Ordered phase of the $\mathrm{O}\left(N\right)$ model within the nonperturbative renormalization group

We analyze nonperturbative renormalization group flow equations for the ordered phase of ${\mathbb{Z}}_2$ and $\mathrm{O}\left(N\right)$ invariant scalar models. This is done within the well-known derivative expansion scheme. For its leading order [local potential approximation (LPA)], we show that not every regulator yields a smooth flow with a convex free energy and discuss for which regulators the flow becomes singular. Then we generalize the known exact solutions of smooth flows in the “internal” region of the potential and exploit these solutions to implement an improved numerical algorithm, which is much more stable than previous ones for $N>1$. After that, we study the flow equations at second order of the derivative expansion and analyze how and when the LPA results change. We also discuss the evolution of the field renormalization factors.

Figure 1

Top: Typical evolution of the potential to the high temperature phase. Center: Typical evolution near the critical point. Bottom: Typical evolution of the potential in the low temperature phase.

Figure 2

Right hand side of Eq. (22) as function of $w_k$.

Figure 3

Solution of Eq. (22) for $w_k\left(\rho \right)$ as a function of $\rho$ for various values of $k={10}^{-2}\mathrm{\Lambda }$ (blue diamonds), $k={10}^{-3}\mathrm{\Lambda }$ (red squares), and ${10}^{-4}\mathrm{\Lambda }$ (black plain line).

Figure 4

Comparison of numerical (black plain line) and semianalytical (red dashed) solutions for $w_k+1$ at large $N$ as a function of $\rho$ for the $\theta$ regulator $\left(d=3\right)$ for $k=0.002$.

Figure 5

$y[1+r(y\left)\right]$ for different regulators. The vertical dashed lines show the minimum of $y[1+r(y\left)\right],y_0$, for each case.

Figure 6

Right hand side of Eq. (20) for the exponential regulator and $\alpha =3 \left(d=3\right)$. The dotted line points out the position of the singularity $-y_0[1+r\left(y_0\right)]$.

Figure 7

Comparison of numerical (black plain line) and semianalytical (red dashed) solutions at large $N$ for $w_k+y_0[1+r\left(y_0\right)]$ as a function of $\rho$ for the exponential regulator $\left(d=3\right)$. Curves for $k=0.08$.

Figure 8

Right hand side of Eq. (20) for the exponential regulator and $\alpha =1 \left(d=3\right)$. The dotted line point out the position of the singularity $-\tilde{R}=-{lim}_{y\to 0}y[1+r\left(y\right)]$.

Figure 9

Derivative of the potential at large $N$ and $d=3$ as a function of $\rho$ for various values of $k$ (lower curves are for larger values of $k$).

Figure 10

$f\left(u\right)$ as a function of $u$ for various values of $N$.

Figure 11

Top: $w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$. Bottom: $W_k\left({\rho }_0\right)$ where ${\rho }_0={lim}_{k\to 0}{\rho }_0\left(k\right)$ as a function of $t=ln(k/\mathrm{\Lambda })$. In both cases, the curves are for $N=4$ and $d=3$. Comparison between the direct numerical solution (red circles) and the improved algorithm (black plain line) in both cases with the $\theta$ regulator.

Figure 12

Top: $w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$. Bottom: $W_k\left({\rho }_0\right)$ where ${\rho }_0={lim}_{k\to 0}{\rho }_0\left(k\right)$ as a function of $t=ln(k/\mathrm{\Lambda })$. In both cases, the curves are for $N=4$ and $d=3$. Comparison between the direct numerical solution (red circles) and the improved algorithm (black plain line) in both cases with the exponential regulator for $\alpha =3$.

Figure 13

$w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$ for $N=1$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =2{\rho }_0$.

Figure 14

Top: $w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$. Bottom: $W_k\left(\rho \right)$ as a function of $\rho$ for various values of $t$ ranging from 0 to $-1$. In both figures, $N=1$ and $d=3$ using the exponential regulator (7) with $\alpha =3$ and normalization fixed at $\rho =0$.

Figure 15

$Z_k\left(\rho \right)$ for $N=1$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =0$. The arrow shows the last position of the minimum reached by the simulation.

Figure 16

${\eta }_k$ for $N=1$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =0$. Observe that, given that $Z_k(\rho =0)$ and $Z_k(\rho =2{\rho }_0)$ are different, the position of the singularity is different from the LPA order.

Figure 17

${\widehat{Z}}_k\left(\rho \right)$ as function of $\rho$ for various values of $t$ for $N=1$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =0$. The arrow shows the last position of the minimum reached by the simulation.

Figure 18

Top: $w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$. Bottom: $W_k\left(\rho \right)$ as a function of $\rho$ for different values of $t$. In both figures, for $N=1$ and $d=2$ using the exponential regulator (7) with $\alpha =3$ and normalization fixed at $\rho =0$.

Figure 19

${\eta }_k$ for $N=1$ and $d=2$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =0$.

Figure 20

Top: $Z_k\left(\rho \right)$ as a function of $\rho$. Bottom: ${\widehat{Z}}_k\left(\rho \right)$ as function of $\rho$. In both cases, for various values of $t$ for $N=1$ and $d=2$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =0$. The arrows show the last position of the minimum reached by the simulation.

Figure 21

Top: $w_k(\rho =0)$ as a function of $t=ln(k/\mathrm{\Lambda })$. Bottom: $W_k\left(\rho \right)$ as a function of $\rho$ for various values of $t$. In both figures, for $N=4$ and $d=3$ using the exponential regulator (7) with $\alpha =3$ and normalization fixed at $\rho =2{\rho }_0$.

Figure 22

${\eta }_k$ as a function of $t$ for $N=4$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =2{\rho }_0$.

Figure 23

Top: $Z_k\left(\rho \right)$ as a function of $\rho$. Bottom: $Z_k\left(\rho \right)+\rho Y_k\left(\rho \right)$ as function of $\rho$. In both cases, for various values of $t$ for $N=4$ and $d=3$ using the exponential regulator (7) with $\alpha =3$. Normalization fixed at $\rho =2{\rho }_0$. The arrows show the last position of the minimum reached by the simulation.

Figure 24

$ln\left[Z_k\left({\rho }_0\right)+{\rho }_0{Y}_k\left({\rho }_0\right)\right]$ as a function of $ln(k/\mathrm{\Lambda })$.