Cutoff and lattice effects in the $\varphy^4$ theory of confined systems

TL;DR

This paper investigates how cutoff and lattice effects influence the finite-size behavior of the $ ext{O}(n)$ symmetric $ ext{phi}^4$ theory in confined geometries, revealing nonuniversal deviations from bulk criticality and emphasizing the need for non-perturbative approaches.

Contribution

It demonstrates that finite cutoff effects cause nonuniversal deviations from finite-size scaling in the $ ext{phi}^4$ theory, contrasting with lattice model results and highlighting the importance of non-perturbative methods.

Findings

Finite cutoff predicts deviations scaling as $( ext{cutoff} imes L)^{-2}$.

Lattice model deviations decay exponentially with system size.

Non-perturbative treatment is necessary for accurate size dependence.

Abstract

We study cutoff and lattice effects in the O(n) symmetric $\phi^4$ theory for a $d$-dimensional cubic geometry of size $L$ with periodic boundary conditions. In the large-N limit above $T_c$, we show that $\phi^4$ field theory at finite cutoff $\Lambda$ predicts the nonuniversal deviation $\sim (\Lambda L)^{-2}$ from asymptotic bulk critical behavior that violates finite-size scaling and disagrees with the deviation $\sim e^{-cL}$ that we find in the $\phi^4$ lattice model. The exponential size dependence requires a non-perturbative treatment of the $\phi^4$ model. Our arguments indicate that these results should be valid for general $n$ and $d > 2$.