Microscopic theory of the long-wavelength modes of two-component plasmas and ionic liquids: III. The space-time correlation functions

The identity between the exact screening length obtained from the static charge density correlation function and the one which appears in the Einstein relation between the transport coefficients of electrical conductivity and mass diffusion is demonstrated from first principles. For the space-time correlation functions of the number densities we show that their long-wavelength behaviour is completely determined by the four hydrodynamical modes of the two-component system of neutral particles. For charged particle systems there are only three hydrodynamical modes while we have moreover to add the two charge relaxation modes in order to exhaust the long-wavelength limit of the first sum-rule. The strengths with which the various modes appear in the space-time correlation functions have been computed exactly in the limit of long wavelengths.     

Chebyshev expansion approach to the AC conductivity of the Anderson model

We propose an advanced Chebyshev expansion method for the numerical calculation of linear response functions at finite temperature. Its high stability and the small required resources allow for a comprehensive study of the optical conductivity of non-interacting electrons in a random potential (Anderson model) on large three-dimensional clusters. For low frequency the data follows the analytically expected power-law behaviour with an exponent that depends on disorder and has its minimum near the metal-insulator transition, where also the extrapolated DC conductivity continuously goes to zero. In view of the general applicability of the Chebyshev approach we briefly discuss its formulation for interacting quantum systems.Received: 6 June 2004, Published online: 12 August 2004PACS: 78.20.Bh Theory, models, and numerical simulation - 72.15.Rn Localisation effects (Anderson or weak localisation) - 05.60.Gg Quantum transport     

Response function of the second kind in many-body systems

We analyze a new type of response function which portrays the properties of a system perturbed by an external field in terms of the perturbed two-point correlations of density fluctuations rather than in terms of perturbed averages of physical quantities. This response function of the second kind satisfies both fluctuation-dissipation-like theorems, relating it to three-point equilibrium functions, and hierarchical relationship linking it to conventional quadratic (rather than linear) response functions. In the equal-time limit, when the two density fluctuations are observed at the same time, the response function of the second kind is intimately connected to the two-particle correlation function of kinetic theory. This linkage opens an avenue for developing novel approximation techniques for correlated many-body systems.     

A Semiclassical Theory of a Dissipative Henon—Heiles System

A semiclassical theory of a dissipative Henon—Heiles system is proposed. Based on -scaling of an equation for the evolution of the Wigner quasiprobability distribution function in the presence of dissipation and thermal diffusion, we derive a semiclassical equation for quantum fluctuations, governed by the dissipation and the curvature of the classical potential. We show how the initial quantum noise gets amplified by classical chaotic diffusion, which is expressible in terms of a correlation of stochastic fluctuations of the curvature of the potential due to classical chaos, and ultimately settles down to equilibrium under the influence of dissipation. We also establish that there exists a critical limit to the expansion of phase space. The limit is set by chaotic diffusion and dissipation. Our semiclassical analysis is corroborated by numerical simulation of a quantum operator master equation.     

Short-distance wavefunction statistics in one-dimensional Anderson localization

We investigate the short-distance statistics of the local density of states in long one-dimensional disordered systems, which display Anderson localization. It is shown that the probability distribution function can be recovered from the long-distance wavefunction statistics, if one also uses parameters that are irrelevant from the perspective of two-parameter scaling theory.Received: 10 July 2003, Published online: 15 October 2003PACS: 72.15.Rn Localization effects (Anderson or weak localization) - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 42.25.Dd Wave propagation in random media - 73.20.Fz Weak or Anderson localization     

Fluctuation conductivity in layered d-wave superconductors near critical disorder

We consider the fluctuation conductivity in the critical region of a disorder induced quantum phase transition in layered d-wave superconductors. We specifically address the fluctuation contribution to the systems conductivity in the limit of large (quasi-two-dimensional system) and small (quasi-three-dimensional system) separation between adjacent layers of the system. Both in-plane and c-axis conductivities were discussed near the point of insulator-superconductor phase transition. The value of the dynamical critical exponent, z = 2, permits a perturbative treatment of this quantum phase transition under the renormalization group approach. We discuss our results for the system conductivities in the critical region as function of temperature and disorder.Received: 10 October 2003, Published online: 23 December 2003PACS: 74.40. + k Fluctuations (noise, chaos, nonequilibrium superconductivity, localization, etc.) - 73.43.Nq Quantum phase transitions     

Effects of electron-hole correlation on ultrasonic attenuation in bismuth in strong magnetic fields

A theory is developed on giant quantum attenuation of ultrasound in bismuth. The present theory successfully explains the following experimental results in strong magnetic fields (H 100 kG): (i) When two attenuation peaks, the one due do electrons and the other due to holes, coincide as a function of magnetic field, the attenuation is exceptionally large at temperatures around 1 K and decreases rapidly with increasing temperatures; (ii) on the contrary, an isolated attenuation peak shows only a weak temperature dependence; (iii) the line shape of an isolated hole peak is highly asymmetric. The theory includes both intraband and interband impurity scatterings, acoustic phonon scattering, and takes account of Coulomb correlation effects via electron-electron, hole-hole and electron-hole two-body distribution functions. As a result, the electron-hole attractive correlation is found to play a crucial role in making the large attenuation mentioned in (i). For (ii), the electron-hole correlation is ineffective because of the large difference in Fermi velocities, and the acoustic phonon scattering is found to be important. Finally, the result (iii) is attributed to the small density of states of the reservoir Landau subbands in the strong magnetic field regime. The present theory assumes no phase transition to account for the result (i) in contrast to previous theories.     

Long time tails in the quantum Hall effect

Velocity correlation functions in the time domain offer a new approach to the dissipative conductivity in the quantum Hall effect at half-filled Landau bands. We describe two methods of calculating these functions directly by wave packet propagation techniques. We address the question whether quantum effects modify the high field diffusion described within the semiclassical percolation picture of the QHE. We investigate a number of random potentials with finite correlation length. Coupling to higher Landau levels leads to damped cyclotron oscillations in the velocity correlation. Besides that, we observe a long time tail (t)t ^– with 2.3 for all potential ranges investigated. Within the statistical uncertainty and the time interval considered, this power law is consistent with the semiclassical result obtained by averaging over all cluster sizes. Our findings imply detectable deviations from Drude's law for the AC conductivity.     

Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections

We determine the universal law for fidelity decay in quantum computations of complex dynamics in presence of internal static imperfections in a quantum computer. Our approach is based on random matrix theory applied to quantum computations in presence of imperfections. The theoretical predictions are tested and confirmed in extensive numerical simulations of a quantum algorithm for quantum chaos in the dynamical tent map with up to 18 qubits. The theory developed determines the time scales for reliable quantum computations in absence of the quantum error correction codes. These time scales are related to the Heisenberg time, the Thouless time, and the decay time given by Fermis golden rule which are well-known in the context of mesoscopic systems. The comparison is presented for static imperfection effects and random errors in quantum gates. A new convenient method for the quantum computation of the coarse-grained Wigner function is also proposed.Received: 13 December 2003, Published online: 18 March 2004PACS: 03.67.Lx Quantum computation - 05.45.Pq Numerical simulations of chaotic systems - 05.45.Mt Quantum chaos; semiclassical methods     

Linear response theory and the KMS condition

The response, relaxation and correlation functions are defined for any vector state ε of a von Neumann algebra \(\mathfrak{M}\) , acting on a Hilbert space ?, satisfying the KMS-condition. An operator representation of these functions is given on a particular Hilbert space . With this technique we prove the existence of the static admittance and the relaxation function. Finally we generalize the fluctuation-dissipation theorem and other relations between the above mentionned functions to infinite systems.     

Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa — and Coulomb systems

We estimate the canonical and grand canonical partition function in a finite volume and prove stability and existence of the thermodynamic limit for the pressure of two component classical and quantum systems of particles with charge ± interacting via two body Yukawa — or Coulomb forces. In the case of Coulomb forces we require neutrality. For the classical system in two dimensions there exists a critical temperatureT _c at and below which the system collapses. For the classical Yukawa system the correlation functions exist for arbitrary fugacity and the general structure of the pure phases can be analyzed completely.     

Electrical Conductivity of Non-ideal Plasmas and the Ion Distribution Function

The theory of electrical conductivity of electron-ion-systems is developed for a density region which reaches from the region of non-ideal plasmas up to the region of liquid metals. The conductivity is expressed by quantum mechanical correlation functions. Different forms of the electron-ion pseudopotentials are considered. The ion distribution function is derived using the mean spherical approximation (MSA) theory or the nonlinear Debye-theory. Higher order scattering effects are treated by introducing scattering phase shifts for the statically screened electron-ion potential. The numerical results for the conductivity show a SPITZER-like behaviour in the low-density non-degenerate limit where higher order scattering is important, and a ZIMAN-like behaviour in the strongly degenerate high-density limit where the ion distribution functions and the form of the electron-ion pseudopotential become more important.     

Correlation functions of the 1-D coulomb gas,solitons and the 1-D Fermi gas

We study various correlation functions of the classical one-dimensional two component Coulomb gas (1-D CG). As a first example the dielectric function _q is considered. Explicit expressions for the first two terms of the long-wavelength expansion of _q ^-1 are rigorously derived by extending the method of Edwards and Lenard. On account of the equivalence between the 1-D CG and the sine-Gordon (SG) model _q can also be viewed as a correlation function of the SG field and the effects of small amplitude oscillations and of solitons can be traced in _q . We show that the nonmetallic nature of the 1-D CG is solely attributable to solitons. A similar analysis is carried out for other correlation functions. We establish a relation between these correlation functions and certain response functions of a one-dimensional Fermi gas by exploiting the known equivalence between the backward-scattering model of the Fermi gas and a 2-D CG on the surface of a cylinder and by investigating the conditions under which this 2-D CG can be considered one-dimensional.     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

QuasiBoolean algebras and simultaneously definite properties in quantum mechanics

We define and characterize a new abstract notion of quasiBoolean algebra, intermediate in nature between an (ortho)lattice and a Boolean algebra. It will turn out that such algebras are natural candidates for representing the simultaneously definite properties of a quantum system. We then prove a general theorem about maximal quasiBoolean subalgebras of an ortholattice which we use to derive a number of different proposals in the literature for what properties of a quantum system should be regarded as simultaneously definite.     

Wavelet Transformation and Wigner-Husimi Distribution Function

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.     

The non-Markovian stochastic Liouville equation and anomalous quantum relaxation kinetics

We analyze within the continuous-time random walk approach, the kinetics of phase and population relaxation in quantum systems induced by noise with the anomalously slowly decaying correlation function P(t) ∝ (wt)^-α, where 0 < α < 1. The relaxation kinetics is shown to be anomalously slow. Moreover, for α < 1, in the limit of a short characteristic time of fluctuations w^-1, the kinetics is independent of w. As α → 1, the relaxation regime changes from the static limit to narrowing of fluctuation. Simple analytical expressions are obtained that describe the specific features of the kinetics.     

Muonium quantum diffusion and localization in cryocrystals

We review our recent study of atomic muonium ( ⁺e^– or Mu, a light isotope of the hydrogen atom) diffusion in the simplest solids-van der Waals cryocrystals. We give experimental evidence of the quantum-mechanical nature of the Mu diffusion in these solids. The results are compared with the current theories of quantum diffusion in insulators. In solid nitrogen bothT ⁷ andT ^–7 temperature dependences of the Mu hop rate are observed directly for the first time, which is taken as a confirmation of a two-phonon scattering mechanism. In solid xenon and krypton, by contrast, the one-phonon interaction is dominant in the whole temperature range under investigation due to extremely low values of the Debye temperatures. Particular attention is dedicated to processes of inhomogeneous quantum diffusion and Mu localization. It is shown that at low temperatures static crystal disorder results in an inhomogeneity of the Mu quantum diffusion which turns out to be inconsistent with diffusion models using a single correlation time _c . Conventional trapping mechanisms are shown to be ineffective at low temperatures in insulators. The localization effects in Mu quantum diffusion are studied in detail in solid Kr. In all the cryocrystals studied muonium atom turns out to be localized at low temperatures.