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arXiv:2305.04980 · Physical review. B./Physical review. B

Decay rates of almost strong modes in Floquet spin chains beyond Fermi's Golden Rule

Hsiu-Chung Yeh, Achim Rosch, Aditi Mitra

Published Aug 4, 2023• 13 citations
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Abstract

The stability and dynamics of almost strong zero and $\ensuremath{\pi}$ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction ${J}_{z}$ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where $T$ is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to ${J}_{z}^{2n}$. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in ${J}_{z}$, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.

DOI
10.1103/physrevb.108.075112

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The stability and dynamics of almost strong zero and \ensuremathπ\ensuremath{\pi}\ensuremathπ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction Jz{J}_{z}Jz​ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo 2\ensuremathπ/T2\ensuremath{\pi}/T2\ensuremathπ/T, where TTT is the period of the Floquet system. If there are 4n\ensuremath−14n\ensuremath{-}14n\ensuremath−1 bulk excitations whose quasienergies add up to zero (or \ensuremathπ/T\ensuremath{\pi}/T\ensuremathπ/T for a \ensuremathπ\ensuremath{\pi}\ensuremathπ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to Jz2n{J}_{z}^{2n}Jz2n​. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in Jz{J}_{z}Jz​, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.
Long-lived π\piπ edge modes of interacting and disorder-free Floquet spin chains
Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively different from their static counterparts by, for example, hosting π\piπ edge modes that show stable period-doubled dynamics. However the stability of these edge modes to interactions has traditionally required the system to be many-body localized in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence of bulk heating, π\piπ edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the π\piπ edge mode manifests as the quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.
Integrable Floquet spin chains are known to host strong zero and $\ensuremath{\pi}$ 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 implies 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 zero and $\ensuremath{\pi}$ 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.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
Results are presented for a Floquet Ising chain with duality twisted boundary conditions, taking into account the role of weak integrability breaking in the form of four-fermion interactions. In the integrable case, a single isolated Majorana zero mode exists which is a symmetry in the sense that it commutes both with the Floquet unitary and the Z2{Z}_{2}Z2​ symmetry of the Floquet unitary. When integrability is weakly broken, both in a manner so as to preserve or break the Z2{Z}_{2}Z2​ symmetry, the Majorana zero mode is still found to be conserved for small system sizes. This is reflected in the dynamics of an infinite-temperature autocorrelation function which, after an initial transient that is controlled by the strength of the integrability breaking term, approaches a plateau that does not decay with time. The height of the plateau agrees with a numerically constructed conserved quantity and is found to decrease with increasing system sizes. It is argued that the existence of the plateau and its vanishing for larger system sizes is closely related to a localization-delocalization transition in Fock space triggered by the integrability-breaking interactions.
The transverse field Ising model (TFIM) on the half-infinite chain possesses an edge zero mode. This work considers an impurity model: TFIM perturbed by a boundary integrability-breaking interaction. For sufficiently large transverse field, but in the ordered phase of the TFIM, the zero mode is observed to decay. The decay is qualitatively different from zero modes where the integrability-breaking interactions are nonzero all along the chain. It is shown that for the impurity model, the zero mode decays by relaxing to a nonlocal quasiconserved operator, the latter being exactly conserved when the opposite edge of the chain has no noncommuting perturbations so as to ensure perfect degeneracy of the spectrum. In the thermodynamic limit, the quasiconserved operator vanishes, and a regime is identified where the decay of the zero mode obeys Fermi's golden rule. A toy model for the decay is constructed in Krylov space and it is highlighted how Fermi's golden rule may be recovered from this toy model.
Abstract Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively different from their static counterparts by, for example, hosting π edge modes that show stable period-doubled dynamics. However the stability of these edge modes to interactions has traditionally required the system to be many-body localized in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence of bulk heating, π edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the π edge mode manifests as the quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.