Anomaly-Free Symmetries with Obstructions to Gauging and Onsiteability

Area of Science:

• Condensed Matter Physics
• Quantum Information Theory
• Lattice Field Theory

Background:

• Traditional understanding posits that internal symmetries must be gauged or on-site to be anomaly-free.
• Anomalous symmetries typically lead to inconsistencies in quantum field theories or lattice models.
• The existence of anomaly-free symmetries with non-standard properties has been a subject of theoretical interest.

Purpose of the Study:

• To challenge the conventional understanding of anomalous symmetries.
• To construct and characterize novel types of anomaly-free internal symmetries in two-dimensional lattice models.
• To demonstrate that symmetries not amenable to gauging or on-site construction can still be consistent.

Main Methods:

• Construction of unitary, internal symmetries for two-dimensional lattice models.
• Analysis of the gaugeability and on-site disentanglement properties of these symmetries.
• Verification of anomaly-freedom by demonstrating the existence of symmetric, gapped Hamiltonians with unique ground states.
• Characterization of these symmetries using a topological index in cohomology.

Main Results:

• We present explicit counterexamples to the lore that non-gaugeable or non-on-site symmetries are necessarily anomalous.
• We construct two-dimensional lattice models with unitary internal symmetries that cannot be coupled to gauge fields or disentangled to on-site operators.
• These constructed symmetries are proven to be anomaly-free, admitting symmetric gapped ground states.
• A new characterization of these anomaly-free symmetries is provided via an index [ω]∈H^{2}(G,Q_{+}).

Conclusions:

• The study introduces a new class of anomaly-free symmetries that defy traditional classification.
• These findings expand the landscape of possible symmetries in quantum many-body systems and lattice field theories.
• The characterization via a cohomology index opens avenues for further theoretical exploration and classification of quantum phases of matter.