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

TL;DR

This work investigates Krylov spread in a non-Hermitian 1D Ising model with a complex transverse field, showing that Krylov dynamics reveals dynamical phases not captured by conventional correlation functions. By exactly diagonalizing the non-Hermitian Hamiltonian and constructing the right Bogoliubov vacuum, the authors derive analytic expressions for the zz spin correlation function and the Krylov spread, connecting static correlations to dynamical spreading through SU(2) per-k structure and elliptic integrals. A key finding is a third-order dynamical transition in the Krylov spread density across the gapped/gapless boundary when evolving from the JW vacuum to the non-Hermitian vacuum, and a finite-time Krylov fidelity that refines the gapped region into distinct dynamical regimes with characteristic timescales. In the gapless phase, the long-time Krylov behavior matches the unitary dynamics, while in the gapped phase the Krylov fidelity reveals multiple decay channels, highlighting Krylov spread as a versatile probe of hidden dynamical transitions in monitored/non-Hermitian quantum systems.

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.

Figures (6)

• Figure 1: Examples of the real, $E(k)$ (a), and the imaginary, $\Gamma(k)$ (b), parts of the spectrum \ref{['eq:x09']} satisfying the convention $\mathfrak{Im}\,\Gamma(k) \leq 0$ for all $k$ in units of $J$, exemplifying cases (I)-(IV) discussed in the main text. (I): $h= 0.5$, $\gamma = 1$ (black curves). (II): $h = 1/2$, $\gamma = 4$ (blue). (III): $h = 1.5$, $\gamma = 4$ (purple). (IV): $h = 0.2$, $\gamma = \gamma_c(h) = 4\sqrt{1-h^2} \approx 3.91$ (red). Panel (c) shows the $h$-$\gamma$ plane with the four regions corresponding to cases (I)-(IV) indicated. The green dashed line separates regions (II) and (III) of the phase where $\Gamma(k)$ is gapped. The critical curve $\gamma=\gamma_c(h)$ corresponding to case (IV) is shown in red.
• Figure 2: Schematic phase diagram displaying the asymptotic behavior of $\texttt{C}^{zz}(x)$ calculated in the non-Hermitian vacuum $\ket{\Omega}$, in the gapped and gapless regions [cf. Fig. \ref{['fig:1']}] separated by the critical curve $\gamma = \gamma_c(h)$. The functions $\mu(x)$ and $\chi(x)$ are given by Eqs. \ref{['eq:mu']} and \ref{['eq:chi1']}, respectively. The correlation length $\xi$ is given by \ref{['xi-z1']} and $\xi_+=\lim_{\gamma \to 0}\xi$. The correlation function is continuous throughout the subregion $0\leq \gamma <\gamma_c(h)$ and the approach to the segment $0\leq h <J$ at $\gamma = 0$ must be understood as a limiting procedure at the level of the non-Hermitian Hamiltonian \ref{['eq:x1']}; that is, the sign-convention for the eigenvalues of $H$ is chosen, and then, one takes the limit $\gamma \rightarrow 0$. As a consequence, $\texttt{C}^{zz}(x)$ does not display an exponential decay that would correspond to the standard (Hermitian, $\gamma\equiv 0$) ferromagnetic phase for $0\leq h<J$.
• Figure 3: Krylov spread complexity density \ref{['eq:x68']} for $\ket{0}$ evolving to $\ket{\Omega}$ via unitary dynamics ($h$ and $\gamma$ are in units of $J$).
• Figure 4: Schematic phase diagram defined by the fidelity \ref{['eq:fi2']}. The white and red regions correspond to the gapless phase, where $\Gamma(\pm q) = 0$, and do not have any associated characteristic time of the form given by Eq. \ref{['eq:time']}. The dynamical phases and their characteristic times $t^*$ are: 1 (gray) characterized by $t^*\simeq \gamma^{-1}$; 2 (green) by $t^*\simeq (4\abs{\Gamma(\bar{k})})^{-1}$; and 3 (blue) by $t^* \simeq \abs{\gamma_Y}^{-1}$, where $\pm \bar{k}$ are the modes given by Eq. \ref{['eq:kbar']}.
• Figure 5: Contours of integration used to evaluate the contractions (a) $\expval{\texttt{A}_0\texttt{A}_{x}}$ and (b) $\expval{\texttt{A}_0\texttt{B}_{\pm x}}$ in the gapped phase.