From explosive to infinite-order transitions on a hyperbolic network

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous (“explosive”) transition in ordinary percolation. To this end, we solve the $q$-state Potts model in the analytic continuation for noninteger $q$ with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in $q$ shows that the discontinuous transition is generic in the regime $q<2$ that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, $q=2$. For $q>2$ the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless type.

Figure 1

Depiction of hyperbolic network MK1 of generation $k=5$. The recursive, hierarchical pattern is evident. The network features a regular geometric structure, in the form of a one-dimensional backbone, and a distinct set of small-world links (shaded green arched lines). Only the backbone bonds renormalize.

Figure 2

Plot of the fixed-point couplings ${\kappa }_{\infty }^{\pm }$ in Eq. (5) as a function of temperature $\mu$ for values of $q=\frac{1}{2},1,2,7$ (black lines, from left to right on the bottom). The blue line locates the branch-point singularity in ${\kappa }_{\infty }^{\pm }$, evaluated at ${\mathcal{D}}_q\left(\mu \right)=0$ in Eq. (6), which starts from $\mu =0$ and $\kappa =-\infty$ at $q=1$ and rises to cross $\kappa =0$ exactly at $\mu =\frac{1}{2}$ for $q=2$, above which it recedes back to $\mu =0$ for $q\to \infty$. The green line marks the raw coupling ${\kappa }_0=\mu$ from which the RG flow in Eq. (4) initiates and flows vertically towards the nearest stable fixed point, ${\kappa }_{\infty }^{+}>0$ or ${\kappa }_{\infty }^0=0$. For all $q>2$, the location of the branch point provides the critical point ${\mu }_c(<\frac{1}{2})$ with an infinite-order BKT transition [27]. For all $q<2$, ${\mu }_c=\frac{1}{2}$ and the transition becomes discontinuous.

Figure 3

Plot of the Jacobian eigenvalues $\lambda$ in Eq. (8) as a function of temperature $\mu$ for the $q$-state Potts model on MK1 for $q=1,1.2,2,7$. For all $q$ at the strong-coupling fixed point ${\kappa }_{\infty }^0=0$ we have ${\lambda }^0=2\mu$, while the two branches ${\lambda }^{\pm }$ belonging to nonzero ${\kappa }_{\infty }^{\pm }$ change dramatically with $q$ and vary nontrivially with $\mu$. At $q=1$, ${\lambda }^{-}=0$ and ${\lambda }^{+}=2-2\mu$ remain disconnected but coalesce for $1<q<2$ (here, $q=1.2$) with a branch point at unity (dashed line). At $q=2$, both branches degenerate into ${\lambda }^{\pm }=(1-\mu )/\mu$. For $q>2$ (here, $q=7$), the branches reopen and resemble the case $1<q<2$. Stable fixed points correspond to $\lambda <1$ (black), unstable ones to $\lambda >1$ (red).

Figure 4

Plot of the exponent $\Psi$ as a function of temperature $\mu$ for $q=1,\frac{3}{2},2,7$. The physical branch is given by the thickened, dark lines. The red or dashed lines mark the unphysical branches of $\Psi$. Note that $\Psi \left(\mu \right)$ is continuous at ${\mu }_c=\frac{1}{2}$ for $q<2$ with a parabolic continuation but degenerates at $q=2$ (i.e., for the Ising model) to a linear continuation. For $q>2$, $\Psi \left(\mu \right)$ is discontinuous at ${\mu }_c={\mu }_B<\frac{1}{2}$ and continues after the drop-off on the lower branch of the root singularity.

Figure 5

Plot of the susceptibility $\chi$ as a function of temperature $\mu$, rescaled with the appropriate power of system size $N$ according to Eq. (16) using $\Psi \left(\mu \right)$ (and $\Psi \equiv 1$ for $\mu <{\mu }_c$). From left to right, panels refer to $q\searrow 1$, $q=1.2$, $q=2$, and $q=7$. For the larger values of $q$, the rescaled $\chi$ are shown for a sequence of increasing system sizes to demonstrate convergence in the thermodynamic limit, while for the case $q\to 1$ we show that $\chi$ requires a further rescaling by $1/(q-1)$ to converge to a finite value throughout (at some fixed, already large system size, here $N=2^{512}$). Note that in this scaling, the susceptibility develops a minimum at the corresponding value of ${\mu }_c$; however, it is sharply peaked at the respective value at which $\Psi$ reaches (and then remains at) zero (see Fig. 4).