Theory of Next-Generation Even-Denominator States

TL;DR

This work develops a unified theory for next-generation even-denominator fractional quantum Hall states by extending composite-fermion theory to second-generation CFs and juxtaposing Bonderson-Slingerland states. It establishes that flux attachment leaves edge-topology and stability unchanged, relates NG states to BS states through identical quasiparticle charges and statistics, and provides a comprehensive framework tying bulk $K$-matrix data to edge signatures. The authors construct NG and BS trial wave functions, analyze their energetics across Landau levels, and predict experimental probes including thermal Hall conductance, tunneling scaling, upstream noise, and coherent conductance, with numerical results showing NG states favored in the zeroth Landau level and BS states in the first excited level. The bilayer graphene phase diagram further demonstrates environmental dependence of NG vs BS order, offering concrete guidance for experiments in heterostructures. Overall, the paper offers a thorough, testable bridge between microscopic CF constructions, topological order, and measurable edge phenomena for NG even-denominator quantum Hall states.

Abstract

Even-denominator quantum Hall states are leading candidates for realizing non-Abelian topological orders, with the $ν=\frac{5}{2}$ plateau in GaAs the first and most-studied example. Recent experiments in GaAs and bilayer graphene (BLG) have observed many `next-generation' even-denominator states at filling factors such as $ν=\frac{3}{4}$, $\frac{3}{8}$, and $\frac{3}{10}$. We develop the theory of these states, including analyses of their bulk quasiparticles, of methods for distinguishing between pairing channels in edge transport measurements, and of their trial wavefunctions. As part of this study, we derive general relations of how flux attachment affects many universal properties of states. In particular, we prove that the topological stability of interface modes is invariant under flux attachment. We compare next-generation paired states to Bonderson-Slingerland states at the same filling factors, and demonstrate that their quasiparticles carry identical charges and obey the same exchange statistics. The next-generation and Bonderson-Slingerland states still describe distinct phases, and we find that the former are energetically favored in the lowest Landau level, while the latter are favored in the first excited level.

Figures (14)

• Figure 1: Composite fermions at the effective filling factor $\nu^* = n + \nu_0^*$ form a state with $n$ fully filled CF Landau levels and a partially filled level at filling $\nu_0^*$. The latter contains $N_e^0 = \nu_0^*(N_\phi^* + 2n + S_0^*)$ CFs and realizes a FQH state characterized by the finite-size shift $S_0^*$. The remaining $n(N_\phi^* + n)$ CFs fully occupy the lower levels.
• Figure 2: Equivalence between particle-hole conjugation of first-generation states and NG states. The particle-hole conjugate of a CF state at effective filling $\bar{\nu}^*=\nu_0^*$ can be equivalently described as a NG CF state at $\nu^* = -1 - \nu_0^*$, corresponding to one filled CF Landau level and a partially filled level at filling $\nu_0^*$ in the negative effective field ($N^*_\phi<0$). Both descriptions yield identical electronic filling factors, shifts, and $K$-matrices, and therefore represent the same topological phase.
• Figure 3: Edge structure at interfaces between NG paired state with pairing channel $\ell$ at filling $\nu^{\ell}_{\text{NG}}$and reference states at $\nu_{\text{Ref}}(q)$. For the latter, we choose NG Jain states corresponding to $\nu^*_0=\frac{q}{2q+1}$ with positive or negative $q$. Both NG states have the same number $n$ of filled levels and the same number $2p$ of fluxes attached. The direction of the magnetic field is chosen to preserve the direction of the edge structure for any sign of $\nu^*$ as specified in the top ($\nu^*>0$) and bottom ($\nu^*<0$) panels.
• Figure 4: The $\pi$-junction setup for NG even-denominator states. The blue lines indicate the net charge flow, while red dashed lines denote Majorana modes.
• Figure 5: The Coulomb energy of the NG Laughlin state at $\nu=\frac{4}{11}$ in the four lowest LLs. The energies of the optimal triangular Wigner crystals are indicated in matching colors. WC$M$ denotes a crystal of $M$-electron bubbles, which we find to have the lowest energy at $\nu=\frac{4}{11}$ in the $(M-1)$th LL.