We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA, the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot--Mend\'ez-Galain--Wschebor approximation scheme). As an application we consider the two-dimensional quantum O( $N$ ) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the ``Higgs'' amplitude mode) and the conductivity. In particular, we show how, using Pad\'e approximants to perform the analytic continuation $i{\ensuremath{\omega}}_{n}\ensuremath{\rightarrow}\ensuremath{\omega}+i{0}^{+}$ of imaginary frequency correlation functions $\ensuremath{\chi}(i{\ensuremath{\omega}}_{n})$ computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA, the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot--Mend\'ez-Galain--Wschebor approximation scheme). As an application we consider the two-dimensional quantum O($N$) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the ``Higgs'' amplitude mode) and the conductivity. In particular, we show how, using Pad\'e approximants to perform the analytic continuation $i{\ensuremath{\omega}}_{n}\ensuremath{\rightarrow}\ensuremath{\omega}+i{0}^{+}$ of imaginary frequency correlation functions $\ensuremath{\chi}(i{\ensuremath{\omega}}_{n})$ computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.

arXiv:1801.03118Physical review. B./Physical review. BMay 16, 201814 citations

Applications of the Functional Renormalization Group

Roberto Percacci A

Jan 1, 20130 citations

Effect of droplet configurations within the functional renormalization group of the Ising model approaching the lower critical dimension

Ivan Balog, Lucija Nora Farkaš, Maroje Marohnić +1 more

We explore the application of the nonperturbative functional renormalization group (NPFRG), within its most common approximation scheme based on truncations of the derivative expansion, to the Z_{2}-symmetric scalar φ^{4} theory as the lower critical dimension d_{lc} is approached. We aim to assess whether the NPFRG-a broad, nonspecialized method which is accurate in d≥2-can capture the effect of the localized (droplet) excitations that drive the disappearance of the phase transition in d_{lc} and control the critical behavior as d→d_{lc}. We extend a prior analysis to the next (second) order of the derivative expansion to check the convergence of the results and the robustness of the conclusions. The study turns out to be much more involved. Through extensive numerical and analytical work we provide evidence that the convergence to d_{lc} is nonuniform in the field dependence and is characterized by the emergence of a boundary layer near the minima of the fixed-point effective potential. This is the mathematical mechanism through which the NPFRG within the truncated derivative expansion reproduces nontrivial features predicted by the droplet theory of Bruce and Wallace [Phys. Rev. Lett. 47, 1743 (1981)0031-900710.1103/PhysRevLett.47.1743; J. Phys. A: Math. Gen. 16, 1721 (1983)0305-447010.1088/0305-4470/16/8/018], namely, the existence of two distinct small parameters as d→d_{lc} that control different aspects of the critical behavior and that are nonperturbatively related. The second order of the derivative expansion fixes several issues that were encountered at the lower level and improves the compatibility with the droplet-theory predictions.We explore the application of the nonperturbative functional renormalization group (NPFRG), within its most common approximation scheme based on truncations of the derivative expansion, to the Z_{2}-symmetric scalar φ^{4} theory as the lower critical dimension d_{lc} is approached. We aim to assess whether the NPFRG-a broad, nonspecialized method which is accurate in d≥2-can capture the effect of the localized (droplet) excitations that drive the disappearance of the phase transition in d_{lc} and control the critical behavior as d→d_{lc}. We extend a prior analysis to the next (second) order of the derivative expansion to check the convergence of the results and the robustness of the conclusions. The study turns out to be much more involved. Through extensive numerical and analytical work we provide evidence that the convergence to d_{lc} is nonuniform in the field dependence and is characterized by the emergence of a boundary layer near the minima of the fixed-point effective potential. This is the mathematical mechanism through which the NPFRG within the truncated derivative expansion reproduces nontrivial features predicted by the droplet theory of Bruce and Wallace [Phys. Rev. Lett. 47, 1743 (1981)0031-900710.1103/PhysRevLett.47.1743; J. Phys. A: Math. Gen. 16, 1721 (1983)0305-447010.1088/0305-4470/16/8/018], namely, the existence of two distinct small parameters as d→d_{lc} that control different aspects of the critical behavior and that are nonperturbatively related. The second order of the derivative expansion fixes several issues that were encountered at the lower level and improves the compatibility with the droplet-theory predictions.

arXiv:2506.23415Physical review. EJan 30, 20260 citations

Optimization of field-dependent nonperturbative renormalization group flows

Léonie Canet

We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $\ensuremath{\nu}=0.628$ and $\ensuremath{\eta}=0.044$.We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $\ensuremath{\nu}=0.628$ and $\ensuremath{\eta}=0.044$.

arXiv:hep-th/0409300Physical Review BJan 31, 200533 citations

N

) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the ``Higgs'' amplitude mode) and the conductivity. In particular, we show how, using Pad\'e approximants to perform the analytic continuation $i{\ensuremath{\omega}}_{n}\ensuremath{\rightarrow}\ensuremath{\omega}+i{0}^{+}$ of imaginary frequency correlation functions $\ensuremath{\chi}(i{\ensuremath{\omega}}_{n})$ computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.