Exact solution of a non-Hermitian $\mathscr{PT}$-symmetric Heisenberg spin chain

TL;DR

This paper provides an exact solution for a non-Hermitian PT-symmetric Heisenberg spin chain, revealing distinct phases with real and complex energies, and analyzing boundary effects on localized states.

Contribution

It introduces the exact solution of a PT-symmetric Heisenberg chain with boundary fields, identifying phase structures and boundary string states.

Findings

Two phases with PT-symmetry unbroken and broken sectors.

Existence of boundary localized states with complex energies.

Wavefunction broadening with increased boundary parameter imaginary part.

Abstract

We construct the exact solution of a non-Hermitian $\mathscr{PT}$-symmetric isotropic Heisenberg spin chain with integrable boundary fields. We find that the system exhibits two types of phases we refer to as $A$ and $B$ phases. In the $B$ type phase, the $\mathscr{PT}$- symmetry remains unbroken and it consists of eigenstates with only real energies, whereas the $A$ type phase contains a $\mathscr{PT}$-symmetry broken sector comprised of eigenstates with only complex energies and a sector of unbroken $\mathscr{PT}$-symmetry with eigenstates of real energies. The $\mathscr{PT}$-symmetry broken sector consists of pairs of eigenstates whose energies are complex conjugates of each other. The existence of two sectors in the $A$ type phase is associated with the exponentially localized bound states at the edges with complex energies which are described by boundary strings. We find that both $A$ and $B$ type phases can be further divided into sub-phases which exhibit different ground states. We also compute the bound state wavefunction in one magnon sector and find that as the imaginary value of the boundary parameter is increased, the exponentially localized wavefunction broadens thereby protruding more into the bulk, which indicates that exponentially localized bound states may not be stabilized for large imaginary values of the boundary parameter.