Asymptotically exact solution of the non-Hermitian disordered interacting Hatano-Nelson chain

Abstract: We present an asymptotically exact solution of a paradigmatic non-Hermitian model: the disordered interacting fermionic Hatano-Nelson model, or equivalently, the non-Hermitian spin-1/2 XXZ model. We use a renormalization group method suited for disordered systems and show that non-Hermitian couplings are relevant perturbations to the Hermitian model, which ultimately leads to a quantum-to-classical crossover. The ground state of the model consists of a collection of strongly coupled pairs of spins of arbitrary size at random positions which, unlike the Hermitian case, do not form singlets, but a mixture of the singlet and the $M=0$ triplet state. As a result, the magnetic susceptibility in the $x,y$-directions becomes negative and diverges at a finite small temperature. Additionally, in sharp contrast to the $\ln(L)$ increase observed in disordered Hermitian chains, the entanglement entropy of a partition of size $L$ saturates for large $L$, as the strongly coupled pairs become classical and stop contributing at large length scales.