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Journal of Physics A Mathematical and Theoretical

Functional truncations for the solution of the nonperturbative RG equations

J. Kaupužs, Roderick Melnik

Published Nov 2, 2022• 2 citations
Publisher

Abstract

Abstract We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of ρ ˜ − ρ ˜ 0 , where ρ ˜ ∝ ϕ 2 , φ is the averaged order parameter, and ρ ˜ 0 corresponds to the minimum of the dimensionless potential u k ( ρ ˜ ) , depending on the infrared cut-off scale k . We propose a new approach of functional truncations, using the expansion u k ( ρ ˜ ) − u k ( 0 ) = ( 1 − s ) − μ u 1 , k s + u 2 , k s 2 + ⋯ , where s = ρ ˜ / ( ρ ˜ 0 + ρ ˜ ) , ρ ˜ 0 is an optimization parameter and µ is the exponent, describing the ρ ˜ → ∞ asymptotic. The newly developed method provides accurate estimates of the critical exponents η , ν and als

DOI
10.1088/1751-8121/ac9f8c

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Abstract We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of ρ ˜ − ρ ˜ 0 , where ρ ˜ ∝ ϕ 2 , φ is the averaged order parameter, and ρ ˜ 0 corresponds to the minimum of the dimensionless potential u k ( ρ ˜ ) , depending on the infrared cut-off scale k . We propose a new approach of functional truncations, using the expansion u k ( ρ ˜ ) − u k ( 0 ) = ( 1 − s ) − μ u 1 , k s + u 2 , k s 2 + ⋯ , where s = ρ ˜ / ( ρ ˜ 0 + ρ ˜ ) , ρ ˜ 0 is an optimization parameter and µ is the exponent, describing the ρ ˜ → ∞ asymptotic. The newly developed method provides accurate estimates of the critical exponents η , ν and als
Abstract The known approximation schemes for the solution of the Wetterich exact renormalization group (RG) equation are critically reconsidered, and a new truncation scheme is proposed. In particular, the equations of the derivative expansion up to the ∂ 2 order for a scalar model are derived in a suitable form, clarifying the role of the off-diagonal terms in the matrix of functional derivatives. The natural domain of validity of the derivative expansion appears to be limited to small values of q / k in the calculation of the critical two-point correlation function, depending on the wave-vector magnitude q and the infrared cut-off scale k . The new approximation scheme has the advantage to be valid for any q / k , and, therefore, it can be auspicious in many current and potential applications of the celebrated Wetterich equation and similar models. Contrary to the derivative expansion, derivatives are not truncated at a finite order in the new scheme. The RG flow equations in the first approximation of this new scheme are derived and approximately solved as an example. It is shown that the derivative expansion up to the ∂ 2 order is just the small- q approximation of our new equations at the first order of truncation.
We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
We present a truncation scheme of the effective-average-action approach of the nonperturbative renormalization group that allows for an accurate description of the critical regime as well as of correlation functions at finite momenta. The truncation is a natural modification of the standard derivative expansion that includes both all local correlations and two-point and four-point irreducible correlations to all orders in the derivatives. We discuss schemes for both the symmetric and the symmetry broken phase of the O(N) model and present results for D=3. All approximations are done directly in the effective average action rather than in the flow equations of irreducible vertices. The approach is numerically relatively easy to implement and yields good results for all N both for the critical exponents and for the momentum dependence of the two-point function.
The determination of the critical exponents by means of the Exact Renormalizion Group approach is still a topic of debate. The general flow equation is by construction scheme independent, but the use of the truncated derivative expansion generates a model depencence in the determination of the universal quantities. We derive new nonperturbative flow equations for the one-component, Z_2 symmetric scalar field to the next-to-leading order of the derivative expansion by means of a class of proper time regulators. The critical exponents eta, nu and omega for the Wilson-Fisher fixed point are computed by numerical integration of the flow equations, without resorting to polynomial truncations. We show that by reducing the width of the cut-off employed, the critical exponents become rapidly insensitive to the cut-off width and converge to values that are in good agreement with the results of entirely different approaches.
We project the Wilson/Polchinski renormalization group equation onto its uniform external field dependent effective free energy and connected Green's functions. The result is a hierarchy of equations which admits a choice of "natural" truncation and closure schemes for nonperturbative approximate solution. In this way approximation schemes can be generated which avoid power series expansions in either fields or momenta. When following one closure scheme the lowest order equation is the mean field approximation, while another closure scheme gives the "local potential approximation." Extension of these closure schemes to higher orders leads to interesting new questions regarding truncation schemes and the convergence of nonperturbative approximations. One scheme, based on a novel "momentum cluster decomposition" of the connected Green's functions, seems to offer new possibilities for accurate nonperturbative successive approximation.