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arXiv:2210.07227 · Physical review. E

Random-bond Ising model and its dual in hyperbolic spaces

Benedikt Placke, Nikolas P. Breuckmann

Published Feb 16, 2023• 10 citations
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Abstract

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.

DOI
10.1103/physreve.107.024125

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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.
We develop a recently proposed mapping of the two-dimensional Ising model with random exchange (RBIM) via the transfer matrix, to a network model for a disordered system of noninteracting fermions. The RBIM transforms in this way to a localization problem belonging to one of a set of nonstandard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalization transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to M3,{M}^{3},M3, and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM the free energy, typical correlation lengths in quasi-one dimension for both the spin and the disorder operators, and the even powers of spin-spin correlation functions and their disorder averages. We examine in detail the square-lattice, nearest-neighbor $\ifmmode\pm\else\textpm\fi{}J$ RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability $1\ensuremath{-}p.$ Studying temperatures T> 0.4J,T>~0.4J,T> 0.4J, we obtain precise coordinates in the $p\ensuremath{-}T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point.
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We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration p∊[0,1]p∊[0,1]p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension LLL up to 700700700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
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We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model. Phase transition temperatures, thermal and magnetic critical exponents are determined for lattice sizes in the range L=20-100 via a sophisticated two-stage numerical strategy of entropic sampling in dominant energy subspaces, using mainly the Wang-Landau algorithm. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the two-dimensional Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections, our findings thus supporting the marginal irrelevance of quenched bond randomness. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with nu=1.30(6) and beta/nu = 0.128(5) . These results amount to a strong violation of universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two sets of results supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of two-dimensional systems with and without quenched disorder. In the conversion by bond randomness of the first-order transition of the pure system to second order, we detect, by introducing and evaluating connectivity spin densities, a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness.