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arXiv:cond-mat/9607144 · Physical review. B, Condensed matter

Comparing conductance quantization in quantum wires and quantum Hall systems

Anton Alekseev, Vadim Cheianov, Jürg Martin Fröhlich

Published Dec 15, 1996• 29 citations
arXivHTMLPDF

Abstract

We suggest a means to calculate the dc conductance of a one-dimensional electron system described by the Luttinger model. Our approach is based on the ideas of Landauer and B\"uttiker on transport in ballistic channels and on the methods of current algebra. We analyze in detail the way in which the system can be coupled to external reservoirs. This determines whether the conductance is renormalized or not. We provide a parallel treatment of a quantum wire and a fractional quantum Hall system on a cylinder with two widely separated edges. Although both systems are described by the same effective theory, the physical electrons are identified with different types of excitations, and hence the coupling to external reservoirs is different. As a consequence, the conductance in the wire is quantized in integer units of $\frac{{e}^{2}}{h}$ per spin orientation whereas the Hall conductance allows for fractional quantization.

DOI
10.1103/physrevb.54.r17320

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•Oct 1, 2019•31 citations
We suggest a means to calculate the dc conductance of a one-dimensional electron system described by the Luttinger model. Our approach is based on the ideas of Landauer and B"uttiker on transport in ballistic channels and on the methods of current algebra. We analyze in detail the way in which the system can be coupled to external reservoirs. This determines whether the conductance is renormalized or not. We provide a parallel treatment of a quantum wire and a fractional quantum Hall system on a cylinder with two widely separated edges. Although both systems are described by the same effective theory, the physical electrons are identified with different types of excitations, and hence the coupling to external reservoirs is different. As a consequence, the conductance in the wire is quantized in integer units of e2h\frac{{e}^{2}}{h}he2​ per spin orientation whereas the Hall conductance allows for fractional quantization.
We examine the effect of interactions between the electrons on the Landauer-B\"uttiker conductances of some systems of quantum wires with different geometries. The systems include a long wire with a stub in the middle, a long wire containing a ring which can enclose a magnetic flux, and a system of four long wires which are connected in the middle through a fifth wire. Each of the wires is taken to be a weakly interacting Tomonaga-Luttinger liquid, and scattering matrices are introduced at all the junctions present in the systems. Using a renormalization group method developed recently for studying the flow of scattering matrices for interacting systems in one dimension, we compute the conductances of these systems as functions of the temperature and the wire lengths. We present results for all three regimes of interest, namely, high, intermediate, and low temperature. These correspond, respectively, to the thermal coherence length being smaller than, comparable to, and larger than the smallest wire length in the different systems, i.e., the lengths of the stub or each arm of the ring or the fifth wire. The renormalization group procedure and the formulas used to compute the conductances are different in the three regimes. In particular, the dimensionality of the scattering matrix effectively changes when the thermal length becomes larger than the smallest wire length. We also present a phenomenologically motivated formalism for studying the conductances in the intermediate regime where there is only partial coherence. At low temperatures, we study the line shapes of the conductances versus the energy of the electrons near some of the resonances; the widths of the resonances are found to go to zero with decreasing temperature. Our results show that the Landauer-B\"uttiker conductances of various systems of experimental interest depend on the temperature and lengths in a non-trivial way when interactions are taken into account.
The transport in a pure one-dimensional quantum wire is investigated for any range of interactions. First, the wire is connected to measuring leads. The transmission of an incident electron is found to be perfect, i.e., the conductance is not renormalized by the interactions. Either Landauer's approach or the Kubo formula can be used as long as the reservoirs impose the boundary conditions. Second, the Kubo formula as a response to the local field is reconsidered in a generic Luttinger liquid: the ``intrinsic'' conductance thus obtained is determined by the same combination of interaction parameters as that which renormalizes the current.
We use the technique of bosonization to understand a variety of recent experimental results on the conductivity of a quantum wire. The quantum wire is taken to be a finite-length Luttinger liquid connected on two sides to semi-infinite Fermi liquids through contacts. The contacts are modeled as (short) Luttinger liquids bounded by localized one-body potentials. We use effective actions and the renormalization group to study the effects of electronic interactions within the wire, the length of the wire, finite temperature, and a magnetic field, on the conductivity. We explain the deviations of the conductivity from 2Ne2/h{2Ne}^{2}/h2Ne2/h in wires that are not too short as arising from renormalization effects caused by the repulsive interactions. We also explain the universal conductance corrections observed in different channels at higher temperatures. We study the effects of an external magnetic field on electronic transport through this system and explain why odd and even spin-split bands show different renormalizations from the universal conductance values. We discuss the case of resonant transmission and of the possibility of producing a spin valve that only allows electrons of one value of the spin to go through. We compare our results for the conductance corrections with experimental observations. We also propose an experimental test of our model of the contact regions.
We develop a detailed theory for spin transport in a one-dimensional quantum wire described by Luttinger-liquid theory. A hydrodynamic description of the quantum wire is supplemented by boundary conditions, taking into account the exchange coupling between the magnetization of ferromagnetic reservoirs and the boundary magnetization in the wire. Spin-charge separation is shown to imply drastic and qualitative consequences for spin-dependent transport. In particular, the spin accumulation effect is quenched except for fine-tuned parameter regimes. We propose several feasible setups involving an external magnetic field to detect this phenomenon in transport experiments on single-wall carbon nanotubes. In addition, electron-electron backscattering processes, which do not have an important effect on thermodynamic properties or charge transport, are shown to modify spin-dependent transport through long quantum wires in a crucial way.
We discuss the conductance of quantum wires in terms of the Tomonaga-Luttinger liquid (TLL) theory. We use explicitly the charge fractionalization scheme which results from the chiral symmetry of the model. We suggest that results of the standard two-terminal (2T) conductance measurement depend on the coupling of TLL with the reservoirs and can be interpreted as different boundary conditions at the interfaces. We propose a three-terminal (3T) geometry in which the third contact is connected weakly to the bulk of TLL subjected to a large bias current. We develop a renormalization-group (RG) analysis for this problem by taking explicitly into account the splitting of the injected electronic charge into two chiral irrational charges. We study in the presence of bulk contact the leading-order corrections to the conductance for two different boundary conditions, which reproduce in the absence of bulk contact, respectively, the standard 2T source-drain (SD) conductance GSD(2)=e2/h{G}_{\mathrm{SD}}^{(2)}{=e}^{2}/hGSD(2)​=e2/h and GSD(2)=ge2/h,{G}_{\mathrm{SD}}^{(2)}{=ge}^{2}/h,GSD(2)​=ge2/h, where g is the TLL charge interaction parameter. We find that under these two boundary conditions for the end contacts the 3T SD conductance GSD(3){G}_{\mathrm{SD}}^{(3)}GSD(3)​ shows an UV-relevant deviation from the above two values, suggesting new fixed points in the Ohmic limit. Nontrivial scaling exponents are predicted as a result of electron fractionalization.
We propose a method by which the quantization of the Hall conductance can be directly measured in the transport of a one-dimensional atomic gas. Our approach builds on two main ingredients: (1) a constriction optical potential, which generates a mesoscopic channel connected to two reservoirs, and (2) a time-periodic modulation of the channel specifically designed to generate motion along an additional synthetic dimension. This fictitious dimension is spanned by the harmonic oscillator modes associated with the tightly confined channel, and hence, the corresponding “lattice sites” are intimately related to the energy of the system. We analyze the quantum-transport properties of this hybrid two-dimensional system, highlighting the appealing features offered by the synthetic dimension. In particular, we demonstrate how the energetic nature of the synthetic dimension combined with the quasienergy spectrum of the periodically driven channel allows for the direct and unambiguous observation of the quantized Hall effect in a two-reservoir geometry. Our work illustrates how topological properties of matter can be accessed in a minimal one-dimensional setup, with direct and practical experimental consequences.