Entanglement spectrum of the Heisenberg XXZ chain near the ferromagnetic point

TL;DR

This paper investigates how the entanglement spectrum of the XXZ spin chain evolves near the ferromagnetic point, revealing divergent levels, boundary condition effects, and combinatorial structures in the spectrum.

Contribution

It provides a detailed analysis of the entanglement spectrum near the ferromagnetic point, including conjectured combinatorial formulas and boundary condition dependencies, using exact diagonalization and Bethe ansatz.

Findings

Most ES levels diverge as   -1^+

ES depends on boundary conditions away from   -1

Conjectured combinatorial formula for ES multiplicities

Abstract

We study the entanglement spectrum (ES) of a finite XXZ spin 1/2 chain in the limit \Delta -> -1^+ for both open and periodic boundary conditions. At \Delta=-1 (ferromagnetic point) the model is equivalent to the Heisenberg ferromagnet and its degenerate ground state manifold is the SU(2) multiplet with maximal total spin. Any state in this so-called "symmetric sector" is an equal weight superposition of all possible spin configurations. In the gapless phase at \Delta>-1 this property is progressively lost as one moves away from the \Delta=-1 point. We investigate how the ES obtained from the states in this manifold reflects this change, using exact diagonalization and Bethe ansatz calculations. We find that in the limit \Delta ->-1^+ most of the ES levels show divergent behavior. Moreover, while at \Delta=-1 the ES contains no information about the boundaries, for \Delta>-1 it depends dramatically on the choice of boundary conditions. For both open and periodic boundary conditions the ES exhibits an elegant multiplicity structure for which we conjecture a combinatorial formula. We also study the entanglement eigenfunctions, i.e. the eigenfunctions of the reduced density matrix. We find that the eigenfunctions corresponding to the non diverging levels mimic the behavior of the state wavefunction, whereas the others show intriguing polynomial structures. Finally we analyze the distribution of the ES levels as the system is detuned away from \Delta=-1.