Momentum lattice simulation of theφ4model close to criticality

Abstract

We present results of a numerical simulation of the 44 model in the symmetric phase close to the critical line, using a momentum lattice and Langevin updating. We discuss how to extract physics from the zero-momentum behavior of the correlation function. As an example, we have computed mR and ZR and compared it with the analytical results of Liischer and Weisz.

Momentum Lattice Simulation on a Small Lattice Using Stochastic Quantization

“…It can be solved using a standard method like Euler's method where the truncation term is O(∆τ 2 ), or a higher order scheme like Heun's method (second order Runge-Kutta) where the truncation term is O(∆τ 3 ). Batrouni et al [1] have shown that a higher order integration scheme has many advantages, and our experience [3] agrees with theirs. For the same level of error, Heun's method is faster; it takes fewer "time" steps.…”

“…This kind of behavior has not been seen in the φ 4 -model. The knowledge of the stability region is very important for simulations of theories like SU(2) and SU (3). For these models, the time needed for generating equilibrium configurations is much larger than for the U(1)-model.…”

“…When we compare the results of Figs. [3,4] we notice a difference which seems to exceed the error O(∆τ 2 ) due to finite ∆τ . We can reduce fluctuations by increasing the number of configurations of fields, but we can also average several values of τ up to τ f inal .…”

“…In Figs. [3] and [4], we computed the expectation value of the Wilson loop operator for different sizes of loops and for two different "time" steps on a lattice of 6 4 points. We can see the fluctuation introduced by the noise fieldη(k, τ ).…”

“…Many other methods have been developed to accelerate convergence of the calculation, e.g., the cluster algorithm [2], suitable for Ising-type systems. When we studied the scalar φ 4 -model near the critical point on a momentum lattice [3], it turned out that quite reasonable results for the renomalized mass, wave function renormalization, etc., could be obtained on relatively small lattices (3 4 -7 4 ). In this note we want to report a more detailed numerical analysis of the convergence behavior on a momentum lattice.…”