Opening Krylov space to access all-time dynamics via dynamical symmetries

Abstract

Solving short and long time dynamics of closed quantum many-body systems is one of the main challenges of both atomic and condensed matter physics. For locally interacting closed systems, the dynamics of local observables can always be expanded into (pseudolocal) eigenmodes of the Liouvillian, so called dynamical symmetries. They come in two classes - transient operators, which decay in time and perpetual operators, which either oscillate forever or stay the same (conservation laws). These operators provide a full characterization of the dynamics of the system. Deriving these operators, apart from a very limited class of models, has not been possible. Here, we present a method to numerically and analytically derive some of these dynamical symmetries in infinite closed systems by introducing a naturally emergent open boundary condition on the Krylov chain. This boundary condition defines a partitioning of the Krylov space into system and environment degrees of freedom, where non-local operators make up an effective bath for the local operators. We demonstrate the practicality of the method on some numerical examples and derive analytical results in two idealized cases. Our approach lets us directly relate the operator growth hypothesis to thermalization and exponential decay of observables in chaotic systems.

Figures (5)

• Figure 1: In Krylov space, dynamics can be mapped to a single particle hoping on a 1D half chain. We split the chain into a system (left) where the hopping coefficients are exactly known and an environment (right). The left part of the chain corresponds to simple operators supported on $k$-local Pauli strings while the environment correspond to complex non-local operators. We wish to describe the dynamics solely based on the system degrees of freedom, but we need to allow for flow into the environment and back which leads to non-hermitian boundary conditions on the left part of the chain.
• Figure 2: Krylov chain dynamics for the ideal cases of $b_n=n$ (bottom) and $b_n=\sqrt n$ (top). We start with $\varphi_n = \delta_{0,n}$ and show $\varphi_n(t)$ for different $t$. Each color corresponds to a particular time, dark is early, yellow is late. Colors are exact results and dashed lines correspond to the open Krylov chain \ref{['eq:open_chain']}. The insets show the time evolution of $\mathop{\mathrm{Tr}}\nolimits(O_1 O_0(t))$. The orange line is obtained with our newly proposed boundary condition while the blue line used standard Dirichlet boundary condition $\varphi_{21}=0$, in the latter case the wavepacket bounces of the edge of the Krylov chain, leading to unphysical revivals.
• Figure 3: Top: Liouvillian spectrum of the ideal case $b_n=n$ (left) and Liouvillian spectrum of the ideal case $b_n=\sqrt n$ (right). Bottom: distribution of $\text{Re}(\omega)$ for the Liouvillian with truncated Krylov chain (orange) or open boundary condition as defined in eq \ref{['eq:open_chain']} (blue). In the case of the simple truncated chain, $\mathcal{L}$ is hermitian and its spectrum is real. When introducing the boundary condition, the spectrum becomes complex but the distribution of the real parts is almost identical.
• Figure 4: (a, b) Time evolution of the initial operator $O_0$ for the $XXZ$ model and chaotic chain respectively. In the XXZ model, any operator that has overlap with the dynamical symmetry will oscillate perpetually. In the chaotic model, all observables decay because there are no perpetual (real valued) dynamical symmetries. These results are obtained by solving for $\varphi_0(t)$ by diagonalizing the open Krylov chain \ref{['eq:open_chain']}. For reference, the dashed lines show results for tensor networks simulations with $N=40$ spins. The orange line is obtained by truncating the operators in the Pauli string space in addition to truncating the Krylov chain. (c, d) Spectrum of the open Krylov chain \ref{['eq:open_chain']} for the $XXZ$ model and chaotic chain. In the XXZ model, we recover the dynamical symmetry at $\textup{re}(\omega)=12$ (black). In the chaotic chain, the grey horizontal line indicates $\textup{im}(\omega)=-2\lambda$ where $\lambda$ is the growth rate of the Lanczos coefficients. The color shows $\psi_0$, the first coefficient of the dynamical symmetry in the $O_n$ basis. Darker markers correspond to more local operators.
• Figure 5: Quench from $\ket{\psi_0}=2^{N/2}\left(\ket{0}+\ket{1}\right)^{\otimes N}$ in the XXZ model. $Q_5$ has overlap with the dynamical symmetry and therefore exhibit long time oscillations. These simulations where performed using our open Krylov chain technique. For comparison, the grey line in the second plot shows results for a truncated Krylov chain without using our boundary conditions. $L$ correspond to the length of the Krylov chain.