We reexamine the two-dimensional linear O(2) model (ϕ 4 theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the KosterlitzThouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension η(TKT) 0.24 and the stiffness jump ρs(T − KT ) 0.64 at the transition temperature TKT, in very good agreement with the exact results η(TKT) = 1/4 and ρs(T − KT ) = 2/π, as well as an essential singularity of the correlation length in the high-temperature phase as T → TKT.

We consider the d-dimensional imperfect (mean-field) Bose gas confined in a slit-like geometry and subject to periodic boundary conditions. Within an exact analytical treatment we first extract the bulk critical properties of the system at Bose-Einstein condensation and identify the bulk universality class to be the one of the classical d-dimensional spherical model. Subsequently we consider finite slit width D and analyze the excess surface free energy and the related Casimir force acting between the slit boundaries. Above the bulk condensation temperature (T > Tc) the Casimir force decays exponentially as a function of D with the bulk correlation length determining the relevant length scale. For T = Tc and for T < Tc its decay is algebraic. The magnitude of the Casimir forces at Tc and for T < Tc is governed by the universal Casimir amplitudes. We extract the relevant values for different d and compute the scaling functions describing the crossover between the critical and low-temperature asymptotics of the Casimir force. The scaling function is monotonous at any d ∈ (2, 4) and becomes constant for d > 4 and T ≤ Tc.

We analyze the interplay of longitudinal and transverse fluctuations in a $U(1)$ symmetric two-dimensional $\phi^4$-theory. To this end, we derive coupled renormalization group equations for both types of fluctuations obtained from a linear (cartesian) decomposition of the order parameter field. Discarding the longitudinal fluctuations, the expected Berezinskii-Kosterlitz-Thouless (BKT) phase characterized by a finite stiffness and an algebraic decay of order parameter correlations is recovered. Renormalized by transverse fluctuations, the longitudinal mass scales to zero, so that longitudinal fluctuations become increasingly important for small momenta. Within our expansion of the effective action, they generate a logarithmic decrease of the stiffness, in agreement with previous functional renormalization group calculations. The logarithmic terms imply a deviation from the vanishing beta-function for the stiffness in the non-linear sigma model describing the phase fluctuations at three-loop order. To gain further insight, we also compute the flow of the parameters characterizing longitudinal and transverse fluctuations from a density-phase representation of the order parameter field, with a cutoff on phase fluctuations. The power-law flow of the longitudinal mass and other quantities is thereby confirmed, but the stiffness remains finite in this approach. We conclude that the marginal flow of the stiffness obtained in the cartesian representation is an artifact of the truncated expansion of momentum dependences.Comment: Updated version. Substantial changes in Title, Abstract, Conclusion. New Section

We extend the Hertz-Millis theory of quantum phase transitions in itinerant electron systems to phases with broken discrete symmetry. By using a set of coupled flow equations derived within the functional renormalization group framework, we compute the second order phase transition line T c ͑␦͒, where ␦ is a nonthermal control parameter, near a quantum critical point. We analyze the interplay and relative importance of quantum and classical fluctuations at different energy scales, and we compare the Ginzburg temperature T G to the transition temperature T c , the latter being associated with a non-Gaussian fixed point.