Criticality of $\mathrm{O}\left(N\right)$ symmetric models in the presence of discrete gauge symmetries

We investigate the critical properties of the three-dimensional antiferromagnetic ${\mathrm{RP}}^{N-1}$ model, which is characterized by a global $\mathrm{O}(N$) symmetry and a discrete ${\mathbb{Z}}_2$ gauge symmetry. We perform a field-theoretical analysis using the Landau-Ginzburg-Wilson (LGW) approach and a numerical Monte Carlo study. The LGW field-theoretical results are obtained by high-order perturbative analyses of the renormalization-group flow of the most general ${\mathrm{\Phi }}^4$ theory with the same global symmetry as the model, assuming a gauge-invariant order-parameter field. For $N=4$ no stable fixed point is found, implying that any transition must necessarily be of first order. This is contradicted by the numerical results that provide strong evidence for a continuous transition. This suggests that gauge modes are not always irrelevant, as assumed by the LGW approach, but they may play an important role to determine the actual critical dynamics at the phase transition of $\mathrm{O}(N$) symmetric models with a discrete ${\mathbb{Z}}_2$ gauge symmetry.

Figure 1

RG flow of the LGW theory (7) for $N=4$, in the $\overline{\mathrm{MS}}$ scheme without $\epsilon$ expansion, for several values of the ratio $s\equiv u_0/\left|v_0\right|$ of the bare quartic parameters. The curves are obtained by solving Eqs. (27) with the initial conditions (28): in the legend we report the value of $s$ and the sign of $v_0$ (“$+$” and “$-$” correspond to $v_0>0$ and $v_0<0$, respectively). The two solid lines represent the boundary of the Borel-summability region, defined by $u+v/4>0$ and $u+b_{4}v=u+7v/12>0$ (see Appendix pp2).

Figure 2

RG flow of the LGW theory (7) for $N=4$, in the MZM scheme, for several values of the ratio $s\equiv u_0/\left|v_0\right|$ of the bare quartic parameters. The curves are obtained by solving Eqs. (27) with the initial conditions (28): in the legend we report the value of $s$ and the sign of $v_0$ (“$+$” and “$-$” correspond to $v_0>0$ and $v_0<0$, respectively). The two solid lines represent the boundary of the Borel-summability region, defined by $u+v/4>0$ and $u+b_{4}v=u+7v/12>0$ (see Appendix pp2).

Figure 3

MC estimates of $R_{\xi }$ (a) and $U$ (b) for the ${\mathrm{ARP}}^3$ lattice model and several lattice sizes $L$ up to $L=100$. For both $R_{\xi }$ and $U$, the data sets for different $L$ show a crossing point for $\beta \approx 6.8$. The dotted lines are drawn to guide the eye.

Figure 4

$R_{\xi }$ versus $X\equiv L^{1/\nu }(\beta -{\beta }_c)$ with ${\beta }_c=6.779$ and $\nu =0.59$. The data approach a scaling curve with increasing $L$, supporting the scaling behavior (38).

Figure 5

Plot of $\chi /L^{2-\eta }$ versus $X\equiv L^{1/\nu }(\beta -{\beta }_c)$, using ${\beta }_c=6.779$, $\nu =0.59$, and $\eta =0.08$.

Figure 6

Histograms of the energy density, defined as $E=\langle H_{\mathrm{RP}}\rangle /L^3$, for three values of $\beta$ close to the critical point, for $L=60$ (a) and $L=100$ (b). To favor comparison, in both figures we use the same range of values of $E$, the same energy step, $\mathrm{\Delta }E={10}^{-5}$, and a similar number of energy measurements, of the order of ${10}^5$. There is no evidence of double peaks. Moreover, the width of the distributions decreases as $L$ increases, as expected at a continuous transition.