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arXiv:1402.5285 · Physical Review B

Density of states of interacting quantum wires with impurities: A Dyson equation approach

Redouane Zamoum, M. Guigou, Cristina Bena B, Adeline Crépieux C

Published Aug 7, 2014• 2 citations
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Abstract

We calculate the density of states for an interacting quantum wire in the presence of two impurities of arbitrary potential strength. To perform this calculation, we describe the Coulomb interactions in the wire within the Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson equation for the fermionic retarded Green's functions, we study how the profile of the local density of states is affected by the interactions in the entire range of impurity potentials. Same as in the noninteracting case, when increasing the impurity strength, the central part of the wire becomes more and more disconnected from the semi-infinite leads, and discrete localized states begin to form; the width and the periodicity of the corresponding peaks in the spectrum depends on the interaction strength. As expected from the Luttinger liquid theory, impurities also induce a reduction of the local density of states at small energies. Two other important aspects are highlighted: the appearance of an extra modulation in the density of states at nonzero Fermi momentum when interactions are present, and the fact that forward scattering must be taken into account in order to recover the Coulomb-blockade regime for strong impurities.

DOI
10.1103/physrevb.90.085408

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We calculate the density of states for an interacting quantum wire in the presence of two impurities of arbitrary potential strength. To perform this calculation, we describe the Coulomb interactions in the wire within the Tomonaga-Luttinger liquid theory. After establishing and solving the Dyson equation for the fermionic retarded Green's functions, we study how the profile of the local density of states is affected by the interactions in the entire range of impurity potentials. Same as in the noninteracting case, when increasing the impurity strength, the central part of the wire becomes more and more disconnected from the semi-infinite leads, and discrete localized states begin to form; the width and the periodicity of the corresponding peaks in the spectrum depends on the interaction strength. As expected from the Luttinger liquid theory, impurities also induce a reduction of the local density of states at small energies. Two other important aspects are highlighted: the appearance of an extra modulation in the density of states at nonzero Fermi momentum when interactions are present, and the fact that forward scattering must be taken into account in order to recover the Coulomb-blockade regime for strong impurities.
We study theoretically the transport properties of an interacting quantum wire in the presence of a time-dependent point impurity. Using the Keldysh formalism within the framework of Luttinger liquid model, we calculate and analyze the dc contribution of the backscattered current and the shot noise of the system with arbitrary Luttinger interaction strength and impurity position. From some numerical examples, we find that the time-dependent point impurity in the wire can enhance the total current in the strongly interaction regime, and the total current enhancement is dependent on both the impurity potential frequency and the equivalent frequency in relation to the drain-source voltage VVV. The dc contribution of the backscattered current oscillates as a function of impurity frequency with the period depending on the Luttinger interaction strength. Furthermore, the oscillation of the dc contribution of the backscattered current is not a finite-length effect, which is different from that in the case of system with a static point impurity, and the oscillation characteristic is also connected with VVV. However, the final current-noise relation is similar to that in the case of a static point impurity.
The screening effect due to a scanning tunneling microscope tip which is placed in the vicinity of an interacting quantum wire is considered. With the help of a bosonization procedure, we are able to determine nonperturbatively the Green's functions of the quantum wire in the presence of both electrostatic screening by the tip and Coulomb interactions in the wire. In our approach we justify that the working Hamiltonian of the whole system is quadratic when Kc>1/2{K}_{c}>1/2Kc​>1/2 and can be solved by integration over the degrees of freedom of the tip. Once the Green's functions are known, we calculate the spectral properties. We show that the spectral function, as well as the tunneling density of states, is affected by the screening and that the local density of states strongly deviates from its unscreened value when the tip gets close to the wire. Moreover, we observe that the spatial extension of the deviation of the local density of states is related to both the Coulomb interactions parameter and the screening strength.
We develop a continuous-time quantum Monte Carlo (CTQMC) method for quantum impurities coupled to interacting quantum wires described by a Tomonaga-Luttinger liquid. The method is negative-sign free for any values of the Tomonaga-Luttinger parameter, which is rigorously proved, and, thus, efficient low-temperature calculations are possible. Duality between electrons and bosons in one-dimensional systems allows us to construct a simple formula for the CTQMC algorithm in these systems. We show that the CTQMC for Tomonaga-Luttinger liquids can be implemented with only minor modifications of previous CTQMC codes developed for impurities coupled to noninteracting fermions. We apply this method to the Kane-Fisher model of a potential scatterer in a spin-less quantum wire and to a single spin coupled with the edge state of a two-dimensional topological insulator assuming an anisotropic XXZ coupling. Various dynamical response functions such as the electron Green's function and spin-spin correlation functions are calculated numerically and their scaling properties are discussed.
We have studied low-temperature properties of interacting electrons in a one-dimensional quantum wire (Luttinger liquid) side-hybridized with a single-level impurity. The hybridization induces a backscattering of electrons in the wire which strongly affects its low-energy properties. Using a one-loop renormalization group approach valid for a weak electron-electron interaction, we have calculated a transmission coefficient through the wire, T(epsilon), and a local density of states, nu(epsilon) at low energies epsilon. In particular, we have found that the antiresonance in T(epsilon) has a generalized Breit-Wigner shape with the effective width Gamma(epsilon) which diverges at the Fermi level.
One-dimensional electron systems interacting with long-range Coulomb forces (quantum wires) show a Wigner crystal structure. We investigate in this paper the transport properties of such a Wigner crystal in the presence of impurities. Contrary to what happens when only short-range interactions are included, the system is dominated by 4kF{\mathit{k}}_{\mathit{F}}kF​ scattering on the impurities. There are two important length scales in such a problem: one is the pinning length above which the (quasi-)long-range order of the Wigner crystal is destroyed by disorder. The other length, ${\ensuremath{\xi}}_{\mathrm{cr}}$, is the length below which Coulomb interactions are not important and the system is behaving as a standard Luttinger liquid with short-range interactions. We obtain the frequency and temperature dependence of the conductivity. We show that such a system is very similar to a classical charge density wave pinned by impurities, but with important differences due to quantum fluctuations and long-range Coulomb interactions. Finally, we discuss our results in comparison with experimental systems.
We study the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several extended time-dependent impurities (barriers). Employing the B\"uttiker-Landauer approach, we first consider the scattering of noninteracting electrons (g=1)(g=1)(g=1) by a rectangularlike barrier and find an exact solution for the backscattering current, as well as a perturbative solution for a weak static potential with an arbitrary shape. We then include electron-electron interactions and use the Keldysh formalism combined with the bosonization technique to study oscillating extended barriers. We show that the backscattering current off time-dependent impurities can be expressed in terms of the current for the corresponding static barrier. Then we determine the backscattering current for a static extended potential, which, in the limit of noninteracting electrons (g=1)(g=1)(g=1), coincides with the result obtained using the B\"uttiker-Landauer formalism. In particular, we find that the conductance can be increased beyond its quantized value in the whole range of repulsive interactions 0<g<10<g<10<g<1 already in the case of a single oscillating extended impurity, in contrast to the case of a pointlike impurity, where this phenomenon occurs only for 0<g<1∕20<g<1∕20<g<1∕2.