Entanglement in fermionic chains with finite range coupling and broken symmetries

TL;DR

This paper derives a formula for entanglement entropy in fermionic chains with complex, symmetry-breaking couplings, revealing a logarithmic scaling behavior linked to discontinuities in the system's mathematical representation.

Contribution

It generalizes existing formulas for entanglement entropy to include complex, symmetry-breaking couplings and introduces a new logarithmic term in the entropy scaling.

Findings

Derived a determinant formula for block Toeplitz matrices with complex couplings.

Discovered a logarithmic scaling term in entanglement entropy due to symbol discontinuities.

Applied the formula to analyze entanglement in Dzyaloshinski-Moriya and Kitaev chains.

Abstract

We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the Renyi entropy of a partial observation to a subsystem consisting of $X$ contiguous sites in the limit of large $X$. The present work generalizes similar results due to Its, Jin, Korepin and Its, Mezzadri, Mo. A striking new feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size of $X$. This logarithmic behaviour originates from certain discontinuities in the symbol of the block Toeplitz matrix. Equipped with this formula we analyse the entanglement entropy of a Dzyaloshinski-Moriya spin chain and a Kitaev fermionic chain with long range pairing.