Cooling stochastic quantization with colored noise

Abstract

Smoothing of field configurations is highly important for precision calculations of physical quantities on the lattice. We present a cooling method based on Stochastic Quantization with a built-in UV momentum cutoff. The latter is implemented via a UV-regularized, hence colored, noise term.Our method is tested in a two-dimensional scalar field theory. We show, that UV modes can be removed systematically without altering the physics content of the theory. The approach has an interpretation in terms of the non-perturbative (Wilsonian) renormalization group that facilitates the physics interpretation of the cutoff procedure. It also can be used to define the maximal colored cooling applicable without changing the theory.

Low energy effective theories on the lattice with coloured noise

“…By employing coloured noise, see, e.g., [11], it is possible to have a hybrid Langevin time evolution, where fluctuations with momentum larger than a certain energy scale Λ follow a gradient flow. Thus, in the long time limit fluctuations with 𝑝 2 > Λ 2 are suppressed, while those below Λ contribute to the quantum expectation values.…”

“…In practice, this is achieved by considering the noise field 𝜂 𝑖 𝑥 in momentum space, applying some regulator function to it, and Fourier transforming it back to position space. In this work we follow [11] and impose a sharp cutoff of the form 𝜃 (Λ 2 − 𝑝 2 ). Since our theory is defined on the lattice within a finite volume, the available momentum states are both discrete and finite.…”

“…That is, 𝑠 controls the fraction of momentum states that contribute to the quantum expectation values. Note that our definition of 𝑠 differs from that of [11].…”

Gradient flow exact renormalization group -- inclusion of fermion fields

“…Possible connections between ERG and the gradient flow or diffusion equations have been studied in Refs. [23][24][25][26][27][28][29][30][31][32][33].…”

Stochastic renormalization group and gradient flow

“…In this work, we will contribute to these discussions by demonstrating the equivalence of certain kinds of FRG transformations with a class of stochastic (Markov) processes on field space. 2 It has been noted before [17][18][19] that the functional RG equations for effective actions, when written in terms of effective Boltzmann weights, are of the form of a Fokker-Planck (FP) equation, whose solution is therefore a probability distribution over effective fields. Taking this observation seriously, and recalling that Fokker-Planck distributions can be thought of as being generated by a Langevin equation on the degrees of freedom appearing in the FP distribution, one may ask what kinds of Langevin equation generate the FRG effective actions.…”