Impurity back-action, inelastic scattering, and finite-size transport in the disordered one-dimensional Peierls conductor: the unresolved questions of the Abrikosov–Dorotheyev model

Abrikosov and Dorotheyev (J. Low Temp. Phys. 46, 53 (1982)) gave a controlled, non-perturbative treatment of a one-dimensional Peierls conductor with Gaussian white-noise elastic impurity disorder, computing the charge-density-wave order parameter $\Delta(T)$, the disorder-suppressed transition temperature, the single-particle density of states, and the static and finite-frequency conductivity of finite samples; their treatment sums all intersecting impurity diagrams and thereby goes beyond the kinetic (nonintersecting-impurity-line) approximation, which is uncontrolled in one dimension. The later literature resolved the elastic single-particle sector exactly: the white-noise density of states and localization properties of the disordered Peierls–Fröhlich chain, including all elastic scattering channels, are now known in closed form, and the source's Bessel-function results are consistent with these exact solutions. What remains open within the source's model is threefold: (1) a controlled treatment of the back-action of the self-consistently formed charge-density wave on the impurity scattering — the renormalization of the impurity interaction in the Peierls state, i.e. the competition between the destructive lifetime effect and CDW-induced ordering of the impurity pattern; (2) the effect of inelastic scattering processes on $\Delta(T)$, the density of states, and low-frequency transport; and (3) validation of the source's finite-sample conductance and low-frequency conductivity formulas.

Background

A quasi-one-dimensional conductor — a crystal whose electronic structure is so anisotropic that it is effectively a weakly coupled bundle of metallic chains — is subject to two competing instabilities. First, because the Fermi surface of a single chain consists of just two points, $\pm k_F$, any electron–phonon coupling renders the lattice unstable at wave vector $2k_F$: a periodic lattice distortion forms together with an electron–hole condensate, a charge-density wave (CDW), whose complex order parameter $\Delta$ opens a single-particle energy gap $2|\Delta|$ at the Fermi level in the clean, mean-field picture. This is the Peierls transition. Second, a static random impurity potential localizes all single-particle states in a strictly one-dimensional system (the localization length — the scale over which an electronic wave function decays exponentially — is finite for any disorder strength), so a disordered chain is insulating at zero temperature however weak the impurities.

A controlled theory of the two effects together must treat electron–hole pairing and impurity scattering in the same approximation. The kinetic (Born) approximation sums only nonintersecting impurity lines in the disorder-averaged Green function; in one dimension, diagrams with intersecting impurity lines contribute at the same order and cannot be dropped, so the kinetic approximation is uncontrolled exactly where it is most used. Abrikosov and Dorotheyev (1982) therefore constructed a self-consistent treatment of a one-dimensional Peierls conductor with Gaussian white-noise (delta-correlated) elastic impurity disorder, including all intersecting impurity diagrams. The impurity scattering enters through a forward-scattering time $\tau_1$ (momentum transfers $\approx 0$ within each Fermi point) and a backward-scattering time $\tau_2$ (transfers $\approx 2k_F$ connecting the two Fermi points). Their main results are: (i) a self-consistent equation for $|\Delta(T)|$ in the presence of scattering, with the suppression of the transition temperature $T_c$ taking the standard digamma-function form familiar from superconductors with paramagnetic impurities; (ii) the single-particle density of states $\nu(\omega)$ in closed form in modified Bessel functions $I_\nu$, showing that no hard gap survives any finite disorder: $\nu(0) = \nu_n/I_0^2(2|\Delta|\tau)$ (with $\nu_n$ the normal-state value and $\tau$ the transport time, $1/\tau = 2/\tau_1 + 1/\tau_2$), exponentially small $\sim 4\pi|\Delta|\tau\, e^{-4|\Delta|\tau}$ for weak disorder but nonzero; (iii) the ensemble-averaged resistance of a finite sample of length $L$, growing as $\exp[(2/l_2 + 4|\Delta|^2 l/v^2)L]$ (with $l_1 = v\tau_1$ and $l_2 = v\tau_2$ the forward- and backward-scattering mean free paths, $l$ the transport mean free path defined by $1/l = 2/l_1 + 1/l_2 = 1/(v\tau)$, and $v$ the Fermi velocity), showing that CDW formation enhances localization; and (iv) static and low-frequency conductivity formulas for finite and infinite samples, with the low-frequency conductivity collapsing rapidly as $T \to T_c$ from below.

Several directions lie outside the model's scope: the impurity scattering times are bare, impurity-concentration-fixed parameters, so no back-action of the self-consistently formed charge-density wave on the impurity scattering is included; exponentially small corrections to the displayed asymptotic results were not controlled; the derived density-of-states forms differ in structure from the earlier one-dimensional white-noise results of Gor'kov and Dorokhov (the source's Ref. [5]), a difference later reconciled by the exact solutions; no independent validation of the finite-sample transport formulas has been found in the literature; and only elastic intravalley (forward and backward) scattering was included, leaving intervalley (umklapp) and inelastic processes unanalyzed.

Subsequent work closed the elastic single-particle sector. The one-dimensional white-noise disordered Peierls (Peierls–Fröhlich) chain — in the continuum a two-band (Dirac-type) Hamiltonian with a static gap plus random scalar, mass, and gauge-potential disorder — was solved exactly. The commensurate (real order parameter) and incommensurate (complex order parameter) white-noise limits were solved in the late-1970s Soviet literature and re-derived and generalized in 1987–1996 by exact-solution techniques — the supersymmetry method, S-matrix summation, and Fokker–Planck equations for the phase of the electron wave function (Hayn and Fischbeck 1989; Fischbeck and Hayn 1990; Hayn and Mertsching 1996; reviewed by Bartosch 2001). The exact general solution includes all three elastic scattering channels (forward, backward, and umklapp — the last connecting the two Fermi points in a commensurate situation), contains the earlier partial results as special cases, and yields the density of states and inverse localization length in closed form; the source's Bessel-function results are consistent with these exact solutions. Numerical Riccati-equation methods (integration of the first-order Riccati equation equivalent to the stationary Schrödinger equation, which yields the local density of states and the Lyapunov exponents along the chain) and instanton (optimal-fluctuation) calculations completed the picture for states deep in the pseudogap — the soft, disorder-filled suppression of the density of states around the Fermi level — and the classification of all cases within the random-mass Dirac family is due to Bunder and McKenzie (2001). A related line, the fluctuating gap model introduced by Lee, Rice, and Anderson (1973), models thermal Peierls fluctuations by a static random backscattering potential; its exact-solution program is the same one that settled the white-noise density-of-states questions. What this exact program does not include: the impurity back-action on the CDW (the disorder parameters are fixed external data), any inelastic scattering, and the source's finite-sample transport predictions.

Problem Statement

Within the Abrikosov–Dorotheyev model — a continuum one-dimensional Peierls (Fröhlich) conductor with a self-consistent charge-density-wave order parameter $\Delta(T)$ and Gaussian white-noise elastic impurity disorder with bare forward/backward scattering times $\tau_1, \tau_2$, treated with all intersecting impurity diagrams — the following questions, left open by the model's scope and not closed by the subsequent exact-solution literature, remain:

(1) Impurity back-action (renormalization of impurity scattering by the CDW). Construct a controlled treatment in which the scattering rates are not fixed external parameters but are renormalized by the self-consistently formed CDW — equivalently, in which the CDW and the impurity distribution influence each other (the competition between the destructive lifetime effect — impurity scattering suppressing the charge-density wave through electron–hole pair breaking, in analogy with paramagnetic impurities in superconductors — and the CDW-induced ordering/pinning of the impurity pattern). Determine the resulting modifications of $\Delta(T)$, the disorder suppression of $T_c$, the density of states $\nu(\omega)$, and the localization length, and establish whether the mean-field transition remains continuous or becomes reentrant — the charge-density wave surviving only in an intermediate window of impurity concentration — as predicted by self-consistent treatments that ignore intersecting-impurity effects.

(2) Inelastic and non-intravalley scattering. Include inelastic processes — e.g. scattering of electrons by phasons, the collective phase oscillations of the charge-density wave, or by ordinary phonons within the Peierls state — and compute their effect on $\Delta(T)$, $\nu(\omega)$, and the low-frequency transport. The elastic umklapp channel is already contained in the exact general solution of Hayn and Mertsching (1996) and need not be redone.

(3) Finite-sample transport validation. Validate the source's finite-length resistance and static/low-frequency conductivity formulas — including the CDW-induced enhancement of localization and the rapid decrease of the low-frequency conductivity as $T \to T_c$ from below — for the disordered, self-consistently distorted chain.

These three components share one scientific objective: completing the controlled, quantitative theory of the one-dimensional Peierls conductor with quenched disorder beyond the exactly solved elastic, no-back-action sector — the sector in which the impurity parameters are bare and all scattering is elastic and intravalley. All three components are required for that objective, and a partial answer resolves only the component it addresses.

References

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