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Fermionic Renormalization Group Approach to Junctions of Strongly Interacting Quantum Wires

Domenico Giuliano, Andrea Kazemi Nava

Published Apr 2, 2015• 0 citations
OpenAlex

Abstract

We construct a generalized Fermionic Renormalization Group approach to describe junctions of strongly interacting spinful one-dimensional quantum wires. Our approach is based on a combined use of bosonization and fermionization techniques to build nonlocal transformations between dual fermion operators. It allows for trading strongly interacting (in the original coordinates) fermionic Hamiltonian for weakly interacting (in the dual coordinates) ones. Using such an approach, on one hand we are able to pertinently complement the results about the phase diagram of the junction obtained within bosonization approach, on the other hand, we map out the full crossover of the conductance tensors between any two fixed points in the phase diagram connected by a renormalization group trajectory.

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We construct a generalized Fermionic Renormalization Group approach to describe junctions of strongly interacting spinful one-dimensional quantum wires. Our approach is based on a combined use of bosonization and fermionization techniques to build nonlocal transformations between dual fermion operators. It allows for trading strongly interacting (in the original coordinates) fermionic Hamiltonian for weakly interacting (in the dual coordinates) ones. Using such an approach, on one hand we are able to pertinently complement the results about the phase diagram of the junction obtained within bosonization approach, on the other hand, we map out the full crossover of the conductance tensors between any two fixed points in the phase diagram connected by a renormalization group trajectory.
Making a combined use of bosonization and fermionization techniques, we build nonlocal transformations between dual fermion operators, describing junctions of strongly interacting spinful one-dimensional quantum wires. Our approach allows for trading strongly interacting (in the original coordinates) fermionic Hamiltonians for weakly interacting (in the dual coordinates) ones. It enables us to generalize to the strongly interacting regime the fermionic renormalization group approach to weakly interacting junctions. As a result, on one hand, we are able to pertinently complement the information about the phase diagram of the junction obtained within the bosonization approach; on the other hand, we map out the full crossover of the conductance tensors between any two fixed points in the phase diagram connected by a renormalization group trajectory.
Junctions of multiple one-dimensional quantum wires of interacting electrons have received considerable theoretical attention as a basic constituent of quantum circuits. While results have been obtained on these models using bosonization and density-matrix renormalization-group (DMRG) methods, another powerful technique is based on direct perturbation theory in the bulk interactions combined with the renormalization group. This technique has so far only been applied to the case in which finite-length interacting wires are attached to noninteracting Fermi liquid leads. We extend this method to cover the case of infinite-length interacting leads, obtaining results on two- and three-lead junctions in good agreement with previous bosonization and DMRG results.
Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase diagram and compute all fixed-point conductances as a function of the three Luttinger parameters.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.
We investigate transport of spinless fermions through a single site dot junction of MMM one-dimensional quantum wires. The semi-infinite wires are described by a tight-binding model. Each wire consists of two parts, the noninteracting leads and a region of finite extent in which the fermions interact via a nearest-neighbor interaction. The functional renormalization group method is used to determine the flow of the linear conductance as a function of a low-energy cutoff for a wide range of parameters. Several fixed points are identified and their stability is analyzed. We determine the scaling exponents governing the low-energy physics close to the fixed points. Some of our results can already be derived using the non-self-consistent Hartree-Fock approximation.
We study the conductance of three or more semi-infinite wires which meet at a junction. The electrons in the wires are taken to interact weakly with each other through a short-range density-density interaction, and they encounter a general scattering matrix at the junction. We derive the renormalization-group equations satisfied by the S-matrix, and we identify its fixed points and their stabilities. The conductance between any pair of wires is then studied as a function of physical parameters such as temperature. We discuss the possibility of observing the effects of junctions in present day experiments, such as the four-terminal conductance of a quantum wire and crossed quantum wires.