Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field

Krylov complexity measures the spread of an evolved state in a natural basis, induced by the generator of the dynamics and the initial state, and serves as an effective probe for detecting various properties of many-body systems. Here, we study the spread in Hilbert space of the state of an Ising chain subject to a complex-valued transverse magnetic field, initialized in a trivial product state with all spins pointing down. We demonstrate that Krylov spread reveals structural features of many-body systems, which remain hidden in correlation functions that are traditionally employed to determine phase diagrams. When the imaginary part of the spectrum of the non-Hermitian Hamiltonian is gapped, the system's state asymptotically approaches the non-Hermitian Bogoliubov vacuum for this Hamiltonian. We find that the spread of this evolution unravels three different dynamical phases based on how the spread reaches its infinite-time value. We also show that the second derivatives of the spread with respect to the real and imaginary components of the complex magnetic field exhibit algebraic divergence at the transition between gapped and gapless imaginary spectra. Furthermore, we establish a connection between the Krylov spread and the static correlation function for the $z$ components of spins in the underlying non-Hermitian Bogoliubov vacuum, providing a full analytical characterization of correlations across the phase diagram. Specifically, for a gapped imaginary spectrum in a finite magnetic field, we find that the correlation function exhibits an oscillatory behavior that decays exponentially in space. Conversely, for a gapless imaginary spectrum, the correlation function displays an oscillatory behavior with an amplitude that decays algebraically in space; the underlying power law depends on the manifestation of two exceptional points within this phase. Our findings, showing the potential of the Krylov spread in exposing dynamical transitions, motivate further studies of Krylov's complexity in open many-body systems, including monitored ones.