Built on

A. Dua, D. J. Williamson, J. Haah, and M. Cheng, Compactifying fracton stabilizer models, Phys. Rev. B 99
Original
1903
Earlier work this paper cites.
W. Shirley, K. Slagle, and X. Chen, Twisted foliated fracton phases, arXiv:1907.09048 (2019c)
Original
1907
Earlier work this paper cites.
G. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212
1988
Earlier work this paper cites.
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Communications in Mathematical Physics 129
1990
Earlier work this paper cites.
A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A - Atomic, Molecular, and Optical Physics 54
1996
Earlier work this paper cites.
A. Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 452
1996
Earlier work this paper cites.
A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303
2003
Earlier work this paper cites.
M. Levin and X.-G. Wen, Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B 67
2003
Earlier work this paper cites.
C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Physical Review Letters 94
2005
Earlier work this paper cites.
A. Hamma, P. Zanardi, and X.-G. Wen, String and membrane condensation on three-dimensional lattices, Phys. Rev. B 72
2005
Earlier work this paper cites.
Kitaev Alexei, Anyons in an exactly solved model and beyond, Annals of Physics 321
2006
Earlier work this paper cites.
H. Bombin and M. A. Martin-Delgado, Exact topological quantum order in D = 3 and beyond: Branyons and brane-net condensates, Phys. Rev. B 75
2007
Earlier work this paper cites.
C. Castelnovo, C. Chamon, and D. Sherrington, Quantum mechanical and information theoretic view on classical glass transitions, Physical Review B 81
Original
2010
Earlier work this paper cites.
S. Bravyi, B. Leemhuis, and B. M. Terhal, Topological order in an exactly solvable 3D spin model, Ann. Phys. 326
Original
2010
Earlier work this paper cites.
I. H. Kim, Local non-CSS quantum error correcting code on a three-dimensional lattice, arXiv:1012.0859 (2010)
Original
2010
Earlier work this paper cites.
J. Haah, Local stabilizer codes in three dimensions without string logical operators, Physical Review A - Atomic, Molecular, and Optical Physics 83
Original
2011
Earlier work this paper cites.
S. Bravyi and J. Haah, Energy landscape of 3D spin hamiltonians with topological order, Physical Review Letters 107
Original
2011
Earlier work this paper cites.
C. Castelnovo and C. Chamon, Topological quantum glassiness, Philosophical Magazine 92
Original
2012
Earlier work this paper cites.
I. H. Kim, 3D local qupit quantum code without string logical operator, arXiv:1202.0052 (2012)
Original
2012
Earlier work this paper cites.
K. Walker and Z. Wang, (3+1)-TQFTs and topological insulators, Frontiers of Physics 7
Original
2012
Earlier work this paper cites.
B. Yoshida, Exotic topological order in fractal spin liquids, Physical Review B 88
Original
2013
Earlier work this paper cites.

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Then

Z. Weinstein, E. Cobanera, G. Ortiz, and Z. Nussinov, Absence of finite temperature phase transitions in the x-cube model and its zp generalization, arXiv:1812.04561 (2018)
Original
2018
Later among the works it cites.
Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Symmetric Fracton Matter: Twisted and Enriched, arXiv:1805.09800 (2018)
Original
2018
Later among the works it cites.
A. T. Schmitz, H. Ma, R. M. Nandkishore, and S. A. Parameswaran, Recoverable information and emergent conservation laws in fracton stabilizer codes, Physical Review B 97
Original
2018
Later among the works it cites.
A. T. Schmitz, Gauge Structures: From Stabilizer Codes to Continuum Models, arXiv:1809.10151 (2018)
Original
2018
Later among the works it cites.
J. Haah, L. Fidkowski, and M. B. Hastings, Nontrivial Quantum Cellular Automata in Higher Dimensions, arXiv:1812.01625 (2018)

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2019

Sorting topological stabilizer models in three dimensions

Dua, Arpit, Kim, Isaac H., Cheng, Meng et al.

Understand

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.

Built on

A. Dua, D. J. Williamson, J. Haah, and M. Cheng, Compactifying fracton stabilizer models, Phys. Rev. B 99
Original
1903
Earlier work this paper cites.
W. Shirley, K. Slagle, and X. Chen, Twisted foliated fracton phases, arXiv:1907.09048 (2019c)
Original
1907
Earlier work this paper cites.
G. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212
1988
Earlier work this paper cites.
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Communications in Mathematical Physics 129
1990
Earlier work this paper cites.
A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A - Atomic, Molecular, and Optical Physics 54
1996
Earlier work this paper cites.
A. Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 452
1996
Earlier work this paper cites.
A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303
2003
Earlier work this paper cites.
M. Levin and X.-G. Wen, Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B 67
2003
Earlier work this paper cites.
C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Physical Review Letters 94
2005
Earlier work this paper cites.
A. Hamma, P. Zanardi, and X.-G. Wen, String and membrane condensation on three-dimensional lattices, Phys. Rev. B 72
2005
Earlier work this paper cites.
Kitaev Alexei, Anyons in an exactly solved model and beyond, Annals of Physics 321
2006
Earlier work this paper cites.
H. Bombin and M. A. Martin-Delgado, Exact topological quantum order in D = 3 and beyond: Branyons and brane-net condensates, Phys. Rev. B 75
2007
Earlier work this paper cites.
C. Castelnovo, C. Chamon, and D. Sherrington, Quantum mechanical and information theoretic view on classical glass transitions, Physical Review B 81
Original
2010
Earlier work this paper cites.
S. Bravyi, B. Leemhuis, and B. M. Terhal, Topological order in an exactly solvable 3D spin model, Ann. Phys. 326
Original
2010
Earlier work this paper cites.
I. H. Kim, Local non-CSS quantum error correcting code on a three-dimensional lattice, arXiv:1012.0859 (2010)
Original
2010
Earlier work this paper cites.
J. Haah, Local stabilizer codes in three dimensions without string logical operators, Physical Review A - Atomic, Molecular, and Optical Physics 83
Original
2011
Earlier work this paper cites.
S. Bravyi and J. Haah, Energy landscape of 3D spin hamiltonians with topological order, Physical Review Letters 107
Original
2011
Earlier work this paper cites.
C. Castelnovo and C. Chamon, Topological quantum glassiness, Philosophical Magazine 92
Original
2012
Earlier work this paper cites.
I. H. Kim, 3D local qupit quantum code without string logical operator, arXiv:1202.0052 (2012)
Original
2012
Earlier work this paper cites.
K. Walker and Z. Wang, (3+1)-TQFTs and topological insulators, Frontiers of Physics 7
Original
2012
Earlier work this paper cites.
B. Yoshida, Exotic topological order in fractal spin liquids, Physical Review B 88
Original
2013
Earlier work this paper cites.