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arXiv:1909.01270 · Journal of Statistical Mechanics Theory and Experiment

Emptiness formation probability and Painlevé V equation in the XY spin chain

Filiberto Ares, Jacopo Viti

Published Jan 1, 2020• 11 citations
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Abstract

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 .

DOI
10.1088/1742-5468/ab5d0b

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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 .
We study an asymptotic behaviour 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 behaviours for n → ∞ 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 behaviour 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 behaviour. We study this crossover using the bosonization approach.
We revisit the study of the emptiness formation probability, the probability of forming a sequence of ` 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 `−1 log ` correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging. 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. ar
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.
We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization, and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a by-product, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both the σ^{z} and σ^{x} basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a different conjecture.
We propose a new correlator in one-dimensional quantum spin chains, the s -emptiness formation probability ( s -EFP). This is a generalization of the emptiness formation probability (EFP), which is the probability that the first n spins of the chain are all aligned downwards. In the s -EFP we let the spins in question be separated by s sites. The usual EFP corresponds to the special case when s = 1. Taking s > 1 allows us to quantify non-local correlations. We express the s -EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.
We study the Emptiness Formation Probability (EFP) for the spin 1/2 XXZ spin chain. EFP P (n) detects a formation of ferromagnetic string of the length n in the ground state. It is expected that EFP decays in a Gaussian way for large strings P (n) n . Here, we propose the explicit expressions for the rate of Gaussian decay C as well as for the exponent . In order to con rm the validity of our formulae, we employed an ab initio simulation technique of the density-matrix renormalization group to simulate XXZ spin chain of sucient length. Furthermore, we performed Monte-Carlo integration of the Jimbo-Miwa multiple integral for P (n). Those numerical results for P (n) support our formulae fairly de nitely.