Emptiness formation probability, Toeplitz determinants, and conformal field theory

TL;DR

This paper investigates the emptiness formation probability in quantum spin chains, employing conformal field theory and Toeplitz determinants to analyze universal scaling behaviors and corrections, with a focus on XXZ and Ising models.

Contribution

It introduces a CFT-based framework to derive universal terms in the emptiness formation probability and explores the effects of boundary conditions and massiveness in different spin chain models.

Findings

Universal logarithmic corrections in the scaling of emptiness formation probability.

Distinct behaviors between XXZ and Ising chains regarding scale invariance.

Numerical evidence supporting the theoretical predictions of universal terms.

Abstract

We revisit the study of the emptiness formation probability, the probability of forming a sequence of $\ell$ spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in free fermionic systems. These are based on numerical simulations or, when possible, mathematical results on the asymptotic behavior of Toeplitz and Toeplitz+Hankel determinants. A perturbed CFT analysis uncovers an interesting $\ell^{-1}\log \ell$ correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging, as scale invariance is broken. We use a simple qualitative picture in which the ferromagnetic sequence of spins freezes all degrees of freedom inside of a certain "arctic" region, that we determine numerically. We also provide numerical evidence for the existence of universal logarithmic terms, generated by the massless field theory living outside of the arctic region.