2023 Simons Collaboration on Cracking the Glass Problem Annual Meeting

The Collaboration on Cracking the Glass Problem had its final annual meeting at the Simons Foundation on Thursday and Friday, March 9–10, 2023. The meeting was a fitting end to the seven years of the collaboration, which were successful in so many ways. Of course, there was success on the scientific and research front, but as was clear from the talks and discussions during the meeting, the collaboration spawned many friendships and one-on-one research endeavors. It will have a lasting effect on all those who were lucky enough to participate in this wonderful effort.

It should be noted here that the work of the collaboration was very well recognized by the external community. The most prestigious award was the Nobel Prize in Physics, as well as the Pomeranchuk and the Wolf Prizes, to Giorgio Parisi. But there were many other wonderful recognitions to our members. It should be noted here that the work of the collaboration was very well recognized by the external community. The most prestigious award was the Nobel Prize in Physics, as well as the Pomeranchuk and the Wolf Prizes, to Giorgio Parisi. But there were many other wonderful recognitions to our members. Andrea Liu was elected to the National Academy of Sciences. Silvio Franz became a senior member of the Institut Universitaire de France. Lisa Manning received the Maria Goeppert Mayer Award from the American Physical Society and the Young Scientist Prize in Statistical Physics from the International Union of Pure and Applied Physics (IUPAP). David Reichman was elected to fellowship in the American Academy of Arts and Sciences. Camille Scalliet received the IUPAP Young Scientist Prize in Statistical Physics and the ‘For Women in Science Award’ from the L’Oréal Foundation and UNESCO. Ada Altieri also won the L’Oréal-UNESCO For Women in Science Award, as well as the Springer Thesis Prize. Valentina Ros was awarded the European Statistical and Nonlinear Physics Early Career Prize. Patrick Charbonneau and Matthieu Wyart were elected to fellowship in the American Physical Society. The awards culminated this year with Sidney Nagel being awarded the America Physical Society’s highest honor, the APS Medal for Exceptional Achievement in Research, and Ludo Berthier being awarded the Silver Medal of the CNRS. The collaboration has certainly produced a great deal of exciting research that has excited the scientific community broadly.

This annual meeting had two specific goals: one was to highlight some of the most exciting science that has overlap with our collaboration. The second was to have some of our own members overview the progress that we have made over the past seven years and give their perspective for the future. In the following, we will first overview the four talks from the external speakers and then describe the overview talks from our own principal investigators.

External Talks to the Collaboration

Andrea Cavagna gave a talk about natural swarms, for example, of insects, bacteria or colonies of cells. A unifying ingredient is the presence of strong correlations: experiments in insect swarms found that the correlation length is significantly larger than the microscopic scales. Dynamic scaling was also observed in which spatial and temporal relaxation are entangled so that the relaxation time scales as a power of the correlation length, thus defining the dynamical critical exponent, z. He reviewed how this can be related to the strong correlations and scaling emerging from renormalization group calculations where fixed points organize into a few universality classes. In his talk, he applied the renormalization group to the dynamics of natural swarms. The results are a significant step towards testing the core idea of renormalization at the biological level: integrating out the short-scale details of a strongly correlated system impacts its large-scale behavior. He concluded that the renormalization group and universality may have an incisive impact in biology.

Xiaoming Mao gave an elegant talk about topological mechanics. This is a new field where concepts of topologically protected states of matter are realized in mechanical systems, leading to robust soft modes and self-stress states protected by topology. So far, most studies of topological mechanics focus on periodic lattices that permit convenient mathematical characterization of topological indices. Xiaoming asked whether topological states can arise in disordered mechanical systems. In her talk, she reviewed research where new experiments and theories are proposed to realize and characterize disordered topological mechanical systems, and she discuss their far-reaching implications on robust properties in natural and engineered materials.

Laura Foini discussed the development of the eigenstate thermalization hypothesis with the aim of explaining the mechanism by which chaotic systems reach thermal equilibrium from a generic state. The hypothesis implies a form for the matrix elements of the local operators between the eigenstates of the Hamiltonian. Numerous studies have led to the characterization of these objects in increasingly fine detail to provide a framework for understanding the thermodynamics of quantum many-body systems. She provided a generalization of the hypothesis in order to take into account the correlations between the elements of the matrix, which are essential to describing the high-order correlation functions. She explained how one can assume a hierarchy between these correlations and discussed how this underlies a relationship between the hypothesis and free probability, a branch of mathematics that studies non-commutative random variables. This unveiled the structure of the time-dependent correlation functions in thermal equilibrium.

Cris Moore discussed mathematical topic: algorithms from tensor networks. He started with a noisy observation of a rank-one tensor where there is a hidden vector v. He observed v’s p-th outer product with itself, giving a tensor of arity p, with Gaussian noise in every entry. Given the resulting tensor T, his goal was to reconstruct v. In physics terms, this is a p-spin model. Computationally, it is a tensor version of PCA. Many algorithms approach such problems by “flattening” the tensor into a matrix and using that matrix in a spectral algorithm. A more natural approach is to form a tensor network with copies of the observed tensor and contract it to obtain a scalar or vector. This might help solve the hypothesis testing problem. He reviewed the use of this type of algorithm and made general observations, including about whether they can succeed all the way down to the conjectured computational phase transition.

Reviews of the Collaboration’s Progress

Giulio Biroli gave a talk on a real-space perspective on the glass transition and glassy dynamics. One of the main aims of our collaboration has been to identify and characterize the structural and dynamical properties associated with the glass transition. The special goal was to provide a theoretical framework to predict and explain this emergent behavior. He presented the recent achievements and discussed the challenges ahead. He explained how we aim to understand how local excitations and local rearrangements combine to lead to relaxation and flow in super-cooled liquids. He discussed the main physical mechanisms, the directions we followed to construct a quantitative theory and the connection with the high-dimensional approach.

Patrick Charbonneau reviewed the collaboration’s research about starting from infinite dimensions to address the glass transition that exists in our 3-dimensional world. One of the three pillars of our collaboration was the exact solution of simple glass models in the limit of infinite spatial dimension. He reviewed how this core solution was refined, enriched and extended to finite-dimensional contexts. The theoretical predictions were carefully compared with numerical results obtained in various dimensions, and some features of the mean-field solution were found to be remarkably robust, while other aspects have required a careful consideration of fluctuations and activated processes. He reviewed the successes and challenges of this approach, notably covering jamming criticality, Gardner physics, out-of-equilibrium quenches and instanton calculations.

In his talk, David Reichman reviewed the progress on addressing the theoretical description of the glass transition based on our proposed multi-pronged approach taking advantage of progress in the description of the jamming transition and the rigorous formulation of a high dimensional theory of glasses. Largely through the lens of computer simulation techniques, which have leveraged crucial advances made under the auspices of the collaboration, he highlighted key progress that has been made and outstanding open problems which have yet to be solved.

In her talk, Andrea Liu reviewed the wonderful progress made by our collaboration on using glass concepts for materials design. These topics were not even in existence at the time the proposal was submitted. The materials design problem is an inverse problem in which we first specify the desired properties and then determine the interactions required to achieve them. She emphasized the connection to machine learning, where classifying images requires a cost function that penalizes incorrect identifications and minimizes that function with respect to “learning degrees of freedom” that specify the neural networks. The cost function can have a complex landscape with many local minima so that learning can face the same challenges as the glass problem, in which a physical cost function with a complex landscape must be minimized with respect to the particle positions. She explained how our collaboration pioneered a new strategy for inverse design. The inverse design problem requires two cost functions — learning and physical ones to be minimized with respect to two coupled sets of degrees of freedom: the learning and physical degrees of freedom. She discussed how ideas from the glass problem have not only led to progress in inverse design but have also led to new approaches to machine learning that uses local rules instead of global gradient descent.
 

This was a stimulating meeting. It was bursting with pride about our previous work and hope, excitement and optimism about the future. We all left the Simons Foundation filled with gratitude for having had such a wonderful opportunity to spend the last seven years working together on such an interesting and important set of scientific problems.

Giulio Biroli
École Normale Supérieure

Glass Transition and Glassy Dynamics: A Real Space Perspective
View Slides (PDF)

One of the main aims of our collaboration has been to identify and characterize the emergent structural and dynamical properties associated with the glass transition, and provide a theoretical framework to predict and explain them. In this talk, Giulio Biroli will present recent achievements along this line of research and discuss the challenges that remain ahead. Biroli will follow a real-space perspective in which the goal is to understand how local excitations and local rearrangements combine together and lead to relaxation and flow in super-cooled liquids. Biroli will discuss the main physical mechanisms identified, the directions being followed to construct a quantitative theory and the connection with the high-dimensional approach.
 

Andrea Cavagna
Sapienza Università di Roma

Natural Swarms in 3.99 Dimensions
View Slides (PDF)

Collective behavior is found in a startling variety of biological systems, from clusters of bacteria and colonies of cells, up to insect swarms, bird flocks and vertebrate groups. A unifying ingredient is the presence of strong correlations: experiments in bird flocks, fish schools, mammal herds, insect swarms, bacterial clusters and proteins have found that the correlation length is significantly larger than the microscopic scales. In the case of natural swarms of insects, another key hallmark of statistical physics has been verified, namely dynamic scaling: spatial and temporal relaxation are entangled into one simple law, so that the relaxation time scales as a power of the correlation length, thus defining the dynamical critical exponent, z. Within statistical physics, strong correlations and scaling laws are the two steppingstones leading to the renormalization group (RG): when we coarse-grain short-scale fluctuations, the parameters of different models flow towards one common fixed point ruling their large-scale behavior. RG fixed points therefore organize into few universality classes the macroscopic behavior of strongly correlated systems, thus providing parameter-free predictions of the collective behavior. Biology is vastly more complex than physics, but the widespread presence of strong correlations and the validity of scaling laws can hardly be considered a coincidence, and they rather call for an exploration of the correlation-scaling-RG path also in collective biological systems. However, to date there is yet no successful test of an RG prediction against experimental data on living systems. In this talk Andrea Cavagna will apply the renormalization group to the dynamics of natural swarms of insects. Swarms of midges in the field are strongly correlated systems, obeying dynamic scaling with an experimental exponent z=1.37 +/- 0.11, significantly smaller than the naive value z = 2 of equilibrium overdamped dynamics. Cavagna will show that this anomalous exponent can indeed be reproduced by an RG calculation to one-loop, provided that off-equilibrium activity and inertial dynamics are both considered; the theory gives z=1.35, a value closer to the experimental exponent than any previous theoretical determination and perfectly in line with the numerical value, z=1.35 +/- 0.04. This successful result is a significant step towards testing the core idea of the RG even at the biological level, namely that integrating out the short-scale details of a strongly correlated system impacts on its large-scale behavior by introducing anomalies in the dimensions of the physical quantities. In the light of this, it is fair to hope that the renormalization group, with its most fruitful consequence — universality — may have an incisive impact also in biology.
 

Patrick Charbonneau
Duke University

Starting from d → ∞ to Crack the Glass Problem
View Slides (PDF)

One of the three pillars of the Simons Collaboration on Cracking the Glass Problem is the exact solution of simple glass models in the limit of infinite spatial dimension, d. Throughout the collaboration, this core solution has been refined and significantly enriched. It has also been variously extended to finite-dimensional contexts and its theoretical predictions have been carefully compared with numerical results obtained in various d. While some features of the d → ∞ solution and its finite-dimensional extension were found to be remarkably robust, others have required a careful consideration of fluctuations and activated processes, and yet others remain works in progress. In this talk, Patrick Charbonneau will review the successes and challenges of this epistemic approach, notably covering jamming criticality, Gardner physics, out-of-equilibrium quenches and instanton calculations.
 

Laura Foini
Centre National de la Recherche Scientifique, Institut de Physique Théorique

The Eigenstate Thermalization Hypothesis and Thermal Correlation Functions in Many-Body Quantum Systems
View Slides (PDF)

The development of the eigenstate thermalization hypothesis (ETH) was conducted with the aim of explaining the mechanism by which chaotic systems reach thermal equilibrium from a generic state. ETH implies a form for the matrix elements of the local operators between the eigenstates of the Hamiltonian, and since then, numerous studies have led to the characterization of these objects in increasingly fine detail, providing a solid framework for understanding the (thermo)dynamics of quantum many-body systems. ETH can be derived by analogy with the theory of random matrices and, in fact, in this ansatz these matrix elements are modeled as pseudo-random variables. In their work, Laura Foini and collaborators have provided a generalization of the ETH ansatz in order to take into account the correlations between the elements of the matrix which are essential to describe the high-order correlation functions.

Laura Foini will explain how, by analogy with random matrix theory, one can assume a certain hierarchy between these correlations and will discuss how this generalized ansatz underlies a relationship between ETH and free probability, a branch of mathematics that studies non-commutative random variables. This relationship allowed us to unveil a particular structure of the time-dependent correlation functions in thermal equilibrium.
 

Andrea Liu
University of Pennsylvania

Glass Concepts for Materials Design
View Slides (PDF)

In many-body systems, we normally specify the interactions and study the resulting properties — this is known as the forwards problem. The materials design problem is an inverse problem in which we first specify the desired properties and then determine the interactions required to achieve them. Similar inverse problems are solved routinely in machine learning with neural networks. For example, to classify images, one first specifies a learning cost function that penalizes incorrect identifications, and then minimizes the cost function with respect to “learning degrees of freedom” that specify the neural networks, such as node weights. A set of node weights that reaches the global minimum of the cost function corresponds to a neural network that correctly classifies a set of images, but the cost function can have a complex landscape with many local minima. Learning in neural networks can therefore face the same challenges as the glass problem, in which a physical cost function with a potentially complex landscape, such as the free energy or energy, must be minimized with respect to physical degrees of freedom, such as the particle positions. Our collaboration has pioneered a new strategy for inverse design of materials in which one specifies a learning cost function that embodies the desired properties, which is minimized with respect to learning degrees of freedom that specify interactions. Because the materials are physical systems, however, one must simultaneously minimize a physical cost function, such as the energy, with respect to physical degrees of freedom. Thus, the inverse design problem requires two different cost functions — a learning and a physical cost function — to be minimized with respect to two sets of degrees of freedom — the learning and physical degrees of freedom — which are coupled to each other. Andrea Liu will discuss how ideas from the glass problem have not only led to progress in inverse design but has also led to new approaches to machine learning that use local rules instead of global gradient descent to learn in a distributed way that is not achievable in artificial neural networks.
 

Xiaoming Mao
University of Michigan

Topological Mechanics in Disordered Networks

Topological mechanics is a new field where concepts of topologically protected states of matter are realized in mechanical systems, leading to robust soft modes and self-stress states protected by topology. So far, most studies of topological mechanics focus on periodic lattices which permit convenient mathematical characterization of topological indices. Can topological states arise in disordered mechanical systems? In this talk, Xiaoming Mao will review current research where new experiments and theories are proposed to realize and characterize disordered topological mechanical systems and discuss their far-reaching implications on robust properties in natural and engineered materials.
 

Cristopher Moore
Santa Fe Institute

Algorithms from Tensor Networks

Suppose we have a noisy observation of a rank-one tensor. That is, there is a hidden vector, v, and we observe v’s p-th outer product with itself, giving a tensor of arity p, with Gaussian noise in every entry. Given the resulting tensor T, our goal is to reconstruct v, as well as to reject the null hypothesis that the tensor consists only of noise. In physics terms, this is a planted p-spin model, where p is the arity of the tensor. Computationally, it is a tensor version of PCA. But we lack a theory of linear algebra for tensors, so we can’t simply look at the dominant eigenvector and hope it is correlated with v. What should we do instead? What is the “spectrum” of a tensor anyway?

Many algorithms approach problems like this by “flattening” the tensor into a matrix, and then using the resulting matrix in a spectral algorithm. But a more natural approach is to form a tensor network with copies of the observed tensor, and contract it to obtain a scalar or vector. This scalar might help us solve the hypothesis testing problem, and the vector might help us reconstruct v. Cristopher Moore will review the use of this type of algorithm in the previous papers of others and make some observations about them in general, including whether they can succeed all the way down to the conjectured computational phase transition.
 

David Reichman
Columbia University

Progress and Possibilities: Where We Stand on Cracking the Glass Problem
View Slides (PDF)

Nearly seven years ago our collaboration put forward a roadmap for addressing the theoretical description of the glass transition, a multi-pronged approach taking advantage of progress in the description of the jamming transition and the rigorous formulation of a high dimensional theory of glasses was put forward. Largely through the lens of computer simulation techniques which have leveraged crucial advances made under the auspices of the collaboration, David Reichman will highlight key progress that has been made and outstanding open problems which have yet to be solved.