We investigate the exact solvability and point-gap topological phase transitions in non-Hermitian lattice models. These models incorporate site-dependent nonreciprocal hoppings $J{e}^{\pm g_n}$, facilitated by a spatially fluctuating imaginary gauge field $i{g}_n\widehat{x}$ that disrupts translational symmetry. By employing suitable imaginary gauge transformations, it is revealed that a lattice characterized by any given $g_n$ 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 $i\overline{g}\widehat{x}$. This framework offers a comprehensive method for analytically predicting spectral topological invariance and associated boundary localization phenomena for bond-disordered nonperiodic lattices, based on gauge-transformed isospectral periodic lattices. Notably, for a lattice with quasiperiodic $g_n=ln|\lambda cos2\pi \alpha n|$ and an irrational $\alpha$, 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 $\lambda <2$ or $\lambda >2$. A phase transition is identified at the critical point $\lambda \approx 2$, at which all eigenstates undergo delocalization. The theory has been shown to be relevant for long-range hopping models and for higher dimensions.

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• Edge states
• Localization
• Synthetic gauge fields
• Topological phase transition
• Topological phases of matter
• Topological materials
• Lattice gauge theory