Phase Diagram, Symmetry Breaking, and Critical Behavior of Three-Dimensional Lattice Multiflavor Scalar Chromodynamics

We study the nature of the phase diagram of three-dimensional lattice models in the presence of non-Abelian gauge symmetries. In particular, we consider a paradigmatic model for the Higgs mechanism, lattice scalar chromodynamics with $N_f$ flavors, characterized by a non-Abelian $\mathrm{SU}(N_c)$ gauge symmetry. For $N_f\ge 2$ (multiflavor case), it presents two phases separated by a transition line where a gauge-invariant order parameter condenses, being associated with the breaking of the residual global symmetry after gauging. The nature of the phase transition line is discussed within two field-theoretical approaches, the continuum scalar chromodynamics, and the Landau-Ginzburg-Wilson (LGW) ${\mathrm{\Phi }}^4$ approach based on a gauge-invariant order parameter. Their predictions are compared with simulation results for $N_f=2$, 3 and $N_c=2–4$. The LGW approach turns out to provide the correct picture of the critical behavior at the transitions between the two phases.

Figure 1

Sketch of the phase diagram of 3D lattice scalar chromodynamics with $N_f\ge 2$. The transition line is continuous for $N_f=2$ and of first order for $N_f\ge 3$. We conjecture that its nature is the same for any finite ${\beta }_g$. The endpoint for ${\beta }_g\to \infty$ is the $\mathrm{O}(N)$ critical point ($N=2N_c{N}_f$).

Figure 2

$R_{\xi }$ vs $(\beta -{\beta }_c)L^{1/\nu }$ for $N_f=2$, $N_c=3$, and ${\beta }_g=0$, up to $L=64$. We use the O(3) value [46, 47] $\nu =0.7117(5)$ and ${\beta }_c=3.7518(2)$. The data collapse on a unique curve confirms the O(3) critical behavior. (Inset) $R_{\xi }$ vs $\beta$. The vertical dashed line corresponds to ${\beta }_c$, while the horizontal one corresponds to the critical value $R_{\xi }^{*}$, which is consistent with the O(3) value $R_{\xi }^{*}=0.5639(2)$.

Figure 3

Binder parameter $U$ vs $R_{\xi }$ for $N_f=2$ models: results for $N_c=3$, ${\beta }_g=0$ (top), for $N_c=3$, ${\beta }_g=3$ (bottom left), and for $N_c=4$, ${\beta }_g=0$ (bottom right). In all cases, the data appear to converge to the O(3) scaling curve, obtained by Monte Carlo simulations of the O(3) vector model [44]. The dotted horizontal and vertical lines correspond to the O(3) critical values [46] $U^{*}=1.1394(3)$ and $R_{\xi }^{*}=0.5639(2)$.

Figure 4

$R_{\xi }$ vs $(\beta -{\beta }_c)L^{1/\nu }$ for $N_f=2$, $N_c=2$, and ${\beta }_g=0$, up to $L=96$. We use the O(5) value [51] $\nu =0.779(3)$ and ${\beta }_c=2.68885(5)$. The data collapse supports the O(5) critical behavior. (Inset) Data vs $\beta$: at ${\beta }_c$ (vertical dashed line), $R_{\xi }$ is consistent with the O(5) value $R_{\xi }^{*}=0.538(1)$ (horizontal dashed line). Plots of $U$ vs $R_{\xi }$ as in Fig. 3 also support the O(5) critical behavior.