Negative-Power Density of States in Chiral-Orthogonal Disordered Systems and the IR Phase of QCD

The recently discovered IR phase of QCD is characterized by an infrared (IR) sector of Dirac modes that separates and dynamically decouples from the bulk of the spectrum, becoming scale invariant in the process [1]. Among key features of this IR sector is the negative power-law behavior of Dirac spectral density near zero with power $p$ close to $-1$. Here we start investigating to what extent can this behavior be mimicked by chiral Anderson models. In particular, we study 3-D chiral-orthogonal systems with logarithmic off-diagonal disorder where negative-power ``spikes'' have been observed [2]. We find that at finite disorder strength $W$ of moderate strength, these models in fact do not have a power-singular density of states (DOS) at zero energy in the thermodynamic limit. Rather, this power $p_1$ vanishes and the spectrum features a very accurate power behavior with $p<0$ on an interval around $E = 1$ spreading increasingly closer to $E=0$ and $E = \infty$ with increasing $W$. We connect this to the fact that random hoppings $t$ in these quantum systems are drawn from a scale-invariant $1/t$ distribution regularized via $t \in [e^{-W/2}, e^{W/2}]$, and DOS with negative power can naturally arise inside that range. We show that $\lim_{W \to \infty} p(W, E)=-1$ for all energies accessible by our data. This indicates that, upon removing the cutoff $W$, quantum mechanics transforms the scale-invariant $1/t$ noise into a scale-invariant $1/E$ distribution of eigenstates, which mimics the interpretation of scale-invariant Dirac spectral densities in the IR phase as being due to scale-invariant glue backgrounds [1]. Our results imply that, to model the IR sector of the IR phase, the IR cutoff in chiral Anderson systems needs to be viewed as descending from the randomness strength $W$ rather than the lattice size $L$.