Effect of Interaction and Proximitized Superconductivity in Luttinger Liquid Junctions

Ratnakar, Amulya (2024) Effect of Interaction and Proximitized Superconductivity in Luttinger Liquid Junctions. PhD thesis, Indian Institute of Science Education & Research Kolkata.

Text (PhD thesis of Amulya Ratnakar (18RS084)) 18RS084.pdf - Submitted Version Restricted to Repository staff only Download (13MB)

Abstract

The study of Luttinger liquids (LLs), which are one-dimensional interacting fermionic systems, has significantly advanced our understanding of quantum many-body physics. LLs are characterized by non-Fermi liquid behavior, making them a rich platform for exploring novel quantum phenomena. A specific class of LLs, known as Chiral Luttinger Liquids (CLLs), are realized as the edge states of a quantum Hall (QH) system. Due to the one-dimensional chiral nature of the edge modes, these excitations exhibit "Luttinger-liquid" behavior, marked by a power-law dependence of correlation functions. This thesis investigates the junctions of such CLLs, either among themselves or with different phases of matter, using the standard bosonization technique to study the low-energy excitations of these systems. CLL junctions have garnered considerable attention due to their potential as fundamental components of dissipationless quantum circuits. The interactions and strong correlations at these junctions can lead to intriguing phenomena such as Andreev reflection, spin-charge separation, and charge fractionalization. Additionally, superconductor-quantum Hall (SC/QH) edge junctions can host non- Abelian anyons like Majorana fermions (for ν = 1) and parafermions (for ν ̸= 1), making them promising candidates for fault-tolerant quantum information processing. The first part of this thesis explores the tunneling density of states (TDOS) behavior of LL junctions, both near and away from the junction. One of the hallmark properties of LLs is the power-law suppression of local single-electron TDOS in the zero-bias limit. Previous studies have shown that an exception to this suppression, resulting in TDOS enhancement, is possible at junctions of three or more LL wires due to the complex structure of the junction’s fixed point. Typically, these fixed points, which show TDOS enhancement, involve Andreev-like processes and are unstable against perturbations at the junction. For the first time, we report a scenario where TDOS enhancement and a stable junction fixed point (in the leading order perturbative Renormalization Group sense) can occur at a simpler junction of two CLLs with generalized sets of inter-electron repulsive in teractions. We consider two types of CLL junctions: one with current-conserving normal (N) boundary conditions (BC) and one with current non-conserving superconducting (SC) BC. We establish a symmetry relation between the normal and superconducting sectors of the theory in the presence of a generalized densitydensity interaction. This model, which allows for such generalized interactions, can be realized as the edge states of a bilayer quantum Hall system with tunnel coupling (or Cooper pair coupling in the case of an SC junction) at a point. This study is inspired by recent experimental developments involving bilayer QH systems and experiments involving SC/QH hybrid junctions where large cross- Andreev reflection is observed. The second part of the thesis includes the study of parafermions, which are Z2m generalizations of Majorana fermions, and the fractional Josephson effect, which provides a clear signature of parafermions (evidenced by a 4πm periodicity in the Josephson current). We envisage an SC/QH hybrid junction that can host such non-Abelian anyons. We demonstrate that the difference in lengths (which may be regulated by external potential gates) of the two counter-propagating chiral edges in the Josephson junction region can induce a spontaneous phase bias driving the Josephson current. For Laughlin filling fractions ν = 1/m, m ∈ 2Z +1, this allows electrical control of either Majorana (m = 1) or parafermion (m ̸= 1) zero modes. We also conduct a detailed investigation of the effect of edge reconstruction in the QH system on the fractional Josephson effect. Specifically, for the QH bulk filling fraction of ν = 1, we show that the periodicity of the Josephson current remains robust (4π) even with edge reconstruction, despite significant differences in the many-body spectrum. We show that the various topological sectors in the manybody spectrum are the consequence of edge reconstruction, and the signatures of these sectors can be found in Josephson current if the velocities of the edge modes are taken to be different. In conclusion, this thesis provides a deep understanding of the interplay between interactions and proximitized superconductivity in CLL junctions, opening new avenues for research in topological phases of matter and contributing to the broader understanding of one-dimensional quantum systems.

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