Doping lattice non-Abelian quantum Hall states

We study quantum phases of a fluid of mobile charged non-Abelian anyons, which arise upon doping the lattice Moore-Read quantum Hall state at lattice filling \nu = 1/2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and its generalizations to the Read-Rezayi ( RR_k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> ) sequence at \nu = k/(k+2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mi>/</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . In contrast to their Abelian counterparts, non-Abelian anyons present unique challenges due to their non-invertible fusion rules and non-Abelian braiding structures. We address these challenges using a Chern-Simons-Ginzburg-Landau (CSGL) framework that incorporates the crucial effect of energy splitting between different anyon fusion channels at nonzero dopant density. For the Moore-Read state, we show that doping the charge e/4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> non-abelion naturally leads to a fully gapped charge- 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> superconductor without any coexisting topological order. The chiral central charge of the superconductor depends on details of the interactions determining the splitting of anyon fusion channels. For general RR_k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> states, our analysis of states obtained by doping the basic non-abelion a_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> with charge e/(k+2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mi>/</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> reveals a striking even/odd pattern in the Read-Rezayi index k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> . We develop a general physical picture for anyon-driven superconductivity based on charge-flux unbinding, and show how it relates to the CSGL description of doped Abelian quantum Hall states. Finally, as a bonus, we use the CSGL formalism to describe transitions between the RR_k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> state and a trivial period- (k+2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> CDW insulator at fixed filling, driven by the gap closure of the fundamental non-Abelian anyon a_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . Notably, for k=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , this predicts a per