Renormalization group analysis of a confined, interacting Bose gas

The renormalization group is not only a powerful method for describing universal properties of phase transitions, but it is also useful for evaluating non-universal thermodynamic properties beyond mean-field theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group, thus changing the thermodynamic properties of the gas. Received: 20 July 2001 / Revised version: 20 August 2001 / Published online: 23 November 2001     

Effective one-dimensional dynamics of elongated Bose-Einstein condensates

By using a variational approach in combination with the adiabatic approximation we derive a new effective 1D equation of motion for the axial dynamics of elongated condensates. For condensates with vorticity ∣q∣ = 0 or 1, this equation coincides with our previous proposal [A. Muñoz Mateo, V. Delgado, Phys. Rev. A 77 (2008) 013617]. We also rederive the nonpolynomial Schrödinger equation (NPSE) in terms of the adiabatic approximation. This provides a unified treatment for obtaining the different effective equations and allows appreciating clearly the differences and similarities between the various proposals. We also obtain an expression for the axial healing length of cigar-shaped condensates and show that, in the local density approximation and in units of the axial oscillator length, it coincides with the inverse of the condensate axial half-length. From this result it immediately follows the necessary condition for the validity of the local density approximation. Finally, we obtain analytical formulas that give the frequency of the axial breathing mode with accuracy better than 1%. These formulas can be relevant from an experimental point of view since they can be expressed in terms only of the axial half-length and remain valid in the crossover between the Thomas-Fermi and the quasi-1D mean-field regimes. We have corroborated the validity of our results by numerically solving the full 3D Gross-Pitaevskii equation.     

Freezing transition in the mean-field approximation model of pedestrian counter flow

We study the freezing transition in the counter flow of pedestrians within the channel numerically and analytically. We present the mean-field approximation (MFA) model for the pedestrian counter flow. The model is described in terms of a couple of nonlinear difference equations. The excluded-volume effect and bi-directionality are taken into account. The fundamental diagrams (current-density diagrams) are derived. When pedestrian density is higher than a critical value, the dynamical phase transition occurs from the free flow to the freezing (stopping) state. The critical density is derived by using the linear stability analysis. Also, the velocity and current (flow) at the steady state are derived analytically. The analytical result is consistent with that obtained by the numerical simulation.     

Dielectric critical behaviour at a metal-insulator transition under lattice compression

We study dielectric critical behaviour around a continuous metal-insulator transition in crystalline Cesium Iodide induced by changing the lattice parameter. The ab initio calculations of band structure and various quantities related to the dielectric response are performed in the transition region, within the local density approximation of the density functional theory. These calculations allow us to establish the power-law singularities of various quantities on two sides of the transition. The exponents obtained here are mean-field like due to the approximation in which interactions and disorder are treated. The critical behaviour is discussed by applying the scaling principle to the wavevector and frequency dependent dielectric function. We further investigate the effect of dielectric anomalies on optical properties by calculating the reflectance around transition region taking the ionic contribution to the dielectric function also into account. We find that the reflectance as a function of frequency shows very different kind of behaviour on both sides of the metal-insulator transition.     

Universal scaling properties of extreme type-II superconductors in magnetic fields

Anisotropic Ginzburg-Landau superconductors of extreme type-II are considered in an approximation where magnetic field fluctuations are neglected. A formulation of the scaling properties is presented for the singular part of the free energy density in the presence of a magnetic field. From the existence of a magnetization, a diamagnetic susceptibility and superconductivity we determine the limiting behavior of the scaling function in the vicinity of the zero field transition temperature, where critical fluctuations dominate. Our predictions for the temperature and field dependence of magnetization, magnetic torque and melting line etc., uncover the universal critical properties and provide an extension of hitherto used mean-field treatments. The results are consistent with experimental data. Received: 24 April 1998 / Accepted: 5 May 1998     

Quantum spin model of a cubic ferromagnet in a magnetic field

The zero temperature phase diagram of a one-dimensional ferromagnet with cubic single ion anisotropy in an external magnetic field is studied. The mean-field approximation and the density-matrix renormalization group method are applied. Two phases at finite magnetic fields are identified: a canted phase with spontaneously broken symmetry and a phase with magnetization along the magnetic field. Both methods predict that the canted phase exists even for the single-ion anisotropy strong enough to destroy the magnetic order at zero magnetic field. In contrast to the mean-field theory, the density-matrix renormalization group predicts a reentrant behavior for the model. The character of the phase transition at finite magnetic field has also been considered and the critical index has been found. Received 9 May 2000 and Received in final form 5 July 2000     

The universal sound velocity formula for the strongly interacting unitary Fermi gas

Due to the scale invariance,the thermodynamic laws of strongly interacting limit unitary Fermi gas can be similar to those of non-interacting ideal gas.For example,the virial theorem between pressure and energy density of the ideal gas P=2E/3V is still satisfied by the unitary Fermi gas.This paper analyses the sound velocity of unitary Fermi gases with the quasi-linear approximation.For comparison,the sound velocities for the ideal Boltzmann,Bose and Fermi gas are also given.Quite interestingly,the sound velocity formula for the ideal non-interacting gas is found to be satisfied by the unitary Fermi gas in different temperature regions.     

Increase of temperature of an ideal nondegenerate quantum gas in a suddenly expanding box due to energy quantization

We show that due to energy quantization the temperature of an ideal nondegenerate quantum gas in a rectangular box always increases after a sudden expansion of the box and a subsequent thermalization. The maximal increment of temperature is proportional to the square root of the product of the initial absolute temperature by the energy of the first discrete quantum level, i.e., it is proportional to the first power of the Planck constant.     

Bose-Einstein transition in a dilute interacting gas

We study the effects of repulsive interactions on the critical density for the Bose-Einstein transition in a homogeneous dilute gas of bosons. First, we point out that the simple mean field approximation produces no change in the critical density, or critical temperature, and discuss the inadequacies of various contradictory results in the literature. Then, both within the frameworks of Ursell operators and of Green's functions, we derive self-consistent equations that include correlations in the system and predict the change of the critical density. We argue that the dominant contribution to this change can be obtained within classical field theory and show that the lowest order correction introduced by interactions is linear in the scattering length, a, with a positive coefficient. Finally, we calculate this coefficient within various approximations, and compare with various recent numerical estimates. Received 15 July 2001     

Power spectrum analysis of the average-fluctuation density separation in interacting particle systems

The power spectrum analysis using the Lomb-Scargle false alarm probability statistic shows a clear separation between the average and fluctuating parts of the state density in embedded two-body random matrix ensembles with a mean-field for both fermion and boson systems as well as in the nuclear shell model.     

F－L模型的状态方程与退禁闭相变图象

用温度场论方法计算了Ｆ－Ｌ模型在有限温度和密度下的状态方程，分析了压强对净重子数密度的等温线．结果表明，在平均场近似下，Ｆ－Ｌ模型所给出的退禁闭相变为一级相变．     

Flow equations for the spin-boson problem

Using continuous unitary transformations recently introduced by Wegner [1], we obtain flow equations for the parameters of the spin-boson Hamiltonian. Interactions not contained in the original Hamiltonian are generated by this unitary transformation. Within an approximation that neglects additional interactions quadratic in the bath operators, we can close the flow equations. Applying this formalism to the case of Ohmic dissipation at zero temperature, we calculate the renormalized tunneling frequency. We find a transition from an untrapped to trapped state at the critical coupling constant α _c =1. We also obtain the static susceptibility via the equilibrium spin correlation function. Our results are both consistent with results known from the Kondo problem and those obtained from mode-coupling theories. Using this formalism at finite temperature, we find a transition from coherent to incoherent tunneling atT ₂ ^* ≈2T ₁ ^* , whereT ₁ ^* is the crossover temperature of the dynamics known from the NIBA.     

Mean-field approximation for the potts model of a diluted magnet in the external field

The Potts model of a diluted magnet with an arbitrary number of states placed in the external field has been considered. Phase transitions of this model have been studied in the mean-field approximation, the dependence of the critical temperature on the external field and the density of magnetic atoms has been found, and the magnetic susceptibility has been calculated. An improved mean-field technique has been proposed, which provides more accurate account of the effects associated with nonmagnetic dilution. The influence of dilution on the first-order phase transition curve and the magnetization jump at the phase transition has been studied by this technique.     

Superconducting and charge-density wave instabilities in ultrasmall-radius carbon nanotubes

We perform a detailed analysis of the band structure, phonon dispersion, and electron-phonon coupling of three types of small-radius carbon nanotubes (CNTs): (5,0), (6,0), and (5,5) with diameters 3.9, 4.7, and 6.8 Å respectively. The large curvature of the (5,0) CNTs makes them metallic with a large density of states at the Fermi energy. The density of states is also strongly enhanced for the (6,0) CNTs compared to the results obtained from the zone-folding method. For the (5,5) CNTs the electron-phonon interaction is dominated by the in-plane optical phonons, while for the ultrasmall (5,0) and (6,0) CNTs the main coupling is to the out-of-plane optical phonon modes. We calculate electron-phonon interaction strengths for all three types of CNTs and analyze possible instabilities toward superconducting and charge-density wave phases. For the smallest (5,0) nanotube, in the mean-field approximation and neglecting Coulomb interactions, we find that the charge-density wave transition temperature greatly exceeds the superconducting one. When we include a realistic model of the Coulomb interaction we find that the charge-density wave is suppressed to very low temperatures, making superconductivity dominant with the mean-field transition temperature around one K. For the (6,0) nanotube the charge-density wave dominates even with the inclusion of Coulomb interactions and we find the mean-field transition temperature to be around five Kelvin. We find that the larger radius (5,5) nanotube is stable against superconducting and charge-density wave orders at all realistic temperatures.     

Nonautonomous solitons of Bose-Einstein condensation in a linear potential with an arbitrary time-dependence

In the presence of a linear potential with an arbitrary time-dependence, Hirota method is developed carefully for applying into the effective mean-field model of quasi-one-dimensional Bose-Einstein condensation with repulsive interaction. We obtain the exact nonautonomous soliton solution (NSS) analytically. These solutions show that the time-dependent potential can affect the velocity of NSS. In some special cases the velocity has the character of both increase and oscillation with time. A detail analysis for the asymptotic behaviour of solutions shows that the collision of two NSSs is elastic.     

The shear viscosity of a trapped Bose-condensed gas

By obtaining Kubo formula type and using nonequilibrium Green’s functions, we calculate the shear viscosity of a trapped Bose-condensed gas below and above the Bose-Einstein condensation temperature (T_BEC). The contributions of the interactions between condensate and noncondensate atoms and between noncondensate atoms take into account to the viscous relaxation time, by evaluating second order self-energies in Beliaev approximation.     

Condensed fraction of an atomic Bose gas induced by critical correlations

We study the condensed fraction of a harmonically trapped atomic Bose gas at the critical point predicted by mean-field theory. The nonzero condensed fraction f(0) is induced by critical correlations which increase the transition temperature T(c) above T(c) (MF). Unlike the T(c) shift in a trapped gas, f(0) is sensitive only to the critical behavior in the quasiuniform part of the cloud near the trap center. To leading order in the interaction parameter a/λ(0), where a is the s-wave scattering length and λ(0) the thermal wavelength, we expect a universal scaling f(0) proportionally (a/λ(0))(4). We experimentally verify this scaling using a Feshbach resonance to tune a/λ(0). Further, using the local density approximation, we compare our measurements with the universal result obtained from Monte Carlo simulations for a uniform system, and find excellent quantitative agreement.     

Langevin dynamics of the glass forming polymer melt: Fluctuations around the random phase approximation

In this paper the Martin-Siggia-Rose (MSR) functional integral representation is used for the study of the Langevin dynamics of a polymer melt in terms of collective variables: mass density and response field density. The resulting generating functional (GF) takes into account fluctuations around the random phase approximation (RPA) up to an arbitrary order. The set of equations for the correlation and response functions is derived. It is generally shown that for cases whenever the fluctuation-dissipation theorem (FDT) holds we arrive at equations similar to those derived by Mori-Zwanzig. The case when FDT in the glassy phase is violated is also qualitatively considered and it is shown that this results in a smearing out of the ideal glass transition. The memory kernel is specified for the ideal glass transition as a sum of all “water-melon” diagrams. For the Gaussian chain model the explicit expression for the memory kernel was obtained and discussed in a qualitative link to the mode-coupling equation. Received: 9 January 1998 / Revised: 24 April 1998 / Accepted: 2 July 1998