Old and New Questions in the Theory of Spin Glasses

We study the properties of the groundstates of the 2D Ising spins on a triangular lattice with quasiperiodic interactions described by the quasiperodic parameter and 2 random angles. Thanks to the planarity of the interaction, groundstates could be computed exactly for each realisation via the Onsager solution/perfect matching up to linear size L=512. We study the groundstate rigidity by computing domain wall energy: the energy difference between groundstates with open and twisted (in 1 drection) boundary conditions. By construction the domain wall energy is always positive, and we observe the growth of the mean and variance of the energy, suggesting existence of an ordered low-T phase (for the system sizes considered). To elucidate the nature of the potential ordering, we analyse structure factors of the groundstates, and find no evidences of an order captured by the structure factor.

We consider the dynamics of domain growth in fractonic systems, which are characterized by the presence of excitations with restricted mobility. First, we set the stage by recalling how, in the absence of conserved quantities and following a quench into the ordered phase of the Ising model, the typical size of ordered domains grows in time as $R(t) \sim t^{1/2}$. Then, we review how the domain growth slows to $R(t) \sim t^{1/3}$ when the order parameter is conserved. Finally, we move to the fractonic setting, by requiring the conservation of the center of mass or higher moments of the order parameter. We show, both analytically and numerically, that conservation of the $m$-th multipole moment causes domains to grow anomalously slowly as $R(t) \sim t^{1/(2m+3)}$. This cascade of dynamical critical exponents characterizes a new family of non-equilibrium universality classes.

We explore the spectral properties of the $4$-fermion Sachdev-Ye-Kitaev model with interaction sourced from a Lévy Stable (fat-tailed) distribution. Lévy random matrices are known to demonstrate non-ergodic behaviour through the emergence of a mobility edge. We study the eigenvalue distribution, focusing on long- and short-range correlations and extreme statistics. This model demonstrates a crossover from chaotic to integrable behaviour (in the spectral correlations) as the distribution becomes increasingly fat-tailed. We investigate this crossover through a hierarchical analysis of the eigenvalue spectrum, based on the multi-fractal hierarchy of the Lévy Stable distribution. The crossover is explained in terms of a genuine many-body effect, distinct from the transition (controlled by a mobility edge) in the Lévy random matrices. We conclude with discussions on the model's solvability (including conformal limit and the large$-q$ solution, to appear) and discussion of possible models with exact transitions.

Kinetically constrained models are a fascinating, yet conceptually simple, platform for the study of non-equilibrium quantum many-body phenomena. In this talk, I will focus on chiral models, which lack inversion symmetry. Starting from the particle-conserving quantum East model, I will discuss its emerging quantum Hilbert space fragmentation and the related anomalous dynamics. I will then introduce the East-West model, and present its fascinating anomalous transport properties. Finally, in the last part of the talk, I will discuss recent results on dissipative kinetically constrained models in two dimensions, where the constrained nature of the dissipation yields robust bistability and interesting dynamical properties.

Work in collaboration with Bertrand Lacroix-A-Chez-Toine We study a simple model of $M$ random linear equations in $N$ variables subject to a spherical constraint. In the high dimensional limit, this model exhibits a SAT/UNSAT transition separating a zero-cost phase, where all constraints can be satisfied, from a phase with positive quadratic cost. Here we investigate the large time dynamics generated by gradient descent, for finite N and M, allowing us to probe the dynamics at scales inaccessible by the standard dynamical mean field theory. Indeed, the energy, correlation and response function, are related to the edge statistics of the spectrum of the constraint matrix, and are therefore not always self averaging. We identify the different time scales for the dynamics, and show that aging effects typically disappear at large times, with a crossover time that grows with the system size. Moreover, the behavior is markedly different in the SAT phase (exponential decay) and UNSAT phase (power-law decay), with exponents that we are able to compute analytically.

Variational neural network models have achieved remarkable success in solving ground-state prob- lems of quantum many-body systems. However, addressing classical and quantum spin glasses re- mains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.

We investigate the efficiency of counterdiabatic (CD) driving in accelerating adiabatic dynamics across the exponentially small gaps typical for spin-glass systems and first-order quantum phase transitions. We analyze both a minimal, analytically tractable spin-glass bottleneck model—exhibiting an exponentially closing gap and a macroscopic rearrangement of spins—and realistic NP-hard spin-glass problems, the 3-regular Max-Cut and 3-XORSAT. Using variational Floquet–Krylov constructions, we find that local CD expansion schemes can suppress excitations but fail to overcome the intrinsic spin-glass bottleneck, yielding only marginal improvements in ground-state fidelity even at substantially increased driving times. To address this limitation, we introduce quantum brachistochrone counterdiabatic driving (QBCD), which employs an approximate full CD coupling between the ground and first excited states at a single parameter value near the critical point. In the minimal spin-glass model, QBCD enables exponentially faster adiabatic evolution than local strategies. To mitigate the experimental and classical complexity of QBCD in realistic NP-hard problems, we exponentially reduce its non-locality by sparsifying the Hamiltonian to the density of local CD expansions. Despite this drastic simplification, sparsified QBCD retains finite ground-state fidelity at driving times exponentially shorter than those achievable with local CD methods, including counterdiabatic optimized local driving (COLD). This demonstrates that even an exponentially small fraction of non-locality near the critical point is sufficient to overcome spin-glass bottlenecks, while simultaneously enabling efficient classical simulation and resource-aware quantum implementations. András Grabarits, Federico Balducci, and Adolfo del Campo, arxiv:2410.02520 (accepted in PRX Quantum)

We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for arbitrary size of the block. For the model with three-spin couplings we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus this model represents a new infinite disorder universality class.

Complex landscapes are thought to describe diverse phenomena, from optimization to glasses. Their geometry and topology has long been connected to out-of-equilibrium dynamics, most notably through the statistics of their metastable states. Recent development of the Overlap Gap Property depends on a new topological measure related to structure of the constant energy level set, and bounds from above the performance of a wide class of algorithms and physical processes. We describe other characterizations of level set topology developed with an eye towards understanding the typical performance of mediocre algorithms like gradient flow. These include the typical Euler characteristic of the level set, and whether typical points in the level set belong to a spanning cluster or are isolated, and speculate on the possible relationship between these properties and simple dynamics. Finally, we discuss the application of these (and other) techniques to understand the topology of the solution manifold in constraint satisfaction problems. References: Structure of solutions to continuous constraint satisfaction problems through the statistics of wedged and inscribed spheres, JK-D, arXiv:2510.12926 (2025) Very persistent random walkers reveal transitions in landscape topology, JK-D, arXiv:2505.16653 (2025) On the topology of solutions to random continuous constraint satisfaction problems, JK-D, SciPost Physics 18, 158 (2025)

In this talk, we show that what kind of energy landscape occurs if the disorder is not Gaussian. First, if the disorder is heavy-tailed (x^{-\alpha} like disorder) with $\alpha<2$ , then we shows that the Gibbs measure, free energy and its limiting spin behavior only depends on its largest interaction. Second, for the free energy of p-spin sphiercal model, the model shows \alpha=2p$ threshold phase transition: For the disoder with finite 2p-th moment, this exhibits same universality for Gaussian, and for $\alpha<2p$ disorder, the free energy only depends on the dominant interaction while the $\alpha=2p$ threshold shows TAP-like variational formula between them.

Disordered systems, such as a spin glasses, are defined from a set of high-dimensional configurations, the degrees of freedom, and a Hamiltonian, which prescribes a random energy to each configuration, resulting from the complex interactions between these. A natural question for this type of system is to find the ground-state energy (GSE), i.e. the lowest energy accessible by any configuration of the system. While the average (and typical) GSE has been studied in detail, the rare fluctuations of the GSE away from its average have, in comparison, received much less attention. I will show how replica computations can be used to derive the large deviation function at speed $N$ of the GSE for spherical spin glasses in terms of a functional optimisation problem, extending Parisi’s formula. This large-deviation function generically displays a rich phase diagram. Most interestingly, I will show that the behaviour of the large deviation function in the vicinity of the average and typical GSE is universal and allows to make precise predictions on the non-trivial tails of the distribution of typical fluctuations of the GSE. This talk is based on an article (J. Stat. Phys. 191 (2), 11 (2024) / arXiv preprint arXiv:2306.11927) in collaboration with Y.V. Fyodorov (King's College London) and P. Le Doussal (LPENS).

Politics around the world exhibits increasing polarization, demonstrated in part by rigid voting configurations in institutions like legislatures or courts. A crux of polarization is separation along a unidimensional ideological axis, but voting behavior is in reality more complex, with other signatures of collective order. We extend a foundational, statistical physics framework, restricted Boltzmann machines, to explain the full complexity of voting. The models we propose are minimal, fit strongly correlated voting data, and have parameters that transparently give vote probabilities. The model accounts for multi-dimensional voter preferences and the context in which such preferences are expressed to disentangle individual from collective contributions; for example, legislative bills can negotiate multiple issues, whose appeals add up or compete for individual votes. With the example of the U.S.~Senate, we find that senators have multi-dimensional preferences, and, as one consequence, non-polarized coalitions coexist with polarized ones. Increasing polarization is predominantly explained by fewer votes that elicit bipartisan coalitions. We show that these accounts can be consistent, if far more parsimonious, than interaction-driven order. The findings highlight the collective choice of the content of and the rules of voting in the ebb and flow of polarization.

Feedbacks between evolution and ecology are ubiquitous, with ecological interactions determining which mutants are successful, and these mutants in turn modifying community structure. We study the evolutionary dynamics of several ecological models with overlapping niches, including consumer resource and Lotka-Volterra models. Evolution is assumed slow and extinctions are permanent, with ecological dynamics reaching a stable fixed point between introductions of invaders or mutants. When new strains are slowly added to the community, the ecosystem converges, after an initial evolutionary transient, to a diverse eco-evolutionary steady state. In this "Red Queen" phase of continual evolution, the biodiversity continues to turn over without the invasion probability of new variants getting any smaller. For resource-mediated interactions, the Red Queen phase obtains for any amount of asymmetry in the interactions between strains, and is robust to "general fitness" differences in the intrinsic growth rates of strains. Via a dynamical mean field theory framework valid for high-dimensional phenotype space, we analytically characterize the Red Queen eco-evolutionary steady state in a particular limit of model parameters. Scaling arguments enable a more general understanding of the steady state and evolutionary transients toward it. This work therefore establishes simple models of continual evolution in an ecological context without host-pathogen arms races, and points to the generality of Red Queen evolution. However, we also find other eco-evolutionary phases in simple models: For generalized Lotka-Volterra models with weakly asymmetric interactions an "oligarch" phase emerges in which the evolutionary dynamics continually slow down and a substantial fraction of the community's abundance condenses into a handful of slowly turning-over strains.

We show how graph theory concepts can provide insight into the origin of slow dynamics and weak ergodicity breaking. In particular, we observe that slow dynamics, i.e., violations to the strong ETH can be captured by carefully defined measures of centrality on the state graph of the quantum Hamiltonian. For this we introduce a "graph energy centrality" (GEC) measure and show that it can be used to characterize regimes of weak ergodicity breaking. Further, we show that using GEC we can also capture the onset of weak ergodicity breaking accurately. Further, the GEC can be always be calculated efficiently without the use of exact diagonalization, in many cases even entirely analytically, allowing the study of weak ergodicity breaking in large systems.

Belief propagation (BP), originally developed in statistical inference, can be lifted naturally to the level of tensor networks, where it provides a unifying lens for studying correlations in both classical and quantum many-body systems. In this perspective, BP acts as a mean-field theory on networks, approximating the complex environment of a tensor by self-consistent message tensors and capturing the dominant low-energy or high-weight subspace of the contraction. We study a rigorous framework that interprets BP as the zeroth-order description of correlations, with systematic corrections organized through a loop or cluster expansion over the underlying graph. This expansion expresses deviations from BP as contributions from connected clusters, yielding explicit conditions under which BP is accurate and a controlled, exponentially convergent route to the exact result, with a tunable trade-off between accuracy and computational cost. Beyond providing a principled and efficiently parallelizable contraction algorithm for high-dimensional tensor networks, this viewpoint opens a path to studying “loop-correlations” in many-body systems.

The replica symmetry breaking (RSB) scheme proposed by Parisi, has been remarkably successful in predicting the quenched free energy and the statistics of overlaps in mean field spin glasses and many related models. In this talk I will discuss joint work with Bernard Derrida where we explore generalisations of the Parisi scheme. The assumption in Parisi's original scheme is that the replica matrix $Q_{a,b}$ is divided into blocks of fixed equal size. We have found a number of simple models where the block sizes appear to fluctuate in a non-trivial way and that this leads to overlap statistics that are different from the predictions of the original Parisi scheme. We have obtained these insights by comparing the exact results for a number of generalizations of the random energy model with the corresponding replica analysis. Examples include finite size corrections, overlaps between two temperatures and models with discrete energy levels. It remains an open question whether this type of generalisation of Parisi's RSB scheme is required for more complex spin glass models when looking at similar problems.

Despite decades of research, the universal nature of fluctuations in disordered quantum systems remains poorly understood. Here, we present extensive numerical evidence that fluctuations in two-dimensional (2D) Anderson localization belongs to the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. In turn, by adopting the KPZ framework, we gain fresh insight into the structure and phenomenology of Anderson localization itself. We analyze both localized eigenstates and time-evolved wave packets, demonstrating that the fluctuation of their logarithmic density follows the KPZ scaling. Moreover, we reveal that the internal structure of these eigenstates exhibits glassy features characteristic of the directed polymer problem, including the emergence of dominant paths together with pinning and avalanche behavior. Anderson localization is not isotropic but organized along preferential branches of weaker confinement, corresponding to these dominant paths. For localized wave packets, we further demonstrate that their spatial profiles obey a stretched-exponential form consistent with the KPZ scaling, while remaining fully compatible with the single-parameter scaling hypothesis. Additionally, we also identify the dominant contribution of a directed path to the wave packet density and find that its transverse fluctuations are characterized by a wandering exponent 2/3, same as the directed polymer problem. Altogether, our results establish a unified KPZ framework for describing fluctuations and microscopic organization in 2D Anderson localization, revealing the glassy nature of localized states and providing new understanding into the universal structure of disordered quantum systems.

Spin-glass physics has proved not only interesting in itself but has also found its way into other fields of research. One place to look for similar behavior is in bosonic particle systems exhibiting superfluidity, while also featuring particle interactions. It remains then to be established how the system behaves when those effects coexist with disorder. A key discovery in this context is the superglass phase, first experimentally observed in solid 4He [1,2] and initially identified as a supersolid [1]. In this phase, the glass and superfluid orders were found to coexist [3]. However, the glassiness studied in this context was closer to the structural rather than the spin-glass type, as it involved particle density fluctuations. It remains to be established whether a similar coexistence is possible for other kinds of glass. In this contribution, we present our studies of a system of strongly interacting bosons with a spin-glass-like disorder in the hopping term [4-7]. We thus focus on off-diagonal disorder in the hopping, rather than the more widely studied diagonal (onsite) disorder responsible for the Bose glass. We determine the phase diagram of the system and characterize the phases, including the quantum glass [4] and superglass [5]. Both phases feature a new type of glassiness, based on the complex U(1) phases of bosons rather than density fluctuations, and thus connect to quantum phase-glass theories [8], while providing a setting in which glassy order and superfluidity can be treated on equal footing. We model the system using the Bose-Hubbard Hamiltonian featuring the onsite interaction, chemical potential, and random hopping terms. The hopping integrals are independent Gaussian-distributed random variables, and the results are integrated over these distributions. We solve the model using a quantum counterpart of the Sherrington-Kirkpatrick derivation: we employ the replica trick and the Trotter-Suzuki expansion to finally arrive at a set of self-consistent equations, which we then solve numerically. We find the phase boundaries, derive conditions for stability of the replica-symmetric solution [6], and characterize the phases. To obtain the results, we work in both the replica-symmetric and 1-step replica-symmetry-breaking cases. We observe the interplay between two orders: glassy and superfluid. Their combinations lead to four phases: disordered, superfluid, glass, and superglass. The latter two are of special interest due to their novelty. We show the existence of quantum phase transitions [4,5] and reentrant phase transitions [7] in the system. We analyze the phases and find that in the superglass phase, the glass and superfluid orders compete with each other [5]. Finally, we outline a concrete cold-atom implementation of our model using ultracold bosons in extended optical traps with a random woodpile geometry [4], thereby providing experimental access to the quantum glass and superglass regimes. [1] E. Kim, M. H. W. Chan, Nature 427, 225 (2004). [2] B. Hunt et al., Science 324, 632 (2009). [3] M. Boninsegni, N. Prokof’ev, B. Svistunov, Phys. Rev. Lett. 96, 105301 (2006); G. Biroli, C. Chamon, F. Zamponi, Phys. Rev. B 78, 224306 (2008); X. Yu, M. Müller, Phys. Rev. B 85, 104205 (2012). [4] A. M. Piekarska, T. K. Kopeć, Phys. Rev. Lett. 120, 160401 (2018). [5] A. M. Piekarska, T. K. Kopeć, Phys. Rev. B 105, 174203 (2022). [6] A. M. Piekarska, T. K. Kopeć, J. Stat. Mech. 2022, 73302 (2022). [7] A. M. Piekarska, T. K. Kopeć, Phys. A 609, 128360 (2023). [8] D. Dalidovich, P. Phillips, Phys. Rev. Lett. 89, 27001 (2002).

We study the strong-to-weak spontaneous symmetry breaking (sw-SSB) of non-Abelian symmetries. Using a group-theoretic approach, we introduce a general class of strongly symmetric quantum channels that symmetrically insert pairs of irreducible representations (irreps). The non-deterministic fusion of irreps from different error chains, leads to extensive coherences in the decohered mixed state. This feature obstructs a direct calculation of the optimal recovering threshold and complicates any straightforward connection to thermal physics. To make progress, we focus on strongly symmetric SU(2) dynamics, where the existence of a finite-time threshold would challenge the applicability of Mermin–Wagner–type arguments. First, we analyze a replicated theory that maps onto coupled O(4) non-linear sigma models, yielding power-law–decaying Rényi-1 correlators in the replica limit. In addition, we propose a fusion-hierarchy structure of measured syndromes to systematically decode and probe the sw-SSB transition. These quantities instead map onto coupled disordered non-linear sigma models. We argue that the phase diagram of such disordered models provides a lower bound on the sw-SSB threshold, models which can be conveniently represented as spin networks and hence numerically simulated using projected entangled pair states.

Quantum kernel methods (QKMs) offer an appealing framework for machine learning on near-term quantum computers. However, QKMs generically suffer from exponential concentration, requiring an exponential number of measurements to resolve the kernel values, with the exception of trivial (i.e., classically simulable) kernels. Here we propose a QKM that is free of exponential concentration, yet remains hard to simulate classically. Our QKM utilizes the weak ergodicity-breaking many-body dynamics in the Rydberg blockade of coherently driven neutral atom arrays. We demonstrate the fundamental properties of our QKM by analytically solving an approximate toy model of its underpinning quantum dynamics, as well as by extensive numerical simulations on randomly generated datasets. We further show that the proposed kernel exhibits effective learning on real data. The proposed QKM can be implemented in current neutral atom quantum computers.

A process of statistical inference can exhibit a phase transition as a function of the quantity or quality of the data used for the inference. In the "reconstructing" phase, the data is rich enough to enable the parameters to be inferred accurately, while in the "non-reconstructing" phase the data is so poor that errors in the inferred parameters grow extensively with the system size. Here we demonstrate this phase transition in a computational imaging technique used for spatial genomics, in which the positions of cells in a biological tissue are reconstructed from single-cell sequencing data by examining correlations between the levels of DNA barcodes present within the cells. We provide analytical and numerical evidence for the transition and its critical properties using both simulated data and real experimental data collected by experiments at the Broad Institute.

Classical low-density parity-check (cLDPC) codes defined on expander graphs are a fundamental ingredient in the construction of good quantum LDPC codes, a recent milestone in quantum error correction. They also define interesting statistical mechanics models in their own right, as they include examples of spin glass order without quenched randomness or frustration. We investigate this via a case study of a cLDPC code on a locally tree-like expander graph. Recursive techniques on trees, made possible by the locally tree-like property, probe a menagerie of stable, incongruent valleys induced by imposing different boundary conditions at low temperature. A complementary numerical study of the valleys on closed finite graphs reveals the inequivalence of the microcanonical and canonical ensemble for certain valleys.

Understanding the fundamental properties of complex systems with disordered microscopic interactions remains a longstanding challenge. Spin glass models provide a systematic framework for studying such systems. Notably, at low levels of thermal activation, these models exhibit a glassy phase characterized by broken ergodicity. The impact of small deviations from Hamiltonian dynamics has been widely debated. In this work, we explore how asymmetrical interactions affect systems with marginally stable local equilibria. Our findings support the prevailing view that a true spin-glass phase cannot persist in the thermodynamic limit when interactions are asymmetric, to any degree. However, we demonstrate that large but finite systems can still show glass-like behavior. Specifically, for the XY-model with infinite-range interactions, we compute the finite-size phase diagram.

Establishing a common framework for ergodicity-breaking transitions has many potential applications and provides insight into the nature of non-ergodic phases. In this work, we show that the softening of fluctuations within the recently established fading ergodicity framework can be derived directly from the emergence of the Fermi Golden Rule (FGR), ultimately classifying fading ergodicity as manifestation of FGR physics in quantum many-body systems. We show that this framework identifies the width of the local density of states and the fractal nature of eigenstates in the unperturbed basis as building blocks for fading ergodicity. Furthermore, we argue that our theory can be also applied to integrability-breaking transitions, where the critical point drifts exponentially with system size to a singular point, providing a common framework for ergodicity breaking in RMT models and integrability-breaking in local Hamiltonians.

The emergence of the arrow of time in quantum many-body systems stems from the inherent tendency of Hamiltonian evolution to scramble quantum information and increase entanglement. While, in principle, one might counteract this temporal directionality by engineering a perfectly inverted Hamiltonian to reverse entanglement growth, such a scenario is fundamentally unstable because even minor imperfections in the backward evolution can be exponentially amplified, a hallmark of quantum many-body chaos. Therefore, successfully reversing quantum many-body dynamics demands a deep understanding of the underlying structure of quantum information scrambling and chaotic dynamics. In this letter, by using solid-state nuclear magnetic resonance on a macroscopic ensemble of randomly interacting spins, we measure the out-of-time-ordered correlator (OTOC) and validate key predictions of scramblon theory, a universal theoretical framework for information scrambling. Crucially, this theory enables us to isolate and mitigate errors in the OTOC caused by imperfections in the backward evolution. As a result, this protocol uncovers the anticipated exponential behavior of quantum many-body chaos and extracts the quantum Lyapunov exponent in a many-body experimental system for the first time. Our results push the fundamental limits of dynamical reversibility of complex quantum systems, with implications for quantum simulation and metrology.