Session Z9: QHE: Quantum Computing

Double quantum dots at the edge of Abelian and non-Abelian factional quantum Hall sates

We theoretically study two quantum dots tunnel coupled to the edge of Abelian and non-Abelian fractional quantum Hall states. We find a number of interesting low-energy fixed points which are a function of inter-dot coupling and separation. We study the stability and structure of the fixed points under various perturbations.   

Density dependence of the 5/2 energy gap

In this talk, we will present results from our recent experiments examining the spin-polarization of the 5/2 state by investigating the competition between the Coulomb and Zeeman energies utilizing a HIGFET (heterojunction insulated-gated field-effect transistor) device. Rather than tuning their ratio in a fixed density specimen by tilt, we keep the $B$-field perpendicular to the 2D electron gas and vary its density. This approach is equivalent to tilting the sample, but it cannot cause a tilted-field induced phase transition. The HIGFET device has a peak electron mobility of $12 \times 10^6$ cm$^2$/Vs, more than a factor of two increase compared to the one used in an earlier study. We observed that in the density range of $1.2 - 3.6 \times 10^{11}$ cm$^{-2}$ the 5/2 state was activated. Therefore, a true 5/2 energy gap was obtained. It increases with increasing electron density. We have fitted the density dependence with various theoretical models and will discuss its implications on the spin polarization of the 5/2 state.   

Activation energies for the $\nu $=5/2 Fractional Quantum Hall Effect at 10 Tesla

We reported on the low-temperature magnetotransport in a high-purity (mobility $\sim $ 1$\times $10$^{7}$cm$^{2}$/Vs) modulation-doped GaAs/AlGaAs quantum well with a high electron density (6$\times $10$^{11}$ cm$^{-2})$. A quantized $\nu $=5/2 Hall plateau is observed at B $\sim $ 10 T, with an activation gap $\Delta _{5/2} \sim $ 125$\pm $10 mK; the plateau can persist up to $\sim $ 25$^{o}$ tilt-field. We determined the activation energies $\Delta $ and quasi-gap energies $\Delta ^{quasi}$ for the $\nu $=5/2, 7/3, and 8/3 fractional quantum Hall states in tilted-magnetic field ($\theta )$. The $\Delta _{5/2} $, $\Delta _{7/3} $ and the $\Delta _{5/2}^{quasi} $, $\Delta _{7/3}^{quasi} $are found to decrease in $\theta $. We will present the systematic data and discuss their implications on the spin-polarization of $\nu $=5/2 states observed at 10 T.\\[4pt] [1] R. Willett, Phys. Rev. Lett. \textbf{59}, 1776 (1987).\\[0pt] [2] W. Pan et al, Solid State Commun. \textbf{119}, 641 (2001).   

Breaking of Particle-Hole Symmetry by Landau Level Mixing and the $\nu=5/2$ Quantized Hall State

The nature of the $nu$=5/2 quantum Hall state has been a puzzle for several decades. Based on a large body of numerical work, the community has been slowly converging to the conclusion that the 5/2 state is the same phase of matter as described by the Moore-Read wavefunction. However, two recent papers [1,2] point out that in fact two inequivalent possibilities still remain--the Moore-Read wavefunction, and its particle-hole conjugate, the so-called Anti-Pfaffian, which are distinct topological phases. In the absence of Landau-level mixing (an approximation used in all prior numerical works) these two possibilities cannot be distinguished. In the current work, we perform numerical studies aimed to determine if the fractional quantum Hall state observed at filling $\nu$=5/2 is the Moore-Read wavefunction or the Anti-Pfaffian. Using a truncated Hilbert space approach we find that for realistic interactions, including Landau-level mixing, the Moore-Read state is strongly favored. We also confirm that the ground state remains polarized even in the absence of Zeeman energy when Landau level mixing is allowed. [1] M. Levin, B. I. Halperin and B. Rosenow, Phys. Rev. Lett. 99, 236806 (2007). [2] S.-S. Lee, S. Ryu, C. Nayak and M. P. A. Fisher, Phys. Rev. Lett. 99, 236807 (2007).   

Local Compressibility of the Fractional Quantum Hall State at Filling Factor 5/2

Understanding the ground state and excitations of the quantum Hall state at filling factor $5/2$ is a subject of great interest due to the possibility of realizing non-abelian braiding statistics. All previous experimental probes of the state have relied on transport, and have therefore only accessed the chiral edges of the system. Here we present the first thermodynamic measurements of bulk properties at $\nu=5/2$. By measuring the local compressibility, $\left( \frac{\partial \mu}{\partial n} \right)^{-1}$, of states in the second Landau level, we can monitor the charging of individual localized states in the bulk. Comparing charging spectra at $\nu = 7/3$ and $\nu = 5/2$, we are able to extract the ratio of local charges in the bulk at these filling factors. Averaged over several disorder configurations and samples, we find this ratio to be $4/3$, suggesting that the local charges for these states are $e^*_{7/3}=e/3$ and $e^*_{5/2}=e/4$. Further, by integrating $\frac{\partial \mu}{\partial n}$, we obtain a value for the thermodynamic gap $\Delta_{5/2}$ without relying on activated transport.   

Clustering in quantum Hall wavefunctions and conformal field theory amplitudes

We consider lowest Landau level wavefunctions for bosons subjected to a magnetic field in the plane. We study the zero-energy eigenstates of a projection Hamiltonian which forbids three particles to come together with relative angular momentum less than six and, in addition, forbids one of two linearly-independent states of relative angular momentum six. The counting of edge excitations of this Hamiltonian agrees with the character formula for the N=1 superconformal Kac vacuum module at generic central charge c. This Hamiltonian is expected to be gapless for all c. For particular c, we try to ``improve'' the Hamiltonian by adding additional terms (related to singular vectors in the modules), so as to obtain a rational theory. We consider specifically states whose wavefunctions are related to the M(8,3) and tricritical Ising CFTs.   

Alternating e/4 and e/2 period interference oscillations as evidence for filling factor 5/2 non-Abelian quasiparticles

It is a theoretical conjecture that 5/2 fractional quantum Hall state charge e/4 excitations may obey exotic non-Abelian statistics. In edge state interference these purported non-Abelian quasiparticles should display period e/4 Aharonov-Bohm oscillations if the interfering quasiparticle encircles an even number of localized e/4 charges, but suppression of oscillations if an odd number is encircled. To test this hypothesis, here we perform swept area interference measurements at 5/2. We observe an alternating pattern of e/4 and e/2 period oscillations in resistance. This aperiodic alternation is consistent with proposed non-Abelian properties: the e/4 oscillations occur for encircling an even number of localized quasiparticles, e/2 oscillations are expressed when encircling an odd number. Aperiodic alternation corresponds to the expected area sweep sampling the localized quasiparticles. Importantly, adding localized quasiparticles to the encircled area by changing magnetic field induces interchange of the e/4 and e/2 oscillation periods, specifically consistent with non-Abelian e/4 quasiparticles.   

Transport in line junctions of $\nu=5/2$ quantum Hall liquids

We calculate the tunneling current through long line junctions of a $\nu=5/2$ quantum Hall liquid and i) another $\nu=5/2$ liquid, ii) an integer quantum Hall liquid and iii) a quantum wire. Momentum resolved tunneling provides information about the number, propagation directions and other features of the edge modes and thus helps distinguish several competing models of the 5/2 state. We investigate transport properties for two proposed Abelian states: $K=8$ state and 331 state, and four possible non-Abelian states: Pfaffian, edge-reconstructed Pfaffian, and two versions of the anti-Pfaffian state. We also show that the non-equilibrated anti-Pfaffian state has a different resistance from other proposed states in the bar geometry.   

Exact Solution for Bulk-Edge Coupling in the Non-Abelian $\nu=5/2$ Quantum Hall Interferometer

It has been predicted that the phase sensitive part of the current through a non-abelian $\nu = 5/2$ quantum Hall Fabry-Perot interferometer will depend on the number of localized charged $e/4$ quasiparticles (QPs) inside the interferometer cell. In the limit where all QPs are far from the edge, the leading contribution to the interference current is predicted to be absent if the number of enclosed QPs is odd and present otherwise, as a consequence of the non-abelian QP statistics. Here, we consider a localized QP which is close enough to the boundary so that it can exchange a Majorana fermion with the edge via a tunneling process. We derive an exact solution for the dependence of the interference current on the coupling strength for this tunneling process, and confirm a previous prediction that for sufficiently strong coupling, the localized QP is effectively incorporated in the edge and no longer affects the interference pattern. We confirm that the dimensionless coupling strength can be tuned by the source-drain voltage, and we find that there is a universal shift in the interference phase as a function of coupling strength.   

A numerical study of the disorder effect on the 5/2 fractional quantum hall system

We study the 5/2 fractional quantum hall (FQH) system in the presence of random disorder using exact diagonalization in the torus geometry. We examine the low-lying spectra with different unit cells and find a persistent and robust spectral gap characterizing the incompressible Pfaffian-like states. This gap narrows with increasing disorder strength. The structure factor also exhibits robust characteristics of a uniform quantum hall liquid. The topologically invariant chern number has been calculated to determine the mobility gap of the FQH states, which is used to compare with the experimentally measured activation gap. The mobility gap tends to collapse as the disorder grows, suggesting a disorder-driven transition from the FQH states to the insulator. We further demonstrate the characteristic features of the ground state of the 5/2 FQH system and Pfaffian states using reduced density matrices.   

The entanglement gap and a new principle of adiabatic continuity

We give a complete definition of the entanglement gap separating low-energy, topological levels, from high-energy, generic ones, in the ``entanglement spectrum'' of Fractional Quantum Hall (FQH) states. By removing the magnetic length inherent in the FQH problem - a procedure which we call taking the ``conformal limit,'' we find that the entanglement spectrum of an incompressible ground-state of a generic (i.e. Coulomb) lowest Landau Level Hamiltonian re-arranges into a low-(entanglement) energy part separated by a full gap from the high energy entanglement levels. As previously observed, the counting of these levels starts off as the counting of modes of the edge theory of the FQH state, but quickly develops finite-size effects which we show can also serve as a fingerprint of the FQH state. As the sphere manifold where the FQH resides grows, the level spacing of the states at the same angular momentum goes to zero, suggestive of the presence of relativistic gapless edge-states. We use the entanglement spectrum in the conformal limit basis to investigate whether two states are topologically connected, by using the adiabatic continuity of the low entanglement energy levels.   

General trial wave functions for a three body interaction

The Pfaffian wave function, which is a candidate for the 5/2 FQHE state, is the exact ground state of a short range three body model interaction, but little is known about the solutions of this model at other filling factors. Our starting point is the observation that the Pfaffian can be obtained by fully anti-symmetrizing a bilayer wave function of Halperin. A more general class of composite fermion wave functions for bilayer systems was constructed by Scarola and Jain. We find that, upon full antisymmetrization, these wave function provide a decent approximation to the low energy solutions of the three body model interaction at filling factors other than 1/2. The charged and neutral excitations of the full state are naturally constructed by creating excitations in one or both ``layers.'' We also investigate how well the ground and excited state wave functions work for the Coulomb interaction, both in the lowest and the second Landau levels. Systems with up to 18 particles are studied by a combination of exact diagonalization and Monte Carlo method.   

Superconducting Order Parameter of the Even-denominator Fractional Quantum Hall Effect

Usually formulated in terms of a trial wave function called the Moore-Read Pfaffian wave function, current leading theories attribute the origin of the 5/2 FQHE to the formation of a new superconducting state. The nature of superconductivity in the 5/2 FQHE is particularly puzzling in the sense that this state apparently coexists with strong magnetic field, which poses an interesting dilemma since the Meissner effect is the most important defining property of superconductivity. To overcome this dilemma, it is crucial to understand what it is that actually forms the superconducting condensate, if any. Here, we develop a numerically exact method to create a Cooper pair in terms of the true (elementary) quasi- particle of the system (identified with composite fermion) and explicitly compute the superconducting order parameter as a function of real space coordinate instead of usual momentum. As results, in addition to direct evidence for superconductivity, we obtain quantitative predictions for superconducting coherence length. Based on our calculation, we propose an experimental setup for demonstrating the 5/2 FQHE counterpart of the Josephson effect and thus, if successful, conclusively proving the existence of superconductivity in the 5/2 FQHE.   

A special relationship between non-unitary non-Abelian and unitary Abelian quantum Hall states

We point out a special relationship between quantum Hall states connected with non-unitary conformal field theories (CFTs) and those connected with unitary bosonic CFTs. They are related via boundary insertions as encoded in their root configurations. We discuss the cases of Gaffnian and Haldane-Rezayi non-unitary theories (i.e. quantum Hall states) which, via boundary insertions, can be transmuted into abelian bosonic unitary theories. The construction mimics a global change of parameters in the phase space of the electron system. We also discuss the cases of permanent non-unitary theory and Pfaffian unitary theory which are resistant to this mechanism described in M.V. Milovanovic, Th. Jolicoeur, and I. Vidanovic, Phys. Rev. B 80, 155324 (2009).