In a weakly disordered, many-channel quasi-one-dimensional (quasi-1D) fermion system, inelastic scattering mediated by short-ranged interactions self-generates a diffusive, strongly non-Markovian noise bath. Davis and Foster [Phys. Rev. B 102, 155101 (2020)] showed that the dephasing of weak localization by this bath can be computed only as a short-virtual-time cumulant expansion: the expansion breaks down at long times, no well-defined dephasing rate emerges, and the self-consistent Born approximation fails to stabilize it. The open problem is the exact long-virtual-time asymptotics of the Cooperon (the particle-particle propagator governing weak localization) for the purely diffusive bath in isolation — equivalently, the solution of the associated strongly coupled self-interacting polymer problem. Specifically: does the Cooperon decay exponentially with a finite, well-defined dephasing rate, does it instead decay non-exponentially, or is dephasing arrested altogether?

Background

In a disordered metal, the quantum correction to the classical conductivity — weak localization — is governed by the Cooperon, the diffusively propagating particle-particle propagator that sums pairs of time-reversed scattering paths. Phase coherence between such paths is destroyed by inelastic electron-electron scattering at a rate 1/τφ1/\tau_\varphi, which provides the infrared cutoff of weak localization in one and two dimensions and is measurable through the temperature dependence of the conductance. Altshuler, Aronov, and Khmelnitsky (AAK) computed this rate exactly for quasi-1D many-channel wires (wires wider than the elastic mean free path but thinner than the dephasing length, so that many transverse channels contribute and transport is diffusive along one axis) with dynamically screened Coulomb interactions, obtaining the famous τφ∝T−2/3\tau_\varphi\propto T^{-2/3} law by mapping the dephasing problem to an exactly solvable single-particle problem; this mapping exploits the effectively Markovian (short-memory) character of the Coulomb noise. A long experimental and theoretical controversy concerns the apparent saturation of τφ\tau_\varphi at low temperatures.

The situation changes qualitatively for short-ranged interactions. In the ergodic phase of a disordered fermion system with quenched disorder, short-ranged inelastic collisions mediate density fluctuations that propagate diffusively, so the bath that dephases the electrons is itself diffusive and strongly non-Markovian: its noise kernel in the quasi-1D geometry is Δt(ω,k)=Γt 2Dtk2/(Dt2k4+ω2)\Delta_t(\omega,k)=\Gamma_t\,2D_tk^2/(D_t^2k^4+\omega^2), with coupling Γt\Gamma_t proportional to the square of the spin-triplet interaction strength γt\gamma_t, and DtD_t the interaction-renormalized diffusion constant, whose ratio to the elastic diffusion constant DD of the noninteracting disordered system is β=Dt/D\beta=D_t/D. This setting describes, for example, ultracold Fermi gases with contact interactions and, together with Coulomb interactions, spin-SU(2)-symmetric quantum wires. Dephasing by such a bath is a candidate analytical window onto precursors of many-body localization (MBL), the hypothesized transition by which an interacting, disordered system fails to self-thermalize: in the MBL phase dephasing itself breaks down.

Davis and Foster (2020) evaluated the Cooperon ct(η)c_t(\eta) — where the virtual time η\eta is the imaginary-time length of the time-reversed diffusion paths summed by the Cooperon — through second order in a cumulant expansion of the influence functional (the Feynman-Vernon action encoding the bath's effect on the time-reversed paths). The first-order term damps the Cooperon superexponentially, ∼exp⁡[−(Γtη3/2/D)G1(β)]\sim\exp[-(\Gamma_t\eta^{3/2}/\sqrt{D})G_1(\beta)], so no finite dephasing rate is defined; the second-order term is rephasing (it acts against dephasing); and inserting a self-consistent mean-field decay rate — the self-consistent Born approximation (SCBA), which resums the non-crossing 'rainbow' diagrams to generate a Cooperon mass — does not stabilize the expansion. The rephasing sign of the second-order term was anticipated by the renormalization-group study of Liao and Foster (2018), which found a nontrivial fixed point in d=4−ϵd=4-\epsilon dimensions signaling a possible instability of the ergodic phase (a 'dephasing catastrophe'); however d=1d=1 lies far below the upper critical dimension d=4d=4, so that expansion is uncontrolled there. Davis and Foster also treated a dual-bath problem — a Markovian Coulomb bath solved exactly by an extension of the AAK technique coexisting with the diffusive bath treated perturbatively — where the Markovian bath provides a physical infrared regulator and the expansion parameter is the temperature-independent coupling ratio Γt/ΓM\Gamma_t/\Gamma_M; this gives the first controlled calculation of quasi-1D diffusive-noise dephasing, but only for Γt/ΓM≪1\Gamma_t/\Gamma_M\ll1. They articulated the outstanding goal as a nonperturbative understanding of dephasing due to the diffusive bath in isolation, castable as a self-interacting polymer problem (the Cooperon as a fluctuating line whose coupling to the bath induces a self-interaction; a gyration radius would set the dephasing length), a strongly coupled, ultraviolet-convergent quantum field theory in one dimension.

Problem Statement

For a weakly disordered, many-channel quasi-one-dimensional fermion system with short-ranged interactions, whose Cooperon ct(η)c_t(\eta) as a function of virtual time η\eta is defined by the influence-functional path integral with the self-generated diffusive noise kernel Δt(ω,k)=Γt 2Dtk2/(Dt2k4+ω2)\Delta_t(\omega,k)=\Gamma_t\,2D_tk^2/(D_t^2k^4+\omega^2) [Eqs. (24)-(27) of Davis and Foster, Phys. Rev. B 102, 155101 (arXiv:2004.07245)], determine the exact nonperturbative behavior of ct(η)c_t(\eta) in the long-virtual-time limit η→∞\eta\to\infty. The single overarching objective is the fate of quantum coherence under self-generated diffusive noise, and an answer must settle, with controlled validity: (1) whether ct(η)c_t(\eta) decays exponentially so that a finite, well-defined dephasing rate 1/τφ1/\tau_\varphi exists as an observable — and if so, its dependence on the coupling Γt\Gamma_t and the diffusion-constant ratio β=Dt/D\beta=D_t/D — or whether the decay is instead non-exponential, or dephasing is arrested (the Cooperon fails to decay, the dephasing-catastrophe scenario); and (2) equivalently, the solution of the associated strongly coupled self-interacting polymer problem, identifying whether a gyration radius (dephasing length) exists in the long-virtual-time limit.

References

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