Microscopic theory of the Andreev gap

We present a microscopic theory of the Andreev gap, i.e. the phenomenon that the density of states (DoS) of normal chaotic cavities attached to superconductors displays a hard gap centered around the Fermi energy. Our approach is based on a solution of the quantum Eilenberger equation in the regime $t_D\ll t_E$, where $t_D$ and $t_E$ are the classical dwell time and Ehrenfest-time, respectively. We show how quantum fluctuations eradicate the DoS at low energies and compute the profile of the gap to leading order in the parameter $t_D/t_E$ .Comment: 4 pages, 3 figures; revised version, more details, extra figure, new titl

What is the Thouless Energy for Ballistic Systems?

The Thouless energy, \Ec characterizes numerous quantities associated with sensitivity to boundary conditions in diffusive mesoscopic conductors. What happens to these quantities if the disorder strength is decreased and a transition to the ballistic regime takes place? In the present analysis we refute the intuitively plausible assumption that \Ec loses its meaning as an inverse diffusion time through the system at hand, and generally disorder independent scales take over. Instead we find that a variety of (thermodynamic) observables are still characterized by the Thouless energy.Comment: 4 pages REVTEX, uuencoded file. To appear in Physical Review Letter

Crossovers between superconducting symmetry classes

We study the average density of states in a small metallic grain coupled to two superconductors with the phase difference $\pi$, in a magnetic field. The spectrum of the low-energy excitations in the grain is described by the random matrix theory whose symmetry depends on the magnetic field strength and coupling to the superconductors. In the limiting cases, a pure superconducting symmetry class is realized. For intermediate magnetic fields or couplings to the superconductors, the system experiences a crossover between different symmetry classes. With the help of the supersymmetric sigma-model we derive the exact expressions for the average density of states in the crossovers between the symmetry classes A-C and CI-C.Comment: 6 page

Phonons in Random Elastic Media and the Boson Peak

We show that the density of states of random wave equations, normalized by the square of the frequency, has a peak - sometimes narrow and sometimes broad - in the range of wave vectors between the disorder correlation length and the interatomic spacing. The results of this letter may be relevant for understanding vibrational spectra and light propagation in disordered solids

A Classification of Non-Hermitian Random Matrices

We present a classification of non-hermitian random matrices based on implementing commuting discrete symmetries. It contains 38 classes. This generalizes the classification of hermitian random matrices due to Altland-Zirnbauer and it also extends the Ginibre ensembles of non-hermitian matrices.Comment: 8 pages, contribution to the proceedings of the NATO Advanced Research Workshop on Statistical Field Theories, Como (Italy), 18-23 June 2001. Compared to our 2001 version, we corrected two misprints in one table that in the previous version led us to miscount the number of classes as 43 whereas it should have been 38. Explicit details of the classification are unchange

The supersymmetric technique for random-matrix ensembles with zero eigenvalues

The supersymmetric technique is applied to computing the average spectral density near zero energy in the large-N limit of the random-matrix ensembles with zero eigenvalues: B, DIII-odd, and the chiral ensembles (classes AIII, BDI, and CII). The supersymmetric calculations reproduce the existing results obtained by other methods. The effect of zero eigenvalues may be interpreted as reducing the symmetry of the zero-energy supersymmetric action by breaking a certain abelian symmetry.Comment: 22 pages, introduction modified, one reference adde

Quantum interference and the formation of the proximity effect in chaotic normal-metal/superconducting structures

We discuss a number of basic physical mechanisms relevant to the formation of the proximity effect in superconductor/normal metal (SN) systems. Specifically, we review why the proximity effect sharply discriminates between systems with integrable and chaotic dynamics, respectively, and how this feature can be incorporated into theories of SN systems. Turning to less well investigated terrain, we discuss the impact of quantum diffractive scattering on the structure of the density of states in the normal region. We consider ballistic systems weakly disordered by pointlike impurities as a test case and demonstrate that diffractive processes akin to normal metal weak localization lead to the formation of a hard spectral gap -- a hallmark of SN systems with chaotic dynamics. Turning to the more difficult case of clean systems with chaotic boundary scattering, we argue that semiclassical approaches, based on classifications in terms of classical trajectories, cannot explain the gap phenomenon. Employing an alternative formalism based on elements of quasiclassics and the ballistic $\sigma$-model, we demonstrate that the inverse of the so-called Ehrenfest time is the relevant energy scale in this context. We discuss some fundamental difficulties related to the formulation of low energy theories of mesoscopic chaotic systems in general and how they prevent us from analysing the gap structure in a rigorous manner. Given these difficulties, we argue that the proximity effect represents a basic and challenging test phenomenon for theories of quantum chaotic systems.Comment: 21 pages (two-column), 6 figures; references adde

Distribution of the local density of states, reflection coefficient and Wigner delay time in absorbing ergodic systems at the point of chiral symmetry

Employing the chiral Unitary Ensemble of random matrices we calculate the probability distribution of the local density of states for zero-dimensional ("quantum chaotic") two-sublattice systems at the point of chiral symmetry E=0 and in the presence of uniform absorption. The obtained result can be used to find the distributions of the reflection coefficent and of the Wigner time delay for such systems.Comment: 4 pages, 3 figure

Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems

In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical simulation of quantum lattice systems of infinite size. This exploration is done mainly in one and two spatial dimensions (1d and 2d). We describe a number of numerical algorithms based on corner matri- ces and tensors to approximate different ground state properties of these systems. The proposed methods make also use of matrix product operators and projected entangled pair operators, and naturally preserve spatial symmetries of the system such as translation invariance. In order to assess the validity of our algorithms, we provide preliminary benchmarking calculations for the spin-1/2 quantum Ising model in a transverse field in both 1d and 2d. Our methods are a plausible alternative to other well-established tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in 2d. The computational complexity of the proposed algorithms is also considered and, in 2d, important differences are found depending on the chosen simulation scheme. We also discuss further possibilities, such as 3d quantum lattice systems, periodic boundary conditions, and real time evolution. This discussion leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of corner transfer matrices and corner tensors. Our paper also offers a perspective on many properties of the corner transfer matrix and its higher-dimensional generalizations in the light of novel tensor network methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on some of the algorithms have been moved to appendices. To appear in PR

Topological dualities via tensor networks

The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable inasmuch as it connects systems commonly associated to different areas of physics -- that of long range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically non-local, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this work, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs.Comment: 19 pages, 19 figure