We determine the size of the critical region of the superfluid transition in the BCS-BEC crossover of a three-dimensional fermion gas, using a renormalization-group approach to a bosonic theory of pairing fluctuations. For the unitary Fermi gas, we find a sizable critical region $[T_G^-,T_G^+]$, of order $T_c$, around the transition temperature $T_c$ with a pronounced asymmetry: $|T_G^+-T_c|/|T_G^--T_c|sim8$. The critical region is strongly suppressed on the BCS side of the crossover but remains important on the BEC side.
We reexamine the two-dimensional linear O(2) model ($varphi^4$ 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 Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $eta(Tkt)simeq 0.24$ and the stiffness jump $rho_s(Tkt^-)simeq 0.64$ at the transition temperature $Tkt$, in very good agreement with the exact results $eta(Tkt)=1/4$ and $rho_s(Tkt^-)=2/pi$, as well as an essential singularity of the correlation length in the high-temperature phase as $Tto Tkt$.
We study the Higgs amplitude mode in the relativistic quantum O($N$) model in two space dimensions. Using the nonperturbative renormalization group we compute the O($N$)-invariant scalar susceptibility in the vicinity of the zero-temperature quantum critical point. In the zero-temperature ordered phase, we find a well defined Higgs resonance for $N=2$ with universal properties in agreement with quantum Monte Carlo simulations. The resonance persists at finite temperature below the Berezinskii-Kosterlitz-Thouless transition temperature. In the zero-temperature disordered phase, we find a maximum in the spectral function which is however not related to a putative Higgs resonance. Furthermore we show that the resonance is strongly suppressed for $Ngeq 3$.
We propose a nonperturbative renormalization-group (NPRG) approach to fermion systems in the two-particle-irreducible (2PI) effective action formalism, based on an exact RG equation for the Luttinger-Ward functional. This approach enables us to describe phases with spontaneously broken symmetries while satisfying the Mermin-Wagner theorem. We show that it is possible to choose the Hartree-Fock--RPA theory as initial condition of the RG flow and argue that the 2PI-NPRG is not restricted to the weak-coupling limit. An expansion of the Luttinger-Ward functional about the minimum of the 2PI effective action including only the two-particle 2PI vertex leads to nontrivial RG equations where interactions between fermions and collective excitations naturally emerge.
We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure $P(mu,T)$ as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model -- we find the same universal scaling function -- and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in a ultracold gas in an optical lattice.
We study the thermodynamics of the relativistic quantum O($N$) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N(T^3/c^2)calF_N(Delta/T)$ where $c$ is the velocity of the excitations at the QCP and $Delta$ is a characteristic zero-temperature energy scale. Using both a large-$N$ approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function $calF_N$. For small values of $N$ ($Nlesssim 10$) we find that $calF_N(x)$ is nonmonotonous in the quantum critical regime ($|x|lesssim 1$) with a maximum near $x=0$. The large-$N$ approach -- if properly interpreted -- is a good approximation both in the renormalized classical ($xlesssim -1$) and quantum disordered ($xgtrsim 1$) regimes, but fails to describe the nonmonotonous behavior of $calF_N$ in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio $Tkt/rho_s(0)$ is very close to $pi/2$, implying that the stiffness $rho_s(Tkt^-)$ at the transition is only slightly reduced with respect to the zero-temperature stiffness $rho_s(0)$. Finally, we briefly discuss the experimental determination of the universal function $calF_2$ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.
We study the thermodynamics near the generic (density-driven) superfluid--Mott-insulator transition in the three-dimensional Bose-Hubbard model using the nonperturbative renormalization-group approach. At low energy the physics is controlled by the Gaussian fixed point and becomes universal. Thermodynamic quantities can then be expressed in terms of the universal scaling functions of the dilute Bose gas universality class while the microscopic physics enters only {it via} two nonuniversal parameters, namely the effective mass $m^*$ and the scattering length $a^*$ of the elementary excitations at the quantum critical point between the superfluid and Mott-insulating phase. A notable exception is the condensate density in the superfluid phase which is proportional to the quasi-particle weight $Zqp$ of the elementary excitations. The universal regime is defined by $m^*a^*{}^2 Tll 1$ and $m^*a^*{}^2|deltamu|ll 1$, or equivalently $|bar n-bar n_c|a^*{}^3ll 1$, where $deltamu=mu-mu_c$ is the chemical potential shift from the quantum critical point $(mu=mu_c,T=0)$ and $bar n-bar n_c$ the doping with respect to the commensurate density $bar n_c$ of the T=0 Mott insulator. We compute $Zqp$, $m^*$ and $a^*$ and find that they vary strongly with both the ratio $t/U$ between hopping amplitude and on-site repulsion and the value of the (commensurate) density $bar n_c$. Finally, we discuss the experimental observation of universality and the measurement of $Zqp$, $m^*$ and $a^*$ in a cold atomic gas in an optical lattice.
We present a non-perturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the RG flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with the quantum Monte Carlo results and reproduces the two universality classes of the superfluid--Mott-insulator transition with a good estimate of the critical exponents. Furthermore, it reveals the crucial role of the Ginzburg length as a crossover length between a weakly- and a strongly-correlated superfluid phase.
In systems with a spontaneously broken continuous symmetry, the perturbative loop expansion is plagued with infrared divergences due to the coupling between transverse and longitudinal fluctuations. As a result the longitudinal susceptibility diverges and the self-energy becomes singular at low energy. We study the crossover from the high-energy Gaussian regime, where perturbation theory remains valid, to the low-energy Goldstone regime characterized by a diverging longitudinal susceptibility. We consider both the classical linear O($N$) model and interacting bosons at zero temperature, using a variety of techniques: perturbation theory, hydrodynamic approach (i.e., for bosons, Popovs theory), large-$N$ limit and non-perturbative renormalization group. We emphasize the essential role of the Ginzburg momentum scale $p_G$ below which the perturbative approach breaks down. Even though the action of (non-relativistic) bosons includes a first-order time derivative term, we find remarkable similarities in the weak-coupling limit between the classical O($N$) model and interacting bosons at zero temperature.
We propose a modification of the non-perturbative renormalization-group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a $d$-dimensional hypercubic lattice and classical spin systems. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the 3D Ising, XY and Heisenberg models to an accuracy of the order of one percent. We show how the local potential approximation can be improved to include a non-zero anomalous dimension $eta$ and discuss the Berezinskii-Kosterlitz-Thouless transition of the 2D XY model on a square lattice.
