Localization of disordered bosons and magnets in random fields

We study localization properties of disordered bosons and spins in random fields at zero temperature. We focus on two representatives of different symmetry classes, hard-core bosons (XY magnets) and Ising magnets in random transverse fields, and contrast their physical properties. We describe localization properties using a locator expansion on general lattices. For 1d Ising chains, we find non-analytic behavior of the localization length as a function of energy at ω=0$\omega =0$, ξ⁻¹(ω)=ξ⁻¹(0)+A|ω|^α${\xi }^{-1}\left(\omega \right)={\xi }^{-1}\left(0\right)+A{\left|\omega \right|}^{\alpha }$, with α$\alpha$ vanishing at criticality. This contrasts with the much smoother behavior predicted for XY magnets. We use these results to approach the ordering transition on Bethe lattices of large connectivity K$K$, which mimic the limit of high dimensionality. In both models, in the paramagnetic phase with uniform disorder, the localization length is found to have a local maximum at ω=0$\omega =0$. For the Ising model, we find activated scaling at the phase transition, in agreement with infinite randomness studies. In the Ising model long range order is found to arise due to a delocalization and condensation initiated at ω=0$\omega =0$, without a closing mobility gap. We find that Ising systems establish order on much sparser (fractal) subgraphs than XY models. Possible implications of these results for finite-dimensional systems are discussed.     

Localization of particles in a two-dimensional random potential

A recently proposed theory of the density response of particles moving in a random potential is applied to two-dimensional systems. The particles are found to be localized for arbitrary small disorder. By decreasing the potential fluctuations we find an abrupt transition from an insulating state to a quasi-conducting state exhibiting exceedingly small values for the inverse susceptibility, the inverse localization length and the excitation gap of the dynamical conductivity.     

The superspin approach to a disordered quantum wire in the chiral-unitary symmetry class with an arbitrary number of channels

We use a superspin Hamiltonian defined on an infinite-dimensional Fock space with positive definite scalar product to study localization and delocalization of noninteracting spinless quasiparticles in quasi-one-dimensional quantum wires perturbed by weak quenched disorder. Past works using this approach have considered a single chain. Here, we extend the formalism to treat a quasi-one-dimensional system: a quantum wire with an arbitrary number of channels coupled by random hopping amplitudes. The computations are carried out explicitly for the case of a chiral quasi-one-dimensional wire with broken time-reversal symmetry (chiral-unitary symmetry class). By treating the space direction along the chains as imaginary time, the effects of the disorder are encoded in the time evolution induced by a single site superspin (non-Hermitian) Hamiltonian. We obtain the density of states near the band center of an infinitely long quantum wire. Our results agree with those based on the Dorokhov–Mello–Pereyra–Kumar equation for the chiral-unitary symmetry class.     

Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential

We present calculations of the localisation length, , for two interacting particles (TIP) in a one-dimensional random potential, presenting its dependence on disorder, interaction strength U and system size. is computed by a decimation method from the decay of the Green function along the diagonal of finite samples. Infinite sample size estimates are obtained by finite-size scaling. For U=0 we reproduce approximately the well-known dependence of the one-particle localisation length on disorder while for finite U, we find that with varying between and . We test the validity of various other proposed fit functions and also study the problem of TIP in two different random potentials corresponding to interacting electron-hole pairs. As a check of our method and data, we also reproduce well-known results for the two-dimensional Anderson model without interaction. Received 19 June 1998 and Received in final form 29 October 1998     

Two interacting particles in a random potential

We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.     

Localization and interaction in two dimensions: The disordered Tomonaga model

The effect of electron-electron interactions on the localization behavior of two-dimensional disordered systems is investigated in the framework of a generalized version of Tomonaga's model. The main result is a closed expression for the dynamical conductivity which allows to predict the critical electron concentration below which localization sets in as a function of microscopic parameters only. Experimental verification is possible, although difficult.     

Localization of light in evanescently coupled disordered waveguide lattices: Dependence on the input beam profile

We investigate how the transverse localization of light in evanescently coupled, disordered, lossless waveguide lattices depends on the shape and size of an input beam. Our detailed numerical study not only reveals waveguide-like propagation of the localized state inside such a disordered discrete medium but also shows that a specific localized state is independent of the spatial profile of the input beam. Dependence of the localized state on input beam parameters and lattice parameters is also reported. Our results should be of interest for engineering light propagation with discrete diffractive optics in practical optical geometries (e.g., microstructured arrays of optical waveguides, fiber arrays, etc.) and for realizing waveguide-like (without any diffractive spread) propagation even in the presence of structural disorders and refractive index perturbations.     

Localization properties of two interacting particles in a quasi-periodic potential with a metal-insulator transition

We study the influence of many-particle interactions on a metal-insulator transition. We consider the two-interacting-particle problem for onsite interacting particles on a one-dimensional quasiperiodic chain, the so-called Aubry-André model. We show numerically by the decimation method and finite-size scaling that the interaction does not modify the critical parameters such as the transition point and the localization-length exponent. We compare our results to the case of finite density systems studied by means of the density-matrix renormalization scheme. Received 28 June 2001     

There are no nice interfaces in (2+1)-dimensional SOS models in random media

We prove that in dimensiond2 translation-covariant Gibbs states describing rigid interfaces in a disordered solid-on-solid (SOS) cannot exist for any value of the temperature, in contrast to the situation ind3. The proof relies on an adaptation of a theorem of Aizenman and Wehr.     

Stability of hierarchical interfaces in a random field model

We study a hierarchical model for interfaces in a random-field ferromagnet. We prove that in dimensionD>3, at low temperatures and for weak disorder, such interfaces are rigid. Our proof uses renormalization group transformations for stochastic sequences.     

Localization for random and quasiperiodic potentials

A survey is made of some recent mathematical results and techniques for Schrödinger operators with random and quasiperiodic potentials. A new proof of localization for random potentials, established in collaboration with H. von Dreifus, is sketched.     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

Violation of the Bell inequality in quantum critical random spin-1/2 chains

We investigate the entanglement and nonlocality properties of two random XX spin-1/2 critical chains, in order to better understand the role of breaking translational invariance to achieve nonlocal states in critical systems. We show that breaking translational invariance is a necessary but not sufficient condition for nonlocality, as the random chains remain in a local ground state up to a small degree of randomness. Furthermore, we demonstrate that the random dimer model does not have the same nonlocality properties of the translationally invariant chain, even though they share the same universality class for a certain range of randomness.     

Stability of interfaces in a random environment. A rigorous renormalization group analysis of a hierarchical model

We study a hierarchical model of domain walls in aD-dimensional bond disordered Ising model at low temperatures. Using a renormalization group method inspired by the work of Bricmont and Kupiainen for the random field Ising model, we prove the existence of rigid interfaces at low enough temperatures in dimensionsD>3.     

Localization in disordered,nonlinear dynamical systems

We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrödinger equations, or randomly coupled rotors.     

Rigorous mean-field model for coherent-potential approximation: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner'sn-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites, but free. Freeness of random variables is an analog of the concept of independence for noncommuting random operators. A possible realization is the ensemble of randomly rotated matrices at different lattice sites. The one- and two-particle Green functions of the proposed Hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is novanishing everywhere inside the band. The long-range behavior and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multisite) model is a rigorous mean-field model for the (single-site) CPA. We show how the Lloyd model is included in our model and treat various kinds of noises.     

Scalar particles in a narrow-band periodic potential

A system of an infinite number of spinless particles in a narrow-band periodic potential is treated. The dimension of the space is arbitrary, the tight-binding approximation is used, and it is assumed that the filling fraction is nearly one electron per atom. After a preliminary discussion of the Hartree approximation, the full Schrödinger equation is considered and a rigorous spectral perturbation theory in the kinetic energy term is set up. To get rid of the lack of relative boundedness of this perturbation, a vacuum state is constructed and its energy renormalized to zero, and passage is made to an excitonic representation in which the quasiparticles appear naturally as local perturbations of the vacuum. In this representation, relative boundedness is recovered and Rayleigh-Schrödinger expansions can be used to find cluster expansions for all local observables.