Perfect Scalars on the Lattice

TL;DR

This work develops a blocking-from-the-continuum renormalization-group approach to produce perfect lattice actions for free scalar fields of any mass. The resulting actions have couplings that decay exponentially and reproduce the continuum spectrum with negligible lattice artifacts, and truncation to a unit hypercube preserves much of this improvement. It also demonstrates practical preconditioning for these actions and extends the framework to non-hypercubic lattices and wavelet-inspired blocking, with implications for gauge fields and a fermionic fixed point. The study provides a concrete, scalable route to artifact-free lattice simulations of scalar theories and informs broader applications in lattice field theory.

Abstract

We perform renormalization group transformations to construct optimally local perfect lattice actions for free scalar fields of any mass. Their couplings decay exponentially. The spectrum is identical to the continuum spectrum, while thermodynamic quantities have tiny lattice artifacts. To make such actions applicable in simulations, we truncate the couplings to a unit hypercube and observe that spectrum and thermodynamics are still drastically improved compared to the standard lattice action. We show how preconditioning techniques can be applied successfully to this type of action. We also consider a number of variants of the perfect lattice action, such as the use of an anisotropic or triangular lattice, and modifications of the renormalization group transformations motivated by wavelets. Along the way we illuminate the consistent treatment of gauge fields, and we find a new fermionic fixed point action with attractive properties.

Figures (7)

• Figure 1: The decay of the couplings $\vert \rho (i,0,0,0)\vert \propto \exp \{ - c_{1} (\alpha ) i\}$ of the perfect scalars for masses 0, 2 and 4. The maxima are very close to the values of $\bar{\alpha} (m)$, 0.167, 0.203 resp. 0.364, which correspond to eq. (\ref{['smear']}).
• Figure 2: The decay of the couplings $\vert \rho (i,i,i,i)\vert$ in the optimally local perfect scalars for masses 0, 1, 2 and 4. The lines are least square fits to $\vert \rho (i,i,i,i) \vert \propto \exp \{ -c_{4}(m)i \}$, with $c_{4}(0) = 6.515, \ c_{4}(1) = 6.775, c_{4}(2) = 7.493$ and $c_{4}(4) = 9.769$.
• Figure 3: The decay of the couplings of the perfect scalars -- measured by $c_{4}$ in $\vert \rho (i,i,i,i) \vert \propto \exp \{ - c_{4}i \}$ -- as a function of the mass $m$. The optimally local perfect scalars follow approximately a parabola, and its decay is clearly faster the one obtained from a $\delta$ function RGT.
• Figure 4: The mass dependence of the hypercubic couplings $\rho_{0} = \rho (0,0,0,0)$ (left), as well as $\rho_{1} = \rho (1,0,0,0)$, $\rho_{2} = \rho(1,1,0,0)$, $\rho_{3} = \rho (1,1,1,0)$ and $\rho_{4}= \rho (1,1,1,1)$ (right).
• Figure 5: The dispersion relation for a scalar of mass $m=0$ (left) and $m=2$ (right) in the (110) direction with the perfect, truncated perfect ("hypercubic") and standard lattice action. For the hypercube action (as well as the standard action) there are no "ghosts" (higher branches) in any direction, in contrast to the Symanzik improved action.