Universal properties of the many-body Lanczos algorithm at finite size

TL;DR

This work establishes that finite-size many-body quantum systems exhibit universal Lanczos coefficient patterns that govern late-time autocorrelation plateaus. By deriving an exact link between $C_L(\infty)$ and the Lanczos data, and formulating three conjectures—hydrodynamic scaling, vanishing plateaus, and strong zero modes—the authors connect hydrodynamic tails to finite-size behavior and validate these ideas through extensive numerics across short-range, long-range, and higher-dimensional models. The results show that, despite finite-size constraints, the Lanczos framework captures robust, model-independent information about operator growth and memory, enabling extraction of universal late-time properties from finite systems. These insights offer practical routes to probe hydrodynamics and zero-mode phenomena with finite numerical data and may inform experimental interpretations in synthetic quantum matter.

Abstract

We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behaviour at finite size, we conjecture that in the large $n$ limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero-modes is also discussed. We support our conjecture with a numerical study of different models.

Figures (10)

• Figure 1: Left: Lanczos coefficients computed for model \ref{['Eq:TFIM']} for several sizes, from $L=10$ to $L=13$, with periodic boundary conditions. The observable is $\sigma^z_1$ and the parameters are $[J,h_x,h_z] = [1,1,1.5]$. The plot clearly shows the presence of an infinite-size behaviour for $n<n^*$ followed by a fluctuating plateau region affected by finite-size effects. Right: Corresponding correlation functions $C_L(t)$ as a function of time $t$; a size-dependent plateau is observed at late times.
• Figure 2: The cumulative product $F(n)$ for the operators $\sigma^z_1$ (left panel) and $\sigma^z_1\sigma^z_2$ (right panel). The data refer to the Ising Hamiltonian \ref{['Eq:TFIM']} with parameters $[J,h_x,h_z] = [1,1,1.5]$, and the dashed black lines show the fit with bi-exponential curves.
• Figure 4: (Left) The cumulative product $F(n)$ for the Ising model \ref{['Eq:TFIM']} and the operator $\sigma^y_1$: the data shows a monotonic increasing trend. (Right) The cumulative product $F(n)$ for Hamiltonian \ref{['eq:H_zm']}, hosting an approximate zero mode, and the operator $\sigma^x_1$: the data are consistent with saturation to a finite value.
• Figure 5: (Left) Lanczos coefficients $b_n$ associated with the operator $\sigma^z_1$ for the long-range Ising Hamiltonian in Eq. \ref{['eq:Hlongrange']} with $\alpha=1.5, h_z=1.5$. (Right) The function $F(n)$ for different system sizes; the colored dots are numerical data, while the dashed black lines are biexponential fits.
• Figure 6: We plot the quantity $-L^{1+a}\log F(n) /n$ for the operator $\sigma^z_1$: we consider $a=1$ (left) and $a=0.5$ (right). The data refer to the long-range Ising Hamiltonian in Eq. \ref{['eq:Hlongrange']} with $\alpha=1.5, h_z=1.5$.