Connecting phase transitions between the 3-d O(4) Heisenberg model and 4-d SU(2) lattice gauge theory

Abstract

SU(2) lattice gauge theory is extended to a larger coupling space where the coupling parameter for horizontal (spacelike) plaquettes, $\beta_H$, differs from that for vertical (Euclidean timelike) plaquettes, $\beta_V$. When $\beta_H \rightarrow \infty$ the system, when in Coulomb Gauge, splits into multiple independent 3-d O(4) Heisenberg models on spacelike hyperlayers. Through consideration of the robustness of the Heisenberg model phase transition to small perturbations, and illustrated by Monte Carlo simulations, it is shown that the ferromagnetic phase transition in this model persists for $\beta_H < \infty$. Once it has entered the phase-plane it must continue to another edge due to its symmetry-breaking nature, and therefore must necessarily cross the $\beta_V = \beta_H$ line at a finite value. Indeed, a higher-order SU(2) phase transition is found at $\beta = 3.18 \pm 0.08$, from a finite-size scaling analysis of the Coulomb gauge magnetization from Monte Carlo simulations, which also yields critical exponents. An important technical breakthrough is the use of open boundary conditions, which is shown to reduce systematic and random errors of the overrelaxation gauge-fixing algorithm by a factor of several hundred. The string tension and specific heat are also shown to be consistent with finite-order scaling about this critical point using the same critical exponents.