Bosonic matrix product state description of Read-Rezayi states and its application to quasihole spins

We study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>k</a:mi><a:mo>=</a:mo><a:mn>3</a:mn></a:mrow></a:math> Read-Rezayi quantum Hall state by means of a purely bosonic matrix product state formulation, as described in detail. We calculate the density profiles in the presence of bulk quasiholes of six different types: one for each <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi mathvariant="double-struck">Z</b:mi><b:mn>3</b:mn></b:msub></b:math> parafermion sector. From the density profiles, we calculate the (local) spins of these quasiholes. By employing a spin-statistics relation, we obtain the exchange statistics parameters. Our results, which are entirely based on properties of the quasiholes, corroborate previous results obtained by explicitly braiding quasiholes, showing that the exchange statistics can be read off from the monodromy properties of the wave functions, i.e., the associated Berry phase vanishes. We also discuss the entanglement spectrum, to show that our bosonic matrix product state formulation correctly captures the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi mathvariant="double-struck">Z</d:mi><d:mn>3</d:mn></d:msub></d:math> parafermionic structure of the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>k</f:mi><f:mo>=</f:mo><f:mn>3</f:mn></f:mrow></f:math> Read-Rezayi states.