Relaxation of antiferromagnetic order and growth of R\'enyi entropy in a generalized Heisenberg star

TL;DR

This paper investigates how a central spin influences magnetic order and entanglement in a generalized Heisenberg star, revealing that weak coupling can relax antiferromagnetic order in gapless phases and that entanglement growth signals quantum criticality.

Contribution

It provides an exact analysis of the long-time dynamics of a generalized Heisenberg star, highlighting the control of magnetic order and entanglement through system-bath interactions in strongly correlated spins.

Findings

Weak coupling causes near-complete relaxation of antiferromagnetic order in gapless phase.

Steady staggered magnetization increases with anisotropy in gapped phase.

Renyi entropy exhibits minima at the XXZ bath's critical point.

Abstract

Interacting central spin systems, in which a central spin is coupled to a strongly correlated spin bath with intrabath interaction, consist of an important class of spin systems beyond the usual Gaudin magnet. These systems are relevant to several realistic setups and serve as an interesting platform to study interaction controlled decoherence and frustration induced instability of magnetic order. Using an equations-of-motion method based on analytical representations of spin-operator matrix elements in the XX chain, we obtain exact long-time dynamics of a generalized Heisenberg star consisting of a spin-$S$ central spin and an inhomogeneously coupled XXZ chain of $N\leq 16$ bath spins. In contrast to previous studies where the central spin dynamics is mainly concerned, we focus on the influence of the central spin on the dynamics of magnetic orders within the spin bath. By preparing the XXZ bath in a N\'eel state, we find that in the gapless phase of the bath even weak system-bath coupling could lead to nearly perfect relaxation of the antiferromagnetic order. In the gapped phase, the staggered magnetization decays rapidly and approaches a steady value that increases with increasing anisotropy parameter. These findings suggest the possibility of controlling internal dynamics of strongly correlated many-spin systems by certain coupled auxiliary systems of even few degrees of freedom. We also study the dynamics of the R\'enyi entanglement entropy of the central spin when the bath is prepared in the ground state. Both the overall profile and initial growth rate of the R\'enyi entropy are found to exhibit minima at the critical point of the XXZ bath.