Topological Phase Classification

• Topological phase classification is the rigorous identification of distinct quantum phases by employing global invariants and symmetry data instead of traditional local order parameters.
• The approach integrates diverse frameworks—K-theory, cohomology, cobordism, and dynamical analysis—to address both non-interacting and interacting systems across various dimensions and symmetries.
• Recent advances including AI-driven methods and experimental techniques enhance our ability to detect and manipulate topological phases in quantum materials and photonic systems.

Topological phase classification is the systematic identification and differentiation of distinct quantum phases of matter that are characterized not by traditional local order parameters but by global, robust topological invariants. This concept emerges across non-interacting and interacting systems, in various dimensions, and encompasses crystalline and quasicrystalline materials, open or thermal systems, and non-Hermitian extensions. Modern developments unify geometric, algebraic, and cohomological approaches, leading to a comprehensive and rigorous understanding of topological phases, their transitions, and stability under symmetries, disorder, or interactions.

1. K-Theoretic, Symmetry-Based, and Tensor Category Approaches

The foundational classification of topological phases in symmetry-protected free-fermion systems is captured by K-theory and operator algebraic frameworks. The "ten-fold way" classifies free-fermion Hamiltonians into ten symmetry classes—A, AI, AII, AIII, BDI, D, DIII, C, CI, CII—using symmetry data such as time-reversal, particle-hole, and chiral symmetries [(Thiang, 2014); (Ludwig, 2015)]. K-theoretic classification encodes all symmetries (including magnetic translations, disorder, spatial symmetries) in a graded, twisted crossed-product C*-algebra, where the relevant K-group or difference-group $K_0(\mathcal{A})$ captures the "stable" phases. Dimension shifts and Bott periodicity yield the periodic table that predicts which topological invariants ($\mathbb{Z}$, $\mathbb{Z}_2$, or $0$) arise in a particular class and dimension.

In the presence of symmetries beyond the standard ten-fold way (crystalline or spatial symmetries), the classification is further refined by mapping the full symmetry group (including projective, graded, or antiunitary elements) into the algebraic structure, often producing richer invariants and protected boundary modes (Thorngren et al., 2016). For interacting (2+1)D phases, modular tensor category theory supplies a categorical classification: phases are specified by a triple of unitary braided fusion categories $\mathcal{E}\subset\mathcal{C}\subset\mathcal{M}$ and the chiral central charge $c$, with $\mathcal{E}$ capturing symmetry, $\mathcal{C}$ encoding all excitations, and $\mathcal{M}$ being the minimal modular extension (the gauged theory) (Lan, 2018).

2. Cohomological, Cobordism, and Homotopy Theoretic Frameworks

For symmetry-protected topological (SPT) phases, group cohomology and spin-cobordism frameworks generalize the classification to interacting many-body systems. The cohomology group $H^{d+1}(BG, U(1))$ (with $BG$ the classifying space of the symmetry group $G$) exhaustively classifies bosonic SPTs in $d$-dimensional systems with internal symmetry, subject to possible refinements from cobordism theory in the presence of orientation-reversing symmetries (such as reflection and inversion) (Thorngren et al., 2016, Xiong, 2019). These approaches are unified by the generalized cohomology hypothesis: invertible SPT phases form the 0-th homotopy groups of an $\Omega$-spectrum, so bulk classifications take the form $h^{\phi+d}_G(\mathrm{pt})$ for a suitable generalized cohomology $h$ and twist $\phi$, and systematic "building up" of crystalline states leverages tools like the Atiyah–Hirzebruch spectral sequence and Mayer–Vietoris sequences (Xiong, 2019). This structure automatically incorporates crystalline, glide, and higher-form symmetries, and elegantly encodes the bulk-boundary correspondence.

Interacting fermionic SPT and SET (symmetry enriched topological) phases require supercohomology and fermionic modular tensor category theory. Fermionic topological symmetry action, symmetry fractionalization, and defect data are subject to a hierarchy of obstructions and are classified, for fixed symmetry group $\mathcal{G}^f$, torsorially by the data of fractionalization for quasiparticles ($H^2$ classes) and vortex extensions, as well as $H^3$-obstruction conditions, with stacking operations governed by the Kitaev $16$-fold way and spin-cobordism groups (Aasen et al., 2021).

3. Mixed-State, Open System, and Dynamical Topological Classification

The clean spectral flattening and ground-state wavefunction approaches do not directly apply to thermal, mixed, or open quantum systems where density matrices replace pure states. Uhlmann's parallel transport provides an extension of geometric phase to density matrices, allowing one to define holonomy matrices $M$ whose eigenvalue phases generalize the Berry phase and yield quantized invariants when the amplitude spectrum is gapped (Huang et al., 2014). This leads to an integer classification (recovering the TKNN or $\mathbb{Z}_2$ invariants at $T=0$) that remains robust below a critical temperature $T_c$, but becomes trivial at high temperature, with a "gapless" intermediate regime lacking a well-defined topological index—mirroring phase transitions in open and thermal quantum systems. The holonomy is computed via

$M = \rho_0 \cdot U_{01} U_{12} \ldots U_{n-1,n}$

where $U_{ab}$ is constructed from polar decomposition of amplitudes $W = \sqrt{\rho} U$. The winding number of the Uhlmann phases along momentum-space cycles generalizes TKNN results to mixed-state and dephased settings.

Dynamical/topological classification extends to nonequilibrium quantum quenches: dynamical phase transitions are associated with non-analyticities in the Loschmidt amplitude, and their occurrence is dictated by changes in bulk topological invariants (winding number in 1D, Chern number in 2D) between initial and final Hamiltonians (Vajna et al., 2014). The number of Fisher zero crossings correlates directly with topological differences (e.g., $\Delta\nu$ in 1D), providing direct topological protection for DPTs.

Further, dynamical detection schemes use quench dynamics to extract topological invariants from (pseudo)spin dynamics on band inversion surfaces (BIS), establishing bulk–surface duality and dynamical bulk–surface correspondence. This enables high-precision classification using only non-equilibrium (time-resolved) signatures, with invariants defined by

$$w_{d-1} = \sum_j \frac{\Gamma(d/2)}{2\pi^{d/2}(d-1)!} \int_{\textrm{BIS}_j} \widetilde{\mathbf{g}}(\mathbf{k}) [d\widetilde{\mathbf{g}}(\mathbf{k})]^{d-1}$$

where the observed time-averaged spin-structure $\widetilde{\mathbf{g}}$ on the BIS uniquely encodes the universal bulk topological invariant (Zhang et al., 2018).

4. Classification with Subsystem, Crystalline, and Quasiperiodic Structure

Spatial and subsystem symmetries produce new phenomena and classification schemes. Crystalline SPT phases are classified (for crystalline topological liquids) using group cohomology over the classifying spaces of the space group, and—by the Crystalline Equivalence Principle—can often be mapped to an equivalent internal symmetry SPT classification (with orientation-reversing symmetries mapped to anti-unitary internal actions) (Thorngren et al., 2016, Xiong, 2019). Explicitly, for space group $G$,

$\text{SPT phases} \cong H^{d+1}(BG, U(1))$

for bosonic crystalline SPTs with orientation-preserving (Sohncke) groups.

Subsystem symmetry-protected phases (SSPTs) introduce subsystem symmetries acting along rigid lines or planes. In 2D, strong SSPT phases (intrinsically 2D, not reducible to stacked 1D SPT chains) are classified by

$$\mathcal{C}[G_s] = \mathcal{H}^2[G_s^2, U(1)] / (\mathcal{H}^2[G_s, U(1)])^3$$

where $G_s$ is the abelian onsite symmetry group. These phases necessarily exhibit spurious topological entanglement entropy on cylinders, which serves as a diagnostic of strong (not weak/stacked) order (Devakul et al., 2018).

In aperiodic systems, such as quasicrystals, classification relies on mapping the system to an extended parameter (higher-dimensional) space—e.g., via the ancestor 2D model for a 1D quasiperiodic chain—and assigning Chern numbers to gaps. The mapping yields robust quantized pumping and edge state protection akin to the quantum Hall effect, even in non-periodic or irrational settings (Zilberberg, 2020).

5. Non-Hermitian Topological Phase Classification and AI-Based Methods

For systems with non-Hermitian Hamiltonians (where eigenvalues can be complex), standard K-theoretic and tenfold symmetry classifications break down, especially given the emergence of new gap types (point, line, or real/imaginary gaps), non-Hermitian skin effects, and exceptions to the bulk-boundary correspondence.

The mapping of non-Hermitian Hamiltonians to enlarged Hermitian Hamiltonians with enforced chiral symmetry permits a systematic adaptation of Clifford algebra and K-theory approaches, producing new periodic tables for 38 non-Hermitian symmetry classes (including spatial/antiunitary symmetries and parity) (Liu et al., 2018). Nontrivial invariants include winding numbers (for point and line gaps) and $\mathbb{Z}_2$ indices. The existence/absence of bulk-edge correspondence is fundamentally altered, as observed for models with or without reflection symmetry.

A significant new direction is AI-driven unsupervised classification. By constructing a similarity function between Hamiltonian spectra under symmetry-preserving interpolations (e.g., for $H_1, H_2$ via $H_\alpha = (1-\alpha)H_1 + \alpha H_2$), and monitoring gap closings/topological transitions, this approach clusters Hamiltonians into distinct topological sectors without pre-assuming an invariant (Long et al., 30 Dec 2024). The topological periodic table is thus constructed by counting emergent clusters via the similarity matrix:

$$\mathcal{K}^{(\text{point/line})}_{ij} = \prod_{k \in \text{BZ}} \left[1 - \exp\left(-\frac{|v_{\text{point/line}}|^2}{\varepsilon^2}\right)\right]$$

where $v$ encodes the essential gap data. This scheme remains robust to the presence of generalized Brillouin zones (GBZs) required by skin effects or OBC-induced transitions. This unsupervised paradigm circumvents omissions associated with conventional invariant-based schemes, extending the applicability to data-driven classification in experiment as well as theory.

6. Experimental, Practical, and Theoretical Implications

From ultracold atoms to photonic meta-materials and graphene superlattices, the proper classification of topological phases underpins the design of robust devices (waveguides, quantum wires, topological lasers). Phase diagrams, transitions, and experimental detection protocols are guided by precise knowledge of the allowed invariants, their transitions, and domain of stability under disorder, interactions, or external driving. Notably, the analysis of half-Heusler compounds underscores the necessity of properly accounting for subtle orbital and symmetry interactions—the three-band mechanism involving $\Gamma_6$, $\Gamma_6^*$, and $\Gamma_8$ orbitals fundamentally alters the phase diagram and robustness of topological phases compared to prior symmetry-based expectations (Araújo et al., 2 Sep 2024).

Furthermore, practical schemes in photonic continuous media now connect the sign of an analytically constructed index $y = \varepsilon^2 - \mu^2$ (with $\varepsilon$, $\mu$ the permittivity and permeability) to the existence of robust gapless edge states and full-range impedance matching, directly linking experimental design principles to topological phase boundaries (He et al., 2023).

In summary, topological phase classification provides a mathematically and physically rigorous decomposition of quantum matter, integrating symmetry, entanglement, geometry, cohomology, and—recently—machine learning. This hierarchy continues to expand, unifying disparate phenomena while enabling both detailed theoretical understanding and practical technological development.