Entanglement entropy in quantum spin chains with broken parity number symmetry

TL;DR

This paper investigates the entanglement entropy in quantum spin chains with broken parity symmetry, using an enlarged quadratic Hamiltonian approach to analyze eigenstates, correlation functions, and entanglement properties.

Contribution

It introduces a method to handle boundary magnetic fields by enlarging the Hamiltonian, revealing universal entanglement features of eigenstates that break parity symmetry.

Findings

Eigenstates are degenerate due to zero modes.

Entanglement structure is universal across Hamiltonians.

Broken parity symmetry affects reduced density matrices.

Abstract

Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in-depth the properties of the eigenstates of the enlarged Hamiltonian. In particular, we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.