Random-matrix theory of Majorana fermions and topological superconductors

The theory of random matrices originated half a century ago as a universal description of the spectral statistics of atoms and nuclei, dependent only on the presence or absence of fundamental symmetries. Applications to quantum dots (artificial atoms) followed, stimulated by developments in the field of quantum chaos, as well as applications to Andreev billiards—quantum dots with induced superconductivity. Superconductors with topologically protected subgap states, Majorana zero modes, and Majorana edge modes, provide a new arena for applications of random-matrix theory. These recent developments are reviewed, with an emphasis on electrical and thermal transport properties that can probe the Majorana fermions.

Figure 1

Quantum-dot Josephson junction formed by a segment of a semiconducting wire (InAs) in a superconducting ring (Al), enclosing a magnetic flux $\mathrm{\Phi }$. A weakly coupled tunnel probe (Au) at bias voltage $V$ measures the excitation spectrum of electron and hole quasiparticles confined to the quantum dot, as a function of the phase difference $\varphi =\mathrm{\Phi }\times 2e/\hslash$ across the junction. Electron micrographs (entire device and enlarged region). From [36].

Figure 2

Model calculation of the excitation spectrum of a Josephson junction similar to Fig. 1, but of larger dimensions in order to increase the flux inside the junction (about $3h/e$ for a disordered normal region of size $1\mu \mathrm{m}\times 2\mu \mathrm{m}$). The spin-orbit coupling length is $l_{\mathrm{so}}=250\mathrm{nm}$, the superconducting gap ${\mathrm{\Delta }}_0=0.4\mathrm{meV}$, and the Fermi energy $E_{\mathit{F}}=2.5\mathrm{meV}$ (corresponding to $N=20$ electronic modes in the nanowire). Level repulsion at nonzero $E$ coexists with level crossings at $E=0$. Here, the number of crossings in a $2\pi$ phase increment is even, but it can be odd in a topological superconductor. Data provided by M. Wimmer (private communication).

Figure 3

Scanning electron microscope image of a device similar to the one studied in [115], where superconductivity is induced in an InSb nanowire by proximity to a NbTiN superconductor. The Au electrodes are normal-metal Ohmic contacts, used to measure the electrical conductance of the wire. Image from V. Mourik (private communication).

Figure 4

Model calculation of a device similar to Fig. 3, with an additional point contact (as shown in the inset) to allow for variation of the number of transmitted modes. The ratio of the spin-orbit coupling energy $E_{\mathrm{so}}$ and superconducting gap ${\mathrm{\Delta }}_0$ is adjusted (at fixed Zeeman energy $E_{\mathit{Z}}=6E_{\mathrm{so}}$), so that the superconductor is in either the topologically trivial phase (${\mathrm{\Delta }}_0/E_{\mathrm{so}}=8$) or the nontrivial phase (${\mathrm{\Delta }}_0/E_{\mathrm{so}}=4$). By varying the Fermi energy inside the point contact, the number $N$ of transmitted electronic modes is varied between 0 and 8. The first conductance plateau in the topologically nontrivial phase remains precisely quantized, notwithstanding the presence of a large amount of disorder in the simulation. Adapted from [174].

Figure 5

Chain of magnetic nanoparticles on a superconducting substrate. If the direction of the magnetization (arrows) varies along the chain, there can be Majorana zero modes at the ends without any spin-orbit coupling in the superconductor.

Figure 6

Right panel: schematic of a quantum dot, created by a gate electrode at the edge of a quantum spin-Hall (QSH) insulator in a perpendicular magnetic field $B$. A current $I$ is passed between metallic and superconducting contacts, and the differential conductance $dI/dV$ is determined as a function of the bias voltage $V$. Results of a model calculation for an InAs/GaSb quantum well are shown in the left panel. The Majorana zero mode produces a resonant peak at $V=0$, which survives the average over disorder realizations. Adapted from [20].

Figure 7

Andreev billiard on the surface of a topological insulator. A sign change of the superconducting pair potential $\pm {\mathrm{\Delta }}_0$ closes the excitation gap inside the billiard without breaking time-reversal symmetry. The statistics of the thermal conductance between two metal electrodes at temperature difference $\delta T$ is governed by a random-matrix ensemble in symmetry class DIII (in zero magnetic field), in class BDI (nonzero field at the Dirac point), or class D (nonzero field away from the Dirac point). Adapted from [46].

Figure 8

Chiral $p$-wave superconductor with two domains of opposite chirality. Arrows indicate the direction of propagation of the Majorana edge modes. If the two modes along the domain wall are uniformly mixed, the statistics of the thermal conductance is determined by the probability distribution of a $2\times 2$ orthogonal random matrix.

Figure 9

Excitation spectrum (solid curves) of two magnetic particles with an angle $\theta =70°$ between their magnetic moments, calculated from the Hamiltonian (1) at fixed ${\mu }_0=|m_n|=2{\mathrm{\Delta }}_0$ as a function of the hopping energy $t_0$. The level crossings at $E=0$ do not split because the ground states at the two sides of the crossing differ in fermion parity $\mathcal{P}$ [dashed curve, calculated from Eq. (6)]. Adapted from [41].

Figure 10

Determinant of the reflection matrix at one end of a chain of magnetic nanoparticles [Hamiltonian from Eq. (1)], ensemble averaged over random and uncorrelated orientations of the magnetic moments (with fixed magnitude $|m_n|=2t_0$ and ${\mathrm{\Delta }}_0=0.9t_0$). The transition into the topologically nontrivial phase becomes sharper with increasing length $M$ of the chain. Adapted from [41].

Figure 11

Dependence of the (inverse) compressibility (13) on the pairing strength $g_0$ of the Kitaev chain, for periodic boundary conditions ($+$ data points) and for antiperiodic boundary conditions ($\circ$ data points), calculated from the particle-number conserving Hamiltonian (b10) for $\mathcal{N}=2N=512$, $L=2048$. The dashed curves result in the thermodynamic limit $\mathcal{N},L\to \infty$ at fixed $\rho =\mathcal{N}/L=0.25$. The inset shows the mean-field phase diagram. Adapted from [122].

Figure 12

(a),(b) The spectrum of a vortex core in a class-D superconductor. The $\pm E$ symmetric spectrum may or may not have an unpaired Majorana zero mode, pinned to $E=0$. The ensemble-averaged density of states (32) is plotted in (c). The delta-function contribution from the zero mode is accompanied by a dip in the smooth part ${\rho }_{-}$ of the density of states. Without the zero mode, there is a midgap spectral peak ${\rho }_{+}$. The density of states in symmetry class C is given by ${\rho }_{-}$ without the zero mode contribution.

Figure 13

Ensemble-averaged density of states in the four Altland-Zirnbauer ensembles, calculated numerically for Hamiltonians of dimensionality $\mathcal{N}=60$ in class C and CI, $\mathcal{N}=60+\nu$ in class D, and $\mathcal{N}=120+2\nu$ in class DIII. [Analytical expressions in the large-$\mathcal{N}$ limit are collected in [83], where class D, $\nu =1$ is called “class B.”] The delta-function singularity of the zero mode for $\nu =1$ is not plotted. Except for class D with $\nu =0$, the density of states vanishes at the Fermi level as $|E{|}^{{\alpha }_E+\nu {\beta }_E}$. Adapted from [112].

Figure 14

Superconducting wire (S) connected at both ends to a normal-metal contact (N), in a geometry similar to Fig. 3. Majorana zero modes may appear at the two NS interfaces. The current $I$ flowing from the normal metal (at voltage $V$) into the grounded superconductor gives the electrical conductance $G=I/V$, determined by the Andreev reflection matrix $r_{he}$.

Figure 15

Probability distribution of the electrical conductance for $N=3$ modes, in the circular ensemble of class D (solid curves) and class BDI (dashed curves), either without any Majorana zero modes ($\nu =0$) or with one zero mode ($\nu =1$). The curves follow by integration of the probability distribution (60) of the Andreev reflection eigenvalues. The histograms are the results of a microscopic calculation for the Rashba-Zeeman Hamiltonian (54), in a three-mode nanowire of width $W=l_{\mathrm{so}}=100\mathrm{nm}$ (in class D, but close to the crossover into class BDI at $W\lesssim l_{\mathrm{so}}/2$). Adapted from [19].

Figure 16

Disorder-averaged differential conductance as a function of bias voltage, calculated for the nanowire shown in the inset. (The solid vertical line indicates the position of the tunnel barrier, transmission probability $T=0.4$ per mode; disordered regions are dotted.) The $\nu =0$ curve is for a weak magnetic field ($E_{\mathit{Z}}=2.5E_{\mathrm{so}}$), when the system is topologically trivial and the zero-bias peak is due to weak antilocalization. (The corresponding peak for a single disorder realization is shown in Fig. 19.) The $\nu =1$ curve, in a stronger magnetic field ($E_{\mathit{Z}}=10.5E_{\mathrm{so}}$), shows the Majorana resonance in the topologically nontrivial regime. Adapted from [125].

Figure 17

Example of a phase-conjugate series of scattering events responsible for the weak antilocalization effect in an NS junction. The same loop is traversed once as an electron ($e$) and once as a hole ($h$), with an intermediate Andreev reflection. Since electron and hole encircle the same flux $\mathrm{\Phi }$ with opposite charge, the total accumulated phase shift vanishes even though time-reversal symmetry is broken by the magnetic field. The systematic constructive interference shows up as a zero-bias conductance peak that survives a disorder average. Adapted from [125].

Figure 18

Amplitude $\delta G$ of the average zero-voltage conductance peak as a function of (mode-independent) transmission probability $T$ through the tunnel barrier, calculated numerically for the class-D circular ensemble. The dashed and solid curves represent, respectively, the topologically trivial and nontrivial superconductor. The dash-dotted curve is the topology-independent large-$N$ limit (65). Adapted from [125].

Figure 19

Voltage and magnetic-field dependence of the conductance in the same nanowire as in Fig. 16, but now for a single disorder realization. The magnetic-field range is in the topologically trivial regime, without Majorana zero modes. The conductance peak indicated by the dotted line is pinned to zero voltage over a range of magnetic-field values because of the accumulation of reflection matrix poles on the imaginary axis in a class-D superconductor; see Fig. 20. Adapted from [125].

Figure 20

Reflection matrix poles $ϵ=E-i\gamma$ in the two Altland-Zirnbauer ensembles with broken time-reversal symmetry (parameters $\mathcal{N}=500$, $N=25$, $T=0.2$). Only a narrow energy range near $E=0$ is shown, to contrast the accumulation of the poles on the imaginary axis in class D and the repulsion in class C. Adapted from [112].

Figure 21

Distribution of the transmittance of a Majorana edge mode in a chain of parallel nanowires on a superconducting substrate. The histograms are calculated numerically for a model Hamiltonian of an anisotropic $p$-wave superconductor, for different lengths $L$ of the edge. Dashed lines show the analytical result (73), with the mean free path $\ell$ as the single fit parameter. With increasing $L$, a bimodal distribution evolves, which produces a slow $1/\sqrt{L}$ decay of the shot noise power $P_{\text{shot}}=\frac{1}{2}T(e^3{V}_2/h)$. Adapted from [54].

Figure 22

Thermal conductance and determinant of reflection matrix of a clean and disordered superconducting nanowire. The curves are calculated for the Rashba-Zeeman Hamiltonian (54), with parameters $W=L/10=l_{\mathrm{so}}$, ${\mathrm{\Delta }}_0=10E_{\mathrm{so}}$, and $E_{\mathit{Z}}=10.5E_{\mathrm{so}}$. The number of propagating modes $N$ varies from 0 to 4 in the range of Fermi energies shown. The thermal conductance $\kappa =I_{\text{heat}}/\delta T$ gives the heat current between the two contacts at temperature $T_0$ and $T_0+\delta T$. Adapted from [3].

Figure 23

Thermal conductance of a five-mode superconducting nanowire, calculated for two disorder strengths (solid and dashed curves) without chiral symmetry (class D) and with chiral symmetry (class BDI). When two peaks merge, $\kappa$ drops below the quantized value ${\kappa }_0$ in class D but rises above it in class BDI. In either symmetry class, the isolated conductance peaks have the same line shape (77). Adapted from [69]

Figure 24

Thermal and electrical transport by chiral edge modes. (a) The chiral Majorana modes at the edge of a topological superconductor, with either spin-triplet $p$-wave or spin-singlet $d$-wave pairing. The shaded strip at the center represents a disordered boundary between two domains of opposite chirality of the order parameter. (b) The chiral edge modes of the quantum Hall effect in graphene, where modes of opposite chirality meet at a bipolar junction between electron-doped and hole-doped regions. (c) The probability distributions (82) of the conductance in the corresponding circular ensembles are plotted. (a) From [146]; (b) from [1]; and (c) from [46].

Figure 25

Geometry to measure the thermopower $\mathcal{S}$ of a semiconductor quantum dot (mean level spacing ${\delta }_0$) coupled to chiral Majorana modes at the edge of a topological superconductor. A temperature difference $\delta T$ induces a voltage difference $V=-\mathcal{S}\delta T$ under the condition that no electrical current flows between the contacts. For a random-matrix theory, we assume that the Majorana modes are uniformly mixed with the modes in the point contact, via quasibound states in the quantum dot. Adapted from [107].

Figure 26

Probability distribution of the dimensionless thermopower in symmetry class C (solid curve, bottom and left axes), and in class D (dashed curve, top and right axes). These are results for the quantum dot of Fig. 25 connecting a single-channel point contact to the Majorana edge modes of a chiral superconductor. Adapted from [107].

Figure 27

Thermal conduction through an Andreev billiard on the surface of a topological insulator. Multiple Majorana zero modes are stabilized by chiral symmetry, when the Fermi level lines up with the Dirac point joining the two cones of the band structure (marked by a dot in the left inset).

Figure 28

Probability distribution of the dimensionless thermal conductance $\kappa /{\kappa }_0=T_1+T_2$ in an Andreev billiard, calculated from Eqs. (94) and (95) in symmetry class D (only particle-hole symmetry) and class BDI (particle-hole with chiral symmetry). In class D, there is no dependence on the number $\nu$ of Majorana zero modes, while in class BDI there is.

Figure 29

Excitation spectrum of an InSb Josephson junction, calculated using the Rashba-Zeeman Hamiltonian (54) in a weak parallel magnetic field [(a) Zeeman energy $V_{\mathit{Z}}=0.35\mathrm{meV}$] and in a stronger field [(b) $V_{\mathit{Z}}=0.52\mathrm{meV}$]. The other parameters are the same in (a) and (b) (${\mathrm{\Delta }}_0=0.4\mathrm{meV}$, $E_{\mathit{F}}=2.4\mathrm{meV}$, $N=19$). The dashed and solid levels $E_{\pm }$ that cross at $E=0$ correspond to different parity $\mathcal{P}=\pm 1$ of the number of electrons in the junction. The phase periodicity of the spectrum at fixed parity is $2\pi$ in (a) (topologically trivial junction) and $4\pi$ in (b) (topologically nontrivial). Data provided by M. Wimmer.

Figure 30

Spacing distribution of level crossings in a Josephson junction. The smooth curve shows the prediction from RMT. The histogram results from a microscopic calculation using the Rashba-Zeeman Hamiltonian (54), for a Josephson junction formed out of a disordered InSb wire in a weak perpendicular magnetic field, sampled over different impurity configurations. The excitation spectrum for one impurity configuration is shown in Fig. 2. This system is topologically trivial ($N_X=4$ crossings in a $2\pi$ phase interval). Adapted from [20].

Figure 31

Discrete vortex in a three-terminal quantum-dot Josephson junction. (a) The geometry. The fluxes ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ produce a $2\pi$ winding of the order parameter within the dashed triangles of (b). The data in (b) and (c) are obtained numerically from the symplectic ensemble of chaotic quantum dots ($N=10$ modes per lead, level spacing $\gg {\mathrm{\Delta }}_0/N$). (b) The energy $E_{\min }$ of the lowest subgap state for a single sample, with level crossings appearing on the colored contour. The ensemble averaged density of states is plotted in (c) for two arrangements of fluxes. The gap closes with a midgap peak when there is a $2\pi$ winding. Adapted from [158].