Stable Real-Space Invariants (SRSIs)

• Stable Real-Space Invariants (SRSIs) are discrete numerical or finite-group quantities derived from localized Wannier orbitals that reveal hidden topological characteristics.
• They are computed using the Smith normal form and band-representation matrices, enabling rigorous classification of atomic insulators and identification of topological obstructions.
• Beyond band theory, SRSIs extend to real-space quantum chemistry, persistence theory, and metric geometry, offering robust tools for classifying both crystalline and amorphous systems.

A stable real-space invariant (SRSI) is a mathematically well-defined, discrete, and stable integer or finite-group–valued quantity constructed from data that is intrinsically localized in real space. In crystalline band theory, SRSIs capture topological properties of band structures that are invisible to momentum-space symmetry indicators (SIs) and provide a complete invariant for the stable equivalence of atomic insulators. Beyond band theory, SRSIs have become a unifying theme across topological quantum chemistry, metric geometry, persistence theory, and topological data analysis, serving as robust, physically and computationally meaningful invariants wherever real-space structure determines the global behavior.

1. Formalism: Definition and Construction

Within the context of topological band theory, SRSIs are defined as linear combinations—over $\mathbb{Z}$ or $\mathbb{Z}_n$—of the multiplicities of Wannier orbitals localized at specific Wyckoff positions and transforming according to irreducible representations (irreps) of their point-group site symmetries: $$\nu_i \;=\;\sum_{p,\alpha} n^{(i)}_{p,\alpha}\,m_{p,\alpha} \quad\Bigl(\bmod\;n_i\Bigr)$$ Here:

• $p$ indexes Wyckoff positions within the unit cell.
• $\alpha$ runs over site-symmetry irreps at $p$.
• $m_{p,\alpha} \in \mathbb{Z}_{\geq 0}$ is the multiplicity of the irrep $\alpha$ at $p$ among the occupied Wannier states.
• $n^{(i)}_{p,\alpha} \in \mathbb{Z}$ determines the $i$th invariant.
• $n_i$ is the modulus: $n_i=1$ for $\mathbb{Z}$ invariants, $n_i>1$ for $\mathbb{Z}_{n_i}$ invariants.

SRSIs can be systematically constructed by analyzing the allowed adiabatic processes (encoded in an integer-valued process matrix $q$) that move or shuffle Wannier centers within a given symmetry setting. The Smith normal form of $q$ partitions the invariants into $\mathbb{Z}$ and finite-group parts, each corresponding to a distinct kind of obstruction to trivialization under symmetry-preserving deformations. This construction guarantees that SRSIs are topological invariants: they can only change when the gap closes or symmetries are explicitly broken (Hwang et al., 14 May 2025).

2. Computational Framework and Symmetry Relations

The mapping from real-space Wannier data to momentum-space irreps is realized via a band-representation matrix $BR$: $B=BR\,p,$ where $p$ encodes all $(m_{p,\alpha})$ and $B$ lists multiplicities of little-group irreps at high-symmetry momenta.

• The $\mathbb{Z}$-valued SRSIs are functionally determined by the momentum-space data $B$, and thus recover all standard symmetry indicators (SIs).
• The $\mathbb{Z}_n$-valued SRSIs, which are invisible to the momentum-space data (i.e., they lie in $\operatorname{ker} BR$), detect topological obstructions not captured by SIs.

If an attempted decomposition of an atomic band representation into sub-manifolds $V$ and $C$ fails the additive consistency for $\mathbb{Z}_n$ SRSIs,

$$\theta_{Z_n}[{\rm total}] \not\equiv \theta_{Z_n}[V] + \theta_{Z_n}[C] \bmod n,$$

then at least one branch is necessarily non-atomic—implying non-symmetry-indicated topology (Hwang et al., 14 May 2025).

3. Classification Across Space Groups

A comprehensive classification of SRSIs in all 230 nonmagnetic space groups (both spinless and spinful) was performed by full Smith normal form analysis. Key findings:

• Each SG admits at least the trivial $\mathbb{Z}$ SRSI ("total filling mod 1").
• Additional $\mathbb{Z}$ SRSIs arise in direct correspondence with standard SIs.
• $\mathbb{Z}_2$ SRSIs appear in many SGs; $\mathbb{Z}_4$ SRSIs emerge only in four spinful SGs.
• The catalogued SRSIs constitute a complete invariant for the stable classification of atomic insulators: two atomic insulators are stably equivalent if and only if all their SRSIs agree.
• The tabulated list of invariants for every space group forms the computational backbone for applications across band theory (Hwang et al., 14 May 2025).

4. Stable Equivalence of Insulators and Beyond Symmetry Indicators

SRSIs provide a necessary and sufficient condition for stable equivalence of atomic (BR-induced) insulators: two insulators $\rho_1 \uparrow G$ and $\rho_2 \uparrow G$ are stably equivalent (i.e., adiabatically deformable to one another with the addition of auxiliary trivial BRs) if and only if their SRSIs—both $\mathbb{Z}$ and $\mathbb{Z}_n$ parts—match. This allows SRSIs to fully capture the stable topological content of band structures, encompassing and extending the formalism of symmetry indicators. Notably, SRSIs diagnose topological phases—including fragile and non-symmetry-indicated ones—that are invisible to all momentum-space or SI-based diagnostics.

Explicitly, in the case of split elementary band representations (EBRs), certain splits can yield topological band sub-manifolds not detectable by SIs. Among all 211 identified cases of split EBRs in 51 space groups, $\mathbb{Z}_n$ SRSIs successfully identify nontrivial topology in all but eight exceptional cases (distributed among five space groups), where the EBR is stably, though not strictly, equivalent to the sum of its components (Hwang et al., 14 May 2025).

5. Physical Applications and Broader Manifestations

SRSIs are not unique to the crystallographic context. Their generalizations and analogues appear in numerous real-space-based classification problems:

• In topological phases of matter beyond non-interacting band theory, SRSIs generalize to many-body real-space invariants, quantifying symmetry-protected irreducible quantum numbers associated with open-boundary subregions or global toroidal ground states in strongly correlated and interacting settings (Herzog-Arbeitman et al., 2022).
• For Hofstadter-type band structures with magnetic flux, SRSIs persist as robust indicators under projective symmetries, controlling phase transitions and higher-order topology even when Bloch theory breaks down (Herzog-Arbeitman et al., 2022).
• In persistence theory and applied topology, SRSIs give rise to stable hierarchical invariants under interleaving distances, proving crucial for robust multiparameter data analysis (Gäfvert et al., 2017, Bauer et al., 2021).
• In the metric geometry of spaces (e.g., coarse geometry), stable invariants constructed from equivalence classes of "go-to-infinity" sequences at all scales fully capture large-scale (coarse) equivalence (DeLyser et al., 2011).
• In continuous geometry and shape analysis, SRSIs as explicit, polynomial (degree 4) invariants of real-space moments enable complete, stable $SE(3)$ classification for generic $C^3$ compact orientable surfaces (Hayut et al., 2021).

6. Algorithms, Computational Properties, and Limitations

The construction of SRSIs via Smith normal form reduction and linear algebraic manipulation of representation multiplicities is algorithmically tractable for band theory applications. The completeness and additive properties of SRSIs enable a fully algorithmic diagnosis of stable equivalence and topological obstructions in band representations (Hwang et al., 14 May 2025).

However, the computation of general stable invariants in multiparameter persistence modules is provably NP-hard as soon as the parameter number exceeds one, aligning with known complexity bounds for multi-filtration persistent homology (Gäfvert et al., 2017).

7. Significance, Impact, and Open Problems

SRSIs provide a rigorous, functorial, and robust framework for encoding topological information accessible through real-space data, continuously extending the scope of bulk-boundary correspondence, stable classification, and data-driven topology. They are foundational for:

• The stable and complete classification of atomic insulators and the detection of non-symmetry-indicated band topology (Hwang et al., 14 May 2025).
• Characterizing fragile topology and distinguishing between trivial, atomic, and robust topological phases in both non-interacting and interacting settings (Herzog-Arbeitman et al., 2022, Kooi et al., 2020).
• Enabling real-space computations in both crystalline and non-crystalline (quasiperiodic, amorphous, disordered) systems (Rodriguez-Vega et al., 15 May 2025, Carvalho et al., 2018).
• Establishing continuous, stable, and universal invariance principles in metric geometry and persistence theory.

Open problems include the search for computationally tractable subclasses of persistence modules allowing efficient stable-invariant computation (Gäfvert et al., 2017), characterization of SRSI completeness in more general symmetry group settings, and generalizing the theory to interacting and non-crystalline contexts with little or no underlying periodic structure.

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Key references:

• "Stable Real-Space Invariants and Topology Beyond Symmetry Indicators" (Hwang et al., 14 May 2025)
• "Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants" (Herzog-Arbeitman et al., 2022)
• "Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries" (Herzog-Arbeitman et al., 2022)
• "Stable Invariants for Multiparameter Persistence" (Gäfvert et al., 2017)
• "A coarse invariant for all metric spaces" (DeLyser et al., 2011)
• "Bulk-corner correspondence of time-reversal symmetric insulators: deduplicating real-space invariants" (Kooi et al., 2020)
• "Invariants of stable maps from the 3-sphere to the Euclidean 3-space" (Huamani et al., 2018)
• "Complete $SE(3)$ invariants for a comeagre set of $C^3$ compact orientable surfaces in $\mathbb{R}^3$" (Hayut et al., 2021)