Low-cost fermions in classical field simulations

Abstract

We discuss the possible extension of the bosonic classical field theory simulations to include fermions. This problem has been addressed in terms of the inhomogeneous mean field approximation by Aarts and Smit. By performing a stochastic integration of an equivalent set of equations we can extend the original 1+1 dimensional calculations so that they become feasible in higher dimensions. We test the scheme in 2 + 1 dimensions and discuss some classical applications with fermions for the first time, such as the decay of oscillons.

Schwinger mechanism revisited

“…In real-time lattice simulations for fermion fields, another method called the low-cost fermion method [73] has been used in several works [74][75][76][77]. In the low-cost fermion method, instead of using one stochastic field (161), one employs two kinds of stochastic fields called "male" and "female" fields:…”

Connecting real-time properties of the massless Schwinger model to the massive case

“…Moreover, we used x ±t ≡ (t ± a t , x) and x ±s ≡ (t, x ± a s ). We do not include a Wilson term for the temporal doubler modes as they are naturally suppressed for suitable initial conditions and for a temporal lattice spacing being much smaller than the spatial one, a t ≪ a s [18,19,24,25]. For practical purposes, a ratio a s /a t ≃ 20 usually suffices to guarantee that temporal doubler modes are not excited during the simulated time interval.…”

Fermion production from real-time lattice gauge theory in the classical-statistical regime

“…it ensures that only low-momentum excitations show a low-energy dispersion relation. We do not include a Wilson term for the temporal doubler modes as they are naturally suppressed for suitable initial conditions and if the temporal lattice spacing is taken to be much smaller than the spatial ones, a 0 ≪ a i [9,39,40]. Moreover, to simplify simulations afterwards, we use the gauge freedom and employ U 0,n = 1, which is the lattice equivalent of the temporal-axial gauge condition A 0,n = 0.…”

“…It can be evaluated without further approximations using a mode function expansion as described in Appendix D or employing a stochastic 'low-cost' fermion algorithm [16,39]. In the former approach, the equations of motion (52) are then regarded as equations of motion for the mode functions Φ u λ,n,q and Φ v λ,n,q .…”