The two-particle-irreducible vertex of the two-dimensional lattice $ϕ^4$ model across the Ising transition

Abstract

We reconstruct the 2PI vertex $Γ(k,p;q)$ from Monte Carlo measurements of the connected two-particle correlator for the two-dimensional single-component $ϕ^4$ lattice field theory and follow it across the Ising transition. Resolving the vertex in the irreducible representations of the point group $C_{4v}$, we find that the instability is driven by the $A_1$ (ferromagnetic) channel at zero transfer, whose leading eigenvalue of the symmetrized Bethe--Salpeter kernel approaches unity. Substantial $B_1$ (nematic) and $B_2$ (diagonal nematic) contributions cooperate with $A_1$ across all system sizes, highlighting that the soft sector is multidimensional. In real space, the vertex is short-ranged away from criticality while it develops a power-law tail at the critical point. In the ordered phase, the $q=0$ eigenvalue collapses because the ferromagnetic weight has condensed into the (one-particle-reducible) order parameter (or collective coordinate for a finite system), although finite-momentum fluctuations persist. By stripping the crossed-channel ladders, we obtain the fully irreducible vertex, which is a local contact -- to a very good approximation. Inserted into the parquet and Schwinger--Dyson equations, this contact reproduces the Monte Carlo self-energy with an accuracy better than one-tenth of a percent. This provides a first-principles benchmark of the dynamical local-vertex approximation (D$Γ$A). Additionally, we demonstrate that in the critical region, the physical solution of the parquet equations behaves as a repulsive fixed point, driven initially by a single order-parameter mode.