Localization of weakly disordered flat band states

Certain tight binding lattices host macroscopically degenerate flat spectral bands. Their origin is rooted in local symmetries of the lattice, with destructive interference leading to the existence of compact localized eigenstates. We study the robustness of this localization to disorder in different classes of flat band lattices in one and two dimensions. Depending on the flat band class, the flat band states can either be robust, preserving their strong localization for weak disorder W, or they are destroyed and acquire large localization lengths ξ that diverge with a variety of unconventional exponents ν, ξ ~ 1 /W ^ν .     

Dimension of an optimum impurity cluster and multiple-occupancy corrections in the theory of scattering of vibrational excitations in a solid solution

A relationship is derived for the correlation length L determining the size of the region in a solid solution in which excitations are scattered coherently. The correlation length depends on the fraction of impurity atoms x in the solid solution and the lattice dimension d. In the physical analysis of single-particle scattering processes in the solid solution and calculations, it is sufficient to take into account clusters with the number of cells n corresponding to the correlation volume L ^d . A theoretical analysis is illustrated by calculations of the spectral functions of the solid solution at different values of x and n. The multiple-occupancy corrections (polynomials in powers of x) to scattering diagrams are calculated using the method of sequential breaking apart of the interaction lines in the diagrams for the self-energy part. The method used was previously applied to the case of scattering by a single impurity. In this paper, the efficiency of the method is checked for scattering by multi-impurity clusters. It is demonstrated that the method can be useful in analyzing and estimating the contributions of scattering diagrams.     

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L?? ^d determines a lattice spin system with potential A _Φ . The pressure P(A _Φ ) and free energy F _AΦ (β)=?(1/β)P(βA _Φ ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Hölder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.     

Non-Fermi liquid theory of quantum Hall effects

Within the Grassmannian U(2N)/U(N) × U(N) nonlinear σ-model representation of localization, one can study the low-energy dynamics of both a free and interacting electron gas. We study the crossover between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.     

On the Problem of Dynamical Localization in the Nonlinear Schrödinger Equation with a Random Potential

We prove a dynamical localization in the nonlinear Schrödinger equation with a random potential for times of the order of O(β ^?2), where β is the strength of the nonlinearity.     

Optical Hall conductivity of systems with gapped spectral nodes

We calculate the optical Hall conductivity within the Kubo formalism for systems with gapped spectral nodes, where the latter have a power-law dispersion with exponent n. The optical conductivity is proportional to n and there is a characteristic logarithmic singularity as the frequency approaches the gap energy. The optical Hall conductivity is almost unaffected by thermal fluctuations and disorder for n = 1, whereas disorder has a stronger effect on transport properties if n = 2.     

Maximum coefficient of light extinction in a nonabsorbing dispersive medium

The maximum value of the light extinction coefficient μ, which can be observed in a dispersive medium with a relative refractive index n of the scattering particles, is studied within the framework of a quasi-crystalline approximation for nonabsorbing dispersive media consisting of monodisperse spherical scatterers. A change in the diffraction parameter x of the scattering particles and their volume concentration c _v is accompanied by nonmonotonic variations of the extinction coefficient, and the function μ(x, c _v ) exhibits several maxima. The dimensions and concentrations of particles are determined, for which the extinction coefficient reaches the absolute maximum μ^max. The μ^max value exhibits a monotonic growth with increasing relative refractive index n of the scattering particles. The conditions of validity of the Ioffe-Regel criterion of radiation localization have been studied. It is established that the localization in nonabsorbing dispersive media can be observed only for n ? 2.7. The intervals of x and c _v in which the criterion of radiation localization is satisfied in dispersive media consisting of particles with n = 3.0 and 3.5 are determined.     

Precision measurements of optical excitation functions for the mercury atom

We describe a computer-based facility for studying the excitation of atoms by ultramonochromatic electrons and give optical excitation functions for the 12 mercury spectral lines that originate from the n ¹ S ₀, n ¹ P ₁, n ¹ D ₂, n ³ S ₁, n ³ P _j , and n ³ D _j levels. We detected about 100 features in the energy dependences measured from the excitation threshold to 15.5 eV. The previously found positions of the features on the energy scale are in good agreement with our results. Most of the resonant features are shown to be mainly attributable to the decay of short-lived states of the negative mercury ion. We detected a postcollision interaction effect in the optical excitation functions of the lines that originate from the n ¹ S ₀ levels at energies of about 11 eV.     

Periodic Striped Ground States in Ising Models with Competing Interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J_c(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.     

The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The V_C,n voltage across each capacitor is selected as a non-linear function of the electric charge q_n, i.e., V_C,n = q_n(1/C_n -ε_n|q_n|²)where C_n is the linear part of the capacitance and ε_n the amplitude of the non-linear term. We follow a binary distribution of values of ε_n, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.     

Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction

We study disorder effects upon the temperature behavior of the upper critical magnetic field in an attractive Hubbard model within the generalized DMFT+Σ approach. We consider the wide range of attraction potentials U—from the weak coupling limit, where superconductivity is described by BCS model, up to the strong coupling limit, where superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures significantly higher than superconducting transition temperature, as well as the wide range of disorder—from weak to strong, when the system is in the vicinity of Anderson transition. The growth of coupling strength leads to the rapid growth of H_c2(T), especially at low temperatures. In BEC limit and in the region of BCS–BEC crossover H_c2(T), dependence becomes practically linear. Disordering also leads to the general growth of H_c2(T). In BCS limit of weak coupling increasing disorder lead both to the growth of the slope of the upper critical field in the vicinity of the transition point and to the increase of H_c2(T) in the low temperature region. In the limit of strong disorder in the vicinity of the Anderson transition localization corrections lead to the additional growth of H_c2(T) at low temperatures, so that the H_c2(T) dependence becomes concave. In BCS–BEC crossover region and in BEC limit disorder only slightly influences the slope of the upper critical field close to T _c . However, in the low temperature region H_c2 (T may significantly grow with disorder in the vicinity of the Anderson transition, where localization corrections notably increase H_c2 (T = 0) also making H_c2(T) dependence concave.     

Disorder effect on the quantum Hall effect in thin films of three-dimensional topological insulators

We numerically study the quantum Hall effect (QHE) in three-dimensional topological insulator (3DTI) thin film in the presence of the finite Zeeman energy g and the hybridization gap Δ under a strong magnetic field and disorder. For Δ = 0 but g ≠ 0, the Hall conductivity remains to be odd-integer quanti-zed σ _xy = ν(e ²/h) , where ν = 2? + 1 with ? being an integer. In the presence of disorder, the Hall plateaus can be destroyed through the float-up of extended levels toward the band center and the higher plateaus disappear first. The two central plateaus with ν = ± 1 around the band center are strongest against disorder scattering. With the increasing of the disorder strength, Hall plateaus are destroyed faster for the system with a weaker magnetic field. If g = 0 but Δ ≠ 0, there is a splitting of the central (n = 0) Landau level, yielding a new plateau with ν = 0, in addition to the original odd-integer plateaus. In the strong-disorder regime, the QHE plateaus can be destroyed due to the float-up of extended levels toward the band center. The ν = 0 plateau around the band center is strongest against disorder scattering, which eventually disappears. For both g ≠ 0 and Δ ≠ 0, the simultaneous presence of nonzero g and Δ causes the splitting of the degenerating Landau levels, so that all integer Hall plateaus ν = ? appear. The ν = 0,1 plateaus are the most stable ones. In the strong-disorder regime, all QHE states are destroyed by disorder, and the system transits into an insulating phase.     

1 / <Emphasis Type="Italic">n</Emphasis> Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy

Using a 1 / n expansion, that is an expansion in descending powers of n, for the number of matchings in regular graphs with 2n vertices, we study the monomer-dimer entropy for two classes of graphs. We study the difference between the extensive monomer-dimer entropy of a random r-regular graph G (bipartite or not) with 2n vertices and the average extensive entropy of r-regular graphs with 2n vertices, in the limit \(n \rightarrow \infty \). We find a series expansion for it in the numbers of cycles; with probability 1 it converges for dimer density \(p < 1\) and, for G bipartite, it diverges as \(|\mathrm{ln}(1-p)|\) for \(p \rightarrow 1\). In the case of regular lattices, we similarly expand the difference between the specific monomer-dimer entropy on a lattice and the one on the Bethe lattice; we write down its Taylor expansion in powers of p through the order 10, expressed in terms of the number of totally reducible walks which are not tree-like. We prove through order 6 that its expansion coefficients in powers of p are non-negative.     

Congratulations to Akusticheskii Zhurnal on the occasion of its 50th anniversary. Vibrations of spherical inclusions in elastic solids

Variational principles are derived for the analysis of dynamical phenomena associated with spherical inclusions embedded in homogeneous isotropic elastic solids. The starting point is Hamilton’s principle, with the material properties assumed to vary only with the radial distance r from the origin. Attention is restricted to disturbances that are symmetric about the polar (z) axis, such that the nonzero displacement components in spherical coordinates, u _r and u_θ, are independent of the polar coordinate φ. The symmetry allows for a decoupling of the polar components, the nth of which is described by U _{r, n} (r, t)P _n (cosθ) and U_{θ, n}(r, t)dP _n /dθ. A variational principle is subsequently derived for the field quantities U _{r, n} and U_{θ, n}. Concepts analogous to those of the theory of matched asymptotic expansions are used to embellish the principle in order to allow for the damping associated with the outward radiation of elastic waves. Examples illustrating the use of the variational principle for formulating plausible lumped-parameter models are given for the cases of n = 0 and n = 1.     

Observational Data Fitting to Constrain Variable Modified Chaplygin Gas in the Background of Horava-Lifshitz Gravity

FRW universe in Horava-Lifshitz (HL) gravity model filled with a combination of dark matter and dark energy in the form of variable modified Chaplygin gas (VMCG) is considered. The permitted values of the VMCG parameters are determined by the recent astrophysical and cosmological observational data. Here we present the Hubble parameter in terms of the observable parameters Ω _{d m0}, Ω _{v m c g0}, H ₀, redshift z and other parameters like α, A, γ and n. From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the χ ² test. The best-fit values of the parameters are obtained by 66 %, 90 % and 99 % confidence levels. Next due to joint analysis with BAO and CMB observations, we have also obtained the bounds of the parameters (A, γ) by fixing some other parameters α and n. The best fit value of distance modulus μ(z) is obtained for the VMCG model in HL gravity, and it is concluded that our model is perfectly consistent with the union2 sample data.     

Doppler three-level (V-scheme) laser without population inversion

The theory of amplification and lasing without population inversion in a three-level medium with inhomogeneous broadening via the formation of an open V configuration is elaborated. The conditions for energy transfer from the infrared into the visible spectral range, i.e., the conditions of up-conversion n _b >n _c >n _a , and the external field required for saturation of the b?a transition are established. Two-photon resonant Raman transitions in ensemble of mobile atoms of a gas-discharge plasma are analyzed. The frequency shift of the probe field spectrum as a whole is shown to be governed by the frequency shift of the pump field multiplied by the ratio of the wave numbers of the probe amplification field and the pump field. The interaction of atoms through Ne transitions with the pump field (λ=1.15 εm, 2p ₄-2s ₂ transition) and the lasing field (λ=0.6328 εm, 3s ₂-2p ₂ transition) with an increase in the lasing frequency by a factor of 1.82 with respect to the absorbed radiation is calculated.     

Eigenfunction structure and scaling of two interacting particles in the one-dimensional Anderson model

The localization properties of eigenfunctions for two interacting particles in theone-dimensional Anderson model are studied for system sizes up to N = 5000 sitescorresponding to a Hilbert space of dimension ≈10⁷ using the Green function Arnoldi method. Theeigenfunction structure is illustrated in position, momentum and energy representation,the latter corresponding to an expansion in non-interacting product eigenfunctions.Different types of localization lengths are computed for parameter ranges in system size,disorder and interaction strengths inaccessible until now. We confirm that one-parameterscaling theory can be successfully applied provided that the condition of N being significantlylarger than the one-particle localization length L₁ is verified.The enhancement effect of the two-particle localization length L₂ behaving asL₂ ~ L₂¹ is clearly confirmed for a certain quite large intervalof optimal interactions strengths. Further new results for the interaction dependence in avery large interval, an energy value outside the band center, and different interactionranges are obtained.     

Lower and upper bounds on the critical temperature for anisotropic three-dimensional Ising model

The Ising model is considered on a simple cubic lattice, with a coupling constant J along one axis and coupling constants J’ along the remaining two axes. The transfer-matrix technique and an extended phenomenological renormalization group theory [18, 19] are applied to obtain two-sided bounds on the critical temperature for the model with J′/J≤1. The bounds monotonically converge with decreasing J′/J and provide improved estimates for the phase-transition temperature in anisotropic three-dimensional Ising model, as compared with those available from the literature.     

Der Grundzustand der Urfermionen im Gitterraum

The energy of the Dirac sea of interacting urfermions in a lattice space withZ ³ points is calculated using Heisenberg's Hamiltonian and a two-particle approximation which is a variational calculation with the test function ¦?〉=e ⁱη¦D ₀〉; ¦D ₀〉 is the Dirac sea without interaction,η=(ψ ^° ψ) a bilinear expression of the urfermion creation and annihilation operators. The same result is obtained by a BCS-calculation. Beyond that, we derive simple lower and upper bounds for the energy. Excited states are considered consisting of a particle-antiparticle pair with the energyE=2√ω ²+M ². The massM and the interaction constantW are connected by the equation (4W)^?1=Z ^?3∑(ω ²+M ²)^?1/2. For usual masses 4W～√Z/1 (1 a nuclear length). Methods are discussed to improve the results.     

Resonant tunneling effect in one-dimensional twinned lattice photonic crystal under total reflection conditions

Under total reflection conditions, it typically seems as though light waves will be reflected completely on the interface; in actuality, the waves can penetrate the medium as evanescent waves. In this paper, we present a twinned lattice photonic crystal with a unit cell composed of AB layers and their mirror. We assume that the refractive index n ₀ of the input and output end is equal to n _B and larger than n _A . We first demonstrate the dependence of band structure on the incidence angle and normalized wavelength, in which the resonant tunneling bands are exposed. We then draw a comparison of bands between ABBA and AB. To conclude, we discuss the resonant tunneling effect in the twinned lattice photonic crystal under the total reflection conditions. As incidence angle increases, the resonant tunneling band ultimately vanishes completely.