For a true phase transition in one dimension, mathematicians had proved that two of these exponents must be greater than 2. But the MBL simulations had found them to be 1 — a major disagreement. In a still-unpublished preprint posted in 2015, Oganesyan and Chandran, together with Christopher Laumann of Boston University, showed that the mismatch was not just a trivial side effect of studying short chains rather than infinite ones. Something more fundamental seemed off.
“They looked into it carefully,” Huse said. “But we couldn’t figure out what was wrong.”
A string of bigger shocks came over the next few years. Imagine the kind of mountainous landscape that would lead to MBL. Now extend that landscape to infinity in all directions. If you randomly explore enough of it, at some point you’re bound to run into an extended flat patch.
Particles in a flat zone can easily find states of similar energy to tunnel to, so they mingle and thermalize. In such a region, energy states abound, increasing the odds that a particle in the neighboring mountains could make contact and become thermalized itself, argued De Roeck of KU Leuven and François Huveneers, who was then at the University of Paris-Dauphine in France. Thus, the flat zone can serve as a source of thermalizing energy.
But could such a tiny patch take down the whole system? The scenario intuitively seemed about as plausible as a hot tub in Denver causing meltdowns in Vail, Breckenridge and Telluride. Physicists didn’t accept it right away. When De Roeck and Huveneers raised the possibility at conferences, their talks provoked angry outbursts from the audience.
“It was a big surprise,” De Roeck said. “A lot of people in the beginning did not believe us.”
In a series of papers starting in 2016, De Roeck, Huveneers and collaborators laid out their case for a process now known as an avalanche. They argued that, unlike a hot tub, what starts as a drop of thermalized particles can snowball into an ocean.
“You have a heat bath, and it recruits neighboring sites into the heat bath,” Imbrie said. “It gets stronger and stronger and pulls in more and more sites. That’s the avalanche.”
The crucial question was whether an avalanche would gain momentum or lose it. With each step, the heat bath would indeed become a bigger and better energy reservoir. But each step also made thermalizing the next site harder. Reminiscent of Anderson’s single-particle localization, the debate came down to a race between two effects: the bath’s improvement versus its difficulty in growing further.
De Roeck and Huveneers argued that avalanches would win in two and three dimensions, because they stockpiled energy states incredibly quickly — at rates related to their rapidly growing area (in 2D) or volume (in 3D). Most physicists came to accept that avalanches in these landscapes were unstoppable, making MBL a remote prospect in sheets or bricks.
But the possibility of MBL in one-dimensional chains survived, because an avalanche sweeping across a line accrues energy states more slowly. In fact, the heat bath grows more powerful at about the same rate at which the difficulty of growth rises. It was a tie. Avalanches might continue in 1D, or they might stop.
Other physicists, meanwhile, grew skeptical that MBL could exist even in a 1D chain. In 2019, a team of Slovenian chaos experts including Tomaž Prosen reanalyzed old numerical data and highlighted the fact that as the landscape got more mountainous, thermalization slowed tremendously but never completely stopped — an inconvenient truth MBL researchers had taken to be an artifact of their small-scale simulations. Anatoli Polkovnikov of Boston University and Dries Sels, now of New York University and the Flatiron Institute, among other researchers, came to similar conclusions. Their arguments directly challenged the central allure of MBL: the promise of eternal life for a quantum sandcastle.
“At the level of theorists talking about MBL,” Chandran said, “there’s an honest-to-God regime where [the thermalization time] is not just age of the universe, and we can’t see it. No, it’s truly infinite.”
A vigorous debate followed, both in the academic literature and in private discussions. Sels and Huse spent hours on Zoom during the depths of the pandemic. They talked past each other at times, but each credits the other with productive insights. The ins and outs of the controversy are extremely technical, and not even the researchers involved can fully articulate all the perspectives. But ultimately, their differences come down to each camp making a different educated — extremely educated — guess as to what you would see if you could watch a chain of particles flip forever.
The two sides still disagree about whether a genuine MBL phase exists in one dimension, but one concrete result of the clash is that it drove researchers to scrutinize the effect that avalanches might have on the presumed onset of MBL.
The skeptical groups “had some very good points, but they took them a little too far,” Huse said. “It really got us motivated.”
Huse, collaborating with a team of MBL veterans including Khemani, cooked up a way to simulate the effect of an avalanche on short chains without actually triggering one. (No one has seen an avalanche, even numerically, because to get a big enough flat spot you might need a chain billions of particles long, Sels estimates, and researchers typically study chains of about 12.) Sels subsequently developed his own avalanche mock-up.
The two groups came to similar conclusions in 2021: The MBL transition, if it existed, required a much more mountainous landscape than researchers had believed. With the ruggedness level previously thought to bring about MBL, thermalization would slow, but would not stop. To give quantum snowmen a fighting chance against avalanches, the landscape would have to be more disordered than Huse and company had suspected. Huse’s group initially found that the mountains would need to be least twice as rugged. Sels’ work pushed that number up to at least six times as rugged, making the mountains more like Himalayas than Rockies. MBL may still occur in those extreme settings, but the theory that had been built around the less rugged transition did indeed have problems.
“We sort of accepted it too thoroughly, and we didn’t look at the subtleties of it,” Huse said.
In the 2021 works, the researchers rewrote and expanded the MBL phase diagram for 1D chains. In Kansas-like flatlands, particles thermalize quickly. In the Rockies, the researchers reclassified the MBL “phase” as a “pre-thermal regime.” This is the seemingly stable regime discovered by BAA, the Princeton simulations, and atomic experiments. But now the researchers had concluded that if one waited an extremely long time — literally billions of years for some setups — particles separated by the Rockies would in fact mingle and thermalize.
Beyond the Rockies lie the Himalayas. What happens there remains an open question. Sels and Prosen are convinced that energy will spread and thermalization will eventually occur, even if it takes eons. Huse and company continue to believe that genuine MBL sets in.
Chief among their reasons for belief in MBL is the 2014 proof. Of the once numerous pillars of evidence supporting the existence of true MBL, Imbrie’s proof is the last one standing. And after a career developing bespoke mathematical tools for just this type of problem, he stands by it.
“It’s not unheard of in mathematics to have an error in a proof,” he said, “but I think I know what I’m doing.”
The proof divides physicists, however, because physicists don’t understand it. It isn’t for lack of trying. Laumann once got Imbrie to teach the proof to him and a handful of researchers over the course of a week in Italy, but they couldn’t follow the steps in detail. That’s not entirely surprising, though, as physicists typically use mathematics in a faster and looser way than mathematicians do. Imbrie’s argument doesn’t depend on any specific level of ruggedness in the landscape, so the recent revisions to the MBL phase diagram in no way undermine it. To determine whether MBL truly exists, researchers will need to buckle down and either find a problem in the proof or verify every line.
Such efforts are underway. Sels and collaborators say they’re finalizing an argument that will contradict Imbrie’s. Meanwhile, De Roeck and Huveneers, the mathematicians who discovered the threat of avalanches, are two years into an effort to rewrite Imbrie’s proof in a more accessible form. De Roeck says they’ve put all the major pieces in place, and so far the logic looks solid.
“MBL, I believe it exists,” De Roeck said. But “we’re doing mathematics here, so any small problem can derail the whole thing.”
Beyond Quantum Angels
In the universe we inhabit, which will itself thermalize in some incomprehensible number of years, permanence is always something of an illusion. Manhattan is sinking under its own weight at 1.6 centimeters per decade. The continents will merge in roughly 250 million years. And while it’s a myth that the bottoms of medieval stained-glass windows have thickened slightly over the centuries, physicists do believe that glass flows over some unknown timescale, likely many billions of years or more.
If MBL proves unstable, a many-body localized system will be at least as durable as any of these examples. So will those quantum phenomena that depend on MBL states. Time crystals, for instance, might lose their textbook designations as “phases of matter,” but they’d still be able to keep ticking for far, far longer than the quantum computers that simulate them (or the humans that operate the computers, for that matter). Many academics do care deeply about the mathematical possibility of defeating thermalization as the beautiful, academic question that it is. But these days, most aren’t losing much sleep over it.
“Maybe it was always angels dancing on the head of a pin,” Chandran said.
Instead, Chandran and others have reveled in the chance to discover a new thermalization-causing phenomenon, one that physicists might actually observe in small systems.
Back in 2018, she and her collaborator Philip Crowley had set out to understand why small chains appeared to slowly thermalize even though they were far too small for flat spots to crop up. The duo determined that groups of particles were occasionally getting lucky and borrowing energy from a neighboring group in the exact amount they needed to flip to a new configuration. They dubbed these coincidences “resonances” and observed how they tended to spread from group to group, leading to a drawn-out thermalization in systems too small for avalanches. In 2020, they showed that resonances can explain the 2015 exponent mismatch and many of the fishy features that have been showing up in numerical experiments, insights that helped Huse and company update the phase diagram for short chains in 2021.
Today, physicists believe that resonances destabilize modest chains with Rockies-level disorder, while avalanches destabilize longer chains at higher levels of disorder.
As Chandran and others improve their simulations and experiments and explore longer, more rugged chains, they wonder what else might lurk in the Himalayas and beyond.
“It seems like there is other physics going on in there,” Huse said. “That would be nicest for me. I like finding new things.”
Editor’s note: A few researchers who appear in this article have received funding from the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage. More details available here.
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