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Quantum dynamics—the study of how particles behave and interact in a quantum system—has long fascinated physicists due to its puzzling and sometimes bizarre behaviors. Unlike classical systems, where particles follow predictable paths, quantum systems can act unpredictably, such as in superposition, where the particle is in multiple quantum states simultaneously.

As particles interact, these systems often evolve towards thermal equilibrium, where the system is equally likely to be found in any configuration with the same total energy. However, in some cases, quantum systems have been conjectured to resist this process and exhibit what’s known as many-body localization (MBL), where, even as particles are allowed to interact with each other, the energy and quantum information remain “trapped” in localized microscopic configurations rather than spreading out among all available configurations over time.

Understanding whether and why MBL happens can help scientists delve into the fundamental laws of nature and unlock new possibilities for technologies like quantum computing, where preventing the loss of quantum information is critical. For years, physicists have debated whether MBL could occur in systems with many interacting particles.

Now, in a published as an Editor’s Suggestion in Physical Review Letters,��CU ��«����Ƶ Physics Associate Professors Andrew Lucas and Rahul Nandkishore, along with graduate student Chao Yin, provide a first-of-its-kind mathematical proof showing how MBL can happen in a many-particle system.

“So I would say the basic result is that our work settles a critical point of principle,” stated Nandkishore. “I think it sort of settles it in a way that's a little more easily understandable and transparent.”

The Mystery of Many-Body Localization

While most quantum systems typically show thermalization, where the energy and particles in the system tend to spread out over space and time evenly, some quantum systems can resist this process and get stuck in a state known as thermalization. Instead of a mouse in a maze exploring every nook and cranny of its new environment, a “localized” mouse becomes stuck and only stays in one part of the maze.

The idea of localization was initially proposed��in 1958 by physicist Phillip Anderson, who showed that localization could occur in single particle systems. This work would later be cited in his 1977 Nobel Prize.

However, whether such localization can occur in many-body systems—systems with many interacting particles—has been a topic of heated debate in physics for the last twenty years.

Andrew Lucas explained, “People are interested in understanding systems where this will not happen. Even at infinite times, you watch the system, and it just doesn’t [thermalize]. Conventional statistical mechanics cannot describe it.”

Computer Science and Physics Meet to Study MBL

Previous studies on many-body localization mainly focused on simple, one-dimensional systems. At least one earlier proof claimed that localization can occur in such systems, but it was long and complex and relied on a plausible but unproven assumption.

The CU ��«����Ƶ team approached the problem from a different angle, taking inspiration from computer science.�� They studied a quantum system inspired by low-density parity check (LDPC) codes—mathematical tools commonly used in error correction for digital communication, such as in 5G communications. Error-correcting codes store information in a redundant way among many bits, such that experts can detect and correct such errors if a low enough number of bits have been corrupted.

Studying quantum systems based on LDPC codes allowed the researchers to bypass some of the complications of the previous research and provide a more accessible and rigorous demonstration of MBL.

Lucas explained the crux of their approach: “We are almost able to analyze this many-body problem as if it was a single-particle problem... because these error-correcting codes have a very complicated energy landscape.”

This energy landscape acts as a sort of maze, where the quantum system remains stuck near one of many low-energy configurations (or “code words”), much like how an error-correcting code helps data stay stable despite noise.

Using the LDPC code, the researchers showed that the system's quantum particles get “trapped” in specific configurations, unable to explore all possible states due to large energy barriers. Their proof demonstrated that for systems governed by LDPC-like structures, quantum particles remain trapped in localized states indefinitely, even in systems with many interactions.

Nandkishore highlighted this significance: “What we were able to do is rigorously establish this via a relatively short and understandable proof.”

This is the first time a fully rigorous proof has shown that many-body localization can occur in a system with many interacting particles and extensive configurations. Currently, the proof only works in an infinite dimensional geometry.

MBL Can Advance Other Fields

Understanding the dynamics of MBL is significant for many fields, including quantum computing, where the goal is to keep quantum states stable long enough to perform calculations. In a thermalizing system, quantum information would be quickly lost as particles spread energy and interact. However, in a localized system, quantum information could be preserved much longer, making error correction more feasible.

“This is the first construction where we have a mathematical proof for this localization phenomenon,” stated Lucas. “It’s suggestive that we can use ideas from error correction to learn things about physics and push the limits of concepts like statistical mechanics and thermodynamics."