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PHYSICAL REVIEW B 102 , 195152 (2020) Mesoscopic conductance fluctuations and noise in disordered Majorana wires Daniil S. Antonenko , 1 , 2 , 3 Eslam Khalaf, 4 Pavel M. Ostrovsky, 5 , 2 and Mikhail A. Skvortsov 1 , 2 1 Skolkovo Institute of Science and Technology, Moscow 121205, Russia 2 L. D.

Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia 3 Moscow Institute of Physics and Technology, Moscow 141700, Russia 4 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 5 Max-Planck-Institut fьr Festkцrperforschung, 70569 Stuttgart, Germany (Received 22 July 2020; revised 28 October 2020; accepted 28 October 2020; published 30 November 2020) Superconducting wires with broken time-reversal and spin-rotational symmetries can exhibit two distinct topological gapped phases and host bound Majorana states at the phase boundaries.

When the wire is tuned to the transition between these two phases and the gap is closed, Majorana states become delocalized leading to a peculiar critical state of the system. We study transport properties of this critical state as a function of the length L of a disordered multichannel wire. Applying a nonlinear supersymmetric sigma model of symmetry class D with two replicas, we identify the average conductance, its variance, and the third cumulant in the whole range of L from the Ohmic limit of short wires to the regime of a broad conductance distribution when L exceeds the correlation length of the system.

In addition, we calculate the average thermal shot noise power and variance of the topological index for arbitrary L . The general approach developed in the paper can also be applied to study combined effects of disorder and topology in wires of other symmetries. DOI: 10.1103/PhysRevB.102.195152 I. INTRODUCTION Disorder, inevitably present in experimental realizations, affects the above picture in several ways. First, it may shift the position of the border between the topological phases [ 17 ]. Second, it breaks translational invariance, compromis- ing classification of topological phases in the momentum representation.

Nevertheless, for a given disorder realiza- tion it is still possible to distinguish topological phases by analyzing the real-space transport properties. The topologi- cal invariant for class-D quantum wires can be expressed in terms of the matrix r of quasiparticle reflection amplitudes as q = sgn det r [ 18 ]. Hence, right at the transition between the topological phases, the wire has a fully open channel with unit transmission. The third complication introduced by disorder is that the topological invariant q depends not only on the mean value of disorder strength but on its particular realization.

As a result, for a given wire length L and for a fixed mean value of dis- order strength, the topological number becomes a statistically distributed variable [ 17 ]. Its mean q gradually varies from 1 to 1 by changing the control parameter μ , which drives the transition from the topological to trivial phase. It is only in the thermodynamic limit L when the transition be- tween these phases becomes a sharp quantum phase transition. This happens due to complete Anderson localization of both topologically trivial and nontrivial insulating states on the two sides of the transition.

Direct experimental observation of quasiparticle transport in quantum wires of class D can be problematic due to shunt- ing effects of the superconducting condensate. A possible way to address quasiparticle dynamics is by measuring thermal rather than electrical transport properties [ 19 ]. The thermal Topological insulators hosting gapless excitations at their boundary have been the subject of intense studies during the last two decades [ 1 ]. One of the most fascinating features of topological materials is the possibility of observing Majorana fermions in a solid state setup.

Manipulation of topologi- cally protected Majorana bound states is believed to be a promising platform for quantum computation and informa- tion processing, since it may overcome the decoherence issue in conventional qubits [ 2 ]. Several realizations of Majorana fermions have been proposed, including vortex bound states in p -wave superconductors [ 3 ], Kitaev chain [ 4 ], and a semi- conducting wire proximized with an s -wave superconductor [ 5 , 6 ]. The latter approach turned out to be the most suitable for experimental implementation, and a number of publica- tions reporting observation of Majorana states in proximized quantum wires have appeared in recent years [ 7 13 ].

Majorana fermion formation requires a superconductor with broken time-reversal and spin-rotation symmetries, a system of symmetry class D in the classification of Ref. [ 14 ]. In one spatial dimension (quantum wires), this class is charac- terized by Z 2 topological quantum number [ 15 , 16 ], indicating that there exist two topologically distinct phases. Both trivial ( q = 1) and topological ( q = − 1) phases have a spectral gap, which by tuning a control parameter μ is closed and then reopens with a different sign, giving rise to a Majorana mode localized at the phase boundary.

In a clean, translationally invariant system, the topological invariant q can be expressed in terms of the Pfaffians of the Hamiltonian in the center and at the corner of the Brillouin zone [ 4 ]. 2469-9950/2020/102(19)/195152(19) 195152-1 ©2020 American Physical Society

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DANIIL S. ANTONENKO et al. PHYSICAL REVIEW B 102 , 195152 (2020) coefficient T (and so of g ) [ 25 ]: P ( T ) = g 2 1 T 1 T , (1) where g is defined by the value of P ( λ = 0): g 2 2 π var λ = 2 ξ π L . (2) Equation ( 1 ) should be corrected at smallest T e 2 / g due to roll-off of P ( λ ). Remarkably, the distribution ( 1 ) formally coincides with the Dorokhov bimodal distribution [ 26 ], which is known to describe transmission eigenvalues density for multichannel disordered wires in the Drude regime, where conductance is a self-averaging quantity. Contrary to that, Eq.

( 1 ) refers to the single-channel case, when conductance strongly fluctuates on the interval 0 g 1, with all its mo- ments g n sharing the scaling of g 1 / FIG. 1. Dependence of the dimensionless conductance g of a wire of length L ξ on a control parameter μ for a number of dis- order realizations (sketch). Conductance reaches its maximal value g = 1 at μ v λ / L (see discussion in the text). The peak has a width that scales as 1 / L , while peak centers are distributed with a typical width scaling as 1 / L . L .

In particular, for the variance var g = ⟨ g 2 g 2 one finds conductance G of a mesoscopic system can be conveniently expressed in units of thermal conductance quantum G 0 = π k 2 B T / 6Ї h . The corresponding dimensionless conductance g = G / G 0 is then given by the standard Landauer formula as a sum of transmission probabilities: g = n T n . Influence of potential disorder on the thermal conductance in Majorana wires was studied in a number of theoretical works [ 17 , 18 , 20 22 ], both analytically and numerically.

The simplest description of quasiparticle transport in disor- dered quantum wires of class D in the vicinity of the topolog- ical transition is achieved for a single propagating transverse mode ( one-channel case ) [ 18 , 23 , 24 ]. Its low-energy physics is governed by a one-dimensional (1D) random-mass Dirac Hamiltonian H = − i v σ z x + [ μ + m ( x )] σ y , where σ i are Pauli matrices, μ is a control parameter, and a position- dependent m ( x ) fluctuates around zero mean (we assume it to be short correlated).

For this model, the Lyapunov exponent λ , which determines the reflection coefficient r = tanh λ and dimensionless conductance g = T = 1 / cosh 2 λ , undergoes a drifted random walk with the increase of the wire length L : λ = μ L / v + λ , where λ = L 0 dx m ( x ) / v . Due to the central limit theorem, λ is a normally distributed random variable that fluctuates around zero with the variance var λ = L /ξ , where ξ = 2 l is the disorder-dependent correlation length , l is elastic mean free path, and the factor of 2 accounts for the presence of two counterpropagating modes.

In an infinite system, the fluctuating component λ becomes irrelevant, and the transition between the topological phases takes place right at μ = 0. For a finite system, the topological transition be- comes smeared, see Fig. 1 . For a given disorder realization m ( x ), ideal transmission takes place at μ v λ / L , with the conductance decaying from unity at μ μ v / L . Sample- to-sample fluctuations of μ are characterized by var μ = v 2 / L ξ . Below we will be mainly interested in the critical regime realized for μ = 0, when the drift term for the Lyapunov exponent vanishes and the localization length ξ loc = v / | μ | diverges.

In this case, random walk for λ results in the zero- centered normal distribution P ( λ ), which in the limit L ξ translates to the following distribution of the transmission var g = 2 3 g . (3) The absence of self-averaging can be also seen by comparing the scaling of g with the stretch-exponential decay of the typical conductance g typ = exp log g exp( 4 L / 2 πξ ). Note that at the critical point considered, half of disorder realizations belongs to the trivial phase and another half be- longs to the topological phase. At large L , the majority of realizations have a small conductance g 1.

However, for relatively rare configurations (probability decreases as 1 / L ) the sample is close to the phase transition with g 1, and these very configurations determine the average conductance g 1 / L , as well as all its higher moments. To understand the scaling g 1 / L , note that in the critical regime ( μ = 0) the probability to get such near-critical configuration in the process of disorder sampling can be assessed as the ratio of the single peak width 1 / L to the width of the peak centers distribution 1 / L (see Fig. 1 ).

Tuning the control parameter μ away from the transition puts all realizations to the same topological phase, with an exponentially small conductance in the long-wire limit. Similar outcomes were obtained in Ref. [ 27 ] with the help of real-space strong-disorder renor- malization group and transfer matrix approaches. In the multichannel case , localization in quantum wires of class D was considered in Refs. [ 20 , 21 ] within the Dorokhov- Mello-Pereyra-Kumar (DMPK) approach [ 26 , 28 ].

Solving the Fokker-Planck equation for the distribution function of trans- mission eigenvalues T n = 1 / cosh 2 λ n in the long-wire limit leads to the usual “crystallization” of the Lyapunov exponents λ n [ 29 ]. However, the peculiarity of class D fine tuned to the critical point (the opposite case requires a special treatment [ 21 ]) is that the lowest exponent λ 1 is distributed normally near zero (ideal transmission) with the variance var λ 1 = L /ξ , where ξ = 2 Nl is the correlation length of the N -channel wire (compare with the N = 1 case considered above).

With the contribution of other channels being exponentially sup- pressed, the conductance g = T 1 is determined by the most transparent channel, making this regime completely analo- gous to the one-channel model discussed above. Hence, in the limit L ξ , Eqs. ( 1 ) and ( 2 ) are applicable for multichannel wires as well, with T = T 1 . Such a behavior is also observed 195152-2

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MESOSCOPIC CONDUCTANCE FLUCTUATIONS AND NOISE PHYSICAL REVIEW B 102 , 195152 (2020) 2.5 0.14 2.0 0.12 1.5 0.08 0.10 1.0 0.06 0.04 0.5 0.02 0.5 1.0 1.5 2.0 2.5 3.0 3.5 10 20 30 40 FIG. 2. Disorder-averaged conductance of quantum wires of class D at criticality as a function of the wire length L . Dashed lines show the leading short- and long-wire asymptotics. FIG. 3. Dependence of the conductance variance for class D at criticality on the wire length L . Dashed lines show the short- and long-wire asymptotics. Calculation of all those quantities requires averaging of four Green’s functions that cannot be done within the one-replica supersymmetric sigma model considered in Refs.

[ 17 , 22 ], forcing us to consider a two-replica ( n = 2) version of the supersymmetric sigma model. The heat kernel for the latter is obtained with the use of the Iwasawa decomposition trick [ 40 42 ]. The main technical achievement of this paper is classifica- tion of radial eigenfunction of a higher-rank (several replicas) supersymmetric sigma model. We find that the straightfor- ward approach for their construction outlined in Refs. [ 40 42 ] produces an incomplete basis due to vanishing of Grass- mann integration of “too symmetric” Iwasawa wave functions.

This problem is solved by including additional subfamilies of eigenfunctions with a smaller amount of quantum numbers, which are intimately related to the radial eigenfunctions of the sigma model with a smaller number of replicas. This finding is expected to be relevant for the heat kernel construction for all higher-rank ( n > 1) sigma models of arbitrary symmetry classes. The paper is organized as follows. In Sec. II we summa- rize new physical results obtained in this work. In Sec. III we introduce the main mathematical ingredients required for construction of the heat kernel of the supersymmetric sigma model. The outlined procedure for class D with two replicas is implemented in Sec.

IV , where we introduce an additional subfamily of radial eigenfunctions and discuss the behavior of eigenfunctions at particular lines needed to extract their normalization and behavior at the origin. The final expressions for the conductance variance, its third cumulant, shot-noise power, and average square of the determinant of the reflection amplitude matrix are obtained in Sec. V . The technique devel- oped and results obtained are discussed in Sec. VI . Important technical details are relegated to numerous Appendixes. in other topological classes tuned to a critical regime in 1D, a phenomenon called “superuniversality” in Ref. [ 21 ].

A weak point of the DMPK approach is that it allows to calculate transport properties in the crossover between the Drude and localization regimes only in special cases. This happens when the underlying Fokker-Planck equation can be solved with the help of the Sutherland transformation: in class A [ 30 32 ], CI and DIII [ 20 , 33 ], and AIII [ 34 ]. In order to trace the dependence of conductance mo- ments on the wire length L for class D, one should resort to a complimentary sigma-model technique [ 35 ]. The average conductance g ( L ) was calculated in this way in Refs.

[ 17 , 22 ] (the difference from our result ( 59 ) by an overall numerical factor is due to an apparently different normalization of the dimensionless conductance [ 36 ]). A striking feature of the symmetry class D is that the target space of the supersym- metric sigma model consists of two connected components [ 37 , 38 ]. Remarkably, localization happens only if one al- lows for jumps (domain walls) between the two components [ 38 , 39 ].

Such processes are described by an additional term in the sigma-model action proportional to log χ , where χ is the so-called kink fugacity [ 17 ], characterizing the deviation from the critical regime and thus resulting in a finite localization length ξ loc = 2 ξ/ χ 2 [ 17 ]. In the following we will focus exclusively on the critical regime with completely suppressed kinks χ = μ = 0). This regime corresponds to the DMPK approach of Ref. [ 20 ] dis- cussed above, which is characterized by an algebraic decay of the average conductance at L ξ given by Eq. ( 2 ) and a rather involved analysis is required to see localization in the DMPK framework [ 21 ].

The critical regime can be defined in terms of physical quantities at arbitrary L as the regime when the average determinant of the reflection matrix is zero: det r 0. The dependence of g ( L ) for class D at criti- cality obtained in Ref. [ 17 ] is shown in Fig. 2 . It smoothly interpolates between the Drude regime with g ξ/ L to the long-wire regime with g given by Eq. ( 2 ). In the present paper we make a next step in the analysis of quasiparticle transport in multichannel quantum wires of class D at criticality and derive exact expressions for the conductance variance var g , its third cumulant, average noise power, and var det r at arbitrary L /ξ in the diffusive regime.

The results of our study are presented graphically in Figs. 3 6 depicts the dependence of the conductance variance on the wire length, illustrating the crossover behavior from the value of var g = 2 / 15 at L ξ in the Drude regime (universal conductance fluctuations (UCF) [ 43 ]) to the superuniversal limit var g = (2 / 3) g at L ξ , as given by Eq. ( 3 ). Contrary II. SUMMARY OF RESULTS 195152-3

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DANIIL S. ANTONENKO et al. PHYSICAL REVIEW B 102 , 195152 (2020) 0.15 0.5 1.0 1.0 0.5 0.0 0.8 0.10 2.0 1.5 1.0 0.6 0.05 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.4 0.2 10 20 30 40 FIG. 4. Dependence of the third cumulant g 3 of the conduc- tance for class D at criticality on the wire length L . Inset: Short-wire part of the dependence. Dashed lines show the short- and long-wire asymptotics. 5 10 15 20 0.0 FIG. 6. Length dependence of the average square of the determi- nant of the reflexion matrix (det r ) 2 var det r . The fact that the curve approaches the asymptotic value of 1 indicates that for every sample det r = ± 1 in the limit L .

to a featureless dependence of g with the crossover around L /ξ 1 (see Fig. 2 ), the dependence of var g on L exhibits a reentrant behavior, with the crossover being strongly dis- placed towards larger wire lengths L /ξ 15. Figure 4 depicts the dependence of the third cumulant of conductance g 3 g 3 3 g 2 g 2 g 3 on the wire length. This quantity determines the asymmetry of the con- ductance distribution (skewness) about its average. In the quasiclassical limit L ξ , the third cumulant is g 3 ( L /ξ ) 2 with a very small numerical coefficient. At longer distances it changes sign twice before approaching the asymp- totic dependence ξ/ L at L 40.

The two-replica sigma model also allows one to calculate the thermal shot noise power, which corresponds to counting quasiparticles irrespective of their charge and differs from a usual electrical shot noise power. We will characterize the former by the pseudo-Fano factor is typical for diffusive metals, where Dorokhov distribution describes transmission eigenvalues density and Fano factor is a self-averaging quantity due to the aggregated contribution of many channels. Contrary to that, for long wires, transport is provided by only one mode with the lowest Lyapunov exponent. Since it is described by the same Eq.

( 1 ), we arrive at the same value of F = 1 / 3, with the actual Fano factor F exhibiting strong sample-to-sample fluctuations. Yet another way to characterize topological properties of the wire is to study moments of the determinant of the reflection amplitudes det r . Though different from the true topological number q = sgn det r , this quantity shows similar behavior of interpolating between ± 1 as the control parameter is driven across the phase transition. Its average value det r as a function of L and kink fugacity χ was calculated in Ref. [ 17 ].

Vanishing right at the critical line χ = 0, det r flows to ± 1 with increasing L for any finite bare kink fugacity χ , in a sense similar to the renormalization-group flow of σ xy in the integer quantum Hall effect [ 44 , 45 ]. Here we calculate the second moment of the determinant det 2 r , and demonstrate that it does not vanish at the critical line (where det r 0). With the increase of the wire length L , it interpolates between 0 (short wires) and 1 (long wires), as shown Fig. 6 . Taking into account that | det r | 1 due to unitarity of quantum mechanics, one concludes that in the limit L , det r takes values + 1 and 1 for any specific sigma-model perspective:

all of the samples are F = n T n (1 T n ) n T n , (4) which is determined by the ratio of the average transport moments instead of the average ratio. The length dependence of F is shown in Fig. 5 . Remarkably, it equals 1 / 3 both in short- and long-wire limits, as both of them are characterized by the Dorokhov distribution ( 1 ). For short wires, the situation 0.4 0.3 0.2 0.1 outline the main steps to- in the sigma-model for- mathematical concepts. Our 40 42 ] and implemented for cal- culation of g in quantum wires of class D in Refs. [ 17 , 22 ].

The peculiarity of class D is that calculation of g 2 already requires the use of the sigma model with two replicas ( n = 2), which significantly complicates the whole analysis. 2 4 6 8 10 FIG. 5. Length dependence of the pseudo-Fano factor ( 4 ), which determines the average thermal shot-noise power of the quasiparticle current. Dashed lines show the short- and long-wire asymptotics. 195152-4 sample with equal probabilities. That about the large-L behavior from a s even in the critical regime, almost insulating and topological/trivial in e III.

MATHEMATICAL PR In this preparatory section we ou wards calculation of ⟨g⟩, ⟨g2⟩, and ⟨g3 malism and introduce necessary mat analysis closely follows a pathway de

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