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arXiv:2003.02621 · Physical Review Letters

Topological Interface between Pfaffian and Anti-Pfaffian Order in ν=5/2 Quantum Hall Effect

W. Zhu, D. N. Sheng, Kun Yang

Published Sep 30, 2020• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.125.146802

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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.
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.
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.
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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.
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.
We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.