heisenberg star