Block renormalization group for Euclidean Fermions

Abstract

Abstract. Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and "6-function" weight functions. For the free field the sequence of actions Dk generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform in k, after rescaling to the unit lattice. It is shown that the two-point function D-1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the "6-function" RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.

Locality of the fourth root of the staggered-fermion determinant: Renormalization-group approach

“…ν | 2 where again c 2 = O(1) [6]. Hence the k (n) µ -summation converges, and |B n µ (p)| = O(2 −n ) for all p. Taking the limit n → ∞ we conclude that B n µ (p) → 0, uniformly in p.…”

“…This simple argument shows that, indeed, the proof given in ref. [6] generalizes to the RG-blocked staggered Dirac operator in its flavor representation.…”

“…[12,6]. Starting from a bilinear fermion action with Dirac operator D 0 , an RG blocking is introduced via the identities…”

“…Central to this discussion is a theorem on the locality of RG-blocked Wilson fermions proved in ref. [6]; only a trivial amendment is needed in order to generalize the theorem to the flavor representation of staggered fermions. The limiting operator is constructed explicitly.…”

Renormaliation-group blocking the fourth root of the staggered determinant

“…As a result [6], the tastebreaking part of the coordinate-space free propagator vanishes exponentially with the separation, with an O(1) decay rate in lattice units. We must keep in mind, though, that for momenta or separations which are O(1) in lattice units, the skewed Wilson term does spoil the diagonal taste structure.…”