Axel Pelster

Recursive graphical construction of Feynman diagrams and their multiplicities in φ 4 and φ 2 A theory

The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation that can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagrams with their proper multiplicities. The procedure is applied to a multicomponent scalar field theory with a φ4 self-interaction and then to a theory of two scalar fields φ and A with an interaction φ2A. All Feynman diagrams with external lines are obtained from functional derivatives of the connected vacuum diagrams with respect to the free correlation function. Finally, the recursive graphical construction is automatized by computer algebra with the help of a unique matrix notation for the Feynman diagrams.

Analytical approach for the Floquet theory of delay differential equations.

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.

Delayed complex systems: an overview

There does not exist a generally accepted definition for the notion of complex systems in science, but it is a common belief that complex systems show features that cannot be explained by just looking at their constituents. Thus, a complex system normally involves interaction of subunits, and

Quantum Phase Diagram of Bosons in Optical Lattices

We work out two different analytical methods for calculating the boundary of the Mott-insulator\char21{}superfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from quantum Monte Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.

Condensation of ideal Bose gas confined in a box within a canonical ensemble

We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of noninteracting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-T behavior in contrast to an earlier theory in Feynmans textbook on statistical mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum-mechanical treatment. Furthermore, we derive the finite-size effect of the system.

Thermodynamical properties of a rotating ideal Bose gas

In a recent experiment, a Bose-Einstein condensate was trapped in an anharmonic potential that is well approximated by a harmonic and a quartic part. The condensate was set into such a fast rotation that the centrifugal force in the corotating frame overcompensates the harmonic part in the plane perpendicular to the rotation axis. Thus, the resulting trap potential becomes sombrero shaped. We present an analysis for an ideal Bose gas that is confined in such an anharmonic rotating trap within a semiclassical approximation, where we calculate the critical temperature, the condensate fraction, and the heat capacity. In particular, we examine in detail how these thermodynamical quantities depend on the rotation frequency.

High-order variational calculation for the frequency of time-periodic solutions

We develop a convergent variational perturbation theory for the frequency of time-periodic solutions of nonlinear dynamical systems. The power of the theory is illustrated by applying it to the Duffing oscillator.

Geometric resonances in Bose-Einstein condensates with two- and three-body interactions

We investigate geometric resonances in Bose?Einstein condensates by solving the underlying time-dependent Gross?Pitaevskii equation for systems with two- and three-body interactions in an axially symmetric harmonic trap. To this end, we use a recently developed analytical method (Vidanovi? et al 2011 Phys. Rev. A 84 013618), based on both a perturbative expansion and a Poincar??Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose?Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.

Thermodynamics of a Bose-Einstein condensate with weak disorder

We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in the presence of weak quenched disorder. Following the zero-temperature approach of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an external random {delta}-correlated potential by means of a Bogoliubov transformation. We extend this approach to finite temperature by combining the Popov and the many-body T-matrix approximations. This approach permits us to include the quasiparticle interactions within this temperature range. We derive the disorder-induced shifts of the Bose-Einstein critical temperature and of the temperature for the onset of superfluidity by approaching the transition points from below, i.e., from the superfluid phase. Our results lead to a phase diagram consistent with that of the finite-temperature theory of Lopatin and Vinokur which was based on the replica method, and in which the transition points were approached from above.

Variational perturbation theory for Markov processes.

We develop a convergent variational perturbation theory for conditional probability densities of Markov processes. The power of the theory is illustrated by applying it to the diffusion of a particle in an anharmonic potential.