Temporal Higher-Order Topological Insulator

• Temporal HOTIs are periodically driven systems that combine spatial and time-domain symmetries to create robust, low-dimensional boundary modes.
• They leverage Floquet theory with time-dependent topological invariants, enabling the realization of both 0-modes and anomalous π-modes.
• These systems offer dynamic control for quantum switching, fault-tolerant circuits, and reconfigurable photonic or mechanical devices.

A temporal higher-order topological insulator (HOTI) generalizes the concept of spatial higher-order topological phases to systems where topology is dynamically induced and protected by symmetries that act jointly in space and time. These phases exhibit protected low-dimensional boundary modes, not only as a consequence of crystalline symmetries but also due to explicit time-periodicity or "Floquet" driving. Temporal HOTIs leverage spatio-temporal symmetry, including combinations of time-reversal symmetry (TRS) and spatial operations (mirror, rotation, inversion), or their dynamic analogs, to yield robust modes (corners, hinges, etc.) at locations or times determined by symmetry constraints. In such systems, topological invariants explicitly include the time domain and may be defined over augmented Brillouin zones or spaces combining momentum and quasi-energy.

1. Spatio-Temporal Symmetry and Higher-Order Topology

The principal mechanisms underlying HOTI phases are dictated by symmetry-broken boundary correspondence. In static HOTIs, the essential protection comes from crystalline spatial symmetries—four-fold rotation (C₄), mirror (M), inversion (I), or combinations thereof with TRS (T). The bulk invariant commonly takes forms such as the quantized Chern–Simons angle,

$$\theta = \frac{1}{4\pi} \int d^3k\, \epsilon^{abc} \text{Tr}\left[A_a \partial_b A_c + \frac{2}{3}A_a A_b A_c\right]$$

where $A_a$ is the Berry connection for occupied bands. For mirror-protected HOTIs, invariants are given by mirror Chern numbers,

$C_m = \frac{C_+ - C_-}{2}$

with $C_+$ ($C_-$) computed in the mirror eigenspaces, and determine the number of Kramers pairs of hinge states.

Temporal HOTIs extend these concepts to periodically driven systems, where micromotion or drive-induced band structure modulations permit the definition of time-dependent symmetry operations. In Floquet systems, symmetry can act not only on spatial degrees of freedom but on time translation itself, enabling invariants defined over the combined space-time torus.

2. Floquet Theory and Temporal HOTI Realization

Floquet theory treats a periodically driven system with

$H(t) = H(t+T)$

by establishing eigenstates

$$|\Psi_a(t)\rangle = e^{-i\varepsilon_a t}|\Phi_a(t)\rangle$$

with time-periodic Floquet modes $|\Phi_a(t+T)\rangle = |\Phi_a(t)\rangle$. The effective Floquet Hamiltonian is constructed from the evolution over one period,

$$U(T) = \mathcal{T} \exp\left(-i \int_0^T H(t) dt\right) \equiv e^{-i H_F T}$$

where $H_F$ encodes the drive-induced topology.

Crucially, the quasi-energy spectrum is periodic modulo the drive frequency, permitting gap openings and topology at both $\varepsilon = 0$ and $\varepsilon = \pi$, i.e., so-called 0-modes (conventional) and anomalous $\pi$-modes (unique to Floquet systems) (Ghosh et al., 2023).

Temporal HOTIs are realized by drive protocols that gap out higher-dimensional boundary states via symmetry-breaking mass terms (often of the Wilson-Dirac form), while leaving domain walls—corners or hinges—where the mass flips sign. Examples include periodic kicks, two-step drives, mass modulation, and circularly polarized light. These can convert topologically trivial or first-order phases into second- or third-order HOTIs with boundary modes pinned at both 0 and $\pi$ quasi-energies.

3. Topological Invariants and Classification in Driven Systems

In the temporal domain, topological invariants generalize static cases by integrating over both momentum and time (or quasi-energy). Key examples include:

• Floquet winding number:

$$W = \oint dt \oint d^{d-k}k_S\, \text{tr}\left[S\left(H_F^{-1}(k_S, t)\, dH_F(k_S, t)\right)^{d-k+1}\right]$$

where $k_S$ is tangent momentum to the symmetry-enforced boundary and $S$ is a chiral operator.

• Floquet quadrupole moment:

$Q_{xy} = \frac{1}{2\pi}\,\text{Im}\{\ln\det U\}$

where $U$ is the Wilson loop of the occupied Floquet modes.

• Bulk multipole moments (quadrupole, octupole, hexadecapole) can be defined in synthetic or real dimensions, discriminating boundaries of codimension $n$ (Dutt et al., 2019).

Driven inversion-symmetric HOTIs retain $Z_2$ classification under dynamic inversion symmetry, where inversion-odd mass terms guarantee gap closings at symmetry-related boundaries in both space and time (Khalaf, 2018).

4. Dimensional Hierarchy: Boundaries, Multipole Moments, and Localized Modes

HOTIs in $d$ dimensions of order $n$ host modes on boundaries of dimension $d_c = d-n$. Periodic driving can engineer transitions between various boundary dimensionalities, e.g.:

• 2D Floquet SOTIs: robust 0D corner modes at $\varepsilon=0$ and $\varepsilon=\pi$.
• 3D Floquet SOTIs: 1D hinge modes protected by mirror/TRS or rotation/TRS (Schindler et al., 2017).
• 3D Floquet TOTIs: 0D corner modes via mass-kick protocols or multi-step drives (Ghosh et al., 2023).

Symmetry-protected sign changes of mass terms are instrumental:

$\text{sgn}(m_I m_{II}) = -1$

at edges or corners yields localized zero modes (Dirac-type domain walls). In time-dependent systems, these nodes can dynamically move or switch under inversion-symmetric perturbations.

5. Experimental Realizations and Temporal Control

Experimental proposals span condensed matter (SnTe nanowires, surface-modified Bi$_2$TeI/BiSe/BiTe (Schindler et al., 2017), weak TIs) and metamaterial platforms (photonic molecules, acoustic crystals). Temporal HOTI phases may be accessed by:

• STM and transport measurements in engineered nanowires with controllable strain or distortion (topological coaxial cable geometry).
• Photonic networks with synthetic dimensions, where phase tuning enables real-time topological switching (Dutt et al., 2019).
• Floquet-engineered electronic systems with circularly polarized fields, demonstrated to host both chiral and corner modes (Ghosh et al., 2019).

Floquet $\pi$-modes are detectable as midgap states at quasi-energy $\varepsilon=\omega/2$ in spectroscopic experiments and provide handles for dynamic topological transitions.

6. Implications and Future Directions

Temporal HOTIs offer a route to dynamically controlled, tunable topological systems:

• Quantum devices with dissipationaless pathways harnessing hinge/corner modes.
• Floquet switching of topological invariants for reconfigurable electronics, robust photonic routing, and adaptive mechanical metamaterials.
• Majorana/parafermion modes in higher-order topological superconductors for quantum computation (Saha et al., 2021).
• Synthetic-dimension architectures permit realization of high-order multipole moments (octupole, hexadecapole) transcending physical spatial dimensions.

A plausible implication is the tailoring, real-time manipulation, and localization of protected states for fault-tolerant quantum information processing and low-power circuit design. The emergence of anomalous $\pi$-modes exclusive to driven systems expands the accessible phase space beyond static higher-order topology. Temporal modulation—both in synthetic and real time—stands as a universal protocol for inducing, tuning, and probing higher-order topological phenomena in materials and engineered platforms.

7. Summary Table: Key Features of Temporal Higher-Order Topological Insulators

Feature	Static HOTI	Temporal (Floquet) HOTI
Protection mechanism	Spatial/crystalline symmetries	Spatio-temporal/Floquet symmetries
Boundary modes	Lower-dimensional (hinges, corners)	Dynamically controlled, 0- and π-modes
Topological invariants	Chern–Simons θ, mirror Chern numbers, multipole moments	Floquet winding, quadrupole/hexadecapole moments
Key materials/platforms	SnTe, Bi$_2$TeI, BiSe, BiTe, acoustic lattices	Periodically driven semimetals, photonic molecules, synthetic lattices
Applications	Quantum transport, photonic routing, topological qubits	Dynamic topological switching, robust energy flows, reconfigurable devices

Temporal higher-order topological insulators unite spatial symmetry protection with time-domain engineering, yielding a tunable taxonomy of phases with robust, symmetry-enforced low-dimensional modes suitable for emerging quantum, electronic, and photonic technologies.