Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology (1403.1467v3)

Abstract: We propose that Symmetry Protected Topological Phases with a finite symmetry group G are classified by cobordism groups of the classifying space of G. This provides an explanation for the recent discovery of bosonic SPT phases which do not fit into the group cohomology classification. We discuss the connection of the cobordism classification of SPT phases to gauge and gravitational anomalies in various dimensions.

Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

In this paper, the researchers explore the classification of Symmetry Protected Topological (SPT) phases using cobordism theory, moving beyond the traditional group cohomology framework. SPT phases are important constructs in condensed matter physics and high-energy physics, characterized by equivalence classes of gapped phases of matter with a symmetry group $G$ and no long-range entanglement. This paper focuses on bosonic SPT phases and proposes a refined classification schema employing the Pontryagin dual of the torsion subgroup of the oriented bordism group, specifically considering the implications of thermal Hall responses.

To contextualize, it has been previously suggested that bosonic SPT phases in space-time dimension $d$ are categorized by the degree-$d$ cohomology group of $BG$ with $U(1)$ coefficients. However, limitations were observed in the classification, especially regarding phases with time-reversal symmetry in four-dimensional frameworks that eluded capture by existing models. This paper revisits these classifications, leveraging cobordism groups to provide a more nuanced perspective.

Key Findings

1. Refined Classification using Cobordism: The researchers propose that bosonic SPT phases, particularly those with finite internal symmetry group $G$ and vanishing thermal Hall response, are classified by the quotient $\Omega^{d}_{SO}(BG, U(1))/\mathrm{im}\, e$. This is shown to be a superior classification method compared to traditional group cohomology, particularly for $d \leq 6$.
2. Role of ’t Hooft Anomalies: The classification of SPT phases is connected to ’t Hooft anomalies, which appear in $d-1$ dimensions. These anomalies, ambiguous phases in gauged systems' partition functions, also receive a comprehensive categorization following this refined approach.
3. Bosonic Time-Reversal Symmetry: The group discusses the classification of bosonic SPT phases with time-reversal symmetry through unoriented cobordism groups $\Omega^{d}_{O}(pt, U(1))$, aligning their findings with Thom’s results and demonstrating that some phases recognized by group cohomology are trivial in cobordism classification, further refining our understanding of these symmetries.
4. Discrepancies in Classifications: Notably, certain SPT phases are inherently detected by cobordism but remain obscure via group cohomology. For example, nontrivial bosonic phases in dimensions like $d = 5$ are identified through this method, showcasing limitations in prior classification schemas.

Implications and Future Directions

The paper's proposed framework redefines understanding in theoretical physics of bosonic SPT phases, specifically how gravitational and gauge anomalies interplay with these phases' classification. By employing cobordism, the authors enhance the methodology, adding pivotal insights into the symmetries potentially affecting SPT phases in dimensions greater than $d = 4$.

This poses implications for future developments in AI and quantum computing since deeper insights into the structural characteristics of quantum phases might inform more robust error-correcting codes and enhance quantum system modeling technologies. Consider extending this classification to fermionic SPT phases using spin cobordism groups, potentially bridging predictions from K-theory in the context of advanced quantum models.

Overall, while addressing noted limitations within group cohomology, the paper advances the community's understanding of SPT phases and the anomalies associated with them, proposing valuable new avenues to explore in both theoretical and applied physics domains.