Topological phase transition in fluctuating imaginary gauge fields

We investigate the exact solvability and point-gap topological phase transitions in non-Hermitian lattice models. These models incorporate site-dependent nonreciprocal hoppings J e± gₙ, facilitated by a spatially fluctuating imaginary gauge field igₙ ~x that disrupts translational symmetry. By employing suitable imaginary gauge transformations, it is revealed that a lattice characterized by any given gₙ is spectrally equivalent to a lattice devoid of fields, under open boundary conditions. Furthermore, a system with closed boundaries can be simplified to a spectrally equivalent lattice featuring a uniform mean field iḡ~x. This framework offers a comprehensive method for analytically predicting spectral topological invariance and associated boundary localization phenomena for bond-disordered nonperiodic lattices. These predictions are made by analyzing gauge-transformed isospectral periodic lattices. Notably, for a lattice with quasiperiodic gₙ= ln |λ cos 2π α n| and an irrational α, a previously unknown topological phase transition is unveiled. It is observed that the topological spectral index W assumes values of $-N$ or $+N$, leading to all N open-boundary eigenstates localizing either at the right or left edge, solely dependent on the strength of the gauge field, where λ<2 or λ>2. A phase transition is identified at the critical point λ≈2, at which all eigenstates undergo delocalization. The theory has been shown to be relevant for long-range hopping models and for higher dimensions.