Advances in Mathematical Physics

Volume 2018, Article ID 4016394, 11 pages

https://doi.org/10.1155/2018/4016394

## Heat Transport and Majorana Fermions in a Superconducting Dot-Wire System: An Exact Solution

Departamento de Física, Laboratório de Física Teórica e Computacional, Universidade Federal de Pernambuco, 50670-901 Recife, Pernambuco, Brazil

Correspondence should be addressed to A. M. S. Macêdo; moc.liamg@odecam.solirum.a

Received 16 August 2018; Accepted 28 October 2018; Published 2 December 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 Oscar Bohórquez and A. M. S. Macêdo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We obtain exact expressions for the first three moments of the heat conductance of a quantum chain that crosses over from a superconducting quantum dot to a superconducting disordered quantum wire. Our analytic solution provides exact detailed descriptions of all crossovers that can be observed in the system as a function of its length, which include ballistic-metallic and metallic-insulating crossovers. The two Bogoliubov-de Gennes (BdG) symmetry classes with time-reversal symmetry are accounted for. The striking effect of total suppression of the insulating regime in systems with broken spin-rotation invariance is observed at large length scales. For a single channel system, this anomalous effect can be interpreted as a signature of the presence of the elusive Majorana fermion in a condensed matter system.

#### 1. Introduction

Random-matrix theory (RMT) has been widely used in the study of phase-coherent complex quantum systems and has been particularly successful in uncovering universal properties of quantum transport in chaotic and disordered systems [1]. Much of the success of RMT in quantum transport is due to the strong correspondence between the statistical properties of random-matrix ensembles and the fluctuations of measured observables of complex quantum systems as a function of some control parameter, such as energy or magnetic field. The universality of transport properties, such as the moments of the conductance, lies in their independence of microscopic details of the scattering source. Nevertheless, random-matrix ensembles are sensitive to certain intrinsic symmetries of the system, such as time-reversal (TR), spin-rotation (SR), electron-hole (e-h), and chiral (Ch). It has been established that these symmetries lead to a classification of RMT ensembles into ten universal classes (the tenfold way) [2, 3], which are divided into three categories: (i) Wigner-Dyson (WD, three classes), appropriate to describe normal disordered conductors, (ii) chiral (Ch, three classes), appropriate for systems with a purely off-diagonal disorder, and (iii) Bogoliubov-de Gennes (BdG, four classes), appropriate for normal-superconducting (NS) hybrid systems.

From the perspective of quantum transport, phase-coherent mesoscopic systems can be classified into two types: (I) disordered conductors, in which impurities generate multiple elastic scatterings with an associated mean free path that is less than the system’s dimensions, and (II) ballistic cavities, where is greater than the system’s dimensions and thus the dominant scattering mechanism is reflection at the border of the cavity. Remarkably, RMT can efficiently describe both types of systems since universal transport properties do not depend on the details of either the impurity potential (for disordered systems) or the shape of the cavity (for ballistic systems) [1]. This insensitivity to microscopic details goes as far as to allow an identification, under certain conditions, of a multichannel disordered quantum wire with a chain of ballistic cavities [4].

Besides RMT, there are other well-developed approaches to quantum transport in both disordered wires and chaotic ballistic cavities: the field-theoretic nonlinear sigma model [5] and the trajectory-based semiclassical approach [6] are the most well known. These three approaches have many advantages and pitfalls, but since they are constructed from different and somewhat unrelated statistical hypothesis on the behavior of the underlying degrees of freedom, they may be considered a complementary, albeit equivalent, physical description of the system. Notwithstanding, a full fledged mathematical proof of the equivalence of these three approaches is still missing, in spite of much effort and some successes in particular systems, such as quantum dots with ideal couplings to external leads [7].

In superconducting systems, quantum transport has very striking and different features in comparison with their normal counterpart, which in part is due to what is known as Andreev reflections (AR). The most remarkable phenomenon is probably the possibility of a condensed matter realization of Majorana fermions as protected bound states at the ends of topological superconducting wires [8]. Transport observables in these systems, such as the electrical conductivity, can give information about topological invariants and topological quantum numbers. The thermal conductivity, on the other hand, despite maybe not containing direct information of topological invariants, as can be seen from their random-matrix description [9], can still provide valuable information about topological phase transitions [10]. It was, however, in the study of disordered quantum wires that evidence of the presence of condensed matter Majorana modes emerged most clearly. These can be traced back to the prediction [11] that for quantum wires in the chiral classes (for odd open scattering channels) and in the superconducting D and DIII classes there is no exponential localization, since, unlike its behavior in the standard classes, the average conductance falls off in the limit of long distances as , which is a kind of super-ohmic behavior. They also found that in these special classes the average density of states (DOS) diverges logarithmically, , as energy (where and the Fermi velocity has been set to unity). A similar singularity had already been found by Dyson in the analysis of disordered linear chains [12]. On the other hand, [13] found through a general analysis using strong-disorder renormalization group (RG) that systems of classes D and DIII exhibit localization and an average DOS that vanishes as a power law with , as . Remarkably, at certain critical points obtained by fine-tuning the disorder, both delocalization and Dyson’s divergence can be present. The authors of [13] were also among the first to relate this special type of criticality to transitions between topological phases and also to point out that it could be a signature of the existence of Majorana zero modes. Reference [14] confirmed these results and presented evidence that the delocalization at critical points is well described by the DMPK equation of superconductors. Furthermore, they claimed that there may be “superuniversality” combining the chiral class and the superconducting classes D and DIII, since they have certain similar characteristics in regard to these critical points. Later, the authors of [15] argued that the diverging nature of the average density of states at the band center is a signature of topologically protected zero modes bound to point defects.

In this paper we employ random-matrix theory to study two classes of superconducting dot-wire systems. We obtain, in the continuum limit of a quantum chain with TR symmetry, an exact description of the crossover in thermal conduction between a superconducting chaotic ballistic cavity (a quantum dot) and a disordered multichannel superconducting quantum wire. The calculations were guided by a recent classification scheme of RMT Brownian motion ensembles [16] and were performed by means of a multivariate integral transform method proposed in [16, 17]. More specifically, we obtain exact expressions for the first three moments of the heat conductance of two classes of disordered superconducting quantum wires with time-reversal symmetry and with a crossover to a quantum dot in the small length limit. The analytic solution describes in detail various types of crossovers as a function of the systems’ length, which include ballistic-metallic and metallic-insulating crossovers. If the system is realized as a single channel topological superconductor with broken spin-rotation invariance, we can interpret the total suppression of the insulating regime as a signature of the presence of a condensed matter Majorana fermion.

#### 2. The Scattering Problem

Consider a confined quantum system ideally coupled to two electron reservoirs via point contacts with and open scattering channels, respectively. According to the Landauer-Büttiker scattering formalism [18], coherent particle transfer through such a device can be efficiently described by its scattering matrix, which can be generically written aswhere and are reflection matrices and and are transmission matrices. The subscripts denote the matrix dimensions and the matrices and are Hermitian and although they have different spectra, the spectrum of the smaller matrix coincides with the nonzero eigenvalues (transmission eigenvalues) of the bigger one . Transport observables can be conveniently written in terms of these transmission eigenvalues. For instance, the thermal conductance of a superconducting system, at low temperature , is given by [19],where , , and is the spin and/or particle-hole degeneracy.

In ballistic chaotic cavities the transmission eigenvalues are strongly correlated random variables, which because of assumption of ergodic dynamics are well described by RMT. According to RMT, the scattering matrix of a ballistic chaotic cavity with ideal contacts is uniformly distributed over its manifold, and thus the probability density is only restricted by the presence or absence of certain symmetries. For the BdG classes the corresponding joint distribution of transmission eigenvalues is given by [19]where and is a normalization constant. The values of the parameters and are solely determined by the symmetries, as shown in Table II of [19]. Here we shall consider only systems in the presence of TR symmetry, which implies . Moreover, we must set for systems in the presence of SR symmetry and for systems with broken SR symmetry. According to the tenfold way of classifying random-matrix ensembles, these classes are denoted by CI and DIII, respectively. We remark that the system described by (3) differs from a normal ballistic cavity, because, in addition to the two normal contacts coupling to the reservoirs, the cavity is geometrically defined by a normal-superconducting interface, which generates Andreev reflections [1]. As a matter of fact, this type of cavity is also known as an Andreev quantum dot [19]. We remark that if we set in (3) we recover the Wigner-Dyson A class for normal systems with broken TR symmetry, which turns out to be a useful way to compare our exact expressions for the moments of the thermal conductance with known results of the literature.

We proceed by combining Andreev quantum dots in a chain geometry, as shown in Figure 1. In such a setup the excitation gap induced by the proximity effect in the inner region is closed by adjusting the superconducting boundaries to have a phase difference of , which also ensures that there is no breaking of TR symmetry [10, 19]. The RMT description of the system can be obtained by appropriately combining the scattering matrices of the Andreev quantum dots, or from the corresponding product of random transfer matrices; see [20] for more details. An equivalent description using the supersymmetric nonlinear sigma model is also possible [21]. Since we want to obtain exact analytical results, we follow [22, 23] and take the continuum limit which leads to a Fokker-Planck equation for the evolution, with the sample’s length, of the joint probability distribution of transmission eigenvalues with a zero-length initial condition given by the RMT description of an Andreev quantum dot. See [21, 23] for the corresponding problem with normal quantum dots.