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arXiv:cond-mat/9710305 · Physical review. B, Condensed matter

Resonant multilead point-contact tunneling

Chetan Nayak, Matthew P. A. Fisher, A. W. W. Ludwig, Hsiu‐Hau Lin

Published Jun 15, 1999• 144 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.59.15694

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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.
We study point contact tunneling between two leads of a Tomonaga-Luttinger liquid through two degenerate resonant levels in parallel. This is one of the simplest cases of a quantum junction problem where the Fermi statistics of the electrons plays a nontrivial role through the Klein factors appearing in bosonization. Using a mapping to a ``generalized Coulomb model'' studied in the context of the dissipative Hofstadter model, we find that any asymmetry in the tunneling amplitudes from the two leads grows at low temperatures, so that ultimately there is no conductance across the system. For the symmetric case, we identify a nontrivial fixed point of this model; the conductance at that point is generally different from the conductance through a single resonant level.
We study a system consisting of a Luttinger liquid coupled to a quantum dot on the boundary. The Luttinger liquid is expressed in terms of fermions interacting via density-density coupling, and the dot is modeled as an interacting resonant level onto which the bulk fermions can tunnel. We solve the Hamiltonian exactly and construct all eigenstates. We study both the zero- and finite-temperature properties of the system, in particular, we compute the exact dot occupation as a function of the dot energy in all parameter regimes. The system is seen to flow from weak to strong coupling for all values of the bulk interaction, with the flow characterized by a nonperturbative Kondo scale. We identify the critical exponents at the weak- and strong-coupling regimes.
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