Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation. One of the simplest examples of such a spin system is Kitaev’s toric code. Naaijkens made mathematically rigorous the treatment of toric code on an infinite planar lattice (the thermodynamic limit), using an operator algebraic approach via algebraic quantum field theory. We adapt his methods to study the case of toric code with gapped boundary. In particular, we recover the condensation results described in Kitaev and Kong and show that the boundary theory is a module tensor category over the bulk, as expected.
REFERENCES
1.
Alicki
, R.
, Fannes
, M.
, and Horodecki
, M.
, “A statistical mechanics view on Kitaev’s proposal for quantum memories
,” J. Phys. A: Math. Theor.
40
(24
), 6451
–6467
(2007
).2.
Bratteli
, O.
and Robinson
, D. W.
, Operator Algebras and Quantum Statistical Mechanics. 1: C∗- and W∗-algebras, Symmetry Groups, Decomposition of States
, 2nd ed.
, Texts and Monographs in Physics
(Springer-Verlag
, New York
, 1987
).3.
Bratteli
, O.
and Robinson
, D. W.
, Operator Algebras and Quantum Statistical Mechanics. 2: Equilibrium States. Models in Quantum Statistical Mechanics
, 2nd ed.
, Texts and Monographs in Physics
(Springer-Verlag
, Berlin
, 1997
).4.
Doplicher
, S.
, Haag
, R.
, and Roberts
, J. E.
, “Local observables and particle statistics I
,” Commun. Math. Phys.
23
, 199
–230
(1971
).5.
Doplicher
, S.
, Haag
, R.
, and Roberts
, J. E.
, “Local observables and particle statistics II
,” Commun. Math. Phys.
35
, 49
–85
(1974
).6.
Etingof
, P.
, Gelaki
, S.
, Nikshych
, D.
, and Ostrik
, V.
, Tensor categories
, Mathematical Surveys and Monographs
(American Mathematical Society
, Providence, RI
, 2015
), Vol. 205
.7.
Fiedler
, L.
and Naaijkens
, P.
, “Haag duality for Kitaev’s quantum double model for abelian groups
,” Rev. Math. Phys.
27
(09
), 1550021
(2015
).8.
Halvorson
, H.
and Mueger
, M.
, “Algebraic quantum field theory
,” in Handbook of the Philosophy of Physics
, edited by Butterfield
, J.
and Earman
, J.
(Kluwer Academic Publishers
, 2006
).9.
Henriques
, A.
, Penneys
, D.
, and Tener
, J.
, “Categorified trace for module tensor categories over braided tensor categories
,” Doc. Math.
21
, 1089
–1149
(2016
).10.
Kadison
, R. V.
and Ringrose
, J. R.
, in Fundamentals of the theory of operator algebras. Vol. II
, Graduate Studies in Mathematics
(American Mathematical Society
, Providence, RI
, 1997
), Vol. 16
, Advanced Theory, Corrected Reprint of the 1986 Original
.11.
Kawahigashi
, Y.
, Longo
, R.
, and Müger
, M.
, “Multi-interval subfactors and modularity of representations in conformal field theory
,” Commun. Math. Phys.
219
(3
), 631
–669
(2001
).12.
Keyl
, M.
, Matsui
, T.
, Schlingemann
, D.
, and Werner
, R. F.
, “Entanglement Haag-duality and type properties of infinite quantum spin chains
,” Rev. Math. Phys.
18
(09
), 935
–970
(2006
).13.
Kitaev
, A.
, “Fault-tolerant quantum computation by anyons
,” Ann. Phys.
303
(1
), 2
–30
(2003
).14.
Kitaev
, A.
, “Anyons in an exactly solved model and beyond
,” Ann. Phys.
321
(1
), 2
–111
(2006
).15.
Kitaev
, A.
and Kong
, L.
, “Models for gapped boundaries and domain walls
,” Commun. Math. Phys.
313
(2
), 351
–373
(2012
).16.
Longo
, R.
, “Index of subfactors and statistics of quantum fields. I
,” Commun. Math. Phys.
126
(2
), 217
–247
(1989
).17.
Matsui
, T.
, “The split property and the symmetry breaking of the quantum spin chain
,” Commun. Math. Phys.
218
(2
), 393
–416
(2001
).18.
Naaijkens
, P.
, “Localized endomorphisms in Kitaev’s toric code on the plane
,” Rev. Math. Phys.
23
(04
), 347
–373
(2011
).19.
Naaijkens
, P.
, “Haag duality and the distal split property for cones in the toric code
,” Lett. Math. Phys.
101
(3
), 341
–354
(2012
).20.
Naaijkens
, P.
, “Kosaki-Longo index and classification of charges in 2D quantum spin models
,” J. Math. Phys.
54
(8
), 081901
(2013
).21.
Nayak
, C.
, Simon
, S. H.
, Stern
, A.
, Freedman
, M.
, and Das Sarma
, S.
, “Non-Abelian anyons and topological quantum computation
,” Rev. Mod. Phys.
80
(3
), 1083
–1159
(2008
).22.
Ogata
, Y.
, “A derivation of braided C∗-tensor categories from gapped ground states satisfying the approximate Haag duality
,” J. Math. Phys.
63
(1
), 011902
(2022
).23.
Ogata
, Y.
, “Type of local von Neumann algebras in abelian quantum double model
,” arXiv:2212.09036 (2022
).24.
Rieffel
, M. A.
and Van Daele
, A.
, “The commutation theorem for tensor products of von Neumann algebras
,” Bull. London Math. Soc.
7
(3
), 257
–260
(1975
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
