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arXiv:cond-mat/0106062 · Journal of the Physical Society of Japan

Emptiness Formation Probability for the One-Dimensional IsotropicXYModel

Masahiro Shiroishi, Minoru Takahashi, Yoshihiro Nishiyama

Published Dec 15, 2001• 79 citations
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Abstract

We study a correlation function for the one-dimensional isotropic ${XY}$ model (${XX0}$ model), which is called the Emptiness Formation Probability (EFP). It is the probability of the formation of a ferromagnetic string in the anti-ferromagnetic ground state. Using the expression of the EFP as a Toeplitz determinant, we discuss its asymptotic behaviors. We also compare the analytical results with numerical calculations as the density-matrix renormalization group and the quantum Monte-Carlo method.

DOI
10.1143/jpsj.70.3535

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We study a correlation function for the one-dimensional isotropic XY{XY}XY model (XX0{XX0}XX0 model), which is called the Emptiness Formation Probability (EFP). It is the probability of the formation of a ferromagnetic string in the anti-ferromagnetic ground state. Using the expression of the EFP as a Toeplitz determinant, we discuss its asymptotic behaviors. We also compare the analytical results with numerical calculations as the density-matrix renormalization group and the quantum Monte-Carlo method.
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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.
We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of a formation of a ferromagnetic string P(n) in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for n≤6 for the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbrary n.
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 spin-1/2 Heisenberg XXX antiferromagnet for which the spectrum of the Hamiltonian was found by Bethe in 1931. We study the probability of the formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P ( n ). This is the most fundamental correlation function. We prove that, for short strings, it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P (5) for the first time. We have also calculated P ( n ) numerically by the density matrix renormalization group. The results agree quite well with the analytical results. Furthermore, we study the asymptotic behaviour of P ( n ) at finite temperature by quantum Monte Carlo simulation. This also agrees with our previous analytical results.