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We study the non-Hermitian (NH) Toda model coupled to fermions throughsoliton theory techniques and the realizations of the pseudo-chiral andpseudo-Hermitian symmetries. The interplay of non-Hermiticity, integrability,nonlinearity, and topology significantly influence the formation and behaviorof a continuum of bound state modes (CBM) and extended waves in the localizedcontinuum (ELC). The non-Hermitian soliton-fermion duality, the complex scalarfield topological charges and winding numbers in the spectral topology areuncovered. The Hermitian bound states/solitons lie on the unit circle $|z|=1$defined by the uniformization parameter $z \in \IC \backslash \{0\}$ related tothe complex energy eigenvalue, whereas the non-Hermitian bound states/solitonslie on the complex plane such that $|z| \neq 1$. The biorthogonal Majorana zeromodes, dual to the NH Toda solitons with topological charges $\pm 1$, appear atthe complex-energy point gap and are pinned at zero energy. The regions of$\IC\backslash \{0\}$ with real eigenvalues are uncovered, and these come inreal pairs $\pm \l_1\, (\l_1 \in \IR)$ preserving the pseudo-chiral symmetry.Our findings improve the understanding of exotic quantum states, but also pavesthe way for future research in harnessing non-Hermitian phenomena fortopological quantum computation, as well as the exploration of integrabilityand NH solitons in the theory of topological phases of matter.the theory oftopological phases of matter.

Biorthogonal Majorana zero modes, extended waves in continuum of bound states and non-Hermitian Toda soliton-fermion duality