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

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.