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Exact solution of Riemann–Hilbert problem for a correlation function of the XY spin chain

Yasuhiro Fujii

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann–Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian. In this letter we study the Riemann–Hilbert problem associated with a correlation function of the XY spin chain. The Hamiltonian of the model is given by

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Exact Solution of Riemann–Hilbert Problem for a Correlation Function of the XY Spin Chain

Yasuhiro Fujii

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian.

arXiv:cond-mat/0009458•Journal of the Physical Society of Japan•Oct 15, 2000•1 citation

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A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann–Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian. In this letter we study the Riemann–Hilbert problem associated with a correlation function of the XY spin chain. The Hamiltonian of the model is given by
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 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 analyze the transverse dynamical two-point correlation function of the XX chain by means of a thermal form factor series. The series is rewritten in terms of the resolvent and the Fredholm determinant of an integrable integral operator. This connects it with a matrix Riemann-Hilbert problem. We express the correlation function in terms of the solution of the matrix Riemann-Hilbert problem. The matrix Riemann-Hilbert problem is then solved asymptotically in the high-temperature limit. This allows us to obtain the leading high-temperature contribution to the two-point correlation function at any fixed space-time separation.
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.