Renormalization group analysis of the competition between distinct order parameters

We perform a detailed renormalization group analysis to study a ($2+1$)-dimensional quantum field theory that is composed of two interacting scalar bosons, which represent the order parameters for two continuous phase transitions. This sort of field theory can describe the competition and coexistence between distinct long-range orders, and therefore plays a vital role in statistical physics and condensed matter physics. We first derive and solve the renormalization group equations of all the relevant physical parameters, and then show that the system does not have any stable fixed point in the lowest energy limit. Interestingly, this conclusion holds in both the ordered and disordered phases, and also at the quantum critical point. Therefore, the originally continuous transitions are unavoidably turned to first order due to ordering competition. Moreover, we examine the impacts of massless Goldstone boson generated by continuous symmetry breaking on ordering competition, and briefly discuss the physical implications of our results.

Figure 1

Schematic phase diagram on $(x,T)$ plane of the order parameters described by complex ($\psi$) and real ($\varphi$) fields. $x$ represents a certain adjustable parameter whose variation tunes phase transitions. $x_1$ and $x_2$ are quantum critical points of the corresponding phase transitions, respectively.

Figure 2

Free propagators for (a) $h$, (b) $\eta$, and (c) $\varphi$.

Figure 3

Vertices: (a) $h^3$; (b) ${\eta }^{2}h$; (c) ${\varphi }^{2}h$; (d) $h^4$; (e) $h^2{\eta }^2$; (f) $h^2{\varphi }^2$; (g) ${\eta }^4$; (h) ${\varphi }^4$; (i) ${\eta }^2{\varphi }^2$.

Figure 4

One-loop corrections to ${\alpha }_h$.

Figure 5

One-loop corrections to ${\alpha }_{\varphi }$.

Figure 6

One-loop corrections to ${\gamma }_h$.

Figure 7

One-loop corrections to ${\gamma }_{\eta h}$.

Figure 8

One-loop corrections to ${\gamma }_{\varphi h}$.

Figure 9

One-loop corrections to ${\beta }_h$.

Figure 10

One-loop corrections to ${\beta }_{\eta }$.

Figure 11

One-loop corrections to ${\beta }_{\varphi }$.

Figure 12

One-loop corrections to ${\lambda }_{\eta h}$.

Figure 13

One-loop corrections to ${\lambda }_{h\varphi }$.

Figure 14

One-loop corrections to ${\lambda }_{\eta \varphi }$.

Figure 15

Initial values of dimensionless parameters: (a) ${\alpha }_0=10^{-7}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-7}$; (b) ${\alpha }_0=10^{-7}$, ${\beta }_0=u_0=10^{-7}$, ${\lambda }_0=10^{-1}$; (c) ${\alpha }_0=10^{-7}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-1}$; (d) ${\alpha }_0=10^{-1}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-1}$.

Figure 16

Initial values of dimensionless parameters: (a) ${\alpha }_0=r_0=10^{-7}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-7}$; (b) ${\alpha }_0=r_0=10^{-7}$, ${\beta }_0=u_0=10^{-7}$, ${\lambda }_0=10^{-7}$; (c) ${\alpha }_0=r_0=10^{-7}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-1}$; (d) ${\alpha }_0=r_0=10^{-1}$, ${\beta }_0=u_0=10^{-1}$, ${\lambda }_0=10^{-1}$.