A Tomography of the GREM: Beyond the REM Conjecture

In a companion paper we proved that in a large class of Gaussian disordered spin systems the local statistics of energy values near levels N^1/2+α with α<1/2 are described by a simple Poisson process. In this paper we address the issue as to whether this is optimal, and what will happen if α=1/2. We do this by analysing completely the Gaussian Generalised Random Energy Models (GREM). We show that the REM behaviour persists up to the level β_cN, where β_c denotes the critical temperature. We show that, beyond this value, the simple Poisson process must be replaced by more and more complex mixed Poisson point processes. Research supported in part by the DFG in the Dutch-German Bilateral Research Group ``Mathematics of Random Spatial Models from Physics and Biology' and by the European Science Foundation in the Programme RDSES.     

Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to be satisfied in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. Research supported in part by the DFG in the Dutch-German Bilateral Research Group ``Mathematics of Random Spatial Models from Physics and Biology' and by the European Science Foundation in the Programme RDSES.     

Effect of the Dzyaloshinskii-Moriya interaction on heat conductivity in one-dimensional quantum Ising chains

The effect of the Dzyaloshinskii-Moriya (DM) interaction on the heat conduction in the quantum Ising chain has been studied by solving the Lindblad master equation. The chain is subject to a uniform transverse field h, while the exchange couplings {J _m } between the nearest-neighbor spins are either uniform, random or quasi-periodic. The average energy-density profile and the average energy current in the non-equilibrium steady state have been numerically calculated. The ballistic transport is observed in the uniform Ising chain with DM interaction. For the random Ising chain with DM interaction, the energy gradient is observed in the bulk of the spin chain whose energy current appears to scale as the system size ⟨Q⟩ ∼ exp(βN) with β < 0. For the quasi-periodic Ising chain with DM interaction, the J _m takes the two values J _A and J _B arranged in the Fibonacci sequence. The energy gradient also exists in the spin chain and the energy current behaves as ⟨Q⟩ ∼ N ^α with α < 0. By increasing the strength of the DM interaction D, a non-trivial transition from the thermal insulator heat transport to anomalous heat conduction is found in the Fibonacci Ising chain with large ratio of couplings λ = J _A /J _B . A rough phase diagram of λ vs. D is given in this paper as well.     

Focal switch of cosine-Gaussian beams

The focal switch of cosine-Gaussian (CsG) beams passing through a system with the aperture and lens separated is studied analytically and numerically. It is shown that the focal switch of CsG beams can appear not only for the apertured case, but also for the unapertured case. The necessary condition for the focal switch is that truncation parameter α > α_c and the beam parameter β > β_c, α_c, β_c being the corresponding critical values. There exists a maximum of the relative transition height Δz _sw as α varies, and Δz _sw increases with increasing β and decreasing N _w. The normalized axial intensity minimum I _min / I _max decreases with an increase of α and β, and I _min / I _max remains unchanged as N _w varies.     

Threshold open charm production in nucleon-nucleon collisions

The associative charm particles production in nucleon-nucleon collisions N + N↦Λ_c(Σ_c) + ˉD + N, is described in a general way and the spin and isospin structure of the corresponding matrix elements are derived. Using an analogy with strange-particle (ΛK) production, the D-meson exchange model is considered in detail. Estimations of the energy behavior of the threshold cross-sections show a large dependence on the form of the phenomenological hadron form factors and indicate that, at threshold, the cross-section is three orders of magnitude smaller than for strange-particle production. Received: 18 October 2002 / Accepted: 5 December 2002 / Published online: 18 March 2003 RID="a" ID="a"Permanent address: National Science Center KFTI, 310108 Kharkov, Ukraine. RID="b" ID="b"e-mail: etomasi@cea.fr Communicated by V.V. Anisovich     

The Effect of Disorder on the Free-Energy for the Random Walk Pinning Model: Smoothing of the Phase Transition and Low Temperature Asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in (Berger and Toninelli (Electron. J. Probab., to appear) and Birkner and Sun (Ann. Inst. Henri Poincaré Probab. Stat. 46:414–441, 2010; ). Given a fixed realization of a random walk Y on ℤ ^d with jump rate ρ (that plays the role of the random medium), we modify the law of a random walk X on ℤ ^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=ò₀^t 1_Xs=Ys dsL_{t}(X,Y)=\int_{0}^{t} \mathbf {1}_{X_{s}=Y_{s}}\,\mathrm {d}s: the weight of the path under the new measure is exp (βL _t (X,Y)), β∈ℝ. As β increases, the system exhibits a delocalization/localization transition: there is a critical value β _c , such that if β>β _c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d≥3, the presence of disorder makes the phase transition at least of second order. This, in dimension d≥4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.     

A Farey Fraction Spin Chain

We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and a phase transition occurs for positive inverse temperature β= 2. The free energy is the same as that of related, non-translation-invariant number-theoretic spin chain. Using a number-theoretic argument, the low-temperature (β > 3) state is shown to be completely magnetized for long chains. The number of states of energy E= log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at β= 2, and suggesting a possible connection with the Riemann ζ-function. The spin interaction coefficients include all even many-body terms and are translation invariant. Computer results indicate that all the interaction coefficients, except the constant term, are ferromagnetic. Received: 20 August 1998/ Accepted: 17 December 1998     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

基于幂次相互作用的二维磁性团簇耦合能研究

在扩散限制凝聚模型基础上,采用Monte Carlo方法模拟了磁耦合作用随粒子间距离幂次变化的磁性粒子动力学凝聚过程.重点研究了在不同幂指数α值下团簇在生长过程中,即随着粒子数N的增加,团簇平均耦合能E_c(N)的演化过程.模拟结果表明:对于α≥5时,E_c(N)随着粒子数N的增加变化较小;当α=2时,E关键词： 扩散限制凝聚模型 幂次相互作用 耦合能     

Dynamics of α-clusters in N = Z nuclei

The systematics for binding energies per α-particle in N = Z nuclei, E _Bα/N _α, are studied up to ¹⁶⁴Pb. It is shown that, although a geometrical model can be used to explain the systematics for light nuclei, the binding energy per α-particle exhibits structures which are due to the well-known shells of the mean field of nucleons in nuclei. The overall dependence of E _Bα/N _α on N _α in N = Z nuclei (for the ground-state masses) can be described in a liquid-drop model of α-particles. Conditions for a phase change with the formation of an α-particle condensate, a dilute Bose gas in excited compound nuclei are discussed for E _Bα/N _α = 0, at the thresholds. This is achieved when the binding energy per nucleon in nuclei is equal to or smaller than in the α-cluster. At somewhat smaller excitation energies the appearance of a Bose gas with a closed-shell core (N = Z, e.g. of ⁴⁰Ca) is proposed within the same concept. The experimental observation of the decay of such condensed α-particle states is proposed with the coherent emission of several correlated α-particles not described by the Hauser-Feshbach approach for compound-nucleus decay. This decay will be observed by the emission of unbound resonances in the form of ⁸Be and ¹²C ^* (0⁺ ₂) clusters.     

EPR of manganese-containing plasticine: Spectral simulation and use as an internal standard for comparison of the signal intensities of various free-radical EPR spectra

Electron paramagnetic resonance spectra of manganese-containing plasticine putty have been obtained both at 9.4 and 94.2 GHz (room temperature). It has been possible to analyze both, quantitatively by computer-based simulation and exact diagonalization of a general spin-Hamiltonian, by use of an appropriate powder technique. A fine fit was obtained for both frequencies with an isotropicg-2.00117, uniaxial⁵⁵Mn hyperfine valuesA _∥/g _c β _c = −93.5 G (where ‖ implies direction (ℤ) andA _⊥/g _c β _c = −94.3 G, a nuclear quadrupole effectP _z(⁵⁵Mn)=0, and a uniaxial electronic quadrupole matrix (D/g _c β _c = 3D _Z/2g _c β _c = −82.8 G). The electronic octupole energy was taken into account via the single parameterB ₄⁰/g _c β _c = −0.024 G. The plasticine has been demonstrated to afford a fine intensity and magnetic-field standard, say, for use with free radicals such as spin-trap adducts. Examples provided are 2,2-diphenyl-l-picrylhydrazyl, and hydroxyl radical generated by the Fenton reaction and spin-trapped with 5,5-dimethyl-1-pyroline-N-oxide.     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

Statistical mechanics of braided Markov chains: I. Analytic methods and numerical simulations

We investigate numerically and analytically the statistics of Markov chains on so-called braid (B _n ) and locally free (ℒℱ _n ) groups. Namely, we compute the mean length 〈μ〉 and the variance 〈μ²〉−〈μ〉² of the shortest word which remains after applying of all group relations to the randomly generatedN-letter word (Markov chain). We express the conjecture (numerically justified) that the mean value 〈μ〉 for the random walk on the groupB _n (n≫1) coincides with high accuracy with the same value for the random walk on the “locally free group weth errors” if the number of errors is of order of 20%.     

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β _c , the inverse-gap is O(1) for β < β _c , polynomial in the surface area for β = β _c and exponential in it for β > β _c . This has been proved for \mathbbZ²{\mathbb{Z}^2} except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β _c and exponential for β > β _c were established, where β _c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β _c , the inverse-gap and mixing-time are both exp[Θ((β − β _c )h)].     

Surfing and generation of cosmic rays in relativistic shock waves

We consider the problem of cosmic-ray generation through the surfing acceleration of charged particles in relativistic magnetosonic shock waves (the branch of fast magnetic sound) propagating in magnetized space plasmas. The dependence of the particle surfing acceleration efficiency on the angle θ _Bn between the normal to the shock front plane and the magnetic field vector in the plasma upstream of the shock is analyzed in detail. We show that for angles satisfying the condition χ = βΓ tan θ _Bn ⩾ 1, where β = U/c, Γ = (1 − β)^{2 −1/2}, U is the shock velocity, and c is the speed of light, the particles can theoretically be accelerated through surfing for an unlimited time and can gain an unlimited energy. For angles satisfying the condition χ < 1, the kinetic energy ℰ of the particles is limited by ℰ = 2mc ²χ²/(1 − χ²) (m is the particle rest mass). Our main conclusion is that the generation of cosmic rays through the surfing acceleration of particles in the front of a relativistic shock wave for Γ ≫ 1 is also efficient when the angle θ _Bn is very small, i.e., it differs significantly from a right angle. Estimates for the energies of particles accelerated through surfing in relativistic jets are provided.     

Spectral Form Factors of Rectangle Billiards

The Berry–Tabor conjecture asserts that local statistical measures of the eigenvalues λ _j of a “generic” integrable quantum system coincide with those of a Poisson process. We prove that, in the case of a rectangle billiard with random ratio of sides, the sum behaves for τ random and N large like a random walk in the complex plane with a non-Gaussian limit distribution. The expectation value of the distribution is zero; its variance, which is essentially the average pair correlation function, is one, in accordance with the Berry–Tabor conjecture, but all higher moments (≥ 4) diverge. The proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces. The Berry–Tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space. Received: 16 February 1998 / Accepted: 24 April 1998     

A Random Surface Wave Equation for Bosonic QCD(<Emphasis Type="Italic">SU</Emphasis>(∞))

We propose a quantum surface wave functional describing the interaction between a colored SU(N _c ) membrane and a quantized Yang-Mills field. Additionally, we deduce its associated wave equation in the t’Hooft N _c →∞ limit. We show that its reproduces the Yang-Mills Field Theory at a large rigid random surface scale.     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

A pseudopotential approach to the superconducting state parameters of (Ni33Zr67)1? x M x (M=Ti, V, Co, Cu) ternary metallic glasses

Superconductivity in ternary metallic glasses has been investigated using the model pseudopotential approach, which has been found quite successful in explaining superconductivity in metals, binary alloys and binary glasses. It is observed that this simple methodology successfully explains superconducting behaviour of ternary glasses without requiring the solution of Dirac equation for a many body problem or estimation of various interactions as required in ab-initio pseudopotential theory. In the present work superconducting state parameters of fourteen metallic glasses of (Ni-Zr)-M system (M=Ti, V, Co, Cu) have been determined in the BCS-Eliashberg-McMillan framework. It is observed that addition of V, Co, and Cu as the third element (M) to a binary metallic glass (Ni₃₃ Zr₆₇) causes the parameters λ,T _c, α, andN ₀ V to decrease, and Coulomb pseudopotential (μ*) to increase with concentration of M, showing that the presence of third element (M) causes suppression of superconducting behaviour of the alloy. The decrease inT _c with increasing concentration of third element (M) may be attributed to the modifications in density of states at the Fermi levelN(E _F), and probable changes in the band structure of the alloy due to addition of the third element (M). Slight difference is noticed when Ti is added to the Ni₃₃ Zr₆₇ alloy. In this caseT _c rises initially and then decreases with concentration of M, showing a peak at aboutx=0.05. This indicates that on addition of Ti, 3d states grow near the Fermi level and hence contribute substantially toN(E _F), favouring superconducting behaviour in this case. The present results forT _c show an excellent agreement with the experimental data. QuadraticT _c equations have been proposed, which provide successfully theT _c values of ternary metallic glasses under consideration. Paper presented at National Conference on Current Trends in Condensed Matter Research, Warangal, India, September 20–22, 2004.     

On the investigation of magnetic critical phenomena in ferromagnets by μSR and PAC

On the basis of current theoretical views on the critical phenomena in isotropic Heisenberg ferromagnets the power temperature behavior Λ=c(τ)λ₀τ^-w has been derived for the muon spin relaxation rate Λ as π-T _c ⁻¹ (T-T _c ) → 0⁺. It is shown that the crossover from an exchange critical regime to a dipolar one is accompanied not only with the change in the critical exponentw in the above law, but also with the reduction of the coefficientc(π). A comparison with the temperature behaviour of the inverse nuclear relaxation timet _R ⁻¹ measured in the PAC experiment is carried out.