Strong and almost strong modes of Floquet spin chains in Krylov subspaces

TL;DR

This work analyzes strong and almost strong edge modes in integrable Floquet spin chains by mapping their Heisenberg dynamics to Krylov subspaces constructed via Lanczos (based on the Floquet Hamiltonian $H_F$) and Arnoldi (based on the Floquet unitary $U$). The resulting Krylov chains are effectively SSH-like 1D lattices whose boundary states correspond to $0$- and $\pi$-modes, with lifetimes set by edge-mode amplitudes and, under interactions, becoming almost-strong modes with system-size independent lifetimes. The authors derive analytic expressions for edge-mode lifetimes within the Arnoldi framework and validate them against exact diagonalization across regimes with and without interactions. The approach links topological edge-mode physics to slow, driven dynamics and points toward a topological classification of Krylov subspaces, with potential extensions to scar states and other slow dynamical phenomena.

Abstract

Integrable Floquet spin chains are known to host strong zero and $π$ 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 $π$ 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.

Figures (6)

• Figure 1: The autocorrelation function $A_{\rm \infty}$ at logarithmically separated stroboscopic times for a chain of length $L=8$ and for the free binary drive. The parameters are $g=0.3$ and $T=2.0,3.5,5.5,8.25$ corresponding respectively to SZM phase, SZM-SPM phase, trivial phase, and a SPM phase.
• Figure 2: Upper panel: The $b_n$s for the binary drive with $g=0.3$ and system size $L=20$. Lower panel: The corresponding spectra with $x$-axis label "index" denoting the location of the eigenvalues arranged in ascending order. For both upper and lower panels, from left to right $T=2.0,3.5,5.5,8.25$. These parameters correspond to SZM, SZM-SPM, trivial, and SPM phases respectively. Horizontal red lines in lower panels correspond to energies $0$ and $\pm \pi$ in units of $T^{-1}$.
• Figure 3: The mod-square of the eigenfunctions of the Krylov chain for the same parameters as Fig. \ref{['fig2']}. Upper (lower) panels correspond to modes at zero ($\pm \pi$) energy. The orange and blue data sets in each panel reflect the fact that these modes appear in pairs. In particular, there are two zero edge modes and two $\pi$ edge modes, the latter occurring at $\pm \pi$ energy. The plots show that $T=5.5$ hosts no edge modes, consistent with a trivial phase. On the other hand $T=2.0$ hosts a pair of edge modes at zero and no edge modes at $\pi$ energy. $T=8.25$ hosts a pair of edge modes at $\pm\pi$ but no edge modes at $0$ energy. $T=3.5$ hosts edge modes at both $0$ and $\pm\pi$ energies.
• Figure 4: Exploring the difference between the spin basis and the Majorana basis for the binary drive with $g=0.3$ and $L=4$. All rows, from left to right $T = 2.0, 3.5,5.5,8.25$. These correspond respectively to SZM, SZM-SPM, trivial, and SPM phases. Top rows show the $b_n$s for the Majorana basis (labeled as "free"), and the two spin bases (labeled as spin$_{1,2}$), where spin$_1$ is the basis that gives the unfolded spectrum, while spin$_2$ is the basis that gives the spectrum folded into the FBZ. Middle rows show the spectra of the corresponding Krylov Hamiltonians, with the $x$-axis label "index" labeling the eigenvalues. The bottom rows plot $A_{\infty}$ obtained from performing the time-evolution using the three different Krylov Hamiltonians. The red data labeled as "ED" in the bottom rows is $A_{\infty}$ from ED.
• Figure 5: Spectrum of $i\ln{W}$ for free example (upper row) and an interacting example with $J_z=0.05$ (lower row). $x$-axis label "index" denotes the location of the eigenvalues arranged in descending order of their magnitude. For all data $L=10$ and $W$ is a $20 \times 20$ matrix. In addition $g=0.3$. The four different values of $T$ are from left to right $T = 2.0, 3.5,5.5,8.25$. These correspond respectively to (A)SZM, (A)SZM-(A)SPM, trivial, and (A)SPM phases. The horizonal red lines indicate $0,\pm \pi$.