Localization via Quasi-Periodic Bulk-Bulk Correspondence

Abstract

We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal insulator transition (MIT). By constructing quasi-periodic transfer matrix equations from the limit of rational approximate projected Green's functions, we relate results from $\text{SL}(2,\mathbb{R})$ co-cycle theory (transfer matrix eigenvalue scaling) to consequences of rational band theory. This reduction links the eigenfunction localization of the MIT to the chiral edge modes of the Hofstadter Hamiltonian, implying the localized phase roots in a topological "bulk-bulk" correspondence, a bulk-boundary correspondence between the 1D AAH system (boundary) and its 2D parent Hamiltonian (bulk). This differentiates quasi-periodic localization from Anderson localization in disordered systems. Our results are widely applicable to systems beyond this paradigmatic model.

arXiv comment6+12 pages, 2+2 figures, 1 table. Companion paper to arXiv:2109.13933 concerning different themes and new results. Supplementary materials shared with arXiv:2109.13933, but kept for readability and completeness. arXiv admin note: substantial text overlap with arXiv:2109.13933

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• Critical phase dualities in 1D exactly-solvable quasiperiodic models
• Renormalization-Group Theory of 1D quasiperiodic lattice models with commensurate approximants
• Theory of polarization textures in crystal supercells
• The fundamental localization phases in quasiperiodic systems: A unified framework and exact results
• Hidden self-duality and exact mobility edges in quasiperiodic network models
• Exact non-Hermitian mobility edges and robust flat bands in two-dimensional Lieb lattices with imaginary quasiperiodic potentials
• Emergent multi-loop nested point gap in a non-Hermitian quasiperiodic lattice
• Localization and mobility edges in non-Hermitian continuous quasiperiodic systems
• Non-equilibrium dynamics of localization phase transition in the non-Hermitian Disorder-Aubry-André model
• Quasiperiodicity protects quantized transport in disordered systems without gaps
• Engineering many-body quantum Hamiltonians with non-ergodic properties using quantum Monte Carlo
• Generic non-Hermitian mobility edges in a class of duality-breaking quasicrystals