Exact statistics of the resistance of disordered one-dimensional conductors beyond the Born approximation: the open core of Abrikosov's static-conductivity paradox

A. A. Abrikosov's 1981 paper 'The paradox with the static conductivity of a one-dimensional metal' derived, for a strictly one-dimensional conductor with Gaussian white-noise disorder in the Born approximation, the exact probability distribution $W(\rho,t)$ of the total (Landauer) resistance $\rho$ at reduced length $t=L/\ell_2$, its moments, and the exact mean $\bar{\rho}=\frac{1}{2}(e^{2t}+1)$, showing that the Landauer-type scale transformation fails at $t\lesssim 1$, that the disorder-averaged conductivity decays only as $t^{-1/2}e^{-t/4}$ (dominated by exponentially rare transparent samples), and that only the log-resistance is self-averaging. The research objective is the extension of this exact statistical theory beyond those limits: stronger and non-Gaussian disorder beyond the Born/random-phase approximation, quasi-one-dimensional multichannel geometry, a systematic bridge between finite-length and asymptotic regimes for general disorder statistics, and the identification of the physically representative average; the frequency-dependent scaling of the infinite-sample conductivity is a known companion result to be reconciled with.

Background

A one-dimensional (1D) metal with randomly positioned impurities is the simplest arena in which disorder and quantum interference compete. For uncorrelated disorder of any strength, every electron eigenstate in 1D is exponentially localized (Anderson localization); exact transfer-matrix treatments (the transfer matrix relates the wave amplitudes at the two ends of a disordered segment, so the whole sample is described by the product of the per-segment matrices), with B. I. Halperin's 1965 exact Green-function solution as an early landmark, make this essentially rigorous, so the dc conductivity of an infinitely long sample must vanish. For a finite sample of length $L$ attached to ideal reservoirs, however, each realization of the disorder has a well-defined resistance, and 'the static conductivity of the sample' becomes a statistical question.

The natural observable is the dimensionless Landauer resistance $\rho$: the total (two-terminal) resistance in units of the single-channel resistance quantum $\pi\hbar/e^{2}$ (Abrikosov works in units $\hbar=1$). For weak disorder it is characterized by the mean free path for backscattering $\ell_2$ and the reduced length $t=L/\ell_2$. In 1970 R. Landauer argued that the resistances of consecutive wire segments add, so the disorder-averaged resistance should grow exponentially with $t$; the scaling theory of localization (Anderson, Thouless, Abrahams and Fisher, 1980) gave a refined 'scale transformation' prediction $\bar{\rho}_L=\frac{1}{2}(e^{2t}-1)$. Abrikosov's 1981 paper addressed the conflict hidden here, which is the paradox of its title: localization forces the typical resistance to grow as $\rho_{typ}=e^{t}$ (equivalently, the typical conductance decays as $e^{-t}$), yet the disorder-averaged conductivity of a finite sample decays only as $t^{-1/2}e^{-t/4}$ - an exponential rate four times smaller than that of the typical sample, and eight times smaller than the $e^{-2t}$ mean-conductance rate corresponding to the scale-transformation prediction $\bar{\rho}_L\sim e^{2t}$ - because the average is dominated by exponentially rare, nearly transparent samples; averaging $\rho$, $\rho^{-1}$, or $\ln\rho$ gives exponentially different 'conductivities', while a naive kinetic (Drude) treatment would suggest a finite, length-independent value.

Abrikosov modeled the impurity potential as Gaussian white noise (a delta-correlated random potential) and treated scattering in the Born approximation (the lowest-order perturbative treatment, in which forward and backscattering enter only through the mean free paths $\ell_1$ and $\ell_2$, to second order in the scattering amplitude). Starting from the per-realization conductivity $\sigma=(e^{2}/\pi)L/\rho$ and the transfer-matrix (S-matrix) evolution along the wire, he derived a Fokker-Planck equation - a diffusion-type evolution equation in $t$ - for the probability distribution $W(\rho,t)$ of the resistance, and solved it exactly. The results include the exact mean $\bar{\rho}=\frac{1}{2}(e^{2t}+1)$ (differing from the scale-transformation value at $t\lesssim 1$); the exact second moment $\overline{\rho^{2}}=\frac{1}{6}e^{6t}+\frac{1}{2}e^{2t}+\frac{1}{3}$; the inverse-moment asymptotics $\overline{\rho^{-n}}\approx\frac{1}{2}\pi^{3/2}\frac{\Gamma^{2}(n-\frac{1}{2})}{\Gamma^{2}(n)}\,t^{-3/2}e^{-t/4}$; and the logarithmic moments $\overline{\ln\rho}=t$, $\operatorname{var}\ln\rho=2t+O(1)$. The physical conclusion: among $\rho$, the conductance $\rho^{-1}$, and $\ln\rho$, only the log-resistance is self-averaging (its relative fluctuations vanish as $(2/t)^{1/2}$ for large $t$), so $\ln\rho$ - equivalently the Lyapunov exponent, the asymptotic exponential growth rate of the transfer matrix with length - is the properly representative quantity, and heuristic scale-transformation predictions are qualitatively correct but quantitatively wrong at finite $t$.

These results are exact but narrow in scope: Born approximation, Gaussian white noise, a single channel, finite samples. The authors' companion review (Abrikosov and Ryzhkin) had developed the same machinery for quasi-1D systems with transverse motion and for the frequency-dependent (ac) conductivity of the infinite localized system. The later framework that grew from this circle of ideas is the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation - the multichannel evolution equation for the transmission eigenvalues (the eigenvalues of the sample's transmission matrix, whose sum gives the conductance) of a quasi-1D wire, whose single-channel reduction reproduces the exact 1D evolution - and with it the modern statistical theory of conductance fluctuations, in which the log-normal distribution of $\rho$ and the random phase approximation (RPA: the assumption that the transfer-matrix phases are uniformly distributed, which underlies both the Born-level and DMPK-level equations) are standard tools. 'Non-ergodicity' in this context means that statistics estimated from finitely many disorder realizations $K$ need not converge to the ensemble distribution before the system length $L$ is taken to infinity, so the order of the limits $L\to\infty$ and $K\to\infty$ can change the answer.

Problem Statement

The exact statistical results of A. A. Abrikosov (Solid State Commun. 37, 997-1000 (1981)) hold only in the Born approximation for Gaussian white-noise disorder, in a strictly one-dimensional single-channel geometry, for finite samples; the task is to construct the exact statistical theory of the resistance and static conductivity of disordered low-dimensional conductors beyond those limits, as one coherent objective - the exact probability distribution of the transport observable and its length evolution, together with the identification of the physically representative average. The components of this extension, which share this common objective and technique family, are:

(i) the resistance/conductance distribution, its moments, and its asymptotics beyond the Born approximation - for stronger disorder and general disorder statistics, in the regime where the random phase approximation underlying the 1D evolution equations is violated and the resistance is coupled to transfer-matrix phase variables;

(ii) the generalization to quasi-one-dimensional (multichannel) systems with transverse channels, i.e. the multichannel evolution equation beyond the diagonal DMPK approximation (the standard DMPK equation retains only the diagonal part of the tensor-valued diffusion coefficient of the general equation);

(iii) the systematic connection between the short/intermediate-length regime (where finite-$t$ corrections matter) and the truly asymptotic regime for general disorder statistics, including broad (heavy-tailed) and correlated disorder where the log-normal law itself may fail;

(iv) the formulation and verification of the frequency-dependent (finite-frequency) scaling of the conductivity of the infinite localized sample, and - the direct descendant of the paper's central question - the determination of which disorder average (mean, typical, or log-mean, and with which order of the limits $L\to\infty$ and $K\to\infty$ for $K$ realizations) represents the measured static conductivity.

The extension sought consists of component (i) (the exact beyond-Born/beyond-random-phase statistics of the resistance for stronger and general disorder statistics), the beyond-diagonal (tensor) part of the multichannel evolution equation in component (ii) and its solution, the general-disorder part of component (iii) (the connection between the finite-length and asymptotic regimes for broad, heavy-tailed and correlated disorder, an independent requirement not discharged by component (i)), and the representative-average/non-ergodicity part of component (iv) (which disorder average, with which order of the limits $L\to\infty$ and $K\to\infty$, represents the measured static conductivity). The remaining parts - the diagonal DMPK limit of component (ii), the white-noise part of component (iii), and the finite-frequency scaling of the conductivity in component (iv) - are settled limits that the general construction reproduces. The extension is to form a single framework - one model class and one set of evolution equations - that specifies the disorder statistics, geometry and reservoir boundary conditions, yields the exact evolution equations and their solutions with a stated domain of validity, and reproduces the settled limits, including Abrikosov's $W(\rho,t)$, $\bar{\rho}=\frac{1}{2}(e^{2t}+1)$, and the inverse-moment asymptotics in the Born limit, and the usual DMPK equation in the diagonal multichannel limit.

References

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