Impurity-induced Shiba bound state in the BCS–BEC crossover regime of two-dimensional Fermi superfluid

For a two-dimensional ultra-cold Fermi superfluid with an effective static magnetic impurity, we theoretically investigated the variation of the Yu–Shiba–Rusinov(YSR) bound state in the Bardeen–Cooper–Schrieffer(BCS) to Bose–Einstein condensation(BEC) crossover regime.Within the framework of mean-field theory, analytical results of the YSR bound state energy were obtained as a function of the interaction parameters.First, when the background Fermi superfluid system stays in the weakly interacting BCS regime, we found that the YSR bound state energy is linearly dependent on the gap parameter with its coefficient slightly different from previous results.Second, we discovered re-entrance phenomena for the YSR state and an upper bound of the strength of the interaction between the paired atoms.By carefully analyzing the bound state energy as a function of the interaction parameters, we obtained a phase diagram showing the existence of the YSR state.Finally, we concluded that the re-entrance phenomena and the critical point can be easily experimentally detected through measurement of radio-frequency spectroscopy and density of states using current experimental techniques.     

Dynamic density and spin responses of a superfluid Fermi gas in the BCS–BEC crossover: Path integral formulation and pair fluctuation theory

We present a standard field theoretical derivation of the dynamic density and spin linear response functions of a dilute superfluid Fermi gas in the BCS–BEC crossover in both three and two dimensions. The derivation of the response functions is based on the elegant functional path integral approach which allows us to calculate the density–density and spin–spin correlation functions by introducing the external sources for the density and the spin density. Since the generating functional cannot be evaluated exactly, we consider two gapless approximations which ensure a gapless collective mode (Goldstone mode) in the superfluid state: the BCS–Leggett mean-field theory and the Gaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our results of the response functions agree with the known results from the random phase approximation. We further consider the pair fluctuation effects and establish a theoretical framework for the dynamic responses within the GPF theory. We show that the GPF response theory naturally recovers three kinds of famous diagrammatic contributions: the Self-Energy contribution, the Aslamazov–Lakin contribution, and the Maki–Thompson contribution. We also show that unlike the equilibrium state, in evaluating the response functions, the linear (first-order) terms in the external sources as well as the induced order parameter perturbations should be treated carefully. In the superfluid state, there is an additional order parameter contribution which ensures that in the static and long wavelength limit, the density response function recovers the result of the compressibility (compressibility sum rule). We expect that the f$f$-sum rule is manifested by the full number equation which includes the contribution from the Gaussian pair fluctuations. The dynamic density and spin response functions in the normal phase (above the superfluid critical temperature) are also derived within the Nozières–Schmitt–Rink (NSR) theory.     

Dependence ofℏω on the mass number of nuclei under the assumption of a trapezoidal average nuclear density

The variation of the harmonic oscillator energy level spacing with the mass number of nuclei is investigated under the assumption of a trapezoidal distribution for the average nuclear density and the results are compared with those obtained with a Fermi distribution. The dependence of the parameters on the mass number is also discussed.     

Vortex lattices in a rotating Fermi superfluid in the BCS–BEC crossover with many Landau levels

We present an explicit analytical analysis of the ground state of vortex lattice structure, based on a minimization of the generalized Gross–Pitaevskii energy functional in a trapped rotating Fermi superfluid gas. By a Bogoliubov-like transformation we find that the coarse-grained average of the atomic density varies as inverted parabola in three dimensional cases; the Fermi superfluid in the BEC regime enters into the lowest Landau level at fast rotation, in which the vortices form an almost regular triangular lattice over a central region and the vortex lattice is expanded along the radial direction in the outer region; the fluid in the unitarity and BCS regimes occupies many low-lying Landau levels, in which a trapped gas with a triangular vortex lattice has a superfluid core surrounded by a normal gas. The calculation is qualitatively consistent with recent numerical and experimental data both in the vortex lattice structure and vortex numbers and in the density profiles versus the stirring frequency in the whole BCS–BEC crossover.     

On the ε-expansion procedure in the RNG theory of turbulence

Rescaling the equations of eliminating small scales, the physical justification for the-expansion procedure in the RNG theory of turbulence is proposed, in terms of that the inertial effects are small comparing with the viscous effects at the vicinity of the Kolmogorov dissipation wavenumber.We are grateful to Professor Chao-Hao Gu for numerous helpful suggestions. We would also like to acknowledge Professor Ke-Lin Wang and Bing-Hong Wang for many stimulating discussions of these problems.     

Kirkwood–Buff integrals of finite systems: shape effects

The Kirkwood–Buff (KB) theory provides an important connection between microscopic density fluctuations in liquids and macroscopic properties. Recently, Krüger et al. derived equations for KB integrals for finite subvolumes embedded in a reservoir. Using molecular simulation of finite systems, KB integrals can be computed either from density fluctuations inside such subvolumes, or from integrals of radial distribution functions (RDFs). Here, based on the second approach, we establish a framework to compute KB integrals for subvolumes with arbitrary convex shapes. This requires a geometric function w(x) which depends on the shape of the subvolume, and the relative position inside the subvolume. We present a numerical method to compute w(x) based on Umbrella Sampling Monte Carlo (MC). We compute KB integrals of a liquid with a model RDF for subvolumes with different shapes. KB integrals approach the thermodynamic limit in the same way: for sufficiently large volumes, KB integrals are a linear function of area over volume, which is independent of the shape of the subvolume.     

Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schrödinger equation

We discuss the zero-temperature hydrodynamics equations of bosonic and fermionic superfluids and their connection with generalized Gross-Pitaevskii and Ginzburg-Landau equations through a single superfluid nonlinear Schrödinger equation.     

Universality at the Edge of the Spectrum¶in Wigner Random Matrices

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n→+∞. As a corollary, we show that, after proper rescaling, the 1^th, 2^nd, 3^rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases. Received: 15 May 1999 / Accepted: 18 May 1999     

The reconstruction of density and Young’s modulus of an inhomogeneous rod

The problems of reconstructing the Young modulus and the density of an inhomogeneous elastic rod by analyzing its longitudinal and bending vibrations are considered. For solving these problems, an iteration algorithm based on the apparatus of the Fredholm integral equations is proposed. Specific examples of reconstruction are presented.     

“Soft” edge states in inhomogeneous 2D electron systems

The structural details of an edge electronic state in external-field-bounded 2D charged (electron or hole) systems at the zero boundary density of the mobile charge carriers are discussed. It is shown that the so-called “soft” edge electronic states (SESs) appear along these boundaries. The details of their structure and spectrum are analyzed, and the possible effects with the SESs are outlined.     

Methods for Attenuation of the Cross-Modulation Effects in the Wigner–Ville Distribution

We consider the characteristics of typical nonlinear distortions in the dynamical spectra of signals, obtained using the Wigner–Ville method, and discuss the existing techniques for attenuating these distortions. Certain methods are proposed to clean Wigner–Ville distributions of products of the nonlinear interaction between the signal component under study and its more powerful harmonic or pulsed components. The proposed techniques are mainly based on a preliminary linear processing (e.g., using the fast Fourier transform) and/or frequency filtering of the signal. A software package for calculating the Wigner–Ville distribution on a PC is developed. Using this package, we approved successfully the proposed algorithms for test and actual signals.     

Calculation of the Kirkwood–Frohlich correlation factor and dielectric constant of methanol using a statistical model and density functional theory

The geometries of methanol monomer and methanol clusters, (CH₃OH) _m , m = 2–10, were optimized using the DFT/B3LYP/6-31++G(d,p) method. For each m > 2, a number of conformers were found to satisfy the optimization condition, showing no imaginary frequency in their calculated IR spectra. With increasing m, five- and six-membered rings begin to appear with open chain branches and the calculated IR spectra approach the experimentally observed IR spectrum of liquid methanol. Using the average energy of formation of one hydrogen bond and a statistical model, the Kirkwood–Frohlich (K–F) correlation factor (g) and dielectric constant (ε) were calculated for each methanol cluster. From a plot of ε versus cluster size (m), the bulk dielectric constant was obtained by extrapolation to m→∞. The value of g averaged over all conformers is in almost complete agreement with the g value obtained in an earlier molecular dynamics simulation study by Fonseca and Ladanyi [J. Chem. Phys. 93, 8148 (1990)]. Using this value of g in the K–F equation, the dielectric constant (ε) of methanol was calculated and found to be in fair agreement with (～17% lower than) the experimental value and also with an earlier molecular dynamics simulation [Mol. Phys. 94, 435 (1998)]. The calculated ε follows the same trend in variation with temperature as the experimental ε in the range 288–318 K.     

Fluctuations in diffusion reaction systems. I: Adiabatic elimination of transport modes from a mesoscopicN-body system and the Ω-expansion

We develop a concise method to compute the corrections to the master equation for chemically reacting systems in particle number space that arise if the system is not a well-stirred tank reactor, but the transport occurs by diffusion. Starting from the master equation in theR ^N space of all reactant particle positions, we expand in inverse powers of the diffusion constant and eliminate all transport modes adiabatically. It is found that the overall effect of spatially nonuniform fluctuations cannot be treated as a mere renormalization of the reaction rate constants. From second order on there appear correction terms with a new structure that corresponds formally to additional virtual reaction paths. An intuitive interpretation along this line is impeded, however, by the formal occurrence of negative reaction rate constants in these terms, i.e., the reaction rate may depend on the concentrations of the final products of the virtual reaction rather than on the ingoing products. We also identify Avogadro's constant as the suitable parameter and extend van Kampen's-expansion systematically, to spatially continuous systems. This secondary expansion then serves to interpret the corrections to the rate equation, and the average and autocorrelation of the density in the stationary state. It is seen that the limitsD and do not commute. The relevant length and time scales are discussed.     

Similarity of the Fermi surface in the hidden order state and in the antiferromagnetic state of URu₂Si₂

Shubnikov-de Haas measurements of high quality URu2Si2 single crystals reveal two previously unobserved Fermi surface branches in the so-called hidden order phase. Therefore, about 55% of the enhanced mass is now detected. Under pressure in the antiferromagnetic state, the Shubnikov-de Haas frequencies for magnetic fields applied along the crystalline c axis show little change compared with the zero pressure data. This implies a similar Fermi surface in both the hidden order and antiferromagnetic states, which strongly suggests that the lattice doubling in the antiferromagnetic phase due to the ordering vector Q(AF)=(001) already occurs in the hidden order. These measurements provide a good test for existing or future theories of the hidden order parameter.     

Another representation of the β form of the inhomogeneous Picard-Fuchs equation

In this letter we give another representation of the β form in the inhomogeneous Picard-Fuchs equation for open topological string for some one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. Furthermore, the corresponding domain wall tensions calculated by using these β forms are consistent with the results that appear in literature. The β form is essential for the calculation of the D-brane domain wall tension, and a convenient choice of β forms should simplify the calculation. The freedom of the choice of β forms shows some symmetries in Calabi-Yau space.