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

TL;DR

The paper investigates the weak-coupling regime of the matrix-QCD$_{2,1}^{adj}$ model, revealing a localization-delocalization quantum phase transition at $g_0^\ast\simeq0.143$ between localized ground states near $A_i=0$ and delocalized configurations, with the transition marked by a non-normalizable state and vanishing chromoelectric field, interpreted as a dual superconducting phase. Using a variational/Ritz approach in singular-value coordinates and finite-size scaling, the authors show that higher excited states exhibit analogous singular points $g_n^\ast$ accumulating at $g=0$, and that turning on a chiral chemical potential $c$ introduces a first-order ground-state transition line $g_0^R(c)$ and a rich $c$-$g$ phase diagram. At $c=1$ the model possesses formal $\mathcal{N}=1$ SUSY, which is spontaneously broken in the localized phase due to disrupted supermultiplets, while delocalized phases preserve SUSY. Collectively, the work provides a tractable, QCD-like setting in which condensate formation, dual superconductivity, and SUSY dynamics emerge from a finite-dimensional matrix model, offering insights into non-perturbative gauge dynamics and phase structure. The findings motivate further exploration of corner-state dynamics on the equal-singular-value surface and connections to improved Born-Oppenheimer descriptions.

Abstract

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 $A_i = \frac{σ_i}{2g}$. 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 $g_0^\ast \pm|ε|$. 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.

Figures (18)

• Figure 1: Equipotential surfaces of $V_{YM}$ in the $a_1$-$a_2$-$a_3$ space.
• Figure 2: a) The energies of the states in $n_F=0$ sector: $E_n^{(0)}$ as function of $g$ for $n=0,1,2$. The dots represent the data with $N_{b}=18$ and the solid lines with $N_{b}=16$. b) The comparison of $E_n^{(0)}$ with the corresponding fit in (\ref{['fit_E0_nf0']}), which is represented by the black dashed line. Here, we have chosen $c=0$.
• Figure 3: a) $E_0^{(2)}$ as a function of $g$ for various $N_{b}$. The solid lines represent the data from Rayleigh-Ritz method. The black dashed lines shows the estimate from perturbation theory as in Eq.(\ref{['pert_gs_E']}). The blue dashed line is the extrapolation to the $N_{b} \to \infty$ limit. b) The region around the kink in $E_0^{(2)}(g)$. The colored curves here follow the same labelling as in Fig. 3a. Here $c=0$ for both the figures.
• Figure 4: $\widetilde{g}_n$ vs $N_{b}$ for $n=0,1,2$. The dots represent the numerical data using Rayleigh-Ritz method and the dashed lines are fits in (\ref{['fit_g0']}) and (\ref{['fit_gn']}) with parameters in Table \ref{['Table-2']}. Here, we have chosen $c=0$.
• Figure 5: The expectation values $\mathcal{O}_{A,0}$ as a function of $g$ for various $N_{b}$ in the $0\leq g \leq 0.5$ regime. Black dashed lines corresponds to the estimate from perturbation theory: $\mathcal{O}_{1,0} \simeq 24 g + O(g^3)$, $\mathcal{O}_{2,0} \simeq \frac{9}{2} + \frac{57}{2} g^2 + O(g^4)$, and $\mathcal{O}_{3,0} \simeq \frac{9}{4} + \frac{51}{4} g^2 + O(g^4)$. Here, we have chosen $c=0$.