[Paper Review] Krylov complexity of many-body localization: Operator localization in Krylov basis

Fabian Ballar Trigueros, Cheng-Ju Lin|arXiv (Cornell University)|Dec 9, 2021

Quantum many-body systems77 references84 citations

TL;DR

This paper investigates operator growth in many-body localized (MBL) systems using the Krylov basis framework, where operator dynamics map to a single-particle hopping problem on a semi-infinite chain with Lanczos coefficients as hopping amplitudes. Despite Lanczos coefficients scaling asymptotically like ergodic systems (∼n/ln(n)), MBL systems exhibit even-odd alternation and effective randomness, leading to localized wavefunctions and bounded Krylov complexity—indicating operator localization in Krylov space.

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.

Motivation & Objective

• To understand operator growth and complexity in many-body localized (MBL) systems using the Krylov basis framework.
• To analyze the behavior of Lanczos coefficients in MBL systems and compare them with ergodic and integrable regimes.
• To investigate whether the emergent single-particle hopping problem in Krylov space is localized or delocalized in MBL systems.
• To explore the feasibility of linear extrapolation of Lanczos coefficients toward the thermodynamic limit in MBL systems.
• To contrast Krylov complexity in a microscopic MBL model with a phenomenological MBL model featuring l-bit operators.

Proposed method

• Apply the Lanczos algorithm to generate a Krylov basis for operator dynamics, mapping the problem to a single-particle hopping Hamiltonian on a semi-infinite chain.
• Use the Lanczos coefficients as hopping amplitudes in the effective single-particle model, with initial state localized at the first site.
• Compute spectral functions, wavefunction profiles, return probabilities, and integrals of motion to probe localization properties.
• Define Krylov complexity as the mean position of the emergent single-particle wavefunction to quantify operator spreading.
• Perform linear extrapolation of Lanczos coefficients to estimate their behavior in the thermodynamic limit, despite known limitations.
• Compare results from the microscopic MBL model (random-field Ising chain) with a phenomenological model featuring l-bit flipping operators.

Experimental results

Research questions

• RQ1 Do Lanczos coefficients in MBL systems exhibit the same asymptotic scaling as in ergodic systems, and what additional structures emerge?
• RQ2 Is the emergent single-particle hopping problem in the Krylov basis localized when initialized at the first site in MBL systems?
• RQ3 How does Krylov complexity evolve in time in MBL systems, and how does it compare to ergodic and integrable regimes?
• RQ4 Can linear extrapolation of Lanczos coefficients provide meaningful insight into the thermodynamic limit of MBL systems?
• RQ5 How does the Krylov complexity differ between a microscopic MBL model and a phenomenological l-bit model?

Key findings

• Lanczos coefficients in MBL systems scale asymptotically as ∼n/ln(n), matching the ergodic regime, but with an additional even-odd alternation and effective randomness.
• The emergent single-particle wavefunction in the Krylov basis remains localized when initialized at the first site, indicating operator localization in Krylov space.
• Krylov complexity is bounded in time in the MBL system, in contrast to the superpolynomial (exponential or stretched exponential) growth expected in ergodic systems.
• The spectral function and return probability show signatures of localization, with no evidence of wavefunction spreading.
• In the phenomenological l-bit model, Lanczos coefficients approach constants with even-odd alternation, and Krylov complexity grows linearly in time with a propagating wavefront.
• Linear extrapolation of Lanczos coefficients suggests a possible path to studying the thermodynamic limit, though the method is likely too simplistic for exact thermodynamic behavior.