Krylov complexity of many-body localization: Operator localization in Krylov basis

TL;DR

The paper develops a Krylov-Lanczos framework to study operator growth in many-body localization, mapping Heisenberg dynamics to a semi-infinite chain whose hopping amplitudes are the Lanczos coefficients. It finds that in MBL the coefficients scale as n/log n but exhibit even-odd modulation and effective randomness, prompting a simple linear extrapolation to approach the thermodynamic limit. Using both original and extrapolated coefficients, it analyzes spectral functions, zero modes, Krylov complexity, wavefunction profiles, and return probabilities, finding localization in Krylov space for the microscopic MBL model. A parallel study of a phenomenological l-bit MBL model shows similar even-odd structure with constants-at-large-n and yields linear growth of Krylov complexity, highlighting a traveling-wavefront picture for operator growth. Overall, the work provides a Krylov-space diagnostic of MBL, discusses extrapolation caveats, and prompts questions about the thermodynamic limit and the MBL-ergodic transition.

Abstract

We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as $\sim n/\ln(n)$ asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.

Figures (18)

• Figure 1: (a)The disorder-averaged Lanczos coefficients $\overline{b_n}$ for several disorder strengths $h$ and system size $L = 13$ in the random field Ising model, starting with the initial operator $Z_{0}$. In the MBL regime $h \gtrsim 3$, $\overline{b_n}$ has the same asymptotic behavior as in the ergodic regime, but with an additional even-odd alteration. (b)Disorder-averaged Lanczos coefficients $\overline{b_n}$ for several system sizes $L$ and $h=7$, with the initial operator $Z_0$. The onset point of the plateau of $\overline{b_n}$ increases with the system size $L$. We therefore conclude that the saturation of $\overline{b_n}$ is due to the finite size. Inset: the disorder averaged Lanczos coefficients starting with the the $X_0$, $Y_0$ and $Z_0$ operators for $L = 13$ and $h=7$. The even-odd alteration of the Lanczos coefficients is independent of the choice of the initial operators.
• Figure 2: (a)-(h) The disorder realizations of $b_n$ for Eq. (\ref{['eqn:IsingH']}) for $h=7$ and $L=13$, with the initial operator $Z_0$. We use $b_n$ in the range of $n= 10$ to $n=40$ to fit the linear-extrapolation formula Eq. (\ref{['eqn:fitformula']}), and then use the fitted results to extrapolate $b_n$ for $n > 40$ for each disorder realization.
• Figure 3: (a)(b) The histograms of the fitted parameters $a_{o}$ and $c_{o}$ from Eq. (\ref{['eqn:fitformula']}) for the odd branch. (c)(d) The histograms of the fitted data $a_{e}$ and $c_{e}$ for the even branch. (e) The histogram of $\Gamma_n$ in Eq. (\ref{['eqn:fitformula']}), obtained by collecting the difference between the data and the fitted results $y-y_{\text{fit}}$ for each realization and $n \in [10,40]$. The Gaussian distributions of the histogram supports the conjectured effective randomness of the parameters. We use the $b_n$ sequence obtained from $L=13$ and $h=7$ in the range of $n=10$ to $40$ for the extrapolation.
• Figure 4: (a) The disorder-averaged cumulative spectral function $1-\overline{\chi(\omega)}$ in the ergodic regime $h = 1.5$ for several system sizes $L$. (b) Same as (a) but in the MBL regime $h=7$. (c) The disorder-averaged cumulative spectral function $1-\overline{\chi(\omega)}$ from the extrapolated $b_n$ with several Krylov cutoff order $n_{\max}$. Insets: The disorder-averaged cumulative spectral function $\overline{\chi(\omega)}$. Note that the flattening of $1-\overline{\chi(\omega)}$ at the high frequency in all of the figures is due to the numerical round-off error.
• Figure 5: (a)The finite-$n_{\max}$ extrapolation of the disorder-averaged amplitude of the zero-frequency delta function for several system sizes $L$ and disorder strengths $h$. (b)The finite-size $L$ extrapolation of the disorder-averaged amplitude of the zero-frequency delta function for several disorder strengths $h$. The results are obtained from the extrapolations in (a). We also benchmark the results of the amplitudes obtained from ED.