Random-bond Ising model and its dual in hyperbolic spaces

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature $T$ and the fraction of antiferromagnetic bonds $p$. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.

Figure 1

(a) Poincaré disk model of the infinite hyperbolic plane ${\mathbb{H}}^2$ with the $\{5,5\}$ lattice. All edges have the same length with respect to the hyperbolic metric, see Eq. (2).

Figure 2

A hyperbolic surface of genus 3 tessellated by the $\{7,3\}$ tessellation (left). If we cut the surface open, then we obtain a flat piece of hyperbolic space (right). The plaquettes are colored to guide they eye.

Figure 3

The left shows a cycle on the dual lattice (blue) and the associate cocycle on the primal lattice (red). The right shows a boundary on the primal lattice (blue) and the associate coboundary on the dual lattice (red).

Figure 4

Schematic phase diagram of the random bond Ising model and its dual on the hyperbolic plane as a function of temperature $T$ and the fraction of antiferromagnetic bonds $p$. The high-temperature paramagnetic (PM) phase at low temperatures gives way either to a ferromagnetic (FM) phase spin-glass (SG) phase at weak and strong disorder, respectively. For the dual model, we only indicated the schematic boundary of the FM phase. Note that although the temperatures $T_c$ and $T_c^{*}$ are related by the Kramers-Wannier relation, the dual model of the hyperbolic Ising model is different from the original model even on self-dual lattices (see main text for details). The phase boundary of the dual model corresponds to the decoding threshold of the hyperbolic surface code under phenomenological noise. The Nishimori line is indicated in dashed gray.

Figure 5

Some small biconnected subgraphs of the $\{5,5\}$ tiling. Removing a vertex and all its incident edges will leave the graphs connected. Only biconnected graphs contribute to the series expansion.

Figure 6

Phase diagram on the $\{5,5\}$ lattice as a function of temperature $T$ and disorder strength $p$. We show both the magnetization $m$ as well as the Edwards-Anderson order parameter $q_{\mathrm{EA}}$ (inset) obtained from Monte Carlo (MC) simulations of a $N=1920$ system. We superimpose this with the phase boundaries obtained from the high-temperature series expansion (HTSE) and MC (see main text for details).

Figure 7

Critical temperature $T_c$ obtained from high-temperature expansion for different tilings of the hyperbolic plane. The inset shows $v_c=tanh(J/T_c)$ as a function of curvature $\kappa$ for the pure model ($p=0$), along the Nishimori line $[(1-p)/p=e^{-2\beta J}]$ and for the spin-glass boundary ($p=1/2$).

Figure 8

Finite-size scaling collapse of the Binder cumulant $g$ [Eq. (38)], the susceptibility [Eq. (41)], and the magnetization $m$ [Eq. (36)] of the random bond Ising model on the $\{5,5\}$ lattice along the Nishimori line [Eq. (23)].

Figure 9

Evidence for a strongly first-order phase transition of the pure dual Ising model [that is, Eq. (18) with $p=0$] on the $\{5,5\}$ lattice. We show the vertex magnetization $m=\langle {\sigma }_j\rangle$, its Binder cumulant $g$, as well as the loop magnetization $m_{\eta }=\langle {\eta }_{\ell }\rangle$ and its Binder cumulant $g_{\eta }$ as a function of temperature.

Figure 10

Free-energy difference [Eq. (46)] between the Ising model and its dual on the $\{5,5\}$ lattice.

Figure 11

Free-energy difference [Eq. (46)] between the RBIM and the dual-RBIM on the $\{5,5\}$ lattice along the Nishimori line. The shaded region indicates the location of the Nishimori point $p_N=0.0228\pm 0.0010$. The inset shows the best data collapse, assuming the same correlation exponent $\mu$ as in the RBIM.