Abstract
In the landscape of approaches toward the simulation of Lattice Models with complex action the Complex Langevin (CL) appears as a straightforward method with a simple, well defined setup. Its applicability, however, is controlled by certain specific conditions which are not always satisfied. We here discuss the procedures to meet these conditions and the estimation of systematic errors and present some actual achievements.
Highlights
For a model defined by a path integral with a complex action S, Complex Langevin (CL) is a stochastic process proceeding on the complexification of the original manifold, e.g., Rn −→ Cn or S U(n) −→ S L(n, C)
E.g., for one variable x → z = x + i y the CL amounts to two real Langevin processes in the process time t δx(t) = Kx(z) δt + ηx(t), Kx = ReK(z), δy(t) = Ky(z) δt + ηy(t), Ky = ImK(z), ηx = 0, ηy = 0, η2x = 2NR δt η2y = 2NI δt, NR − NI = 1
We present results in terms of μ for a targeted 4-dim. lattice model with Nτ = 8 and κ = 0.12
Summary
For a model defined by a path integral with a complex action S , CL is a stochastic process proceeding on the complexification of the original manifold, e.g., Rn −→ Cn or S U(n) −→ S L(n, C). It involves a drift term K = −∇S and a suitably normalized random noise η [1], cf [2]. The process realises a real probability distribution P(x, y; t) accompanying the complex distribution ρ(x, t) (with asymptotic limit ρ(x) = exp(−S (x))).
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