Subspace Property for Hamiltonians

• Subspace property for Hamiltonians is a structural constraint where H(k) maps a fixed subspace 𝓜 into another 𝓜′, enabling unique topological invariants independent of traditional symmetries.
• It produces distinct boundary modes such as zero-energy unpaired modes or skin modes, established through a robust bulk-boundary correspondence in both Hermitian and non-Hermitian systems.
• This framework provides new classification tools with winding numbers and ℤ₂ invariants for engineered quantum systems, influencing designs in circuitry, graphene, and heavy fermion platforms.

A subspace property for Hamiltonians refers to a structural constraint in which the action of the Hamiltonian H(k) on a particular (crystal momentum–independent) subspace 𝓜 of the Hilbert space 𝓗 maps it into another fixed subspace 𝓜′, i.e., H(k)𝓜 ⊆ 𝓜′ for all k. This selection rule gives rise to a new class of topological invariants and phases—termed subspace-protected topological phases—which are not dictated by conventional internal symmetries but by the architecture of the Hilbert space itself. The presence of this subspace property generates distinctive boundary phenomena and establishes an associated form of bulk-boundary correspondence, with implications for both Hermitian and non-Hermitian systems (Shimomura et al., 28 Aug 2025).

1. Mathematical Formulation of the Subspace Property

The subspace property is formalized as follows: for Bloch Hamiltonians H(k) acting on a finite-dimensional Hilbert space 𝓗, there exist two subspaces 𝓜, 𝓜′ ⊆ 𝓗 (not depending on k) such that

$$H(k)\,\mathcal{M}\,\subseteq\,\mathcal{M}^{\prime} \qquad\forall\,k,$$

or equivalently, for any |ψ⟩∈𝓜 and |ψ′⟩∈𝓜′⊥,

$\langle \psi'| H(k) | \psi \rangle = 0.$

This relation is tantamount to a selection rule, similar in spirit to that imposed by symmetry constraints, but originating purely from the structural features of the Hilbert space decomposition rather than group-theoretic considerations. This formalism decouples topological protection from the necessity of internal (e.g., symmetry) constraints, relying instead on subspace mapping (Shimomura et al., 28 Aug 2025).

2. Subspace Dimensionality and Classification

The essential case is when dim 𝓜 = dim 𝓜′ ≡ m, rendering the restriction H(k)|𝓜 : 𝓜 → 𝓜′ a generically invertible map when H(k) is point-gapped at zero energy (i.e., det H(k) ≠ 0 ∀k). There are two structurally distinct regimes:

• 𝓜 = 𝓜′: The subspace property is "stable" against global energy shifts; H(k) can be shifted by a scalar without violating the property. This leads to "zero-winding skin modes" with boundary-localized states whose energy is not pinned to zero.
• 𝓜 ≠ 𝓜′ (often 𝓜′ = 𝓜⊥): Here, the property is destroyed by an energy shift, so topological boundary phenomena are strictly pinned to zero energy, most prominently through the emergence of "unpaired zero modes" at a single boundary.

If dim 𝓜 > dim 𝓜′, then H(k) necessarily supports a zero-energy flat band by linear dependence; if dim 𝓜 < dim 𝓜′, the constraint is vacuous (Shimomura et al., 28 Aug 2025).

3. Topological Invariants Derived from the Subspace Property

Provided det H(k) ≠ 0 for all k, one can define a winding number on the restricted Hamiltonian H(k)|𝓜. In odd spatial dimensions (d = 2n+1), in the absence of additional symmetries, the invariant is

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

where the BZ denotes the Brillouin zone and Tr₍𝓜₎ is the trace over 𝓜. This integer invariant is stable under continuous deformations of H(k) that preserve the point gap and the subspace property. A nonzero value signals a subspace-protected topological phase, independent of symmetry protection (Shimomura et al., 28 Aug 2025).

4. Bulk-Boundary Correspondence and Characteristic Boundary Modes

The nontrivial invariant w^𝓜 induces robust boundary phenomena under open (e.g., semi-infinite) boundary conditions. By constructing a "doubled" Hermitian Hamiltonian,

$$\tilde{H}(k) = \begin{pmatrix} 0 & H|_{\mathcal{M}}^\dagger \ H|_{\mathcal{M}} & 0 \end{pmatrix},$$

which possesses an artificial chiral symmetry, the index theorem applies: the difference N₊ – N₋ of zero modes with chiralities ±1 equals w^𝓜. For 𝓜 = 𝓜′, boundary states are "skin modes" (energetically unpinned), while for 𝓜 ≠ 𝓜′ (e.g., 𝓜′ = 𝓜⊥), one obtains robustly localized, unpaired zero-energy modes confined to a single boundary. In both cases, the boundary phenomena reflect a bulk-boundary correspondence that does not rely on traditional symmetry-protected mechanisms (Shimomura et al., 28 Aug 2025).

5. Interplay with Internal Symmetries and Derived ℤ₂ Invariants

When the restricted Hamiltonian H(k)|𝓜 additionally satisfies internal symmetries (e.g., chiral, time-reversal, particle-hole), the structure admits further classification. For instance, when H(k)|𝓜 falls into class BDI, a ℤ₂ invariant arises: $$(-1)^{\nu} = \mathrm{sgn}\left[\frac{\mathrm{Pf}(H(\pi)|_{\mathcal{M}}\sigma_x)}{\mathrm{Pf}(H(0)|_{\mathcal{M}}\sigma_x)}\right],$$ where σ_x acts on the internal degrees of freedom of 𝓜 and Pf refers to the Pfaffian. This distinguishes nontrivial (ν=1) and trivial (ν=0) subspace-protected topological phases, with degeneracy and localization properties of the boundary zero modes accordingly (Shimomura et al., 28 Aug 2025).

6. Physical Realizations and Platform Proposals

Several platforms manifest the subspace property:

• 𝓜 = 𝓜′: Realizations via engineered one-way couplings leading to upper-triangular Hamiltonians, already achieved in mechanical and electrical circuit lattices (observing zero-winding skin modes).
• Open Quantum Systems: The effective self-energy in Lindblad or Keldysh settings naturally implements the subspace property through its block-triangular structure.
• 𝓜 ≠ 𝓜′: Engineered by partial symmetry breaking in bipartite systems, as in bilayer graphene on boron nitride and in heavy fermion systems with hybridization between dispersive and flat bands.

Such setups enable the observation of both skin modes and unpaired zero modes, depending on the relation between 𝓜 and 𝓜′ (Shimomura et al., 28 Aug 2025).

7. Broader Significance and Outlook

The framework of subspace-protected topological phases extends the taxonomy of topological states beyond symmetry-based classifications. The bulk-boundary correspondence persists in this setting, but with boundary phenomena (such as unpaired zero modes or skin effects) rooted in the subspace structure rather than symmetry. This suggests new design principles for topological phases in both Hermitian and non-Hermitian systems. The ability to define and compute topological invariants on subspaces provides new tools for exploring exotic boundary physics in engineered and natural quantum systems (Shimomura et al., 28 Aug 2025).