Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach

Abstract: We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e. with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer Quantum Hall states, topological insulators and superconductors. We adapt the well-known Chern-Simons {K}-matrix description of Quantum Hall states to classify such `integer' topological phases. Our main result is a general formalism that incorporates symmetries into the {{K}}-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder include (i) bosonic insulators with a Hall conductance quantized to even integers (ii) a bosonic analog of quantum spin Hall insulators and (iii) a bosonic analog of a chiral topological superconductor, whose K matrix is the Cartan matrix of Lie group E$_8$. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly we construct microscopic models of these phases from coupled one-dimensional systems.

• The paper extends the K-matrix formalism to classify interaction-driven topological phases in 2D using a Chern-Simons framework.
• It incorporates symmetries to systematically categorize bosonic and fermionic phases in alignment with group cohomology classifications.
• The study provides detailed edge theories and microscopic models that offer clear pathways for experimental realizations.

Review of "Theory and Classification of Interacting 'Integer' Topological Phases in Two Dimensions: A Chern-Simons Approach"

The paper "Theory and Classification of Interacting 'Integer' Topological Phases in Two Dimensions: A Chern-Simons Approach" by Yuan-Ming Lu and Ashvin Vishwanath presents a comprehensive theoretical framework for understanding and classifying topological phases of matter, specifically focusing on interacting systems in two dimensions. This work is particularly notable for its use of the Chern-Simons K-matrix formalism, a technique traditionally applied to the study of quantum Hall states, to investigate and classify non-chiral topological phases without topological order.

Summary of Key Contributions

The authors build on the well-established K-matrix description of quantum Hall states, adapting it to accommodate symmetries and interactions in a systematic manner. The paper's primary contributions include:

1. Extension of K-Matrix Formalism: The adaptation of the K-matrix formalism to explore 'integer' topological phases that appear only due to interactions, analogous to integer quantum Hall states, but distinguished by their lack of topological order (e.g., no fractional excitations).
2. Symmetry Incorporation: A significant achievement of the paper is the incorporation of symmetries into the K-matrix framework, allowing for precise classification of topological phases. The authors emphasize the role of the K-matrix in describing bosonic topological phases which are exclusively defined by interactions.
3. Classification Scheme: The paper presents a classification scheme for these phases that is consistent with group cohomology classifications, thus validating the K-matrix approach. It categorizes various bosonic and fermionic phases based on symmetry groups, including U(1), time-reversal (Z_2^T), and spin-related symmetries.
4. Physical Insights and Edge Theories: Beyond classification, the work provides field theories and explicit edge state descriptions for these phases. The analysis emphasizes that the stability and properties of edge states in these systems are intimately tied to the realized symmetries and interactions.
5. Microscopic Models: The authors also construct microscopic models, particularly using quasi-one-dimensional systems, to support their theoretical framework. This approach not only validates their theoretical predictions but also suggests potential experimental realizations.

Implications and Speculations on Future Directions

This work is pivotal in providing a unified framework for understanding a broad class of topological phases that emerge from interactions and symmetries. Specifically, it underscores the significance of incorporating symmetries in the study of topological phases, thus opening new avenues for using symmetry-protected topological (SPT) phases to develop novel quantum computing paradigms and materials with robust edge states.

• Advancing Beyond Free-Fermion Models: By focusing on interacting systems, this work moves beyond the constraints of free-fermion models, providing a more comprehensive understanding of topological behaviors in correlated systems.
• Towards Higher-Dimensional Analogues: Although the investigation is limited to two dimensions, the theoretical insights could inspire further studies on analogous phases in higher dimensions, potentially leading to the discovery of new classes of topological materials.
• Experimental Pursuits: The identification of interaction-driven topological phases provides a clear direction for experimentalists to seek materials and systems that exhibit these properties, potentially uncovering new physical phenomena.
• Theoretical Extensions: Future work could explore the extension of these methods to non-Abelian states and systems with more complex symmetry groups, thereby enriching the landscape of topological quantum matter.

In conclusion, the paper by Lu and Vishwanath represents a significant step in the theoretical exploration of topological phases, emphasizing the intricate interplay between symmetry, interactions, and topology. The insights and methodologies presented pave the way for future research aimed at uncovering the rich diversity of phases in strongly correlated materials.