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arXiv:1512.06087

Network complexity and topological phase transitions

Felipe Torres, Jose Antonio Rogan, Miguel Kiwi, J. A. Valdivia

Published Dec 18, 2015• 0 citations
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Abstract

A new type of collective excitations, due exclusively to the topology of a complex random network that can be characterized by a fractal dimension $D_F$, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the $D_F>1$ complex network. An Ising system, with long range interactions over such a network, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. This way we compute the magnetization distribution, magnetization loops, and the two point correlation function; and relate them to the network topology. In summary, the removal of the infrared divergences leads to an unconventional phase transition, where spin correlations are robust against thermal fluctuations.

DOI
10.48550/arxiv.1512.06087

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A new type of collective excitations, due exclusively to the topology of a complex random network that can be characterized by a fractal dimension DFD_FDF​, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the DF>1D_F>1DF​>1 complex network. An Ising system, with long range interactions over such a network, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. This way we compute the magnetization distribution, magnetization loops, and the two point correlation function; and relate them to the network topology. In summary, the removal of the infrared divergences leads to an unconventional phase transition, where spin correlations are robust against thermal fluctuations.
A new type of collective excitations, due to the topology of a complex random network that can be characterized by a fractal dimension DF, is investigated. We show analytically that these excitations generate phase transitions due to the non-periodic topology of the DF > 1 complex network. An Ising system, with long range interactions, is studied in detail to support the claim. The analytic treatment is possible because the evaluation of the partition function can be decomposed into closed factor loops, in spite of the architectural complexity. The removal of the infrared divergences leads to an unconventional phase transition, with spin correlations that are robust against thermal fluctuations.
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Can the topology of a network that consists of many particles interacting with each other change in complexity when a phase transition occurs? The answer to this question is particularly interesting to understand the nature of the phase transitions if the distinct phases do not break any symmetry, such as topological phase transitions. Here we present a novel theoretical framework established by complex network analysis for demonstrating that across a transition point of the topological superconductors, the network space experiences a homogeneous-heterogeneous transition invisible in real space. This transition is nothing but related to the robustness of a network to random failures. We suggest that the idea of the network robustness can be applied to characterizing various phase transitions whether or not the symmetry is broken.
Abstract The critical behavior of spin systems is fundamentally governed by dimensionality and connectivity. Moving beyond translationally invariant lattices, we explore a new paradigm where fractality itself becomes a tunable parameter, engineering magnetic order and critical behavior. By implementing the classical Ising model on a three-dimensional fractal lattice---a static realization of a spin cluster with Hausdorff dimension dH=2.5d_{\rm H}=2.5dH​=2.5 and boundary dimension d=2d=2d=2---we demonstrate how fractal geometry dictates unique critical phenomena. Using the higher-order tensor renormalization group (HOTRG) method, we identify a finite-temperature phase transition at Tc≈2.65231T_c \approx 2.65231Tc​≈2.65231 with exotic critical exponents (α≈0.09\alpha\approx 0.09α≈0.09, α′≈0.06\alpha'\approx 0.06α′≈0.06, β≈0.059\beta \approx 0.059β≈0.059, δ≈35\delta \approx 35δ≈35) and a diverging specific heat---a hallmark absent in lower-dimensional fractals. This work establishes that fractal geometry serves as a powerful and untapped degree of freedom for spintronics. It provides a blueprint for designing materials with programmable magnetic phase transitions, paving the way for next-generation, geometry-driven devices in magnonics and neuromorphic computing.
Disordered systems are very rich laboratories for exploring complex systems. In particular, disordered magnetic systems have been extremely important in the last five decades for understanding a wide range of phenomena. In this work, we use the Edwards-Anderson Hamiltonian to obtain the thermodynamic properties of disordered magnetic systems. In this way, we conduct a systematic investigation of magnetization, correlation functions, order parameter, and fractal dimensions, in function of disorder. In this context, the autocorrelation function for order--parameter fluctuations, introduced by Fisher ( Journal of Mathematical Physics 5, 944322 (1964)), provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension ddd, and an exponent ηηη is introduced to correct the spatial behavior of the correlation function at T=TcT=T_cT=Tc​. In recent work, Lima et al ( Phys. Rev. E 110, L062107 (2024)) demonstrated that at TcT_cTc​ a fractal analysis is necessary for a complete description of the correlation function. We use Monte Carlo simulations to validate analytical results and to show how disorder alters critical exponents , giving rise to different universality classes. On the other hand, there is a subgroup of critical exponents and fractal dimensions that are invariant with disorder. This subgroup heralds a strong universality class.
Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.