Nonanalytic Dependence Research Articles

Singular response of bistable systems driven by telegraph noise

We show that weak periodic driving can exponentially strongly change the rate of escape from a potential well of a system driven by telegraph noise. The analysis refers to an overdamped system, where escape requires that the noise amplitude θ exceed a critical value θ(c). For θ close to θ(c), the exponent of the escape rate displays a nonanalytic dependence on the amplitude of an additional low-frequency modulation. This leads to giant nonlinearity of the response of a bistable system to periodic modulation. Also studied is the linear response to periodic modulation far from θ(c). We analyze the scaling of the logarithm of the escape rate with the distance to the saddle-node and pitchfork bifurcation points. The analytical results are in excellent agreement with numerical simulations.

Generalizing the ADM computation to quantum field theory

The absence of recognizable, low energy quantum gravitational effects requires that some asymptotic series expansion be wonderfully accurate, but the correct expansion might involve logarithms or fractional powers of Newton’s constant. That would explain why conventional perturbation theory shows uncontrollable ultraviolet divergences. We explore this possibility in the context of the mass of a charged, gravitating scalar. The classical limit of this system was solved exactly in 1960 by Arnowitt, Deser and Misner, and their solution does exhibit nonanalytic dependence on Newton’s constant. We derive an exact functional integral representation for the mass of the quantum field theoretic system, and then develop an alternate expansion for it based on a correct implementation of the method of stationary phase. The new expansion entails adding an infinite class of new diagrams to each order and subtracting them from higher orders. The zeroth-order term of the new expansion has the physical interpretation of a first quantized Klein–Gordon scalar which forms a bound state in the gravitational and electromagnetic potentials sourced by its own probability current. We show that such bound states exist and we obtain numerical results for their masses.

Flows of Rényi entropies

We demonstrate that the condensed matter quantum systems encompassing two reservoirs connected by a junction permit a natural definition of flows of conserved measures, i.e., R\'enyi entropies. Such flows are similar to the flows of physical conserved quantities such as charge and energy. We develop a perturbation technique that permits efficient computation of R\'enyi entropy flows and analyze second- and fourth-order contributions. Second-order approximation was shown to correspond directly to the transition events in the system and thereby to possess a set of intuitive features. The analysis of fourth-order corrections reveals a more complicated picture: The intuitive relations do not hold anymore, and the corrections exhibit divergencies in low-temperature limit, manifesting an intriguing nonanalytical dependence of the flows on coupling strength in the limit of weak couplings and vanishing temperatures.

Interference between bulk and boundary scattering in high quality films

Quasiclassical interference between bulk and boundary scattering channels in thin metal films with rough surfaces is discussed. The effective transport time, which is calculated beyond Mathiessen's approximation, exhibits a nonanalytical dependence on the bulk relaxation time. Interference effects strongly affect the temperature (phonon scattering in the bulk) or concentration (impurity-scattering) dependencies of the conductivity. The results for large bulk free paths ${\mathcal{L}}_{b}$ and large correlation radii (lateral sizes) $R$ of surface inhomogeneities are described by simple analytical equations. At ${R}^{2}\ensuremath{\sim}a{\mathcal{L}}_{b}$ we predict a crossover between two asymptotic regimes for interference contributions that are characterized by distinct temperature/concentration dependencies. Experimental implications of our results are discussed.

ENTANGLEMENT IN TWO QUBIT MAGNETIC MODELS WITH DM ANTISYMMETRIC ANISOTROPIC EXCHANGE INTERACTION

In the present paper, an influence of the anisotropic antisymmetric exchange interaction, the Dzialoshinskii–Moriya (DM) interaction, on entanglement of two qubits in various magnetic spin models, including the pure DM model and the most general XYZ model, are studied. We find that the time evolution generated by DM interaction can implement the SWAP gate and discuss realistic quasi-one-dimensional magnets where it can be realized. It is shown that inclusion of the DM interaction to any Heisenberg model creates, when it does not exist, or strengthens, when it exists, the entanglement. We give physical explanation of these results by studying the ground state of the systems at T = 0. Nonanalytic dependence of the concurrence on the DM interaction and its relation with quantum phase transition is indicated. Our results show that spin models with the DM coupling have some potential applications in quantum computations and the DM interaction could be an efficient control parameter of entanglement.

Effect of electric field on diffusion in disordered materials. II. Two- and three-dimensional hopping transport

In the previous paper [Nenashev et al., arXiv:0912.3161] an analytical theory confirmed by numerical simulations has been developed for the field-dependent hopping diffusion coefficient D(F) in one-dimensional systems with Gaussian disorder. The main result of that paper is the linear, non-analytic field dependence of the diffusion coefficient at low electric fields. In the current paper, an analytical theory is developed for the field-dependent diffusion coefficient in three- and two-dimensional Gaussian disordered systems in the hopping transport regime. The theory predicts a smooth parabolic field dependence for the diffusion coefficient at low fields. The result is supported by Monte Carlo computer simulations. In spite of the smooth field dependences for the mobility and for the longitudinal diffusivity, the traditional Einstein form of the relation between these transport coefficients is shown to be violated even at very low electric fields.

Effect of electric field on diffusion in disordered materials. I. One-dimensional hopping transport

An exact analytical theory is developed for calculating the diffusion coefficient of charge carriers in strongly anisotropic disordered solids with one-dimensional hopping transport mode for any dependence of the hopping rates on space and energy. So far such a theory existed only for calculating the carrier mobility. The dependence of the diffusion coefficient on the electric field evidences a linear, non-analytic behavior at low fields for all considered models of disorder. The mobility, on the contrary, demonstrates a parabolic, analytic field dependence for a random-barrier model, being linear, non-analytic for a random energy model. For both models the Einstein relation between the diffusion coefficient and mobility is proven to be violated at any finite electric field. The question on whether these non-analytic field dependences of the transport coefficients and the concomitant violation of the Einstein's formula are due to the dimensionality of space or due to the considered models of disorder is resolved in the following paper [Nenashev et al., arXiv:0912.3169], where analytical calculations and computer simulations are carried out for two- and three-dimensional systems.

Bound States in the Quantum Scalar Electrodynamics

The next relativistic correction to α to for bound state mass of two charged scalar particles is calculated in the quantum scalar electrodynamics by the functional integral method. Contribution of the “nonphysical” time variable turned out to be important and leads to nonanalytic dependence of the bound state mass on α.

10.1007/s11448-008-2009-1

A theory of photoresonance transitions for the electrons localized above the solid H2 surface is proposed on the basis of the experimental data obtained by V. V. Zav’yalov and I. I. Smol’yaninov [Pis’ma Zh. Éksp. Teor. Fiz. 44, 142 (1986); Zh. Éksp. Teor. Fiz. 92, 339 (1987); Zh. Éksp. Teor. Fiz. 94, 307 (1988)] and a simple model of condensed hydrogen, H2. A new explanation is offered for the strong dependence of the transition frequency v on the hydrogen vapor pressure revealed in the above-cited works. In contrast to the notions that were proposed by V. V. Zav’yalov and I. I. Smol’yaninov [Zh. Éksp. Teor. Fiz. 92, 339 (1987); Zh. Éksp. Teor. Fiz. 94, 307 (1988)]; and V. B. Shikin and S. N. Nazin [Pis’ma Zh. Éksp. Teor. Fiz. 82, 752 (2005)] and based on the effects of the quantum refraction of electrons by hydrogen vapor atoms, the temperature dependence of the frequency ν = ν (T) is attributed here to the H2 vapor inhomogeneity. The nonanalytic dependence of ν on the vapor density n ν , ν(n ν ) − ν(0) ∼ n γ ν ∼ e−Δ/T, γ ≅ 0.56 has been revealed from the experimental data. The activation energy Δ corresponds to the “incomplete evaporation” of solid hydrogen [see J. I. Frenkel, Z. Phys. 26, 37 (1924); Kinetic Theory of Liquids (Nauka, Leningrad, 1945; Clarendon, Oxford, 1946)]. The parameter Δ is lower than the energy of the “complete evaporation” of solid H2 equal to 92.6 K.

Entanglement entropy in lattice gauge theories

We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories. It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way. In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions. Such an extension leads to a reduction procedure which can easily be implemented in lattice simulations by constructing lattices with special topology. This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations. On the simplest example of Z2 lattice gauge theory in dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region. We also comment on the holographic interpretation of the entanglement entropy.

Nonanalytic Spin Susceptibility of a Fermi Liquid: The Case of Fe-Based Pnictides

We propose an explanation of the peculiar linear temperature dependence of the uniform spin susceptibility chi(T) in ferropnictides. We argue that the linear in T term appears to be due to the nonanalytic temperature dependence of chi(T) in a two-dimensional Fermi liquid. We show that the prefactor of the T term is expressed via the square of the spin-density-wave (SDW) amplitude connecting nested hole and electron pockets. Because of an incipient SDW instability, this amplitude is large, which, along with a small value of the Fermi energy, makes the T dependence of chi(T) strong. We demonstrate that this mechanism is in quantitative agreement with the experiment.

Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition

We study nonanalytic paramagnetic response of an interacting Fermi system both away and in the vicinity of a ferromagnetic quantum phase transition (QCP). Previous studies found that (i) the spin susceptibility $\ensuremath{\chi}$ scales linearly with either the temperature $T$ or magnetic field $H$ in the weak-coupling regime; (ii) the interaction in the Cooper channel affects this scaling via logarithmic renormalization of prefactors of the $T$, $\ensuremath{\mid}H\ensuremath{\mid}$ terms, and may even reverse the signs of these terms at low enough energies. We show that Cooper renormalization becomes effective only at very low energies, which get even smaller near a QCP. However, even in the absence of such renormalization, generic (non-Cooper) higher-order processes may also inverse the sign of $T$, $\ensuremath{\mid}H\ensuremath{\mid}$ scaling. We derive the thermodynamic potential as a function of magnetization and show that it contains, in addition to regular terms, a nonanalytic ${\ensuremath{\mid}M\ensuremath{\mid}}^{3}$ term, which becomes ${M}^{4}∕T$ at finite $T$. We show that regular $({M}^{2},{M}^{4},\dots{})$ terms originate from fermions with energies of order of the bandwidth, while the nonanalytic term comes from low-energy fermions. We consider the vicinity of a ferromagnetic QCP by generalizing the Eliashberg treatment of the spin-fermion model to finite magnetic field, and show that the ${\ensuremath{\mid}M\ensuremath{\mid}}^{3}$ term crosses over to a non-Fermi-liquid form ${\ensuremath{\mid}M\ensuremath{\mid}}^{7∕2}$ near a QCP. The prefactor of the ${\ensuremath{\mid}M\ensuremath{\mid}}^{7∕2}$ term is negative, which indicates that the system undergoes a first-order rather than a continuous transition to ferromagnetism. We compare two scenarios of the breakdown of a continuous QCP: a first-order instability and a spiral phase; the latter may arise from the nonanalytic dependence of $\ensuremath{\chi}$ on the momentum. In a model with a long-range interaction in the spin channel, we show that the first-order transition occurs before the spiral instability.

QCD vacuum properties in a magnetic field from AdS/CFT: chiral condensate and Goldstone mass

Chiral condensate and $\eta^\prime$ meson mass spectrum are studied under the influence of an external Abelian magnetic field. We work within the D3/D7 Karch--Katz model of flavoured AdS/CFT with supersymmetry broken by the Constable--Myers deformation of the metric. It is shown that this setting yields a quadratic dependence of condensate on field, rather than the non-analytic (linear in field) dependence, typical for chiral perturbation theory in the exact chiral limit. We argue that the analytic (quadratic) result must be put into correspondence with the leading-order in the $1/N_c$ decomposition for the condensate, whereas the existing chiral perturbation theory result, which is linear in field strength, is $1/N_c$ suppressed.

On 1D diffusion problems with a gradient-dependent diffusion coefficient

0021-9991/$ see front matter 2008 Elsevier Inc doi:10.1016/j.jcp.2008.06.032 * Corresponding author. Tel.: +1 609 243 2635; fa E-mail address: jardin@pppl.gov (S.C. Jardin). In solving the 1D (flux surface averaged) transport equations for the temperatures, magnetic fields, and densities in the ‘‘evolving equilibrium” description of a tokamak [1], one increasingly encounters highly nonlinear thermal conductivity and diffusivity functions, such as GLF23 [2], that have a strong and non-analytic dependence on the temperature gradients. These arise from a subsidiary microstability based calculation in which the growth rates and hence transport coefficients are sensitive functions of these gradients [3]. When these nonlinear functions are interfaced with an existing transport framework that uses a standard implicit time advancement algorithm such as Crank-Nicolson or backward Euler [4], large non-physical oscillations can develop and, as a result, non-convergent solutions can occur. Here we describe a relatively simple modification to these implicit algorithms that cures this difficulty. To illustrate the method, we start with a simple diffusion equation in cylindrical polar coordinates:

Temperature dependence of the spectrum of electrons levitating above solid hydrogen

A theory of photoresonance transitions for the electrons localized above the solid H2 surface is proposed on the basis of the experimental data obtained by V. V. Zav’yalov and I. I. Smol’yaninov [Pis’ma Zh. Eksp. Teor. Fiz. 44, 142 (1986); Zh. Eksp. Teor. Fiz. 92, 339 (1987); Zh. Eksp. Teor. Fiz. 94, 307 (1988)] and a simple model of condensed hydrogen, H2. A new explanation is offered for the strong dependence of the transition frequency v on the hydrogen vapor pressure revealed in the above-cited works. In contrast to the notions that were proposed by V. V. Zav’yalov and I. I. Smol’yaninov [Zh. Eksp. Teor. Fiz. 92, 339 (1987); Zh. Eksp. Teor. Fiz. 94, 307 (1988)]; and V. B. Shikin and S. N. Nazin [Pis’ma Zh. Eksp. Teor. Fiz. 82, 752 (2005)] and based on the effects of the quantum refraction of electrons by hydrogen vapor atoms, the temperature dependence of the frequency ν = ν (T) is attributed here to the H2 vapor inhomogeneity. The nonanalytic dependence of ν on the vapor density nν, ν(nν) − ν(0) ∼ nνγ ∼ e−Δ/T, γ ≅ 0.56 has been revealed from the experimental data. The activation energy Δ corresponds to the “incomplete evaporation” of solid hydrogen [see J. I. Frenkel, Z. Phys. 26, 37 (1924); Kinetic Theory of Liquids (Nauka, Leningrad, 1945; Clarendon, Oxford, 1946)]. The parameter Δ is lower than the energy of the “complete evaporation” of solid H2 equal to 92.6 K.

Cooper channel and the singularities in the thermodynamics of a Fermi liquid

We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T$ in a two-dimensional Fermi liquid. We show that $A(T)$ is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.

Effect of interaction range on phonon relaxation in Fermi-Pasta-Ulamβchain

We study the effect of increasing the range of interactions on phonon relaxation in a chain of atoms with quartic anharmonicity. The study is motivated by recent numerical studies, showing that the value of the exponent alpha characterizing the divergence of conductivity with system size apparently depends on the presence of second neighbor couplings. We perform a quantum calculation of the wave-vector (q) dependent relaxation rate gamma(q) in the second order perturbation theory. The nonanalytic dependence of gamma(q) arises due to small-q singularity of the collision integral. We find that gamma(q) proportional to Aq(5/3) + Bq2. This gives rise to an asymptotic value alpha = 0.4, but the q2 terms lead to a higher apparent value of alpha at small sizes of the chain.

Decoherence Dynamics in Low-Dimensional Cold Atom Interferometers

We report on a study of the dynamics of decoherence of a matter-wave interferometer, consisting of a pair of low-dimensional cold atom condensates at finite temperature. We identify two distinct regimes in the time dependence of the coherence factor of the interferometer: quantum and classical. Explicit analytical results are obtained in both regimes. In particular, in the two-dimensional case in the classical (long time) regime, we find that the dynamics of decoherence is universal, exhibiting a power-law decay with an exponent, proportional to the ratio of the temperature to the Kosterlitz-Thouless temperature of a single 2D condensate. In the one-dimensional case in the classical regime we find a universal nonanalytic time dependence of decoherence, which is a consequence of the nonhydrodynamic nature of damping in 1D liquids.

Detecting Non-Abelian Statistics with an Electronic Mach-Zehnder Interferometer

Fractionally charged quasiparticles in the quantum Hall state with a filling factor nu=5/2 are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a nonanalytic dependence on the tunneling amplitudes which can be controlled by gate voltages.

Temperature dependence of the spin susceptibility of a clean Fermi gas with repulsion

Spin susceptibility of a clean Fermi gas with repulsion in any dimension is considered using a supersymmetric low energy theory of interacting spin excitations and renormalization scheme recently proposed by Aleiner and Efetov. We generalize this method to include the coupling to the magnetic field. As a result, we obtain for the correction $\ensuremath{\delta}\ensuremath{\chi}$ to the Pauli susceptibility a nonanalytic temperature dependence of the form ${T}^{d\ensuremath{-}1}{\ensuremath{\gamma}}_{b}^{2}(T)$ in dimensions $d=2$, 3, where ${\ensuremath{\gamma}}_{b}(T)$ is an effective $d$-dependent logarithmically renormalized backscattering amplitude. In one dimension, $\ensuremath{\delta}\ensuremath{\chi}$ is proportional to ${\ensuremath{\gamma}}_{b}(T)$, and we reproduce a well known result obtained long ago by a direct calculation.