Anomaly Obstructions to Symmetry Preserving Gapped Phases

Abstract

Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the $2d$ Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in general spacetime dimension with various types of discrete symmetries including higher-form global symmetries. As a particular application, we use our result to prove that certain $4d$ non-abelian gauge theories at $\theta =\pi$ cannot flow at long distances to a phase which simultaneously, preserves time-reversal symmetry, is confining, and is gapped. We also apply our obstruction to $4d$ adjoint QCD and constrain its dynamics.

35 pages, 9 figures; references added

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• Condensates and anomaly cascade in vector-like theories
• Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Remarks on effects of projective phase on eigenstate thermalization hypothesis
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• Lattice Analysis of $SU(2)$ with 1 Adjoint Dirac Flavor
• Unnecessary quantum criticality in $SU(3)$ kagome magnets
• The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles
• Generalized symmetry enriched criticality in (3+1)d