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arXiv:1908.08049 · Physical review. B./Physical review. B

Sorting topological stabilizer models in three dimensions

Arpit Dua, Isaac H. Kim, Meng Cheng, Dominic J. Williamson

Published Oct 23, 2019• 49 citations
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Abstract

The $S$-matrix invariant is known to be complete for translation invariant topological stabilizer models in two spatial dimensions, as such models are phase equivalent to some number of copies of toric code. In three dimensions, much less is understood about translation invariant topological stabilizer models due to the existence of fracton topological order. Here we introduce bulk commutation quantities inspired by the 2D $S$-matrix invariant that can be employed to coarsely sort 3D topological stabilizer models into qualitatively distinct types of phases: topological quantum field theories, foliated or fractal type-I models with rigid string operators, or type-II models with no string operators.

DOI
10.1103/physrevb.100.155137

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The SSS-matrix invariant is known to be complete for translation invariant topological stabilizer models in two spatial dimensions, as such models are phase equivalent to some number of copies of toric code. In three dimensions, much less is understood about translation invariant topological stabilizer models due to the existence of fracton topological order. Here we introduce bulk commutation quantities inspired by the 2D SSS-matrix invariant that can be employed to coarsely sort 3D topological stabilizer models into qualitatively distinct types of phases: topological quantum field theories, foliated or fractal type-I models with rigid string operators, or type-II models with no string operators.
We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. Our stabilizer codes generalize the Pauli codes: they are extracted from local commuting projector Hamiltonians whose terms are made of local alphabets more general than the Pauli operators. In particular, not all such codes can be entangled/disentangled with Clifford Quantum Cellular Automata. We compute their Witt-equivalence classes in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. In four spatial dimensions we obtain a group of order two, conjecturally corresponding to the non-trivial invertible phase. Additionally, we propose a representing spectrum for the generalized cohomology theory corresponding to the invertible stabilizer codes.
We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields ZN\mathbb{Z}_NZN​ gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.
Quantum error-correcting codes with two-dimensional translation invariance are known to be equivalent to copies of the two-dimensional toric code. Such a simple classification is not possible for quantum codes with higher dimensional translation invariance due to the existence of multiple types of toric codes and infinite families of fracton codes. Here, we focus on D-dimensional translation-invariant quantum codes based on length-D chain complexes. This includes multivariate multicycle codes where the number of variables equals the number of cycles. We show that such codes are equivalent to copies of D-dimensional toric codes. This directly generalizes the classification result for two-dimensional translation-invariant codes.
In this work, we develop a coupled layer construction of fracton topological orders in d=3d=3d=3 spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple d=2d=2d=2 topological phases and from stacks of d=3d=3d=3 fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.
The authors generalize here three-dimensional \ensuremath{\mathbb{Z}}2{}_{2}2​ stabilizer models---the X-cube, toric code, and Haah's code---to their \ensuremath{\mathbb{Z}}N{}_{N}N​ counterparts, revealing size-dependent ground-state degeneracy and novel excitations. They demonstrate that the generalized X-cube hosts quasi-fractons that remain immobile under local operations but become mobile via global ones. In the toric code, both closed-loop and open-string excitations emerge, while Haah's code features new fracton tripole and monopole excitations. These findings uncover unconventional topological orders and broaden the understanding of fracton phases beyond the conventional \ensuremath{\mathbb{Z}}2{}_{2}2​ framework.