Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants
arXiv:1108.1355 · doi:10.1088/1751-8113/44/48/485001
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.
17 pages, 2 figures, Latex, iopart.cls
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