Emptiness formation probability and Painlev\'e V equation in the XY spin chain

TL;DR

This paper links the Emptiness Formation Probability in the XY spin chain to a Painlevé V equation, providing exact asymptotics and a conformal field theory expansion, validated by numerical tests.

Contribution

It establishes an exact description of the crossover behavior of the Emptiness Formation Probability using Painlevé V and conformal blocks, advancing understanding of spin chain correlations.

Findings

Exact Painlevé V characterization of the crossover behavior.

Power series expansion of the $ au$ function via conformal blocks.

Excellent agreement between analytical results and numerical calculations.

Abstract

We reconsider the problem of finding $L$ consecutive down spins in the ground state of the XY chain, a quantity known as the Emptiness Formation Probability. Motivated by new developments in the asymptotics of Toeplitz determinants, we show how the crossover between the critical and off-critical behaviour of the Emptiness Formation Probability is exactly described by a $\tau$ function of a Painlev\'e V equation. Following a recent proposal, we also provide a power series expansion for the $\tau$ function in terms of irregular conformal blocks of a Conformal Field Theory with central charge $c=1$. Our results are tested against lattice numerical calculations, showing excellent agreement. We finally rediscuss the free fermion case where the Emptiness Formation Probability is characterized by a Gaussian decay for large $L$.