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arXiv:0708.0433 · Journal of Statistical Mechanics Theory and Experiment

On the emptiness formation probability of the openXXZspin-1/2 chain

Karol Kajetan Kozlowski

Published Feb 13, 2008• 16 citations
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Abstract

This paper is devoted to the study of the emptiness formation probability τ( m ) of the open XXZ chain. We derive a closed form for τ( m ) at Δ = 1/2 when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitrary boundary field at the free fermion point. Finally, we compute the first term of the asymptotics of ln(τ( m )) for the whole massless regime −1<Δ<1.

DOI
10.1088/1742-5468/2008/02/p02006

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This paper is devoted to the study of the emptiness formation probability τ( m ) of the open XXZ chain. We derive a closed form for τ( m ) at Δ = 1/2 when the boundary field vanishes. Moreover we obtain its leading asymptotics for an arbitrary boundary field at the free fermion point. Finally, we compute the first term of the asymptotics of ln(τ( m )) for the whole massless regime −1<Δ<1.
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We revisit the study of the emptiness formation probability, the probability of forming a sequence of ℓ\ellℓ 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 ℓ−1log⁡ℓ\ell^{-1}\log \ellℓ−1logℓ 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.
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 function of a Painlevé V equation. Following a recent proposal, we also provide a power series expansion for the 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 discuss the free fermion case where the emptiness formation probability is characterized by a Gaussian decay for large L .
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