Fetching the paper…

Reading the bibliography…

Spin Topological Field Theory and Fermionic Matrix Product States · Around

2016

Spin Topological Field Theory and Fermionic Matrix Product States

Kapustin, Anton, Turzillo, Alex, You, Minyoung

Original
PDF
Post on X

Understand

We study state-sum constructions of G-equivariant spin-TQFTs and their relationship to Matrix Product States.

We show that in the Neveu-Schwarz, Ramond, and twisted sectors, the states of the theory are generalized Matrix Product States.
We apply our results to revisit the classification of fermionic Short-Range-Entangled phases with a unitary symmetry G and determine the group law on the set of such phases.
Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.

Built on

M. Atiyah, “Riemann Surfaces and Spin Structures,” Ann. Sci. Ecole Norm. Sup. 4 (1971), p.47-62
1971
Earlier work this paper cites.
V. Ostrik “Module Categories, Weak Hopf Algebras and Modular Invariants,” Transformation Groups Vol.2, no. 2, 177-206 (2003)
2003
Earlier work this paper cites.
G. Segal, “The Definition of a Conformal Field Theory,” in Topology, Geometry and Quantum Field Theory, ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. (2004), p.421-577
2004
Earlier work this paper cites.
F. Verstraete, J. I. Cirac, V. Murg, “Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems,” Adv. Phys. 57
Original
2008
Earlier work this paper cites.

Similar

X. Chen, Z. Gu, X. Wen, “Classification of Gapped Symmetric Phases in 1D Spin Systems”, Phys. Rev. B 83, 035107 (2011)
2011
Cited alongside, same era.
L. Fidkowski, A. Kitaev, “Topological Phases of Fermions in One Dimension,” Phys. Rev. B, 83
2011
Cited alongside, same era.
X. Chen, Z. Gu, X. Wen, “Complete Classification of 1D Gapped Quantum Phases in Interacting Spin Systems”, Phys. Rev. B 84
2011
Cited alongside, same era.
A. Kapustin, “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,” arXiv:1403.1467
Original
Cited in the paper.
D. S. Freed, “Short-Range Entanglement and Invertible Field Theories,” arXiv:1406.7278[cond-mat.str-el]
Original
Cited in the paper.
D. Freed, M. Hopkins, “Reflection positivity and invertible topological phases,” arXiv:1604.06527 [hep-th]
Original
Cited in the paper.
K. Shiozaki, S. Ryu, “Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in 1+1 dimensions,” arXiv:1607.06504 [cond-mat.str-el]
Original
Cited in the paper.
A. Kapustin, A. Turzillo, M. You, “Topological Field Theory and Matrix Product States,” arXiv:1607.06766 [cond-mat.str-el]
Original
Cited in the paper.
D. Gaiotto, A. Kapustin, “Spin TQFTs and Fermionic Phases of Matter,” arXiv:1505.05856 [cond-mat.str-el]
Original
Cited in the paper.
N. Bultinck, D. J. Williamson, J. Haegeman, and F. Verstraete, “Fermionic Matrix Product States and One-Dimensional Topological Phases,” arXiv:1610.07849 [cond-mat.str-el]
Original
Cited in the paper.
G. Moore, G. Segal, “D-branes and K-theory in 2D Topological Field Theory,” arXiv:hep-th/0609042
Cited in the paper.
N. Schuch, D. Perez-García, I. Cirac, “Classifying Quantum Phases using MPS and PEPS,” arXiv:1010.3732 [cond-mat.str-el]
Original
Cited in the paper.

Then

Z. Gu, X. Wen, “Symmetry-protected topological orders for interacting fermions: fermionic topological nonlinear sigma models and a special group supercohomology theory,” Phys. Rev. B 90
2014
Later among the works it cites.
A. Kapustin, R. Thorngren, A. Turzillo, Z. Wang, “Fermionic symmetry protected topological phases and cobordisms,” JHEP 12
2015
Later among the works it cites.
S. Novak, I. Runkel, “State Sum Construction of Two-Dimensional TQFTs on Spin Surfaces,” J. Knot Theory Ramifications 24 no.05, 1550028 (2015)
2015
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…