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arXiv:0807.0804 · Physical Review B

Duality between normal and superconducting junctions of multiple quantum wires

Sourin Das, Sumathi Rao

Published Nov 14, 2008• 29 citations
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Abstract

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 $g$ 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 $g<1$ (repulsive electrons) within the superconducting sector of the theory which makes them experimentally relevant.

DOI
10.1103/physrevb.78.205421

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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↔1/gg\ensuremath{\leftrightarrow}1/gg\ensuremath↔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 \mathbsfD\mathbsfP{\mathbsf{D}}_{\mathbsf{P}}\mathbsfD\mathbsfP​ 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 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.
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.
We demonstrate the equivalence between two distinct Luttinger liquid impurity problems. The first concerns a one-dimensional topological superconductor coupled at one end to the ends of two single channel Luttinger liquids. The second concerns a point contact in the quantum spin Hall effect, where four helical Luttinger liquids meet at a point. Both problems have been studied previously and exhibit several stable phases depending on the Luttinger parameter K, that can be characterized in terms of simple conformally invariant boundary conditions describing perfect normal (or Andreev) transmission or reflection. In addition, both problems exhibit critical points that are described by "intermediate" fixed points similar to those found in earlier studies of an impurity in a Luttinger liquid with spin. Though these two models have different symmetries and numbers of modes, we show they are equivalent and are related by a duality transformation, and we show that the non-trivial intermediate critical points are the same. In the non-interacting limit, K=1, the duality involves two distinct free fermion representations that are related by a non-local transformation that derives from the triality of SO(8). Using the explicit translation between the two theories, we translate results from one problem to the other and vice versa. This allows us to make new predictions about the topological superconductor-Luttinger liquid junction, including predictions about the global behavior of the critical conductance G*(K), as well predictions for the critical exponents and universal crossover scaling functions. In this paper we introduce both models from scratch, using a common notation that facilitates their comparison, and we discuss in detail the dualities that relate them, along with their free fermion limits. We close with a discussion of open problems and future directions.
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.
Andreev scattering and the Josephson current through a one-dimensional interacting electron liquid sandwiched between two superconductors are reexamined. We first present some apparently new results on the noninteracting case by studying an exactly solvable tight-binding model rather than the usual continuum model. We show that perfect Andreev scattering (i.e., zero normal scattering) at the Fermi energy can only be achieved by fine-tuning junction parameters, a fine-tuning which is possible even with bandwidth mismatch between superconductor and normal metal. We also obtain exact results for the Josephson current, which is generally a smooth function of the superconducting phase difference except when the junction parameters are adjusted to give perfect Andreev scattering, in which case it becomes a sawtooth function. We then observe that, even when interactions are included, all low-energy properties of a junction $(E\ensuremath{\ll}\ensuremath{\Delta},$ the superconducting gap) can be obtained by ``integrating out'' the superconducting electrons to obtain an effective Hamiltonian describing the metallic electrons only with a boundary pairing interaction. This boundary model provides a suitable starting point for bosonization-renormalization group-boundary conformal field theory analysis. We argue that total normal reflection and total Andreev reflection correspond to two fixed points of the boundary renormalization group. For repulsive bulk interactions the Andreev fixed point is unstable and the normal one stable. However, the reverse is true for attractive interactions. This implies that a generic junction Hamiltonian (without fine-tuned junction parameters) will renormalize to the normal fixed point for repulsive interations but to the Andreev one for attractive interations. An exact mapping of our tight-binding model to the Hubbard model with a transverse magnetic field is used to help understand this behavior. We calculate the critical exponents, which are different at these two different fixed points.
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.