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Perfect Lattice Perturbation Theory: A Study of the Anharmonic Oscillator

As an application of perfect lattice perturbation theory, we construct an O(λ) perfect lattice action for the anharmonic oscillator analytically in momentum space. In coordinate space we obtain a set of 2-spin and 4-spin couplings \propto λ, which we evaluate for various masses. These couplings never involve variables separated by more than two lattice spacings. The O(λ) perfect action is simulated and compared to the standard action. We discuss the improvement for the first two energy gaps ΔE_1, ΔE_2 and for the scaling quantity ΔE_2 / ΔE1 in different regimes of the interaction parameter, and of the correlation length.