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Higher-Form Anomalies on Lattices.

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We developed a method to define

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Area of Science:

  • Theoretical Physics
  • Condensed Matter Physics
  • Quantum Information

Background:

  • Higher-form symmetries in tensor product Hilbert spaces are emergent.
  • Topological nature of symmetry generators requires energetic enforcement of Gauss law.
  • Understanding 't Hooft anomalies is crucial for characterizing quantum systems.

Purpose of the Study:

  • To present a general method for defining 't Hooft anomalies of higher-form symmetries.
  • To construct an index characterizing 't Hooft anomalies in (2+1)D lattice models.
  • To generalize anomaly characterization to arbitrary dimensions and p-form symmetries.

Main Methods:

  • Constructing an index representing a cohomology class in H^{4}(B^{2}G,U(1)) for (2+1)D models.
  • Utilizing finite-depth circuits to realize Gauss-law operators for 1-form G symmetry.
  • Generalizing the construction to arbitrary d spatial dimensions and p-form G symmetries.

Main Results:

  • A general method for defining 't Hooft anomalies of higher-form symmetries in lattice models.
  • An index characterizing the anomaly in (2+1)D, generalizing Else-Nayak.
  • A formula for anomaly characterization in d-dimensions: H^{d+2}(B^{p+1}G,U(1)).

Conclusions:

  • The developed method provides a universal framework for 't Hooft anomalies of higher-form symmetries.
  • The constructed index successfully characterizes anomalies in specific dimensions and generalizes known results.
  • This work offers a powerful tool for studying topological phases and quantum field theories.