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arXiv:1306.5017 · Journal of Physics A Mathematical and Theoretical

Fisher–Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions

Dmitri A. Ivanov, Alexander G Abanov

Published Sep 2, 2013• 13 citations
arXivHTMLPDF

Abstract

We study the spectrum of the Toeplitz matrix with a sine kernel, which corresponds to the single-particle reduced density matrix for free fermions on the one-dimensional lattice. For the spectral determinant of this matrix, a Fisher--Hartwig expansion in the inverse matrix size has been recently conjectured. This expansion can be verified order by order, away from the line of accumulation of zeros, using the recurrence relation known from the theory of discrete Painleve equations. We perform such a verification to the tenth order and calculate the corresponding coefficients in the Fisher-Hartwig expansion. Under the assumption of the validity of the Fisher-Hartwig expansion in the whole range of the spectral parameter, we further derive expansions for an equation on the eigenvalues of this matrix and for the von Neumann entanglement entropy in the corresponding fermion problem. These analytical results are supported by a numerical example.

DOI
10.1088/1751-8113/46/37/375005

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We revisit the problem of finding the probability distribution of a fermionic number of one-dimensional spinless free fermions on a segment of a given length.The generating function for this probability distribution can be expressed as a determinant of a Toeplitz matrix.We use the recently proven generalized Fisher-Hartwig conjecture on the asymptotic behavior of such determinants to find the generating function for the full counting statistics of fermions on a line segment.Unlike the method of bosonization, the Fisher-Hartwig formula correctly takes into account the discreteness of charge.Furthermore, we numerically check the precision of the generalized Fisher-Hartwig formula, find that it has a higher precision than rigorously proven so far and conjecture the form of the next-order correction to the existing formula.
We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher–Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painlevé VI differential equation that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k −4 , that is proved to be true for all couplings in the Lieb–Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.
Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
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