Subspace-Protected Topological Phases and Bulk-Boundary Correspondence

TL;DR

This paper introduces subspace-protected topological phases in free-fermionic systems, defining new invariants and boundary phenomena, and establishes a bulk-boundary correspondence for these phases, expanding understanding beyond symmetry-based protection.

Contribution

It proposes a novel subspace property for Hamiltonians that leads to new topological invariants and boundary phenomena, and demonstrates the bulk-boundary correspondence in these phases.

Findings

Discovery of subspace-protected topological boundary modes

Definition of new topological invariants based on subspace properties

Establishment of bulk-boundary correspondence for subspace-protected phases

Abstract

While tremendous research has revealed that symmetry enriches topological phases of matter, more general principles that protect topological phases have yet to be explored. In this Letter, we elucidate the roles of subspaces in free-fermionic topological phases. A subspace property for Hamiltonians enables us to define new topological invariants. It results in peculiar topological boundary phenomena, i.e., the emergence of an unpaired zero mode or zero-winding skin modes, characterizing subspace-protected topological phases. We establish and demonstrate the bulk-boundary correspondence in subspace-protected topological phases. We further discuss the interplay of the subspace property and internal symmetries. Toward application, we also propose possible platforms possessing the subspace property.

Equations

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

$$⟨{\psi }^{\prime }|H(\mathbf{k})|\psi ⟩=0,|\psi ⟩\in \mathcal{M},|{\psi }^{\prime }⟩\in {\mathcal{M}}^{\prime \perp }.$$

$$\forall \mathbf{k},\det H(\mathbf{k})\ne 0,$$

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

$$\tilde{H}(\mathbf{k}):=\left(\begin{array}{cc}0& H(\mathbf{k}){|}_{\mathcal{M}}^{†}\\ H(\mathbf{k}){|}_{\mathcal{M}}& 0\end{array}\right)\curvearrowright \mathcal{M}\oplus {\mathcal{M}}^{\prime }$$

$$N_{+}-N_{-}=w^{\mathcal{M}},$$

$$\tilde{H}\left(\begin{array}{c}{\varphi }_n\\ 0\end{array}\right)=0,\text{i.e.,}H{|}_{\mathcal{M}}^{\mathrm{SIBC}}{\varphi }_n=0.$$

$$H^{\mathrm{SIBC}}{\varphi }_n=H{|}_{\mathcal{M}}^{\mathrm{SIBC}}{\varphi }_n=0,$$

$$\#(\text{zero modes of }{H}^{\mathrm{SIBC}})\ge N_{+}\ge w^{\mathcal{M}}.$$

$$\#(\text{zero modes of }{H}^{-\mathrm{SIBC}})\ge -w^{\mathcal{M}},$$

$$H(\mathbf{k})=\left(\begin{array}{cc}ah(\mathbf{k})& H_{12}(\mathbf{k})\\ bh(\mathbf{k})& H_{22}(\mathbf{k})\end{array}\right).$$

$$H(\mathbf{k})=\left(\begin{array}{cc}h(\mathbf{k})& H_{12}(\mathbf{k})\\ 0& H_{22}(\mathbf{k})\end{array}\right).$$

$$H(k)=\left(\begin{array}{cc}h(k)& r(k)\\ 0& h(k{)}^{\ast }\end{array}\right),h(k)=t\cos k+\mathrm{i}g\sin k+\mathrm{i}V,$$

$$w=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{BZ}}\mathrm{Tr}[H(k{)}^{-1}\mathrm{d}H(k)]=0,$$

$$w^{\mathcal{M}}=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{BZ}}h(k{)}^{-1}\mathrm{d}h(k)=\mathrm{sgn}\left(\frac{g}{t}\right)\ne 0.$$

$$H(\mathbf{k})=\left(\begin{array}{cc}0& H_{12}(\mathbf{k})\\ h(\mathbf{k})& H_{22}(\mathbf{k})\end{array}\right).$$

$$H(k)=\left(\begin{array}{cc}0& h(k{)}^{\ast }\\ h(k)& r(k)\end{array}\right),h(k)=u+v{e}^{\mathrm{i}k},$$

$$\displaystyle w^{\mathcal{M}}=\frac{1}{2\pi\mathrm{i}}\int_{\mathrm{BZ}}h(k)^{-1}\mathrm{d}h(k)=\begin{dcases*}1&if $|u|<|v|$\\ 0&if $|u|>|v|$\end{dcases*},$$

$$H(k)=\left(\begin{array}{cc}0& H_1(k{)}^{†}\\ H_1(k)& H_2(k)\end{array}\right),$$

$$H_1(k)=\mathrm{i}g\sin k+(h+t\cos k){\sigma }_z+\mathrm{i}{r}_x\sin k{\sigma }_x+r_y\sin k{\sigma }_y,$$

$$H_2(k)=d_0+d_0^{\prime }\sin k+\sum _{i=x,y,z}(d_i+d_i^{\prime }\sin k){\sigma }_i$$

$$(-1{)}^{\nu }:=\mathrm{sgn}\left[\frac{\mathrm{Pf}(H(\pi ){|}_{\mathcal{M}}{\sigma }_x)}{\mathrm{Pf}(H(0){|}_{\mathcal{M}}{\sigma }_x)}\right]=\mathrm{sgn}\left(\frac{h-t}{h+t}\right)$$

$$H_{\text{stack}}(k)=\left(\begin{array}{cc}H(k)& {\Delta }^{†}\\ \Delta & H(k)\end{array}\right),\Delta =\left(\begin{array}{cc}0& c{I}_2\\ -c{I}_2& 0\end{array}\right),c\in \mathbb{R},$$

Full text

Symmetry is a fundamental principle to characterize and classify topological phases of matter [1]. In particular, various types of symmetry provide free-fermion systems with rich topological structure, as in topological insulators and superconductors [2, 3, 4, 5, 6]. A unitary symmetry resolves the Hilbert space into sectors, in which topological phases are individually classified on the basis of internal symmetries [7, 8, 9, 10, 11, 12] or crystalline symmetries [13, 14, 15, 16, 17, 18, 19, 20, 21]. A significant characteristic of topological phases is the bulk-boundary correspondence: the nontrivial topological phases in gapped bulk involves the presence of gapless boundary modes [22, 23, 24, 25, 26, 27, 28, 29]. Nontrivial topological insulators protected by time-reversal symmetry (TRS) exhibit helical gapless modes as the Kramers pairs [30, 31, 32, 33, 34, 35, 36, 37, 38, 39], and nontrivial topological superconductors can host Majorana zero modes reflecting particle-hole symmetry [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52].

The scheme of topological phases is extended into non-Hermitian systems with symmetries and gap structures enlarged from Hermitian cases [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114]. Above all, point-gapped topological phases are intrinsic in non-Hermitian bulk systems [74] and can cause anomalous boundary behaviors such as non-Hermitian skin effects [59, 67, 85, 84] or the emergence of exceptional boundary states [89, 91, 100]. Symmetry also enriches the point-gapped topological phases; for instance, $\mathbb{Z}_{2}$ skin effects occur protected by transpose-type time-reversal symmetry (TRS*†*) [85].

However, several topological phenomena in free-fermion systems challenge explanation beyond conventional notion of symmetry. For instance, recent studies have observed topological zero modes protected by a so-called sub-symmetry in some Hermitian systems with broken sublattice symmetry [115, 116, 117, 118, 119, 120]. In non-Hermitian cases, trivial point-gapped systems have been reported to host skin modes even with no symmetry [121, 122]. Thus, exploring general principles protecting topological phenomena is an elusive but urgent issue.

In this Letter, we reveal that subspaces of Hilbert spaces serve as a principle to introduce a novel type of free-fermion topological phases in both Hermitian and non-Hermitian systems beyond the conventional notion of symmetry. Specifically, we consider the following condition, which we call the subspace property, for a Bloch Hamiltonian $H(\boldsymbol{k})$ acting on a Hilbert space $\mathcal{H}$ of internal degrees of freedom,

[TABLE]

where $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are subspaces of $\mathcal{H}$ independent of the wave vector $\boldsymbol{k}$, as illustrated in Fig. 1 (a). Note that the subspace property is not reduced to group-theoretical symmetry operations. We propose the concept of topological phases protected by the subspace property (1), dubbed subspace-protected topological phases, and establish the bulk-boundary correspondence in the phases. Furthermore, we clarify the interplay of subspace-protected topological phases and internal symmetries. Toward application, we provide searching guidelines for physical platforms possessing the subspace property.

Consequences of the subspace property.— Prior to the investigation of subspace-protected topological phases, let us explain a physical interpretation of the subspace property. Although the subspace property in Eq. (1) is apparently mathematical, it is equivalent to a selection rule,

[TABLE]

As well as familiar selection rules rooted in symmetry, Eq. (2) may prohibit particular backscattering processes, allowing boundary modes to stabilize against perturbations. Therefore, we should have a topological invariant protected by the subspace property to clarify the stable boundary modes. Later, we will rigorously justify this intuition.

To define topological phases, we identify a condition that systems with the subspace property have gapped spectra. For this purpose, we classify the subspace property according to the relation between the dimensions of $\mathcal{M}$ and $\mathcal{M}^{\prime}$ as shown in Fig. 1 (b).

If $m\coloneqq\dim\mathcal{M}>\dim\mathcal{M}^{\prime}$, then $H(\boldsymbol{k})$ hosts at least a flat band at zero energy in the bulk spectrum: For linearly independent states $\ket{e_{1}},\ldots,\ket{e_{m}}\in\mathcal{M}\setminus\{0\}$, $m$ states $H(\boldsymbol{k})\ket{e_{1}},\ldots,H(\boldsymbol{k})\ket{e_{m}}\in\mathcal{M}^{\prime}$ are linearly dependent, and thus, there is a particular superposition $H(\boldsymbol{k})(c_{1}(\boldsymbol{k})\ket{e_{1}}+\cdots+c_{m}(\boldsymbol{k})\ket{e_{m}})=c_{1}(\boldsymbol{k})H(\boldsymbol{k})\ket{e_{1}}+\cdots+c_{m}(\boldsymbol{k})H(\boldsymbol{k})\ket{e_{m}}=0$ for every $\boldsymbol{k}$, giving a flat band. This flat band hinders defining any gapped topological invariant to detect zero modes, so we exclude this case from our consideration hereafter.

The case of $\dim\mathcal{M}<\dim\mathcal{M}^{\prime}$ includes the situation of $\mathcal{M}^{\prime}=\mathcal{H}$, for which the subspace property $H(\boldsymbol{k})\mathcal{M}\subseteq\mathcal{H}$ always holds for generic Hamiltonian $H(\boldsymbol{k})$. In this case, hence, the subspace property may bring nothing.

Thus, we focus on the case of $m\coloneqq\dim\mathcal{M}=\dim\mathcal{M}^{\prime}$. This case is further divided into two cases, $\mathcal{M}=\mathcal{M}^{\prime}$ and $\mathcal{M}\neq\mathcal{M}^{\prime}$. A difference between these cases appears in the stability of the subspace property with respect to energy shifts. If $\mathcal{M}=\mathcal{M}^{\prime}$, the subspace property $H(\boldsymbol{k})\mathcal{M}\subseteq\mathcal{M}$ for $H(\boldsymbol{k})$ leads to that for the shifted Hamiltonian $(H(\boldsymbol{k})-c)$, as $(H(\boldsymbol{k})-c)\mathcal{M}\subseteq\mathcal{M}$. By contrast, if $\mathcal{M}\neq\mathcal{M}^{\prime}$, the energy shift $H(\boldsymbol{k})\mapsto H(\boldsymbol{k})-c$ breaks the subspace property due to $c\mathcal{M}\neq\mathcal{M}^{\prime}$. As we will see below, this different behavior results in distinct topological phenomena.

Subspace-protected topological invariant and bulk-boundary correspondence.— To introduce topological invariant under the subspace property, we assume both the presence of point gap in $H(\boldsymbol{k})$ at zero energy [74],

[TABLE]

and the subspace property (1) with $\dim\mathcal{M}=\dim\mathcal{M}^{\prime}$. Here, we do not suppose $H(\boldsymbol{k})$ is Hermitian; The discussion below is valid for both Hermitian and non-Hermitian systems.

Our strategy to construct the topological invariant is utilizing the restriction $H(\boldsymbol{k})|_{\mathcal{M}}\colon\mathcal{M}\to\mathcal{M}^{\prime}$ of $H(\boldsymbol{k})$ onto the subspace $\mathcal{M}$. We can show that $H(\boldsymbol{k})|_{\mathcal{M}}$ is also point-gapped: From Eq. (3), $H(\boldsymbol{k})$ is invertible, then from $\dim\mathcal{M}=\dim\mathcal{M}^{\prime}$, $H(\boldsymbol{k})|_{\mathcal{M}}$ is also invertible. Thus, $H(\boldsymbol{k})|_{\mathcal{M}}$ is point-gapped at zero energy, i.e. $\det H(\boldsymbol{k})|_{\mathcal{M}}\neq 0$.

Therefore, we can introduce an additional point-gap topological invariant for $H(\boldsymbol{k})|_{\mathcal{M}}$. Specifically, in the absence of any symmetry, we have the winding number for odd spatial dimension $d=2n+1$ as

[TABLE]

where BZ is the $d$-dimensional Brillouin zone [74]. The quantity $w^{\mathcal{M}}$ is quantized in an integer and invariant under continuous deformation of $H(\boldsymbol{k})$ preserving the point gap and the subspace property with fixed $\mathcal{M}$ and $\mathcal{M}^{\prime}$. Thus, $w^{\mathcal{M}}$ is a subspace-protected topological invariant.

The subspace-protected topological invariant in the bulk gives rise to characteristic boundary behaviors under open boundary conditions. To elucidate this bulk-boundary correspondence, we consider a semi-infinite system on $X=\mathbb{Z}_{\geq 0}\times\mathbb{Z}^{d-1}$ with a single boundary in one direction. Let $H^{\mathrm{SIBC}}\curvearrowright\mathcal{H}\otimes\ell^{2}(X)$ be the real space Hamiltonian under the semi-infinite boundary condition (SIBC) corresponding to the Bloch Hamiltonian $H(\boldsymbol{k})\curvearrowright\mathcal{H}$, where $\ell^{2}(X)$ is the Hilbert space of square-summable functions on $X$ 111 Here, the symbol $H\curvearrowright\mathcal{K}$ denotes $H$ is an operator mapping a space $\mathcal{K}$ to itself, $H:\mathcal{K}\to\mathcal{K}$. . In the same way, we make the real space SIBC Hamiltonian $H|_{\mathcal{M}}^{\mathrm{SIBC}}\colon\mathcal{M}\otimes\ell^{2}(X)\to\mathcal{M}^{\prime}\otimes\ell^{2}(X)$ corresponding to the restricted Hamiltonian $H(\boldsymbol{k})|_{\mathcal{M}}$. As shown below, $H|_{\mathcal{M}}^{\mathrm{SIBC}}$ can host zero eigenvalues, the multiplicity of which is bounded by the subspace-protected topological invariant $w^{\mathcal{M}}$. To see this, we introduce the Hermitian Hamiltonian

[TABLE]

and the corresponding SIBC Hamiltonian $\tilde{H}^{\mathrm{SIBC}}\curvearrowright(\mathcal{M}\oplus\mathcal{M}^{\prime})\otimes\ell^{2}(X)$. Because $\tilde{H}(\boldsymbol{k})$ respects the sublattice symmetry (SLS), $\Gamma\tilde{H}(\boldsymbol{k})\Gamma^{-1}=-\tilde{H}(\boldsymbol{k})$, with $\Gamma=\begin{pmatrix}I_{\mathcal{M}}&0\\ 0&-I_{\mathcal{M}^{\prime}}\end{pmatrix}$ and $w^{\mathcal{M}}$ is the topological invariant for $\tilde{H}(\boldsymbol{k})$ protected by the SLS, the index theorem tells [26, 124, 29]

[TABLE]

where $N_{\pm}$ is the number of zero modes of $\tilde{H}^{\mathrm{SIBC}}$ with chirality $\pm 1$. Hence, we get $N_{+}$ linearly independent eigenfunctions $\phi_{n}\in\mathcal{M}\otimes\ell^{2}(X)$ ($n=1,\ldots,N_{+}$) satisfying

[TABLE]

The zero modes $\phi_{n}$ of the restricted Hamiltonian $H|_{\mathcal{M}}^{\mathrm{SIBC}}$ are nothing but zero modes of $H^{\mathrm{SIBC}}$,

[TABLE]

so $H^{\mathrm{SIBC}}$ hosts at least $N_{+}$ zero modes. Therefore, we have the bulk-boundary correspondence

[TABLE]

Apparently, Eq. (9) makes no sense if $w^{\mathcal{M}}<0$. This is because we are considering SIBC systems on the positive side with respect to the boundary. By taking another SIBC Hamiltonian $H^{-\mathrm{SIBC}}$ on $\mathbb{Z}_{\leq 0}\times\mathbb{Z}^{d-1}$, we also have

[TABLE]

which makes sense for $w^{\mathcal{M}}<0$.

So far we assume the subspace property with $\dim\mathcal{M}=\dim\mathcal{M}^{\prime}$. As mentioned above, we can further divide it into two cases, $\mathcal{M}\neq\mathcal{M}^{\prime}$ and $\mathcal{M}=\mathcal{M}^{\prime}$. If $\mathcal{M}\neq\mathcal{M}^{\prime}$, the subspace property is unstable against energy shifts and thus protects only zero modes under the SIBC among all the boundary modes. By contrast, if $\mathcal{M}=\mathcal{M}^{\prime}$, we can shift the origin of energy respecting the subspace property, so $H^{(-)\mathrm{SIBC}}$ has at least $(-)w^{\mathcal{M}}$ modes at any energy $E\in\mathbb{C}$ in a neighborhood of zero, which are nothing but skin modes in one dimension [85]. Ultimately, a nonzero subspace-protected topological invariant $w^{\mathcal{M}}$ implies two distinct boundary phenomena: the non-Hermitian skin effect for $\mathcal{M}=\mathcal{M}^{\prime}$ and the presence of boundary zero modes for $\mathcal{M}\neq\mathcal{M}^{\prime}$.

We expect the boundary phenomena in nontrivial phases to occur under the open boundary condition (OBC) as well as the SIBC. We illustrate this expectation with several examples of $\mathcal{H}=\mathbb{C}^{2}$ and $\dim\mathcal{M}=\dim\mathcal{M}^{\prime}=1$. Without loss of generality, we take a specific basis where the basis vector of $\mathcal{M}$ is $\begin{pmatrix}1\\ 0\end{pmatrix}$. Then, let $\begin{pmatrix}a\\ b\end{pmatrix}$ be the basis vector of $\mathcal{M}^{\prime}$. Now the subspace property reads $\begin{pmatrix}H_{11}(\boldsymbol{k})\\ H_{21}(\boldsymbol{k})\end{pmatrix}\propto\begin{pmatrix}a\\ b\end{pmatrix}$, so there is a scalar function $h(\boldsymbol{k})$ of $\boldsymbol{k}$ such that

[TABLE]

The restricted Hamiltonian is $H(\boldsymbol{k})|_{\mathcal{M}}=h(\boldsymbol{k})$.

In the case of $\mathcal{M}=\mathcal{M}^{\prime}$ with $a=1,b=0$, $H(\boldsymbol{k})$ becomes a triangular matrix as

[TABLE]

As a simple model, we consider a one-dimensional non-Hermitian system given by

[TABLE]

with $t,g,V\in\mathbb{R}^{\times}$, $|V|<|g|$ and an arbitrary function $r(k)$. Physically, this is a one-way coupled model of two Hatano-Nelson chains [125, 126] without disorder. The spectrum of $H(k)$ is $\{h(k),h(k)^{*}\}$, which forms two ellipses. While the prime point-gap topological invariant (the winding number) of $H(k)$ is zero

[TABLE]

the subspace-protected topological invariant is non-zero

[TABLE]

Thereby, we expect the presence of the skin effect, exhibiting macroscopic number of boundary-localized eigenmodes under the OBC and distinct spectra under the periodic boundary condition (PBC) and OBC. In Fig. 2 (a), we provide the eigenvectors under the OBC and the PBC and OBC spectra of the model. As shown in the figure, the skin modes appear under the OBC, despite $w=0$. The zero winding skin modes are consistent with the nonzero $w^{\mathcal{M}}$, and thus are protected by the subspace property, beyond the conventional notion of symmetry.

As an example of $\mathcal{M}\neq\mathcal{M}^{\prime}$, suppose that $a=0,b=1$, i.e., $\mathcal{M}^{\prime}=\mathcal{M}^{\perp}$. Then, $H(\boldsymbol{k})$ is of the form

[TABLE]

In particular, we consider a one-dimensional Hermitian Hamiltonian given by

[TABLE]

with $|u|\neq|v|$ to be point-gapped at zero energy. For $r=0$ this model is just the Su-Schrieffer-Heeger (SSH) chain [127, 128, 129, 130, 131] with the SLS, but for nonzero $r$ it no longer has any symmetry. Still, we can define the subspace-protected topological invariant $w^{\mathcal{M}}$ for this model as

[TABLE]

independent of $r(k)$. Corresponding to the topological phase transition with respect to $w^{\mathcal{M}}$, the boundary zero mode appears under the OBC, protected by the subspace property, as seen in Fig. 2 (b). In sharp contrast to the conventional SSH chain, the nontrivial phase supports an unpaired zero mode (a localized zero mode only at a single boundary), and the bulk gap never closes even at the transition point.

The bulk-boundary correspondence between $w^{\mathcal{M}}$ and boundary zero modes is always valid for Hermitian Hamiltonians with $\mathcal{M}\neq\mathcal{M}^{\prime}$, because there is no skin effect. However, if the system suffers from skin effects, say due to non-zero $w$ [85, 84], $w^{\mathcal{M}}$ could fail to bound the number of boundary zero modes. In this case, we should use a subspace-protected topological invariant defined on the generalized Brillouin zone (GBZ) to recover the bulk-boundary correspondence [67].

Interplay of subspace-protected topological phases and internal symmetries.— So far, we have considered subspace-protected topological phases in the absence of any symmetries. Still, the subspace property together with additional symmetry brings a new topological phase. We realize this by imposing additional symmetry to the restricted Hamiltonian $H(\boldsymbol{k})|_{\mathcal{M}}$. Fixing the basis of $\mathcal{H}$ with a particular one, we obtain the matrix representation of $H(\boldsymbol{k})|_{\mathcal{M}}$, for which the Bernard-LeClair type of symmetry condition [132, 74] can be defined.

To be concrete, we take an example where $H(k)|_{\mathcal{M}}$ satisfies symmetry conditions of class BDI [133] as follows:

[TABLE]

with $h,t,g,r_{x},r_{y}\in\mathbb{R}$, $d_{\mu},d^{\prime}_{\mu}\in\mathbb{C}$ ($\mu=0,x,y,z$). The Hermitian Hamiltonian (19) has no symmetry due to the presence of $H_{2}(k)$ but satisfies the subspace property with $\mathcal{M}^{\prime}=\mathcal{M}^{\perp}=0\oplus 0\oplus\mathbb{C}\oplus\mathbb{C}$. The restricted Hamiltonian is $H(k)|_{\mathcal{M}}=H_{1}(k)$, which is non-Hermitian and has time reversal symmetry (TRS) $H(-k)|_{\mathcal{M}}^{*}=H(k)|_{\mathcal{M}}$, particle-hole symmetry (PHS) $\sigma_{x}H(-k)|_{\mathcal{M}}^{\mathsf{T}}\sigma_{x}=-H(k)|_{\mathcal{M}}$, and chiral symmetry (CS) $\sigma_{x}H(k)|_{\mathcal{M}}^{\dagger}\sigma_{x}=-H(k)|_{\mathcal{M}}$. Whereas the PHS or CS for $H(k)|_{\mathcal{M}}$ makes the subspace-protected topological invariant $w^{\mathcal{M}}$ in Eq. (4) trivial, the symmetries enable us to define another point-gap topological invariant $\nu\in\{0,1\}$ for $H(k)|_{\mathcal{M}}$ as [74]

[TABLE]

if $H(k)|_{\mathcal{M}}$ is point-gapped around $E=0$.

The bulk-boundary correspondence is still confirmed for the current case by constructing $\tilde{H}(k)$ in Eq. (5) again. As numerical evidence, we provide spectra of the model (19) resolved by the parameter $h$ in Fig. 3 (a). Topological phase transitions occur at $h=\pm t$, and the model hosts two-degenerate zero modes in the nontrivial phases ($|h|<|t|$) of $\nu=1$.

The presence of the $\mathbb{Z}_{2}$ invariant $\nu$ suggests that stacked models of $H(k)$ always belong to the trivial phase of $\nu=0$. To check this, we stack two $H(k)$’s and add a coupling term $\Delta$ to construct the following model,

[TABLE]

with respecting the subspace property $H_{\text{stack}}(k)\mathcal{M}\oplus\mathcal{M}\subseteq\mathcal{M}^{\prime}\oplus\mathcal{M}^{\prime}$ and the symmetry conditions of class BDI for the restricted Hamiltonian $H_{\text{stack}}(k)|_{\mathcal{M}\oplus\mathcal{M}}=\begin{pmatrix}H_{1}(k)&cI_{2}\\ -cI_{2}&H_{1}(k)\end{pmatrix}$. As shown in Fig. 3 (b), the four-degenerate zero modes at $c=0$ are gapped out as the coupling strength $c$ increases. Thus, the coexistence of the subspace property and symmetry condition for the restricted Hamiltonian $H(\boldsymbol{k})|_{\mathcal{M}}$ guarantees a $\mathbb{Z}_{2}$ topological phase, without any symmetry of the whole Hamiltonian $H(\boldsymbol{k})$.

Guideline for the search of physical platforms possessing the subspace property.— The subspace property may manifest itself in various physical systems. In what follows, we give an overview of relevant physical platforms for cases $\mathcal{M}=\mathcal{M}^{\prime}$ and $\mathcal{M}\perp\mathcal{M}^{\prime}$.

One can realize the case $\mathcal{M}=\mathcal{M}^{\prime}$ by coupling a system to another system unidirectionally as in Eq. (13) [134, 135, 136]. Relevant experiments have already been carried out in an active mechanical lattice [137] and an electrical circuit [122]. Such a system can host zero-winding skin modes with exceptional deficiency [137]. In open quantum systems, furthermore, the self-energy of a fermionic Green function in the Keldysh formalism [138, 139, 140] or the third-quantization form of a quadratic fermionic Lindbladian [141, 142, 143] is written by a triangular matrix, which has a subspace property with $\mathcal{M}=\mathcal{M}^{\prime}$. Thereby, we can define a subspace-protected topological invariant for the self-energy or Lindbladian, which depends only on the retarded and advanced Green functions or the effective Hamiltonian.

Moving on to the case $\mathcal{M}\perp\mathcal{M}^{\prime}$, we have two approaches to find it in materials. First, by partially breaking the sublattice symmetry of a bipartite system, the desired subspace property is achieved. This approach shares the idea with the subsymmetry-protected topological phases [116, 120]. An example is a bilayer graphene on a boron nitride substrate, where couplings to the substrate break the sublattice symmetry of a graphene in the lower layer [144]. Second approach is to hybridize a flat band with dispersive bands. For instance, the effective Hamiltonian in equilibrium heavy fermion systems may possess an approximate subspace property with $\mathcal{M}\perp\mathcal{M}^{\prime}$ if both the self-energy and the $d$- or $f$-bands have little dispersion.

Discussion.— In summary, we have discovered subspace-protected topological phases and established the bulk-boundary correspondence. Our work reveals that the new principle beyond symmetry may play an essential role in topological phases.

Conventional topological phases have a close relation to quantum anomalies [145, 146, 147, 148, 28, 149, 150, 151, 94, 152], so we expect that some anomaly corresponds to subspace-protected topological phases from field-theoretical viewpoints. To clarify a subspace anomaly, we could refer to noninvertible symmetry [153, 154, 155, 156], in which not a group but a category describes symmetry operations due to the absence of an inversion operation. In fact, the subspace property can be regarded as a noninvertible generalization of some symmetries, such as unitary symmetry for $\mathcal{M}=\mathcal{M}^{\prime}$ and sublattice symmtery for $\mathcal{M}^{\prime}=\mathcal{M}^{\perp}$. A subspace anomaly might yield novel phases of matter in strongly-correlated systems such as invertible phases or topological ordered phases.

Through the Letter, we have assumed in the subspace property (1) that $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are complex subspaces. Taking $\mathcal{M}$ and $\mathcal{M}^{\prime}$ as real or quaternionic subspaces may enrich subspace-protected topological phases. Furthermore, $K$-theoretical classification of the phases is also an open problem.

Note added.— A part of this work was presented in [157]. After completing this work, we became aware of a recent work [158]. Although Ref. [158] introduces a similar terminology ”subspace symmetry”, it is an emergent symmetry of a Hamiltonian, not a property of a subspace. Thus, the concept is entirely different from ours.