The Ferromagnetic Heisenberg XXZ chain in a pinning field

TL;DR

This paper studies how a magnetic field at a single lattice site influences the low-energy spectrum of the ferromagnetic Heisenberg XXZ chain, revealing critical field effects and the impact of field orientation on ground states and excitations.

Contribution

It provides a rigorous analysis of the spectral changes induced by localized magnetic fields, including the identification of a critical field strength and the effects of field direction and boundary conditions.

Findings

Existence of a critical magnetic field B_c depending on anisotropy and spin.

Different ground state behaviors for fields aligned with or perpendicular to the z-axis.

Rigorous lower bounds for excitation gaps under non-z-axis magnetic fields.

Abstract

We investigate the effect of a magnetic field supported at a single lattice site on the low-energy spectrum of the ferromagnetic Heisenberg XXZ chain. Such fields, caused by impurities, can modify the low-energy spectrum significantly by pinning certain excitations, such as kink and droplet states. We distinguish between different boundary conditions (or sectors), the direction and also the strength of the magnetic field. E.g., with a magnetic field in the z-direction applied at the origin and ++ boundary conditions, there is a critical field strength B_c (which depends on the anisotropy of the Hamiltonian and the spin value) with the following properties: for B < B_c there is a unique ground state with a gap, at the critical value, B_c, there are infinitely many (droplet) ground states with gapless excitations, and for B>B_c there is again a unique ground state but now belonging to the continuous spectrum. In contrast, any magnetic field with a non-vanishing component in the xy-plane yields a unique ground state, which, depending on the boundary conditions, is either an (anti)kink, or an (anti)droplet state. For such fields, i.e., not aligned with the z-axis, excitations always have a gap and we obtain a rigorous lower bound for that gap.