# Finite Size Scaling Analysis with Linked Cluster Expansions

## TL;DR

This paper extends linked cluster expansions to finite volumes to analyze phase transitions in scalar $O(N)$ models, proposing a new criterion to distinguish first and second order transitions and applying it to locate tricritical points.

## Contribution

It introduces a finite volume linked cluster expansion method and a new criterion for identifying the order of phase transitions, applicable to Monte Carlo simulations.

## Key findings

- Successfully localized the tricritical line in a $\
- Proposed a volume-dependent response function criterion for transition order
- Demonstrated applicability to electroweak transition studies

## Abstract

Linked cluster expansions are generalized from an infinite to a finite volume on a $d$-dimensional hypercubic lattice. They are performed to 20th order in the expansion parameter to investigate the phase structure of scalar $O(N)$ models for the cases of $N=1$ and $N=4$ in 3 dimensions. In particular we propose a new criterion to distinguish first from second order transitions via the volume dependence of response functions for couplings close to but not at the critical value. The criterion is applicable to Monte Carlo simulations as well. Here it is used to localize the tricritical line in a $\Phi^4 + \Phi^6$ theory. We indicate further applications to the electroweak transition.