Exact Mappings in Condensed Matter Physics

Condensed matter systems are complex yet simple. Amidst their complexity, one often find order specified by not more than a few parameters. Key to such a reductionistic description is an appropriate choice of basis, two of which I shall describe in this thesis. The first, an exact mapping known as the Wannier State Representation (WSR), provides an exact Hilbert space correspondence between two intensely-studied topological systems, the Fractional Quantum Hall (FQH) and Fractional Chern Insulator (FCI) systems. FQH states exist within the partially filled Landau levels of interacting 2D electron gases under strong magnetic fields, where quasiparticles exhibit topologically nontrivial braiding statistics. FCI systems, which are novel lattice realizations of FQH systems without orbital magnetic field, are still not completely understood and will benefit from a basis that explicitly connects them to the much better understood FQH systems. The second basis mapping, the Exact Holographic Mapping (EHM), maps any lattice system to a holographic 'bulk' with an additional emergent dimension representing energy scale. Devised in the spirit of the highly popular AdS-CFT correspondence, it attempts to understand the relationship between a given system and its equivalent dual geometry. In particular, we found excellent theoretical agreement of certain dual geometries from EHM with those expected from the Ryu-Takanayagi formula relating the bulk geometry with entanglement entropy. Additionally, the EHM also proves useful in providing a link between different topological quantities, such as relating the Chern number of the abovementioned Chern Insulators with the Axion angle of 3D topological insulators.