Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

Abstract

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds), such that each model can be obtained from the dual of the other after freezing $k$ spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with $k$ being the rank of the first homology group. Our focus is on the case where $k$ is extensive, that is, scales linearly with the number of bonds $n$. Flipping any of these additional spins introduces a homologically non-trivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects have no effect on the free energy density $f(T)$ in the thermodynamical limit, and of a high-temperature region where in the ferromagnetic case an extensive homological defect does not affect $f(T)$. We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting $f(T)$ at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all $\{f,d\}$ tilings of the hyperbolic plane, where $df/(d+f)>2$. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, $T_c^{(\mathrm{f})}<T_c^{(\mathrm{w})}$.

arXiv comment18 pages, 6 figures

References in corpus (36)

• Statistical mechanics of complex networks
• Fault-tolerant quantum computation by anyons
• Good Quantum Error-Correcting Codes Exist
• Resilience of the Internet to random breakdowns
• Topological quantum memory
• Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications
• Stabilizer Codes and Quantum Error Correction
• Simple Quantum Error Correcting Codes
• The frustration-based approach of supercooled liquids and the glass transition: a review and critical assessment
• Networks and the Epidemiology of Infectious Disease
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• Tensor network states and geometry
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Tradeoffs for reliable quantum information storage in 2D systems
• Vortices in Superfluid Films on Curved Surfaces
• Crystallography on Curved Surfaces
• Constructions and Noise Threshold of Hyperbolic Surface Codes
• Quantum "hyperbicycle" low-density parity check codes with finite rate
• Percolation in the Hyperbolic Plane
• Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
• Phase Transitions on Nonamenable Graphs
• Uniqueness and non-uniqueness in percolation theory
• Critical behavior in spherical and hyperbolic spaces
• Periodic boundary conditions on the pseudosphere
• Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
• Crystalline Order On Riemannian Manifolds With Variable Gaussian Curvature And Boundary
• Tensor Network and Black Hole
• Tradeoffs for reliable quantum information storage in surface codes and color codes
• Higher-dimensional quantum hypergraph-product codes
• Two-point correlation functions of scaling fields in the Dirac theory on the Poincare disk
• Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes
• One-loop effective potential on hyperbolic manifolds
• Numerical and analytical bounds on threshold error rates for hypergraph-product codes
• Three Dimensional Gross-Neveu Model on Curved Spaces
• A homological upper bound on critical probabilities for hyperbolic percolation
• Mod-phi convergence I: Normality zones and precise deviations

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