• *Physical* 17, 24

The theoretical work sheds light on why some quantum many-body systems become locally trapped and fail to reach thermal equilibrium – a phenomenon known as many-body localization.

A fundamental principle of statistical physics is the ergodic hypothesis, which states that macroscopic systems eventually explore all allowable microscopic configurations. Its far-reaching consequences include the impossibility of perpetual motion and the arrow of time. However, not all physical systems are ergodic. Systems with microscopic heterogeneity can become trapped in certain configurations, with marked consequences for their macroscopic properties. In both classical and quantum systems, the theoretical description of these non-ergodic systems is incomplete. Simple models exhibit complex phases with varying degrees of non-ergodicity, but it is unclear how much they teach us about real-world systems. Two years ago, Alan Morningstar, now at Global Quantitative Strategies in Chicago, and colleagues identified several important regimes in inhomogeneous quantum systems and reconciled the tensions between theoretical predictions and numerical observations. [1]. Since then, his research has stimulated probes of non-ergodicity and revitalized attempts to connect quantum and classical glasses.

In 1958, physicist Philip Anderson showed that in a sufficiently disordered potential landscape, a single quantum particle, such as an electron, becomes spatially localized within a small region in the neighborhood of its initial position. [2]. What about multiple particles in the same potential landscape? It is conceivable that they could disperse, opening new paths to escape location. But if, despite these pathways, the system remains spatially localized, it is said to exhibit many-body localization (MBL). MBL systems constitute a non-ergodic phase of matter. They are perfect insulators at non-zero temperatures.

Over the past 70 years, a growing body of theoretical work has shown that in one-dimensional systems, the MBL is stable and resists thermalization. In higher dimensions, large and rare regions with low spatial disorder destabilize the MBL. These regions behave like poor heat baths, with low specific heats and granularity in the heat they can absorb. However, they can become better heat baths by balancing degrees of freedom located at their boundaries and absorbing them. Only in one dimension can this runaway process (dubbed quantum avalanche) be stopped, allowing the possibility of stable MBL.

Attempts to empirically confirm the MBL and the broader dynamical phase diagram of disordered quantum systems have been surrounded by controversy. Quantum simulators can mimic the idealized dynamic experiment of letting particles enter a disordered potential and observing what happens, but they are limited to relatively short times. These experiments observe a regime similar to MBL in all dimensions. On the other hand, numerically exact studies are restricted not to time but to a small number of particles (about ten particles at 20 locations). The results they produce are inconsistent with a simple two-phase theoretical framework: an MBL phase with strong disorder and an ergodic phase with weak disorder. Divergent opinions subsequently emerged. In 2020, Jan Šuntajs of the Jožef Stefan Institute, Slovenia, and his collaborators argued that the MBL phase does not exist; In sufficiently large systems, particles always move spatially [3].

Morningstar and colleagues elucidated how empirical findings can be reconciled with theory. They emphasized that the phase diagram has three regimes: an ergodic phase, an MBL phase (where even large systems remain non-ergodic) and, significantly, an intervening “MBL regime” (where accessible systems are non-ergodic, but systems large enough are ergodic). . Numerical simulations probed the transition from the ergodic regime to the MBL regime. At this intersection, number systems are too small to host the poor baths that generate quantum avalanches. Instead, the crossing is determined by an entirely distinct physics – the proliferation of many-body resonances [4, 5]. Loosely speaking, in a many-body resonance, multiple particles spread across multiple locations alternate between two configurations. Resonances detach particles and can cause delocalization. In the crossover region, the probability of forming a resonance at the top of a localized configuration of particles changes dramatically, from being a rare event in the MBL regime to being common across all length scales in the ergodic phase.

If number systems are too small to accommodate poor baths, how can one probe the crossover to the true MBL phase? The main technical innovation developed by Morningstar, together with Dries Sels [6] at New York University, is reformulating the criteria for quantum avalanches: if, when connected to a perfect bath, the equilibration time of a localized piece of the system is too long, then it is not possible for a poor bath to trigger the quantum avalanche. balance. Importantly, this perfect bath equilibrium time is numerically measurable and provides a lower limit to the strength of disorder required for the MBL phase. Morningstar and Sels found that the MBL phase requires much greater disorder strengths than previously predicted: at least 7 times the strength at which the MBL regime develops in models studied numerically.

Because the relevant time scales are exponentially sensitive to the strength of the disturbance, the investigators’ results suggest that the MBL phase is physically inaccessible. In ultracold atomic experiments investigating MBL [7]for example, the dynamics of the MBL phase and the MBL regime become distinct only after approximately 10^{18} seconds. More than twice the age of the Universe!

The work of Morningstar and colleagues and others has resolved several puzzles about the dynamical phase diagram of disordered systems. Looking to the future, the characterization of the physically accessible MBL regime is critical. One issue concerns their dynamic signatures. For large sizes, recent work predicts a large hierarchy of timescales for different microscopic configurations to reach local thermal equilibrium. This property is reminiscent of classic glasses, so exploring the precise connection between these glasses and MBL is an exciting front.

Another application is in optimization problems. Solutions to these problems can be reformulated as searches for the ground states of disordered many-particle systems. Heuristic quantum algorithms that search for such ground states may go through the MBL regime before finding them. In the MBL regime, the system tunnels between resonant configurations; It remains to be seen whether access to these resonances makes heuristic quantum algorithms faster than their classical counterparts.

## References

- A. Morningstar
*and others.*“Avalanches and many-body resonances in localized many-body systems,” Physical. Rev.**105**174205 (2022). - PW Anderson, Local Moments and Localized States, Nobel Lecture, Nobel Prize in Physics (1977).
- J. Šuntajs
*and others.*“Quantum chaos defies many-body localization,” Physical. Rev.**102**062144 (2020). - B. Villalonga and B. K. Clark, “Characterizing the many-body location transition through correlations,” arXiv:2007.06586.
- P.J.D. Crowley and A. Chandran, “A Constructive Theory of Numerically Accessible Many Bodies Localized at Thermal Crossing,” SciPost Physics.
**12**201 (2022). - D. Sels, “Bath-induced delocalization in interacting disordered spin chains,” Physical. Rev.
**106**L020202 (2022). - J.-Y. Choi
*and others.*“Exploring the location transition of many bodies in two dimensions,” Science**352**1547 (2016).