Localization-delocalization transition at weak coupling in two-color matrix QCD

We numerically investigate the matrix model of two-color one-flavor adjoint QCD (matrix-QCD$_{2,1}^{\text{adj}}$) in the weak coupling regime (small $g$) and in the chiral limit. The Yang-Mills potential has two distinct gauge invariant minima: one at $A_i=0$ and the other at $A_i = \frac{\sigma_i}{2g}$. We show that when the chiral chemical potential $c \leq \frac{3}{2}$, there is a quantum phase transition at $g_0^\ast \simeq 0.143$: for $g<g_0^\ast$, the ground state wavefunction is localized near $A_i=0$, while for $g>g_0^\ast$, the ground state is delocalized over the gauge configuration space. The transition between these two phases is singular, with the ground state at $g_0^\ast$ being distinctly different from that of $g_0^\ast \pm|\epsilon|$. At $g_0^\ast$, we show that the square of the chromoelectric field vanishes, strongly suggesting that the system is in a ``dual superconductor" phase. Numerical evidence shows that the localization-delocalization phenomenon holds for the 1st and 2nd excited states as well, leading us to conjecture that there are an infinite number of isolated singular points $g_0^\ast> g_1^\ast>g_2^\ast> \cdots$ accumulating to $g=0$. For $c=1$, the model formally possesses $\mathcal{N}=1$ supersymmetry. We show that in the localized phase (i.e. for $g<g_0^\ast$) the supermultiplet structure is disrupted and SUSY is spontaneously broken.