Effects of boundary conditions and gradient flow in ( 1+1 )-dimensional lattice ϕ4 theory

Abstract

In this work we study the effects of boundary condition and gradient flow in 1+1 dimensional lattice φ 4 theory. Simulations are performed with periodic (PBC) and open (OPEN) boundary conditions in the temporal direction and the lattice fields are then smoothed by applying gradient flow. Our results with observables such as the | φ | and the susceptibility indicate that at a given volume, the phase transition point is shifted towards a lower value of lattice coupling λ0 for fixed m 2 0 in the case of OPEN as compared to the PBC, with this shift found to be diminishing as the volume increases. We have employed the finite size scaling (FSS) analysis to obtain the true critical behavior, mainly to emphasize the necessity of an FSS formalism incorporating the surface effect in the case of the open boundary. Above features have been found to be illuminated more clearly by the application of gradient flow. Finally 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, finite volume effects become dominant in the latter. The surface effect seems to be resulted in a less sharper phase transition for OPEN compared to the PBC for all observables studied here.

Critical Coupling for Two-dimensional $ϕ^4$ Theory in Discretized Light-Cone Quantization

“…Can one demonstrate the vanishing of the wave function renormalization constant at the critical point [44,45] in the continuum limit? Can one compute the critical exponents?…”