Superconductivity in a quasi-one-dimensional metal with impurities beyond the ladder approximation: parquet renormalization with disorder, impurity-dressed kernels, disordered dielectric response, and the localization-pairing competition

In a classic paper motivated by irradiation experiments on quasi-one-dimensional organic superconductors, Abrikosov (J. Low Temp. Phys. 53, 359 (1983)) studied superconductivity of a quasi-one-dimensional metal with ordinary (spin-independent) impurities in a self-consistent-field (ladder/BCS) approximation, justified only where impurity forward scattering dominates and destroys the Peierls logarithm while backward scattering stays weak. He concluded that the singlet critical temperature is impurity-insensitive (an Anderson-theorem analogue), that triplet superconductivity is destroyed by backward scattering through an Abrikosov-Gor'kov-type law with a critical scattering rate, that increasing disorder can replace a triplet phase by a singlet one, and he computed the order-parameter amplitudes and the pure-limit longitudinal dielectric constant near the transition. The paper itself left open the theory beyond this ladder regime: the parquet problem in which Cooper and Peierls logarithmic renormalizations coexist with impurity scattering; the exact impurity-dressed Cooper-channel kernels for arbitrary impurity content; the longitudinal dielectric constant of the disordered system; and the strong-disorder regime where localization, including two-electron bound states in impurity wells, competes with delocalized superconductivity. The ladder-level results were refined by Suzumura and collaborators and the pair-breaking phenomenology was quantitatively confirmed in Bechgaard salts, where the strong-disorder regime is realized as granular superconductivity; the Cooper-Peierls interference (parquet) problem of the pure system was treated by the two-loop renormalization-group program; single-chain disorder renormalization and random-Josephson-coupling models treat adjacent localization physics. No published work treats the parquet problem with impurity scattering in the source's model class or the localized-pair mechanism quantitatively, so the problem remains open.

Background

A quasi-one-dimensional (quasi-1D) metal is a crystal built from well-conducting chains with much weaker electron hopping between chains; its Fermi surface consists of two nearly flat sheets at momenta $\pm p_0$, and the low-energy theory keeps only right-moving electrons near $+p_0$ and left-moving electrons near $-p_0$. In such a metal both the electron-electron (Cooper) channel at zero total momentum and the electron-hole (Peierls) channel at momentum $\approx 2p_0$ acquire logarithmic infrared divergences, and these divergences interfere with one another at two-loop order. Resumming both divergent series simultaneously is called the parquet problem (the name comes from the crossed, floor-parquet-like diagram structure of the Russian diagrammatic school); a ladder or self-consistent-field (BCS-like) approximation sums only one channel and is not generically justified in one dimension. Ordinary, spin-independent impurities in a quasi-1D conductor produce two distinct elastic rates: forward scattering (small momentum transfer, rate $\tau_1^{-1}$) and backward scattering (momentum transfer $\approx 2p_0$, rate $\tau_2^{-1}$). Forward scattering dephases the Peierls electron-hole propagators and destroys the Peierls logarithm, while leaving the Cooper logarithm essentially intact, because a forward-scattering event nearly preserves a zero-total-momentum pair. This is the quasi-1D counterpart of Anderson's theorem, the statement that nonmagnetic impurities do not change the critical temperature $T_c$ of an isotropic s-wave BCS superconductor (Anderson, 1959). The suppression of the Peierls channel by a smooth random potential in quasi-1D conductors was established by Abrikosov and Dorotheyev (1982).

The experimental context was the series of quasi-1D organic conductors $(TMTSF)_2X$ (the Bechgaard salts), which become superconducting under pressure but whose $T_c$ is strongly suppressed by irradiation defects (Bouffard et al., 1982). Abrikosov (1983) asked how ordinary impurities affect the superconducting transition of a quasi-1D metal. Projecting a short-range potential interaction $U_1$ and exchange interaction $U_2$ onto the two Fermi-surface patches yields a singlet pairing coupling $g_0=U_1(0)+U_1(2p_0)-3U_2(0)-3U_2(2p_0)$ and a triplet coupling $g_1=U_1(0)-U_1(2p_0)+U_2(0)-U_2(2p_0)$ (positive means repulsive), so that, depending on their signs and magnitudes, the pure system is a singlet or a triplet superconductor with $T_{c0}=(2\gamma\Lambda/\pi)\exp(-2\pi v/|g_0|)$ or $T_{c1}^{(0)}=(2\gamma\Lambda/\pi)\exp(-2\pi v/|g_1|)$, where the superconducting channel is the one with an attractive (negative) coupling and $|g|$ denotes its magnitude, $v$ is the Fermi velocity, $\Lambda$ a high-energy cutoff of order the Fermi or Debye energy, and $\gamma$ the Euler constant. Within the ladder treatment, backward impurity scattering leaves the singlet $T_{c0}$ unchanged but suppresses the triplet $T_{c1}$ through a pair-breaking relation of the Abrikosov-Gor'kov type, $\ln(T_{c1}^{(0)}/T_{c1})=\psi(1/2+\rho)-\psi(1/2)$ with $\rho=1/(2\pi\tau_2 T_{c1})$ and $\psi$ the digamma function (the analogue, for anisotropic gaps and potential impurities, of the magnetic-impurity pair-breaking theory of Abrikosov and Gor'kov, 1961), so that triplet superconductivity disappears once $\tau_2^{-1}$ exceeds a critical rate of order $T_{c1}^{(0)}$; increasing impurity content can also drive a replacement of a triplet phase by a singlet one. Abrikosov furthermore computed the cubic-order order-parameter amplitudes near $T_c$ and, in the pure limit, the superconducting current near the transition and hence the longitudinal dielectric constant along the chains at not too high frequencies $\omega\ll g\Delta$, where $g$ is the relevant pairing coupling ($g_0$ for singlet, $g_1$ for triplet) and $\Delta$ the corresponding order-parameter magnitude.

The paper flagged what this approximation does not cover, and these limitations define the present problem. First, the ladder treatment is justified only in the impurity regime $\tau_1^{-1}\gg\tau_2^{-1}$ where forward scattering alone destroys the Peierls logarithm; the full parquet problem, in which both logarithmic channels coexist with impurity scattering, was not solved, so it remained unknown whether and how the conclusions change when the Cooper and Peierls renormalizations are treated together (nonperturbatively in the logarithms). Second, the triplet-suppression analysis evaluated asymptotic forms and limiting expressions of the frequency integrals, so the exact impurity-dressed Cooper-channel kernels (denoted $F_{-}(\beta,t)$ with $\beta=4\omega\tau_2$) for arbitrary impurity content were not obtained. Third, the dielectric response was computed only in the pure limit, the disorder average of the relevant four-Green-function combinations having been deemed technically prohibitive, so the longitudinal dielectric constant of the impure quasi-1D superconductor near $T_c$ was left uncalculated. Fourth, at large impurity concentrations the formation of localized two-electron bound states in impurity wells was noted qualitatively as a mechanism that may preempt delocalized singlet superconductivity, but no quantitative treatment of this competition between localization and Cooper pairing in the quasi-1D disordered model was given.

Problem Statement

Construct the theory of superconductivity in Abrikosov's model of a quasi-one-dimensional metal with ordinary impurities beyond the self-consistent-field (ladder) treatment, and determine which of the ladder-regime conclusions survive. The common objective is a controlled treatment of the same microscopic model — two-patch electrons with singlet and triplet couplings $g_0$ and $g_1$ and elastic impurity rates $\tau_1^{-1}$ (forward) and $\tau_2^{-1}$ (backward) — in the regimes and for the quantities the 1983 analysis left open. The requested results and their relationship are:

1. Parquet with disorder (a regime extension): treat the Cooper and Peierls logarithmic renormalizations simultaneously in the presence of impurity forward and backward scattering, i.e. outside the regime $\tau_1^{-1}\gg\tau_2^{-1}$ in which forward scattering alone destroys the Peierls logarithm, and establish whether and how the ladder conclusions are modified — the impurity-insensitivity of the singlet $T_{c0}$, the Abrikosov-Gor'kov-type destruction of triplet $T_{c1}$ with its critical backward-scattering rate, and the impurity-driven replacement of a triplet phase by a singlet one.

2. Impurity-dressed kernels and crossover (a within-regime refinement): obtain the exact Cooper-channel kernels for arbitrary impurity content (finite $\beta=4\omega\tau_2$ in the source's notation), beyond the asymptotic and limiting forms used in the source.

3. Disordered dielectric response (a within-regime gap): compute the longitudinal dielectric constant of the impure quasi-1D superconductor near $T_c$ by disorder-averaging the relevant combinations of four Green functions, whose pure-limit form is known from the source.

4. Localization versus pairing (a regime extension): give a quantitative theory of the strong-disorder regime of the same model, in which localization — including the formation of localized two-electron bound states in impurity wells proposed in the source — competes with or preempts delocalized singlet superconductivity.

These four results are successive regimes and untreated quantities of one microscopic model and one approximation program.

References

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