Quench dynamics via recursion method

Abstract

The recursion method provides a powerful framework for studying quantum many-body dynamics in the Lanczos basis recursively constructed within the Krylov space of operators. Recently, it has been demonstrated that the recursion method, when supplemented by the universal operator growth hypothesis, can effectively compute autocorrelation functions and transport coefficients at infinite temperature. We extend the scope of the recursion method to far-from-equilibrium quench dynamics. We apply the method to spin systems in one, two, and three spatial dimensions. In one dimension its usefulness is limited: although it remains accurate at moderate times, it eventually experiences an abrupt breakdown and subsequently yields results that deviate strongly from the true dynamics. In contrast, in two and three dimensions the method proves far more effective, providing a reasonably accurate description of the evolution across all time scales -- from the initial transient regime to thermalization.

Figures (5)

• Figure 1: Time evolution of the total polarization along $z$ direction in the 1D Ising model \ref{['H 1D Ising']} with $h_x=h_z=1$. The initial state is a product translation-invariant state \ref{['initial state']} with the polarization $(p_x,p_y,p_z)$ indicated on each plot. Orange line -- essentially exact results of the numerical time evolution (see text for details). Blue line - results of the recursion method. The recursion method reproduces initial evolution but abruptly departs from the exact numerical result at the breakdown time $t_*$ (vertical dashed line). This time appears to be independent of the initial state.
• Figure 2: Quench dynamics of nonintegrable Ising models on (a) one-dimensional, (b) square, and (c) cubic lattices. Shown is the expectation value of the total magnetization in the $z$-direction. The initial state is the pure product state \ref{['initial state']} polarized along the same $z$ direction. Blue solid line is the result of the recursion method. The chosen time span is sufficient for the latter to equilibrate. (a) One-dimensional Ising model \ref{['H 1D Ising']} with $h_x=h_z=1$ (the same as in Fig. \ref{['fig 1D']} but for a larger time span). Orange is the exact numerical result. The horizontal dashdotted line is the numerically time-averaged post-quench value of the observable. (b) and (c): The transverse-filed Ising model \ref{['H 2D Ising']} with $h_z=1$ on the square and cubic lattices, respectively. Points are the results of the sparse Pauli dynamics method Begusic_2025_Real-time. Insets show explicitly computed Lanczos coefficients (points) and their extrapolation according to eqs. \ref{['b_n extrapolation 1D']}, \ref{['b_n extrapolation 2D']} (black lines).
• Figure S1: Functions $(-i)^n\phi^n_t$ for $n=0,1,\dots,n_{\rm max}$. Shown are plots for (a) one-dimensional, (b) two-dimensional and (c) three-dimensional Ising models considered in the main text.
• Figure S2: Values of $i^n \langle O^n \rangle$ for the one-dimensional (left column) and two-dimensional (right column) Ising models considered in the main text. Polarizations of the initial state are indicated in each plot.
• Figure S3: Time evolution of the total polarization along $z$ direction in the 1D Ising model \ref{['H 1D Ising']} for various values of $h_z$ and $h_x$. The initial state is a product translation-invariant state \ref{['initial state']} polarized in the $z$-direction. Orange line -- essentially exact results of the numerical time evolution (see text for details). Blue line -- results of the recursion method. Pink line -- the rate function $\lambda(t)$ whose cusps mark the dynamical quantum phase transitions.