On the origin of exponential operator growth in Hilbert space

Abstract

The question of thermalization in quantum many-body systems has long been studied through the properties of matrix elements of operators corresponding to local observables. More recently, the focus has shifted to the dynamics of operators, which lead to seminal works proposing universal bounds on the rate of operator growth. In this work, we unify these two approaches: we show that exponential operator growth in Hilbert space, as measured by Krylov complexity, is governed by an exponential off-diagonal decay of the operator matrix elements in the system eigenbasis. When this decay is algebraic or slower, the growth rate saturates the universal bound, thereby establishing a microscopic origin of operator growth which is independent of chaos, dimensionality or the presence of many-body interactions.

Figures (3)

• Figure 1: Illustration of (a) power-law, (c) exponential and (e) Gaussian decay of off-diagonal matrix elements of an operator in the energy eigenbasis of a random-matrix ensemble. (b), (d) and (f) The corresponding Lanczos sequence averaged over the ensemble tang2023operator. The red dashed line indicates the maximal growth as in \ref{['alpha-bound']}.
• Figure 2: Plots of Lanczos sequence for (a) $\hat{O} =\hat{x}^{100}$ in a harmonic oscillator $V(x)=x^2/2$ (blue dots); the dashed line corresponds to a fictional operator with approximate matrix elements as in \ref{['xk-gauss-approx']} indicating Gaussian decay of matrix elements. (b) $\hat{O}=\hat{x}$ in a quartic anharmonic oscillator $V(x)=x^4$ (yellow) - the artificial transition in the growth rate at $n\sim 25$ is due to the limit in numerical precision while computing position matrix elements; blue dots correspond to a fictional operator with (sub)exponentially decaying matrix elements as in \ref{['landau-mat']}. (c) $\hat{O}=\hat{x}$ in a one-dimensional infinite box (length $L=10$) at various temperatures. Inset: Plot of the growth rate against temperature indicating maximal growth.
• Figure 3: (a) Lanczos sequence for $\hat{O}=\hat{x}$ in a stadium billiard (dots) with area $1$ and the parameter length-radius ratio $a/R=1.0$ as in camargo2024spectral at various temperatures. The dashed lines correspond to the Lanczos sequence for a rectangular box of similar dimensions as shown in the inset. (b) Lanczos sequence for the flip-flop operator $\hat{B}$ in \ref{['B-flip-flop']} for the XXZ model with NN (green) and NNN (blue) coupling. The yellow dots corresponds to the operator truncated to an eigenbasis of size $N=2000$. Inset: Plots of the structure function at $E=0$ for each of the cases, showing exponential (NNN) and Gaussian (NN) decay.