Exact results and open questions in first principle functional RG

Abstract

Some aspects of the functional RG (FRG) approach to pinned elastic manifolds (of internal dimension d) at finite temperature T > 0 are reviewed and reexamined in this much expanded version of Le Doussal (2006) [67]. The particle limit d = 0 provides a test for the theory: there the FRG is equivalent to the decaying Burgers equation, with viscosity ν ∼ T-both being formally irrelevant. An outstanding question in FRG, i.e. how temperature regularizes the otherwise singular flow of T = 0 FRG, maps to the viscous layer regularization of inertial range Burgers turbulence (i.e. to the construction of the inviscid limit). Analogy between Kolmogorov scaling and FRG cumulant scaling is discussed. First, multi-loop FRG corrections are examined and the direct loop expansion at T > 0 is shown to fail already in d = 0, a hierarchy of ERG equations being then required (introduced in Balents and Le Doussal (2005) [36]). Next we prove that the FRG function R( u) and higher cumulants defined from the field theory can be obtained for any d from moments of a renormalized potential defined in an sliding harmonic well. This allows to measure the fixed point function R( u) in numerics and experiments. In d = 0 the beta function (of the inviscid limit) is obtained from first principles to four loop. For Sinai model (uncorrelated Burgers initial velocities) the ERG hierarchy can be solved and the exact function R( u) is obtained. Connections to exact solutions for the statistics of shocks in Burgers and to ballistic aggregation are detailed. A relation is established between the size distribution of shocks and the one for droplets. A droplet solution to the ERG functional hierarchy is found for any d, and the form of R( u) in the thermal boundary layer is related to droplet probabilities. These being known for the d = 0 Sinai model the function R( u) is obtained there at any T. Consistency of the ϵ = 4 - d expansion in one and two loop FRG is studied from first principles, and connected to shock and droplet relations which could be tested in numerics.