Stability of fixed points and generalized critical behavior in multifield models

We study models with three coupled vector fields characterized by $O\left(N_1\right)\oplus O\left(N_2\right)\oplus O\left(N_3\right)$ symmetry. Using the nonperturbative functional renormalization group, we derive $\beta$ functions for the couplings and anomalous dimensions in $d$ dimensions. Specializing to the case of three dimensions, we explore interacting fixed points that generalize the $O\left(N\right)$ Wilson-Fisher fixed point. We find a symmetry-enhanced isotropic fixed point, a large class of fixed points with partial symmetry enhancement, as well as partially and fully decoupled fixed-point solutions. We discuss their stability properties for all values of $N_1,N_2$, and $N_3$, emphasizing important differences to the related two-field models. For small numbers of field components, we find no stable fixed-point solutions, and we argue that this can be attributed to the presence of a large class of possible (mixed) couplings in the three-field and multifield models. Furthermore, we contrast different mechanisms for stability interchange between fixed points in the case of the two- and three-field models, which generically proceed through fixed-point collisions.

Figure 1

We show a sketch of the value of the third largest critical exponent, ${\theta }_3$, of three different FPs (solid, dotted, and dashed lines) in the $O\left(N_1\right)\oplus O\left(N_2\right)$ model at fixed $N_1$ as a function of $N_2$. The regime where a FP is stable is indicated by the labels FP1, FP2, and FP3. As the coordinates of two different FPs coincide at $N_2^{*}$ and $N_2^{**}$, these FPs exchange their stability properties, and ${\theta }_3$ changes its sign if evaluated beyond the stable regime. An explicit calculation showing this situation can be found in Ref. [25].

Figure 2

This illustration of the two-dimensional subspace of the two-field coupling space shows that the Wilson-Fisher theory space is a one-dimensional subspace, corresponding to the ${\lambda }_{2,0}={\lambda }_{0,2}$ line. The critical exponent along this direction corresponds to the largest Wilson-Fisher critical exponent. Another eigendirection of the stability matrix does not respect the enhanced symmetry of the FP.

Figure 3

We show the fourth-largest critical exponent at the IFP as a function of $N_1$, with $N_2=N_3=1$ (upper panel) and as a function of $N_1=N_2=N_3$ (lower panel), with the LPA 4 result (green circles), LPA 6 (blue squares), and LPA 8 (red diamonds). The LPA clearly converges rapidly. As this exponent is positive beyond the LPA 4 and shows a threefold degeneracy, the IFP is characterized by a total of six relevant directions.

Figure 4

We show the fourth-largest critical exponent at the DFP as a function of $N_1=N_2=N_3$, with the LPA 4 result (green circles), LPA 6 (blue squares) and LPA 8 (red diamonds). The LPA clearly converges rapidly.

Figure 5

We show the stable DIFP (large blue dots), the stable DFP (small orange dots), and points without a stable FP (small gray dots) using the LPA $12+\eta$ results. We also include points where the DBFP is conjectured to be stable (middle-sized green dots).

Figure 6

To illustrate the mechanism of how stability properties are exchanged between FPs, we examine the $\beta$ function $\beta =g(c+g)$ for varying parameter $c$ describing the position of the nontrivial FP. As the FPs pass by one another, the local derivatives $\theta =-{\beta }^{\prime }\left(g\right)$ exchange their sign indicating an interchange of their IR stability properties.

Figure 7

We plot ${\lambda }_{1,0,1},{\lambda }_{0,1,1}$, and ${\lambda }_{1,1,0}$ as a function of $N_I$ for the three DBFPs. At $N_I=1.244$, they each pass through the origin of that coordinate system where the DFP sits. For $N_I>1.244$, the three BFPs move away from each other toward more negative values of the mixed couplings.

Figure 8

We plot the fifth-largest critical exponent at the DBFPs (blue points of increasing value) and the corresponding critical exponent at the DFP (red points of decreasing value) as a function of $N_I$ (upper panel). Below, we show the coordinates of the couplings ${\lambda }_{2,0,0}$ and ${\lambda }_{0,2,0}$ at the DBFP (blue diamonds and black squares) and the couplings ${\lambda }_{2,0,0}={\lambda }_{0,2,0}={\lambda }_{0,0,2}$ at the DFP (red dots) in the vicinity of the collision point.

Figure 9

Here, we show the couplings ${\lambda }_{2,0,0},{\lambda }_{0,2,0}$, and ${\lambda }_{1,1,0}$ at the DBFP (blue circles, purple squares, cyan diamonds) and ${\lambda }_{2,0,0}={\lambda }_{0,2,0}={\lambda }_{1,1,0}$ (red triangles) at the DIFP (uppermost panel). We show the fourth- and fifth-largest critical exponent (middle panel) at the DBFP as a function of $N_1$ for $N_2=1$ and $N_3=4$. Around $N_1\approx 1.16$, the three couplings are clearly degenerate, as expected for a collision with the DIFP. At the same point, the fourth critical exponent crosses zero and becomes negative. Simultaneously, the fifth critical exponent at the DIFP (lower panel) crosses zero and becomes positive.

Figure 10

Here, we show the couplings ${\lambda }_{1,0,1}$ and ${\lambda }_{0,1,1}$ at the two anisotropic FPs. As a function of $N_1$ they converge toward zero, which is where the DBFP is sitting. The collision occurs at $N_1\approx 1.2$, which is where the fifth critical exponent of these two FPs (which numerically is nearly the same for both) approaches zero.