Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain

TL;DR

This paper analyzes the asymptotic decay of the Emptiness Formation Probability in the XY spin chain, revealing exponential decay in general and power-law or Gaussian decay at critical points, using Toeplitz determinant theory and bosonization.

Contribution

It provides the first comprehensive asymptotic analysis of EFP in the XY model across the phase diagram, including critical behavior and crossover phenomena.

Findings

Exponential decay of EFP in non-critical regions.

Power-law prefactor with universal exponent at critical lines.

Gaussian decay at the isotropic critical line.

Abstract

We study an asymptotic behavior of a special correlator known as the Emptiness Formation Probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length $n$ in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY Spin Chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviors for $n \to \infty$ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behavior holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behavior. We study this crossover using the bosonization approach.