Finite-Size Effects in the $\phi^{4}$ Field Theory Above the Upper Critical Dimension

TL;DR

This paper reveals that the standard $ ext{O}(n)$ symmetric $ ext{phi}^4$ field theory fails to accurately describe finite-size effects near criticality in high-dimensional lattice spin systems, emphasizing the need for lattice Hamiltonian approaches.

Contribution

It demonstrates the inadequacy of the $ ext{phi}^4$ field theory for finite-size effects above the upper critical dimension and provides explicit results for specific cases.

Findings

Finite-size effects require a lattice Hamiltonian description.

Explicit susceptibility and Binder cumulant results for $n o \infty$ and $n=1$.

Recent Monte Carlo analyses of 5D Ising model are inconclusive due to theoretical limitations.

Abstract

We demonstrate that the standard O(n) symmetric $\phi^{4}$ field theory does not correctly describe the leading finite-size effects near the critical point of spin systems on a $d$-dimensional lattice with $d > 4$. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For $n \to \infty$ and $n=1$ explicit results are given for the susceptibility and for the Binder cumulant. They imply that recent analyses of Monte-Carlo results for the five-dimensional Ising model are not conclusive.