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Physical review. B./Physical review. B

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

E. Medina-Guerra, I. V. Gornyi, Yuval Gefen

Published May 22, 2025• 13 citations
Publisher

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, 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.

DOI
10.1103/physrevb.111.174207

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In high-energy physics, confinement denotes the tendency of fundamental particles to remain bound together, preventing their observation as free, isolated entities. Interestingly, analogous confinement behavior emerges in certain condensed matter systems, for instance, in the Ising model with both transverse and longitudinal fields, where domain walls become confined into meson-like bound states as a result of a longitudinal field-induced linear potential. In this work, we employ the Ising model to demonstrate that Krylov state complexity--a measure quantifying the spread of quantum information under the repeated action of the Hamiltonian on a quantum state--serves as a sensitive and quantitative probe of confinement. We show that confinement manifests as a pronounced suppression of Krylov complexity growth following quenches within the ferromagnetic phase in the presence of a longitudinal field, reflecting slow correlation dynamics. In contrast, while quenches within the paramagnetic phase exhibit enhanced complexity with increasing longitudinal field, reflecting the absence of confinement, those crossing the critical point to the ferromagnetic phase reveal a distinct regime characterized by orders-of-magnitude larger complexity and display trends of weak confinement. Notably, in the confining regime, the complexity oscillates at frequencies corresponding to the meson masses, with its power-spectrum peaks closely matching the semiclassical predictions.

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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 zzz 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.
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.
We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.
We investigate a quantum dynamical phase transition induced by the competition between local unitary evolution and dissipation in a qubit chain with a strong, on-site Z2\mathbb{Z}_2Z2​ symmetry. While the steady-state of this evolution is always maximally-mixed, we show that the dynamical behavior of certain non-local observables on the approach to this steady-state is dictated by a quantum Ising model with a complex transverse-field (cTFIM). A crucial feature of our mapping to the cTFIM is an extensive number of local, weak, Z2\mathbb{Z}_2Z2​ symmetries possessed by the dynamics, allowing us to investigate these observables analytically and uncover a dynamical phase transition as the relative rate of unitary evolution and dissipation is tuned. We show that the weak-dissipation limit corresponds to a cTFIM with a large magnitude of the imaginary transverse-field, for which the many-body "ground-state" (with smallest real eigenvalue) is gapless, exhibiting quasi-long-range correlations of the local magnetization with a continuously-varying exponent. Correspondingly, the dynamics of the non-local observables show oscillatory behavior with an amplitude decaying exponentially in time. The strong-dissipation limit corresponds to a gapped ferromagnetic phase of the cTFIM, and non-local observables show exponential decay on the approach to equilibrium. This transition in (1+1)-dimensions has a peculiar, "two-sided" nature appearing as either first- or second-order depending on the phase from which the transition is approached, an analytic result which is corroborated by numerical studies. In higher dimensions, we present a field-theoretic understanding of the first-order nature of this transition, when approaching from the ferromagnetic phase of the cTFIM, though the nature of the phase with large imaginary transverse-field remains to be understood.
We investigate phase transitions in the extended Su-Schrieffer-Heeger (SSH) model with next-nearest-neighbor hoppings and an imaginary staggered chemical potential. In the presence of small non-Hermiticity, exceptional points emerge in pairs from the gap-closing momenta near the topological phase boundaries of the Hermitian limit. Utilizing the Krylov spread complexity and entanglement entropy, we analyze two dynamical protocols: (i) preparing the non-Hermitian ground state via a unitary transformation, and (ii) evolving the system under the non-Hermitian Hamiltonian. We show that the spread complexity, and long-time spread complexity as well as entanglement entropy can effectively signal phase transitions in the first and second protocols, respectively. To unravel the detailed structure of the transitions, we introduce the momentum-resolved complexity that identifies the characteristic modes and tracks their evolution with the driving parameter. In the regime where the system possesses a purely imaginary spectrum, we further identify dynamical phases based on the saturation behavior of the spread complexity. The entanglement entropy is also found to exhibit similar saturation behavior, thereby providing a more experimentally accessible probe of the dynamical phases.
We characterize the Many-Body Localization (MBL) phase transition using the dynamics of spread complexity and inverse participation ratio in the Krylov space starting from different initial states. Our analysis of the disordered Heisenberg spin-1/2 chain unravels that the ergodic-to-MBL transition can be determined from the transition of the pre-saturation peak in the thermofield double state (TFD) spread complexity. On the other hand, if an initially ordered state or a superposition of a small number of such states is chosen, then the saturation value of spread complexity and Krylov inverse participation ratio (KIPR) can distinguish the ergodic phase from the integrable phases, with no sharp difference between the integrable phases. Interestingly, the distinction between the disorder-free integrable and the MBL integrable phase is established by the spread complexity study of random states chosen from unitary and orthogonal Haar ensembles. We also study the complexity dynamics by coupling the system to a bath, which shows distinctive profiles in different phases. A stretched exponential decay of KIPR is observed when the MBL system is connected to the bath, with the decay starting at an earlier time for a greater value of environmental dephasing. Our work sheds light on the efficacy of Krylov space dynamics in understanding phase transitions in quantum many-body systems.
In high-energy physics, confinement denotes the tendency of fundamental particles to remain bound together, preventing their observation as free, isolated entities. Interestingly, analogous confinement behavior emerges in certain condensed matter systems, for instance, in the Ising model with both transverse and longitudinal fields, where domain walls become confined into meson-like bound states as a result of a longitudinal field-induced linear potential. In this work, we employ the Ising model to demonstrate that Krylov state complexity--a measure quantifying the spread of quantum information under the repeated action of the Hamiltonian on a quantum state--serves as a sensitive and quantitative probe of confinement. We show that confinement manifests as a pronounced suppression of Krylov complexity growth following quenches within the ferromagnetic phase in the presence of a longitudinal field, reflecting slow correlation dynamics. In contrast, while quenches within the paramagnetic phase exhibit enhanced complexity with increasing longitudinal field, reflecting the absence of confinement, those crossing the critical point to the ferromagnetic phase reveal a distinct regime characterized by orders-of-magnitude larger complexity and display trends of weak confinement. Notably, in the confining regime, the complexity oscillates at frequencies corresponding to the meson masses, with its power-spectrum peaks closely matching the semiclassical predictions.