We numerically investigate the matrix model of two-color one-flavor adjoint QCD (matrix-${\mathrm{QCD}}_{2,1}^{\mathrm{adj}}$) 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 $c\le \frac{3}{2}$ there is a quantum phase transition (QPT) at $g_0^{*}\simeq 0.143$: for $g<g_0^{*}$, the ground-state wave function is localized near $A_i=0$, while for $g>g_0^{*}$, the ground state is delocalized over the gauge configuration space. The transition between these two phases is singular, with the ground state at $g_0^{*}$ being distinctly different from that of $g_0^{*}\pm |\epsilon |$. At $g_0^{*}$, 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 first and second excited states as well, leading us to conjecture that there are an infinite number of isolated singular points $g_0^{*}>g_1^{*}>g_2^{*}>\cdots$ 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 $g<g_0^{*}$) the supermultiplet structure is disrupted and supersymmetry is spontaneously broken.

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• Hypothetical particle physics models
• Nonperturbative effects in field theory
• Quantum phase transitions
• Supersymmetry
• Variational approach