Variational theory for correlated lattice fermions in high dimensions

A diagrammatic approach to the evaluation of correlated variational wave functions for strongly interacting fermions is presented. Diagrammatic rules for the calculation of the one-particle density matrix and the Hubbard interaction are derived which are valid for arbitraryd-dimensional lattices. An exact evaluation of expectation values is performed in the limitd=. The wellknown Gutzwiller approximation is seen to become the exact result for the expectation value of the Hubbard Hamiltonian in terms of the Gutzwiller wave function ind=. An efficient procedure to correct the Gutzwiller approximation in finite dimensions is developed. A detailed discussion of expectation values ind= in terms of explicit antiferromagnetic wave functions is given. Thereby an approximate result for the ground state energy of the Hubbard model, obtained recently within a slave-boson approach, is recovered.     

Characteristic Properties of the Scattering Data for the mKdV Equation on the Half-Line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<)×[0,T), T. We suppose that this solution has a C initial function vanishing as x, and C boundary values, vanishing as t when T=. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for x0 and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.     

Generalization ability and information gain of clock-model perceptrons

We study the generalization abilityg _Q ofQ-state Clock-model perceptrons for (i) Hebbian and for certain Non-Hebbian learning procedures, namely (ii) learning with maximal stability, (iii) zero stability and (iv) optimal generalization, for the case of random training sets. Among other results we find thatg _Q behaves quite different in the Hebbian and in the Non-Hebbian cases in the limitQ. E.g. in the Hebbian case for finite ,g _Q vanishes always 1/Q, whereas in the Non-Hebbian cases considered,g _Q converges forQ to a non-trivial continuous functiong (), which vanishes for <2, but increases rapidly for >2. This means that for (ii), (iii) and (iv), as a function of atQ=, there is a 2nd-order phase transition from a non-generalizing phase for 2 to a generalizing phase for >2. Different behaviour of the Hebbian and Non-Hebbian cases, respectively, is also observed for the information gain obtained through learning. For the particular case of AdaTron Learning, which is identical to case (ii), we find a geometrical formulation forg _Q (), which is applicable to more general models.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Calculation and optimization of parameters of radiating distributed feedback structures

A simple method is suggested for calculation of reflection, radiation and transmission coefficients for the distributed feedback structure in the second diffraction order. The method is based on a slight difference between coefficients of reflectionR and radiationI of the surface wave for = (where is the light wavelength corresponding to a precise resonance for the grating length I) and those for =_l (where _l is the light wavelength corresponding to the resonance for the finite grating length). The simplicity of the method makes it possible to use it for optimization of the distributed feedback structure by a number of parameters. The technique can be used in the case of thin-film and diffused waveguides for both TE and TM modes.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

D-Dimensional ideal gas in parastatistics: Thermodynamic properties

We consider a parastatistics ideal gas with energy spectrum ¦k¦ (>0) or even more generally in ad-dimensional box with volumeV (periodic boundary conditions), the numberN of the gas particles being well determined (real particles) or not (quasiparticles). We calculate the main thermodynamic quantities (chemical potential, internal energy, specific heatC, equation of state, latent heat, average numbers of particles) for arbitraryd, ,T (temperature), andp (maximal number of particles per state allowed in the parastatistics). The main asymptotic regimes are worked out explicitly. In particular, the Bose-Einstein condensation for fixed densityN/V appears as a nonuniform convergence in thep limit, in complete analogy with the standard critical phenomena that appear in interacting systems in theN limit. The system behaves essentially like a Fermi-Dirac one forall finite values ofp, and reveals a Bose-Einstein behavioronly in thep limit. For instance, at low temperaturesC T ifp< andC T ^d/ ifp. Finally, the Sommerfeld integral and its expansion are generalized to an arbitrary, finitep.     

The size effect correction of d. c. resistance in metal wires

A simple method of determining the bulk value of the d.c. resistance r, of the residual resistance ratiok , and of the mean free path of electrons in isotropic metal wires is presented. For thick wires the linear correction can be applied under the condition that the experimentally found mean free path _e is smaller than the wire diameterd. In both thick and thin wires another type of correction can be used, based on the difference, between the precise and linear correction as a function of /d, carried out by a graphical method.The author acknowledges useful discussions with P.Svoboda, to whom he is indebted for stimulating remarks. Also the kind advice of L.Smrka and P.Steda was welcome.     

A one-parameter family of cylindrically symmetric perfect fluid cosmologies

Non-stationary cylindrically symmetric one-parameter solutions to Einstein's equations are given for a perfect fluid. There is a time singularity (t=0) at which the pressurep and density are equal to + throughout the radial coordinate range 0 r < , but the solutions are well behaved fort > 0,p and decreasing steadily to zero asr increases through the range 0r<, or as t increases through the range 0<t<. The motion is irrotational with shear, expansion and acceleration. The family of solutions, of Petrov type I, are generally spatially inhomogeneous, of class B(ii), having two spacelike Killing vectors which are mutually orthogonal and hypersurface orthogonal, associated with an orthogonally transitive groupG ₂. The particular members for which there are equations of statep=/3 andp= are specially considered.     

Spatial fluctuations in reaction-limited aggregation

A general method is used for describing reaction-diffusion systems, namely van Kampen's method of compounding moments, to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areK _ij=ij (i, j=1, 2,). For thediffusion constants D _j of clusters of sizej the following class of models is considered:D _j=D if 1Js andD _j=0 ifj>s. The casess= ands< are studied separately. For the models= the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs< the density fluctuations remain finite att _c, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isd _c=2.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

Combining Multifractal Additive and Multiplicative Chaos

The purpose of this article is the study of the new class of multifractal measures, which combines additive and multiplicative chaos, defined bywhere is any positive Borel measure on [0,1] and b is an integer 2. The singularities analysis of the measures _, involves new results on the mass distribution of when describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure.Under suitable assumptions on , the multifractal spectrum of _, is linear on [0,h_, ] for some critical value h_, . Then is strictly concave on the right of h_, , and on this part it is deduced from the multifractal spectrum of by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. They open interesting perspectives in modeling discontinuous phenomena.     

Intersection properties of simple random walks: A renormalization group approach

We study estimates for the intersection probability,g(m), of two simple random walks on lattices of dimensiond=4, 4– as a problem in Euclidean field theory. We rigorously establish a renormalization group flow equation forg(m) and bounds on the -function which show that, ind=4,g(m) tends to zero logarithmically as the killing rate (mass)m tends to zero, and that the fixed point,g*, ind=4– is bounded by const' g*const. Our methods also yield estimates on the intersection probability of three random walks ind=3, 3–. For =0, these results were first obtained by Lawler [1].     

On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

The paper considers the wave equation, with constant or variable coefficients in ⁿ , with odd n3. We study the asymptotics of the distribution _t of the random solution at time t as t . It is assumed that the initial measure ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of _t to a Gaussian measure as t , which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.     

The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .