Volume 38, Issue 4, April 1997
 SPECIAL ISSUE: QUANTUM PROBLEMS IN CONDENSED MATTER PHYSICS


INTRODUCTION
View Description Hide DescriptionCondensed Matter Physics has been one of the most important sources of technology during the past fifty years. But until the beginning of the eighties most theoretical advances did not need the help of sophisticated mathematical methods, and various important discoveries such as Anderson’s localization, the quantum Hall effect, Kondo’s effect,quasicrystals, high superconductors, and persistent currents in mesoscopic systems brought about many problems requiring the use of the most advanced theoretical and mathematical procedures. This special issue aims at focusing on a few of the problems posed in this field, summarizing the present understanding, tracing the methods used, and the relations among them.
 Top

 ELECTRONIC TRANSPORT


Spectral theory of thermal relaxation
View Description Hide DescriptionWe review some results obtained in a recent series of papers on thermal relaxation in classical and quantum dissipative systems. We consider models where a small system I, with a finite number of degrees of freedom, interacts with a large environment R in thermal equilibrium at positive temperature T. The zeroth law of thermodynamics postulates that, independently of its initial configuration, the system I approaches a unique stationary state as t→∞. By definition, this limiting state is the equilibrium state of I at temperature T. Statistical mechanics further identifies this state with the Gibbs canonical ensemble associated with I. For simple models we prove that the above picture is correct, provided the equilibrium state of the environment R is itself given by its canonical ensemble. In the quantum case we also obtain an exact formula for the thermal relaxation time.

Twisted boundary conditions and transport in disordered systems
View Description Hide DescriptionCondensed matter systems are usually characterized by their response to external fields. The Kubo linear response theory is the standard theoretical tool used to analyze it. In order to obtain transport quantities with this method, we generally need to know both the eigenvalues and the eigenfunctions. This article discusses another methodological approach which in some cases allows one to obtain some characteristics of the dc dissipative transport from the response to a gauge field in a multiply connected geometry. This avoids the need of the eigenstates to characterize the transport properties which are directly read off the behavior of the energy spectrum. This method is applied to the problem of transport in metals and Anderson insulators.

Electronic transport properties of quasicrystals
View Description Hide DescriptionWe present a review of some results concerning electronic transport properties of quasicrystals. After a short introduction to the basic concepts of quasiperiodicity, we consider the experimental transport properties of electrical conductivity with particular focus on the effect of temperature, magnetic field, and defects. Then, we present some heuristic approaches that tend to give a coherent view of different, and to some extent complementary, transport mechanisms in quasicrystals. Numerical results are also presented and in particular the evaluation of the linear response Kubo–Greenwood formula of conductivity in quasiperiodic systems in the presence of disorder.

Topological equivalence of tilings
View Description Hide DescriptionWe introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.

Schrödinger operators with RudinShapiro potentials are not palindromic
View Description Hide DescriptionWe prove a conjecture of A. Hof, O. Knill and B. Simon [Commun. Math. Phys. 174, 149–159 (1995)] by showing that the RudinShapiro sequence is not palindromic, i.e., does not contain arbitrarily long palindromes. We prove actually this property for all paperfolding sequences and all RudinShapiro sequences deduced from paperfolding sequences. As a consequence and as guessed by the above authors, their method cannot be used for establishing that discrete Schrödinger operators with RudinShapiro potentials have a purely singular continuous spectrum.

Excitations in one dimension: A geometrical view of the transfer matrix method
View Description Hide DescriptionThe transfer matrix formalism is a useful tool for the study of excitations in onedimensional chains. The geometrical approach presented here maps the dynamic of transfer matrix products onto a dynamic on the Poincaré disc, a suitable model for the hyperbolic plane. This mapping uses a fibration of the three dimensional manifold on which the transfer matrices lie: the transfer matrix dynamic is mapped onto a polygonal trajectory on the base of this fibration. The link between the behavior of these trajectories and the spectrum of the system under study is discussed. We particularly focus on approximants of quasiperiodic systems. We also give a geometrical construction of the socalled tracemap for quasiperiodic systems.

Taming of the wild group of magnetic translations
View Description Hide DescriptionWe use a theorem of Auslander and Kostant on the representation theory of solvable Liegroups for the study of some groups necessary for the description of certain quasiperiodic systems of solidstate physics. We show that the magnetic translation group is tame (Type I) if the magnetic field is not constant but fluctuating.
 Top

 RANDOM MATRIX THEORY, SUPERSYMMETRY AND MESOSCOPIC PHYSICS


Spectral correlations in the crossover between GUE and Poisson regularity: On the identification of scales
View Description Hide DescriptionMotivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter suitably defined on the unfolded scale. Using results for the spectral twopoint correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For the correlations are given by Dawson’s integral while for we derive a novel analytical formula for the twopoint function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticeable, can be identified. We also derive an exact expansion of the local level density for a finite level number.

Spatial structure of anomalously localized states in disordered conductors
View Description Hide DescriptionThe spatial structure of wave functions of anomalously localized states (ALS) in disordered conductors is studied in the framework of the –model approach. These states are responsible for slowly decaying tails of various distribution functions. In the quasionedimensional case, properties of ALS governing the asymptotic form of the distribution of eigenfunction amplitudes are investigated with the use of the transfer matrix method, which yields an exact solution to the problem. Comparison of the results with those obtained in the saddlepoint approximation to the problem shows that the saddlepoint configuration correctly describes the smoothed intensity of an ALS. On this basis, the properties of ALS in higher spatial dimensions are considered. We also study the ALS responsible for the asymptotic behavior of distribution functions of other quantities, such as relaxation time, local and global density of state. It is found that the structure of an ALS may be different, depending on the specific quantity, for which it constitutes an optimal fluctuation. Relations between various procedures of selection of ALS, and between asymptotics of corresponding distribution functions, are discussed.

Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken timereversal invariance
View Description Hide DescriptionAssuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the socalled stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken timereversal invariance coupled to continua via open channels; . A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the distribution of resonance widths (regime of isolated resonances) to a broad powerlike distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer–Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases of the matrix (taken modulo . We use it to find the distribution of derivatives of these eigenphases with respect to the energy (“partial delay times”) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner–Smith time delay at two different energies and as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.

Timereversal symmetry breaking and the field theory of quantum chaos
View Description Hide DescriptionRecent studies have shown that the quantum statistical properties of systems which are chaotic in their classical limit can be expressed in terms of an effective field theory. Within this description, spectral properties are determined by low energy relaxation modes of the classical evolution operator. It is in the interaction of these modes that quantum interferenceeffects are encoded. In this paper we review this general approach and discuss how the theory is modified to account for timereversal symmetry breaking. To keep our discussion general, we will also briefly describe how the theory is modified by the presence of an additional discrete symmetry such as inversion. Throughout, parallels are drawn between quantum chaotic systems and the properties of weakly disordered conductors.

Toward a theory of the integer quantum Hall transition: Continuum limit of the Chalker–Coddington model
View Description Hide DescriptionAn channel generalization of the network model of Chalker and Coddington is considered. The model for is known to describe the critical behavior at the plateau transition in systems exhibiting the integer quantum Hall effect. Using a recently discovered equality of integrals, the network model is transformed into a lattice field theory defined over Efetov’s model space with unitary symmetry. The transformation is exact for all no saddlepoint approximation is made, and no massive modes have to be eliminated. The naive continuum limit of the lattice theory is shown to be a supersymmetric version of Pruisken’s nonlinear model with couplings and at the symmetric point. It follows that the model for which describes a spin degenerate Landau level and the random flux problem, is noncritical. On the basis of symmetry considerations and inspection of the Hamiltonian limit, a modified network model is formulated, which still lies in the quantum Hall universality class. The prospects for deformation to a Yang–Baxter integrable vertex model are briefly discussed.
 Top

 ELECTRONIC CORRELATIONS


Density matrices for itinerant and localized electrons with and without external fields
View Description Hide DescriptionForms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a oneelectron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution to is expressed in terms of first and secondorder density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the firstorder density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.

Offdiagonal long range order and superconductive electrodynamics
View Description Hide DescriptionWe present a general, modelindependent, quantum statistical derivation of superconductive electrodynamics from the assumptions of offdiagonal long range order (ODLRO), local gauge covariance, and thermodynamic stability. On this basis, we obtain the Meissner and Josephson effects, the quantization of trapped magnetic flux, and the metastability of supercurrents. A key to these results is that the macroscopic wave function, specified by the ODLRO condition, enjoys the rigidity property that London [Superfluids, Vol. 1 (Wiley, London, 1950)], envisaged for the microstate of a superconductor.

Accuracy of the Hartree–Fock approximation for the Hubbard model
View Description Hide DescriptionA lower bound 0⩾( E_{gs}(n)−E_{hf}(n) )/Λ⩾−const[n^{2/3}U^{4/3} (ln U+1) +Un^{1/2}Λ^{−1/2d} (ln(Λ^{−1/2})1)] to the difference of the ground state and the Hartree–Fock energy of the Hubbard model is derived. Here Λ is the lattice size, U is the coupling parameter, and n is the electron density per site. This estimate holds for all dimensions d⩾2 and all densities. Thus the Hartree–Fock approximation becomes exact (even beyond terms of order U) for small U and large Λ.

Ground states and lowtemperature phases of itinerant electrons interacting with classical fields: A review of rigorous results
View Description Hide DescriptionWe review, from a unified point of view, a general class of models of itinerant electrons interacting with classical fields. Applications to the static Holstein, Kondo, and Hubbard models are discussed. The ground state structure of the classical field is investigated when the electron band is halffilled. Some of the results are also valid when there is a Hubbard interaction between spin up and spin down electrons. It is found that the ground states are either homogeneous or period two Néel configurations, depending on the geometry of the lattice and on the magnetic fluxes present in the system. In the specific models, Néel configurations correspond to Peierls, magnetic or superconducting instabilities of the homogeneous state. The effect of small thermal and quantum fluctuations of the classical fields are reviewed in the context of the Holstein model. Many of the results described here originate from the work of Elliott Lieb and collaborators.

Finite coherence length for equilibrium states of generalized adiabatic Holstein models
View Description Hide DescriptionWe prove that the equilibrium states of the adiabatic Holstein model, and a wide range of generalizations, have finite coherence length as long as they have electronic gap and phonon gap.
