Impurities, Junctions, and Transport in One Dimension

A point impurity in a one-dimensional interacting liquid is a boundary critical problem. Repulsive interactions amplify weak backscattering until the wire is effectively cut, while attractive interactions heal a weak defect. The dual weak-barrier and weak-link descriptions expose this flow; transport depends additionally on leads, contacts, temperature, and whether momentum can relax.

Required background. Luttinger Liquids supplies the bulk fields and dimensions; Kondo Screening and Renormalization-Group Flow supplies boundary fixed points and crossover scales; Sources, Linear Response, and Kubo Formulae supplies the conductance definition. Helpful background. Free Bosons and Vertex Operators supplies boundary scaling dimensions.

A single impurity and its dual

For a spinless Luttinger liquid, a local potential contains forward scattering and $2k_F$ backscattering. Forward scattering shifts the field locally and does not by itself change the dc fixed point. At $x=0$, backscattering is

$$H_{\mathrm{bs}}=\lambda\cos[2\phi(0)].$$

As a boundary operator it has scaling dimension $K$, giving

$$\frac{d\lambda}{d\ell}=(1-K)\lambda+O(\lambda^3).$$

Thus a weak impurity grows for $K<1$ and shrinks for $K>1$. At strong backscattering the appropriate degrees of freedom are two open ends. Electron tunneling across the cut is

$$H_{\mathrm{link}}=t\cos[2\theta(0)], \qquad \frac{dt}{d\ell}=(1-K^{-1})t+O(t^3).$$

Its dimension $1/K$ is dual to the weak-barrier dimension. The two flows imply that the cut wire is stable for repulsion and unstable for attraction. Kane and Fisher derived this boundary flow and its universal conductance powers Kane and Fisher 1992, §§ II–IV.

Crossover and conductance exponents

The relevant coupling defines a crossover scale, parametrically

$$T_B\sim E_0\left(\frac{|\lambda|}{E_0}\right)^{1/(1-K)}$$

for weak bare backscattering and $K<1$, up to a nonuniversal factor. Above $T_B$, the correction to an otherwise transmitting channel scales as

$$\delta G\propto-\lambda^2T^{2K-2}.$$

Deep in the cut-wire regime, tunneling gives

$$G\propto t^2T^{2/K-2}.$$

These are asymptotic powers. Their prefactors, the interpolation near $T_B$, and nonlinear bias scaling depend on the ultraviolet definition; at special values of $K$, integrability or refermionization gives more detailed crossover functions.

The conductance normalization requires care. A spatially uniform, isolated Luttinger liquid has intrinsic charge stiffness corresponding to $K e^2/h$ per spinless channel. A clean interacting wire attached adiabatically to noninteracting Fermi-liquid reservoirs instead has two-terminal dc conductance $e^2/h$: the leads fix the asymptotic impedance. This contact result was established independently by Maslov and Stone 1995, pp. R5539–R5542 and Safi and Schulz 1995, pp. R17040–R17043. The impurity correction retains interaction-dependent exponents.

Junctions and boundary conditions

At an open end, incoming and outgoing chiral fields are related; “folding” maps the problem to chiral fields on a half-line. A junction of several wires is specified by a boundary condition that conserves the chosen currents. Candidate fixed points include disconnected wires, perfect transmission, and symmetry-allowed mixed boundary conditions. Their stability is decided by the dimensions of boundary tunneling and backscattering operators, computed with the full multichannel interaction matrix.

A single scalar $K$ is insufficient for a generic junction. Charge conservation, time-reversal symmetry, channel anisotropy, superconducting conversion, and Klein-factor algebra restrict the allowed operators. An orthogonal current-splitting matrix can encode a conformal boundary condition, but not every algebraic matrix is reachable from a local microscopic junction. Boundary entropy and conductance tensors provide complementary checks Oshikawa, Chamon, and Affleck 2006, §§ II–V.

Disorder, finite size, and interpretation

One impurity is not a random potential. Distributed disorder generates replicated or stochastic bulk backscattering and has a different relevance threshold. A finite wire also stops the RG flow at the largest of $T$, bias, level spacing $u/L$, and dephasing rate. If $T_B$ lies below that cutoff, the asymptotic cut-wire regime will not be reached.

Measured resistance may include reservoirs, multiple imperfect contacts, phonons, and parallel channels. A power law over a limited interval is therefore not sufficient to identify a Luttinger parameter. Strong evidence combines the exponent with compressibility or spectral estimates of $K$, the predicted crossover under gate tuning, and the appropriate lead geometry.

Exercises

1. For $K=3/4$, find the low-temperature conductance exponent in the cut-wire regime.

Solution

$2/K-2=8/3-2=2/3$, so $G\propto T^{2/3}$ to leading order in the weak-link amplitude. The prefactor is nonuniversal.

2. Determine the fate of both weak backscattering and a weak link at $K=2$.

Solution

Backscattering has eigenvalue $1-K=-1$ and is irrelevant, so a nearly clean wire heals. A weak link has eigenvalue $1-1/K=1/2$ and is relevant, so a nearly cut wire also heals. Both descriptions flow toward perfect transmission.

References

• Kane, C. L., and M. P. A. Fisher. “Transmission through Barriers and Resonant Tunneling in an Interacting One-Dimensional Electron Gas.” Physical Review B 46 (1992): 15233–15262. DOI.
• Maslov, D. L., and M. Stone. “Landauer Conductance of Luttinger Liquids with Leads.” Physical Review B 52 (1995): R5539–R5542. DOI.
• Oshikawa, M., C. Chamon, and I. Affleck. “Junctions of Three Quantum Wires.” Journal of Statistical Mechanics 2006 (2006): P02008. DOI.
• Safi, I., and H. J. Schulz. “Transport in an Inhomogeneous Interacting One-Dimensional System.” Physical Review B 52 (1995): R17040–R17043. DOI.