Local Floquet Drive in Quantum Systems

• Local Floquet drive is a time-periodic, spatially modulated protocol applied to quantum many-body systems to induce robust non-equilibrium phases.
• Theoretical frameworks using l–bit transformations and spectral π-pairing reveal absolute stability and period-doubled dynamics under local perturbations.
• Experimental detection employs quench dynamics and correlation measurements, offering insights into spatiotemporal order and potential quantum information applications.

A local Floquet drive is a time-periodic modulation applied in a spatially local or spatially modulated fashion to a quantum many-body system, as opposed to a uniform global drive. This concept is central to the paper of periodically driven systems exhibiting robust, non-equilibrium phases—such as many-body localized (MBL) phases, time crystals, and spatiotemporal order—that can survive local perturbations and give rise to emergent symmetries or exotic dynamical phenomena. The theoretical framework for understanding the stability and ordering mechanisms resulting from local Floquet drives has been developed in the context of disordered spin chains, Floquet-MBL phases, and models with emergent integrals of motion.

1. Absolute Stability under Local Floquet Drives

The defining property of certain Floquet-MBL phases—such as the π spin glass (πSG)—is their absolute stability: essential phase features, including spectral multiplet structure, long-range spatial order, and period-doubled temporal dynamics, are robust against arbitrary weak local deformations of the drive, even those that explicitly break global symmetries originally used to define the phase.

Consider a prototypical Floquet unitary at the fixed point

$$U_{f0} = P_x \exp\left[ -i \sum_{r=1}^{L-1} J_r \sigma^z_r \sigma^z_{r+1} \right]$$

with Ising symmetry $P_x = \prod_r \sigma^x_r$. Upon introducing a weak, local deformation (to $U_{f\lambda}$), a local “dressing” unitary $\mathcal{V}_\lambda$ transforms the physical spins into emergent $l$-bits,

$$\tau^\beta_{r,\lambda} = \mathcal{V}_\lambda \sigma^\beta_r \mathcal{V}^\dagger_\lambda, \quad \beta = x, y, z.$$

Consequently, the Floquet unitary attains a canonical form,

$$U_{f\lambda} = P^{(\lambda)} \exp\left[ -i f(\{D_r^{(\lambda)}\}) \right], \quad D_r^{(\lambda)} \equiv \tau^z_{r,\lambda} \tau^z_{r+1,\lambda},$$

where $f$ is an even, local function of these dressed domain-wall operators and $P^{(\lambda)} = \prod_r \tau^x_{r,\lambda}$. The key criterion is that $\tau^z_{r,\lambda}$ anticommutes with $U_{f\lambda}$ up to corrections that are exponentially small in system size, i.e.,

$\{ \tau^z_{r,\lambda}, U_{f\lambda} \} \approx 0,$

which protects spectral π-pairing and period doubling even in the absence of explicit global symmetry.

2. Spatiotemporal Long-Range Order

Floquet-MBL phases that are stable to local perturbations exhibit unconventional order in both space and time:

• Spatial order: Floquet eigenstates are “cat states” with long-range connected correlations in the order parameter (the emergent $\tau^z$), reflecting spin-glass-type long-range order. The l–bit basis retains a nonzero overlap with physical spin operators.
• Temporal order: The dressed order parameter flips sign stroboscopically, leading to period-doubled dynamics, formalized as

$$\langle \tau^z(nT) \rangle = (-1)^n \langle \tau^z(0) \rangle.$$

Correlation functions exhibit an antiferromagnetic-like temporal oscillation:

$$C(nT;r,s) \approx c_0(r)c_0(s) + c_1(r)c_1(s)(-1)^n,$$

signaling simultaneous spatial and discrete time-translation symmetry breaking.

The symmetry breaking is “spatiotemporal”: glassy in space and alternating (antiferromagnetic) in time, while preserving a composite symmetry of time translation followed by an emergent Ising flip.

3. Emergent Symmetries and Spectral Multiplet Structure

Even if explicit microscopic symmetries are broken by local drives, the phase retains emergent, Hamiltonian-dependent symmetries derived from the local dressing transformation, e.g., $P^{(\lambda)}$. Correspondingly, each Floquet eigenstate with parity $p$ has a partner of $-p$ with quasienergy splitting exactly π/T (the canonical π-pairing multiplicity):

$u_\lambda(\{d\},p) = p \exp[ -i f(\{d\}) ]$

In more general scenarios (with $\mathbb{Z}_n$ or non-Abelian symmetries), the multiplet spectral splitting is quantized (in simple cases as π/T, or generically as $2\pi g/(nT)$ where $g = \gcd(n,k)$). Importantly, the spectral pairing structure is robust to local perturbations, being tied to the emergent, not microscopic, symmetry of the drive.

4. Experimental Detection of Spatiotemporal Order

Local Floquet driving phases can be experimentally identified by:

• Quench dynamics from product or short-range correlated initial states: The expectation value of local observables (with support on $\tau^z$) exhibits robust π/T-period oscillations over timescales much longer than microscopic ones.
• One-point and two-time correlation functions: Observables such as $\langle \sigma^z(nT) \rangle$ and nonlocal stroboscopic correlation functions directly reveal period-doubling and long-range spatial order.
• Persistence of the oscillatory component: Numerical studies confirm that only the quantized peak at π/T survives in the thermodynamic limit, offering a “fingerprint” for spatiotemporal order.

Platforms such as ultracold atoms or engineered quantum spin chains are suitable for these protocols, as the detection does not require the preparation of individual Floquet eigenstates.

5. Implications for Quantum Phases and Applications

• Enlarged Classification of Phases: Discrete time-translation symmetry becomes a dynamical symmetry, analogous to equilibrium global symmetries, and can be spontaneously broken in concert with spatial symmetry in a robust manner.
• Quantum Information Storage: The absolute stability and emergent symmetry-protected spectral structure suggest mechanisms for robust storage and manipulation of quantum information. Macroscopic superpositions (“cat states”) and quantized spectral pairings are reminiscent of time crystal fingerprints and may be harnessed for quantum control.
• Universality and Future Directions: These robust phases open avenues for realizing spatiotemporal order in dimensions beyond one, exploring connections to SPT phases, pumped boundary charges, and integrating topologically nontrivial orders. They also invite future exploration of higher-dimensional generalizations and the role of emergent one-form symmetries.

6. Methods and Theoretical Framework

The results for local Floquet drives are established by combining:

• Canonical transformation to l–bit basis: Local unitaries $\mathcal{V}_\lambda$ enable a rigorous description of dressed operators and emergent parity.
• Stability criterion: The persistence of the canonical form and anticommution relations under arbitrary local perturbations (not just symmetry-respecting ones) proves “absolute stability.”
• Multiplet structure analysis: Quasienergy splittings are deduced from transformed unitaries, and their quantization tied to emergent symmetry, not microscopic structure.
• Numerical simulations: Robustness and observability of period-doubling in correlation functions were corroborated by large-scale simulations, validating analytical arguments.

7. Broader Context

The paper of local Floquet drives, as established in Floquet-MBL phases, reveals a new landscape of quantum matter where spatiotemporal order, emergent symmetries, and protected multiplet structures are rendered absolutely stable against weak local deformations. The interplay of locality, disorder, periodic driving, and emergent symmetry breaking provides a platform for realizing phenomena—such as time crystals and robust quantum memory—that are fundamentally inaccessible in equilibrium closed-system dynamics (Keyserlingk et al., 2016).