Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures

TL;DR

The paper addresses universal spectral and transport statistics of mesoscopic normal-superconducting structures by classifying BdG Hamiltonians into four symmetry classes (D, C, DIII, CI). It employs maximum-entropy random-matrix ensembles and exact mappings to free 1D Fermi gases or Laguerre ensembles to derive $n$-level correlations and one-point functions, revealing a doubling of slow modes that enhances universal conductance fluctuations. It provides explicit weak-localization corrections and, for certain regimes, exact S-matrix-based variances, showing that superconducting proximity alters the mode structure and yields new universal behavior in the ergodic limit. The results establish robust, symmetry-driven predictions for spectral statistics and transport, with significant implications for mesoscopic NS devices and a framework for exploring crossovers and higher-energy regimes.

Abstract

Normal-conducting mesoscopic systems in contact with a superconductor are classified by the symmetry operations of time reversal and rotation of the electron's spin. Four symmetry classes are identified, which correspond to Cartan's symmetric spaces of type C, CI, D, and DIII. A detailed study is made of the systems where the phase shift due to Andreev reflection averages to zero along a typical semiclassical single-electron trajectory. Such systems are particularly interesting because they do not have a genuine excitation gap but support quasiparticle states close to the chemical potential. Disorder or dynamically generated chaos mixes the states and produces novel forms of universal level statistics. For two of the four universality classes, the n-level correlation functions are calculated by the mapping on a free 1D Fermi gas with a boundary. The remaining two classes are related to the Laguerre orthogonal and symplectic random-matrix ensembles. For a quantum dot with an NS-geometry, the weak localization correction to the conductance is calculated as a function of sticking probability and two perturbations breaking time-reversal symmetry and spin-rotation invariance. The universal conductance fluctuations are computed from a maximum-entropy S-matrix ensemble. They are larger by a factor of two than what is naively expected from the analogy with normal-conducting systems. This enhancement is explained by the doubling of the number of slow modes: owing to the coupling of particles and holes by the proximity to the superconductor, every cooperon and diffuson mode in the advanced-retarded channel entails a corresponding mode in the advanced-advanced (or retarded-retarded) channel.

Figures (5)

• Figure 1: Metallic quantum dot (N) in contact with two superconducting regions (S). The dot is separated from the leads (L) by a tunnel barrier (T).
• Figure 2: Diagrams contributing to the average single-particle Green's function. The light-shaded regions represent a $D$-type cooperon mode, the dark-shaded one a nonsingular $\Pi_A^{{\rm d}0}$-ladder.
• Figure 3: Andreev quantum dot (N) coupled to a single lead (L) via a tunnelling barrier (T). The flux loop on the right is introduced to adjust the difference of the order parameter phases of the superconducting regions (S) to the value $\phi_1-\phi_2 = \pi$ (cf. the discussion of Sec. \ref{['sec:dyn_input']}).
• Figure 4: Pair of semiclassical paths contributing to the $A$-type weak localization process.
• Figure 5: Pair of semiclassical paths contributing to the $D$-type weak localization process. The triangles represent Andreev reflections.