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arXiv:cond-mat/0010492

Comment on "Analytical Structure of One Dimensional Localization Theory: Re-Examining Mott's Law"

F. T. Evers, B. M. Gammel

Published Oct 31, 2000• 0 citations
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Abstract

In Phys. Rev. Lett. 84, 1760 (2000) A.O.Gogolin has challenged the established point of view that Mott's law for the dynamical conductivity of a 1 dimensional insulator is correct. We present the result of a numerical solution of Berezinskii's recursion relations that demonstrate unambigously that the earlier solutions are indeed valid in contrast to Gogolin's claim.

DOI
10.48550/arxiv.cond-mat/0010492

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In Phys. Rev. Lett. 84, 1760 (2000) A.O.Gogolin has challenged the established point of view that Mott's law for the dynamical conductivity of a 1 dimensional insulator is correct. We present the result of a numerical solution of Berezinskii's recursion relations that demonstrate unambigously that the earlier solutions are indeed valid in contrast to Gogolin's claim.
The low-frequency conductivity of a disordered Fermi gas in one spatial dimension is governed by the Mott-Berezinskii law σ(ω)∝ω2ln⁡ω2σ(ω) \propto ω^2 \ln ω^2σ(ω)∝ω2lnω2. In a recent Letter [Phys. Rev. Lett. 84, 1760 (2000)] A. O. Gogolin claimed that this law is invalid, challenging our basic understanding of disordered systems and a massive amount of previous theoretical work. We point out two calculational errors in Gogolin's paper. Once we correct them, the Mott-Berezinskii formula is fully recovered. We also present numerical results supporting the Mott-Berezinskii formula but ruling out that of Gogolin.
In their comment Fabrizio and Gogolin dispute our result of the enhancement of the tunneling density of states in a Tomonaga-Luttinger liquid at the location of a backward scattering defect [Phys. Rev. Lett. 76, 4230(1996); cond-mat/9601020]. They state that the anticommutativity of the fermion operators of the left and right moving electrons was not considered properly in the Letter. We show in the Reply that the result of the Letter can be reproduced following the Comment when its calculations are performed correctly. This clearly indicates that the question about the anticommutation relations was raised by Fabrizio and Gogolin without serious grounds.
A reexamination of the results of my earlier paper [Phys. Rev. E 50, 3116 (1994)] is supplied using entirely straightforward methods, i.e., without appealing to operator formalism. This reexamination confirms the consistency of the work in that paper by extracting a term proportional to time that Dembrinski et al. [Phys. Rev. E 53, 4243 (1996)] claim is not present. The arguments of Dembinski et al. are not refuted, but it is pointed out that they are based upon methods within which one readily encounters paradoxes such [p${\mathrm{^}}^{2}$,p^]\ensuremath{\ne}0 and [x^,p^]\ensuremath{\ne}i^. The present exposition avoids such methods. Unfortunately I am not prepared to address their numerical results. This will have to await further investigation. However, an example of numerical results of my own is presented in order to illustrate the richness of the dynamics of the model. \textcopyright{} 1996 The American Physical Society.
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
This is a comment on the work of Kolomeisky et al., Phys. Rev. Lett. 85, 1146 (2000). We point out that they are using the wrong form of the energy functional for one-dimensional fermions. We point out two possible forms of the energy functional, both of which can be derived from first principles but using different methods. One is obtained from the collective field theory method, while the other is derived from the extended Thomas-Fermi method. These two forms of the energy functional do not support the soliton solutions which are obtained by Kolomeisky et al.