The Nobel Prize in physics this year was a bit of a surprise, at least to me. As one friend described it, it's a bit of a Frankenprize, stitched together out of rather disparate components. (Apologies for the slow post - work was very busy today.) As always, it's interesting to read the more in-depth scientific background of the prize. I was unfamiliar with the climate modeling of Manabe and Hasselmann, and this was a nice intro.

The other prize recipient was Giorgio Parisi, a statistical mechanician whose key cited contribution was in the theory of spin glasses, but was generalizable to many disordered systems with slow, many-timescale dynamics including things like polymers and neural networks.

The key actors in a spin glass are excess spins - local magnetic moments that you can picture as little magnetic dipoles. In a generic spin glass, there is both disorder (as shown in the upper panel of the cartoon, spins - in this case iron atoms doped into copper - are at random locations, and that leads to a broad distribution of spin-spin interactions in magnitude and sign) and frustration (interactions such that flipping spin A to lower its interaction energy with spin B ends up raising the interaction energy with spin C, so that there is no simple configuration of spins that gives a global minimum of the interaction energy). One consequence of this is a very complicated energy landscape, as shown in the lower panel of the cartoon. There can be a very large number of configurations that all have about the same total energy, and flipping between these configurations can require a lot of energy such that it is suppressed at low temperatures. These magnetic systems then end up having slow, "glassy" dynamics with long, non-exponential relaxations, in the same way that structural glasses (e.g., SiO2 glass) can get hung up in geometric configurations that are not the global energetic minimum (crystalline quartz, in the SiO2 case).The standard tools of statistical physics are difficult to apply to the glassy situation. A key assumption of equilibrium thermodynamics is that, for a given total energy, a system is equally likely to be found in any microscopic configuration that has that total energy. Being able to cycle through all those configurations is called ergodicity. In a spin glass at low temperatures, the potential landscape means that the system can get easily hung up in a local energy minimum, becoming non-ergodic.

An approach that Parisi took to this problem involved "replicas", where one considers the whole system as an ensemble of replica systems, and a key measure of what's going on is the similarity of configurations between the replicas. Parisi himself summarizes this in this pretty readable (for physicists) article. One of Parisi's big contributions was showing that the Ising spin glass model of Sherrington and Kirkpatrick is exactly solvable.

I learned about spin glasses as a doctoral student, since the interacting two-level systems in structural glasses at milliKelvin temperatures act a lot like a spin glass (TLS coupled to each other via a dipolar elastic interaction, and sometimes an electric dipolar interaction), complete with slow relaxations, differences between field-cooled and zero-field-cooled properties, etc.

Parisi has made contributions across many diverse areas of physics. Connecting his work to that of the climate modelers is a bit of a stretch thematically - sure, they all worry about dynamics of complex systems, but that's a really broad umbrella. Still, it's nice to see recognition for the incredibly challenging problem of strongly disordered systems.

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I agree that the connection between spin glasses and climate models seems tenuous, at first glance. At the same time, I can't shake off the intuitive hunch that there is some deeper unifying link at work here, perhaps a link that has yet to be fully appreciated by the broader community.

What particularly strikes me here is the essential and critical role of timescale separation in determining the statistical and dynamical properties of such systems. Indeed, if you read Hasselmann's paper (https://www.tandfonline.com/doi/pdf/10.3402/tellusa.v28i6.11316), one of the biggest points that he emphasizes is that it is inherently the separation of weather and climate time scales that makes his predictive models possible.

What is interesting is that if you in fact read some of the original spin glass papers by Parisi and others, one of the critical points is that replica symmetry and its breaking is fundamentally connected to the ultrametricity of states (https://hal.archives-ouvertes.fr/jpa-00209816/document). This ultrametricity, in turn, seems to be deeply related to timescale separation. A good summary of the argument can be found in this PRL: https://pure.tue.nl/ws/files/2083005/Metis213395.pdf. In particular, the 2nd paragraph says: "Ultrametricity states a very striking property for a physical system: essentially it says that the equilibrium configurations of a large system can be classified in a taxonomic (hierarchical) way (as animal in different taxa): configurations are grouped in states, states are grouped in families, families are grouped in superfamilies. This equilibrium ultrametricity has a correspondence in the existence of widely separated time scales in the dynamics, typically of a glassy system."

I wonder, then, if the phenomenon of timescale separation - including its underlying origins in replica symmetry / ergodicity, as well as its implications for the 'predictability' of the complex dynamics of strongly disordered systems, such as the climate or spin glasses, might serve as a sort of unifying principle behind this year's prizes, and the physics of complex systems more generally.

@Pizza Perusing Physicist - A rugged landscape can have states not necessarily organized ultrametrically. Ultrametricity probably should not be viewed as a feature related to glassy dynamics in any way, but a symmetry of a typical disordered system where the distance between states has broken replica symmetry.

Anon@12:59 - interesting, thank you for clarifying. If you don't mind I do have some followup questions:

- In the paragraph from the PRL I quoted, the authors gave me the impression that an ultrametric organization of equilibrium states always implies timescale separation in the dynamics. Is my understanding of the authors' claims correct, and if so, is their statement a correct statement in your opinion? If not, could you please give a counterexample, where an ultrametric organization of states does not lead to timescale separation in the dynamics?

- Is it correct to say that even if the above statement were true, meaning ultrametricity implied timescale separation (which I am not claiming it does, I am not an expert, that's why I am asking you), the converse is definitely NOT true? In other words, timescale separation in the dynamics does not require ultrametrically organized states? If so, is timescale separation just a generic feature of rugged landscapes? Are there any well known theoretical results relating the degree of timescale separation to the amount of ruggedness in a landscape (or to other features of its structure in general)?

@Pizza Perusing Physicist - Yes, timescale separation is a generic feature of rugged landscapes. If your states are valleys then, in order to reach other states, you have to cross barriers and the times scales are just different distances between these valleys. In principle, you can imagine any geometry where given three states the two largest distances are not equal like in the Parisi picture. But this hierarchical organization seems to be typical maybe because these system are high dimensional and this type of symmetry is rather universal.

Anon, thanks again for your thoughts. So I am gleaning from our conversation that perhaps the issue of timescale separation and glassy dynamics is not as central as I originally thought. Nevertheless, I would imagine that ultrametricity does, ultimately, have implications for the 'predictability' of the complex statistical dynamics of the spin glass, as it imposes additional symmetry/structure constraints on the system. I saw a specific example of such a result in this paper: https://iopscience.iop.org/article/10.1088/1367-2630/aae566/meta.

In a related vein, I wonder if the degree of 'predictability' of the weather-climate nonlinear high dimensional dynamical system can likewise be connected in some deep way to the existence of hierarchies of model structures (see, e.g., https://agupubs.onlinelibrary.wiley.com/doi/10.1002/2017MS001038).

Either way, pretty cool that we can make predictions on complex systems, whether spin glasses or the climate!

I think the issue of timescale separation is also central because it is equivalent to replica symmetry breaking. Ultrametricity is a particular ansatz that also allowed Parisi to calculate things explicitly.

Ok, so from a physical viewpoint, the great utility of ultrametricity in spin glasses is that it is a specification realization of RSB, and furthermore, one that is amenable to detailed analytical solutions? But the key general insight gleaned - namely, that RSB and timescale separation have consequences for the predictability of complex dynamics - that is what can be extrapolated beyond the particular ultra metric ansatz?

> Ok, so from a physical viewpoint, the great utility of ultrametricity in spin glasses is that it is a specification realization of RSB, and furthermore, one that is amenable to detailed analytical solutions?

Yes. But I don't know what "the predictability of complex dynamics" means.

I think of it in a way to similar the way Hasselman defines it in his original paper on stochastic climate models. There, he defines predictability as: given the value of the 'slow' climate variable y = y0 (so, having integrated out the 'fast' weather variables x) at time t = 0, how precisely can you predict the probability distribution of y at future times t, p(y,t|y0)? Hasselman defines predictability based on the rate of decay of these temporal correlations.

I guess the analogous notion for the spin glass would be how well we can predict Parisi overlap order parameter given its value at t = 0...

But at this point I am starting to spitball, maybe I better call it a day.

Great discussion. Anon, can you say more explicitly how timescale separation is equivalent to replica symmetry breaking? I'll admit that my intuition for what's really going on in replica symmetry breaking is not great.

Parisi's order parameter is the distribution of the distance (or overlap) between pure states and replica symmetry breaking means that this distribution is nontrivial and that, indeed, there are (infinitely) many pure states. Then the picture is that these states are energy valleys separated by extensive energy barriers at distances of order the size of the system. Dynamically, I guess different time scales correspond to different possible distances.

Anon, another follow up question, if you don't mind. My understanding is that any spontaneously broken continuous symmetry must also necessarily be accompanied by a corresponding massless Nambu-Goldstone boson. So for every order parameter there is corresponding boson. For example, magnetization comes with magnons, the finite volume of a fluid come with phonons, etc...

What is the physical interpretation of the corresponding Nambu-Goldstone boson in spin glasses with spontaneous RSB?

I don't know. In spin glasses it is the overlap symmetry under the permutation of replicas that is broken and not some symmetry on the spin configurations themselves affecting the ground state. So I am not sure what the analogy would be.

It is actually interesting that Parisi has a very well-cited paper (1450 times) on stochastic resonance in climate change, but of course this is not mentioned in the citation as it is not really what he earned the prize for. Also, as Doug mentioned, Parisi has made numerous contributions to physics (the KPZ equation, the creation of the simulated tempering algorithm, thw work with Altarelli on asymptotic freedom, showing that gauge conditions can be understood without explicit gauge fixing via Euclidian field theory, the description of collective behavior in bird flocks, as well as a large body of work on structural glasses) in addition to the exact solution of the SK spin-glass model. Much of this work is on complex classical systems, hence the vague connection to the climate modeling.

To pizza 617, Goldstone modes arise upon breaking continuous symmetries. Not sure if RSB qualifies

jonah - thanks. I did some further research and it appears that, in the limit of an infinite number of steps of RSB, the symmetry does become continuous and you can derive Ward identities, Goldstone modes, so forth. See https://arxiv.org/pdf/cond-mat/9802166.pdf for the specific article I found. Admittedly, I did not have a very clear physical intuition for what the Goldstone mode corresponded to after looking at this paper, but it is nice to know that someone has thought about it!

A couple of paragraphs from the Scientifc Background on the Nobel Prize

https://www.nobelprize.org/uploads/2021/10/advanced-physicsprize2021.pdf

"In order to realize the infinitude of states, Parisi’s great leap was to introduce a new order parameter; q_αβ =1/N\sum_i _α _β (9) wherein α and β are replicas. ... Namely, they describe the average overlap between states belonging to replica solution α and those belonging to replica solution β."

"A key concept it called ultrametricity, which is a functional version of the triangle inequality, which we all know from early school days; the sum of the lengths of any two sides of a triangle are greater than or equal to the length of the third. Here, ultrametricity can be characterized using a network describing the states of the system and one finds that upon choosing any three states at random, at least two overlaps are equal so that the disorder-average distribution of overlaps is

P(q) = \sum_{α,β} w_α w_β δ(q − q_αβ), (10) where the w’s are Boltzmann weights."

α and β are not replicas or replica solutions but state indices. Ultrametricity is not a functional version of the triangle inequality and (10) is the definition of the order parameter valid with or without ultrametricity. Who wrote this?

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