Renormalization group methods

Simone Montangero

Maria Chiara Angelini, Giorgio Parisi A, Federico Ricci‐Tersenghi

Abstract In the paper Angelini M C et al (2013 Phys. Rev. B 87 134201) we introduced a real-space renormalization group called ensemble renormalization group (ERG) and we applied it to the Edwards–Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper Castellana M (2019 J. Phys. A: Math. Theor. 52 445002) re-examined the ERG method from a different perspective, concluding that the previous results were wrong, and claiming that the ERG method predicts trivially wrong critical exponents. In this comment we explain why the conclusions reached by Castellana are wrong, as they are based on a misinterpretation of finite-size effects. We conclude that the ERG method remains a good RG method to obtain critical exponents in strongly disordered models (if properly used).

Martin Andersson

DMITRY V. SHIRKOV

We start with a simple introduction into the renormalization group (RG) in quantum field theory and give an overview of the renormalization group method. The third section is devoted to essential topics of the renorm-group use in the QFT. Here, some fresh results are included. Then we turn to the remarkable proliferation of the RG ideas into various fields of physics. The last section summarizes an impressive recent progress of the "QFT renormalization group" application in mathematical physics.

Evgeny Barkhudarov

Pascal Chauve, Pierre Le Doussal

We develop a method, the exact multilocal renormalization group (EMRG) which applies to a broad set of theories. It is based on the systematic multilocal expansion of the Polchinski-Wilson exact renormalization group (ERG) equation together with a scheme to compute correlation functions. Integrating out explicitly the nonlocal interactions, we reduce the ERG equation obeyed by the full interaction functional to a flow equation for a function, its local part. This is done perturbatively around fixed points, but exactly to any given order in the local part. It is thus controlled, at variance with projection methods, e.g., derivative expansions or local potential approximations. Our EMRG method is well-suited to problems such as the pinning of disordered elastic systems, previously described via functional renormalization group (FRG) approach based on a hard cutoff scheme. Since it involves arbitrary cutoff functions, we explicitly verify universality to O(epsilon=4-D), both of the T=0 FRG equation and of correlations. Extension to finite temperature T yields the finite size (L) susceptibility fluctuations characterizing mesoscopic behavior (Deltachi)2 approximately L(straight theta)/T, where straight theta is the energy exponent. Finally, we obtain the universal scaling function to O(epsilon(1/3)) which describes the ground state of a domain wall in a random field confined by a field gradient, compare with exact results and variational method. Explicit two loop exact RG equations are derived and the application to the FRG problem is sketched.