Floquet operator dynamics and orthogonal polynomials on the unit circle

TL;DR

This work develops a comprehensive link between Floquet operator dynamics and orthogonal polynomials on the unit circle (OPUC) by mapping the operator Krylov space to the CMV basis and linking Verblunsky coefficients to Krylov angles in the inhomogeneous Floquet Ising model (ITFIM). It shows how the stroboscopic evolution of Hermitian operators can be analyzed analytically through Szegő recurrences and Bernstein–Szegő approximations, enabling analytic OPUC for persistent $m$-period dynamics and a continued-fraction representation of autocorrelation Laplace transforms. The authors provide detailed results for the $m=1,2,3,4,6$ cases and present numerical OPUC for the $Z_3$ clock model and a six-sublattice Majorana chain, uncovering spatial periodicities in the Krylov angles and corresponding long-lived edge modes. This framework yields solvable toy models that mimic a range of dynamical behaviors and offers a robust route to engineerKrylov-space dynamics with targeted spectral properties, with potential applications in understanding operator spreading and edge modes in driven quantum systems.

Abstract

Operator spreading under stroboscopic time evolution due to a unitary is studied. An operator Krylov space is constructed and related to orthogonal polynomials on a unit circle (OPUC), as well as to the Krylov space of the edge operator of the Floquet transverse field Ising model with inhomogeneous couplings (ITFIM). The Verblunsky coefficients in the OPUC representation are related to the Krylov angles parameterizing the ITFIM. The relations between the OPUC and spectral functions are summarized and several applications are presented. These include derivation of analytic expressions for the OPUC for persistent $m$-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the $Z_3$ clock model. The numerically obtained Krylov angles of the $Z_3$ clock model with long-lived period tripled autocorrelations show a spatial periodicity of six, and this observation is used to develop an analytically solvable model for the ITFIM that mimics this behavior.

Figures (7)

• Figure 1: The numerical results for $1/|P^{(m)}_k(e^{i\omega})|^2$ for $m = 1,2,4$ (\ref{['Eq: P m=1']}, \ref{['Eq: P m=2']}, \ref{['Eq: P m=4']}) in the top panels and $m = 3,6$ (\ref{['Eq: P m=3']}, \ref{['Eq: P m=6']}) in the bottom panels with $k = 10,20,30,40$ and $A=0.8$. As $k$ increase, the peaks become sharper indicating the delta function behavior of the spectral function. This agrees with the Bernstein–Szegő approximation \ref{['Eq: Bernstein large k']}. Besides the delta function, the background approaches $(1-A)$ as expected from \ref{['Eq: spectral function m=1']}.
• Figure 2: Numerical results of for the Krylov angles, autocorrelation and OPUC for $Z_3$ clock model \ref{['Eq: Z3 clock model']} with initial operator $(\sigma_1 + \sigma_1^\dagger)/\sqrt{2}$ and couplings $T=2$, $J=e^{i\phi}$, $g = \epsilon + i2\pi/(3\sqrt{3})$. First row: the left panel shows Krylov angles for $L = 8$ for long-lived ($T=2, \phi = \pi/2, \epsilon=0.1$) and decaying ($T=2, \phi = 0, \epsilon=0.001$) cases. The Krylov angles of the 3-period case \ref{['Eq: m=3 Krylov angle']} are also plotted for comparison and we set its first Krylov angle to match the $Z_3$ long-lived result. The Krylov angles for both the long-lived and 3-period cases show convergence to $\pi/2$ for large $k$ along with a 6-periodic spatial oscillation in $k$. However, no simple pattern is observed for the decaying case. The corresponding autocorrelation of $Z_3$ clock model are shown in the middle and right panels separately. The upper (lower) envelope in the upper middle and upper right panels correspond to $n+2\, (n,n+1)$ where $n$ equally slices the time in the logarithmic-scale and $n = 1\ \text{mod}\ 3$. Interestingly, the decaying autocorrelation shows a different periodicity at late-times (see inset). Second row: The numerical results for $1/|P_k(e^{i\omega})|^2$ for long-lived (left panel) and decaying (right panel) cases for $k = 10, 20, 30, 40$. The peaks at $\pm 2\pi/3$ reflect the 3-period oscillation of the autocorrelation in both cases. There are peaks at other frequencies for the decaying case as indicated by the late-time oscillation of the autocorrelation. Numerically, it is subtle to conclude if the peaks will eventually lead to delta functions in the spectral function, see discussion in the main text.
• Figure 3: The numerical results of $1/|P_k(e^{i\omega})|^2$ for the 6-sublattice Majorana chain with the Krylov angles \ref{['Eq: 6-sublattice angles']}. The approximate spectral function captures the dynamics of $\gamma_1$ with its peaks corresponding to long-lived modes that overlap with $\gamma_1$ The first row shows results for the case (i): $\theta_1 = \theta_2 = \theta_4 = \theta_5 = \pi/2$ with $\theta_3, \theta_6$ being free parameters. From left to right panels, we set different values of $\theta_3, \theta_6$ such that it shows four different scenarios: $0,\ \pm 2\pi/3$-modes; $\pi,\ \pm\pi/3$-modes; all six modes; no modes. The $0$ and $\pm 2\pi/3$-modes ($\pi$ and $\pm \pi/3$-mode) always appear together because of the threefold degeneracy of the eigenvalue spectrum \ref{['Eq: K R commutation']}. The second and third rows respectively show results for cases (ii): $\theta_2 = \theta_3 = \theta_5 = \pi/2$ with $\theta_1=\theta_4, \theta_6$ being free parameters, and (iii): $\theta_1 = \theta_3 = \theta_4 = \pi/2$ and $\theta_2=\theta_5, \theta_6$ being free parameters. In contrast to case (i), cases (ii) and (iii) do not have a threefold degenerate spectrum (see Fig. \ref{['Fig: Folded spectrum']}), which leads to the absence of the $\pm \pi/3$-mode. The Krylov angles for cases (ii) and (iii) from left to right panels in second and third rows are evaluated for four different situations: $0,\ \pm 2\pi/3$-modes; only $\pi$-modes; $0,\ \pi, \pm2\pi/3$-modes; absence of $0, \pm2\pi/3, \pi$-modes. Note that there are other modes for cases (ii) and (iii) as indicated by the peaks not located at $\omega = 0,\pm2\pi/3, \pi$ in the second and third rows. These new modes can be viewed as deformation of $\pm\pi/3$-mode from case (i) (see Fig. \ref{['Fig: Deformation spectrum']}).
• Figure 4: The single particle eigenvalue spectrum from ED of the 6-sublattice Majorana chain with the Krylov angles \ref{['Eq: 6-sublattice angles']} and for a system size of 60-Majoranas. We choose the parameters to be the same as the leftmost three columns of Fig. \ref{['Fig: 6-sublattice OPUC']}. The full ED spectra is presented in the first rows and the corresponding folded spectra are in the corresponding lower panels. The threefold degeneracy of case (i) agrees with the algebraic relation \ref{['Eq: K R commutation']}. In contrast, cases (ii) and (iii) do not have such a property. Therefore, it leads to the missing $\pm \pi/3$-modes in the second and third rows of Fig. \ref{['Fig: 6-sublattice OPUC']}. Note that the spectra in the middle and right panels are exactly the same because cases (ii) and (iii) are related by reflection of the Majorana chain.
• Figure 5: The eigenvalue spectrum from ED of the 6-sublattice Majorana chain with a system size of 60-Majoranas and for case (ii): $\theta_2 = \theta_3 = \theta_5 = \pi/2$, $\theta_1=\theta_4$. We consider $\theta_1 = \pi/2, \pi/3, \pi/4$ separately and fixed $\theta_6 = 3\pi/4$. Changing $\theta_1$ from $\pi/2$ to $\pi/4$ can be viewed as a deformation from the third panel of case (i) to the leftmost panel of case (ii) in Fig. \ref{['Fig: 6-sublattice OPUC']}. At $\theta_1 = \pi/2$, $\pm \pi/3$-modes exist. As $\theta_1$ decreases, the eigenvalues at $\pm \pi/3$ split and turn into new modes at different frequencies.