Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

Abstract

Fractional quantum Hall (FQH) system at Landau level filling fraction $\ensuremath{\nu}=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=5{e}^{2}/2h$. Thermal Hall conductances of the Pf and APf states are quantized at ${\ensuremath{\kappa}}_{xy}=7/2$ and ${\ensuremath{\kappa}}_{xy}=3/2$, respectively, in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\ensuremath{\nu}=5/2$ FQH state is ${\ensuremath{\kappa}}_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a ${\ensuremath{\kappa}}_{xy}=5/2$ phase. In this paper, we do perturbative and nonperturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold ${\mathrm{\ensuremath{\Lambda}}}_{1}$, which is lowered to an emergent $U(1)\ifmmode\times\else\texttimes\fi{}U$(1) symmetry at energy scales between ${\mathrm{\ensuremath{\Lambda}}}_{1}$ and a higher value ${\mathrm{\ensuremath{\Lambda}}}_{2}$, and is finally lowered to the composite fermion parity symmetry ${\mathbb{Z}}_{2}^{F}$ above ${\mathrm{\ensuremath{\Lambda}}}_{2}$. Based on the emergent symmetries, we propose a phase diagram of the disordered $\ensuremath{\nu}=5/2$ FQH system and show that a ${\ensuremath{\kappa}}_{xy}=5/2$ phase arises at disorder energy scales $\mathrm{\ensuremath{\Lambda}}>{\mathrm{\ensuremath{\Lambda}}}_{1}$. Furthermore, we show the gapped double-semion sector of ${N}_{D}$ compact domain walls contributes nonlocal topological degeneracy ${2}^{{N}_{D}\ensuremath{-}1}$, causing a low-temperature peak in the heat capacity. We implement a nonperturbative method to bootstrap generic topological $1+1\mathrm{D}$ domain walls (two-surface defects) applicable to any $2+1\mathrm{D}$ non-Abelian topological order. We also identify potentially relevant spin topological quantum field theories (TQFTs) for various $\ensuremath{\nu}=5/2$ FQH states in terms of fermionic version of $U{(1)}_{\ifmmode\pm\else\textpm\fi{}8}$ Chern-Simons $\mathrm{theory}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$-class TQFTs.