2026/07/23 by Kansei Inamura, Oskar Wojdel, Lukasz Fidkowski +1
#quant-ph #cond-mat.str-el #hep-th #math.QA
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a ℤp one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality S, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT T. In the continuum, they form a central extension of PSL(2,ℤ4) for p=2, and of SL(2,ℤp) for odd primes p, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a ℤp 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For p=2 they are generated by the non-trivial semion QCA, and for odd prime p they are generated by the non-trivial ℤp Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.