Hidden localization transitions in generalized Aubry-André models

TL;DR

The paper investigates hidden localization transitions in generalized Aubry–André models built from pairs of canonically conjugate operators $\hat X,\hat Y$ with $[\hat X,\hat Y]=i\omega$, revealing a transition that occurs in a rotated basis rather than in position or momentum. It introduces a rotated Aubry–André model with off-diagonal, quasiperiodic hopping, showing a localization transition at $\widetilde J=0$ or $\widetilde K=0$ and detailing how signals emerge in the rotated basis via IPR/NPR, while a remnant NPR signature appears in the position basis. At the critical point, the rotated Hamiltonian maps exactly to the lattice Hamiltonian of a massless Dirac fermion in a curved spacetime metric, establishing a surprising link between localization physics and analog gravity. The work also discusses a duality/triality among Aubry–André, Harper–Hofstadter, and the curved-space Dirac model, suggesting new avenues for quantum simulation of disorder, localization phenomena, and spacetime geometry effects in engineered lattices.

Abstract

Anderson localization is a phase transition between a metallic phase, where wavefunctions are extended and delocalized in space, and an insulating phase, where wavefunctions are completely localized. These transitions are driven by uncorrelated disorder or quasiperiodic disorder, e.g., in the case of the Aubry-André model. Here, I consider a family of Hamiltonians that generalizes the Aubry-André model obtained when position and momentum operators are replaced by an arbitrary couple of canonically conjugate operators. In these models, a hidden localization transition occurs between metallic/insulating phases with wavefunctions delocalized/localized with respect to one of the two canonically conjugate operators. If the canonically conjugate operators coincide with a linear combination of position and momentum, the phase transition is signaled by a zero in the normalized participation ratio in the usual position space. Surprisingly, I found that at the phase transition, this model Hamiltonian coincides with the lattice Hamiltonian of a massless Dirac fermion in a curved spacetime background, indicating an unexpected relation between many-body localization and analog gravity.

Figures (1)

• Figure 1: IPR and NPR of the Hamiltonian $\mathcal{H}_\text{RAA}$ with wavelength $\omega/2\pi\approx\Phi-1$, where $\Phi$ is the golden ratio and $\phi/2\pi\in\mathbb{R}-\mathbb{Q}$. The IPR and NPR are calculated in the basis $\ket{\widetilde{m}^-}$ of eigenmodes of the operator $\frac{1}{2}\left(\omega\hat{x}+\phi\right)-\hat{p}$. (a) Average IPR and NPR as a function of $\widetilde{J}$. (b) IPR as a function of energy and $\widetilde{J}$. (c) NPR as a function of energy and $\widetilde{J}$. The incommensurate wavelength is approximated by $p/q=F_{20}/F_{21}=6765/10946$ on a lattice of $N=F_{21}$ sites (where $F_n$ are the Fibonacci numbers). The phase transition occurs at $\widetilde{J}=0$ separating localized states with $\langle\widetilde{\text{IPR}}\rangle>0$ and $\langle\widetilde{\text{NPR}}\rangle\approx0$ for $\widetilde{J}>0$ and delocalized states with $\langle\widetilde{\text{IPR}}\rangle\approx0$ and $\langle\widetilde{\text{NPR}}\rangle>0$ for $\widetilde{J}<0$. The IPR and NPR calculated in the basis $\ket{\widetilde{m}^+}$ (not shown) are identical after reflection on the horizontal axis $\widetilde{J}\to-\widetilde{J}$.