Nonthermal fixed points represent basic properties of quantum field theories, in addition to vacuum or thermal equilibrium fixed points. The functional renormalization group on a closed real-time path provides a common framework for their description. For the example of an O(N) symmetric scalar theory it reveals a hierarchy of fixed point solutions, with increasing complexity from vacuum and thermal equilibrium to nonequilibrium.
The $SU(2)_A \times U(2)_V$-symmetric chiral linear sigma model in the presence of the axial anomaly is studied in the local-potential approximation of the Functional Renormalization Group (FRG). The renormalization group (RG) flow is investigated in a truncation which reproduces recent results for the $U(2)_A \times U(2)_V$-symmetric model in the limit of vanishing axial anomaly strength. We search for the conjectured O(4) fixed point in the presence of the $U(1)_A$ anomaly and analyze its stability properties.
We employ the functional renormalization group approach formulated on the Schwinger-Keldysh contour to calculate real-time correlation functions in scalar field theories. We provide a detailed description of the formalism, discuss suitable truncation schemes for real-time calculations as well as the numerical procedure to self-consistently solve the flow equations for the spectral function. Subsequently, we discuss the relations to other perturbative and non-perturbative approaches to calculate spectral functions, and present a detailed comparison and benchmark in $d=0+1$ dimensions.
We present first results on the calculation of fermionic spectral functions from analytically continued flow equations within the Functional Renormalization Group approach. Our method is based on the same analytic continuation from imaginary to real frequencies that was developed and used previously for bosonic spectral functions. In order to demonstrate the applicability of the method also for fermionic correlations we apply it here to the real-time quark propagator in the quark-meson model and calculate the corresponding quark spectral functions in the vacuum.
The existence of an U(2)xU(2)-symmetric fixed point in the chiral linear sigma model is confirmed using the Functional Renormalization Group (FRG). Its stability properties and the implications for the order of the chiral phase transition of two-flavor quantum chromodynamics (QCD) are discussed. Furthermore, several technical conclusions are drawn from the comparison with the results of resummed loop expansions.
We compute the low-energy limit of the $O(4)$-symmetric quark-meson model as an effective field theory for Quantum Chromodynamics within the Functional Renormalization Group (FRG) approach. In particular, we analyze the renormalization group flow of momentum-dependent pion self-interactions beyond the local potential approximation. The numerical results for these couplings obtained from the FRG are confronted with a recent tree-level study. Additionally, their effect on the wave-function renormalization and the curvature masses is investigated.
We present a viable method to obtain real-time quantities such as spectral functions or transport coefficients at finite temperature and density within a non-perturbative Functional Renormalization Group approach. Our method is based on a thermodynamically consistent truncation of the flow equations for 2-point functions with analytically continued frequency components in the originally Euclidean external momenta. We demonstrate its feasibility by calculating the mesonic spectral functions in the quark-meson model at different temperatures and quark chemical potentials, in particular around th…
The gauge dependence problem of the effective action for general gauge theories in the framework of a modified functional renormalization group approach proposed recently is studied. It is shown that the effective action remains gauge-dependent on-shell.
We prove that the functional renormalization group flow equation admits a perturbative solution and show explicitly the scheme transformation that relates it to the standard schemes of perturbation theory. We then define a universal scheme within the functional renormalization group.
We study the quantum gravitational system coupled to a charged scalar, Dirac fermions, and electromagnetic fields. We use the "exact" or "functional" renormalization group equation to derive the effective action $Γ_0$ by integrating the flow equation from the ultraviolet scale down to $k=0$. The resulting effective action consists of local terms and nonlocal terms with unique coefficients.
Deriving accurate energy density functional is one of the central problems in condensed matter physics, nuclear physics, and quantum chemistry. We propose a novel method to deduce the energy density functional by combining the idea of the functional renormalization group and the Kohn-Sham scheme in density functional theory. The key idea is to solve the renormalization group flow for the effective action decomposed into the mean-field part and the correlation part. Also, we propose a simple practical method to quantify the uncertainty associated with the truncation of the correlation part. By …
In this paper we propose a method to study the Functional Renormalization Group at finite chemical potential. The method consists of mapping the FRG equations within the Fermi surface into a differential equation defined on a rectangle with zero boundary conditions. To solve this equation we use an expansion of the potential in a harmonic basis. With this method we determined the phase diagram of a simple Yukawa-type model; as expected, the bosonic fluctuations decrease the strength of the transition.
Motivated by the construction of the cMERA for interacting field theories, we derive a non-perturbative functional differential equation for wave functionals in scalar field theories from the exact renormalization group equation. We check the validity of the equation using the perturbation theory. We calculate the wave functional up to the first-order perturbation and verify that it satisfies the equation.
We study higher derivative extension of the functional renormalization group (FRG). We consider FRG equations for a scalar field that consist of terms with higher functional derivatives of the effective action and arbitrary cutoff functions. We show that the epsilon expansion around the Wilson-Fisher fixed point is indeed reproduced by the local potential approximation of the FRG equations.
We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is "exact" in two ways. First, the solution of the resulting flow equation is independent of the choice of…
We propose an approach to describe Spontaneous Symmetry Breaking (SSB) that does not rely on the order parameter dependent free energy (Landau theory). We use the Functional Renormalization Group (FRG) evolution of the explicitly broken theory, using a truncation scheme that is compatible with the Ward identities. To represent the symmetry breaking, we propose to use the "Ward ratio" which is zero in the symmetric phase and unity in the broken phase. In this approach a unified scale evolution of the effective potential is applicable in both phases. It is peculiar that the scale evolution is ac…
We show an application of the functional-renormalization-group aided density functional theory to the homogeneous electron gas with arbitrary spin polarization, which gives the energy density functional in the local spin density approximation. The correlation energy per particle is calculated at arbitrary Wigner-Seitz radius $ r_{\rm s} $ and spin polarization $ ζ$. In the high-density region, our result shows good agreement with Monte Carlo (MC) data. The agreement with MC data is better in the case of small spin polarization, while the discrepancy increases as the spin polarization increases…
Equivalence criteria are established for an effective Yukawa-type theory of composite fields representing two-particle fermion bound states with the original "microscopic" theory of interacting fermions based on the spectral decomposition of the 2-to-2 fermion scattering amplitude. Functional renormalisation group equations of the effective theory are derived exploiting relations expressing the equivalence. The effect of truncating the spectral decomposition is investigated quantitatively on the example of the non-relativistic bound states of two oppositely charged fermi particles.
We reanalyze the construction of the gravitational functional renormalization group equation tailored to the Arnowitt-Deser-Misner formulation emphasizing its connection to the covariant formulation. The results obtained from projecting the renormalization group flow onto the Einstein-Hilbert action are reviewed in detail and we provide a novel example illustrating how the formalism may be connected to the Causal Dynamical Triangulations approach to quantum gravity.
We investigate the order of the color superconducting phase transition using the functional renormalization group approach. We analyze the Ginzburg-Landau effective theory of color superconductivity and more generic scalar $SU(N_c)$ gauge theories by calculating the $β$ function of the gauge coupling in arbitrary dimension $d$ based on two different regularization schemes. We find that in $d=3$, due to gluon fluctuation effects, the $β$ function never admits an infrared fixed point solution. This indicates that, unlike the ordinary superconducting transition, color superconductivity can only s…
We show an $\textit{ab initio}$ construction of the energy density functional (EDF) for electron systems using the functional renormalization group. The correlation energies of the homogeneous electron gas given in our framework reproduce the exact behavior at high density and agree with the Monte-Carlo data in a wide range of densities. Our analytic technique enables us to get the correlation energies efficiently for various densities, which realizes the determination of EDF in the local density approximation (LDA) without any fitting for physically relevant densities. Applied to the Kohn-Sha…
We apply the functional renormalization group equation to a massive Fierz-Pauli action in curved space and find that, even though a massive term is a modification in the infrared sector, the mass term modifies the value of the non-gaussian fixed point in the UV sector.We obtained the beta function for the scale dependent mass parameter and found that the massive Fierz-Pauli case still seems to be an asymptotically safe theory.
The functional-renormalization-group aided density-functional theory (FRG-DFT) is applied to the two-dimensional homogeneous electron gas (2DHEG). The correlation energy of the 2DHEG is derived as a function of the Wigner-Seitz radius $ r_{\rm s} $ directly. We find that our correlation energy completely reproduces the exact behavior at the high-density limit. For finite density, the result of FRG-DFT shows good agreement with the Monte Carlo (MC) results in the high-density region, although the discrepancy between FRG-DFT and MC results becomes larger as the system becomes more dilute. Our st…
We formulate a perturbation expansion for the effective action in a new approach to the functional renormalization group method based on the concept of composite fields for regulator functions being their most essential ingredients. We demonstrate explicitly the principal difference between the properties of effective actions in these two approaches existing already on the one-loop level in a simple gauge model.
The running of the non-minimal parameter (ξ) of the interaction of the real scalar field and scalar curvature is explored within the non-perturbative setting of the functional renormalization group (RG). We establish the RG flow in curved space-time in the scalar field sector, in particular derive an equation for the non-minimal parameter. The RG trajectory is numerically explored for different sets of initial data.
We apply the functional renormalization group approach to a $\mathcal{N}=1$ supersymmetric gauge model with one chiral superfield coupled to a vector $U(1)$ superfield. We find that the nonrenormalization theorem still works at leading order in the supercovariant derivative expansion of the fields. We also find the beta functions and we study the behavior of its fixed points in the local potential approximation. Regulators are also discussed.
The gauge dependence problem of alternative flow equation for the functional renormalization group is studied. It is shown that the effective two-particle irreducible effective action depends on gauges at any value of IR parameter $k$. The situation with gauge dependence is similar to the standard formulation based on the effective one-particle irreducible effective action.
The Lifshitz critical behavior for a single component field theory is studied for the specific isotropic case in the framework of the Functional Renormalization Group. Lifshitz fixed point solutions of the flow equation, derived by using a Proper Time regulator, are searched at lowest and higher order in the derivative expansion. Solutions are found when the number of spatial dimensions d is contained within the interval 5.5 < d < 8.
We investigate the massive Schwinger model in $d=1+1$ dimensions using bosonization and the nonperturbative functional renormalization group. In agreement with previous studies we find that the phase transition, driven by a change of the ratio $m/e$ between the mass and the charge of the fermions, belongs to the two-dimensional Ising universality class. The temperature and vacuum angle dependence of various physical quantities (chiral density, electric field, entropy density) are also determined and agree with results obtained from density matrix renormalization group studies. Screening of fra…
Wetterich's equation provides a powerful tool for investigating the existence and universal properties of renormalization group fixed points exhibiting quantum scale invariance. Motivated by recent works on asymptotically safe scalar-tensor theories, we develop a novel approximation scheme which projects the functional renormalization group equation onto functions of the kinetic term. Applying this projection to scalars and gauge fields, our analysis identifies a new universality class with a very special spectrum of stability coefficients. The implications of our findings in the context of as…
Working with scalar field theories, we discuss choices of regulator that, inserted in the functional renormalization group equation, reproduce the results of dimensional regularization at one and two loops. The resulting flow equations can be seen as nonperturbative extensions of the $\overline{\text{MS}}\,$ scheme. We support this claim by recovering all the multicritical models in two dimensions. We discuss a possible generalization to any dimension. Finally, we show that the method also preserves nonlinearly realized symmetries, which is a definite advantage with respect to other regulators…
We discuss the relation between functional renormalization group (FRG) and local renormalization group (LRG), focussing on the two dimensional case as an example. We show that away from criticality the Wess-Zumino action is described by a derivative expansion with coefficients naturally related to RG quantities. We then demonstrate that the Weyl consistency conditions derived in the LRG approach are equivalent to the RG equation for the $c$-function available in the FRG scheme. This allows us to give an explicit FRG representation of the Zamolodchikov-Osborn metric, which in principle can be u…
We present the first successful functional renormalization group(FRG)-aided density-functional (DFT) calculation of the equation of state (EOS) of an infinite nuclear matter (NM) in (1+1)-dimensions composed of spinless nucleons. We give a formulation to describe infinite matters in which the 'flowing' chemical potential is introduced to control the particle number during the flow. The resultant saturation energy of the NM coincides with that obtained by the Monte-Carlo method within a few percent. Our result demonstrates that the FRG-aided DFT can be as powerful as any other methods in quantu…
We discuss a method to analytically continue functional renormalization group equations from imaginary Matsubara frequencies to the real frequency axis. In this formalism, we investigate the analytic structure of the flowing action and the propagator for a theory of scalar fields with $O(N)$ symmetry. We go on to show how it is possible to derive and solve flow equations for real-time properties such as particle decay widths. Our treatment is fully Lorentz-invariant and enables an improved, self-consistent derivative expansion in Minkowski space.
