Monte Carlo simulation of O(2) phi(4) field theory in three dimensions

From Asymptotic Series to Self-Similar Approximants

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

Nonperturbative Infrared Finiteness in a Superrenormalizable Scalar Quantum Field Theory

We present a study of the IR behavior of a three-dimensional superrenormalizable quantum field theory consisting of a scalar field in the adjoint of SU(N) with a φ^{4} interaction. A bare mass is required for the theory to be massless at the quantum level. In perturbation theory, the critical mass is ambiguous due to IR divergences, and we indeed find that at two loops in lattice perturbation theory the critical mass diverges logarithmically. It was conjectured long ago in [R. Jackiw et al., Phys. Rev. D 23, 2291 (1981)PRVDAQ0556-282110.1103/PhysRevD.23.2291, T. Appelquist et al., Phys. Rev. D 23, 2305 (1981)PRVDAQ0556-282110.1103/PhysRevD.23.2305] that superrenormalizable theories are nonperturbatively IR finite, with the coupling constant playing the role of an IR regulator. Using a combination of Markov Chain Monte Carlo simulations of the lattice-regularized theory, frequentist and Bayesian data analysis, and considerations of a corresponding effective theory, we gather evidence that this is indeed the case.

Possible increased critical temperature Tc in anisotropic bosonic gases

A finite thermal anisotropy, if maintained for times longer than thermal relaxation times, may have a positive effect on the critical temperature in Bose-Einstein condensation of a dilute boson gas not in thermal equilibrium or quasi-particle fermi fluid consisting of spin-compensated electron pairs. It raises the transition temperature while increasing the condensate density.

Quench dynamics of the three-dimensional U(1) complex field theory: Geometric and scaling characterizations of the vortex tangle

We present a detailed study of the equilibrium properties and stochastic dynamic evolution of the U(1)-invariant relativistic complex field theory in three dimensions. This model has been used to describe, in various limits, properties of relativistic bosons at finite chemical potential, type II superconductors, magnetic materials, and aspects of cosmology. We characterize the thermodynamic second-order phase transition in different ways. We study the equilibrium vortex configurations and their statistical and geometrical properties in equilibrium at all temperatures. We show that at very high temperature the statistics of the filaments is the one of fully packed loop models. We identify the temperature, within the ordered phase, at which the number density of vortex lengths falls off algebraically and we associate it to a geometric percolation transition that we characterize in various ways. We measure the fractal properties of the vortex tangle at this threshold. Next, we perform infinite rate quenches from equilibrium in the disordered phase, across the thermodynamic critical point, and deep into the ordered phase. We show that three time regimes can be distinguished: a first approach toward a state that, within numerical accuracy, shares many features with the one at the percolation threshold; a later coarsening process that does not alter, at sufficiently low temperature, the fractal properties of the long vortex loops; and a final approach to equilibrium. These features are independent of the reconnection rule used to build the vortex lines. In each of these regimes we identify the various length scales of the vortices in the system. We also study the scaling properties of the ordering process and the progressive annihilation of topological defects and we prove that the time-dependence of the time-evolving vortex tangle can be described within the dynamic scaling framework.

Renormalization group flow equations with full momentum dependence

After a short elementary introduction to the exact renormalization group for the effective action, I discuss a particular truncation of the hierarchy of flow equations that allows for the determination of the full momentum of the n-point functions. Applications are then briefly presented, to critical O(N) models, to Bose-Einstein condensation and to finite-temperature field theory.

Thermal conductivity, relaxation and low-frequency vibrational mode anomalies in glasses: a model using the Fermi-Pasta-Ulam nonlinear Hamiltonian

We present a nonlinear model that allows exploration of the relationship between energy relaxation, thermal conductivity and the excess of low-frequency vibrational modes (LFVMs) that are present in glasses. The model is a chain of the Fermi-Pasta-Ulam (FPU) type, with nonlinear second neighbour springs added at random. We show that the time for relaxation is increased as LFVMs are removed, while the thermal conductivity diminishes. These results are important in order to understand the role of the cooling speed and thermal conductivity during glass transition. Also, the model provides evidence for the fundamental importance of LFVMs in the FPU problem.

Nonperturbative renormalization group and momentum dependence of n-point functions. II

In a companion paper [Blaizot, Phys. Rev. E 74, 051116 (2006)], we have presented an approximation scheme to solve the nonperturbative renormalization group equations that allows the calculation of the n-point functions for arbitrary values of the external momenta. The method was applied in its leading order to the calculation of the self-energy of the O(N) model in the critical regime. The purpose of the present paper is to extend this study to the next-to-leading order of the approximation scheme. This involves the calculation of the four-point function at leading order, where interesting features arise, related to the occurrence of exceptional configurations of momenta in the flow equations. These require a special treatment, inviting us to improve the straightforward iteration scheme that we originally proposed. The final result for the self-energy at next-to-leading order exhibits a remarkable improvement as compared to the leading order calculation. This is demonstrated by the calculation of the shift DeltaTc, caused by weak interactions, in the temperature of Bose-Einstein condensation. This quantity depends on the self-energy at all momentum scales and can be used as a benchmark of the approximation. The improved next-to-leading order calculation of the self-energy presented in this paper leads to excellent agreement with lattice data and is within 4% of the exact large N result.

Nonperturbative renormalization group and momentum dependence of n-point functions. I

We present an approximation scheme to solve the nonperturbative renormalization group equations and obtain the full momentum dependence of the n-point functions. It is based on an iterative procedure where, in a first step, an initial ansatz for the n-point functions is constructed by solving approximate flow equations derived from well motivated approximations. These approximations exploit the derivative expansion and the decoupling of high momentum modes. The method is applied to the O(N) model. In leading order, the self-energy is already accurate both in the perturbative and the scaling regimes. A stringent test is provided by the calculation of the shift DeltaTc in the transition temperature of the weakly repulsive Bose gas, a quantity which is particularly sensitive to all momentum scales. The leading order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%.

Reentrant phenomenon in the quantum phase transitions of a gas of bosons trapped in an optical lattice

We calculate the location of the quantum phase transitions of a Bose gas trapped in an optical lattice as a function of effective scattering length a(eff) and temperature T. Knowledge of recent high-loop results on the shift of the critical temperature at weak couplings is used to locate a nose in the phase diagram above the free Bose-Einstein critical temperature T((0))(c), thus predicting the existence of a reentrant transition above T((0))(c), where a condensate should form when increasing a(eff). At zero temperature, the transition to the normal phase produces the experimentally observed Mott insulator.

Monte Carlo studies of three-dimensional O1 and O4 phi4 theory related to Bose-Einstein condensation phase transition temperatures

The phase transition temperature for the Bose-Einstein condensation (BEC) of weakly interacting Bose gases in three dimensions is known to be related to certain nonuniversal properties of the phase transition of three-dimensional O(2) symmetric phi(4) theory. These properties have been measured previously in Monte Carlo lattice simulations. They have also been approximated analytically, with moderate success, by large N approximations to O(N) symmetric phi(4) theory. To begin investigating the region of validity of the large N approximation in this application, the same Monte Carlo technique developed for the O(2) model [P. Arnold and G. Moore, Phys. Rev. E 64, 066113 (2001)] to O(1) and O(4) theories has been applied. The results indicate that there might exist some theoretically unanticipated systematic errors in the extrapolation of the continuum value from lattice Monte Carlo results. The final results show that the difference between simulations and next-to-leading order large N calculations does not improve significantly from N=2 to N=4. This suggests that one would need to simulate yet larger N's to see true large N scaling of the difference. Quite unexpectedly (and presumably accidentally), the Monte Carlo result for N=1 seems to give the best agreement with the large N approximation among the three cases.

Convergent resummed linear delta expansion in the critical O(N) (phi 2 i) 2 3d Model

The linear delta expansion (LDE) is applied to the critical O(N) phi4 three-dimensional field theory which has been widely used to study the temperature of condensation of dilute weakly interacting homogeneous Bose gases. We study the higher order convergence of the LDE as it is usually applied to this problem. We show how to improve both the large N and finite N=2 LDE results with an efficient resummation technique which accelerates convergence. In the large N limit, it reproduces the known exact result within numerical integration accuracy. In the finite N=2 case, our improved results support the recent numerical Monte Carlo estimates for the transition temperature.