From ballistic motion to localization: a phase space analysis

We introduce phase space concepts to describe quantum states in a disordered system. The merits of an inverse participation ratio defined on the basis of the Husimi function are demonstrated by a numerical study of the Anderson model in one, two, and three dimensions. Contrary to the inverse participation ratios in real and momentum space, the corresponding phase space quantity allows for a distinction between the ballistic, diffusive, and localized regimes on a unique footing and provides valuable insight into the structure of the eigenstates. Received 5 March 2002     

Infinite chain of different deltas: A simple model for a quantum wire

We present the exact diagonalization of the Schr?dinger operator corresponding to a periodic potential with N deltas of different couplings, for arbitrary N. This basic structure can repeat itself an infinite number of times. Calculations of band structure can be performed with a high degree of accuracy for an infinite chain and of the correspondent eigenlevels in the case of a random chain. The main physical motivation is to modelate quantum wire band structure and the calculation of the associated density of states. These quantities show the fundamental properties we expect for periodic structures although for low energy the band gaps follow unpredictable patterns. In the case of random chains we find Anderson localization; we analize also the role of the eigenstates in the localization patterns and find clear signals of fractality in the conductance. In spite of the simplicity of the model many of the salient features expected in a quantum wire are well reproduced. Received 24 June 2002 Published online 29 November 2002     

High-order vibrational relaxation: a nonperturbative theory

A nonperturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the positive frequency part of the time-dependent non-stationary displacement correlation function of atoms. The nonlinear integral equation for this function is derived and solved numerically. We have found that the rate exhibits a critical behavior: it sharply increases near a specific (critical) value(s) of the interaction; the corresponding dependence is characterized by the critical index k - 1, where k is the number of the created phonons. Received 2 May 2002 Published online 31 July 2002     

Coherent backscattering and localization in a self-attracting random walk model

Intensity propagation of waves in dilute 2D and 3D disordered systems is well described by a random walk path-model. In strongly scattering media, however, this model is not quite correct because of interference effects like coherent backscattering. In this letter, coherent backscattering is taken into account by a modified, self-attracting random walk. Straightforward simulations of this model essentially reproduce the results of current theories on “non-classical” transport behavior, i.e. Anderson localization in 1D and 2D for any amount of disorder and a phase transition from weak to strong localization in 3D. However, in the strongly scattering regime corrections are necessary to account for the finite number of light modes due to their non-vanishing lateral extention. Within our model this correction leads to the observation that strong localization does not take place. Received 17 September 2001     

Delocalization effects and charge reorganizations induced by repulsive interactions in strongly disordered chains

We study the delocalization effect of a short-range repulsive interaction on the ground state of a finite density of spinless fermions in strongly disordered one dimensional lattices. The density matrix renormalization group method is used to explore the charge density and the sensitivity of the ground state energy with respect to the boundary condition (the persistent current) for a wide range of parameters (carrier density, interaction and disorder). Analytical approaches are developed and allow to understand some mechanisms and limiting conditions. For weak interaction strength, one has a Fermi glass of Anderson localized states, while in the opposite limit of strong interaction, one has a correlated array of charges (Mott insulator). In the two cases, the system is strongly insulating and the ground state energy is essentially invariant under a twist of the boundary conditions. Reducing the interaction strength from large to intermediate values, the quantum melting of the solid array gives rise to a more homogeneous distribution of charges, and the ground state energy changes when the boundary conditions are twisted. In individual chains, this melting occurs by abrupt steps located at sample-dependent values of the interaction where an (avoided) level crossing between the ground state and the first excitation can be observed. Important charge reorganizations take place at the avoided crossings and the persistent currents are strongly enhanced around the corresponding interaction value. These large delocalization effects become smeared and reduced after ensemble averaging. They mainly characterize half filling and strong disorder, but they persist away of this optimal condition. Received 5 July 2000 and Received in final form 8 November 2000     

Diffusion, localization and dispersion relations on “small-world” lattices

The spectral properties of the Laplacian operator on “small-world” lattices, that is mixtures of unidimensional chains and random graphs structures are investigated numerically and analytically. A transfer matrix formalism including a self-consistent potential à la Edwards is introduced. In the extended region of the spectrum, an effective medium calculation provides the density of states and pseudo relations of dispersion for the eigenmodes in close agreement with the simulations. Localization effects, which are due to connectivity fluctuations of the sites are shown to be quantitatively described by the single defect approximation recently introduced for random graphs. Received 23 March 1999     

Finite-size scaling of the level compressibility at the Anderson transition

We compute the number level variance Σ ₂ and the level compressibility χ from high precision data for the Anderson model of localization and show that they can be used in order to estimate the critical properties at the metal-insulator transition by means of finite-size scaling. With N, W, and L denoting, respectively, linear system size, disorder strength, and the average number of levels in units of the mean level spacing, we find that both χ(N, W) and the integrated Σ ₂ obey finite-size scaling. The high precision data was obtained for an anisotropic three-dimensional Anderson model with disorder given by a box distribution of width W/2. We compute the critical exponent as ν≈ 1.45±0.12 and the critical disorder as W _c≈ 8.59±0.05 in agreement with previous transfer-matrix studies in the anisotropic model. Furthermore, we find χ≈ 0.28±0.06 at the metal-insulator transition in very close agreement with previous results. Received 1st November 2001 and Received in final form 8 March 2002 Published online 6 June 2002     

Competition of disorder and interchain hopping in a two-chain Hubbard model

We study the interplay of Anderson localization and interaction in a two chain Hubbard ladder allowing for arbitrary ratio of disorder strength to interchain coupling. We obtain three different types of spin gapped localized phases depending on the strength of disorder: a pinned 4k _F Charge Density Wave (CDW) for weak disorder, a pinned 2k _F CDW^π for intermediate disorder and two independently pinned single chain 2k _F CDW for strong disorder. Confinement of electrons can be obtained as a result of strong disorder or strong attraction. We give the full phase diagram as a function of disorder, interaction strength and interchain hopping. We also study the influence of interchain hopping on localization length and show that localization is enhanced by a small interchain hopping but suppressed by a large interchain hopping. Received 6 April 2001     

Persistent non-ergodic fluctuations in mesoscopic insulators: The NSS model in the unitary and symplectic ensembles

We give a detailed picture of the mesoscopic conductance fluctuations in the deep insulating regime (DIR) within the Nguyen, Spivak and Shklovskii model in the unitary and symplectic ensembles. Slutski's theorem is invoked to rigorously state the ergodic problem for conductance fluctuations in the DIR, in contrast with previous studies. A weakly decaying behavior of the log-conductance correlation function, even weaker when spin-orbit scatterers are included, is established on the relevant field scale of the model. Such a slow decay implies that the stochastic process, defined by the fluctuations of the log-conductance, is non-ergodic in the mean square sense in the ensembles with the reported symmetries. The results can be interpreted in terms of the effective number of samples within the available magnetic scale. Using the replica approach, we derive the strong localisation counterparts of the well known 'cooperon' and 'diffuson' which permit analyzing quantitatively the decaying behavior of the correlation function and reveal its symmetry related properties in agreement with the numerical results. Received 11 April 2002 / Received in final form 27 August 2002 Published online 19 November 2002     

Generalizations of Heisenberg uncertainty relation

A survey on the generalizations of Heisenberg uncertainty relation and a general scheme for their entangled extensions to several states and observables is presented. The scheme is illustrated on the examples of one and two states and canonical quantum observables, and spin and quasi-spin components. Several new uncertainty relations are displayed. Received 10 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: dtrif@inrne.bas.bg     

Persistent currents for Coulomb interacting electrons on 2d disordered lattices

A Wigner crystal structure of the electronic ground state is induced by strong Coulomb interactions at low temperature in clean or disordered two-dimensional (2d) samples. For fermions on a mesoscopic disordered 2d lattice, being closed to a torus, we study the persistent current in the regime of strong interaction at zero temperature. We perform a perturbation expansion starting from the Wigner crystal limit which yields power laws for the dependence of the persistent current on the interaction strength. The sign of the persistent current in the strong interaction limit is independent of the disorder realization and strength. It depends only on the electro-statically determined configuration of the particles in the Wigner crystal. Received 14 March 2000     

Time evolution of wave-packets in quasi-1D disordered media

We have investigated numerically the quantum evolution of a -like wave-packet in a quenched disordered medium described by a tight-binding Hamiltonian with long-range hopping (band random matrix approach). We have obtained clean data for the scaling properties in time and in the bandwidth b of the packet width and its fluctuations with respect to disorder realizations. We confirm that the fluctuations of the packet width in the steady-state show an anomalous scaling [0pt] with [0pt] . This can be related to the presence of non-Gaussian tails in the distribution of [0pt]. Finally, we have analysed the steady state probability profile and we have found 1/b corrections with respect to the theoretical formula derived by Zhirov in the limit, except at the origin, where the corrections are . Received 6 August 1999 and Received in final form 22 October 1999     

Delocalization and wave-packet dynamics in one-dimensional diluted Anderson models

We study the nature of one-electron eigen-states in a one-dimensional diluted Anderson model where every Anderson impurity is diluted by a periodic function f(l). Using renormalization group and transfer matrix techniques, we provide accurate estimates of the extended states which appear in this model, whose number depends on the symmetry of the diluting function f(l). The density of states (DOS) for this model is also numerically obtained and its main features are related to the symmetries of the diluting function f(l). Further, we show that the emergence of extended states promotes a sub-diffusive spread of an initially localized wave-packet.Received: 7 July 2003, Published online: 19 November 2003PACS: 63.50. + x Vibrational states in disordered systems - 63.22. + m Phonons or vibrational states in low-dimensional structures and nanoscale materials - 62.30. + d Mechanical and elastic waves; vibrations     

Delocalization in disordered chains with random symmetrical impurities

We study in this paper, with the context of a tight-binding on-side model, the electronic properties of one-dimensional random lattices with correlated impurities. We show that, when symmetrical impurities are inserted in a host chain of site energy and a constant hopping interaction V, diffusion will occur even when is random. We provide analytic expressions for the transmittance and confirm the theoretical results by a great deal of numerical calculations. When = V, we find that the mean-square displacement (MSD) follows the law m ² ∝t ^β with β = 2.0 for = constant and β = 1.0 for = = random, respectively. Received 15 January 2001 and Received in final form 30 April 2001     

Delocalization of two-particle ring near the Fermi level of 2d Anderson model

We study analytically and numerically the problem of two particles with a long range attractive interaction on a two-dimensional (2d) lattice with disorder. It is shown that below some critical disorder the interaction creates delocalized coupled states near the Fermi level. These states appear inside well localized noninteracting phase and have a form of two-particle ring which diffusively propagates over the lattice. Received 29 September 2000 and Received in final form 15 January 2001     

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice II

In this second paper, using N = 3 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional L×L square lattice with periodic boundary conditions and nearest neighbor hopping t, we show that a single unpaired fermion can co-exist with a correlated two particle Wigner molecule for intermediate values of the Coulomb energy to kinetic energy ratio r _s = UL/(2t ). This supports in an ultimate mesoscopic limit a possibility proposed by Andreev and Lifshitz for the thermodynamic limit: a quantum crystal may have delocalized defects without melting, the number of sites of the crystalline array being smaller than the total number of particles. When L = 6, the ground state exhibits four regimes as r_s increases: a Hartree-Fock regime, a first supersolid regime where a correlated pair co-exists with a third fully delocalized particle, a second supersolid regime where the third particle is partly delocalized, and eventually a correlated lattice regime. Received 22 October 2002 Published online 23 May 2003 RID="a" ID="a"e-mail: jpichard@cea.fr     

Andreev-Lifshitz supersolid revisited for a few electrons on a square lattice. I

In 1969, Andreev and Lifshitz have conjectured the existence of a supersolid phase taking place at zero temperature between the quantum liquid and the solid. In this and a succeeding paper, we re-visit this issue for a few polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion on a two dimensional L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. This paper is restricted to the magic number of particles N = 4 for which a square Wigner molecule is formed when U increases and to the size L = 6 suitable for exact numerical diagonalizations. When the Coulomb energy to kinetic energy ratio r _s = UL/(2t ) reaches a value r _s ^F ≈ 10, there is a level crossing between ground states of different momenta. Above r _s ^F, the mesoscopic crystallization proceeds through an intermediate regime ( r _s ^F < r _s < r _s ^W ≈ 28) where unpaired fermions with a reduced Fermi energy co-exist with a strongly paired, nearly solid assembly. We suggest that this is the mesoscopic trace of the supersolid proposed by Andreev and Lifshitz. When a random substrate is included, the level crossing at r _s ^F is avoided and gives rise to a lower threshold r _s ^F(W) < r _s ^F where two usual approximations break down: the Wigner surmise for the distribution of the first energy excitation and the Hartree-Fock approximation for the ground state. Received 21 June 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: jpichard@cea.fr     

T-matrix analysis of biexcitonic correlations in the nonlinear optical response of semiconductor quantum wells

A detailed numerical analysis of exciton-exciton interactions in semiconductor quantum wells is presented. The theory is based on the dynamics-controlled truncation formalism and evaluated for the case of resonant excitation of 1s-heavy-hole excitons. It is formulated in terms of standard concepts of scattering theory, such as the forward-scattering amplitude (or T-matrix). The numerical diagonalization of the exciton-exciton interaction matrix in the 1s-approximation yields the excitonic T-matrix. We discuss the role of the direct and exchange interaction in the effective two-exciton Hamiltonian, which determines the T-matrix, evaluated within the 1s-subspace, and also analyze the effects of the excitonic wave function overlap matrix. Inclusion of the latter is shown to effectively prevent the 1s-approximation from making the Hamiltonian non-hermitian, but a critical discussion shows that other artefacts may be avoided by not including the overlap matrix. We also present a detailed analysis of the correspondence between the excitonic T-matrix in the 1s-approximation and the well-known T-matrix governing two-particle interactions in two dimensional systems via short-range potentials. Received 3 August 2001 and Received in final form 26 December 2001     

Antiferromagnetism and superconductivity in a model with extended pairing interactions

The competition between antiferromagnetism and the d + id superconducting state is studied in a model with near and next near neighbour interactions in the absence of any on-site repulsion. A mean field study shows that it is possible to have simultaneous occurrence of an antiferromagnetic and a singlet d + id superconducting state in this model. In addition, such a coexistence generates a triplet d + id superconducting order parameter with centre of mass momentum Q = (π,π) dynamically having the same orbital symmetry as the singlet superconductor. Inclusion of next nearest neighbour hopping in the band stabilises the d_xy superconducting state away from half filling, the topology of the phase diagram, though, remains similar to the near neighbour model. In view of the very recent observation of a broad region of coexistence of antiferromagnetic and unconventional superconducting states in organic superconductors, the possibility of observation of the triplet state has been outlined. Received 30 November 2000 and Received in final form 27 March 2001