Entanglement Spectrum for the XY Model in One Dimension

TL;DR

This paper derives the exact entanglement spectrum of a large block in the ground state of the 1D XY spin chain, revealing a geometric sequence of eigenvalues and their degeneracies, depending on magnetic field and anisotropy.

Contribution

It provides an exact analytical derivation of the entanglement spectrum for the XY model, including eigenvalues and degeneracies, using modular functions and Renyi entropy.

Findings

Eigenvalues form an exact geometric sequence.

Degeneracy of eigenvalues increases sub-exponentially.

Spectrum depends on magnetic field and anisotropy.

Abstract

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues \lambda_n form an exact geometric sequence. For example, for strong magnetic field \lambda_n = C \exp{(-\pi \tau_0 n)}, here \tau_0>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n \sim \exp{(\pi \sqrt{n/3})}. For weak magnetic field expressions are similar.