Quantum fluctuations of one-dimensional free fermions and Fisher–Hartwig formula for Toeplitz determinants

Abstract

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 check numerically 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. PACS numbers: 05.30.Fk, 05.40.-a, 02.50.-r arXiv:1108.1355v1 [cond-mat.str-el]

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