Critical exponents of the $O(N)$ model in the infrared limit from functional renormalization

We determined the critical exponent $\nu$ of the scalar $O(N)$ model with a strategy based on the definition of the correlation length in the infrared limit. The functional renormalization group treatment of the model shows that there is an infrared fixed point in the broken phase. The appearing degeneracy induces a dynamical length scale there, which can be considered as the correlation length. It is shown that the IR scaling behavior can account either for the Ising type phase transition in the three-dimensional $O(N)$ model, or for the Kosterlitz-Thouless type scaling of the two-dimensional $O(2)$ model.

Figure 1

The phase space of the 3D $O(N)$ model. The flows belonging to the symmetric (broken) phase tend to right (left), respectively.

Figure 2

The evolution of the anomalous dimension $\eta$ is presented as the function of the scale $k$ (top) and the shifted scale $k-k_c$ (bottom). The flow of $\eta$ has a strong singularity as the function of the scale $k$ at $k=k_c$, but it shows a power-law-like behavior as the function of the shifted scale $k-k_c$.

Figure 3

The scaling of the degeneracy condition $1+2\kappa \lambda$ as the function of the scale $k$. The diverging curves correspond to the symmetric phase, while the curves tending sharply to zero correspond to the broken phase.

Figure 4

The scaling of the correlation length $\xi$ as the function of the reduced temperature $t$, for various values of $N$.

Figure 5

The value of the critical exponent $\nu$ for various dimension $d$ (filled squares). The continuous curve obtained from $ϵ$ expansion [24]. The empty squares denote the exact values at integer dimensions.

Figure 6

The KT-type scaling of the correlation length in the $O(2)$ model in $d=2$. The lines correspond to flows belonging to different UV values of $\lambda$.