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

TL;DR

This paper investigates how Krylov complexity reveals hidden structural features and phase distinctions in a non-Hermitian Ising chain with a complex magnetic field, connecting dynamical spread to static correlations.

Contribution

It introduces the use of Krylov spread to analyze non-Hermitian many-body systems and establishes a link between dynamical complexity and static correlation functions across phases.

Findings

Krylov spread uncovers three dynamical phases.

Correlation functions show oscillatory decay patterns.

Connection established between spread and static correlations.

Abstract

Krylov complexity measures the spread of an evolved state in a natural basis, induced by the generator of the dynamics and the initial state. 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 that remain hidden in correlation functions that are traditionally employed to determine the phase diagram. When the imaginary part of the spectrum of the non-Hermitian Hamiltonian is gapped, the system 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. 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.