# Critical Phenomena with Linked Cluster Expansions in a Finite Volume

## TL;DR

This paper extends linked cluster expansions to finite volumes to analyze critical phenomena, proposing a new criterion for phase transition order and applying it to scalar models, with implications for various physical systems.

## Contribution

It introduces a finite-volume linked cluster expansion method and a new criterion for distinguishing phase transition orders, demonstrated on scalar O(N) models.

## Key findings

- Successfully performed 20th order expansions approaching the critical region.
- Proposed a criterion to differentiate 1st and 2nd order transitions in finite size scaling.
- Localized the tricritical line in scalar O(N) models.

## Abstract

Linked cluster expansions are generalized from an infinite to a finite volume. They are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase. A new criterion is proposed to distinguish 1st from 2nd order transitions within a finite size scaling analysis. The criterion applies also to other methods for investigating the phase structure such as Monte Carlo simulations. Our computational tools are illustrated at the example of scalar O(N) models with four and six-point couplings for $N=1$ and $N=4$ in three dimensions. It is shown how to localize the tricritical line in these models. We indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity.