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

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.