The Unreasonable Effectiveness of the Ising Model

The Unreasonable Effectiveness of the Ising Model

(part 1 of 2 really this time I swear)

The wild eyed and white haired physicist archetype sits on your television screen—the backdrop, a rapidly revolving Milky Way animation—gesticulating with his hands as if molding the very fiber of the universe, “Physics, you see, is about the symmetry!  It’s all about symmetry!” he exclaims.

Morgan Freeman kindly translates as platonic solids dance around the screen: “The universe is full of symmetry.”

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The most intuitive sort of symmetry comes from the study of spontaneous symmetry in phase transitions. It’s not particularly intuitive, however, why matter should exist in these three particular states: solid, liquid, and gaseous. Why not a smooth transition from a dense, hard solid to a more jelly-like intermediate phase before becoming a liquid?
Photo credit Bryan Alberstat

Science has a handful of words that are often coopted by popularizers to grab attention and to lend credibility-by-association to their work.  “Coherence” is one of these words, as well as the wildly popular “Quantum” (God forbid you be reading about quantum coherences).

The word “symmetry” is a great go-to because it immediately evokes strong visuals.  Lots of things are symmetrical!  But this isn’t what makes symmetry profound.  As one might expect, these buzzwords do, in fact, have highly technical definitions (sometimes several!).

It should come as no surprise, then, that there are many sorts of ‘symmetry’ that often come up in popular physics.  The primary case, which involves the criminally undertold story of Emmy Noether and her study of mathematical objects called Lagrangians, requires a fair amount of mathematical background, and is nonetheless given a surprisingly decent treatment in popularizations reminiscent of the caricature of the introduction.

The most intuitive sort of symmetry—the sort that requires the least mathematical tomfoolery to understand—comes from the study of spontaneous symmetry breaking or, more generally, phase transitions.

When I was younger, after learning about solids, liquids, and gasses, I wondered why matter happened to like these three particular states.  It’s not particularly intuitive—why not a smooth transition from a dense, hard solid to a more jelly-like intermediate phase before becoming a liquid?  Why do liquids suddenly become completely gaseous and not just get less and less dense?  And Jello is just really confusing to a child.

For some number of intervening years plus a decade, I forgot about this confusion, and just came to terms with matter happening to like being in these three states.  Of course, I picked up some extra tools to understand these phases—that one needs to put a certain amount of energy into a substance, say water, before it can hop from this roiling liquid phase to the free expanding gaseous phase—that intermolecular forces are somehow the culprit behind this hierarchy of order.

But the details were still fuzzy.  Even if we zoom down to individual molecules of water and think about these intermolecular forces, it’s not entirely clear that a given ensemble of water molecules would like to self-assemble into a crystal structure versus just wiggling about somewhat close to each other versus being completely free and zipping about a room.

It turns out water is pretty complicated.  The Chandler and Geissler groups (among others) in our very own Chemistry department actually have active research programs into the nuances of water.

To get a sense of where this difficulty might be, let’s turn to a cartoon of a model system studied by and named after Ernst Ising from the 1920s.  You are to imagine a number of tiny magnets, all lined up in a row.  If you tried this with real magnets, you might find that the north and south poles might want to spontaneously align along whichever axis you laid out the magnets upon.  To prevent the headache of having to account for all the different directions these little magnets can point, we can enforce that each little magnet’s north end points either “up” or “down”, perpendicular to the axis on which the magnets rest.

Further suppose that we can control just how neighboring magnets interact with one another.  By tuning a knob, we can make magnets “want” to align with their neighbors, or we can make neighboring magnets “want” to anti-align with their neighbors.  In practice, this amounts to assigning some energy scale to this interaction—that is, two magnets aligned with each other have amount of energy, call it J, whereas two magnets which are anti-aligned have an energy –J.  Tuning the sign of the quantity J amounts to controlling the flavor of this interaction.  For simplicity, let’s assume the magnets “want” to align with their neighbors.

Now, it might seem like cheating to suddenly claim that we can ascribe anything like a temperature to this toy system.  Formally, this takes a little bit of work and relies on a more nuanced definition of temperature that isn’t just ‘average kinetic energy’.  But your intuition should at least suggest that, if we really are allowed to do this, “hot” little magnets want to wiggle and flip and jump back and forth between pointing up and down relatively quickly, while “cold” magnets essentially just sit still and do very little, flipping every once and a while.

But with this notion of temperature comes something of a puzzle.  For very high temperatures, you’re imagining this line of magnets just rapidly flipping back and forth between up and down.  Formally, we can think of this as the temperature giving rise to fluctuations in energy which are much greater than the energy scale set by J, almost as if you were on a tiny boat in a violently stormy sea, and whether or not your boat was capsized or not encoded whether you—the magnet—are up or down.

But as the temperature is reduced—as those waves settle to an infinite expanse of calm—the magnets start to feel the tug of their neighbors, and it seems like they might want to preferentially align either all pointing “up” or all pointing “down”.  After all, and to cheat a little, we know real life magnets exist, which absolutely have a preferential direction to their magnetization.

But the symmetry of the system, at face, seems to suggest nothing particularly interesting will happen.  After all, if you take any configuration of magnets and flip the orientation of every magnet, you have a new configuration with exactly the same energy.  The energy of your system is symmetric with respect to exchange of all the spins.  If the system just decided to spontaneously orient either “up” or “down”, and remain that way, it would be as if the system just decided to ignore half of the configurations available to it.

In the next installment, we will resolve this dilemma and meet one of the giants of 20th century physics, and the worst lecturer ever: Lars Onsager.

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