Quarter- and half-filled quantum Hall states and their topological orders revealed by daughter states in bilayer graphene

Even-denominator fractional quantum Hall states are promising candidates for fault-tolerant quantum computing due to their underlying non-Abelian topological order. However, the topological order of these states remains hotly debated. Here, we report transport measurements on ultra-clean bilayer graphene heterostructures, where we observed four quarter-filled states and their corresponding Levin-Halperin daughter states, constraining their topological order. Moreover, we complete the sequence of half-filled plateaus by detecting states at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:mfrac> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:mfrac> </mml:math> whose daughters suggest an alternating sequence of non-Abelian orders. This pattern suggests a universal origin supporting their use in identifying topological order at even-denominator fillings, though further confirmation is needed via direct measurements. The observed quarter- and half-filled states appear in N = 0 and N = 1 Landau levels, respectively, and thus highlight a competition between interactions favoring paired states of either four- or two-flux composite fermions. Additionally, we observe several 'next-generation' quantum Hall states that require strong interactions between composite fermions.