Highly Accurate Critical Exponents from Self-Similar Variational Perturbation Theory

We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by re-calculating the critical exponents of O(N)-symmetric \vp^4 theory. From only three-loop perturbation expansions in $4- \epsilon$ dimensions we obtain {\em analytic results for the exponents, with practically the same accuracy as those derived recently from ordinary field-theoretic variational perturbational theory to seventh order. In particular, the theory explains the best-measured exponent \al\approx-0.0127 of the specific heat peak in superfluid helium, found in a satellite experiment with a temperature resolution of nanoKelvin. In addition, our analytic expressions reproduce also the exactly known large-N behaviour of the exponents $\nu$ and $\gamma= \nu (2- \eta)$ with high precision.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/kleiner_re349/preprint.htm

Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift

For an O(N) symmetric scalar field theory with Euclidean action integral d^3x [1/2 |nabla phi|^2 + 1/2 r phi^2 + 1/4! u phi^4], where phi = (phi_1,...,phi_N) is a vector of N real field components, variational perturbation theory through seven loops is employed for N = 0,1,2,3,4 to compute the renormalized value of r/(N+2)u^2 at the phase transition. Its exact large-N limit is determined as well. We also extend an earlier computation of the interaction-induced shift Delta/Nu for N = 1,2,4 to N = 0,3. For N = 2, the results for the two quantities are used to compute the second-order shift of the condensation temperature of a dilute Bose gas, both in the homogenous case and for the wide limit of a harmonic trap. Our results are in agreement with earlier Monte Carlo simulations for N = 1,2,4. The appendix contains previously unpublished numerical seven-loop data provided to us by B.Nickel.Comment: 19 page

Bose-Einstein Condensation Temperature of Homogenous Weakly Interacting Bose Gas in Variational Perturbation Theory Through Seven Loops

The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length `a' is computed for given particle density `n.' Variational perturbation theory is used to resum the corresponding perturbative series for Delta/Nu in a classical three-dimensional scalar field theory with coupling `u' and where the physical case of N=2 field components is generalized to arbitrary N. Our results for N=1,2,4 are in agreement with recent Monte-Carlo simulations; for N=2, we obtain Delta T_c/T_c = 1.27 +/- 0.11 a n^(1/3). We use seven-loop perturbative coefficients, extending earlier work by one loop order.Comment: 8 pages; typos and errors of presentation fixed; beautifications; results unchange

Extrapolation of power series by self-similar factor and root approximants

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on self-similar factor and root approximants, suggested earlier by the authors. It is shown that these approximants and their combinations can effectively extrapolate power series to the region of large variables, even up to infinity. Several examples from quantum and statistical mechanics are analysed, illustrating the approach.Comment: 21 pages, Latex fil

Nonequilibrium Bose systems and nonground-state Bose-Einstein condensates

The theory of resonant generation of nonground-state Bose-Einstein condensates is extended to Bose-condensed systems at finite temperature. The generalization is based on the notion of representative statistical ensembles for Bose systems with broken global gauge symmetry. Self-consistent equations are derived describing an arbitrary nonequilibrium nonuniform Bose system. The notion of finite-temperature topological coherent modes, coexisting with a cloud of noncondensed atoms, is introduced. It is shown that resonant generation of these modes is feasible for a gas of trapped Bose atoms at finite temperature.Comment: Latex file, 16 pages, no figure

Representative Ensembles in Statistical Mechanics

The notion of representative statistical ensembles, correctly representing statistical systems, is strictly formulated. This notion allows for a proper description of statistical systems, avoiding inconsistencies in theory. As an illustration, a Bose-condensed system is considered. It is shown that a self-consistent treatment of the latter, using a representative ensemble, always yields a conserving and gapless theory.Comment: Latex file, 18 page

Critical and Tricritical Points for the Massless 2d Gross-Neveu Model Beyond Large N

Using optimized perturbation theory, we evaluate the effective potential for the massless two dimensional Gross-Neveu model at finite temperature and density containing corrections beyond the leading large-N contribution. For large-N, our results exactly reproduce the well known 1/N leading order results for the critical temperature, chemical potential and tricritical points. For finite N, our critical values are smaller than the ones predicted by the large-N approximation and seem to observe Landau's theorem for phase transitions in one space dimension. New analytical results are presented for the tricritical points that include 1/N corrections. The easiness with which the calculations and renormalization are carried out allied to the seemingly convergent optimized results displayed, in this particular application, show the robustness of this method and allows us to obtain neat analytical expressions for the critical as well as tricritical values beyond the results currently known.Comment: 29 pages, 14 figure

Thermodynamics and Phase Structure of the Two-Flavor Nambu--Jona-Lasinio Model Beyond Large-N_c

The optimized perturbation theory (OPT) method is applied to the $SU(2)$ version of the Nambu--Jona-Lasinio (NJL) model both at zero and at finite temperature and/or density. At the first nontrivial order the OPT exhibits a class of 1/N_c corrections which produce nonperturbative results that go beyond the standard large-N_c, or mean-field approximation. The consistency of the OPT method with the Goldstone theorem at this order is established, and appropriate OPT values of the basic NJL (vacuum) parameters are obtained by matching the pion mass and decay constant consistently. Deviations from standard large-N_c relations induced by OPT at this order are derived, for example, for the Gell--Mann-Oakes-Renner relation. Next, the results for the critical quantities and the phase diagram of the model, as well as a number of other thermodynamical quantities of interest, are obtained with OPT and then contrasted with the corresponding results at large N_c.Comment: 29 pages, 20 figures, revtex. Minor corrections. In press Phys. Rev.

Three-loop HTL Free Energy for QED

We calculate the free energy of a hot gas of electrons and photons to three loops using the hard-thermal-loop perturbation theory reorganization of finite-temperature perturbation theory. We calculate the free energy through three loops by expanding in a power series in m_D/T, m_f/T, and e^2, where m_D and m_f are thermal masses and e is the coupling constant. We demonstrate that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, e ~ 2. The reorganization is gauge invariant by construction, and due to cancellation among various contributions, we obtain a completely analytic result for the resummed thermodynamic potential at three loops. Finally, we compare our result with similar calculations that use the Phi-derivable approach.Comment: 23 pages, 10 figures; v3 - typos corrected, additional discussions of systematics added; corresponds with published versio

Basics of Bose-Einstein Condensation

The review is devoted to the elucidation of the basic problems arising in the theoretical investigation of systems with Bose-Einstein condensate. Understanding these challenging problems is necessary for the correct description of Bose-condensed systems. The principal problems considered in the review are as follows: (i) What is the relation between Bose-Einstein condensation and global gauge symmetry breaking? (ii) How to resolve the Hohenberg-Martin dilemma of conserving versus gapless theories? (iii) How to describe Bose-condensed systems in strong spatially random potentials? (iv) Whether thermodynamically anomalous fluctuations in Bose systems are admissible? (v) How to create nonground-state condensates? Detailed answers to these questions are given in the review. As examples of nonequilibrium condensates, three cases are described: coherent modes, turbulent superfluids, and heterophase fluids.Comment: Review articl