Operator growth and Krylov construction in dissipative open quantum systems

TL;DR

The paper tackles how operator growth and chaos manifest in open quantum systems by extending Krylov-based methods to Lindblad dynamics. It contrastively analyzes Lanczos and Arnoldi approaches in a TFIM with boundary damping and bulk dephasing, finding that Lanczos breaks down under non-Hermitian evolution while Arnoldi remains informative, distinguishing integrable from chaotic behavior and encoding non-Hermiticity through imaginary diagonal elements of the Hessenberg form. The key contribution is demonstrating that Arnoldi iteration is a robust, physically meaningful tool for studying open-system operator growth, even with environmental decoherence. The work suggests broad applicability to other open models (like open XXZ or open SYK) and motivates connections to Krylov complexity and non-Hermitian/PT-symmetric physics in dissipative quantum dynamics.

Abstract

Inspired by the universal operator growth hypothesis, we extend the formalism of Krylov construction in dissipative open quantum systems connected to a Markovian bath. Our construction is based upon the modification of the Liouvillian superoperator by the appropriate Lindbladian, thereby following the vectorized Lanczos algorithm and the Arnoldi iteration. This is well justified due to the incorporation of non-Hermitian effects due to the environment. We study the growth of Lanczos coefficients in the transverse field Ising model (integrable and chaotic limits) for boundary amplitude damping and bulk dephasing. Although the direct implementation of the Lanczos algorithm fails to give physically meaningful results, the Arnoldi iteration retains the generic nature of the integrability and chaos as well as the signature of non-Hermiticity through separate sets of coefficients (Arnoldi coefficients) even after including the dissipative environment. Our results suggest that the Arnoldi iteration is meaningful and more appropriate in dealing with open systems.

Figures (10)

• Figure 1: A schematic diagram for the open system dynamics. In the L.H.S, we have the system plus the environment. R.H.S shows the system dynamics only, with operator growing within system Hilbert space under the action of system Hamiltonian $H_\mathrm{S}$ and the Lindblad operators $\sum_j \alpha_j L_j$, encoding the interaction with the environment.
• Figure 2: Growth of $b_n$'s of the operator $Z_3$ in (a) integrable and (b) chaotic limits by implementing the Lanczos algorithm. We individually compare the growth for (c) $\alpha =0$ (d) $\alpha = 0.15$ in both integrable (marked with diamond) and chaotic limits (marked with triangle). We choose lattice size $N=6$ and the bulk dephasing (on all sites) parameter is chosen as $\gamma = 0.1$. The blue curves are for closed system i.e., for $\alpha = 0$ and $\gamma = 0$.
• Figure 3: Growth of Arnoldi coefficients $h_{n,n-1}$ of the operator $Z_3$ in (a) integrable and (b) chaotic limits for bulk dephasing $\gamma = 0.1$ using Arnoldi iteration. Growth of the difference $|h_{n,n-1} - h_{n-1,n}|$'s of the operator $Z_3$ in (c) integrable and (d) chaotic limits for bulk dephasing $\gamma = 0.1$ using Arnoldi iteration. The blue curves are for closed system i.e., for $\alpha = 0$ and $\gamma = 0$. We choose $N=6$.
• Figure 4: The Liouvillian ($\alpha = \gamma = 0)$ in tridiagonal form in (a) integrable and (b) chaotic limit. The Lindbladian in upper-Hessenberg form in (c) integrable and (d) chaotic limit for boundary Lindblad operators with $\alpha = 0.1$. We choose the lattice size $N=6$.
• Figure 5: Growth of $|h_{n,n}|$'s of the operator $Z_3$ in (a) integrable and (b) chaotic limits for bulk dephasing $\gamma = 0.1$ using Arnoldi iteration. ($\gamma=0$, for $\alpha=0$) (c) integrable and (d) chaotic for boundary Lindblad $\alpha = 0.01$. We choose $N=6$. The blue curves are for closed system i.e., for $\alpha = 0$ and $\gamma = 0$.