Multicritical behavior in models with two competing order parameters

We employ the nonperturbative functional renormalization group to study models with an $\mathrm{O}\left(N_1\right)\oplus \mathrm{O}\left(N_2\right)$ symmetry. Here different fixed points exist in three dimensions, corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters. We discuss the critical behavior of the symmetry-enhanced isotropic, the decoupled and the biconical fixed point, and analyze their stability in the $N_1,N_2$ plane. We study the fate of nontrivial fixed points during the transition from three to four dimensions, finding evidence for a triviality problem for coupled two-scalar models in high-energy physics. We also point out the possibility of noncanonical critical exponents at semi-Gaussian fixed points and show the emergence of Goldstone modes from discrete symmetries.

Figure 1

Here we plot schematic phase diagrams containing a bicritical point (left panel) and a tetracritical one (right panel), at which a mixed phase exists which is dominated by two condensing order parameters. Solid lines denote second-order and dashed lines first-order phase transitions.

Figure 2

Here we plot the potential $u(\varphi ,\chi )$ in eighth-order LPA at a point with ${\Delta }^{\prime }>0$ (upper two panels), ${\Delta }^{\prime }=0$ (middle two panels), and ${\Delta }^{\prime }<0$ (lower two panels). The second case exhibits an O(2) rotation symmetry. For ${\Delta }^{\prime }>0$ the minima lie at $\varphi \ne 0$ and $\chi \ne 0$, whereas they lie along one of the axes in field space in the case ${\Delta }^{\prime }<0$.

Figure 3

For LPA 12 including $\eta \ne 0$, the IFP is stable for $N_1+N_2<2.32$ (thick blue dashed line), and the DFP is stable for values to the upper right of the red full line. The thin lines denote the stability boundaries in LPA 8 with $\eta =0$.

Figure 4

Here we plot the fixed-point values (upper panel) of ${\lambda }_{2,0}$ (green solid line), ${\lambda }_{0,2}$ (red dashed line), and ${\lambda }_{1,1}$ (blue dotted line) as a function of $N_2$ for $N_1=1$ at the BFP. The third largest critical exponent, deciding about the stability of the fixed point, is negative for the BFP between $1.17<N_2<1.5$ (purple disks in the lower panel). For $N_2\lesssim 1.17$, the IFP is stable (blue squares), and for $N_2>1.5$, the IFP becomes stable (orange diamonds). Note that ${\lambda }_{1,1}<0$ does not automatically yield an unstable potential, as long as this coupling does not exceed a critical value.

Figure 5

Here we show several selected trajectories in the symmetry-reduced setting with $\varphi \leftrightarrow \chi$ in LPA 4, which connect the fixed points. Due to the symmetry requirement, the AFP does not appear. The trajectories start in the UV (purple color in online version) and flow toward the IR (red color in online version). None of the shown trajectories cross the ${\Delta }^{\prime }=0$ plane.

Figure 6

Here we plot the two mass eigenvalues of the $u^{\left(2\right)}$ matrix, corresponding to the dimensionless masses $m_i\left(k\right)=\frac{{\overline{m}}_i}{k}$ of physical excitations as a function of the RG scale $t=\ln (k/\Lambda )$ on a trajectory connecting the DFP in the UV (high $t$) with the IFP in the IR (low $t$).

Figure 7

Here we show the fixed point values (upper panel) for ${\lambda }_{2,0}$ (magenta disks), ${\lambda }_{1,1}$ (blue squares), and ${\lambda }_{0,2}$ (green diamonds) at the AFP as a function of $d$ in LPA fourth order. Additionally, we show the real part of the five largest critical exponents (lower panel) that approach the asymptotic values 2 and 0, as expected from canonical scaling arguments.