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arXiv:0901.0126 · Physical Review B

Systematic stability analysis of the renormalization group flow for the normal-superconductor-normal junction of Luttinger liquid wires

Sourin Das, Sumathi Rao, Arijit Saha

Published Apr 13, 2009• 9 citations
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Abstract

We study the renormalization group flows of the two terminal conductance of a superconducting junction of two Luttinger liquid wires. We compute the power laws associated with the renormalization group flow around the various fixed points of this system using the generators of the $SU(4)$ group to generate the appropriate parametrization of an $\mathbb{S}$ matrix representing small deviations from a given fixed point $\mathbb{S}$ matrix [obtained earlier in S. Das, S. Rao, and A. Saha, Phys. Rev. B 77, 155418 (2008)], and we then perform a comprehensive stability analysis. In particular, for the nontrivial fixed point which has intermediate values of transmission, reflection, Andreev reflection, and crossed Andreev reflection, we show that there are eleven independent directions in which the system can be perturbed, which are relevant or irrelevant, and five directions which are marginal. We obtain power laws associated with these relevant and irrelevant perturbations. Unlike the case of the two-wire charge-conserving junction, here we show that there are power laws which are nonlinear functions of $V(0)$ and $V(2{k}_{F})$ [where $V(k)$ represents the Fourier transform of the interelectron interaction potential at momentum $k$]. We also obtain the power law dependence of linear response conductance on voltage bias or temperature around this fixed point.

DOI
10.1103/physrevb.79.155416

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We study the renormalization group flows of the two terminal conductance of a superconducting junction of two Luttinger liquid wires. We compute the power laws associated with the renormalization group flow around the various fixed points of this system using the generators of the SU(4)SU(4)SU(4) group to generate the appropriate parametrization of an S\mathbb{S}S matrix representing small deviations from a given fixed point S\mathbb{S}S matrix [obtained earlier in S. Das, S. Rao, and A. Saha, Phys. Rev. B 77, 155418 (2008)], and we then perform a comprehensive stability analysis. In particular, for the nontrivial fixed point which has intermediate values of transmission, reflection, Andreev reflection, and crossed Andreev reflection, we show that there are eleven independent directions in which the system can be perturbed, which are relevant or irrelevant, and five directions which are marginal. We obtain power laws associated with these relevant and irrelevant perturbations. Unlike the case of the two-wire charge-conserving junction, here we show that there are power laws which are nonlinear functions of V(0)V(0)V(0) and V(2kF)V(2{k}_{F})V(2kF​) [where V(k)V(k)V(k) represents the Fourier transform of the interelectron interaction potential at momentum kkk]. We also obtain the power law dependence of linear response conductance on voltage bias or temperature around this fixed point.
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.
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 Y-junction setup of Luttinger liquid wires threaded by a magnetic flux, allowing for a different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. The scattering matrix and the matrix of conductances are parametrized by three variables. For these we derive coupled renormalization group (RG) equations, up to second order in the interaction, within the scattering state formalism. The fixed point structure of these equations is analyzed in detail. For a 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 an attractive interaction (g<0g<0g<0 and/or g3<0{g}_{3}<0g3​<0) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a) g=0g=0g=0, g3<0{g}_{3}<0g3​<0 or (b) g3+g2/2=0{g}_{3}+{g}^{2}/2=0g3​+g2/2=0, g<0g<0g<0 we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the ggg-g3{g}_{3}g3​ plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.
We study junctions of single-channel spinless Luttinger liquids using bosonization. We generalize earlier studies by allowing the junction to be superconducting and find new charge nonconserving low-energy fixed points. We establish the existence of $g\ensuremath{\leftrightarrow}1/g$ duality (where ggg is the Luttinger liquid parameter) between the charge-conserving (normal) junction and the charge nonconserving (superconducting) junction by evaluating and comparing the scaling dimensions of various operators around the fixed points in both the normal and superconducting sectors of the theory. For the most general two-wire junction, we show that there are two conformally invariant one-parameter families of fixed points which are also connected by a duality transformation. We also show that the stable fixed point for the two-wire superconducting junction corresponds to the situation where the crossed Andreev reflection (an incoming electron is transmitted as an outgoing hole) is perfect between the wires. For the three-wire junction, we study, in particular, the superconducting analogs of the chiral ${\mathbsf{D}}_{\mathbsf{P}}$ and the disconnected fixed points obtained earlier in the literature in the context of charge-conserving three-wire junctions. We show that these fixed points can be stabilized for g<1g<1g<1 (repulsive electrons) within the superconducting sector of the theory which makes them experimentally relevant.
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g1 , g2 , connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on g1 , g2 . The regions of stability of the two fixed points corresponding to conductance G = 0 and G = 1,respectively, are determined.
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.