Martin R. Zirnbauer

NONSTANDARD SYMMETRY CLASSES IN MESOSCOPIC NORMAL-SUPERCONDUCTING HYBRID STRUCTURES

Normal-conducting mesoscopic systems in contact with a superconductor are classified by the symmetry operations of time reversal and rotation of the electron’s spin. Four symmetry classes are identified, which correspond to Cartan’s symmetric spaces of type C, CI, D, and DIII. A detailed study is made of the systems where the phase shift due to Andreev reflection averages to zero along a typical semiclassical single-electron trajectory. Such systems are particularly interesting because they do not have a genuine excitation gap but support quasiparticle states close to the chemical potential. Disorder or dynamically generated chaos mixes the states and produces forms of universal level statistics different from Wigner-Dyson. For two of the four universality classes, the n-level correlation functions are calculated by the mapping on a free one-dimensional Fermi gas with a boundary. The remaining two classes are related to the Laguerre orthogonal and symplectic random-matrix ensembles. For a quantum dot with a normal-metal‐superconducting geometry, the weaklocalization correction to the conductance is calculated as a function of sticking probability and two perturbations breaking time-reversal symmetry and spin-rotation invariance. The universal conductance fluctuations are computed from a maximum-entropy S-matrix ensemble. They are larger by a factor of 2 than what is naively expected from the analogy with normal-conducting systems. This enhancement is explained by the doubling of the number of slow modes: owing to the coupling of particles and holes by the proximity to the superconductor, every cooperon and diffusion mode in the advanced-retarded channel entails a corresponding mode in the advanced-advanced ~or retarded-retarded! channel. @S0163-1829~97!04001-0#

Grassmann integration in stochastic quantum physics: The case of compound-nucleus scattering

Using astochasticmodel for N compound—nucleusresonances coupledto thechannels,we calculatein the limit N—* theensembleaverageof theS-matrix (the “one-point function”),andof theproductof an S-matrixelementwith thecomplexconjugateof another,both takenat different energies(the “two-point function”). Using a generatingfunction involving both commutingand anticommutingintegrationvariables,we evaluate the ensembleaveragestrivially. The problem of carrying Out the remaining integrationsis solved with the help of theHubbard—Stratonovitch transformation.We put specialemphasison theconvergencepropertiesof this transformation,andon the underlyingsymmetriesof thestochastic model for thecompoundnucleus.Thesetwo featurestogethercompletelydefinetheparametrizationof thecompositevariablesin termsof agroup of transformations.Thisgroupis compactin the“Fermion—Fermionblock” andnon-compactin the“Boson—Bosonblock”. The limit N-. x is taken with thehelp of thesaddle-pointapproximation.After integrationover the“massive modes”,we showthat thetwo-point function can be expressed in terms of thetransmissioncoefficients. In this way we prove that thefluctuation propertiesof thenuclearS-matrix arethesameover theentire spectrumof the random Hamiltonian describingthe compoundnucleus.The integration over the saddle-pointmanifold is carried out using symmetry propertiesof the randomHamiltonian. We finally obtain a closed-formexpressionfor the two-point function in terms of a threefold integralover realvariables.This expressioncan be easilyevaluatednumerically.

Riemannian symmetric superspaces and their origin in random‐matrix theory

Gaussian random‐matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they describe the universal ergodic limit of disordered and chaotic single‐particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,M r ), where G/H is a complex‐analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and M r is a maximal Riemannian submanifold of the support of G/H.

Symmetry Classes of Disordered Fermions

Building upon Dyson’s fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(), a bilinear form on which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.

Critique of the replica trick

It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S2 of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.

Random matrix theory of a chaotic Andreev quantum dot.

A new universality class distinct from the standard Wigner-Dyson class is identified. This class is realized by putting a metallic quantum dot in contact with a superconductor, while applying a magnetic field so as to make the pairing field effectively vanish on average. A random-matrix description of the spectral and transport properties of such a quantum dot is proposed. The weak-localization correction to the tunnel conductance is nonzero and results from the depletion of the density of states due to the coupling with the superconductor. Semiclassically, the depletion is caused by a singular mode of phase-coherent long-range propagation of particles and holes. {copyright} {ital 1996 The American Physical Society.}

Anderson localization and non-linear sigma model with graded symmetry

Abstract A model of disordered single-particle systems is studied with regard to properties of the localized phase. Defined over a graded coset space, this model represents the correct non-perturbative extension of a non-linear sigma model introduced into localization theory by Schafer and Wegner. An integral theorem is proven which allows us to change variables and execute the Grassmann integrations rather easily. In the localized phase, the invariant two-point functions are singular on the real axis. It is shown how to extract the singular contribution before evaluation of the functional integral. This is used to derive Efetovs solution of the Cayley tree model in a simple and transparent manner. Finally, a Monte Carlo algorithm is outlined which makes it possible to study Anderson localization in d > 2 dimensions.

Supersymmetry for systems with unitary disorder: circular ensembles

A generalized Hubbard - Stratonovitch transformation relating an integral over random unitary matrices to an integral over Efetovs unitary -model manifold, is introduced. This transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a Hamiltonian but by their scattering matrix or time-evolution operator. In contrast to the standard method, no saddle-point approximation is made, and no massive modes have to be eliminated. This first paper on the subject applies the generalized Hubbard - Stratonovitch transformation to Dysons circular unitary ensemble. It is shown how a supersymmetric variant of the Harish-Chandra - Itzykson - Zuber formula can be used to compute, in the large-N limit, the n-level correlation function for any n. Non-trivial applications to random network models, quantum chaotic maps, and lattice gauge theory, are expected.

Theories of low-energy quasi-particle states in disordered d-wave superconductors

Abstract The physics of low-energy quasi-particle excitations in disordered d -wave superconductors is a subject of ongoing intensive research. Over the last decade, a variety of conceptually and methodologically different approaches to the problem have been developed. Unfortunately, many of these theories contradict each other, and the current literature displays a lack of consensus on even the most basic physical observables. Adopting a symmetry-oriented approach, the present paper attempts to identify the origin of the disagreement between various previous approaches, and to develop a coherent theoretical description of the different low-energy regimes realized in weakly disordered d -wave superconductors. We show that, depending on the presence or absence of time-reversal invariance and the microscopic nature of the impurities, the system falls into one of four different symmetry classes. By employing a field-theoretical formalism, we derive effective descriptions of these universal regimes as descendants of a common parent field theory of Wess–Zumino–Novikov–Witten type. As well as describing the properties of each universal regime, we analyse a number of physically relevant crossover scenarios, and discuss reasons for the disagreement between previous results. We also touch upon other aspects of the phenomenology of the d -wave superconductor such as quasi-particle localization properties, the spin quantum Hall effect, and the quasi-particle physics of the disordered vortex lattice.

Fourier analysis on a hyperbolic supermanifold with constant curvature

The Fourier inversion theorem is proved for a rank-one noncompact homogeneous space, the hyperbolic superplane. The proof makes use of some novel features of perfectly graded superspaces, which are not encountered in classical geometric analysis. An application to quasi-one-dimensional disordered one-electron systems is given.