1112.2530 (Dmitri A. Ivanov et al.)

Counting free fermions on a line: a Fisher-Hartwig asymptotic expansion
for the Toeplitz determinant in the double-scaling limit    [PDF]

Dmitri A. Ivanov, Alexander G. Abanov, Vadim V. Cheianov

We derive an asymptotic expansion for a Fredholm determinant arising in the problem of counting one-dimensional free fermions on a line segment at zero temperature. This asymptotic expansion was conjectured previously from numerical evidence. It is explicitly periodic in the "counting parameter" and describes the nonanaltic dependence of the asymptotic behavior of the determinant on this parameter. The derived expansion is an extension of the result in the theory of Toeplitz determinants known as the generalized Fisher-Hartwig conjecture. We present two ways to derive our result: the matrix Riemann-Hilbert problem and the Painleve V equation. We prove that the expansion coefficients are polynomials in the counting parameter, provide an algorithm for their calculation order by order and list explicitly first several coefficients.

View original: http://arxiv.org/abs/1112.2530