Random-matrix ensembles in p-wave vortices

In disordered vortices in p-wave superconductors the two new random-matrix ensembles may be realized: B and DIII-odd (of so(2N + 1) and so(4N + 2)/u(2N + 1) matrices respectively). We predict these ensembles from an explicit analysis of the symmetries of Bogoliubov– deGennes equations in three examples of vortices with different p-wave order parameters. A characteristic feature of the novel symmetry classes is a quasiparticle level at zero energy. Class B is realized when the time-reversal symmetry is broken, and class DIII-odd when the timereversal symmetry is preserved. We also suggest that the main contribution to disordering the vortex spectrum comes from the distortion of the order parameter around impurities. Since Wigner’s modeling Hamiltonians of complex nuclei by random matrices [1], the randommatrix theory (RMT) has played an important role in studying mesoscopic systems. In many cases, a chaotic (non-integrable) mesoscopic system may be accurately described by RMT. The supersymmetric technique by Efetov [2] provides a microscopic explanation of the RMT approximation for disordered systems. According to the RMT approximation, the only characteristics of the system affecting the eigenvalue correlations at small energy scales are its symmetries. Therefore, a problem arises of classifying symmetries of random-matrix ensembles. It has been suggested by several authors [3, 4, 5] that random-matrix ensembles may be classified as corresponding symmetric spaces. The symmetric spaces may be divided into twelve infinite series reviewed in Table 1 (we split class DIII in two subclasses: DIII-even and DIII-odd) [7]. Each of the symmetry classes may occur in one of the three forms: positive-curvature, negative-curvature and flat [5]. The corresponding Jacobians in the matrix space are expressed in terms of trigonometric, hyperbolic, and polynomial functions respectively. In the present paper we shall only discuss the RMT for Hamiltonians forming a linear space and therefore described by zero-curvature (flat) versions of RMT. The simplest examples of the RMT symmetry classes are the three Wigner-Dyson classes: unitary, orthogonal and symplectic (A, AI, and AII, respectively, in Cartan’s notation) [8]. In these classes, the energy level correlations are invariant under translations in energies (at energy scales much smaller than the spectrum width), and the joint probability distribution of the energy levels ωi is dP{ωi}