On the large order expansion for the anharmonic oscillators

Using functional approaches, we investigate the large-K behaviour of theK ^th coefficientE _k in the perturbation expansion for the ground-state energyE(g) of the generalized anharmonic oscillatorX ^2N with internalO(_n)-symmetry. We establish the equivalence between the pure functional approach and the method of Collins-Soper at any order in 1 /K. For that purpose, we first develop an algebraic treatment of perturbation series and prove a theorem on Borelsummable functions. Finally, we compute analytically the 1 /K and 1 /K ² corrections to the leading term forN=2.     

Effects of finite size and shape on scattering and thermodynamic quantities at 2D Ising critical points

Using recent results by Cardy based on the conformal invariance of critical correlation functions we calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular and rectangular shapes and free boundary conditions. Our results specify the effect of shape on these quantities at the critical point. In particular, the half-width and lineshape of the scattering function is found to be strongly influenced by geometry. For a circle, S(k) follows the infinite system behavior 1/k^2−η, η = 0.25 only for very large k. For a substantial range of intermediate k values it is well represented by 1/k^2−η_app, with an “apparent” exponent η_app. We also discuss the probable influence of end, edge and domain wall effects on the correlation lengths, susceptibilities and specific heat correction terms. The application of our results to experimental systems and other theoretical models is considered.     

Transport and equilibrium characteristics of γ-lithium vanadium bronze

The diffusion coefficient of Li⁺ in the γ-lithium vanadium bronze (Li_1+xV₃O₈) has been measured with the long-pulse galvanostatic technique. Values ranging from 1.7×10⁻⁷ cm²s⁻¹, at x=0.3, to 2.2×10⁻⁸ cm²s⁻¹, at x= 1.4, have been measured. The thermodynamic factors, d ln a/d ln c, determined from the OCV/x curve and from voltage relaxation after the current pulse, have a mean value of ∼15. The pseudo two-phase region observed in the OCV/x curve at high Li⁺ concentrations seems attributable to ordering of Li⁺ in specific sites and to alteration of the unit cell. This process is reversible as shown by X-ray diffractometry. Finally, from OCV/t plots at different x, the partial molar entropy of Li⁺ was determined. The values, on account of the large dE(x)/dt measured, are higher than those found for V₆O₁₃ or TiS₂.     

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e^|k|/k_d{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e^{-C(k/k_d) ln(|k|/k_d)}{{{\rm e}^{-C(k/k_{\rm d})\,{\rm ln}(|k|/k_{\rm d})}}} for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C _* = 1/ ln 2. The same behavior with a universal constant C _* is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.     

Spectrum of Kelvin-wave turbulence in superfluids

We derive a type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature, that describes step-by-step (local) energy cascade over scales caused by 4-wave interactions. Resulting new energy spectrum E _LN(k) ∝ k ^−5/3 must replace in future theory (e.g., in finding the quantum turbulence decay rate) the previously used spectrum E _KS(k) ∝ k ^−7/5, which was recently shown to be inconsistent due to nonlocality of the 6-wave energy cascade.     

Applications of exact solution for strongly interacting one-dimensional Bose-Fermi mixture: Low-temperature correlation functions, density profiles, and collective modes

We consider one-dimensional interacting Bose-Fermi mixture with equal masses of bosons and fermions, and with equal and repulsive interactions between Bose-Fermi and Bose-Bose particles. Such a system can be realized in current experiments with ultracold Bose-Fermi mixtures. We apply the Bethe ansatz technique to find the exact ground state energy at zero temperature for any value of interaction strength and density ratio between bosons and fermions. We use it to prove the absence of the demixing, contrary to prediction of a mean-field approximation. Combining exact solution with local density approximation in a harmonic trap, we calculate the density profiles and frequencies of collective modes in various limits. In the strongly interacting regime, we predict the appearance of low-lying collective oscillations which correspond to the counterflow of the two species. In the strongly interacting regime, we use exact wavefunction to calculate the single particle correlation functions for bosons and fermions at low temperatures under periodic boundary conditions. Fourier transform of the correlation function is a momentum distribution, which can be measured in time-of-flight experiments or using Bragg scattering. We derive an analytical formula, which allows to calculate correlation functions at all distances numerically for a polynomial time in the system size. We investigate numerically two strong singularities of the momentum distribution for fermions at k_f and k_f + 2k_b. We show, that in strongly interacting regime correlation functions change dramatically as temperature changes from 0 to a small temperature ∼E_f/γ ? E_f, where E_f = (π?n)²/(2m), n is the total density and γ = mg/(?²n) ? 1 is the Lieb-Liniger parameter. A strong change of the momentum distribution in a small range of temperatures can be used to perform a thermometry at very small temperatures.     

The long-time behaviour of the electric-field autocorrelation function in a finite photon gas

We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC _E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C _E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC ₀(t)=At^?4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k _BT?L and at timest withct?L, C _E(t) is described by an asymptotic expansion in powers ofL ^?1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC _E(t) in terms of exp(?L ^?1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.     

Bulk and surface properties of ionic salt CsH5(PO4)2

We report a comparative study of transport and thermodynamic properties of single-crystal and polycrystalline samples of the ionic salt CsH₅(PO₄)₂ possessing a peculiar three-dimensional hydrogen-bond network. The observed potential of electrolyte decomposition ≈ 1.3 V indicates that the main charge carriers in this salt are protons. However, in spite of the high proton concentration, the conductivity appears to be rather low with a high apparent activation energy E_a ≈ 2 eV, implying that protons are strongly bound. The transport anisotropy though is not large, correlates with the crystal structure: the highest conductivity is found in the [001] direction (σ_130 °C ∼ 5.6 × 10^− 6 S cm^− 1) while the minimal conductivity is in the [100] direction (σ_130 °C ∼ 10 ⁻⁶ S cm^− 1). The conductivity of polycrystalline samples appears to exceed the bulk one by 1–3 orders of magnitude with a concomitant decrease of the activation energy (E_a ≈ 1.05 eV), which indicates that a pseudo-liquid layer with a high proton mobility is formed at the surface of grains. Infrared and Raman spectroscopy used to study the dynamics of the hydrogen-bond system in single-crystal and polycrystalline samples have confirmed the formation of such a modified surface layer in the latter. However, no bulk phase transition into the superionic disordered phase is observed in CsH₅(PO₄)₂ up to the melting point T_melt ∼ 151.6 °C, in contrast to its closest relative compound CsH₂PO₄.     

On perturbation theory at finite temperature

We show that the largest temperature-dependent contribution to the fermion self-energy of order α² in QED goes as (α²T⁴/p²)ln[(E + p)/(E ? p)], where p is the 3-momentum and E is the energy, and we speculate on the form of higher-order terms in the perturbation expansion.     

Description of uniform many-particle systems in density operator function space

Assuming the ground state wavefunction, ψ₀, of a boson fluid is known, and writing the excited state wavefunctions in the form Fψ₀, a linear eigenvalue equation of the form $\text{H}\text{F = EF}$ is obtained, where E₀ + $\text{E}$ is the excited state energy, E₀ is the ground state energy, and H is a non-hermitian operator which depends in a simple way upon U ≡ ln ψ₀² instead of the potential energy function. An extremum principle is derived in terms of an auxiliary hermitian Hamiltonian operator, H′. The many-body boson plane-wave basis, ?_n(k₁ … k_n) is used to express U in terms of its Fourier components (ordered conveniently in terms of the number of nonzero arguments), making it possible to calculate matrix elements of ovcirc|H and H′ in that basis. A perturbation theory similar to Brillouin-Wigner perturbation theory is developed for the non-hermitian eigenvalue problem. Nonorthogonal perturbation theory is developed for the correlated basis ?_nψ₀. The requirement that these two perturbation theories be consistent produces useful relationships between the components of U and the static structure functions of ψ₀. These relationships are shown to reduce to previous results in the extreme case of low density and weak interactions.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.     

Renormalization group analysis of polymer cyclization with non-equilibrium initial conditions

We develop a renormalization group approach for cyclizing polymers for the case when chain ends are initially close together (ring initial conditions). We analyze the behavior at times much shorter than the longest polymer relaxation time. In agreement with our previous work (Europhys. Lett. 73, 621 (2006)) we find that the leading time dependence of the reaction rate k(t) for ring initial conditions and equilibrium initial conditions are related, namely k _ring(t) ∝ t ^-δ and k _eq(t) ∝ t ^1-δ for times less than the longest polymer relaxation time. Here δ is an effective exponent which approaches δ = 5/4 for very long Rouse chains. Our present analysis also suggests a “sub-leading” term proportional to (ln t)/t which should be particularly significant for smaller values of the renormalized reaction rate and early times. For Zimm dynamics, our RG analysis indicates that the leading time dependence for the reaction rate is k(t) ∼ 1/t for very long chains. The leading term is again consistent with the expected relation between ring and equilibrium initial conditions. We also find a logarithmic correction term which we “exponentiate” to a logarithmic form with a Landau pole. The presence of the logarithm is particularly important for smaller chains and, in the Zimm case, large values of the reaction rate.     

Superconducting fluctuations,weak anti-localization and interaction effects in Nb0.53Ti0.47-Ge multilayers

The effect of weak anti-localization, electron interactions and superconducting fluctuations on the transport properties of disorder Nb_0.53Ti_0.47-Ge multilayers were studied. The temperature dependence of the inelastic scattering time was found to be τ_in ∼ T^−1±0.25 and τ_in ∼ T^{−1.75±0.25} for temperatures lower and higher than ∼ 6K, repectively for a sample with thick Ge layers. The effect of the Ge thickness on the prefacter A in the expression R□ ∼ ln T may arise from an interlayer electron-phonon process.     

Exact two-point correlation functions of turbulence without pressure in three dimensions

We investigate exact results of isotropic turbulence in three dimensions when the pressure gradient is negligible. We derive exact two-point correlation functions of the density in three dimensions and show that the density-density correlator behaves as , where $\text{α}{}_3\text{=2+}\text{1}\text{6}\text{√33}$. It is shown that, in three dimensions, the energy spectrum E(k) in the inertial range scales with exponent $\text{2−}\text{1}\text{12}\text{√33⋍1.5212}$. We also discuss the time scale for which our exact results are valid for strong 3D turbulence in the presence of pressure. We confirm our predictions by using the recent results of numerical calculations and experiment.     

Millimeter-wave spectrum of the C4v molecule IOF5: l-type doubling of the kl = −1 transitions in the v11(E) = 1 state spectrum

Measurements of the rotational spectrum of the C_4v molecule IOF₅ are reported for the excited vibrational state v₁₁(E) = 1 for the transitions J13 ← 12, 14 ← 13, 16 ← 15, and 17 ← 16 (55–72 GHz) including the observation of the kl = −1 (q⁻), l-doubling effect. Detailed assignments of the E-state spectrum are presented based on the overlapping quadrupole structure. These data are analyzed together with earlier results for the excited vibrational state v₆(B₁) = 1 to give information concerning the ν₆(B₁)-ν₁₁(E) Coriolis interaction and the (Δl, Δk) = (2, 2) (q⁺) and (2, −2) (q⁻)l-resonance interactions. It is found that q₁₁⁻ = −2.57(10) MHz, |q₁₁⁺| = 0.094(20) MHz, Δ = ν₆ − ν₁₁ = 45.2(7) cm⁻, ζ_11,11^z = +0.18(1) and |ζ_6,11^y| = 0.73(4).     

Solution of functional equations of restricted An−1(1) fused lattice models

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A_n−1⁽¹⁾ lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra A_n−1⁽¹⁾ with GKO pair A_n−1⁽¹⁾ ⊕ A_n−1⁽¹⁾ ⊃ A_n−1⁽¹⁾.     

Development of high vorticity in incompressible 3D Euler equations: Influence of initial conditions

The incompressible three-dimensional ideal flows develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling ω_max(t) ∝ l(t)^-2/3 between the vorticity maximum and pancake thickness, and provide the leading contribution to the energy spectrum, where the gradual formation of the Kolmogorov interval E _k ∝ k^?5/3 is observed for some initial flows. With the massive numerical simulations, we study the influence of initial conditions on the processes of pancake formation and the Kolmogorov energy spectrum development.     

Calculation of the wave functions of the ground and weakly excited states of helium II

The wave functions of the ground (Ψ₀) and the first excited (Ψ_k) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ₀ and Ψ_k = ψ_kΨ₀ are determined as the eigenfunctions of the N-particle Schrödinger equation from a system of coupled equations for Ψ₀, Ψ_k, and the quasiparticle spectrum E(k) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ₀ and ψ_k are derived from a model potential of interaction between He⁴ atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V ₀ of the potential barrier is a free parameter. Except for V ₀, the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E ₀, and the quasiparticle spectrum E(k) of He II are in agreement with the experimental values for V ₀ ≈ 100 K. The second-order correction to the logarithm of Ψ₀ significantly affects the value of E ₀ and provides the asymptotics E(k → 0) = ck, while the second-order correction to ψ_k slightly affects the E(k). The second-order corrections to Ψ₀ and ψ_k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ₀ and ψ_k well describe the microstructure of He II. Thus, the series for Ψ₀ and Ψ_k can be truncated in spite of the fact that the expansion parameter is not very small (～1/2).     

Sustained turbulence in the three-dimensional Gross-Pitaevskii model

We study the three-dimensional forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form k^−α. Our numerical results show that the exponent α strongly depends on how the inverse particle cascade is attenuated at ks lower than the forcing wave-number. If the inverse cascade is arrested by a friction at low ks, we observe an exponent which is in good agreement with the weak wave turbulence prediction k⁻¹. For a hypo-viscosity, a k⁻² spectrum is observed which we explain using a critical balance argument. In simulations without any low k dissipation, a condensate at k=0 is growing and the system goes through a strongly turbulent transition from a 4-wave to a 3-wave weak turbulence acoustic regime with evidence of k^−3/2 Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov k^−5/3, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.