Topological Symmetry Breaking in Frustrated Spin Systems - Giulio Iannelli - CREF

Topological Symmetry Breaking in Frustrated Spin Systems: An Information-Diffusion Framework on Signed Graphs​ Giulio Iannelli, Enrico Fermi Research Center (CREF)​ Abstract:  Frustration and competing interactions underlie some of the most enduring puzzles in the physics of disordered systems, from the elusive nature of the spin-glass phase to the proliferation of non-equilibrium and metastable states and the breakdown of ergodicity. Field-theoretical and replica-based descriptions have long provided the dominant lens, yet they often obscure the role played by the discrete connectivity of the underlying topological structure. In this talk I will introduce a framework in which the structure of frustrated systems is probed through a particular kind of information-diffusion process it supports. After a general introduction to spectral methods on graphs, the diffusion kernel of a network, and its connection to critical phenomena, I will show that a surprisingly rich phenomenology of competing dynamics is encoded in the spectrum of a single linear operator. The various signatures of frustration are distributed in inverse space, and different non-ergodic regimes correspond to different ways in which spectral content reorganizes: closures of the spectral gap, slow modes acquiring a non-trivial sign structure and losing their non-degeneracy, the ultraviolet end of the spectrum losing its uniqueness, and diffusive modes localizing information à la Anderson. We interpret this concurrent reorganization as a topological symmetry breaking -- a structural transition that occurs at the level of the network operator, prior to and independently of any dynamics placed on top of it. Because this transition lives in the diffusion kernel and not in a specific Hamiltonian, the same spectral diagnosis predicts spin-glass thresholds in arbitrary signed architectures, but also non-ergodic phases that are not of spin-glass type -- including a multistable regime of a specifically disordered Ising model in which the system supports a finite set of noise-resilient, locally ordered states up to the Ising critical temperature; the proliferation of slow timescales characteristic of Griffiths phases in excitation-inhibition models; and the characterization of dynamical regimes in the fully frustrated triangular Ising lattice. Building on the same operator, I will finally show that its binarized eigenmodes -- which we call Chladni states -- form a complete topological basis for the metastable attractors of general Ising dynamics, enabling the spectral reconstruction of arbitrary frozen configurations and behaving as a training-free, geometry-driven autoencoder.