Spin TQFTs and fermionic phases of matter

Abstract: We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spin-statistics relation.

• The paper demonstrates that incorporating spin structures resolves fermionic ambiguities in lattice models, linking SPT phases to invertible spin-TQFTs.
• It details how fermionic toric codes and Turaev-Viro constructions require spin structures to properly embed essential sign conventions.
• The authors develop a robust framework using quadratic refinements and higher cup products that bridges theoretical physics with spin cobordism classifications.

Analysis of "Spin TQFTs and Fermionic Phases of Matter"

The paper "Spin TQFTs and Fermionic Phases of Matter" by Davide Gaiotto and Anton Kapustin proposes an insightful examination of lattice constructions for gapped fermionic phases of matter and their associated topological quantum field theories (TQFTs). The authors provide a detailed theoretical treatment linking these phases to spin-TQFTs, highlighting a critical dependence on spin structures in resolving ambiguities encountered in such systems.

Key Contributions and Results

The authors focus on two primary lattice model constructions: the fermionic Symmetry Protected Topological (SPT) phases and the fermionic analogues of the string-net constructions in 2D. These constructions traditionally struggle with ambiguities unless additional geometric structures are considered. The pivotal argument put forth is that incorporating spin structures can resolve these ambiguities; thus, the fermionic SPT phases are not merely topological but specifically tied to spin-TQFTs. It's shown that the dependence on spin structures is essential for the proper definition of partition functions, echoing the spin-statistics relation in continuum field theories.

1. Fermionic SPT Phases: The study revisits the Gu-Wen fermionic SPT construction, pinpointing that these phases naturally give rise to invertible spin-TQFTs. The authors elucidate the need for spin structures by demonstrating how otherwise persistent fermionic ambiguities (termed obstructions) manifest in complex topologies.
2. Fermionic Toric Codes and Turaev-Viro Construction: The paper extends its analysis to include the nature of fermionic toric codes and general adaptations of Turaev-Viro constructions to fermionic settings. Again, it identifies the necessity for spin structures, explicating how these structures naturally incorporate necessary fermionic sign conventions into lattice models.
3. Mathematical Structures: The paper provides detailed discussions on quadratic refinements of cohomological pairings in lattice systems, utilizing the formal structure of higher cup products to articulate sign conventions and obstructions in change of triangulations (changes in mesh or lattice).
4. Theoretical Implications: Beyond giving a framework for existing construction methods of fermionic phases, this work makes parallel connections to theoretical concepts in condensed matter physics and quantum field theory, suggesting a conjectural relationship between these lattice models and spin cobordism classifications in topology.

Implications and Future Directions

The real strength of this paper is its theoretical implications, suggesting that many constructs in fermionic condensed matter systems inherently predict or require a fermionic topological quantum field theory (spin-TQFT) at low energy. This indicates that spin structures are fundamental to correctly describing relevant phenomena in fermionic lattice models and that existing bosonic analogues may be inadequate for representing these systems' full richness.

The discussions in this paper provide a compelling argument that fermionic phases and their associated quantum field theories might be more deeply intertwined with geometry than previously acknowledged, promoting further inquiry into generalized TQFT frameworks that can accommodate such complexities universally.

For future explorations, the linkages posited between fermionic phases of matter and complex topological structures like spin cobordism could be extended through more experimental validations or by expanding existing mathematical frameworks, perhaps via computational modelling or simulations. The exploration of these theoretical landscapes could foster new approaches to understanding rich phenomena in both mathematical physics and condensed matter systems with stronger ties to quantum information theory and topological phase transitions.

In conclusion, the paper elegantly places lattice models of fermionic phases of matter within the broader context of spin-topological field theoretical frameworks, offering extensive foundational insights that catalyze potential advancements in both theoretical and applied physics domains.