Fluctuation effects in disordered Peierls systems

We review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐T_c superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.     

Interaction effects and phase relaxation in disordered systems

This paper is intended to demonstrate that there is no need to revise the existing theory of the transport properties of disordered conductors in the so-called weak localization regime. In particular, we demonstrate explicitly that recent attempts to justify theoretically that the dephasing rate (extracted from the magnetoresistance) remains finite at zero temperature are based on a profoundly incorrect calculation. This demonstration is based on a straightforward evaluation of the effect of the electron-electron interaction on the weak localization correction to the conductivity of disordered metals. Using well controlled perturbation theory with the inverse conductance g as the small parameter, we show that this effect consists of two contributions. The first contribution comes from the processes with energy transfer smaller than the temperature, and is responsible for setting the energy scale for the magnetoresistance. The second contribution originates from the virtual processes with energy transfer larger than the temperature. It is shown that the latter processes have nothing to do with the dephasing, but rather manifest the second-order (in 1/g) correction to the conductance. This correction is calculated for the first time. The paper also contains a brief review of the existing experiments on the dephasing of electrons in disordered conductors and an extended qualitative discussion of the quantum corrections to the conductivity and to the density of electronic states in the weak localization regime.     

Localisation in disordered systems—I: The disordered spectrum

Starting from a definition of the localisation-delocalisation of electronic, wavefunctions in disordered systems based on the nature of the disordered spectrum, a delocalisation criterion identical to that of Abou-Chacraet al is recovered. The new derivation on provides a very clear picture of the mechanism of delocalisation and brings out its incompatibility with the normalisability of wavefunctions at the transition.     

Ordering dynamics in Disordered systems

We review analytical and numerical studies of phase ordering dynamics or domain growth in systems with quenched disorder. These studies are usually based on kinetic versions of the random-exchange Ising model (REIM) or random-field Ising model (RFIM). We also present detailed numerical results which clarify the nature of domain growth in random magnets. These results demonstrate that domain walls are trapped by disorder barriers with a logarithmic dependence on the domain size.     

Discontinuities in disordered systems

The entropyS _T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk _M (ω ²) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω ² corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek _M (ω ²) has a Lifshitz singularity.     

Wave functions in disordered systems

Particular solutions of the stationary Schrödinger equation for ad-dimensional disordered tight binding model are found. The particular solution is defined by boundary conditions on one face of the system. The determination of the rate of growth of the mean square wave function leads to an exactly soluble eigenvalue problem ind – 1 dimensions. Ford 2 there are three types of particular wave functions in which the mean square amplitude (a) grows exponentially (b) decays exponentially (c) does not grow or decay but oscillates.Supported in part by the National Science Foundation under grant No. DMR 78-10276.     

Instanton in disordered Peierls systems

We study disordered Peierls systems described by the fluctuating gap model. We show that the typical electron states with energies lying deep inside the pseudogap are localized near large disorder fluctuations (instantons), which have the form of a soliton-antisoliton pair. Using the “saddle-point” method we obtain the average density of states and the average optical absorption coefficient at small energy.     

Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

Off-perturbative states in disordered systems

The systematic approach for the off-perturbative calculations in disordered systems is developed. The proposed scheme is applied for the random temperature and the random field ferromagnetic Ising models. It is shown that away from the critical point, in the paramagnetic phase of the random temperature model, and in the ferromagnetic phase of the random field one, the free energy contains non-analytic contributions which have the form of essential singularities. It is demonstrated that these contributions appear due to localized in space instanton-like excitations.     

Ordering kinetics in quasi-one-dimensional Ising-like systems

We present results of a Monte Carlo simulation of the kinetics of ordering in the two-dimensional nearest-neighbor Ising model in anL xM geometry with two free boundaries of length ML. This model can be viewed as representing an adsorbant on a stepped surface with mean terrace widthL. We follow the ordering kinetics after quenches to temperatures 0.25 T/T_c 1 starting from a random initial configuration at a coverage of=0.5 in the corresponding lattice gas picture. The systems evolve in time according to a Glauber kinetics with nonconserved order parameter. The equilibrium structure is given by a one-dimensional sequence of ordered domains. The ordering process evolves from a short initial two-dimensional ordering process through a crossover region to a quasi-one-dimensional behavior. The whole process is diffusive (inverse half-width of the structure factor peak 1/q_{¦¦ t}), in contrast to a model proposed by Kawasakiet al., where an intermediate logarithmic growth law is expected. All results are completely describable in the picture of an annihilating random walk (ARW) of domain walls.     

Electronic localization in disordered systems

A brief review is given of the current understanding of the electronic structure, transport properties and the nature of the electronic states in disordered systems. A simple explanation for the observed exponential behaviour in the density of states (Urbach tails) based on short-range Gaussian fluctuations is presented. The theory of Anderson localization in a disordered system is reviewed. Basic concepts, and the physics underlying the effects of weak localization, are discussed. The scaling as well as the self-consistent theory of localization are briefly reviewed. It is then argued that the problem of localization in a random potential within the so-called ladder approximation is formally equivalent to the problem of finding a bound state in a shallow potential well. Therefore all states are exponentially localized in d=1 and d=2. The fractal nature of the states is also discussed. Scaling properties in highly anisotropic systems are also discussed. A brief presentation of the recently observed metal-to-insulator transition in dequals;2 is given and, finally, a few remarks about interaction effects in disordered systems are presented.     

Stark resonances in disordered systems

We study further the metastable behavior of Metropolis dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model, with positive and small external field, in the limit as the temperature vanishes (see [NS]). We focus on the typical features of the escape (nucleation) from the (metastable) configuration with all spins –1, to the (stable) configuration with all spins +1. Using the reversibility of the process as the main tool, we prove (for the discrete time version of the model) that the first step of a typical escaping path is the time reverse of a typical time evolution of a shrinking subcritical rectangular droplet, which is one slice smaller than a critical droplet. This subcritical droplet then evolves in a time of order 1 to a critical droplet, which finally grows with features described in [NS].Work partially supported by the Brazilian CNPq and by the American NSF, under grant DMS91-00725     

c-Theorem for disordered systems

We find an analog of Zamolodchikov's c-theorem for disordered two-dimensional non-interacting systems in their supersymmetric field theory representation. We show that the energy momentum tensor of such field theories must be a part of a supermultiplet, and that a new parameter b can be introduced with the help of that multiplet. b flows along the renormalization group trajectories much like the central charge for unitary two-dimensional field theories. While it has not been established if this flow is irreversible, that is, if b always flows down to lower values, it does so for all the cases worked out so far. b gives a new way to label different conformal field theories for disordered systems whose central charge is always 0. b turns out to be related to the central extension of a certain algebra, a generalization of the Virasoro algebra, which we show may be present at the critical points of these theories. b is also related to the finite size corrections of the physical free energy of disordered systems. We discuss possible applications by computing b for two-dimensional Dirac fermions with random gauge potential, in other words, for U(1∣1) Kac-Moody algebra.     

Magnetic scattering effects on magnetoresistance in a quasi-two-dimensional disordered electron system

Magnetic-impurity-scattering effects in a quasi-2D disordered electron system have been investigated theoretically with the diagrammatic techniques in perturbation theory. The analytical expressions for magnetoconductivities due to weak-localization effects have been obtained as functions of elastic, inelastic and magnetic scattering times. The relevant dimensional crossover behavior from 3D to 2D with decreasing the interlayer coupling has been discussed, and the condition for the crossover has been obtained. Received 20 March 2001 and Received in final form 28 June 2001     

Diffusion of the electronic excitation in disordered systems

The kinetic equations describing the spreading ot the electronic excitation at zero temperature as a stochastic Brownian motion have been found. In the case of a structural disorder they imply the diffusion ot the excitation into the ranges of the maximum density of atoms. The relaxation time is also density-dependent having the minimum value in the ranges of maximum density of atoms and vice versa. In the case of a substitutional disorder the master equation in the “k”-space imply the spreading of excitation as a decaying wave packet. The relaxation lifetime in the first case of disorder and the lifetime of the coherence state in the second one are much smaller than the radiation lifetimes.