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# Metal-insulator phase transition in a non-Hermitian Aubry-Andr\'e-Harper Model

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### Abstract

Non-Hermitian extensions of the Anderson and Aubry-Andr\'e-Harper models are attracting a considerable interest as platforms to study localization phenomena, metal-insulator and topological phase transitions in disordered non-Hermitian systems. Most of available studies, however, resort to numerical results, while few analytical and rigorous results are available owing to the extraordinary complexity of the underlying problem. Here we consider a parity-time ($$\mathcal{PT}$$) symmetric extension of the Aubry-Andr\'e-Harper model, undergoing a topological metal-insulator phase transition, and provide rigorous analytical results of energy spectrum, symmetry breaking phase transition and localization length. In particular, by extending to the non-Hermitian realm the Thouless$$^{\prime}$$s result relating localization length and density of states, we derive an analytical form of the localization length in the insulating phase, showing that -- like in the Hermitian Aubry-Andr\'e-Harper model-- the localization length is independent of energy.

### Most cited references33

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### Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields

(1976)
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### Anderson localization of a non-interacting Bose-Einstein condensate.

(2008)
Anderson localization of waves in disordered media was originally predicted fifty years ago, in the context of transport of electrons in crystals. The phenomenon is much more general and has been observed in a variety of systems, including light waves. However, Anderson localization has not been observed directly for matter waves. Owing to the high degree of control over most of the system parameters (in particular the interaction strength), ultracold atoms offer opportunities for the study of disorder-induced localization. Here we use a non-interacting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice-a system that features a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. This controllable system may be used to investigate the interplay of disorder and interaction (ref. 7 and references therein), and to explore exotic quantum phases.
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### Bandwidths for a quasiperiodic tight-binding model

(1983)
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### Author and article information

###### Journal
09 August 2019
1908.03371