Zero-frequency Limit Research Articles

Full counting statistics of quantum phase slips

We work out a microscopic theory describing complete statistics of voltage fluctuations generated by quantum phase slips (QPS) in superconducting nanowires. We evaluate the cumulant generating function and demonstrate that shot noise of the voltage as well as the third and all higher voltage cumulants differ from zero only due to the presence of QPS. In the zero-frequency limit voltage fluctuations in superconducting nanowires are described by Poisson statistics just as in a number of other tunneling-like problems. However, at non-zero frequencies quantum voltage fluctuations in superconducting nanowires become much more complicated and are not anymore accounted for by Poisson statistics. In the case of short superconducting nanowires we explicitly evaluate all finite-frequency voltage cumulants and establish a non-trivial relation between these cumulants and the current-voltage characteristics of our system.

Ladderlike optical conductivity in the spin-fermion model

In the nested limit of the spin-fermion model for the cuprates, one-dimensional physics in the form of half-filled two-leg ladders emerges. We show that the renormalization group flow of the corresponding ladder is towards the d-Mott phase, a gapped spin-liquid with short-ranged d-wave pairing correlations, and reveals an intermediate SO(5)$\times$SO(3) symmetry. We use the results of the renormalization group in combination with a memory-function approach to calculate the optical conductivity of the spin-fermion model in the high-frequency regime, where processes within the hot spot region dominate the transport. We argue that umklapp processes play a major role. For finite temperatures, we determine the resistivity in the zero-frequency (dc) limit. Our results show an approximate linear temperature dependence of the resistivity and a conductivity that follows a non-universal power law. A comparison to experimental data supports our assumption that the conductivity is dominated by the antinodal contribution above the pseudogap.

Power series for normal stress differences of polymeric liquids in large-amplitude oscillatory shear flow

Exact solutions for normal stress differences in polymeric liquids subjected to large-amplitude oscillatory shear flow (LAOS) contain many Bessel functions, each appearing in infinite sums. For the simplest relevant model of a polymeric liquid, the corotational Maxwell fluid, Bessel functions appear 38 times in the exact solution. By relevant, we mean that higher harmonics are predicted in LAOS. By contrast, approximate analytical solutions for normal stress differences in LAOS often take the form of the first few terms of a power series in the shear rate amplitude, and without any Bessel functions at all. Perhaps the best example of this, from continuum theory, is the Goddard integral expansion (GIE) that is arrived at laboriously. There is thus practical interest in extending the GIE to an arbitrary number of terms. However, each term in the GIE requires much more work than its predecessor. For the corotational Maxwell fluid, for instance, the GIE for the normal stress differences has yet to be taken beyond the fifth power of the shear rate amplitude. In this paper, we begin with the exact solution for normal stress difference responses in corotational Maxwell fluids, then perform an expansion by symbolic computation to confirm up to the fifth power, and then to continue the GIE. In this paper, for example, we continue the GIE to the 41st power of the shear rate amplitude. We use Ewoldt grids to show that our main result is highly accurate. We also show that, except in its zero-frequency limit, the radius of convergence of the GIE is infinite. We derive the pattern for the common denominators of the GIE coefficients and also for every numerator for the zeroth harmonic coefficients. We also find that the numerators of the other harmonics appear to be patternless.

Magnetic tidal Love numbers clarified

In this brief paper, we clarify certain aspects related to the magnetic (i.e., odd parity or axial) tidal Love numbers of a star in general relativity. Magnetic tidal deformations of a compact star had been computed in 2009 independently by Damour and Nagar [1] and by Binnington and Poisson [2]. More recently, Landry and Poisson [3] showed that the magnetic tidal Love numbers depend on the assumptions made on the fluid, in particular they are different (and of opposite sign) if the fluid is assumed to be in static equilibrium or if it is irrotational. We show that the zero-frequency limit of the Regge-Wheeler equation forces the fluid to be irrotational. For this reason, the results of Damour and Nagar are equivalent to those of Landry and Poisson for an irrotational fluid, and are expected to be the most appropriate to describe realistic configurations.

Helmholtz Decomposition and Boundary Element Method Applied to Dynamic Linear Elastic Problems

The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.

Optical conductivity and transparency in an effective model for graphene

Motivated by experiments confirming that the optical transparency of graphene is defined through the fine structure constant and that it could be fully explained within the relativistic Dirac fermions in 2D picture, in this article we investigate how this property is affected by next-to-nearest neighbor coupling in the low-energy continuum description of graphene. A detailed calculation within the linear response regime allows us to conclude that, somewhat surprisingly, the zero-frequency limit of the optical conductivity that determines the transparency remains robust up to this correction.

Quantum critical points in ferroelectric relaxors: Stuffed tungsten bronze K3Li2Ta5O15 and lead pyrochlore ( Pb2Nb2O7 )

We have synthesised ceramic specimens of the tetragonal tungsten bronze K3Li2Ta5O15 (KLT) and characterized its phase transition via X-ray, dielectric permittivity, ultrasonic spectroscopy and heat capacity measurements. The space group of KLT is reported as both P4/mbm or Cmmm with the orthorhombic distortion occurring when there are higher partial pressures of volatile K and Li used within the closed crucibles for the solid state synthesis. The data show strong relaxor behaviour, with the temperature at which the two dielectric relative permittivity peaks decreasing with 104 K $\geqslant$ Tm1 $\geqslant$ 69 K and 69 K $\geqslant$ Tm2 $\geqslant$ 46 K as probe frequency f is reduced from 1 MHz to 316 Hz. The data satisfy a Vogel-Fulcher model with an extrapolated freezing temperature for {\epsilon}' and {\epsilon}" of Tf1 = + 15.8 and - 11.8 K and Tf2 = - 5.0 and - 15.0 K for f $\rightarrow$ 0 (tending to dc). Therefore by tuning frequency, the transition could be shifted to absolute zero suggesting KLT has a relaxor-type quantum critical point. In addition, we have reanalysed the conflicting literature for Pb2Nb2O7 pyrochlore which suggests that this is also a relaxor-type quantum critical point as the freezing temperature from Vogel-Fulcher fitting is below absolute zero. Since the transition temperature evidenced in the dielectric data at ca. 100 kHz shifts below zero Kelvin for very low frequencies, heat capacity data collected in the zero-frequency (dc) limit, should not indicate a transition. Both of these materials show promise as possible new relaxor-type quantum critical points within non-perovskite based structures as multiple compounds are reported with low-temperature transitions.

Microscopic Linear Response Theory of Spin Relaxation and Relativistic Transport Phenomena in Graphene

We present a unified theoretical framework for the study of spin dynamics and relativistic transport phenomena in disordered two-dimensional Dirac systems with pseudospin-spin coupling. The formalism is applied to the paradigmatic case of graphene with uniform Bychkov-Rashba interaction and shown to capture spin relaxation processes and associated charge-to-spin interconversion phenomena in response to generic external perturbations, including spin density fluctuations and electric fields. A controlled diagrammatic evaluation of the generalized spin susceptibility in the diffusive regime of weak spin-orbit interaction allows us to show that the spin and momentum lifetimes satisfy the standard Dyakonov-Perel relation for both weak (Gaussian) and resonant (unitary) nonmagnetic disorder. Finally, we demonstrate that the spin relaxation rate can be derived in the zero-frequency limit by exploiting the SU(2) covariant conservation laws for the spin observables. Our results set the stage for a fully quantum-mechanical description of spin relaxation in both pristine graphene samples with weak spin-orbit fields and in graphene heterostructures with enhanced spin-orbital effects currently attracting much attention.

Semidiurnal thermal tides in asynchronously rotating hot Jupiters

Context. Thermal tides can torque the atmosphere of hot Jupiters into asynchronous rotation, while these planets are usually assumed to be locked into spin-orbit synchronization with their host star. Aims. In this work, our goal is to characterize the tidal response of a rotating hot Jupiter to the tidal semidiurnal thermal forcing of its host star by identifying the structure of tidal waves responsible for variation of mass distribution, their dependence on the tidal frequency, and their ability to generate strong zonal flows. Methods. We develop an ab initio global modelling that generalizes the early approach of Arras & Socrates (2010, ApJ, 714, 1) to rotating and non-adiabatic planets. We analytically derive the torque exerted on the body and the associated timescales of evolution, as well as the equilibrium tidal response of the atmosphere in the zero-frequency limit. Finally, we numerically integrate the equations of thermal tides for three cases, including dissipation and rotation step by step. Results. The resonances associated with tidally generated gravito-inertial waves significantly amplify the resulting tidal torque in the range 1–30 days. This torque can globally drive the atmosphere into asynchronous rotation, as its sign depends on the tidal frequency. The resonant behaviour of the tidal response is enhanced by rotation, which couples the forcing to several Hough modes in the general case, while the radiative cooling tends to regularize it and diminish its amplitude.

Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits.

We present one- and two-breather solutions of the fourth-order nonlinear Schrödinger equation. With several parameters to play with, the solution may take a variety of forms. We consider most of these cases including the general form and limiting cases when the modulation frequencies are 0 or coincide. The zero-frequency limit produces a combination of breather-soliton structures on a constant background. The case of equal modulation frequencies produces a degenerate solution that requires a special technique for deriving. A zero-frequency limit of this degenerate solution produces a rational second-order rogue wave solution with a stretching factor involved. Taking, in addition, the zero limit of the stretching factor transforms the second-order rogue waves into a soliton. Adding a differential shift in the degenerate solution results in structural changes in the wave profile. Moreover, the zero-frequency limit of the degenerate solution with differential shift results in a rogue wave triplet. The zero limit of the stretching factor in this solution, in turn, transforms the triplet into a singlet plus a low-amplitude soliton on the background. A large value of the differential shift parameter converts the triplet into a pure singlet.

Hydrodynamic mobility of a solid particle near a spherical elastic membrane. II. Asymmetric motion.

In this paper, we derive analytical expressions for the leading-order hydrodynamic mobility of a small solid particle undergoing motion tangential to a nearby large spherical capsule whose membrane possesses resistance toward shearing and bending. Together with the results obtained in the first part [Daddi-Moussa-Ider and Gekle, Phys. Rev. E 95, 013108 (2017)2470-004510.1103/PhysRevE.95.013108], where the axisymmetric motion perpendicular to the capsule membrane is considered, the solution of the general mobility problem is thus determined. We find that shearing resistance induces a low-frequency peak in the particle self-mobility, resulting from the membrane normal displacement in the same way, although less pronounced, to what has been observed for the axisymmetric motion. In the zero-frequency limit, the self-mobility correction near a hard sphere is recovered only if the membrane has a nonvanishing resistance toward shearing. We further compute the in-plane mean-square displacement of a nearby diffusing particle, finding that the membrane induces a long-lasting subdiffusive regime. Considering capsule motion, we find that the correction to the pair-mobility function is solely determined by membrane shearing properties. Our analytical calculations are compared and validated with fully resolved boundary integral simulations where a very good agreement is obtained.

Theory of the dynamical thermal conductivity of metals

The Mori's projection method, known as memory function method is an important theoretical formalism to study various transport coefficients. In the present work, we calculate the dynamical thermal conductivity in the case of metals using the memory function formalism. We introduce thermal memory functions for the first time and discuss the behavior of thermal conductivity in both zero frequency limit and in the case of non-zero frequencies. We compare our results for the zero frequency case with the results obtained by the Bloch-Boltzmann kinetic approach and find that both approaches agree with each other. Motivated by some recent experimental advancements, we obtain several new results for the ac or the dynamical thermal conductivity.

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).

Influence of dipolar interactions on the magnetic susceptibility spectra of ferrofluids.

The frequency-dependent magnetic susceptibility of a ferrofluid is calculated under the assumption that the constituent particles undergo Brownian relaxation only. Brownian-dynamics simulations are carried out in order to test the predictions of a recent theory [A. O. Ivanov, V. S. Zverev, and S. S. Kantorovich, Soft Matter 12, 3507 (2016)1744-683X10.1039/C5SM02679B] that includes the effects of interparticle dipole-dipole interactions. The theory is based on the so-called modified mean-field approach and possesses the following important characteristics: in the low-concentration, noninteracting regime, it gives the correct single-particle Debye-theory results; it yields the exact leading-order results in the zero-frequency limit; it includes particle polydispersity correctly from the outset; and it is based on firm theoretical foundations allowing, in principle, systematic extensions to treat stronger interactions and/or higher concentrations. The theory and simulations are compared in the case of a model monodisperse ferrofluid, where the effects of interactions are predicted to be more pronounced than in a polydisperse ferrofluid. The susceptibility spectra are analyzed in detail in terms of the low-frequency behavior, the position of the peak in the imaginary (out-of-phase) part, and the characteristic decay time of the magnetization autocorrelation function. It is demonstrated that the theory correctly predicts the trends in all of these properties with increasing concentration and dipolar coupling constant, the product of which is proportional to the Langevin susceptibility χ_{L}. The theory is in quantitative agreement with the simulation results as long as χ_{L}≲1.

Gravitational radiation from massless particle collisions

We compute classical gravitational bremsstrahlung from the gravitational scattering of two massless particles at leading order in the (centre of mass) deflection angle . The calculation, although non-perturbative in the gravitational constant, is surprisingly simple and yields explicit formulae—in terms of multidimensional integrals—for the frequency and angular distribution of the radiation. In the range , the GW spectrum behaves like , is confined to cones of angular sizes (around the deflected particle trajectories) ranging from to , and exactly reproduces, at its lower end, a well-known zero-frequency limit. At the radiation is confined to cones of angular size of order resulting in a scale-invariant () spectrum. The total efficiency in GW production is dominated by this ‘high frequency’ region and is formally logarithmically divergent in the UV. If the spectrum is cutoff at the limit of validity of our approximations (where a conjectured bound on GW power is also saturated), the fraction of incoming energy radiated away turns out to be at leading logarithmic accuracy.

Gravity-dominated unequal-mass black hole collisions

We continue our series of studies of high-energy collisions of black holes investigating unequal-mass, boosted head-on collisions in four dimensions. We show that the fraction of the center-of-mass energy radiated as gravitational waves becomes independent of mass ratio and approximately equal to $13\%$ at large energies. We support this conclusion with calculations using black hole perturbation theory and Smarr's zero-frequency limit approximation. These results lend strong support to the conjecture that the detailed structure of the colliding objects is irrelevant at high energies.

Derivative expansion for the electromagnetic Casimir free energy at high temperatures

We study the contribution of the thermal zero modes to the Casimir free energy, in the case of a fluctuating electromagnetic (EM) field in the presence of real materials described by frequency-dependent, local and isotropic permittivity ($\epsilon$) and permeability ($\mu$) functions. Those zero modes, present at any finite temperature, become dominant at high temperatures, since the theory is dimensionally reduced. Our work, within the context of the Derivative Expansion (DE) approach, focusses on the emergence of non analyticities in that dimensionally reduced theory. We conclude that the DE is well defined whenever the function $\Omega(\omega)$, defined by $[\Omega(\omega)]^2 \equiv \omega^2\epsilon(\omega)$, vanishes in the zero-frequency limit, for at least one of the two material media involved.

Field-Cycling Relaxometry as a Molecular Rheology Technique: Common Analysis of NMR, Shear Modulus and Dielectric Loss Data of Polymers vs Dendrimers

Linear poly(propylene glycol) (PPG) as well as a poly(propyleneimine) (PPI) dendrimer with different molar masses (M) are investigated by field-cycling (FC) 1H NMR, shear rheology (G) and dielectric spectroscopy (DS). The results are compared in a reduced spectral density representation: the quantity R1(ωτα)/R1α(0), where R1(ωτα) is the master curve of the frequency dependent spin–lattice relaxation rate with τα denoting the local correlation time, is compared to the rescaled dynamic viscosity η′(ωτα)/η′α(0). The quantities R1α(0) and η′α(0), respectively, are the zero-frequency limits of a simple liquid reference system. Analogously, the dielectric loss data can be included in the methodological comparison. This representation allows quantifying the sensitivity of each method with respect to the polymer-specific relaxation contribution. Introducing a “cumulative mode ratio” Fi(M) for each technique i, which measures the zero-frequency plateau of the rescaled spectral density, characteristic power-law beh...

Thermodynamics of the Bose-Hubbard model in aBogoliubov+Utheory

We derive the $\mathrm{Bogoliubov}+U$ formalism to study the thermodynamical properties of the Bose-Hubbard model. The framework can be viewed as the zero-frequency limit of bosonic dynamical mean-field theory (B-DMFT), but equally well as an extension of the mean-field decoupling approximation in which pair creation and annihilation of depleted particles is taken into account. The self-energy on the impurity site is treated variationally, minimizing the grand potential. The theory containing just three parameters that are determined self-consistently reproduces the $T=0$ phase diagrams of the three-dimensional and two-dimensional Bose-Hubbard model with an accuracy of $1%$ or better. The superfluid to normal transition at finite temperature is also reproduced well and only slightly less accurately than in B-DMFT.

Anomalous diffusion and griffiths effects near the many-body localization transition.

We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i)local magnetization fluctuations relax subdiffusively; (ii)the ac conductivity vanishes near zero frequency as a power law; and (iii)the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.