Phase structure of the $O(2)$ ghost model with higher-order gradient term

The phase structure and the infrared behavior of the Euclidean three-dimensional $O(2)$ symmetric ghost scalar field model with a higher-order derivative term has been investigated in Wegner and Houghton’s renormalization group framework. The symmetric phase in which no ghost condensation occurs and the phase with restored symmetry but with a transient presence of a ghost condensate have been identified. Finiteness of the correlation length at the phase boundary hints to a phase transition of first order. The results are compared with those for the ordinary $O(2)$ symmetric scalar field model.

Figure 1

The blocked potential at various scales $k$.

Figure 2

Our numerical result for ${\rho }_k(\mathrm{\Phi })$ in the IR limit $k\to 0$ (solid line) in comparison with the one reported in [2] (dashed line).

Figure 3

Scaling of the dimensionless couplings ${\tilde{g}}_2={\tilde{v}}_1$, ${\tilde{g}}_4=3{\tilde{v}}_2$, and ${\tilde{g}}_6=15{\tilde{v}}_3$.

Figure 4

Determining the value ${\tilde{g}}_2={\tilde{v}}_1$ in the limit $k\to 0$ at effectively zero step size from extrapolation.

Figure 5

Comparison of the function ${\rho }_k(\mathrm{\Phi })$ obtained by us numerically (solid line) to the one given in Ref. [27] (dashed line) for the molecular phase of the SG model for ${\beta }^2=4\pi$.

Figure 6

The scale-dependence of the dimensionless running couplings of the SG model in the deep IR region for ${\beta }^2=4\pi$.

Figure 7

Phase diagrams in the plane $({\tilde{v}}_1(\mathrm{\Lambda }),{\tilde{v}}_2(\mathrm{\Lambda }))$ for various given values $Y$ for the ghost (to the left) and the ordinary (to the right) $O(2)$ models. The empty regions correspond to the symmetric phase I, and the dotted and shadowed regions correspond to regions IIA and IIB of phase II, respectively.

Figure 8

The parameters $v_1(0)=at+b$ (to the left) and $v_2(0)$ (to the right) vs. the “distance” $t={\tilde{v}}_1(\mathrm{\Lambda })-{\tilde{v}}_u$ from the phase boundary for $v_2(\mathrm{\Lambda })=0.01$. The dots, boxes, and rombs correspond to $Y=0.7$, 1.0, 1.5, respectively, and the lines are for guiding the eyes. The dependence of the coefficient $b$ on the higher-derivative coupling $Y$ is shown in the inset.

Figure 9

The dependence of mean values $\overline{v_1(0)}$ on the higher-derivative coupling $Y$ in phase II. The lines are only to guide the eyes.

Figure 10

Thr TLR flow of $v_1(k)$ and the corresponding ${\rho }_k(\mathrm{\Phi }=0)$ for $t={\tilde{v}}_2(\mathrm{\Lambda })=0.1$ and $Y=0$, 0.7, 1.0, and 1.5 (from the top to the bottom) in phase II of the ghost $O(2)$ model.

Figure 11

Scaling of the correlation length $\xi \sim 1/k_c$ with the reduced temperature $t={\tilde{v}}_u-{\tilde{v}}_1(\mathrm{\Lambda })$ (on a lin-lin plot) at the boundary of phases I and II of the ghost $O(2)$ model for $Y=1.5$ and ${\tilde{v}}_2(\mathrm{\Lambda })=0.1$.

Figure 12

The coefficient $\kappa$ in expression (29) of the correlation length vs. the higher-derivative coupling $Y$. The points correspond to various RG trajectories, and the line is for guiding the eyes.

Figure 13

Typical scaling of the correlation length $\xi \sim 1/k_c$ with the reduced temperature $t={\tilde{v}}_u-{\tilde{v}}_1(\mathrm{\Lambda })$ (on a log-log plot) at the boundary of phases I and II of the ordinary $O(2)$ model for $Y=1.0$ and ${\tilde{v}}_2(\mathrm{\Lambda })=0.1$. The critical exponent is $\nu \approx 0.45\pm 0.02$ in the range $Y\in [0,1.5]$.