Instanton approach to the conductivity of a disordered solid

Hayn, R.; John, W.

doi:10.1016/0550-3213(91)90214-i

Cited by 12 publications

(30 citation statements)

References 14 publications

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“…Among our other results, we mention high peaks of some pair correlation functions (see (4.1) and (4.2) below), appearing in a neighborhood of the origin and on the "resonating" distance, determined by the frequency of the external field. Analogous peaks were found before in the one-dimensional case for strong localization [14] as well as in the weak localization regime [11]. However, in these cases, the peaks are of the order ρ 2 (E F ), while in the general d-dimensional case, the peaks are of the order ρ 2 (E F ) (log ν 0 /ν) d−1 , i.e., much bigger in the regime (1.4) (see also [15] for a similar result).…”

Section: Introductionsupporting

confidence: 86%

“…The asymptotic behavior of the low frequency conductivity in the strong localization regime of the one-dimensional Gaussian white noise potential, defined by the relations V (x) = 0, V (x)V (y) = 2Dδ(x − y), (5.1) was studied in [14]. The potential is often used in the theory of one-dimensional disordered systems (see [21] for results and references).…”

Section: Asymptotically Exact One-dimensional Resultsmentioning

confidence: 99%

“…Formula (1.3) was discussed in many works (see e.g. [7,8,14,10,11,21,25,15] and Section 5). However, a consistent "first principle" derivation of the formula is still not available in a general multi-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

Kirsch

Lenoble

Pastur

2003

J. Phys. A: Math. Gen.

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We present a method that allows us to find the asymptotic form of various characteristics of disordered systems in the strong localization regime, i.e., when either the random potential is big or the energy is close to a spectral edge. The method is based on the hypothesis that the relevant realizations of the random potential in the strong localization regime have the form of a collection of deep random wells that are uniformly and chaotically distributed in space with a sufficiently small density. Assuming this and using the density expansion, we show first that the density of wells coincides in leading order with the density of states. Thus the density of states is in fact the small parameter of the theory in the strong localization regime. Then we derive the Mott formula for the low frequency conductivity and the asymptotic formulas for certain two-point correlators when the difference of the respective energies is small.

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Section: Introductionsupporting

confidence: 86%

Section: Asymptotically Exact One-dimensional Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

Kirsch

Lenoble

Pastur

2003

J. Phys. A: Math. Gen.

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“…For example, in the problem of a disordered single-particle system one has to average observables over disorder, and this can be achieved either using the replica trick or supersymmetry. In the last case one introduces a superfield with bosonic and fermionic components, so that integrating out fluctuations in the fermionic and bosonic sectors one obtains a functional determinant in numerator and denominator, respectively [42]. When using replicas, the ratio of determinants appears in the limit of zero replica number [7].…”

Section: Application To Instanton Calculations: Cancellation Of Overlmentioning

confidence: 99%

On functional determinants of matrix differential operators with multiple zero modes

Falco¹,

Fedorenko²,

Gruzberg³

2017

J. Phys. A: Math. Theor.

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We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of r × r matrix second order differential operators O with 0 < n 2r linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r − n or 2r − n (depending on the boundary conditions) solutions of the homogeneous equation Oh = 0, in the spirit of Gel'fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.

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“…The last expression is what is called the Mott formula in ref. [21]. Deviations of the numerically exact Σ(ω) from Σ MB (ω) are demonstrated in fig.…”

mentioning

confidence: 91%

Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law

Falco

Fedorenko

Gruzberg

2017

EPL

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In one-dimensional disordered wires electronic states are localized at any energy. Correlations of the states at close positive energies and the AC conductivity σ(ω) in the limit of small frequency are described by the Mott-Berezinskii theory. We revisit the instanton approach to the statistics of wave functions and AC transport valid in the tails of the spectrum (large negative energies). Applying our recent results on functional determinants, we calculate exactly the integral over gaussian fluctuations around the exact two-instanton saddle point. We derive correlators of wave functions at different energies beyond the leading order in the energy difference. This allows us to calculate corrections to the Mott-Berezinskii law (the leading small frequency asymptotic behavior of σ(ω)) which approximate the exact result in a broad range of ω. We compare our results with the ones obtained for positive energies.

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Instanton approach to the conductivity of a disordered solid

Cited by 12 publications

References 14 publications

On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

On functional determinants of matrix differential operators with multiple zero modes

Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law

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