# Effects of boundary conditions and gradient flow in 1+1 dimensional lattice $\phi^4$ theory

## TL;DR

This paper investigates how boundary conditions and gradient flow influence the phase transition, mass extraction, and boundary artifacts in 1+1 dimensional lattice  theory, highlighting the importance of finite volume analysis.

## Contribution

It provides a detailed comparison of open and periodic boundary conditions and their effects on phase transition points and mass measurements in lattice  theory.

## Key findings

- Open boundary conditions shift the phase transition point to lower coupling values.
- Boundary artifacts dominate in the critical region when using open boundaries.
- Finite volume effects are significant and require detailed analysis in the critical region.

## Abstract

In this work we study the effects of gradient flow and open boundary condition in the temporal direction in 1+1 dimensional lattice $\phi^4$ theory. Simulations are performed with periodic (PBC) and open (OPEN) boundary conditions in the temporal direction. The Effects of gradient flow and open boundary on the field $\phi$ and the susceptibility are studied in detail along with the finite size scaling analysis. In both cases, at a given volume, the phase transition point is shifted towards a lower value of lattice coupling $\lambda_0$ for fixed $m_0^2$ in the case of OPEN as compared to PBC with this shift found to be diminishing as volume increases. We compare and contrast the extraction of the boson mass from the two point function (PBC) and the one point function (OPEN) as the coupling, starting from moderate values, approaches the critical value corresponding to the vanishing of the mass gap. In the critical region, boundary artifacts become dominant in the latter. Our studies point towards the need for a detailed finite volume (scaling) analysis of the effects of OPEN in the critical region.