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Physical review. B, Condensed matter

Quantum Brownian motion in a periodic potential and the multichannel Kondo problem

Hangmo Yi, C. L. Kane

Published Mar 1, 1998• 64 citations
Publisher

Abstract

We study the motion of a particle in a periodic potential with Ohmic dissipation. In $D=1$ dimension it is well known that there are two phases depending on the dissipation: a localized phase with zero temperature mobility $\ensuremath{\mu}=0$ and a fully coherent phase with $\ensuremath{\mu}$ unaffected by the periodic potential. However, for nonsymmorphic lattices with $D>1$, such as the honeycomb lattice, there is an intermediate phase with a universal mobility ${\ensuremath{\mu}}^{*}$. This intermediate phase is relevant to resonant tunneling experiments in strongly coupled Coulomb-blockade structures as well as multichannel Luttinger liquids. We relate this problem to the Toulouse limit of the $D+1$ channel Kondo problem, which allows us to compute ${\ensuremath{\mu}}^{*}$ exactly using results known from conformal field theory.

DOI
10.1103/physrevb.57.r5579

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We study the motion of a particle in a periodic potential with Ohmic dissipation. In D=1D=1D=1 dimension it is well known that there are two phases depending on the dissipation: a localized phase with zero temperature mobility μ=0μ=0μ=0 and a fully coherent phase with μμμ unaffected by the periodic potential. For D>1D>1D>1, we find that this is also the case for a Bravais lattice. However, for non symmorphic lattices, such as the honeycomb lattice and its DDD dimensional generalization, there is a new intermediate phase with a universal mobility μ∗μ^*μ∗. We study this intermediate fixed point in perturbatively accessible regimes. In addition, we relate this model to the Toulouse limit of the D+1D+1D+1 channel Kondo problem. This mapping allows us to compute μ∗μ^*μ∗ exactly using results known from conformal field theory. Experimental implications are discussed for resonant tunneling in strongly coupled Coulomb blockade structures and for multi channel Luttinger liquids.
We study quantum Brownian motion (QBM) models for a particle in a dissipative environment coupled to a periodic potential. We review QBM for a particle in a one-dimensional periodic potential and extend the study to that for a particle in two-dimensional (2D) periodic potentials of four Bravais lattice types: square, rectangular, triangular (hexagonal), and centered rectangular. We perform perturbative renormalization group analyses to derive the zero temperature flow diagrams and phase boundaries for a particle in these potentials, and observe localization behavior dependent on the anisotropy of the lattice parameters.
We analyze a model of resonant point-contact tunneling between multiple Luttinger-liquid leads. The model is a variant of the multichannel Kondo model and can be related to the quantum Brownian motion of a particle on lattices with $\ensuremath{\pi}$ flux through each plaquette (in the three-lead case, it is a honeycomb lattice with $\ensuremath{\pi}$ flux). By comparing the perturbative and instanton gas expansions, we find a duality property of the model. At the boundary, this duality exchanges Neumann and Dirichlet boundary conditions on the Tomonaga-Luttinger bosons, which describe the leads; in the bulk, it exchanges the ``momentum'' and ``winding'' modes of these bosons. Over a certain range of Luttinger-liquid parameter g, a nontrivial intermediate coupling fixed-point controls the low-energy physics. The finite conductance at this fixed point can be exactly computed for two special values of g. For larger values of g, there is a stable fixed point at strong coupling that has enhanced conductance resulting from an analogue of Andreev reflection at the point contact.
We study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. For a range of repulsive interactions, we find that for symmetric barriers there exist resonances with a universal peak conductance 2g∗e2/h2g^* e^2/h2g∗e2/h that are controlled by a non-trivial intermediate fixed point. This fixed point is also a quantum critical point separating symmetry-protected topological phases. By tuning the system through resonance, all SPT phases can be accessed. For a particular interaction strength with Luttinger parameters gρ=1/3g_ρ=1/3gρ​=1/3 and gσ=1g_σ=1gσ​=1, we show that the problem is equivalent to a two channel SU(3)SU(3)SU(3) Kondo problem(SU(3)2SU(3)_2SU(3)2​ CFT). At the Toulouse limit, both problems can be mapped to a quantum Brownian motion model on a Kagome lattice, which in turn is related to the quantum Brownian motion on a honeycomb lattice and the three-channel SU(2)SU(2)SU(2) Kondo problem(SU(2)3SU(2)_3SU(2)3​ CFT). "Level-rank duality" in the quantum Brownian motion model relating SU(2)kSU(2)_kSU(2)k​ CFT to SU(k)2SU(k)_2SU(k)2​ CFT is also explored. Utilizing the boundary conformal field theory, the on-resonance conductance of our resonant tunneling problem is calculated as well as the scaling dimension of the leading relevant operator. This allows us to compute the scaling behavior of the resonance line-shape as a function of temperature.
Multichannel Kondo lattice models are examples of strongly correlated electronic systems that exhibit non-Fermi-liquid behavior due to the presence of a continuous channel symmetry. Mean-field analyses have predicted that these systems undergo channel symmetry breaking at low temperature. We use the dynamical large- N technique to study temporal and spatial fluctuations of the multichannel Kondo model on a honeycomb lattice and find that this prediction is not generally true. Rather, we find a ( 2+1 )-dimensional conformally invariant fixed point, governed by critical exponents that are found numerically. When we break time-reversal symmetry by adding a Haldane mass to the conduction electrons, three phases, separated by continuous transitions, are discernible: one characterized by dynamic mass generation and spontaneous breaking of the channel symmetry, one where topological defects restore channel symmetry but preserve the gap, and one with a Kondo-coupled chiral spin liquid. We argue that the last phase is a fractional Chern insulator with anyonic excitations. Published by the American Physical Society 2024
A conformal-field-theory approach has recently been developed for quantum-impurity problems including the ``overscreened'' multichannel Kondo model. We present the details of our calculation of the single-fermion Green's function. The universal zero-temperature resistivity and leading temperature-dependent term are derived. Effects of particle-hole symmetry breaking are included. We also give our perturbative proof of the ``g theorem'' governing the change in residual entropy under a renormalization-group flow between two boundary fixed points.