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arXiv:2106.07667 · Physical Review Letters

Identifying the ν=52 Topological Order through Charge Transport Measurements

Misha Yutushui, Ady Stern, David F. Mross

Published Jan 3, 2022• 16 citations
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Abstract

We propose an experiment to identify the topological order of the ν=5/2 state through a measurement of the electric conductance of a mesoscopic device. Our setup is based on interfacing ν=2,5/2, and 3 in the same device. Its conductance can unambiguously establish or rule out the particle-hole symmetric Pfaffian topological order, which is supported by recent thermal measurements. Additionally, it distinguishes between the Moore-Read and anti-Pfaffian topological orders, which are favored by numerical calculations.

DOI
10.1103/physrevlett.128.016401

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We propose an experiment to identify the topological order of the ν=5/2 state through a measurement of the electric conductance of a mesoscopic device. Our setup is based on interfacing ν=2,5/2, and 3 in the same device. Its conductance can unambiguously establish or rule out the particle-hole symmetric Pfaffian topological order, which is supported by recent thermal measurements. Additionally, it distinguishes between the Moore-Read and anti-Pfaffian topological orders, which are favored by numerical calculations.
A recent thermal Hall experiment triggered renewed interest in the problem of ν=5/2 quantum Hall effect, which motivated novel interpretations based on the formation of mesoscopic puddles made of Pfaffian and anti-Pfaffian topological orders. Here, we study an interface between the Pfaffian and anti-Pfaffian states, which may play crucial roles in thermal transport, by means of state-of-the-art, density-matrix renormalization group simulations. We demonstrate that an intrinsic electric dipole moment emerges at the interface, similar to the "p-n" junction sandwiched between N-type and P-type semiconductor. Importantly, we elucidate the topological origin of this dipole moment, whose formation is to counterbalance the mismatch of guiding-center Hall viscosity of bulk Pfaffian and anti-Pfaffian states. In addition, these results imply that the formation of a dipole moment could be helpful to stabilize the puddles made of Pfaffian and anti-Pfaffian states in experimental conditions.
After decades since its discovery, a complete understanding of the topological phase of matter at 52\frac{5}{2}25​ filling appears to be lacking. Previous numerical studies have narrowed the candidates to the Moore-Read Pfaffian or the anti-Pfaffian. Recent thermal transport measurements point to another variant, the particle-hole Pfaffian, which has a nonholomorphic pairing part that complicates comparison with other competing states. Here, the authors have constructed the exact holomorphic particle-hole Pfaffian wave function with full particle-hole symmetry and compared its energetics with the Moore-Read Pfaffian and the anti-Pfaffian.
The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.
Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.
A successful probing of the neutral Majorana mode in recent thermal Hall conductivity measurements opines in favor of the particle-hole symmetric Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous quantized phase that is found to be gapped in the thermodynamic limit, distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an intermediate strength of Landau level mixing. Our proposed wave function consistent with the PH-Pf shift in spherical geometry rightly captures the topological order of this phase, as its overlap with the exact ground state is very high and it reproduces low-lying entanglement spectra. A unique topological order, irrespective of the flux shifts, found for this phase, possibly corroborates the experimentally found topological order.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.