Lattice models that realize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>-1 symmetry-protected topological states for even <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>
Lokman Tsui, Xiao-Gang Wen
Abstract
Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high-energy scales while other topological excitations have low energies. The low-energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry-protected topological order. This motivates us, as a simplest example, to study a lattice model of ${\mathbb{Z}}_{n}$-1 symmetry-protected topological (1-SPT) states in $3+1$ dimensions for even $n$. We write an exactly solvable lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with nontrivial self-statistics. For the $n=2$ case, where the bulk classification is given by an integer $m$ mod 4, we show that the boundary can be gapped with double-semion topological order for $m=1$ and toric code for $m=2$. The bulk ground-state wave-function amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.