Peculiarities in the spectrum of the adjoint scalar kinetic operator in Yang-Mills theory

We study the spectrum of low-lying eigenmodes of the kinetic operator for scalar particles, in the color adjoint representation of Yang-Mills theory. The kinetic operator is the covariant Laplacian, plus a constant which serves to renormalize mass. In the pure gauge theory, our data indicates that the interval between the lowest eigenvalue and the mobility edge tends to infinity in the continuum limit. On these grounds, it is suggested that the perturbative expression for the scalar propagator may be misleading even at distance scales that are small compared to the confinement scale. We also measure the density of low-lying eigenmodes, and find a possible connection to multicritical matrix models of order $m=1$.

Figure 1

$⟨{\mathrm{IPR}}_n⟩/⟨{\mathrm{IPR}}_0⟩$ vs $\Delta {\lambda }_n=⟨{\lambda }_n-{\lambda }_0⟩/a^2$ at $\beta =2.35$, on lattice volumes from ${14}^4$ to ${24}^4$. The data points at each volume are connected by lines.

Figure 2

Same as Fig. 1, except lines connecting data points at different volumes are omitted. The straight line is a best fit through the combined all-volumes data in the interval $\Delta \lambda \in [0:0.045]$.

Figure 3

$⟨{\mathrm{IPR}}_n⟩/⟨{\mathrm{IPR}}_0⟩$ vs $\Delta {\lambda }_n=⟨{\lambda }_n-{\lambda }_0⟩/a^2$ at $\beta =2.5$, on lattice volumes from ${16}^4$ to ${24}^4$. The data points at each volume are connected by lines.

Figure 4

Same as Fig. 3, except lines connecting data points at different volumes are omitted. The straight line is a best fit through the combined all-volumes data in the interval $\Delta \lambda \in [0:0.06]$.

Figure 5

Cumulative number of eigenmodes $n(\lambda )$ with eigenvalue less than $\lambda$ at $\beta =2.5$. Data shown is for lattice volumes from ${14}^4$ to ${24}^4$.

Figure 6

Cumulative eigenvalue number $n(z)$ vs the scaling variable $z=V^{1/(1+\alpha )}(\lambda -{\lambda }_0)$ at $\beta =2.5$ and $\alpha =1$.

Figure 7

Same as Fig. 6, with $\alpha =2$.

Figure 8

Same as Fig. 6, with $\alpha =4$.

Figure 9

Best fit of $n(z)=\frac{1}{3}\kappa z^3$ to the $\beta =2.5$ data (combined volumes) at $\alpha =2$.

Figure 10

Unfolded eigenvalue spacing distribution for $\beta =2.35$, $L=18$.

Figure 11

Cumulative eigenvalue number $n(z)$ vs $z$ at $\beta =2.5$, $\alpha =2$ on a ${16}^4$ lattice volume. The data shown includes all calculated eigenmodes, which extend beyond the mobility edge. For reference, the vertical line intersects the $x$-axis at a value of $z$ corresponding to $\lambda -{\lambda }_0=0.05$ (see text).

Figure 12

Same as Fig. 11, on a ${20}^4$ lattice volume.

Figure 13

Dependence of $(\frac{\mathrm{IPR}}{L^4}{)}^{0.25}$ on $\lambda$ for $\beta =2.40$, $L=20$.

Figure 14

Dependence of $(\frac{\mathrm{IPR}}{L^4}{)}^{0.25}$ on $\lambda$ for various $L$ and $\beta =2.40$.