The fine structure of the entanglement entropy in the classical XY model

TL;DR

This paper compares tensor renormalization group and worm algorithm methods to analyze the particle density and entanglement entropy in the classical XY model, revealing a fine structure in entropy behavior across superfluid phases.

Contribution

It introduces a detailed comparison of TRG and worm algorithms for the XY model and uncovers a fine structure in entanglement entropy related to particle number changes.

Findings

Particle number and winding number increase stepwise with chemical potential.

Entanglement entropy peaks at half-filling, mirroring the thermal entropy behavior.

The entropy patterns suggest an approximate fermionic nature of the system.

Abstract

We compare two calculations of the particle density in the superfluid phase of the classical XY model with a chemical potential $\mu$ in 1+1 dimensions.The first relies on exact blocking formulas from the Tensor Renormalization Group (TRG) formulation of the transfer matrix. The second is a worm algorithm. We show that the particle number distributions obtained with the two methods agree well. We use the TRG method to calculate the thermal entropy and the entanglement entropy. We describe the particle density, the two entropies and the topology of the world lines as we increase $\mu$ to go across the superfluid phase between the first two Mott insulating phases. For a sufficiently large temporal size, this process reveals an interesting fine structure: the average particle number and the winding number of most of the world lines in the Euclidean time direction increase by one unit at a time. At each step, the thermal entropy develops a peak and the entanglement entropy increases until we reach half-filling and then decreases in a way that approximately mirror the ascent. This suggests an approximate fermionic picture.