Rigorous nonperturbative fate of a single weak link in the one-channel Luttinger liquid: the Kane–Fisher impurity renormalization-group flow

Kane and Fisher (1992) argued that in a spinless one-channel Luttinger liquid a single local impurity (weak link) is a relevant perturbation for repulsive interactions, so that at zero temperature the conductance vanishes for any nonzero bare impurity strength, while for attractive interactions the link flows to perfect transmission with conductance $ge^{2}/h$ (in the isolated-liquid convention). The global renormalization-group flow underlying this dichotomy was conjectured in the original work but not proven. Thirty years of later work supports the picture — exact Bethe-ansatz solutions of the boundary field theory and of the Kane–Fisher Hamiltonian, fermionic renormalization-group flows validated against density-matrix renormalization group, and resonant-level duality results. What survives is the original core: a rigorous, nonperturbative derivation of the impurity flow, valid for all bare link strengths and all interaction strengths $g$ in the microscopic interacting model, including the strongly repulsive regime where 'conductance reemergence' has been claimed and the standard perturbative renormalization-group treatment has been disputed.

Background

A one-dimensional interacting electron gas with a linearized spectrum is a Luttinger liquid: its low-energy physics is described by bosonic density waves rather than fermionic quasiparticles. Bosonization — the exact rewriting of the 1D fermion degrees of freedom in terms of a phase field $\phi$ — shows that a short-ranged density–density interaction is fully characterized by a single dimensionless parameter $g$ (with $g<1$ for repulsive and $g>1$ for attractive interactions; $g=1$ is the noninteracting limit). Kane and Fisher (1992) studied transport of such a liquid through a single weak link — a local scatterer of bare strength $V$, equivalently a link of nonzero bare transmission. In the bosonized theory the impurity generates a backscattering operator $\cos 2\phi(0)$ whose scaling dimension is $1-g$: for repulsive interactions ($g<1$) this operator is relevant, so the effective impurity strength grows without bound as the energy scale is lowered, while for $g>1$ it is irrelevant and the impurity effectively disappears. The two stable fixed points are therefore the perfectly transmitting chain and the 'cut' chain — two semi-infinite Luttinger liquids with open boundary conditions joined only by a weak tunneling term. A poor man's (perturbative, single-coupling) renormalization-group (RG) analysis of the two limits (weak backscattering and weak tunneling), combined with the exact duality between them, led Kane and Fisher to conjecture the global RG flow: at $T=0$ the two-terminal conductance is exactly zero for any nonzero bare impurity strength when $g<1$, and equals the perfect-transmission value $ge^{2}/h$ for any finite barrier when $g>1$ (a convention-dependent statement — with Fermi-liquid reservoirs attached, Safi and Schulz (1995) and Ponomarenko (1995) showed the clean two-terminal conductance is $e^{2}/h$ independent of $g$, while the impurity-induced suppression survives). At finite temperature and bias the conductance crosses over with characteristic power laws. The weak-backscattering side of the problem maps to the boundary sine-Gordon model, a massless integrable quantum field theory solvable by Bethe-ansatz techniques — exact wave-function constructions available for special one-dimensional Hamiltonians; the thermodynamic Bethe ansatz, their thermodynamic limit, computes the free energy exactly from the infinite set of scattering data. Numerically, the density-matrix renormalization group (DMRG) — a variational ground-state method for one-dimensional systems — and fermionic RG schemes track the impurity flow over the full energy range. The source paper treated a single spinless channel and a single barrier perturbatively near the two fixed points, acknowledged the role of Fermi-liquid leads, and left open the rigorous nonperturbative justification of the global flow, the extension to two barriers (resonant tunneling through a finite island), to spin and multiple channels, and to realistic experimental systems; the first of these remains the core open problem addressed here.

Problem Statement

Establish, nonperturbatively and with mathematical rigor, the global renormalization-group flow of a single local impurity (weak link) in the spinless one-channel Luttinger liquid, as conjectured by Kane and Fisher (1992): prove (or rigorously refute) that, for the microscopic interacting model with a density-density interaction parametrized by $g$ and an impurity of arbitrary bare strength (with the microscopic model, ultraviolet regularization and bare-strength definition fixed as in the source papers, Refs. 1–2), the zero-temperature conductance flows to zero for every nonzero bare impurity strength when $g<1$ (the chain is cut into two weakly tunnel-coupled halves) and to the perfect-transmission fixed point for every finite barrier when $g>1$, with the corresponding exact finite-temperature and finite-bias crossover functions, and with the conductance conventions fixed relative to Fermi-liquid reservoirs. A complete solution must establish the global flow as a whole: the rigorous $T=0$ dichotomy together with the corresponding finite-temperature and finite-bias crossover functions, with the strongly repulsive regime as the hardest special case of the same all-$g$ requirement and not a separate optional question; a rigorous refutation of either branch of the dichotomy by itself resolves the task, and otherwise the affirmative all-$g$ flow together with the crossover functions is required. The proof must be uniform in the bare link strength, bridging the weak-backscattering and weak-tunneling limits (equivalently, the boundary sine-Gordon model and its dual), and must in particular settle the strongly repulsive regime, where Afonin and Petrov (JETP Letters, 2015) have claimed conductance reemergence and a breakdown of the standard perturbative (poor man's) renormalization-group treatment. The derived companion questions raised in the same source — resonant tunneling through a double-barrier island, spin and multichannel generalizations, and quantitative comparison with experimental systems — are outside the scope of this problem and do not replace this core objective.

References

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