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arXiv:2002.04907 · Physical review. B./Physical review. B

Tunneling into a Luttinger liquid coupled to acoustic phonons out of equilibrium

P. A. Nosov, R. A. Niyazov, D. N. Aristov

Published May 4, 2020• 1 citation
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Abstract

The renormalization of conductances in a $Y$ junction of spinless Luttinger-liquid wires additionally coupled to acoustic longitudinal phonons is investigated in fermionic representation. This system corresponds to geometry of a tunneling experiment and exhibits the interplay between the Coulomb repulsion and the attractive retarded interaction mediated by phonons. The retardation effects related to the propagation of phonons through the junction with arbitrary transmission and reflection amplitudes are taken into account. The appearing logarithmic corrections to conductances of the junction are treated in a renormalization group approach, and scaling exponents are calculated up to infinite order in the interaction after RPA-type summation. The fixed points and corresponding scaling exponents are considered in various nonequilibrium regimes. We show that the boundary exponent and the bulk anomalous dimension of fermion operator are characterized by two different Luttinger parameters, referring to the main wire, thanks to nonlocal character of phonon-mediated interaction. In the limiting case of the junction of only two wires, the scaling exponents found by our method are in exact correspondence with previous bosonization analysis.

DOI
10.1103/physrevb.101.195404

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The renormalization of conductances in a YYY junction of spinless Luttinger-liquid wires additionally coupled to acoustic longitudinal phonons is investigated in fermionic representation. This system corresponds to geometry of a tunneling experiment and exhibits the interplay between the Coulomb repulsion and the attractive retarded interaction mediated by phonons. The retardation effects related to the propagation of phonons through the junction with arbitrary transmission and reflection amplitudes are taken into account. The appearing logarithmic corrections to conductances of the junction are treated in a renormalization group approach, and scaling exponents are calculated up to infinite order in the interaction after RPA-type summation. The fixed points and corresponding scaling exponents are considered in various nonequilibrium regimes. We show that the boundary exponent and the bulk anomalous dimension of fermion operator are characterized by two different Luttinger parameters, referring to the main wire, thanks to nonlocal character of phonon-mediated interaction. In the limiting case of the junction of only two wires, the scaling exponents found by our method are in exact correspondence with previous bosonization analysis.
We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction (``tunneling tip'') is different from that in the other two arms (which together form a ``main wire''). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the limit of weak interactions. The solution shows that, in general, the conductances scale with the bias voltages in an essentially different way compared to their scaling with the temperature TTT. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing TTT with the ``corresponding'' bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero TTT and a zero source-drain voltage.
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 calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction (g,g3>0g,{g}_{3}>0g,g3​>0) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (g<0g<0g<0 and/or g3<0{g}_{3}<0g3​<0) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line g=g3g={g}_{3}g=g3​, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.
We develop the renormalization group theory of the conductances of NNN-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to NNN independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $\ensuremath{\beta}$ functions are derived up to second order.
We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.
In a previous paper [Bischoff and Jeckelmann, Phys. Rev. B 96, 195111 (2017)] we introduced a density-matrix renormalization group method for calculating the linear conductance of one-dimensional correlated quantum systems and demonstrated it on homogeneous spinless fermion chains with impurities. Here we present extensions of this method to inhomogeneous systems, models with phonons, and the spin conductance of electronic models. The method is applied to a spinless fermion wire-lead model, the homogeneous spinless Holstein model, and the Hubbard model. Its capabilities are demonstrated by comparison with the predictions of Luttinger liquid theory combined with Bethe ansatz solutions and other numerical methods. We find a complex behavior for quantum wires coupled to interacting leads when the sign of the interaction (repulsive/attractive) differs in the wire and leads. The renormalization of the conductance given by the Luttinger parameter in purely fermionic systems is shown to remain valid in the Luttinger liquid phase of the Holstein model with phononic degrees of freedom.