Models with on-site symmetry protected topological order in two spatial dimension

Details Models with on-site symmetry protected topological order in two spatial dimension Models with on-site symmetry protected topological order in two spatial dimension

2011-06-23

arxiv.org/abs/1106.4752v1

Physics

Condensed Matter

Strongly Correlated Electrons

14 pages, 11 figures

Scientific paper

Topological order in one dimensional systems has been well characterized and classified. However in higher dimensions, what kind of topological order exist is still an open question, especially in interacting systems. In this paper, we focus on short range entangled systems and present an exactly solvable spin model on two dimensional square lattice with nontrivial on-site symmetry protected topological order (SPT order). We find a close connection between SPT order and the 3-cocycle of the on-site symmetry group, which we expect to generalize into a more complete understanding of SPT order in any dimension. Specifically, our model has an on-site $Z_2$ symmetry which is not broken in the ground state. The ground state is short range entangled but is inequivalent to a product state if the on-site $Z_2$ symmetry is kept. The nontrivialness of the system shows up on the boundary where the effective degrees of freedom has a symmetry which does not take an on-site form and cannot have a gapped symmetric ground state. We show this by developing the tool of Matrix Product Unitary Operators and through its relation to the 3-cocycles of the on-site symmetry group. We use similar ideas to generalize this model to a two-dimensional fermionic system with on-site $Z_2$ symmetry protected topological order.

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