The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.
Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function ${\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))$ once the mass
m and the
s-wave scattering length
ad of the bosons are known (
d is the space dimension). Here $\ensuremath{\mu}$ and
T denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ is a function of $m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}$. We compute the scaling function
F2 using a nonperturbative renormalization-group approach and find that both the $\ensuremath{\mu}/{k}_{B}T$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.
We present a detailed investigation of the momentum-dependent self-energy $\ensuremath{\Sigma}(k)$ at zero frequency of weakly interacting bosons at the critical temperature
Tc of Bose-Einstein condensation in dimensions $3\ensuremath{\leqslant}D<4$. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime
k⪡kc and the short-wavelength regime
k⪢kc, where
kc is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2\ensuremath{-}\ensuremath{\eta}}$, and in the short-wavelength regime the behavior is $\ensuremath{\Sigma}(k)\ensuremath{\propto}{k}^{2(D\ensuremath{-}3)}$ in
D>3. In
D=3, we recover the logarithmic divergence $\ensuremath{\Sigma}(k)\ensuremath{\propto}\mathrm{ln}(k∕{k}_{c})$ encountered in perturbation theory. Our approach yields the crossover scale
kc as well as a reasonable estimate for the critical exponent $\ensuremath{\eta}$ in
D=3. From our scaling function we find for the interaction-induced shift in
Tc in three dimensions, $\ensuremath{\Delta}{T}_{c}∕{T}_{c}=1.23a{n}^{1∕3}$, where
a is the
s-wave scattering length and
n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in
D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension $\ensuremath{\eta}\ensuremath{\approx}0.0513$. We further calculate the constant ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}\phantom{\rule{0.2em}{0ex}}\ensuremath{\Sigma}(k)∕{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and find good agreement with recent Monte Carlo data.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length
a which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density
n, and find for the critical temperature
Tc a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
We investigate Bose-Einstein condensation for ultracold bosonic atoms in two-dimensional systems. The functional renormalization group for the average action allows us to follow the effective interactions from molecular scales (microphysics) to the characteristic extension of the probe
l (macrophysics). In two dimensions the scale dependence of the dimensionless interaction strength $\ensuremath{\lambda}$ is logarithmic. Furthermore, for large
l the frequency dependence of the inverse propagator becomes quadratic. We find an upper bound for $\ensuremath{\lambda}$, and for large $\ensuremath{\lambda}$ substantial deviations from the Bogoliubov results for the condensate depletion, the dispersion relation and the sound velocity. The melting of the condensate above the critical temperature
Tc is associated to a phase transition of the Kosterlitz-Thouless type. The critical temperature in units of the density
Tc∕n vanishes for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ logarithmically.
We study the superfluid transition of weakly interacting Bose gases, classical fields and related systems in two dimensions, where it is a Berezinskii-Kosterlitz-Thouless transition, and in three dimensions, where it is a U(1) transition accompanied by the onset of Bose-Einstein condensation. We show that for small interaction strength U, these systems can be described by a :Ψ:4 model based on classical complex fields and we establish universal behavior among systems of different microscopical details, such as classical and quantum, continuous and lattice. By numerical simulations of the classical lattice field, using Worm Algorithm, we solve all relevant thermodynamical functions of this system, which are density, condensate density (in 3D), superfluid density and quasi-condensate. Through universality considerations, we can translate these results to all systems which can be described by a :Ψ:4 model based on a classical complex field. Our results cover the fluctuation region and extend into mean field and critical region, where we connect them to results of mean field theory and renormalization group theory respectively. We also establish that the quasicondensate, which exists in two and three dimensions, is a basis for a more accurate mean field theory treatment than the condensate, which exists only in three dimensions.