Applications and generalizations of Fisher–Hartwig asymptotics

Abstract

Fisher-Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above T c , while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the FisherHartwig asymptotic form known for Toeplitz determinants.

Random Matrix Theory and Entanglement in Quantum Spin Chains

“…After all, two-dimensional classical spin chains are mathematically equivalent to one-dimensional quantum spin chains. Furthermore, these type of random matrix averages already appear in the calculation of the ground state density matrices for an impenetrable Bose gas in an interval of finite length [9].…”

“…The averages that occur can be expressed either as Toeplitz determinants, in the case of U(N), or as determinants of specific combinations of Toeplitz and Hankel matrices for the other compact groups. Recently, Basor and Ehrhardt [8] and Forrester and Frankel [9] have computed asymptotic formulae for such determinants, and these allow us to write down the leading-order and next-to-leading-order terms in the asymptotics of the entanglement in the limit as the total number of spins tends to infinity and then as N → ∞. We find that in the proximity of a critical point the entanglement grows logarithmically with N, in agreement with the prediction of Korepin [5] and Calabrese and Cardy [6].…”

Random Matrices with Merging Singularities and the Painlevé V Equation

“…The following asymptotic results as n → ∞ were obtained by Krasovsky [25] (see also [3,16,21] for α j 's integers): 31) where k = 1, 2 and…”

Nonequilibrium dynamics of a noisy quantum Ising chain: Statistics of work and prethermalization after a sudden quench of the transverse field

“…(91); in this case, some additional manipulations on the generating function are required, following Ref. [39], we note that…”

“…The integral involed in the Toeplitz Determinant is defined over a circle of radius 1, encircling the origin, (92), but applying Cauhy's theorem inside the anulus 1/g < |z| < g we can move the integration from the circle of radius 1 to the circle of radius g = 1/λ; this is equivalent to make the substituion z → z/λ in (92), and to keep the integration over the circle |z| = 1, as shown in [39] (for a technical remark on this point see [41]). …”