[Paper Review] Strong and almost strong modes of Floquet spin chains in Krylov subspaces

Daniel J. Yates, Aditi Mitra|arXiv (Cornell University)|May 27, 2021

Quantum many-body systemsPhysics and Astronomy53 references62 citations

TL;DR

This paper introduces a Krylov subspace approach to study almost strong zero and π modes in driven (Floquet) spin chains, mapping the dynamics of these modes to effective single-particle chains via Lanczos and Arnoldi iterations. The Arnoldi-based method yields a compact analytical expression for the lifetime of almost strong modes, valid for finite systems and with interactions, validated against exact diagonalization.

ABSTRACT

Integrable Floquet spin chains are known to host strong zero and $\pi$ modes which are boundary operators that respectively commute and anticommute with the Floquet unitary generating stroboscopic time-evolution, in addition to anticommuting with a discrete symmetry of the Floquet unitary. Thus the existence of strong modes imply a characteristic pairing structure of the full spectrum. Weak interactions modify the strong modes to almost strong modes that almost commute or anticommute with the Floquet unitary. Manifestations of strong and almost strong modes are presented in two different Krylov subspaces. One is a Krylov subspace obtained from a Lanczos iteration that maps the time-evolution generated by the Floquet Hamiltonian onto dynamics of a single particle on a fictitious chain with nearest neighbor hopping. The second is a Krylov subspace obtained from the Arnoldi iteration that maps the time-evolution generated directly by the Floquet unitary onto dynamics of a single particle on a fictitious chain with longer range hopping. While the former Krylov subspace is sensitive to the branch of the logarithm of the Floquet unitary, the latter obtained from the Arnoldi scheme is not. The effective single particle models in the Krylov subspace are discussed, and the topological properties of the Krylov chain that ensure stable $0$ and $\pi$ modes at the boundaries are highlighted. The role of interactions is discussed. Expressions for the lifetime of the almost strong modes are derived in terms of the parameters of the Krylov subspace, and are compared with exact diagonalization.

Motivation & Objective

• To understand the stability and dynamics of almost strong zero and π modes in interacting, periodically driven spin chains.
• To map the time evolution of these modes to effective single-particle dynamics on a Krylov chain using iterative methods.
• To derive analytical expressions for the lifetime of almost strong modes that remain valid beyond the non-interacting limit.
• To compare the Lanczos and Arnoldi methods in preserving topological features and spectral structure of the original system.
• To validate the Krylov-based lifetime predictions against exact diagonalization results.

Proposed method

• Uses the Lanczos iteration on the Floquet Hamiltonian to generate a Krylov subspace mapping time evolution to a single-particle chain with nearest-neighbor hopping.
• Applies the Arnoldi iteration directly on the Floquet unitary to construct a Krylov subspace with longer-range hopping, avoiding branch-cut ambiguities.
• Derives effective Krylov chain Hamiltonians in the free limit, highlighting topological invariants that protect 0 and π modes at the boundaries.
• Constructs a compact analytical expression for the lifetime of almost strong modes based on the Krylov subspace parameters.
• Compares the Krylov method’s predictions with exact diagonalization results to validate the lifetime estimates.
• Analyzes the role of interactions by studying how they modify the commutation/anticommutation relations of the modes with the Floquet unitary.

Experimental results

Research questions

• RQ1 How do almost strong zero and π modes emerge in interacting, periodically driven spin chains?
• RQ2 What is the role of the Krylov subspace in capturing the topological protection of edge modes in driven systems?
• RQ3 How do the Lanczos and Arnoldi methods differ in their ability to preserve the spectral and topological structure of the original Floquet dynamics?
• RQ4 Can a compact analytical expression for the lifetime of almost strong modes be derived from the Krylov subspace parameters?
• RQ5 How do the Krylov-based predictions for mode lifetimes compare with exact diagonalization results?

Key findings

• The Arnoldi method produces a Krylov subspace with longer-range hopping that is independent of the logarithm branch choice, unlike the Lanczos method.
• The effective Krylov chain Hamiltonians derived via Arnoldi iteration preserve the topological structure necessary for stable 0 and π modes at the boundaries.
• A compact analytical expression for the lifetime of almost strong modes is derived in terms of Krylov subspace parameters, valid for finite-size systems and with interactions.
• The predicted lifetimes from the Krylov method show good agreement with exact diagonalization results across different parameter regimes.
• Interactions reduce strong modes to almost strong modes, with lifetimes that are non-perturbatively long and captured accurately by the Krylov approach.
• The method enables analytic exploration of mode stability in generic interacting and driven quantum systems beyond integrability.