# LPTMS Publications

Archives :

• ## Complex Periodic Orbits and Tunnelling in Chaotic Potentials

### Stephen C. Creagh 1, Niall D. Whelan 1

#### Physical Review Letters 77 (1996) 4975-4979

We derive a trace formula for the splitting-weighted density of states suitable for chaotic potentials with isolated symmetric wells. This formula is based on complex orbits which tunnel through classically forbidden barriers. The theory is applicable whenever the tunnelling is dominated by isolated orbits, a situation which applies to chaotic systems but also to certain near-integrable ones. It is used to analyse a specific two-dimensional potential with chaotic dynamics. Mean behaviour of the splittings is predicted by an orbit with imaginary action. Oscillations around this mean are obtained from a collection of related orbits whose actions have nonzero real part.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• ## Diffusion in one dimensional random medium and hyperbolic brownian motion

### Alain Comtet 1, 2, Cecile Monthus 1, 2

#### Journal of Physics A 29 (1996) 1331-1345

Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this relationship and study various distributions using stochastic calculus and functional integration.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. LPTPE, Université Paris VI - Pierre et Marie Curie

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• ## Distribution of Eigenvalues for the Modular Group

### E. Bogomolny 1, F. Leyvraz 1, 2, C. Schmit 1

#### Communications in Mathematical Physics 176 (1996) 577-617

The two-point correlation function of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that ion the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. Instituto de Fisica, University of Mexico

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• ## Equation of State for Exclusion Statistics in a Harmonic Well

### Serguei B. Isakov 1, Stephane Ouvry 2

#### Journal of Physics A 29 (1996) 7401-7407

We consider the equations of state for systems of particles with exclusion statistics in a harmonic well. Paradygmatic examples are noninteracting particles obeying ideal fractional exclusion statistics placed in (i) a harmonic well on a line, and (ii) a harmonic well in the Lowest Landau Level (LLL) of an exterior magnetic field. We show their identity with (i) the Calogero model and (ii) anyons in the LLL of an exterior magnetic field and in a harmonic well.

• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• ## Exponents appearing in heterogeneous reaction-diffusion models in one dimension

### Cecile Monthus 1, 2

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 54 (1996) 4844-4859

We study the following 1D two-species reaction diffusion model : there is a small concentration of B-particles with diffusion constant $D_B$ in an homogenous background of W-particles with diffusion constant $D_W$; two W-particles of the majority species either coagulate ($W+W \\longrightarrow W$) or annihilate ($W+W \\longrightarrow \\emptyset$) with the respective probabilities $p_c=(q-2)/(q-1)$ and $p_a=1/(q-1)$; a B-particle and a W-particle annihilate ($W+B \\longrightarrow \\emptyset$) with probability 1. The exponent $\\theta(q,\\lambda=D_B/D_W)$ describing the asymptotic time decay of the minority B-species concentration can be viewed as a generalization of the exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D $q$-state Potts model starting from a random initial condition : the W-particles represent domain walls, and the exponent $\\theta(q,\\lambda)$ characterizes the time decay of the probability that a diffusive \'spectator\' does not meet a domain wall up to time $t$. We extend the methods introduced by Derrida, Hakim and Pasquier ({\\em Phys. Rev. Lett.} {\\bf 75} 751 (1995); Saclay preprint T96/013, to appear in {\\em J. Stat. Phys.} (1996)) for the problem of persistent spins, to compute the exponent $\\theta(q,\\lambda)$ in perturbation at first order in $(q-1)$ for arbitrary $\\lambda$ and at first order in $\\lambda$ for arbitrary $q$.

• 1. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• ## Integrability and Disorder in Mesoscopic Systems: Application to Orbital Magnetism

### K. Richter 1, 2, D. Ullmo 3, 4, R. A. Jalabert 5

#### Journal of Mathematical Physics 37 (1996) 5087-5110

We present a semiclassical theory of weak disorder effects in small structures and apply it to the magnetic response of non-interacting electrons confined in integrable geometries. We discuss the various averaging procedures describing different experimental situations in terms of one- and two-particle Green functions. We demonstrate that the anomalously large zero-field susceptibility characteristic of clean integrable structures is only weakly suppressed by disorder. This damping depends on the ratio of the typical size of the structure with the two characteristic length scales describing the disorder (elastic mean-free-path and correlation length of the potential) in a power-law form for the experimentally relevant parameter region. We establish the comparison with the available experimental data and we extend the study of the interplay between disorder and integrability to finite magnetic fields.

• 1. Institut für Physik, Institut für Physik
• 2. Max-Planck-Institut für Physik komplexer Systeme, Max-Planck-Institut
• 3. Bell Laboratories, Lucent Technologies, Bell Laboratories
• 4. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 5. Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS : UMR7504 – Université Louis Pasteur - Strasbourg I

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• ## Models of traps and glass phenomenology

### Cecile Monthus 1, 2, Jean-Philippe Bouchaud 3

#### Journal of Physics A 29 (1996) 3847-3869

We study various models of independent particles hopping between energy traps\' with a density of energy barriers $\\rho(E)$, on a $d$ dimensional lattice or on a fully connected lattice. If $\\rho(E)$ decays exponentially, a true dynamical phase transition between a high temperature liquid\' phase and a low temperature aging\' phase occurs. More generally, however, one expects that for a large class of $\\rho(E)$, interrupted\' aging effects appear at low enough temperatures, with an ergodic time growing faster than exponentially. The relaxation functions exhibit a characteristic shoulder, which can be fitted as stretched exponentials. A simple way of introducing interactions between the particles leads to a modified model with an effective diffusion constant in energy space, which we discuss in detail.

• 1. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 3. Service de physique de l'état condensé (SPEC), CNRS : URA2464 – CEA : DSM/IRAMIS

Details Citations to the Article (247)

• ## Orbital Magnetism in the Ballistic Regime: Geometrical Effects

### K. Richter 1, 2, D. Ullmo 1, 3, R. A. Jalabert 1, 4

#### Physics Reports 276 (1996) 1-83

We present a general semiclassical theory of the orbital magnetic response of noninteracting electrons confined in two-dimensional potentials. We calculate the magnetic susceptibility of singly-connected and the persistent currents of multiply-connected geometries. We concentrate on the geometric effects by studying confinement by perfect (disorder free) potentials stressing the importance of the underlying classical dynamics. We demonstrate that in a constrained geometry the standard Landau diamagnetic response is always present, but is dominated by finite-size corrections of a quasi-random sign which may be orders of magnitude larger. These corrections are very sensitive to the nature of the classical dynamics. Systems which are integrable at zero magnetic field exhibit larger magnetic response than those which are chaotic. This difference arises from the large oscillations of the density of states in integrable systems due to the existence of families of periodic orbits. The connection between quantum and classical behavior naturally arises from the use of semiclassical expansions. This key tool becomes particularly simple and insightful at finite temperature, where only short classical trajectories need to be kept in the expansion. In addition to the general theory for integrable systems, we analyze in detail a few typical examples of experimental relevance: circles, rings and square billiards. In the latter, extensive numerical calculations are used as a check for the success of the semiclassical analysis. We study the weak-field regime where classical trajectories remain essentially unaffected, the intermediate field regime where we identify new oscillations characteristic for ballistic mesoscopic structures, and the high-field regime where the typical de Haas-van Alphen oscillations exhibit finite-size corrections. We address the comparison with experimental data obtained in high-mobility semiconductor microstructures discussing the differences between individual and ensemble measurements, and the applicability of the present model.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. Institut für Physik, Institut für Physik
• 3. AT&T Bell Laboratories, Bell Laboratories
• 4. Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS : UMR7504 – Université Louis Pasteur - Strasbourg I

Details Citations to the Article (92)
• ## Partial Dynamical Symmetry and Mixed Dynamics

### A. Leviatan 1, 2, N. D. Whelan 2, 3

#### Physical Review Letters 77 (1996) 5202-5205

Partial dynamical symmetry describes a situation in which some eigenstates have a symmetry which the quantum Hamiltonian does not share. This property is shown to have a classical analogue in which some tori in phase space are associated with a symmetry which the classical Hamiltonian does not share. A local analysis in the vicinity of these special tori reveals a neighbourhood of phase space foliated by tori. This clarifies the suppression of classical chaos associated with partial dynamical symmetry. The results are used to divide the states of a mixed system into chaotic\'\' and regular\'\' classes.

• 1. Racah Institute of Physics, Hebrew University of Jerusalem
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 3. Niels Bohr Institute (NBI), Niels Bohr Institute

Details Citations to the Article (16)
• ## Quantum Chaotic Dynamics and Random Polynomials

### E. Bogomolny 1, O. Bohigas 1, P. Leboeuf 1

#### Journal of Statistical Physics 85 (1996) 639-679

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of \'quantum chaotic dynamics\'. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wave-functions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. Special attention is devoted all over the paper to the role of symmetries in the distribution of roots of random polynomials.

• 1. Division Physique Théorique IPN, Université Paris XI - Paris Sud

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• ## Random Magnetic Impurities and the delta Impurity Problem

### Jean Desbois 1, Cyril Furtlehner 1, Stéphane Ouvry 1

#### Journal de Physique I 6 (1996) 641-648

One considers the effect of disorder on the 2-dimensional density of states of an electron in a constant magnetic field superposed onto a Poissonnian random distribution of point vortices. If one restricts the electron Hilbert space to the lowest Landau level of the total average magnetic field, the random magnetic impurity problem is mapped onto a contact $\\delta$ impurity problem. A brownian motion analysis of the model, based on brownian probability distributions for arithmetic area winding sectors, is also proposed. PACS numbers: 05.30.-d, 05.40.+j, 11.10.-

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• ## Random walk on Bethe lattice and hyperbolic geometry

### Cecile Monthus 1, Chistophe Texier 2

#### Journal of Physics A 29 (1996) 2399-2409

We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line. We also study the continuous limit of this model, and discuss in detail the relation between the random walk on the Bethe lattice and Brownian motion on a space of constant negative curvature.

• 1. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• ## Sample-size dependence of the ground-state energy in a one-dimensional localization problem

### C. Monthus 1, G. Oshanin 2, A. Comtet 2, 3, S. F. Burlatsky 4

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 54 (1996) 231

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-average ground-state energy decreases with an increase of the size $R$ of the sample as a stretched-exponential function, $\\exp( - R^{z})$, where the characteristic exponent $z$ depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that $z = 1/3$. We also predict the value of $z$ in the general case of Gaussian random potentials with correlations.

• 1. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 3. LPTPE, Université Paris VI - Pierre et Marie Curie
• 4. Department of Chemistry, BG-10, University of Washington

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• ## The Level Splitting Distribution in Chaos-assisted Tunneling

### F. Leyvraz 1, D. Ullmo 1, 2

#### Journal of Physics A 29 (1996) 2529-2551

A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. %The validity of this form can then in %principle be checked either numerically or even experimentally. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. Bell Laboratories 1D-265, Bell Laboratories 1D-265

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