Definition and Classification of Fermi Surface Anomalies

Abstract: We propose that the Fermi surface anomaly of symmetry group $G$ in any dimension is universally classified by $G$-symmetric interacting fermionic symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the non-commutative nature of the phase space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a $(0+1)$-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-$p$ Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation.

• The paper introduces a novel framework that reduces the classification of Fermi surface anomalies to a (0+1)D SPT problem.
• It utilizes non-commutative geometry and emergent loop group symmetry to relate gapless fermionic modes to topological boundary states.
• The classification provides actionable insights on when symmetric mass generation is possible, illustrated through concrete examples like Z4-symmetric interactions.

Definition and Classification of Fermi Surface Anomalies

The paper "Definition and Classification of Fermi Surface Anomalies" addresses the intriguing challenge of classifying Fermi surface anomalies using principles from symmetry-protected topological (SPT) phases. It provides a comprehensive framework for understanding the intricate relation between Fermi surface anomalies and interacting fermionic SPT phases, casting these familiar condensed matter concepts in a novel mathematical light.

The authors propose a classification system for Fermi surface anomalies based on their correspondence with fermionic SPT phases in the phase space. They argue that despite the seemingly complex nature of phase spaces, the classification of these anomalies can be boiled down to a problem in $(0+1)$-dimensional spacetime. This insight arises from considering the phase space dimensions—characterized by both position and momentum—as possessing non-trivial commutation relations that effectively reduce the dimensionality of the classification problem. Therefore, momentum dimensions are treated as "negative", a consideration that simplifies the classification task to a one-dimensional SPT problem.

The methodology presented involves a phase-space perspective on Fermi liquids, suggesting that gapless fermions on the Fermi surface are analogous to topological boundary modes of Chern insulators in the phase space. This is a departure from conventional views and relies heavily on a detailed understanding of non-commutative geometry and synthetic dimension reduction.

Notably, the paper introduces the notion of an emergent loop group symmetry ($\mathrm{L}_{\partial\Omega}U(1)$) at the Fermi surface, which broadens the symmetry classification of these anomalies. This group captures the pointwise transformations possible at each point on the Fermi surface, aligning well with the anomalous features intrinsic to the Fermi liquid states.

From a practical perspective, the classification indicates when symmetric mass generation (SMG) of the Fermi surface—avoiding gapless fermions via interactions—is either possible or hindered by anomalies. The authors explore codimension-specific cases and examine non-perturbative measures such as the role of interstitial defects, emphasizing their utility in defining and measuring Fermi surface anomalies.

Concrete examples are employed to illustrate various scenarios, such as the capability for $Z_4$-symmetric Fermi surfaces to undergo SMG in the presence of specific four-fermion interactions, or $Z_{4n+2}$ classifications yielding conditions for anomaly-free gapping with odd flavors, which aligns with recent SMG developments. Additionally, discussions on how dimensional considerations resolve classification discrepancies—for example, the distinction between $(4+1)$D and $(0+1)$D SPT phases—highlight the depth of theoretical contributions in this paper.

In terms of broader implications, this framework bridges condensed matter physics with topological and geometric insights, contributing to a stronger theoretical basis for understanding exotic phases of matter. Future research might expand these concepts to complex Fermi surfaces with more challenging symmetry and interaction constraints, potentially elucidating SMG transitions and other non-trivial gap phenomena in correlated electron systems.

In summary, this work contributes a significant theoretical advance in defining and classifying Fermi surface anomalies by elegantly leveraging concepts from topology and condensed matter physics, and sets the groundwork for future explorations into interacting systems with complex symmetry structures.