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 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.