Exact solutions of few-magnon problems in the spin-$S$ periodic XXZ chain

TL;DR

This paper develops an exact solution method for few-magnon problems in finite-size spin-$S$ XXZ chains, revealing bound states, zero-energy states, and dynamics relevant to quantum simulations.

Contribution

The authors introduce a novel formalism converting few-magnon problems into effective single-particle problems, enabling exact solutions and analysis of bound states and dynamics.

Findings

Identification of multimagnon bound states as edge states

Derivation of conditions for zero-energy states without anisotropy

Calculation of dynamic structure factor and real-time magnon dynamics

Abstract

We solve few-magnon problems for a finite-size spin-$S$ periodic Heisenberg XXZ chain with single-ion anisotropy through constructing sets of exact Bloch states achieving block diagonalization of the system. Concretely, the two-magnon (three-magnon) problem is converted to a single-particle one on a one-dimensional (two-dimensional) effective lattice whose size depends linearly (quadratically) on the total number of sites. For parameters lying within certain ranges, various types of multimagnon bound states are manifested and shown to correspond to edge states on the effective lattices. In the absence of the single-ion anisotropy, we reveal the condition under which exact zero-energy states emerge. As applications of the formalism, we calculate the transverse dynamic structure factor for a higher-spin chain near saturation magnetization and find signatures of the multimagnon bound states. We also calculate the real-time three-magnon dynamics from certain localized states, which are relevant to cold-atom quantum simulations, by simulating single-particle quantum walks on the effective lattices. This provides a physically transparent interpretation of the observed dynamics in terms of propagation of bound state excitations. Our method can be directly applied to more general spin or itinerant particle systems possessing translational symmetry.