Topological orders and duality mappings via symmetry and algebraic considerations

Abstract: We show how symmetries can lead to a dimensional reduction and topological order and discuss the effects of thermal fluctuations.

We will further show how exact dualities as well as duality relations that appear only in a sector of certain theories ("emergent dualities") can be systematically derived. Our method relies on the use of bond algebras wherein the algebraic relations between all terms in the Hamiltonian are examined. This method enables us to solve exactly several quantum spin systems in high dimensions and to determine the exact self dual points of all Z_N lattice gauge theories in 3+1 dimensions. Implications for quantum information will be discussed.