We investigate the small frequency and momentum structure of the weakly interacting Bose gas in two dimensions using a functional renormalization group approach. The flow equations are derived within a derivative approximation of the effective action up to second order in spatial and temporal variables and investigated numerically. The truncation we employ is based on the perturbative structure of the theory and is well described as a renormalization group enhanced perturbation theory. It allows to calculate corrections to the Bogoliubov spectrum and to investigate the damping of quasiparticles. Our approach allows to circumvent the divergences which plague the usual perturbative approach.
We investigate the small frequency and momentum structure of the weakly interacting Bose gas in two dimensions using a functional renormalization group approach. The flow equations are derived within a derivative approximation of the effective action up to second order in spatial and temporal variables and investigated numerically. The truncation we employ is based on the perturbative structure of the theory and is well described as a renormalization group enhanced perturbation theory. It allows to calculate corrections to the Bogoliubov spectrum and to investigate the damping of quasiparticles. Our approach allows to circumvent the divergences which plague the usual perturbative approach.
We investigate the single-particle spectral density of interacting bosons within the nonperturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where the perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation, which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasifree bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozi\`eres and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.
We employ the functional renormalization group to study dynamical properties of the two-dimensional Bose gas. Our approach is free of infrared divergences, which plague the usual diagrammatic approaches, and is consistent with the exact Nepomnyashchy identity, which states that the anomalous self-energy vanishes at zero frequency and momentum. We recover the correct infrared behavior of the propagators and present explicit results for the spectral line shape, from which we extract the quasiparticle dispersion and damping.
We present a new approximation scheme to solve the Non-Perturbative Renormalization Group equations and obtain the full momentum dependence of n -point functions. The core of the method consists in deriving approximate flow equations that are then solved exactly. The results thus obtained can next be improved through an iteration procedure. The approximations proposed in this paper lead to approximate equations that are accurate both in the perturbative and the scaling regimes. The scheme is tested with the calculation of the shift Δ T c in the transition temperature of the weakly repulsive Bose gas, a quantity which is very sensitive to all momentum scales. The leading-order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%. The next-to-leading order differs by about 10% from the best accepted result.
Wilson’s renormalization-group approach to the weakly-interacting single-component Bose gas is discussed within the symmetry-broken, condensate phase. Extending upon the work by Bijlsma and Stoof, wave-function renormalization of the temporal derivative contributions to the effective action is included in order to capture sound-like quasiparticle excitations with wave lengths larger than the healing-length scale. By means of a suitable rescaling scheme we achieve convergence of the coupling flows, which serve as a means to determine the condensate depletion in accordance with Bogoliubov theory, as well as the interaction-induced shift of the critical temperature.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3, where a is the s-wave scattering length and n is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.