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arXiv:1611.07301 · Physical review. E

Frequency regulators for the nonperturbative renormalization group: A general study and the model A as a benchmark

Charlie Duclut, Bertrand Delamotte

Published Jan 5, 2017• 45 citations
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Abstract

We derive the necessary conditions for implementing a regulator that depends on both momentum and frequency in the nonperturbative renormalization-group flow equations of out-of-equilibrium statistical systems. We consider model A as a benchmark and compute its dynamical critical exponent z. This allows us to show that frequency regulators compatible with causality and the fluctuation-dissipation theorem can be devised. We show that when the principle of minimal sensitivity (PMS) is employed to optimize the critical exponents η, ν, and z, the use of frequency regulators becomes necessary to make the PMS a self-consistent criterion.

DOI
10.1103/physreve.95.012107

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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 $\ensuremath{\eta}=0.044$.
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We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.
The use of nonperturbative Wilsonian functional renormalization group (FRG) equations within an unavoidable approximation scheme produces physical results that depend on the choice of the coarse-graining regulator. This dependence of physical observables on the regulator can be minimized according to the principle of minimal sensitivity (PMS). For the 1PI formulation based on the Wetterich-Morris equation, we propose to investigate the optimal choice within a space of compactly supported polynomial regulators parametrized by a set of variables. We test this approach using the derivative expansion of the effective average action up to fourth order, applying it to the scalar field theory representing the three-dimensional Ising universality class and minimizing the dependence of the anomalous dimension on the regulator. At fourth order, we find appreciable corrections to the critical exponents that improve the predictions, compared to previous one parameter analyses. We also observe families of compact regulators that rapidly converge towards an optimal functional form as a small number of parameters is progressively introduced.