Subspace-Protected Topological Phases

• Subspace-protected topological phases are quantum states defined by selection rules on specific Hilbert subspaces that yield unique topological invariants and robust boundary phenomena.
• They are classified by restricted winding numbers and Pfaffian invariants, leading to effects such as the zero-winding skin effect and unpaired boundary modes.
• These phases are experimentally realized in systems like non-Hermitian chains, bilayer graphene, and open quantum setups, broadening applications in engineered quantum materials.

Subspace-protected topological phases are phases of quantum matter whose topological properties are protected not purely by global symmetries, but by structural restrictions such as subspace constraints or subsystem symmetries—partial symmetries or selection rules that apply to certain subspaces of the Hilbert space or to localized spatial regions. These mechanisms generate new classes of topological invariants, distinctive boundary phenomena, and extend bulk-boundary correspondence beyond conventional symmetry-protected paradigms.

1. Definition and Fundamental Properties

In subspace-protected topological phases, topological invariants and robust boundary phenomena result from imposing selection rules that restrict the Hamiltonian’s action to particular subspaces. Formally, a Bloch Hamiltonian $H(k)$ acting on Hilbert space $\mathcal{H}$ satisfies a subspace property if for two $k$-independent subspaces $\mathcal{M},\mathcal{M}'$,

$H(k)\mathcal{M} \subset \mathcal{M}'$

for all $k$, leading to selection rules such as $\langle \psi' | H(k) | \psi \rangle = 0$ for $$|\psi\rangle \in \mathcal{M},~|\psi'\rangle \in (\mathcal{M}')^\perp$$ (Shimomura et al., 28 Aug 2025).

Unlike strict global symmetries, subspace properties may not be preserved under arbitrary energy shifts, and their violation may only impact some sectors of the spectrum or spatial regions. This “subspace selection rule” enables the construction of topological invariants not visible in the full spectrum, leading to phenomena distinctive from conventional symmetry-protected or intrinsically topological states.

2. Classification and Topological Invariants

Topological invariants for subspace-protected phases are defined by restricting $H(k)$ to subspace $\mathcal{M}$ (with $\dim(\mathcal{M}) = \dim(\mathcal{M}')$) and requiring a point gap (i.e., $\det H(k)|_{\mathcal{M}} \neq 0$ for all $k$). For odd $d = 2n+1$ spatial dimensions, a $\mathbb{Z}$-valued winding number can be constructed as

$$w^{\mathcal{M}} = \frac{(-1)^n n!}{(2\pi i)^{n+1} (2n+1)!} \int_{\text{BZ}} \operatorname{Tr}_{\mathcal{M}}\left[ (H|_{\mathcal{M}}^{-1} d H|_{\mathcal{M}})^{2n+1} \right]$$

(Shimomura et al., 28 Aug 2025). This winding number, determined by the topology of $H(k)$ on $\mathcal{M}$, remains quantized so long as both the point gap and the subspace property are preserved.

For systems with additional internal symmetries (such as effective time-reversal, particle-hole, or chiral symmetries on the restricted Hamiltonian), the classification can instead be $\mathbb{Z}_2$, exemplified via Pfaffian invariants: $$(-1)^\nu = \operatorname{sgn} \left[ \frac{\operatorname{Pf}(H(\pi)|_{\mathcal{M}}\sigma_x)}{\operatorname{Pf}(H(0)|_{\mathcal{M}}\sigma_x)} \right]$$ (Shimomura et al., 28 Aug 2025).

These invariants are robust to any perturbation that does not close the point gap or break the subspace property, and can distinguish phases not detectable by standard topological classification of the full Hamiltonian.

3. Boundary Phenomena and Bulk-Boundary Correspondence

The presence of a subspace-protected topology leads to distinctive boundary phenomena. Rigorous bulk-boundary correspondence is achieved by constructing a doubled Hermitian Hamiltonian (if necessary): $$\tilde{H}(k) = \begin{pmatrix} 0 & H(k)|_{\mathcal{M}}^\dagger \ H(k)|_{\mathcal{M}} & 0 \end{pmatrix}$$ which, owing to its effective sublattice symmetry, guarantees that the number of boundary (or skin) zero modes is bounded below by the winding number $w^{\mathcal{M}}$: $N_+ - N_- = w^{\mathcal{M}}$ with $N_\pm$ the numbers of zero modes of definite chirality (Shimomura et al., 28 Aug 2025).

This leads to two major scenarios:

• For $\mathcal{M} = \mathcal{M}'$, the topology results in a “zero-winding skin effect”: a macroscopic number of vanishing eigenstates localize near one boundary—akin to the non-Hermitian skin effect—when the conventional (global) invariant vanishes.
• For $\mathcal{M} \neq \mathcal{M}'$, unpaired zero boundary modes arise, meaning a single (not paired) robust state appears at only one boundary, a marked contrast with the two-fold boundary degeneracies of conventional SPTs.

The presence and quantization of these boundary modes are intrinsic consequences of the subspace selection rule, not of symmetry alone.

4. Interplay with Internal and Subsystem Symmetries

The intersection of subspace properties with internal symmetries leads to a richer landscape of topological phases. Internal symmetries acting within the restricted subspace can enhance or trivialize the invariant, depending on their structure.

Papers on related phenomena demonstrate further extension of the concept:

• SubSymmetry-protected (SubSy or SSP) topological states are those where boundary modes are protected by a symmetry acting only on a sublattice (or subsystem), even if the bulk topological invariant associated with the full symmetry is destroyed (Wang et al., 2022, Kang et al., 3 Jun 2024).
• In such contexts, so long as the sub-symmetry is preserved (for instance, by avoiding perturbations that couple specific sublattices), the corresponding edge or corner modes—quantified, for example, by the projected polarization—remain protected and quantized at one boundary but may be destroyed at the other.
• In more complex lattice settings, subsystem symmetries acting along rigid lines or planes yield SSPT phases with decorated domain walls and membrane order parameters that cannot be destroyed even if the global symmetry is explicitly broken (1803.02369, Devakul et al., 2018).

A plausible implication is that subspace protection constitutes a generalization of symmetry protection: the relevant “protecting structure” can be a selection rule, a sub-lattice symmetry, or a combination of system-enforced connectivity and partial symmetry acting on a restricted Hilbert space sector.

5. Experimental Platforms and Theoretical Realizations

Realization of subspace-protected topological phases is predicted in a variety of systems:

• Unidirectional coupled systems: Lattices with unidirectional hopping, exemplified by coupled non-Hermitian Hatano–Nelson chains, as well as active mechanical lattices and electrical circuits, provide natural settings for the winding skin effect due to their built-in subspace structure (Shimomura et al., 28 Aug 2025).
• Partially symmetry-broken materials: Systems such as gapped bilayer graphene on boron nitride, where a flat band is partially decoupled via substrate effects, can achieve approximate subspace constraints.
• Open quantum systems: Quadratic Lindblad master equations and Keldysh-formalism systems often feature triangular self-energy structures, enforcing subspace selection rules in the effective non-Hermitian Hamiltonian and leading to subspace-protected topology.
• Hybridized flat-band systems: Strongly correlated electron systems and heavy fermion compounds, where hybridization can create subspace decoupling at low energy.

Table: Examples of Platforms and Associated Phenomena

System Type	Subspace/Selection Mechanism	Topological Phenomenon
Unidirectional coupled Hatano–Nelson	Unidirectional hopping subspace	Zero-winding skin effect
Bilayer graphene on boron nitride	Sublattice decoupling by potential	Robust unpaired zero mode
Open quantum systems (Keldysh, Lindblad)	Triangular self-energy structure	Non-Hermitian subspace topology
Flat-band/dispersive-band hybrids	Band hybridization and decoupling	Possible protected edge mode

In all cases, maintaining the relevant selection rules is crucial for the protection of the boundary modes.

6. Implications and Extensions

Subspace protection pushes the frontier of topological matter beyond conventional symmetry principles. Notably,

• Selection rules or reduced symmetry operations can enforce nontrivial topology, resulting in boundary states with robustness against broader classes of perturbations than previously recognized.
• The framework is compatible with further generalizations, such as subsystem/parachiral symmetries, noninvertible symmetries, or selection rules arising from gauge constraints.
• The construction of invariants via restriction and doubling lends itself naturally to extension in open, non-Hermitian, or driven systems, suggesting that subspace protection is a unifying principle throughout both equilibrium and non-equilibrium contexts.

A plausible implication is that subspace-protected phases will emerge as a major organizing principle in the systematic search for robust edge phenomena in synthetic materials, quantum devices, and complex engineered systems.

7. Summary

Subspace-protected topological phases are characterized by the presence of a selection rule or restricted symmetry acting on a $k$-independent subspace, which stabilizes new topological invariants and distinctive boundary phenomena, such as unpaired zero modes and skin effects. These features are distinct from and complementary to symmetry-protected topology. Their realization in engineered quantum and classical systems is supported by concrete models and robust bulk-boundary correspondence, broadening the landscape of possible topological matter and inspiring future research in both theory and experiment (Shimomura et al., 28 Aug 2025).