Fermion Path Integrals And Topological Phases (1508.04715v2)

Abstract: Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic $\theta$-angle is equal to $\pi$.

• The paper presents an anomaly-based perspective on symmetry-protected topological phases, predicting protected gapless modes at system boundaries.
• It rigorously analyzes free fermion topological phases in time-reversal invariant insulators and superconductors, highlighting quantized magneto-electric responses.
• It employs advanced topological field theory techniques, including the Atiyah-Patodi-Singer index theorem, to clarify fermion path integrals in complex quantum systems.

Fermion Path Integrals and Topological Phases

The paper "Fermion Path Integrals and Topological Phases" by Edward Witten delivers a comprehensive examination of symmetry-protected topological (SPT) phases, particularly focusing on those associated with fermionic systems. The significance of SPT phases lies in their classification through anomalies—a concept rooted deeply in theoretical physics. This paper extends the understanding of such phases in systems describable by band theory, particularly free fermions, and elucidates how these phases emerge in time-reversal invariant topological insulators and superconductors within two or three spatial dimensions.

Core Contributions

1. Anomaly-Based Perspective: The paper proposes viewing SPT phases through the lens of anomalies. This view predicts characteristic phenomena, such as gapless modes at boundaries, which are protected by symmetry characteristics of the bulk.
2. Exploration of Free Fermion Topological Phases: The paper is primarily concentrated on time-reversal invariant topological insulators and superconductors. These systems, distinct in two and three-dimensional spaces, illustrate how symmetry and topological aspects interplay to give rise to protected surface states.
3. Electromagnetic $\theta$-Angle in Topological Insulators: Witten examines the assertion that in the bulk of three-dimensional time-reversal invariant topological insulators, the electromagnetic $\theta$-angle is effectively $\pi$. This characteristic distinction underpins the topological nature of these phases and leads to specific observable phenomena, such as a quantized magneto-electric response.

Analytical Framework

The paper leverages advanced topological field theory techniques to understand fermionic systems:

• Atiyah-Patodi-Singer Index Theorem: This is employed to evaluate fermion anomalies, particularly focusing on systems with boundaries where these anomalies can manifest as physical surface states. By extending the notion of anomalies from gauge theories to condensed matter systems, it provides a framework to identify SPT phases.
• Spin and Pin Structures: The paper elucidates on utilizing sTQFT (spin topological quantum field theory) concepts, as fermions necessitate a spin structure on spacetime, particularly when it's essential to account for both orientable and non-orientable manifolds.

Results and Implications

• Global Anomalies with Spectral Flow: The paper highlights that global anomalies in defining path integrals are connected to spectral flow, where eigenvalues of associated Dirac operators cross zero in parameter spaces. This concept offers insights into the stability and robustness of surface states in topological systems.
• Bulk-Boundary Correspondence: The interplay between bulk properties, marked by the $\theta$-term or Stieffel-Whitney classes, and their influence on boundary behaviors, effectively illustrates a crucial aspect of topological matter—while the bulk houses latent topological invariants, the boundary exhibits very real physical phenomena protected by these invariants.
• Reduced Classification with Interactions: The work examines how traditional integer invariants (protecting topological phases) reduce in the presence of interactions. Specifically, interactions can alter classifications, like reducing a $\mathbb{Z}$ classification to $\mathbb{Z}_{16}$, which aligns with broader understandings in interacting phases of matter.

Future Directions

The paper paves the way for deeper exploration on several fronts:

• Investigating interacting fermionic systems, potentially leading to novel classification regimes.
• Developing mathematical tools for defining fermion path integrals in broader contexts, including strong coupling limits and systems with disorder or non-standard symmetries.
• Applying these foundational concepts to new materials, possibly incorporating time-reversal and reflection symmetries differently than considered here.

In conclusion, "Fermion Path Integrals and Topological Phases" situates itself as a critical reference for understanding symmetry-protected phases in fermionic systems through an anomaly-based approach. By bridging high-level mathematical constructs with tangible physical predictions, the paper strengthens the conceptual and practical framework linking topological phenomena to measurable physical phenomena in quantum systems.