Quantum particles in complex environments have only a few options: spread freely as waves, become trapped in place, or settle into a delicate critical state with a multifractal, self-similar wave function. Combinations of these three behaviors yield seven fundamentally different localization phases of matter—but until now, no single theory could account for all of them.
A team led by Prof. Xiong-Jun Liu at the International Center for Quantum Materials and School of Physics, Peking University, together with collaborators, has now closed that gap. The researchers established the first complete, exactly solvable framework that unifies all seven fundamental localization phases in quasiperiodic systems—structures that are perfectly ordered yet not strictly periodic. The framework also points to a concrete route for realizing the predicted physics with ultracold atoms. The study was recently published online in Science Bulletin [DOI: 10.1016/j.scib.2026.03.002].
The problem traces back to Anderson localization, the foundational discovery that disorder can halt quantum diffusion. For ordinary random disorder, the nonlinear sigma model has long supplied a universal theoretical language; for quasiperiodic systems, however, no comparable framework for the localization transition had been established. Deterministic yet lacking translational periodicity, quasiperiodic systems support a much richer set of phases—extended, localized and critical states can each appear as a pure phase, or coexist in pairs or all three at once, producing seven fundamental localization phases. Critical states, with their distinctive multifractal structure and unusual dynamics, have been a focus of Liu's group, with prior results published in Phys. Rev. Lett., Nature Physics and other journals. Yet an exactly tractable model containing all localization phases had remained elusive.
The new framework is built on a generic class of spin-1/2 quasiperiodic systems that unifies the major existing one-dimensional spinful and spinless quasiperiodic models. Within it, the team proved three theorems. First, when chiral symmetry is preserved, mobility edges disappear—revealing the symmetry origin of the distinction between pure and coexisting phases. Second, a newly identified mechanism generates critical states in spinful systems, greatly expanding the routes for engineering critical localization beyond those available in spinless ones. Third, when the hopping-coupling matrix is singular, the spinful system reduces to a spinless quasiperiodic chain solvable in closed form—giving a systematic recipe for constructing exactly solvable models.
“Building on these results, we built two new exactly solvable models,” said Xin-Chi Zhou, the first author of the paper. “The first is a spin-selective quasiperiodic model that realizes all three basic types of mobility edges separating extended-localized, extended-critical and localized-critical states. The second is a quasiperiodic optical Raman lattice that yields the first complete phase diagram containing all fundamental localization phases, with closed-form solutions at multiple exactly solvable points.”
“This framework not only unifies what had been a fragmented picture, but also predicts new localization physics that can be tested directly in experiment,” said Prof. Xiong-Jun Liu. “Related experiments with ultracold atoms in optical Raman lattices are already under way, and the framework also lays a foundation for extending the study of localization to systems with SU(N) symmetry, higher dimensions, and many-body interactions.”
Science Bulletin
Computational simulation/modeling