Percolation on hyperbolic lattices

What are the limits of universality?

It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents on Z 4 and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent τ . This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A 382 , 12–19 ( doi:10.1016/j.physleta.2017.10.035 )).

Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models

Circuit quantum electrodynamics is one of the most promising platforms for efficient quantum simulation and computation. In recent groundbreaking experiments, the immense flexibility of superconducting microwave resonators was utilized to realize hyperbolic lattices that emulate quantum physics in negatively curved space. Here we investigate experimentally feasible settings in which a few superconducting qubits are coupled to a bath of photons evolving on the hyperbolic lattice. We compare our numerical results for finite lattices with analytical results for continuous hyperbolic space on the Poincaré disk. We find good agreement between the two descriptions in the long-wavelength regime. We show that photon-qubit bound states have a curvature-limited size. We propose to use a qubit as a local probe of the hyperbolic bath, for example, by measuring the relaxation dynamics of the qubit. We find that, although the boundary effects strongly impact the photonic density of states, the spectral density is well described by the continuum theory. We show that interactions between qubits are mediated by photons propagating along geodesics. We demonstrate that the photonic bath can give rise to geometrically frustrated hyperbolic quantum spin models with finite-range or exponentially decaying interaction.

Quantum phase transitions of interacting bosons on hyperbolic lattices

The effect of many-body interaction in curved space is studied based on the extended Bose-Hubbard model on hyperbolic lattices. Using the mean-field approximation and quantum Monte Carlo simulation, the phase diagram is explicitly mapped out, which contains the superfluid, supersolid and insulator phases at various fillings. Particularly, it is revealed that the sizes of the Mott lobes shrink and the supersolid is stabilized at smaller nearest-neighbor interaction asqin the Schläfli symbol increases. The underlying physical mechanism is attributed to the increase of the coordination number, and hence the kinetic energy and the nearest-neighbor interaction. The results suggest that the hyperbolic lattices may be a unique platform to study the effect of the coordination number on quantum phase transitions, which may be relevant to the experiments of ultracold atoms in optical lattices.

Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry

We show how quantum many-body systems on hyperbolic lattices with nearest-neighbor hopping and local interactions can be mapped onto quantum field theories in continuous negatively curved space. The underlying lattices have recently been realized experimentally with superconducting resonators and therefore allow for a table-top quantum simulation of quantum physics in curved background. Our mapping provides a computational tool to determine observables of the discrete system even for large lattices, where exact diagonalization fails. As an application and proof of principle we quantitatively reproduce the ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincaré disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying interactions and disorder on hyperbolic graphs in the future. Importantly, our analysis reveals that even relatively small discrete hyperbolic lattices emulate the continuous geometry of negatively curved space, and thus can be used to experimentally resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity.

Critical properties of the Ising model in hyperbolic space

The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two- and three-dimensional hyperbolic spaces using Monte Carlo and high- and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the "asymptotic" limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three-dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.

Constraint percolation on hyperbolic lattices

Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices, and they are interesting in their own right, with ordinary percolation exhibiting not one but two phase transitions. We study four constraint percolation models-k-core percolation (for k=1,2,3) and force-balance percolation-on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggest that all of the k-core models, even for k=3, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the k-core percolation models.

Percolation thresholds in hyperbolic lattices

We use invasion percolation to compute numerical values for bond and site percolation thresholds p_{c} (existence of an infinite cluster) and p_{u} (uniqueness of the infinite cluster) of tesselations {P,Q} of the hyperbolic plane, where Q faces meet at each vertex and each face is a P-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on P and Q and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for p_{c} and p_{u} that can be used to find the scaling of both thresholds as a function of P and Q.

Viscous fluid fingering on a negatively curved surface

Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. We examine the development of viscous fluid fingering on a two-dimensional surface of constant negative Gaussian curvature, the hyperbolic plane H(2). A perturbative mode-coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system: fingertip splitting and finger competition. We also report on a time-dependent control strategy placed on the injection rate, which is able to minimize viscous fingering growth on H(2).

Characterizing Social Interaction in Tobacco-Oriented Social Networks: An Empirical Analysis

Social media is becoming a new battlefield for tobacco "wars". Evaluating the current situation is very crucial for the advocacy of tobacco control in the age of social media. To reveal the impact of tobacco-related user-generated content, this paper characterizes user interaction and social influence utilizing social network analysis and information theoretic approaches. Our empirical studies demonstrate that the exploding pro-tobacco content has long-lasting effects with more active users and broader influence, and reveal the shortage of social media resources in global tobacco control. It is found that the user interaction in the pro-tobacco group is more active, and user-generated content for tobacco promotion is more successful in obtaining user attention. Furthermore, we construct three tobacco-related social networks and investigate the topological patterns of these tobacco-related social networks. We find that the size of the pro-tobacco network overwhelms the others, which suggests a huge number of users are exposed to the pro-tobacco content. These results indicate that the gap between tobacco promotion and tobacco control is widening and tobacco control may be losing ground to tobacco promotion in social media.

Universal statistics of the knockout tournament

We study statistics of the knockout tournament, where only the winner of a fixture progresses to the next. We assign a real number called competitiveness to each contestant and find that the resulting distribution of prize money follows a power law with an exponent close to unity if the competitiveness is a stable quantity and a decisive factor to win a match. Otherwise, the distribution is found narrow. The existing observation of power law distributions in various kinds of real sports tournaments therefore suggests that the rules of those games are constructed in such a way that it is possible to understand the games in terms of the contestants' inherent characteristics of competitiveness.

Bounds of percolation thresholds on hyperbolic lattices

We analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative Gaussian curvature. The quantity of our main concern is p(c2), the value of occupation probability where a unique unbounded cluster begins to emerge. By applying the substitution method to known bounds of the order-5 pentagonal tiling, we show that p(c2) ≥ 0.382508 for the order-5 square tiling, p(c2) ≥ 0.472043 for its dual, and p(c2)≥ 0.275768 for the order-5-4 rhombille tiling.

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

We study the Ising model in a hierarchical small-world network by renormalization group analysis and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities, including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point.

Upper transition point for percolation on the enhanced binary tree: a sharpened lower bound

Hyperbolic structures are obtained by tiling a hyperbolic surface with negative Gaussian curvature. These structures generally exhibit two percolation transitions: a system-wide connection can be established at a certain occupation probability p = pc1, and there emerges a unique giant cluster at pc2 > pc1. There have been debates about locating the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree (EBT), which is constructed by adding loops to a binary tree. This work presents its lower bound as pc2 ≳ 0.55 by using phenomenological renormalization-group methods and discusses some solvable models related to the EBT.

Crossing on hyperbolic lattices

We divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p. We consider the {7,3} (heptagonal), enhanced or extended binary tree (EBT), the EBT-dual, and the {5,5} (pentagonal) lattices. We find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p_{l} to the upper p_{u} critical values. We find bounds and estimates for the values of p_{l} and p_{u} for these lattices and identify the self-duality point p corresponding to where the crossing probability equals 1/2. Comparison is made with recent numerical and theoretical results.

Ising model on a hyperbolic plane with a boundary

A hyperbolic plane can be modeled by a structure called the enhanced binary tree. We study the ferromagnetic Ising model on top of the enhanced binary tree using the renormalization-group analysis in combination with transfer-matrix calculations. We find a reasonable agreement with Monte Carlo calculations on the transition point, and the resulting critical exponents suggest the mean-field surface critical behavior.

Exact solution of the nonconsensus opinion model on the line

The nonconsensus opinion model (NCO) introduced recently by Shao et al. [Phys. Rev. Lett. 103, 018701 (2009)] is solved exactly on the line. Although, as expected, the model exhibits no phase transition in one dimension, its study is interesting because of the possible connection with invasion percolation with trapping. The system evolves exponentially fast to the steady state, rapidly developing long-range correlations: The average cluster size in the steady state scales as the square of the initial cluster size, of the (uncorrelated) initial state. We also discuss briefly the NCO model on Bethe lattices, arguing that its phase transition diagram is different than that of regular percolation.

Analytic results for the percolation transitions of the enhanced binary tree

Percolation for a planar lattice has a single percolation threshold, whereas percolation for a negatively curved lattice displays two separate thresholds. The enhanced binary tree (EBT) can be viewed as a prototype model displaying two separate percolation thresholds. We present an analytic result for the EBT model which gives two critical percolation threshold probabilities, p(c1) = 1/2 square root(13) - 3/2 and p(c2) = 1/2, and yields a size-scaling exponent Φ = ln[(p(1+p))/(1-p(1-p))]/ln 2. It is inferred that the two threshold values give exact upper limits and that pc1 is furthermore exact. In addition, we argue that p(c2) is also exact. The physics of the model and the results are described within the midpoint-percolation concept: Monte Carlo simulations are presented for the number of boundary points which are reached from the midpoint, and the results are compared to the number of routes from the midpoint to the boundary given by the analytic solution. These comparisons provide a more precise physical picture of what happens at the transitions. Finally, the results are compared to related works, in particular, the Cayley tree and Monte Carlo results for hyperbolic lattices as well as earlier results for the EBT model. It disproves a conjecture that the EBT has an exact relation to the thresholds of its dual lattice.

Surface and bulk criticality in midpoint percolation

The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular d-dimensional hypercubic lattices are investigated in the present work using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to d=10 by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension d=6, it is found that the percolation cluster contains only a finite number of surface points in the infinite-size limit. This is in accordance with the expectation from studies of lattices with negative curvature. It is also found that the number of surface points, reached by the percolation cluster in the infinite limit, approaches 2d for large dimensions d. We also note that the size dependence in proliferation of percolating clusters for d>or=7 can be obtained by solely counting surface points of the midpoint cluster.

Majority-vote model on hyperbolic lattices

We study the critical properties of a nonequilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents 1/nu , beta/nu , and gamma/nu are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation 2beta/nu+gamma/nu=D(eff), where D(eff) is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.

Survival of short-range order in the Ising model on negatively curved surfaces

We examine the ordering behavior of the ferromagnetic Ising lattice model defined on a surface with a constant negative curvature. Small-sized ferromagnetic domains are observed to exist at temperatures far greater than the critical temperature, at which the inner-core region of the lattice undergoes a mean-field phase transition. The survival of short-range order at such high temperatures can be attributed to strong boundary-spin contributions to the ordering mechanism as a result of which boundary effects remain active even within the thermodynamic limit. Our results are consistent with the previous finding of disorder-free Griffiths phase that is stable at temperatures lower than the mean-field critical temperature.

Phase transition of q-state clock models on heptagonal lattices

We study the q-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every q>or=2. The persistence of the third phase for all q is in contrast with the disappearance of the counterpart phase in a planar system for small q, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects break the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.