An isotropic to anisotropic transition in a fractional quantum Hall state

          @misc{ pirsa_PIRSA:10050090,
            doi = {10.48660/10050090},
            url = {https://pirsa.org/10050090},
            author = {},
            keywords = {},
            language = {en},
            title = {An isotropic to anisotropic transition in a fractional quantum Hall state},
            publisher = {Perimeter Institute},
            year = {2010},
            month = {may},
            note = {PIRSA:10050090 see, \url{https://pirsa.org}}
          }
          

Abstract

I describe a novel abelian gauge theory in 2+1 dimensions which has surprising theoretical and phenomenological features. The theory has a vanishing coefficient for the square of the electric field $e_i^2$, characteristic of a quantum critical point with dynamical critical exponent $z=2$, and a level-$k$ Chern-Simons coupling, which is marginal at this critical point. For $k=0$, this theory is dual to a free $z=2$ scalar field theory describing a quantum Lifshitz transition, but $k \neq 0$ renders the scalar description non-local. The $k \neq 0$ theory exhibits properties intermediate between the (topological) pure Chern-Simons theory and the scalar theory. For instance, the Chern-Simons term does not make the gauge field massive. Nevertheless, there are chiral edge modes when the theory is placed on a space with boundary, and a non-trivial ground state degeneracy $k^g$ when it is placed on a finite-size Riemann surface of genus $g$. The coefficient of $e_i^2$ is the only relevant coupling; it tunes the system through a quantum phase transition between an isotropic fractional quantum Hall state and an anisotropic fractional quantum Hall state. I describe zero-temperature transport coefficients in both phases and at the critical point, and comment briefly on the relevance of the results to recent experiments.