Resurrecting the coherent state variational algorithm for large $N$ gauge theories

Abstract

The feasibility of studying, numerically, properties of infinite volume QCD-like theories in the large $N$ limit using coherent state variational methods is reassessed. An entirely new implementation of this approach is described, applicable to SU($N$) lattice gauge theories, with or without fundamental representation fermions, on cubic lattices of up to four dimensions. In addition to various test cases, initial results are presented for Hamiltonian Yang-Mills theory on an infinite two-dimensional spatial lattice.

Figures (20)

• Figure 1: Left: Free energy of the Euclidean one-plaquette model. Shown is the exact result (solid black line) and results from variational calculations using 1 through 5 generators. Results with three or more generators are not visually distinguishable from the exact curve. Right: Semi-log plot of the absolute error magnitude between exact and variational results with 1 (blue), 2 (orange), 3 (green), 4 (red), or 5 (purple) generators; curves with progressively longer dashes have an increasing number of generators.
• Figure 2: Left: Winding one loop expectation value $w_1 \equiv \langle x \rangle$ in the Euclidean one-plaquette model. Shown is the exact result (solid black line) and results from variational calculations using 1 through 5 generators. Results with three generators are barely visually distinguishable from the exact curve. Right: Semi-log plot of the absolute error magnitude between exact and variational results with the number of generators ranging from 1 to 5. (The same line styles as in Fig. \ref{['fig:ym1e-f']} are used.)
• Figure 3: Left: Winding three loop expectation $w_3 \equiv \langle xxx \rangle$ in the Euclidean one-plaquette model. Shown is the exact result (solid black line) and results from variational calculations using 1 through 5 generators. Results with four generators are barely visually distinguishable from the exact curve. Right: Semi-log plot of the absolute error magnitude between exact and variational results with the number of generators ranging from 1 to 5. (The same line styles as in Fig. \ref{['fig:ym1e-f']} are used.)
• Figure 4: Left: Ground state energy of the Hamiltonian one-plaquette model. Shown is the exact result (solid black line) and results from variational calculations using 1, 2, 4 and 8 generators. (The eight generator curve is indistinguishable from the exact curve.) Right: Semi-log plot of the absolute error magnitude between exact and variational results with 1 (blue), 2 (orange), 4 (green) and 8 (red) generators; curves with progressively longer dashes have an increasing number of generators. The black triangle on the $x$-axis marks the position of the phase transition.
• Figure 5: Left: Single winding loop expectation value $w_1 \equiv \langle x \rangle$ in the Hamiltonian one-plaquette model. Shown is the exact result (solid black line) and results from variational calculations using 1, 2, 4 and 8 generators. (The eight generator curve is indistinguishable from the exact curve.) Right: Semi-log plot of the absolute error magnitude between exact and variational results with 1, 2, 4 and 8 generators. (The same line styles as in Fig. \ref{['fig:ym1h-h']} are used.)