A new theory of dualities and dimensional reduction: Applications to phase transitions, topological quantum order, and quantum information processing

A new unified theory of quantum and classical dualities is developed based on bond algebras of interactions. Its applications include discrete lattice, continuum field, and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits systematic searches for (self-)dualities in systems of any complexity. Non-local transformations like dual variables (topological excitations) and Jordan-Wigner dictionaries follow automatically from local mappings of bond algebras. The bond-algebraic theory of dualities provides a solution to the non-Abelian duality problem, which is illustrated with a non-Abelian duality for SU(2) principal chiral field. Bond-algebraic dualities constrain and realize Fermionization in an arbitrary number of dimensions. Interacting Majorana systems are analyzed on arbitrary lattices, and from them various universal spin duals are derived. The existence of topological quantum order and bounds on autocorrelation times can be inferred from symmetries, and it is possible to engineer quantum simulators out of these Majorana networks. Finally, the theoretical basis of dimensional reduction and holographic correspondence is investigated. It is shown that exact dimensional reduction is encoded in dualities and that effective dimensional reduction is encoded in inequalities. These inequalities link quantum systems of different spatial dimensionality by establishing bounds on correlation functions, gaining potency in the presence of special symmetries. The implications of dimensional reduction for the storage of quantum information are discussed.