Emptiness formation probability for the anisotropic XY spin chain in a magnetic field

Abstract

We study an asymptotic behavior of the probability of formation of a ferromagnetic string (referred to as EFP) of length n in a ground state of the one-dimensional anisotropic XY model in a transversal magnetic field as n → ∞. We find that it is exponential everywhere in the phase diagram of the XY model except at the critical lines where the spectrum is gapless. One of those lines corresponds to the isotropic XY model where EFP decays in a Gaussian way, as was shown in Ref. [1]. The other lines are at the critical value of the magnetic field. There, we show that EFP is still exponential but acquires a non-trivial power-law prefactor with a universal exponent.

An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

“…and F [τ ] are defined as in (A.6) and τ ± come from the decomposition 11) so that τ + (τ − ) is analytic and non-zero inside (outside) the unit circle on which τ is defined. They also satisfy the boundary conditions τ + (0) = τ − (∞) = 1.…”

“…More complicated correlators like the Emptiness Formation Probability [10,11,12] and the Von Neumann [13,14] and Renyi [15] entanglement entropies were calculated more recently, as well as several out-of-equilibrium properties [16]. Virtually all static correlation functions of the model can be expressed as determinants of matrices with a special structure, known as Toeplitz matrices [17].…”

Boundary emptiness formation probabilities in the six-vertex model at $\mathbf{\Delta }=-\frac{\mathbf{1}}{\mathbf{2}}$

“…We define a new matrix M (3) , wherein each column is a linear combination of two columns of M (2) : The determinant is given by . . .…”

The emptiness formation probability correlation in homogeneous and dimerized XX chains

“…The XX model discussed above corresponds to the line γ = 0, where the ground-state EFP is asymptotically Gaussian [6]. Away from that line, however, the EFP is asymptotically exponential [7,15,16]. For nonzero temperature the EFP is asymptotically exponential for arbitrary γ and h [16].…”

“…In the anisotropic model, in contrast, the shape of the single-particle spectrum depends on both h and γ, the two energy bands contain different numbers of states, the lower quasiparticle band is always filled, and the upper band is always empty, see [17] for more details about the ground states of the two models. Thus, it is of interest to study the EFP correlation for the dimerized model and to compare the results to those obtained [7,15,16] for the anisotropic model. Our results show that the EFP of the dimerized system is asymptotically Gaussian at T = 0 and exponential at T > 0.…”