Translation-symmetry-protected topological orders in quantum spin systems
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Wen_Translation-symmetry.pdf
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Author(s)
Kou, Su-Peng
Wen, Xiao-Gang
Date Issued
December 2009
Journal
Physical Review B
Publisher
American Physical Society
Citation
Kou, Su-Peng, and Xiao-Gang Wen. “Translation-Symmetry-Protected Topological Orders in Quantum Spin Systems.” Phys. Rev. B 80, no. 22 (December 2009). © 2009 The American Physical Society
Version
Final published version
Abstract
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four Z[subscript 2] topological indices ζ[subscript k] = 0,1 at k = (0,0), (0,π), (π,0), and (π,π) to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2A topological orders for a system on three-dimensional cubic lattice.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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DOI of Published Version
https://doi.org/10.1103/PhysRevB.80.224406
