aqora

Luttinger liquids with multiple Fermi edges: Generalized Fisher-Hartwig conjecture and numerical analysis of Toeplitz determinants

I. V. Protopopov, D. B. Gutman, A. D. Mirlin

It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.

arXiv:1203.6418Lithuanian Journal of PhysicsJan 1, 201211 citations

Affine Quantum Groups and Quantum Spin Chains

Akishi Kato

Mar 20, 19940 citations

Luttinger liquids with multiple Fermi edges: Generalized Fisher-Hartwig conjecture and numerical analysis of Toeplitz determinants

Ivan V. Protopopov, D. B. Gutman, A. D. Mirlin

It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.It has been shown that solutions of a number of many-body problems out of equilibrium can be expressed in terms of Toeplitz determinants with Fisher-Hartwig (FH) singularities. In the present paper, such Toeplitz determinants are studied numerically. Results of our numerical calculations fully agree with the FH conjecture in an extended form that includes a summation over all FH representations (corresponding to different branches of the logarithms). As specific applications, we consider problems of Fermi edge singularity and tunneling spectroscopy of Luttinger liquid with multiple-step energy distribution functions, including the case of population inversion. In the energy representation, a sum over FH branches produces power-law singularities at multiple edges.

Lithuanian Journal of PhysicsJan 1, 201211 citations

The One-Dimensional Quantum Spin Chain

Naoto Nagaosa

Springer eBooksJan 1, 19990 citations

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

Alexander G Abanov, Dmitri A. Ivanov, Yachao Qian

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.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.

arXiv:1108.1355Journal of Physics A Mathematical and TheoreticalNov 8, 201148 citations

Quantum Spin Chains

Jill C. Bonner, Henk W. J. Blöte, Hans Beck +1 more

Springer series in solid-state sciencesJan 1, 198146 citations

Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture

Related papers

Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture

Related papers