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arXiv:2506.20611 · Physical review. B./Physical review. B

Floquet operator dynamics and orthogonal polynomials on the unit circle

Hsiu-Chung Yeh, Aditi Mitra

Published Dec 15, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.1103/jp5f-wpkq

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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 mmm-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the Z3Z_3Z3​ clock model. The numerically obtained Krylov angles of the Z3Z_3Z3​ 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.
It is shown that the stroboscopic time evolution under a Floquet unitary, in any spatial dimension, and of any Hermitian operator, can be mapped to an operator Krylov space, which is identical to that generated by the edge operator of the noninteracting Floquet transverse-field Ising model (TFIM) in one-spatial dimension, and with inhomogeneous Ising and transverse field couplings. The latter has four topological phases reflected by the absence (topologically trivial) or presence (topologically nontrivial) of edge modes at 0 and/or $\ensuremath{\pi}$ quasienergies. It is shown that the Floquet dynamics share certain universal features characterized by how the Krylov parameters vary in the topological phase diagram of the Floquet TFIM with homogeneous couplings. These results are highlighted through examples, all chosen for numerical convenience to be in one spatial dimension: nonintegrable Floquet spin 1/21/21/2 chains and Floquet Z3{Z}_{3}Z3​ clock model where the latter hosts period-tripled edge modes.
Recursion methods such as Krylov techniques map complex dynamics to an effective noninteracting problem in one dimension. For example, the operator Krylov space for Floquet dynamics can be mapped to the dynamics of an edge operator of the one-dimensional Floquet inhomogeneous transverse field Ising model (ITFIM), where the latter, after a Jordan-Wigner transformation, is a Floquet model of noninteracting Majorana fermions and the couplings correspond to Krylov angles. We present an application of this showing that a moment method exists where given an autocorrelation function, one can construct the corresponding Krylov angles and from that the corresponding Floquet ITFIM. Consequently, when no solutions for the Krylov angles are obtained, it indicates that the autocorrelation is not generated by unitary dynamics. We highlight this by studying certain special cases: stable mmm-period dynamics derived using the method of continued fractions, exponentially decaying, and power-law decaying stroboscopic dynamics. Remarkably, our examples of stable mmm-period dynamics correspond to mmm-period edge modes for the Floquet ITFIM where, deep in the chain, the couplings correspond to a critical phase. Our results pave the way to engineer Floquet systems with desired properties of edge modes and also provide examples of persistent edge modes in gapless Floquet systems.
The problem of Anderson localization, as well as the single particle localization-delocalizaton transition of the Aubry-André model, is studied employing operator Krylov space methods. It is shown that even when the dynamics is generated by a Hamiltonian, studying the dynamics at stroboscopic rather than continuous times has its advantages. In particular, mapping the dynamics to an effective Floquet problem results in an operator Krylov space description where quantities such as the spectral function can be computed with fewer computational resources, while a moment method exists that allows for the extraction of Krylov parameters directly from the discrete time autocorrelation function. For stroboscopic dynamics, the operator Krylov space corresponds to the dynamics of an edge operator of an inhomogeneous Floquet transverse field Ising model, with the parameters of this effective model generated recursively. The Krylov parameters show disorder-averaged renormalization with their distribution narrowing as the recursion step increases. It is shown that a more physical spectral function is obtained from the Krylov parameters obtained from the disorder-averaged autocorrelation function, rather than the disorder-averaged Krylov parameters. The delocalized (localized) phase is shown to correspond to the appearance (absence) of a Porter-Thomas distribution, a ballistically propagating (localized) wavefront in operator Krylov space, and a smooth (discrete) Berstein-Szegö power-spectrum. The localization-delocalization transition is also demonstrated in operator Krylov space. A Porter-Thomas distribution is also observed at the critical point. The long-time dynamics and the inverse participation ratio at the critical point is shown to exhibit behavior consistent with a multi-fractal scaling with system size.
Recursion methods such as Krylov techniques map complex dynamics to an effective non-interacting problem in one dimension. For example, the operator Krylov space for Floquet dynamics can be mapped to the dynamics of an edge operator of the one-dimensional Floquet inhomogeneous transverse field Ising model (ITFIM), where the latter, after a Jordan-Wigner transformation, is a Floquet model of non-interacting Majorana fermions, and the couplings correspond to Krylov angles. We present an application of this showing that a moment method exists where given an autocorrelation function, one can construct the corresponding Krylov angles, and from that the corresponding Floquet-ITFIM. Consequently, when no solutions for the Krylov angles are obtained, it indicates that the autocorrelation is not generated by unitary dynamics. We highlight this by studying certain special cases: stable mmm-periodic dynamics derived using the method of continued fractions, exponentially decaying and power-law decaying stroboscopic dynamics. Remarkably, our examples of stable mmm-periodic dynamics correspond to mmm-period edge modes for the Floquet-ITFIM where deep in the chain, the couplings correspond to a critical phase. Our results pave the way to engineer Floquet systems with desired properties of edge modes and also provide examples of persistent edge modes in gapless Floquet systems.
We generalize the Krylov construction to periodically driven (Floquet) quantum systems using the theory of orthogonal polynomials on the unit circle. Compared to other approaches, our method works faster and maps any quantum dynamics to a one-dimensional tight-binding Krylov chain, which is efficiently simulated on both classical and quantum computers. We also suggest a classification of chaotic and integrable Floquet systems based on the asymptotic behavior of Krylov chain hopping parameters (Verblunsky coefficients). We illustrate this classification with random matrix ensembles, kicked top, and kicked Ising chain.
Periodically driven quantum many-body systems play a central role for our understanding of nonequilibrium phenomena. For studies of quantum chaos, thermalization, many-body localization and time crystals, the properties of eigenvectors and eigenvalues of the unitary evolution operator, and their scaling with physical system size L L are of interest. While for static systems, powerful methods for the partial diagonalization of the Hamiltonian were developed, the unitary eigenproblem remains daunting. % In this paper, we introduce a Krylov space diagonalization method to obtain exact eigenpairs of the unitary Floquet operator with eigenvalue closest to a target on the unit circle. Our method is based on a complex polynomial spectral transformation given by the geometric sum, leading to rapid convergence of the Arnoldi algorithm. We demonstrate that our method is much more efficient than the shift invert method in terms of both runtime and memory requirements, pushing the accessible system sizes to the realm of 20 qubits, with Hilbert space dimensions \geq 10^6 ≥ 10 6 .