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Haah codes on general three manifolds · Around

2018

Haah codes on general three manifolds

Tian, Kevin T., Samperton, Eric, Wang, Zhenghan

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Haah codes represent a singularly interesting gapped Hamiltonian schema that has resisted a natural generalization, although recent work shows that the closely related type I fracton models are more commonplace.

These type I siblings of Haah codes are better understood, and a generalized topological quantum field theory framework has been proposed.
Following the same conceptual framework, we outline a program to generalize Haah codes to all 3-manifolds using Hastings' LR stabilizer codes for finite groups.

Built on

W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.)
1982
Earlier work this paper cites.
J. Hempel, Residual finiteness for 3 3 -manifolds, Combinatorial group theory and topology (Alta, Utah, 1984), Ann. of Math. Stud., vol. 111, pp. 379–396, Princeton Univ. Press, Princeton, NJ, 1987
1987
Earlier work this paper cites.
L. Funar, TQFT for general Lie algebras and applications to open 3-manifolds, Journal of Mathematical Sciences-University of Tokyo
1997
Earlier work this paper cites.
A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002
2002
Earlier work this paper cites.
J. Haah, Local stabilizer codes in three dimensions without string logical operators, Physical Review A
2011
Earlier work this paper cites.

Similar

J. Haah, Commuting Pauli Hamiltonians as maps between free modules, Communications in Mathematical Physics
2013
Cited alongside, same era.
M. B. Hastings, LR codes, private communication, 2014
2014
Cited alongside, same era.
S. Vijay, J. Haah, and L. Fu, A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations, Physical Review B
2015
Cited alongside, same era.
J. Haah, Algebraic Methods for Quantum Codes on Lattices, Revista Colombiana de Matemáticas
2016
Cited alongside, same era.

Then

R. M. Nandkishore, and M. Hermele, Fractons, arXiv:1803.11196 2018
Original
2018
Closest in time.
W. Shirley, K. Slagle, Z. Wang, and X. Chen, Fracton models on general three-dimensional manifolds
2018
Closest in time.
E. Samperton, K. T. Tian, and Z. Wang, On fracton phases of matter, In preparation, 2018
2018
Closest in time.
M. Shokrian-Zini, and Z. Wang, Conformal field theories as scaling limit of anyonic chains, Communications in Mathematical Physics
2018
Closest in time.

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