Surprises in $O(N)$ Models: Nonperturbative Fixed Points, Large $N$ Limits, and Multicriticality

We find that the multicritical fixed point structure of the $O(N)$ models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions ($d=3$) as well as at $N=\infty$. These fixed points come together with an intricate double-valued structure when they are considered as functions of $d$ and $N$. Many features found for the $O(N)$ models are shared by the $O(N)\otimes O(2)$ models relevant to frustrated magnetic systems.

Figure 1

Summary of the common wisdom: Below each critical dimension $d_n=2+2/n$ a new FP emerges from the Gaussian FP $G$. $T_2$ and $T_3$ stand or the tricritical and tetracritical FPs. The BMB FPs exist only at $N=\infty$ and in the critical dimensions $d_{n>1}$.

Figure 2

The two curves $N_c(d)$ and $N_c^{\prime }(d)$, respectively, defined by $T_2=C_3$ and $C_2=C_3$ and the curve $3.6/(3-d)$. $N_c(d)$ is calculated with the LPA (red circles) and at order 2 of the derivative expansion (blue squares). We show a path joining the point $Q$ located at ($d=3^{-}$, $N=40$) to the point at $N=\infty$ and $d=2.8$.

Figure 3

Singular point $S$ and the two lines $N_c(d)$ (red squares) and $N_c^{\prime }(d)$ (blue stars). Starting from $P$, the FP $T_2$ is followed along a clockwise (left) or anticlockwise (right) closed path surrounding $S$. On the clockwise path, $T_2$ becomes $C_2$ after a full rotation. On the anticlockwise path, $T_2$ collides with $C_3$ on $N_c(d)$ and disappears. It actually becomes complex-valued and remains so all along the dashed path. On $N_c^{\prime }(d)$ it becomes real again but is now $C_2$. The path joining $N_c(d)$ and $N_c^{\prime }(d)$ at fixed $N=33$ is also shown in panel (a). In panel (b), we indicate which (real) multicritical FPs exist in each region. In the white region, there is only one multicritical FP with two directions of instability that can be continuously followed from both $T_2$ and $C_2$ depending on the path followed.

Figure 4

$O(N)\otimes O(2)$ model. In the gray region, starting in $d=4$ at $N=21.8$, no FP at all is found. Above this region and for $d$ close to 4, both the critical $C_{+}$ and the tricritical $C_{-}$ FPs are found. The line on the right joining the squares indicates the region where two nonperturbative FPs, $M_2$ and $M_3$, appear. On the line joining the crosses, $C_{-}$ and $M_3$ collapse. In each region, we indicate the FPs that are present.