Since their discovery, it has been suggested that pairing in pnictides can be mediated by spin fluctuations between hole and electron bands. In this view, multiband superconductivity would substantially differ from other systems like MgB 2 , where pairing is predominantly intraband. Indeed, interband-dominated pairing leads to the coexistence of bonding and antibonding superconducting channels. Here, we show that this has profound consequences on the nature of the low-energy superconducting collective modes. In particular, the so-called Leggett mode for phase fluctuations is absent in the usual two-band description of pnictides. On the other hand, when also the repulsion between the hole bands is taken into account, a more general three-band description should be used, and a Leggett mode is then allowed. Such a model, which has been proposed for strongly hole-doped 122 compounds, can also admit a low-temperature s + is phase that breaks the time-reversal symmetry. We show that the (quantum and thermal) transition from the ordinary superconductor to the s + is state is accompanied by the vanishing of the mass of Leggett-like phase fluctuations, regardless the specific values of the interaction parameters. This general result can be obtained by means of a generalized construction of the effective action for the collective degrees of freedom that allows us also to deal with the nontrivial case of dominant interband pairing.

The quasi-bound states of a superconducting quantum dot that is weakly coupled to a normal metal appear as resonances in the Andreev reflection probability, measured via the differential conductance. We study the evolution of these Andreev resonances when an external parameter (such as magnetic field or gate voltage) is varied, using a random-matrix model for the N × N scattering matrix. We contrast the two ensembles with broken time-reversal symmetry, in the presence or absence of spin-rotation symmetry (class C or D). The poles of the scattering matrix in the complex plane, encoding the center and width of the resonance, are repelled from the imaginary axis in class C. In class D, in contrast, a number ∝ √ N of the poles has zero real part. The corresponding Andreev resonances are pinned to the middle of the gap and produce a zero-bias conductance peak that does not split over a range of parameter values (Y-shaped profile), unlike the usual conductance peaks that merge and then immediately split (X-shaped profile). Contribution for the JETP special issue in honor of A.F. Andreev's 75th birthday.

We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to M conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states ρ 0 at the Fermi level. The ensemble average ρ 0 = δdeviates from the bulk value 1/δ 0 by an amount depending on the Altland-Zirnbauer symmetry indices α,β. The divergent average for M = 1,2 in symmetry class D (α = −1, β = 1) originates from the midgap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes.

In many of the experimental systems that may host Majorana zero modes, a so-called chiral symmetry exists that protects overlapping zero modes from splitting up. This symmetry is operative in a superconducting nanowire that is narrower than the spin-orbit scattering length, and at the Dirac point of a superconductor-topological insulator heterostructure. Here we show that chiral symmetry strongly modifies the dynamical and spectral properties of a chaotic scatterer, even if it binds only a single zero mode. These properties are quantified by the Wigner-Smith time-delay matrix Q ¼ −iℏS † dS=dE, the Hermitian energy derivative of the scattering matrix, related to the density of states by ρ ¼ ð2πℏÞ −1 TrQ. We compute the probability distribution of Q and ρ, dependent on the number ν of Majorana zero modes, in the chiral ensembles of random-matrix theory. Chiral symmetry is essential for a significant ν dependence. DOI: 10.1103/PhysRevLett.114.166803 PACS numbers: 73.23.-b, 73.63.-b, 74.78.Na In classical mechanics the duration τ of a scattering process can be defined without ambiguity, for example as the energy derivative of the action. The absence of a quantum mechanical operator of time complicates the simple question, "By how much is an electron delayed?" [1,2]. Since the action, in units of ℏ, corresponds to the quantum mechanical phase shift ϕ, the quantum analogue of the classical definition is τ ¼ ℏdϕ=dE. In a multichannel scattering process, described by an N × N unitary scattering matrix SðEÞ, one then has a set of delay times τ 1 ; τ 2 ; …; τ N , defined as the eigenvalues of the so-called Wigner-Smith matrixThis dynamical characterization of quantum scattering processes goes back to work by Wigner and others [3][4][5] in the 1950s. Developments in the random-matrix theory of chaotic scattering from the 1990s [6,7] allowed for a universal description of the statistics of the delay times τ n in an ensemble of chaotic scatterers. The inverse delay matrix Q −1 turns out to be statistically equivalent to a socalled Wishart matrix [8]: the Hermitian positive-definite matrix product WW † , with W a rectangular matrix having independent Gaussian matrix elements. The corresponding probability distribution of the inverse delay times γ n ≡ 1=τ n > 0 (measured in units of the Heisenberg time τ H ¼ 2πℏ=δ 0 , with mean level spacing δ 0 ), takes the form [9,10] Pðfγ n gÞ ∝The symmetry index β ∈ f1; 2; 4g distinguishes real, complex, and quaternion Hamiltonians. This connection between delay-time statistics and the Wishart ensemble is the dynamical counterpart of the connection between spectral statistics and the Wigner-Dyson ensemble [11,12] -discovered several decades later although the Wishart ensemble [13] is several decades older than the WignerDyson ensemble. The delay-time distribution (2) assumes ballistic coupling of the N scattering channels to the outside world. It has been generalized to coupling via a tunnel barrier [14,15], and has been applied to a variety of transport properties (such as thermopower...