CyberSec Research

We study the stability of ground states in the Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. After reviewing the concepts of critical droplets, flexibilities and metastates, we show that, in any dimension, a certain kind of critical droplet with space-filling (i.e., positive spatial density) boundary does not exist in ground states generated by coupling-independent boundary conditions. Using this we show that if incongruent ground states exist in any dimension, the variance of their energy difference restricted to finite volumes scales proportionally to the volume. This in turn is used to prove that a metastate generated by (e.g.) periodic boundary conditions is unique and supported on a single pair of spin-reversed ground states in two dimensions. We further show that a type of excitation above a ground state, whose interface with the ground state is space-filling and whose energy remains O(1) independent of the volume, as predicted by replica symmetry breaking, cannot exist in any dimension.

The Krylov subspace expansion is a workhorse method for sparse numerical methods that has been increasingly explored as source of physical insight into many-body dynamics in recent years. We revisit the venerable Anderson model of localization in dimensions $d=1, 2, 3, 4$ to construct local integrals of motion (LIOM) in Krylov space. These appear as zero eigenvalue edge states of an effective hopping problem in Krylov superoperator subspace, and can be analytically constructed, given the Lanczos coefficients. We exploit this idea, focusing on $d=3$, to study the manifestation of the disorder driven Anderson transition in the anatomy of LIOMs. We find that the increasing complexity of the Krylov operators results in a suppression of the fluctuations of the Lanczos coefficients. As such, one can study the phenomenology of the integrals of motion in the disorder averaged Krylov chain. We find edge states localized on vanishing fraction of Krylov space (of dimension $D_K=V^2$ for cubes of volume $V$), both in localized and extended phases. Importantly, in the localized phase, disorder induces powerlaw decaying dimerization in the (Krylov) hopping problem, producing stretched exponential decay of the LIOMs (in Krylov space) with a stretching exponent $1/2d$. Metallic LIOMs are completely delocalized albeit across only $\propto \sqrt{D_K}$ states). Critical LIOMs exhibit powerlaw decay with an exponent matching the expected fractal exponent.

Accurately describing the ground state of strongly correlated systems is essential for understanding their emergent properties. Neural Network Backflow (NNBF) is a powerful variational ansatz that enhances mean-field wave functions by introducing configuration-dependent modifications to single-particle orbitals. Although NNBF is theoretically universal in the limit of large networks, we find that practical gains saturate with increasing network size. Instead, significant improvements can be achieved by using a multi-determinant ansatz. We explore efficient ways to generate these multi-determinant expansions without increasing the number of variational parameters. In particular, we study single-step Lanczos and symmetry projection techniques, benchmarking their performance against diffusion Monte Carlo and NNBF applied to alternative mean fields. Benchmarking on a doped periodic square Hubbard model near optimal doping, we find that a Lanczos step, diffusion Monte Carlo, and projection onto a symmetry sector all give similar improvements achieving state-of-the-art energies at minimal cost. By further optimizing the projected symmetrized states directly, we gain significantly in energy. Using this technique we report the lowest variational energies for this Hamiltonian on $4\times 16$ and $4 \times 8$ lattices as well as accurate variance extrapolated energies. We also show the evolution of spin, charge, and pair correlation functions as the quality of the variational ansatz improves.

We study the ability of Transformer models to learn sequences generated by Permuted Congruential Generators (PCGs), a widely used family of pseudo-random number generators (PRNGs). PCGs introduce substantial additional difficulty over linear congruential generators (LCGs) by applying a series of bit-wise shifts, XORs, rotations and truncations to the hidden state. We show that Transformers can nevertheless successfully perform in-context prediction on unseen sequences from diverse PCG variants, in tasks that are beyond published classical attacks. In our experiments we scale moduli up to $2^{22}$ using up to $50$ million model parameters and datasets with up to $5$ billion tokens. Surprisingly, we find even when the output is truncated to a single bit, it can be reliably predicted by the model. When multiple distinct PRNGs are presented together during training, the model can jointly learn them, identifying structures from different permutations. We demonstrate a scaling law with modulus $m$: the number of in-context sequence elements required for near-perfect prediction grows as $\sqrt{m}$. For larger moduli, optimization enters extended stagnation phases; in our experiments, learning moduli $m \geq 2^{20}$ requires incorporating training data from smaller moduli, demonstrating a critical necessity for curriculum learning. Finally, we analyze embedding layers and uncover a novel clustering phenomenon: the model spontaneously groups the integer inputs into bitwise rotationally-invariant clusters, revealing how representations can transfer from smaller to larger moduli.

Artificial Intelligence (AI) has become an exceptionally powerful tool for analyzing scientific data. In particular, attention-based architectures have demonstrated a remarkable capability to capture complex correlations and to furnish interpretable insights into latent, otherwise inconspicuous patterns. This progress motivates the application of AI techniques to the analysis of strongly correlated electrons, which remain notoriously challenging to study using conventional theoretical approaches. Here, we propose novel AI workflows for analyzing snapshot datasets from tensor-network simulations of the two-dimensional (2D) Hubbard model over a broad range of temperature and doping. The 2D Hubbard model is an archetypal strongly correlated system, hosting diverse intriguing phenomena including Mott insulators, anomalous metals, and high-$T_c$ superconductivity. Our AI techniques yield fresh perspectives on the intricate quantum correlations underpinning these phenomena and facilitate universal omnimetry for ultracold-atom simulations of the corresponding strongly correlated systems.

Delayed generalization, termed grokking, in a machine learning calculation occurs when the training accuracy approaches its maximum value long before the test accuracy. This paper examines grokking in the context of a neural network trained to classify 2D Ising model configurations.. We find, partially with the aid of novel PCA-based network layer analysis techniques, that the grokking behavior can be qualitatively interpreted as a phase transition in the neural network in which the fully connected network transforms into a relatively sparse subnetwork. This in turn reduces the confusion associated with a multiplicity of paths. The network can then identify the common features of the input classes and hence generalize to the recognition of previously unseen patterns.

Self-similarity, where observables at different length scales exhibit similar behavior, is ubiquitous in natural systems. Such systems are typically characterized by power-law correlations and universality, and are studied using the powerful framework of the renormalization group (RG). Intriguingly, power laws and weak forms of universality also pervade real-world datasets and deep learning models, motivating the application of RG ideas to the analysis of deep learning. In this work, we develop an RG framework to analyze self-similarity and its breakdown in learning curves for a class of weakly non-linear (non-lazy) neural networks trained on power-law distributed data. Features often neglected in standard treatments -- such as spectrum discreteness and lack of translation invariance -- lead to both quantitative and qualitative departures from conventional perturbative RG. In particular, we find that the concept of scaling intervals naturally replaces that of scaling dimensions. Despite these differences, the framework retains key RG features: it enables the classification of perturbations as relevant or irrelevant, and reveals a form of universality at large data limits, governed by a Gaussian Process-like UV fixed point.

This study investigates the emergence of quantum spin liquid phases in pyrochlore oxides with non-Kramers ions, driven by structural randomness that effectively acts as a transverse field, introducing quantum fluctuations on top of the spin ice manifold. This is contrary to the naive expectation that disorder favors phases with short-range entanglement by adjusting the spins with their local environment. Given this unusual situation, it is essential to assess the stability of the spin-liquid phase with respect to the disorder. To perform this study, a minimal model for disordered quantum spin ice, the transverse-field Ising model, is analyzed using a formulation of gauge mean-field theory (GMFT) directly in real space. This approach allows the inclusion of disorder effects exactly and provides access to non-perturbative effects. The analysis shows that the quantum spin ice remains remarkably stable with respect to disorder up to the transition to the polarized phase at high fields, indicating that it can occur in real materials. Moreover, the Griffiths region of enhanced disorder-induced fluctuations appears tiny and restricted to the immediate vicinity of this transition due to the uniqueness of the low-energy excitations of the problem. For most of the phase diagram, an average description of the disorder captures the physical behavior well, indicating that the inhomogeneous quantum spin ice behaves closely to its homogeneous counterpart.

Disordered systems subject to a fluctuating environment can self-organize into a complex history-dependent response, retaining a memory of the driving. In sheared amorphous solids, self-organization is established by the emergence of a persistent system of mechanical instabilities that can repeatedly be triggered by the driving, leading to a state of high mechanical reversibility. As a result of self-organization, the response of the system becomes correlated with the dynamics of its environment, which can be viewed as a sensing mechanism of the system's environment. Such phenomena emerge across a wide variety of soft matter systems, suggesting that they are generic and hence may depend very little on the underlying specifics. We review self-organization in driven amorphous solids, concluding with a discussion of what self-organization in driven disordered systems can teach us about how simple organisms sense and adapt to their changing environments.

Rydberg atom arrays are a promising platform for quantum optimization, encoding computationally hard problems by reducing them to independent set problems with unit-disk graph topology. In Nguyen et al., PRX Quantum 4, 010316 (2023), a systematic and efficient strategy was introduced to encode multiple problems into a special unit-disk graph: the King's subgraph. However, King's subgraphs are not the optimal choice in two dimensions. Due to the power-law decay of Rydberg interaction strengths, the approximation to unit-disk graphs in real devices is poor, necessitating post-processing that lacks physical interpretability. In this work, we develop an encoding scheme that can universally encode computationally hard problems on triangular lattices, based on our innovative automated gadget search strategy. Numerical simulations demonstrate that quantum optimization on triangular lattices reduces independence-constraint violations by approximately two orders of magnitude compared to King's subgraphs, substantially alleviating the need for post-processing in experiments.

Topological phases of matter have garnered significant interest over the past two decades for two main reasons: their identification, via topological invariants, relies on the quantum geometry of the Bloch states, bringing attention to an aspect of electronic band structure overlooked up to their discovery. Secondly, these classes of materials present electronic states with unusual properties, leading to exotic phenomena and making them relevant for potential applications. In this thesis we explore both fundamental and technological aspects of the first discovered topological phase: the topological insulator. To this end, we consider different models of topological insulators with a particular emphasis on Bismuth compounds, evaluating their viability for photovoltaic applications, and separately, the impact of structural disorder on their properties.

In this work, we examine the conditions for the emergence of chimera-like states in Ising systems. We study an Ising chain with periodic boundaries in contact with a thermal bath at temperature T, that induces stochastic changes in spin variables. To capture the non-locality needed for chimera formation, we introduce a model setup with non-local diffusion of spin values through the whole system. More precisely, diffusion is modeled through spin-exchange interactions between units up to a distance R, using Kawasaki dynamics. This setup mimics, e.g., neural media, as the brain, in the presence of electrical (diffusive) interactions. We explored the influence of such non-local dynamics on the emergence of complex spatiotemporal synchronization patterns of activity. Depending on system parameters we report here for the first time chimera-like states in the Ising model, characterized by relatively stable moving domains of spins with different local magnetization. We analyzed the system at T=0, both analytically and via simulations and computed the system's phase diagram, revealing rich behavior: regions with only chimeras, coexistence of chimeras and stable domains, and metastable chimeras that decay into uniform stable domains. This study offers fundamental insights into how coherent and incoherent synchronization patterns can arise in complex networked systems as it is, e.g., the brain.

Recent research has shown that memory, in the form of slow degrees of freedom, can induce a phase of long-range order (LRO) in locally-coupled fast degrees of freedom, producing power-law distributions of avalanches. In fact, such memory-induced LRO (MILRO) arises in a wide range of physical systems. Here, we show that MILRO can be transferred to coupled systems that have no memory of their own. As an example, we consider a stack of layers of spins with local feedforward couplings: only the first layer contains memory, while downstream layers are memory-free and locally interacting. Analytical arguments and simulations reveal that MILRO can indeed drag across the layers, enabling downstream layers to sustain intra-layer LRO despite having neither memory nor long-range interactions. This establishes a simple, yet generic mechanism for propagating collective activity through media without fine tuning to criticality, with testable implications for neuromorphic systems and laminar information flow in the brain cortex.

We investigate long-range resonances in quasiperiodic many-body localized (MBL) systems. Focusing on the Heisenberg chain in a deterministic Aubry-Andr\'{e} potential, we complement standard diagnostics by analyzing the structure of long-distance pairwise correlations at high energy. Contrary to the expectation that the ergodic-MBL transition in quasiperiodic systems should be sharper due to the absence of Griffiths regions, we uncover a broad unconventional regime at strong quasiperiodic potential, characterized by fat-tailed distributions of longitudinal correlations at long distance. This reveals the presence of atypical eigenstates with strong long-range correlations in a regime where standard diagnostics indicate stable MBL. We further identify these anomalous eigenstates as quasi-degenerate pairs of resonant cat states, which exhibit entanglement at long distance. These findings advance the understanding of quasiperiodic MBL and identify density-correlation measurements in ultracold atomic systems as a probe of long-range resonances.

For three decades statistical physics has been providing a framework to analyse neural networks. A long-standing question remained on its capacity to tackle deep learning models capturing rich feature learning effects, thus going beyond the narrow networks or kernel methods analysed until now. We positively answer through the study of the supervised learning of a multi-layer perceptron. Importantly, (i) its width scales as the input dimension, making it more prone to feature learning than ultra wide networks, and more expressive than narrow ones or with fixed embedding layers; and (ii) we focus on the challenging interpolation regime where the number of trainable parameters and data are comparable, which forces the model to adapt to the task. We consider the matched teacher-student setting. It provides the fundamental limits of learning random deep neural network targets and helps in identifying the sufficient statistics describing what is learnt by an optimally trained network as the data budget increases. A rich phenomenology emerges with various learning transitions. With enough data optimal performance is attained through model's "specialisation" towards the target, but it can be hard to reach for training algorithms which get attracted by sub-optimal solutions predicted by the theory. Specialisation occurs inhomogeneously across layers, propagating from shallow towards deep ones, but also across neurons in each layer. Furthermore, deeper targets are harder to learn. Despite its simplicity, the Bayesian-optimal setting provides insights on how the depth, non-linearity and finite (proportional) width influence neural networks in the feature learning regime that are potentially relevant way beyond it.

Evolutionary systems must learn to generalize, often extrapolating from a limited set of selective conditions to anticipate future environmental changes. The mechanisms enabling such generalization remain poorly understood, despite their importance to predict ecological robustness, drug resistance, or design future-proof vaccination strategies. Here, we demonstrate that annealed population heterogeneity, wherein distinct individuals in the population experience different instances of a complex environment over time, can act as a form of implicit regularization and facilitate evolutionary generalization. Mathematically, annealed heterogeneity introduces a variance-weighted demographic noise term that penalizes across-environment fitness variance and effectively rescales the population size, thereby biasing evolution toward generalist solutions. This process is indeed analogous to a variant of the mini-batching strategy employed in stochastic gradient descent, where an effective multiplicative noise produces an inductive bias by triggering noise-induced transitions. Through numerical simulations and theoretical analysis we discuss the conditions under which variation in how individuals experience environmental selection can naturally promote evolutionary strategies that generalize across environments and anticipate novel challenges.

Topolectrical circuits provide a versatile platform for exploring and simulating modern physical models. However, existing approaches suffer from incomplete programmability and ineffective feature prediction and control mechanisms, hindering the investigation of physical phenomena on an integrated platform and limiting their translation into practical applications. Here, we present a deep learning empowered programmable topolectrical circuits (DLPTCs) platform for physical modeling and analysis. By integrating fully independent, continuous tuning of both on site and off site terms of the lattice Hamiltonian, physics graph informed inverse state design, and immediate hardware verification, our system bridges the gap between theoretical modeling and practical realization. Through flexible control and adiabatic path engineering, we experimentally observe the boundary states without global symmetry in higher order topological systems, their adiabatic phase transitions, and the flat band like characteristic corresponding to Landau levels in the circuit. Incorporating a physics graph informed mechanism with a generative AI model for physics exploration, we realize arbitrary, position controllable on board Anderson localization, surpassing conventional random localization. Utilizing this unique capability with high fidelity hardware implementation, we further demonstrate a compelling cryptographic application: hash based probabilistic information encryption by leveraging Anderson localization with extensive disorder configurations, enabling secure delivery of full ASCII messages.

I discuss the consequences of the constraints imposed on the Dirac spectrum by the restoration of chiral symmetry in the chiral limit of gauge theories with two light fermion flavors, with particular attention to the fate of the anomalous $\mathrm{U}(1)_A$ symmetry. Under general, physically motivated assumptions on the spectral density and on the two-point eigenvalue correlation function, I show that effective $\mathrm{U}(1)_A$ breaking in the symmetric phase requires specific spectral features, including a spectral density effectively behaving as $m^2\delta(\lambda)$ in the chiral limit, a two-point function singular at zero, and delocalized near-zero modes, besides an instanton gas-like behavior of the topological charge distribution. I then discuss a $\mathrm{U}(1)_A$-breaking scenario characterized by a power-law divergent spectral peak tending to $O(m^4)/|\lambda|$ in the chiral limit and by a near-zero mobility edge, and argue that the mixing of the approximate zero modes associated with a dilute gas of topological objects provides a concrete physical mechanism producing the required spectral features, and so a viable mechanism for effective $\mathrm{U}(1)_A$ breaking in the symmetric phase of a gauge theory.