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arXiv:1901.00046 · Physical review. B./Physical review. B

Edge-induced topological phase transition of the quantum Hall state at half filling

Bo Yang, Na Jiang, Xin Wan, Jie Wang, Zi-Xiang Hu

Published Apr 8, 2019• 1 citation
arXivHTMLPDF

Abstract

We show that in quantum Hall systems at half filling, edge potentials alone can drive transitions between the Pfaffian and anti-Pfaffian topological phases. We conjecture this is true in realistic systems even in the presence of bulk interactions that weakly break the particle-hole symmetry. The strong effects of edge potentials could be understood from different topological shifts of competing phases at half filling, manifested on the disk geometry as the variation of orbital numbers at a fixed number of particles. In particular, we show analytically that the particle-hole conjugation of Hamiltonians on the disk is equivalent to the tuning of edge potentials, which allows us to explicitly demonstrate the phase transition numerically. The importance of edge potentials in various experimental contexts, including the recently discovered particle-hole symmetric phase, is also discussed.

DOI
10.1103/physrevb.99.161108

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We show that in quantum Hall systems at half filling, edge potentials alone can drive transitions between the Pfaffian and anti-Pfaffian topological phases. We conjecture this is true in realistic systems even in the presence of bulk interactions that weakly break the particle-hole symmetry. The strong effects of edge potentials could be understood from different topological shifts of competing phases at half filling, manifested on the disk geometry as the variation of orbital numbers at a fixed number of particles. In particular, we show analytically that the particle-hole conjugation of Hamiltonians on the disk is equivalent to the tuning of edge potentials, which allows us to explicitly demonstrate the phase transition numerically. The importance of edge potentials in various experimental contexts, including the recently discovered particle-hole symmetric phase, is also discussed.
We discuss the implications of approximate particle-hole symmetry in a half-filled Landau level in which a paired quantum Hall state forms. We note that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit of vanishing Landau-level mixing, in which particle-hole transformation is an exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a particle-hole conjugate state, which we call the anti-Pfaffian, which is degenerate with the Pfaffian in this limit. We observe that strong Landau-level mixing should favor the Pfaffian, but it is an open problem which state is favored for the moderate Landau-level mixing which is present in experiments. We discuss the bulk and edge physics of the anti-Pfaffian. We analyze a simplified model in which transitions between analogs of the two states can be studied in detail. Finally, we discuss experimental implications.
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.
We report on the study of the fractional quantum Hall effect at the filling factor 5/2 using exact diagonalization method with torus geometry. The particle-hole symmetry breaking effect is considered using an additional three-body interaction. Both Pfaffian and anti-Pfaffian states can be the ground state depending on the sign of the three-body interaction. The results of the low-energy spectrum, the wave function overlap, and the particle-hole parity evolution, have shown clear evidence of a direct sharp transition (possibly first order) from the Pfaffian to the anti-Pfaffian state at the Coulomb point. A quantum phase diagram is established, where one finds further transitions from the Pfaffian or anti-Pfaffian state to the nearby compressible phases induced by a change of the pseudopotential.
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.
We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).
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.