Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

Abstract

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary ${\mathbb{Z}}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $\pi$ statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary ${\mathbb{Z}}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.

60 pages, 14 figures, 3 tables; v2: typos corrected, references added; v3: typos corrected

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• Modified Villain formulation of abelian Chern-Simons theory
• Gauge Enhanced Quantum Criticality Beyond the Standard Model
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
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• Subsystem Symmetry Fractionalization and Foliated Field Theory
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• Anomalies of global symmetries on the lattice
• Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes
• Loops in 4+1d Topological Phases