Infinitely robust order and local order-parameter tulips in Apollonian networks with quenched disorder

For a variety of quenched random spin systems on an Apollonian network, including ferromagnetic and antiferromagnetic bond percolation and the Ising spin glass, we find the persistence of ordered phases up to infinite temperature over the entire range of disorder. We develop a renormalization-group technique that yields highly detailed information, including the exact distributions of local magnetizations and local spin-glass order parameters, which turn out to exhibit, as function of temperature, complex and distinctive tulip patterns.

Figure 1

(Color) Local-order-parameter tulips: probability distributions of local magnetization (left panels) and spin-glass (right panels) order parameters of the interior sites on an Apollonian network with Ising spin-glass interactions, as a function of temperature, for three different antiferromagnetic bond concentrations $p$.

Figure 2

(Color online) (a) Construction of an Apollonian network. (b) Star-triangle transformation. (c) Two successive RG transformations of an Apollonian network.

Figure 3

(Color online) Phase diagram of the Ising spin glass on an Apollonian network, in temperature $1/J$ versus antiferromagnetic bond concentration $p$. The boundary between the ferromagnetic and spin-glass phases is first order. The paramagnetic phase appears with a first-order phase transition at $p=1$.

Figure 4

(Color online) Magnetization $m$ and spin-glass order parameter $q$ for the Ising spin glass on an Apollonian network. In the left panels, the curves, consecutively leftmost from the top, are for constant temperature $1/J=0.2$, 1.0, 2.0, and 10.0. In the right panels, the curves, consecutively from the top, are for antiferromagnetic bond concentration $p=0.10$, 0.38, 0.50, and 0.90.