Old and New Questions in the Theory of Spin Glasses

We investigate the stability of ground states in the classical Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. We will discuss how ground state stability is related to fluctuations in the energy difference, restricted to finite volumes, between two infinite-volume spin glass ground states. We find that if incongruent ground states exist in any dimension, these fluctuations grow with the volume. These results are used to prove that in the appropriate setting the two-dimensional Edwards-Anderson Ising spin glass has just a single pair of globally spin-reversed ground states. 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) independently of the volume, cannot exist in any dimension.

The nature of the ordered phase of spin glasses in physical dimensions has been controversial for four decades. There is universal agreement that the Edwards-Anderson (EA) model is the appropriate model for studying this question. In the limit when the dimensionality d of the system goes to infinity, there is also universal agreement that its ordered state is that described by the Parisi replica symmetry breaking solution (RSB) of the Sherrington-Kirkpatrick (SK) model. This theory predicts that in the presence of an applied field there is a phase transition to a state with RSB at the de Almeida-Thouless line hAT(T). Monte Carlo simulations are very challenging in high-dimensions so we have studied the existence of such a line for the one-dimensional long-range diluted proxy model for classical Heisenberg spins and found that as d→6, hAT(T)2 ∼(d−6), implying that the nature of the ordered state changes below six dimensions. In the one-dimensional proxy model, the probability that two spins separated by a distance r interact with each other, decays as 1/r2σ. Tuning the exponent σ is equivalent to changing the space dimension of a short-range model Edwards-Anderson model: the relation is d = 2/(2σ−1). The non-existence of the de Almeida-Thouless line when d < 6 has been suspected since 1980. I will briefly describe some of the non-numerical arguments which have been advanced to support its vanishing below 6 dimensions. What then is the appropriate picture of the ordered phase of spin glasses below six dimensions in zero field and in particular, for d = 3? I will argue that it is that of the droplet picture. This picture is a scaling picture which focusses on the energy of droplets of reversed spins. In the droplet picture, droplets of reversed spins have an interface fractal dimension ds 50 lattice spacings in three dimensions and approaching infinity as d→6−) so that nearly all numerical studies have been done in the regime where LL∗ will P(q) go over to the replica symmetric form predicted by droplet scaling.

Spin glasses have long been considered paradigmatic complex systems, combining a rich phenomenology with the existence of well-defined theoretical models. Despite this relevance, experimental and theoretical research on spin glasses largely progressed along separate lines for decades. The starkest example of this disconnect is the study of memory and rejuvenation, which remained unassailable by numerical simulations for 20 years despite being the most striking experimental features. In this talk I will review the first reproduction of the memory and rejuvenation effects on the computer, which was made possible by the Janus II special-purpose supercomputer, and by the insight on the nature of the spin-glass phase obtained in the last 15 years by the Janus Collaboration. In the context of spin glasses, rejuvenation is the observation that when the system is left to evolve at a temperature T1 < Tc for a time, and then cooled to a sufficiently lower T2, the spin glass reverts apparently to the same state it would have achieved had it been cooled directly to T2. That is, its apparent state is independent of its having approached equilibrium (aged) at temperature T1. However, when the spin glass is then warmed back to temperature T1, it appears to return to its aged state, hence memory. Rejuvenation is closely linked to another notoriously elusive property of the spin-glass phase: temperature chaos. In this talk I will first explain how chaos can be quantified in a three-dimensional spin glass. This study will reveal that the coherence length —the size of the growing glassy domains— is, by itself, insufficient to describe the state of a spin-glass sample. Instead, by introducing two additional length scales —a macroscopic correlation length related to the Zeeman effect and a two-time correlation length between rearrangements— we can gain an understanding of chaos, rejuvenation and memory. Finally, I will introduce memory coefficients that enable a quantitative study of memory, both in experiment and simulation. Refs: Janus Collaboration, Nat. Phys. 19, 978-985 (2023). Janus Collaboration, Phys. Rev. Lett. 133, 256704 (2024). E. D. Dahlberg et al., Rev. Mod. Phys. 97, 045005 (2025).

We consider the effect of perturbing a single bond on ground states of nearest-neighbor Ising spin glasses on a wide range of lattices and graphs, revealing that the ground states are strikingly fragile with respect to such changes. The resulting excitations are extensive with fractal boundaries, but bounded in energy, hence combining properties usually attributed to the droplet and replica-symmetry breaking pictures of the spin-glass transition. The existence of threshold instabilities triggering such excitations is connected to the 'local hardness' of finding spin-glass ground states. We consider the efficiency of local predictors of relative spin orientations and find it to be closely related to the occurrence of such instabilities. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the more computationally intensive global minimization. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem, thus suggesting the existence of various classes of local hardness.

Dense Associative Memories are recurrent neural networks with fixed point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks, which were popular in the 1980s, Dense Associative Memories have a very large memory storage capacity, which makes them appealing tools for many problems in AI and neuroscience. In this talk, I will provide an intuitive understanding and a mathematical framework for this class of models, and will give examples of problems in AI that can be tackled using these new ideas. Specifically, I will explore the relationship between Dense Associative Memories and two prominent generative AI models: transformers and diffusion models. I will present a neural network, called the Energy Transformer, which unifies energy-based modeling, associative memories, and transformers in a single architecture. Furthermore, I will discuss an emerging perspective that views diffusion models as Dense Associative Memories operating above the critical memory storage capacity. This insight opens up interesting avenues for leveraging associative memory theory to analyze the memorization-generalization transition in diffusion models, which is closely related to spin glass transition in Dense Associative Memories.

We develop a gauge-theoretic formulation of disordered locally interacting systems with O(N) spins and random two-body couplings. Our starting premise is that the relevant physics of a quenched disorder ensemble is controlled only by gauge-invariant frustration data. We show that the infrared description can be organized as nonlinear sigma models coupled to a background gauge field whose effective action is determined by the frustration distribution of the disorder ensemble. We then construct replicated Ginzburg–Landau field theories that make this dictionary explicit and provide a controlled large-N framework for the infrared phase diagram. We illustrate our formulation for two examples. In two dimensions, the disorder-averaged theory maps to a scalar chromodynamics with replicated matter fields. In the large-N limit, we obtain an effective action for two gauge-invariant inter-replica meson fields. We derive the exact saddle-point equations, analyze the stability of replica-symmetric saddles, and compute gauge-invariant observables to diagnose phases of the underlying disordered model. If time permits, we will discuss a quantum disordered model in three dimensions with topological terms.

At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-tuned ferromagnet-to-paramagnet transition governed by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix---the Hamiltonian of a noninteracting quantum problem---this RG procedure corresponds to an iterative diagonalization of the Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size: $\log \emin^{-1} \sim L^\psi$. The tunneling exponent $\psi$ is equal to the spin stiffness exponent $\theta_c$ characterizing the zero-temperature fixed point.

Progress towards silicon-based quantum technologies has yielded the capability to create ultrathin («delta») layers of dopants in group IV semiconductors, presenting novel and extremely well specified realizations of two-dimensional Hubbard models. Here we describe fabrication, X-ray metrology, angle-resolved photoemission and magnetoelectrical characterization of such layers, with particular attention to charge and spin glass-like features near the metal-insulator transition. References P. Constantinou et al., Momentum-space imaging of ultra-thin electron liquids in δ-doped silicon, Adv. Sci.10, 2302101 (2023) N. D’Anna et al., Non-destructive X-ray imaging of patterned delta-layer devices in silicon, Adv. Electron. Mater. 9, 2201212 (2023) P. Constantinou et al., “EUV-induced hydrogen desorption as a step towards large-scale silicon quantum device patterning” Nature Communications, vol. 15: no. 1, 694 (2024) Guy Matmon et al. Physical Review B, vol. 97: no. 15, 155306 (2018).

While the physics of classical and quantum Ising spin glasses has been rather thoroughly understood, glasses of Heisenberg (vector) spins have remained a difficult problem, including especially its quantum version. I will present the numerically exact mean field solution of quantum and classical Heisenberg spin glasses, based on the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time Quantum Monte Carlo. We find Heisenberg (vector) spin glasses to have a rougher energy landscape than their Ising analogues, which affects their avalanche response to external stimuli. The short time quantum dynamics and collective excitations exhibit a surprisingly slow temperature evolution, that, at asymptotically low temperatures, tend to the superuniversal form found in all insulating mean field glasses, for which we give a simple physical interpretation. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays much slower spin dynamics than the insulating counter part. This behavior extends all the way to the quantum critical melting point. Finally I will discuss interesting aspects of quantum glass transitions in finite dimensional systems that have so far received little attention.

Disorder in quantum many-body systems can drive transitions between ergodic and non-ergodic phases, yet the nature and even existence of these transitions remains intensely debated. Using a two-dimensional array of superconducting qubits, we study an interacting spin model at finite temperature in a disordered landscape, tracking dynamics both in real space and in Hilbert space. Over a broad disorder range, we observe an intermediate non-ergodic regime with glass-like characteristics: physical observables become broadly distributed and some -- but not all -- degrees of freedom are effectively frozen. The Hilbert-space return probability shows slow power-law decay, consistent with finite-temperature quantum glassiness. In the same regime, we detect the onset of a finite Edwards-Anderson order parameter and the disappearance of spin diffusion. By contrast, at lower disorder, spin transport persists with a nonzero diffusion coefficient. Our results show that there is a transition out of the ergodic phase in two-dimensional systems.

Hard combinatorial optimization and inference problems defined on sparse random graphs represent key benchmarks for algorithms. On the one hand, they are extremely challenging. On the other hand, the corresponding disordered model can be solved, and the phase diagram may be connected to the algorithms' performances. I will present results on the performance of the well-known Simulated Annealing and its algorithmic thresholds when it is used as a solver for hard inference and combinatorial optimization problems.

I will introduce "Canyon Landscapes models" [1] as the simplest models of: - the loss surface of overparametrized neural networks. - models of confluent tissues (aka random tesselations of space under tension) - the Smale's 17th problem. These models are exactly solvable and one can study gradient descent dynamics on them [3]. In this talk I will: (1) Discuss how these models can be used to solve the central puzzle in machine learning, namely how feature learning and overfitting can co-exist in overparametrized neural networks, see [2]. (2) Discuss a quantum version of the models which has a natural interpretation as a high-dimensional quantum billiard with connections to SYK physics [4]. Based on: [1] Urbani, A continuous constraint satisfaction problem for the rigidity transition in confluent tissues, 2023 J. Phys. A: Math. Theor. 56 115003 [2] Montanari, Urbani, Dynamical Decoupling of Generalization and Overfitting in Large Two-Layer Networks, arXiv:2502.21269 (NeurIPS 2025, oral) [3] Kamali, Urbani, Dynamical mean field theory for models of confluent tissues and beyond, SciPost Phys. 15, 219 (2023) [4] Urbani, Quantum exploration of high dimensional canyon landscapes.

I will describe a simple method for rigorous analysis of many-body eigenstates in k-local quantum models with energy barriers. First, I will study a model of good classical LDPC codes perturbed by typical k-local quantum fluctuations, and prove that with overwhelming probability, every eigenstate below a critical (finite) energy density is localized near a single low-energy region (as measured by Hamming distance on the classical configuration space). This establishes the existence of many-body Anderson localization and provides a counterexample to the eigenstate thermalization hypothesis. Second, I will present a new proof that the ground state of quantum ferromagnets in d >= 2 spatial dimensions exhibits spontaneous symmetry breaking. In contrast to the existing theory of the stability of quantum phases of matter, our new proof relies only on the presence of energy barriers, and is therefore applicable to gapless systems. As an example, we establish that the ground state of certain quantum random bond Ising models is ferromagnetic, even when they are gapless and frustrated.

Glassiness has been the guiding principle for many average-case hardness results. In the classical setting, glassiness characterizes many natural problems (e.g., random k-SAT) and obstructs a family of "stable" algorithms (e.g., Langevin dynamics). In this work, we develop analogous quantum results. Our techniques, based on channel complexity in quantum optimal transport, differ significantly from classical probabilistic approaches due to the sign problem in the absence of a known eigenbasis. We define stable quantum algorithms and show that glassiness obstructs Lindbladian dynamics and other algorithms analogously to classical results. Contrary to prior Lindbladian runtime lower bounds that only apply to evolution from worst-case initial states, our results hold even when starting from the maximally mixed state. Using the replica trick, we find that random 3-local Pauli Hamiltonians are quantumly average-case hard. Moreover, the phase diagram of glassiness for random k-local Pauli Hamiltonians is distinct from both the classical Ising p-spin model and the fermionic SYK model. Based on https://arxiv.org/abs/2510.08497.

Statistical mechanics assumes that a quantum many-body system at low temperature can be effectively described by its Gibbs state. However, many complex quantum systems exist only as metastable states of dissipative open system dynamics, which appear stable and robust yet deviate substantially from true thermal equilibrium. In this work, we model metastable states as approximate stationary states of a quasi-local, (KMS)-detailed-balanced master equation representing Markovian system-bath interaction, and unveil a universal structural theory: all metastable states satisfy an area law of mutual information and a Markov property. The more metastable the states are, the larger the regions to which these structural results apply. Therefore, the hallmark correlation structure and noise resilience of Gibbs states are not exclusive to true equilibrium but emerge dynamically. Behind our structural results lies a systematic framework encompassing sharp equivalences between local minima of free energy, a non-commutative Fisher information, and approximate detailed balance conditions. Our results build towards a comprehensive theory of thermal metastability and, in turn, formulate a well-defined, feasible, and repeatable target for quantum thermal simulation.

We describe a novel experimental technique for investigating the low-frequency magnetic properties of spin glasses. The technique is based on interfacing a spin glass material with an ensemble of nitrogen-vacancy (NV) centers in diamond, operating as radio-frequency stray magnetic field sensors. This close proximity allows the sensors to detect stray fields that directly reflect the behavior of a small, localized ensemble of spins within the glass. We develop various detection protocols, including access to the second and fourth cumulants of the magnetic noise generated by these spins. These experimental techniques offer unprecedented access to previously inaccessible quantities in spin glasses, creating opportunities to challenge existing theory and uncover new phenomena.

Quantum annealing is a quantum-mechanical analogue to the classical optimization algorithm called simulated annealing, wherein one uses tunable quantum fluctuations to guide a collection of qubits into the ground state of a target spin-glass Hamiltonian. Although the vast majority of work has focused on the time required to reliably prepare the ground state, the complementary question --- how low in energy can be reached in a fixed amount of time --- is equally important. The traditional (mean-field) theory of spin glasses makes a very underwhelming prediction, that there exists a unique "threshold" energy which all classical and quantum annealing protocols converge to on O(1) timescales. However, recent work by Folena et al (PRX 2020) has shown that this prediction is generically incorrect as far as simulated annealing is concerned. Here we examine the consequences of this surprising result for **quantum** annealing, using the well-known mixed p-spin model as a tractable case study. On the one hand, we show that the energy reached by quantum annealing is far less sensitive to the annealing protocol than its classical counterpart. Yet this is of little concern, for we demonstrate that straightforward quantum annealing is capable of reaching energies far below the naive threshold, and we even present evidence suggesting that it can be better at doing so than the optimal simulated annealing protocol.

I will discuss recent experiments realizing a spin glass phase of bosons in multimode cavities (Kroeze et al., Science 389, 1122 (2025); Marsh et al. arXiv:2509.12202). These experiments enable the real-time visualization of ergodicity breaking (and the quantitative extraction of overlap distributions among quasi-steady states) through the pattern of light emitted from the cavity. I will present experimental and numerical evidence that these overlaps are hierarchically distributed. I will also discuss the "basins of attraction" of these steady states, and present evidence that the open-system dynamics leads to larger basins than one finds in standard Metropolis dynamics.