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We investigate the spectral topology of one-dimensional lattices where thenonreciprocal hoppings within the nearest $r_d$ neighboring sites are the same.For the purely off-diagonal model without onsite potentials, the energyspectrum of the lattice under periodic boundary conditions (PBCs) forms aninseparable loop that intertwines with itself in the complex energy plane andis characterized by winding numbers ranging from 1 up to $r_d$. Thecorresponding spectrum under open boundary conditions (OBCs), which is real inthe nearest neighboring model, will ramify and take the shape of an$(r_d+1)$-pointed star with all the branches connected at zero energy. If wefurther introduce periodic onsite modulations, the spectrum will graduallydivide into multiple separable bands as we vary the parameters. Mostimportantly, we find that a different kind of band gap called loop gap canexist in the PBC spectrum, separating an inner loop from an outer one with eachcomposed by part of the spectrum. In addition, loop structures also exist inthe OBC spectra of systems with onsite potentials. We further study thelattices with power-law decaying long-range nonreciprocal hopping and foundthat the intertwined loops in the PBC spectrum will be untangled. Finally, wepropose an experimental scheme to realize the long-range nonreciprocal modelsby exploiting electrical circuits. Our work unveils the exotic spectraltopology in the long-range nonreciprocal lattices.

Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices