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arXiv:2507.07161

Majorana edge reconstruction and the $ν=5/2$ non-Abelian thermal Hall puzzle

Tevž Lotrič, Taige Wang, Michael P. Zaletel, Steven H. Simon, S. A. Parameswaran

Published Jul 9, 2025• 0 citations
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Abstract

Pioneering thermal transport measurements on two-dimensional electron gases in high magnetic fields have demonstrated that the quantized Hall state at filling factor $ν=5/2$ has a thermal Hall conductance $κ$ quantized in half-integer multiples of $κ_0 = {π^2 k_B^2 T}/{3h}$. Half-integer $κ/κ_0$ is a signature of neutral Majorana edge modes, in turn linked to the presence of non-Abelian anyon excitations in the bulk. However, the experimentally observed value of $κ$ corresponds to the 'PH-Pfaffian' state, in tension with numerical studies which instead favor either the Pfaffian or the AntiPfaffian. A variety of mechanisms have been invoked to explain this discrepancy, but have been either ruled out by further experiments or else involve fine-tuning. Building on density-matrix-renormalization group studies of physically realistic edges and analytic calculations of edge structure, we propose an alternative resolution of this puzzle involving an 'edge reconstruction' solely involving the neutral Majorana sector of the theory. Such a Majorana edge reconstruction can "screen'' a Pfaffian or AntiPfaffian bulk, so that transport signatures become indistinguishable from those of the PH-Pfaffian. We argue that this physically natural scenario is consistent with experiment.

DOI
10.48550/arxiv.2507.07161

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Theory of the disordered ν=52 quantum thermal Hall state: Emergent symmetry and phase diagram

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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.

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Majorana edge reconstruction and the ν=5/2ν=5/2ν=5/2 non-Abelian thermal Hall puzzle
Pioneering thermal transport measurements on two-dimensional electron gases in high magnetic fields have demonstrated that the quantized Hall state at filling factor ν=5/2ν=5/2ν=5/2 has a thermal Hall conductance κκκ quantized in half-integer multiples of κ0=π2kB2T/3hκ_0 = {π^2 k_B^2 T}/{3h}κ0​=π2kB2​T/3h. Half-integer κ/κ0κ/κ_0κ/κ0​ is a signature of neutral Majorana edge modes, in turn linked to the presence of non-Abelian anyon excitations in the bulk. However, the experimentally observed value of κκκ corresponds to the 'PH-Pfaffian' state, in tension with numerical studies which instead favor either the Pfaffian or the AntiPfaffian. A variety of mechanisms have been invoked to explain this discrepancy, but have been either ruled out by further experiments or else involve fine-tuning. Building on density-matrix-renormalization group studies of physically realistic edges and analytic calculations of edge structure, we propose an alternative resolution of this puzzle involving an 'edge reconstruction' solely involving the neutral Majorana sector of the theory. Such a Majorana edge reconstruction can "screen'' a Pfaffian or AntiPfaffian bulk, so that transport signatures become indistinguishable from those of the PH-Pfaffian. We argue that this physically natural scenario is consistent with experiment.
In the Kitaev honeycomb model, a weak magnetic field gaps the gapless Majorana cone, driving the system into a gapped non-Abelian phase with Ising topological order. The edge supports a single chiral Majorana mode with chiral central charge c−=1/2c_- = 1/2c−​=1/2, yielding a half-quantized thermal Hall conductance κxy/T=π2kB2/6h\kappa_{xy}/T = \pi^2 k_B^2/6hκxy​/T=π2kB2​/6h. We rigorously investigate whether observing this half-quantized value uniquely identifies the non-Abelian Ising anyon sector, or whether Abelian topological orders or extrinsic mechanisms can produce the same signature. By combining the edge-bulk correspondence with the classification of strictly two-dimensional bosonic topological orders, we show that a chiral central charge c−=1/2c_-=1/2c−​=1/2 (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer c−c_-c−​, with the minimal nonzero value being c−=1c_-=1c−​=1. Thus, a robustly half-quantized κxy/T\kappa_{xy}/Tκxy​/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible E8E_8E8​ stacking or phonon contamination. ## 1. Introduction The Kitaev honeycomb model is an exactly solvable model of a quantum spin liquid, characterized by fractionalized excitations known as anyons. In the absence of a magnetic field, the model hosts a gapless Majorana cone. A weak magnetic field in the [111] direction gaps this cone and induces a non-Abelian phase with Ising topological order. This phase supports a single chiral Majorana edge mode, which contributes to a quantized thermal Hall conductance. The central question addressed in this preprint is whether the observation of a half-quantized thermal Hall conductance, corresponding to a chiral central charge c−=1/2c_- = 1/2c−​=1/2, uniquely identifies the non-Abelian Ising anyon sector, or if Abelian topological orders can mimic this signature. ## 2. Background The thermal Hall conductance $\kappa_{x
Fractional quantum Hall (FQH) system at Landau level filling fraction $\ensuremath{\nu}=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=5{e}^{2}/2h$. Thermal Hall conductances of the Pf and APf states are quantized at ${\ensuremath{\kappa}}_{xy}=7/2$ and ${\ensuremath{\kappa}}_{xy}=3/2$, respectively, in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\ensuremath{\nu}=5/2$ FQH state is ${\ensuremath{\kappa}}_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a ${\ensuremath{\kappa}}_{xy}=5/2$ phase. In this paper, we do perturbative and nonperturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold ${\mathrm{\ensuremath{\Lambda}}}_{1}$, which is lowered to an emergent $U(1)\ifmmode\times\else\texttimes\fi{}U$(1) symmetry at energy scales between ${\mathrm{\ensuremath{\Lambda}}}_{1}$ and a higher value ${\mathrm{\ensuremath{\Lambda}}}_{2}$, and is finally lowered to the composite fermion parity symmetry Z2F{\mathbb{Z}}_{2}^{F}Z2F​ above ${\mathrm{\ensuremath{\Lambda}}}_{2}$. Based on the emergent symmetries, we propose a phase diagram of the disordered $\ensuremath{\nu}=5/2$ FQH system and show that a ${\ensuremath{\kappa}}_{xy}=5/2$ phase arises at disorder energy scales $\mathrm{\ensuremath{\Lambda}}>{\mathrm{\ensuremath{\Lambda}}}_{1}$. Furthermore, we show the gapped double-semion sector of ND{N}_{D}ND​ compact domain walls contributes nonlocal topological degeneracy ${2}^{{N}_{D}\ensuremath{-}1}$, causing a low-temperature peak in the heat capacity. We implement a nonperturbative method to bootstrap generic topological 1+1D1+1\mathrm{D}1+1D domain walls (two-surface defects) applicable to any 2+1D2+1\mathrm{D}2+1D non-Abelian topological order. We also identify potentially relevant spin topological quantum field theories (TQFTs) for various $\ensuremath{\nu}=5/2$ FQH states in terms of fermionic version of $U{(1)}_{\ifmmode\pm\else\textpm\fi{}8}$ Chern-Simons $\mathrm{theory}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$-class TQFTs.
Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction ν=5/2ν={5/2}ν=5/2 fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from g=1/4g=1/4g=1/4 to g=1/2g=1/2g=1/2 for the edge reconstructed Pfaffian state, and changes from g=1/2g=1/2g=1/2 to g=0.55−0.75g=0.55-0.75g=0.55−0.75 for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed ν=5/2ν={5/2}ν=5/2 state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.
Co-propagating chiral Majorana edge modes are predicted to exist in proximity-coupled quantum anomalous Hall systems. They combine the exciting prospects of edge vortices as flying non-Abelian Ising anyons, with their fully electrical detection through anyon fusion processes. In this setup, electrical transport arises from the interference of a pair of Majorana edge states. By bosonizing the model with an additional replica sector, we generalize previous treatments to fully asymmetric configurations. We analyze a Mach-Zehnder interferometer and show that while the DC conductance does not exhibit signatures of edge-vortex interference, at low frequencies, this interference manifests as an effective capacitance. This quantity is sensitive to the non-Abelian anyon statistics of edge vortices and, in particular, carries signatures of their topological spin and nontrivial conformal dimension. We demonstrate that the thermal length lT=ℏv/(πkBT)l_T = \hbar v/(πk_B T)lT​=ℏv/(πkB​T) is the relevant scale at low temperatures TTT, analogous to quasiparticle transport in conventional quantum Hall edge states with velocity vvv. As long as the interferometer arm lengths match within this length scale, transport signatures of the non-Abelian statistics of edge vortices remain visible.
The search for Majorana excitations has seen tremendous efforts in recent years, ultimately aiming for their individual controllability in future topological quantum computers. A promising framework to realize such exotic Majorana fermions are topologically ordered non-Abelian phases of matter, such as certain fractional quantum Hall states. Quantum simulators provide unprecedented controllability and versatility to investigate such states, and developing experimentally feasible schemes to realize and identify them is of immediate relevance. Motivated by recent experiments, we consider bosons on coupled chains, subjected to a magnetic flux and experiencing Hubbard repulsion. At magnetic filling factor ν = 1 , similar systems on cylinders have been found to host the non-Abelian Moore-Read Pfaffian state in the bulk. Here, we address the question of whether more realistic few-leg ladders can host this exotic state and its chiral Majorana edge states. To this end, we perform extensive density-matrix renormalization-group simulations and determine the central charge of the ground state. While we do not find any evidence of gapless Majorana edge modes in systems of up to six legs, exact diagonalization of small systems reveals evidence for the Pfaffian state in the entanglement structure. By systematically varying the number of legs and monitoring the appearance and disappearance of this signal, our work highlights the importance of finite-size effects for the realization of exotic states in experimentally realistic systems.
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.