Abstract
We study glide protected topological (GSPT) phases of interacting bosons and fermions in three spatial dimensions with certain on-site symmetries. They are crystalline SPT phases, which are distinguished from a trivial product state only in the presence of non-symmorphic glide symmetry. We classify these GSPT phases with various on-site symmetries such as U(1) and time reversal, and show that they can all be understood by stacking and coupling two-dimensional (2D) short-range-entangled phases in a glide-invariant way. Using such a coupled layer construction we study the anomalous surface topological orders of these GSPT phases, which gap out the 2D surface states without breaking any symmetries. While this framework can be applied to any non-symmorphic SPT phase, we demonstrate it in many examples of GSPT phases including the non-symmorphic topological insulator with ‘hourglass fermion’ surface states.
Highlights
After the discovery of topological insulators (TIs) with surface Dirac fermions[1,2,3], a large class of symmetryprotected topological (SPT) phases[4, 5] have been revealed to exist beyond Landau’s paradigm
While systematic classifications of on-site symmetry (G0) protected topological phases of interacting bosons have been obtained from group cohomology[15] and cobordism[32, 33], there is no general classification of spatial symmetry protected phases especially for interacting fermions
Here we focus on SPT phases protected by glide symmetry, it’s straightforward to see that all above discussions and fixed-point wavefunctions (8) apply to any 2-fold non-symmorphic symmetries, such as 2-fold screw symmetry where each layer is perpendicular to the screw axis
Summary
After the discovery of topological insulators (TIs) with surface Dirac fermions[1,2,3], a large class of symmetryprotected topological (SPT) phases[4, 5] have been revealed to exist beyond Landau’s paradigm. Take the two-dimensional (2d) surface of a three-dimensional (3d) SPT phase for example, though usually gapless in a weakly-interacting fermion system like TIs, strong interactions can fully gap out the surface states in a symmetric way by developing intrinsic topological orders on the 2d surface[6,7,8,9,10,11,12,13] These gapped surface topological orders are anomalous, in the sense that anyons carry certain symmetry quantum numbers that are not allowed in a pure 2d state[14]. In this work we focus on crystalline SPT phases protected by non-symmorphic glide symmetry and other global symmetries, such as U (1) charge/spin symmetry and time reversal symmetry T These “weak” SPT phases are distinguished from a trivial product state only in the presence of glide symmetry, coined glide symmetry protected (GSPT) phases. We explicitly construct these GSPT phases by coupling an array of 2d SPT layers[19, 26, 29, 30] preserving the global symmetries, in a glide-invariant fashion
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