YM-Thermodynamics 2024, 42: Thermodynamical Quantities & Lattice Gauge Theory

For proper lattices of finite volume, the integral method approaches the limit of infinite volume from the very start. The partition function is defined through the differential

By using an integral approach for P(T) yields a large dependence of the spatial size and time extent of the lattice. The deviation from the assumed TD limit is considerable. By a coupling factor for a correction factor, find a spatial anisotropy coefficient, effectively the β-function. The correction factor accounts for finite-size effects. From the 2-loop perturbative β-function, negative pressures for temperatures close to the critical temperature, as well as quickly approaching energy density and pressures to their free-gas limits emerge.

Lattice simulation can measure effects of the adjoint scalar fields directly. Since its emergence is self-consistently tied to its maximal resolution with constant lattice spacing, associated with much higher spatial resolution. This defines the field over-resolved, so that it doesn't play a direct role in lattice simulations, which seems like a problem. However, averaging over the trace of the field strength tensors' integral after singular gauge fixing yields a sine-function with τ-dependency. In deconfining phase there was a perimeter law based on Wilson action, implying that it obeys an area law at high temperatures. Certain aspects of emerging monopoles might be missing from the simulation.

With inverse spatial lattice spacing, at several times critical temperature is considerably larger than natural resolution. Due to topologically nontrivial field contributions to the partition function, the lattice action is more complicated than the Wilson action of the continuum YM-action. Using the Wilson action, the fundamental coupling is barely affected by bulk properties of the nonperturbative ground state physics. Perfect lattice action is very expensive to compute though.

The lattice partition function is not invariant under changes of resolution scale at high temperatures, due to the resolution dependence of the Wilson action in the fundamental gauge coupling. The magnetic flux then is measured through the spatial contour in lattice simulations. Effects of the large holonomy dissociation in infinite-volume YM-TD on a spacetime continuum are probably lost in computation.

Finite spatial extent of the lattice cuts off long-range correlations of screened and stable magnetic (anti-)monopoles becoming massless around critical temperature. The fact that the formation of a stable monopole-antimonopole condensate takes place at some temperature (even below critical), shrinks the effective region. In simulations, this leads to much exaggerated condensation detection without resolution of the preceeding step of stable (anti-)monopole condensation. Compared to perfect lattice action, the Wilson action is likely a unreliable approximation.