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arXiv:1010.5645 · Journal of Physics A Mathematical and Theoretical

Non-equilibrium 1D many-body problems and asymptotic properties of Toeplitz determinants

Dmitri B. Gutman, Yuval Gefen, A. D. Mirlin

Published Mar 30, 2011• 45 citations
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Abstract

Non-equilibrium bosonization technique facilitates the solution of a number of important many-body problems out of equilibrium, including the Fermiedge singularity, the tunneling spectroscopy and full counting statistics of interacting fermions forming a Luttinger liquid.We generalize the method to non-equilibrium hard-core bosons (Tonks-Girardeau gas) and establish interrelations between all these problems.The results can be expressed in terms of Fredholm determinants of the Toeplitz type.We analyze the long time asymptotics of such determinants, using Szegő and Fisher-Hartwig theorems.Our analysis yields dephasing rates as well as power-law scaling behavior, with exponents depending not only on the interaction strength but also on the non-equilibrium state of the system.

DOI
10.1088/1751-8113/44/16/165003

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Non-equilibrium bosonization technique facilitates the solution of a number of important many-body problems out of equilibrium, including the Fermiedge singularity, the tunneling spectroscopy and full counting statistics of interacting fermions forming a Luttinger liquid.We generalize the method to non-equilibrium hard-core bosons (Tonks-Girardeau gas) and establish interrelations between all these problems.The results can be expressed in terms of Fredholm determinants of the Toeplitz type.We analyze the long time asymptotics of such determinants, using Szegő and Fisher-Hartwig theorems.Our analysis yields dephasing rates as well as power-law scaling behavior, with exponents depending not only on the interaction strength but also on the non-equilibrium state of the system.
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.
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions and for the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interacting quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electron Green's functions, which can be measured via tunneling spectroscopy technique and carry the information about energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinants of single-particle ``counting'' operators. The corresponding time-dependent scattering phase is found to be intrinsically related to ``fractionalization'' of electron-hole excitations in the tunneling process and at boundaries with leads. Results are generalized to the case of spinful particles as well to Green's functions at different spatial points (relevant to the problem of dephasing in Luttinger liquid interferometers). For double-step distributions, the dephasing rates are oscillatory functions of the interaction strength.
We develop a bosonization technique for one-dimensional fermions out of equilibrium. The approach is used to study a quantum wire attached to two electrodes with arbitrary energy distributions. The non-equilibrium electron Green function is expressed in terms of functional determinants of a single-particle"counting" operator with a time-dependent scattering phase. The result reveals an intrinsic relation of dephasing and energy redistribution in Luttinger-liquids to "fractionalization" of electron-hole excitations in the tunneling process and at boundaries with leads.
We study the relaxation dynamics of a nonequilibrium Tomonaga-Luttinger model after a sudden interaction switch-on (``quench''), focusing on a double-step initial momentum distribution function. In the framework of the nonequilibrium bosonization, the results are obtained in terms of singular Fredholm determinants that are evaluated numerically and whose asymptotics are found analytically. While the quasiparticle weights decay exponentially with time after the quench, this is not a relaxation into a thermal state, in view of the integrability of the model. The steady-state distribution emerging at infinite times retains two edges which support Luttinger-liquid-like power-law singularities smeared by dephasing. The obtained critical exponents and the dephasing length are found to depend on the initial nonequilibrium state.
We revisit the problem of full counting statistics of particles on a segment of a one-dimensional gas of free fermions. Using a combination of analytical and numerical methods, we study the crossover between the counting of discrete particles and of the continuous particle density as a function of smoothing in the counting procedure. In the discrete-particle limit, the result is given by the Fisher-Hartwig expansion for Toeplitz determinants, while in the continuous limit we recover the bosonization results. This example of full counting statistics with smoothing is also related to orthogonality catastrophe, Fermi-edge singularity and non-equilibrium bosonization.