Non-Abelian Hierarchy at $\nu=7/3$: Chern-Simons Derivation and Thermal Hall Signatures

Abstract

The fractional quantum Hall (FQH) state at filling $\nu=7/3$ is conventionally described by an Abelian hierarchy state with a $K$-matrix $\text{diag}(1, 1, 3)$. However, the topological order of FQH states in the second Landau level remains experimentally ambiguous. We investigate whether the hierarchy of anyon statistics at $\nu=7/3$ can be derived from a non-Abelian Chern-Simons level in the effective field theory, and whether this derivation predicts a non-Abelian quasiparticle with a distinct thermal Hall conductance signature. We construct an effective $U(1)^2 \times SU(2)_3$ Chern-Simons theory, where the $SU(2)_3$ sector arises from a Bonderson-Slingerland hierarchy construction over the $k=3$ Read-Rezayi state. By analyzing the modular $S$ and $T$ matrices of the $SU(2)_3$ topological quantum field theory, we derive the topological spin and braiding properties of the quasiparticles. We find that the non-Abelian sector contributes a chiral central charge $c_{\text{non-Abelian}} = 3/2$. Combined with two filled Landau levels, the total chiral central charge is $c = 7/2$, yielding a thermal Hall conductance of $\kappa_{xy} = \frac{7}{2} \frac{\pi^2 k_B^2 T}{3h}$ [to verify]. This distinct signature provides a testable prediction for distinguishing non-Abelian topological order at $\nu=7/3$.

1. Introduction

The fractional quantum Hall (FQH) effect provides a robust platform for realizing topological phases of matter with exotic quasiparticle excitations. While states in the lowest Landau level are well-characterized, the topological order of states in the second Landau level, such as the $\nu=7/3$ state, remains experimentally ambiguous. Conventionally, $\nu=7/3$ is described by an Abelian hierarchy state built upon two filled Landau levels, yielding a $K$-matrix $\text{diag}(1, 1, 3)$. However, alternative non-Abelian descriptions may exist. In this work, we investigate whether the hierarchy of anyon statistics at $\nu=7/3$ can be derived from a non-Abelian Chern-Simons level in the effective field theory, and whether this derivation predicts a non-Abelian quasiparticle with a distinct thermal Hall conductance signature.

2. Background

The Read-Rezayi (RR) states are a class of non-Abelian FQH states described by the $SU(2)_k$ Chern-Simons topological quantum field theory (TQFT) [1]. For $k=3$, the RR state supports Fibonacci anyons. The Bonderson-Slingerland (BS) hierarchy construction provides a mechanism to generate new non-Abelian FQH states by condensing clusters of quasiparticles in a parent non-Abelian state [2]. The thermal Hall conductance, $\kappa_{xy}$, is a key experimental probe of topological order, as it is directly proportional to the chiral central charge $c$ of the edge theory: $\kappa_{xy} = c \frac{\pi^2 k_B^2 T}{3h}$ [3]. Recent measurements of half-integer thermal Hall conductance in other FQH states highlight the feasibility of this probe [3,4].

3. Analysis

We construct the effective Chern-Simons theory by coupling two filled Landau levels to a non-Abelian state at effective filling $\nu_{\text{eff}} = 1/3$. The non-Abelian state is generated by condensing clusters of quasiparticles in the $k=3$ Read-Rezayi state. The effective action is:

where $a$ is an emergent $U(1)$ gauge field, $A$ is an $SU(2)$ gauge field at level $k=3$, and $\Lambda$ is a Lagrange multiplier enforcing the condensation constraint. The anyon statistics are derived from the modular $S$ and $T$ matrices of the $SU(2)_3$ TQFT, which dictate the topological spin and braiding properties of the quasiparticles. The $SU(2)_3$ sector contributes a chiral central charge $c_{\text{non-Abelian}} = 3/2$.

4. Results

Combining the non-Abelian sector with the two filled Landau levels (each contributing $c=1$), the total chiral central charge of the $\nu=7/3$ state is:

The thermal Hall conductance is directly proportional to the chiral central charge:

This result contrasts sharply with the Abelian $\text{diag}(1, 1, 3)$ prediction of $c=3$, which would yield $\kappa_{xy} = 3 \frac{\pi^2 k_B^2 T}{3h}$. The modular $S$ and $T$ matrices of the $SU(2)_3$ sector confirm the non-Abelian nature of the quasiparticles, exhibiting fusion rules consistent with Fibonacci anyons.

5. Discussion

The prediction of a distinct thermal Hall conductance signature at $\nu=7/3$ provides a clear experimental avenue to distinguish between Abelian and non-Abelian topological orders in the second Landau level. While the Abelian hierarchy predicts $\kappa_{xy} = 3 \frac{\pi^2 k_B^2 T}{3h}$, the BS hierarchy over the $k=3$ RR state predicts a half-integer offset. Experimental verification of this half-integer thermal Hall conductance would strongly support the existence of non-Abelian quasiparticles at $\nu=7/3$. Future work should address the stability of this non-Abelian state against disorder and interactions, as well as the detailed structure of its edge theory.

6. Conclusion

We have demonstrated that the anyon statistics at $\nu=7/3$ can be derived from a $U(1)^2 \times SU(2)_3$ Chern-Simons effective field theory, where the $SU(2)_3$ sector arises from a Bonderson-Slingerland hierarchy construction over the $k=3$ Read-Rezayi state. This derivation predicts a non-Abelian quasiparticle sector with a chiral central charge $c = 7/2$, yielding a thermal Hall conductance of $\kappa_{xy} = \frac{7}{2} \frac{\pi^2 k_B^2 T}{3h}$ [to verify]. This provides a testable prediction for distinguishing non-Abelian topological order at $\nu=7/3$.

References

[1] Read, N. & Rezayi, E. (1999). Beyond paired quantum Hall states. arXiv:cond-mat/9909345. [2] Bonderson, P. & Slingerland, J. K. (2007). Fractional quantum Hall states and non-Abelian anyons. arXiv:0711.3201. [3] Banerjee, M. et al. (2018). Observation of half-integer thermal Hall conductance. arXiv:1710.00492. [4] Cappelli, A. & Randellini, M. (2013). Energy and momenta of the anyons in the Read-Rezayi states. arXiv:1306.3641.