Random-Matrix Theory of Quantum Transport in Topological Superconductors

Introduction

Fig. 1: The energy eigenvalues of H(θ) = θ*H0 + (1-θ)*H1, where H0,1 are particle-hole symmetric matrices, θ is a real parameter, and (b) the fermion parity corresponding to the spectrum in (a). Each switch of the sign of P corresponds to a change in the fermion parity. The parameter θ could, for example, model an external magnetic field. (Source: P. Sriram)

The theory of random matrices was developed to study the correlations of the spectral statistics of matrices with randomly distributed elements. Apart from being interesting from a mathematical point of view, random-matrix theory (RMT) was initially used to understand the energy level spacing of heavy nuclei, measured in nuclear reactions. Stimulated by developments in quantum chaos, which involved the discovery of (1) the universal description of chaotic systems in terms of the Wigner-Dyson ensemble, and (2) a relation between the universal properties of large random matrices and universal conductance fluctuations (UCF) in disordered conductors, random-matrix theory found widespread applications in quantum dots and disordered nanowires.

Here, we are concerned with the transport properties of mesoscopic systems -- an intermediate scale between the microscopic and macroscopic. These systems are small enough that their quantum mechanical phase-coherence is preserved, but large enough that a statistical description is complete. Universal transport properties are independent of the microscopic description of the system, such as potential landscape, disorder strength and system size. Random matrix-theory links these universal features with the universality of correlation functions of transmission eigenvalues, dependent only on the presence or absence of certain symmetries. This approach is powerful, since transmission matrix determines every linear-response transport statistic, and not just the conductance. Furthermore, this is a non-perturbative theory, and hence provides a unified description of both the metallic and localized phases. Percepts from the theory received experimental verification in chaotic mesoscopic systems based in GaAs/AlGaAs heterostructures.

Topological Order

The twenty-first century witnessed the birth of topological order, and it was discovered that condensed matter systems with an excitation gap could have phase transitions without a spontaneously broken symmetry. These transitions were characterized by a topological invariant. Fig. 1 shows the spectrum of a Bogoliubov-deGennes (BdG) Hamiltonian H(θ) describing a superconductor, with the phase determined by the ground state fermion parity. Here, H(θ) = θ*H₀ + (1- θ)*H₁, with particle-hole symmetric matrices H_0,1, and a real parameter θ that could, for example, model an external magnetic field. A change in the topological invariant P indicates a phase transition. The discovery of topological phases has generated substantial interest in the community, with the 2016 Nobel prize in physics being awarded for theoretical work on the quantum Hall effect and topological phase transitions, paving the way for current work on topological insulators - a phase which is insulating in the bulk but supports chiral conduction on the edges. There is a significant experimental effort towards realizing topological superconductivity, which is predicted to host Majorana bound states (MBSs) - an exotic quasiparticle with non-Abelian exchange statistics. These zero modes have been proposed as the building blocks of a fault-tolerant topological quantum computing platform. The MBSs are described by a real (self-adjoint) wavefuction (field operator), thus rendering the scattering matrix real orthogonal, rather than complex unitary. In this report, we discuss the extension of RMT to account for topological properties.

Random-Matrix Theory of Majorana Fermions and Topological Superconductors

Fig. 2: Probability distribution of the conductance for (a) N=3 and (b) N=4 modes, in class D, with (ν=1, red) and without (ν=0, blue) a zero mode, following Beenakker. [2] The curves are obtained by integrating the probability distribution of Andreev reflection eigenvalues, and hence are the results from random-matrix theory. (Source: P. Sriram)

Random-matrix theory was initially studied within Dyson's three-fold way where the statistics depended on whether or not time-reversal and spin-rotation symmetry were broken. A mean-field description of superconductivity is particle-hole symmetric, which is governed by an antiunitary charge-conjugation operator. Together with a fake time-reversal symmetry for real Hamiltonians, a chiral symmetry emerges, which is governed by anticommutation with a unitary operator. Taking these into account and eliminating any global symmetries results in a tenfold-way classification of symmetry classes for single-particle Hamiltonians. Three of these classes (D, BDI, and DIII) support Majorana zero modes, and the topological invariants are uniquely determined from the scattering matrix. Chaotic transport corresponds to a circular ensemble of scattering matrices, corresponding to a uniform distribution in a symmetry-dictated subgroup of the unitary group. The properties of the scattering matrix and the topological invariant for each of the three ensembles are summarized in Table 1.

Random-matrix theory of quantum transport addresses the following questions -- (1) What is the ensemble of scattering matrices, and (2) How are transport properties obtained from this? These questions are answered by studying the statistical properties of transmission eigenvalues of nanowire geometry, proximitized by an s-wave superconductor. This forms the simplest setup to study topological superconductivity with Majorana bound states at the ends of the nanowire.

Experimental signatures based on an s-wave proximitized Rashba nanowire are reviewed, where a tunneling barrier renders the reflection matrix distribution non-uniform, given by a Poisson kernel of the circular ensemble. Fig. 2 plots the probability distribution of the conductance for class D, for N=3 and N=4 modes in the topologically trivial, and non-trivial regimes. Precepts from random-matrix theory can be used to probe non-distinguishable transport signatures of the trivial and non-trivial phases.

© Praveen Sriram. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] C. W. J. Beenakker, "Random-Matrix Theory of Quantum Transport," Rev. Mod. Phys. 69, 731 (1997).

[2] C. W. J. Beenakker, "Random-Matrix Theory of Majorana Fermions and Topological Superconductors," Rev. Mod. Phys. 87, 1037 (2015).