PIRSA:13100086

Geometry and Topology in the Fractional Quantum Hall Effect

Haldane, D. (2013). Geometry and Topology in the Fractional Quantum Hall Effect. Perimeter Institute. https://pirsa.org/13100086

Haldane, Duncan. Geometry and Topology in the Fractional Quantum Hall Effect. Perimeter Institute, Oct. 07, 2013, https://pirsa.org/13100086 

          @misc{ pirsa_PIRSA:13100086,
            doi = {10.48660/13100086},
            url = {https://pirsa.org/13100086},
            author = {Haldane, Duncan},
            keywords = {Condensed Matter},
            language = {en},
            title = {Geometry and Topology in the Fractional Quantum Hall Effect},
            publisher = {Perimeter Institute},
            year = {2013},
            month = {oct},
            note = {PIRSA:13100086 see, \url{https://pirsa.org}}
          }

Duncan Haldane Princeton University

DOI 10.48660/13100086
Collection
Talk Type Scientific Series
Subject

Abstract

The FQHE is exhibited by electrons moving on a 2D surface through which a magnetic flux passes, giving rise to flat bands with extensive degeneracy (Landau levels). The degeneracy of a partially-filled Landau level is lifted by Coulomb repulsion between the electrons, which at certain rational fillings, leads to gapped incompressible topologically-ordered fluid states exhibiting the FQHE. Successful model wavefunctions for FQHE states, such as the Laughlin and Moore-Read states, are surprisingly related to Euclidean conformal field theory, even though they are gapped incompressiible quantum fluids with a fundamental unit of area set by the area per magnetic flux quantum h/e. The model wavefunctions are parametrized by a continuously-variable Euclidean metric, just like the Euclidean conformal group of the cft to which they are related. This metric is fixed locally both by the form of the projected Coulomb interaction within the partially-filled Landau level, and by local gradients of the tangential electric field on the 2D surface, promoting it from a static flat metric fixed globally by the cft, to a dynamic local physical degree of freedom of the FQHE fluid with area-preserving zero-point fluctuations that leave an imprint in the ground-state structure function. The curious connection to cft appears to be that the Virasoro algebra plays a fundament role in both cft and FQHE, for apparently-unrelated reasons. In the FQHE it derives from a chiral "gravitional" (geometric) topologically-protected anomaly at the edge of the fluid that is also revealed in the entanglement spectrum of a cut through the bulk fluid.