Session Z19: Invited Session: Non-Abelian States in the 1st Excited Landau Level: Experimental Status and Theoretical Outlook

Interferometric evidence for non-Abelian quasiparticles at filling factor 5/2

The 5/2 fractional quantum Hall state charge e/4 excitations are proposed to follow non-Abelian statistics [1]. 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 [2-3]. To test this, we have performed swept area interference measurements at 5/2 [4-5]. We observe an alternating pattern of e/4 and e/2 period oscillations in resistance for a large change in the interferometer area, with the area sweep changing the enclosed localized e/4 quasiparticle number. This observed aperiodic alternation is consistent with proposed non-Abelian e/4 properties: the e/4 oscillations occur for encircling an even number of localized quasiparticles over their aperiodic spatial distribution, and the lower amplitude e/2 oscillations are observed when encircling an odd number as the e/4 oscillations are suppressed, allowing observation of the persistent smaller Abelian e/2 oscillations. Importantly, adding localized quasiparticles to the encircled area by changing magnetic field can change the parity of the enclosed quasiparticle number and should induce interchange of the expressed e/4 and e/2 periods: such interchange is observed in these measurements. In further experiments with the goal of understanding specific e/4 edge propagation properties, a series of interferometers of different sizes have been tested. The range of device dimensions has allowed measurement of the e/4 quasiparticle propagation attenuation length, demonstrating that small interferometric pathlengths are necessary to observe the interference oscillations. The stability in phase and amplitude of the e/4 oscillations has been tested with respect to sample dimensions, time, and temperature using this set of interferometers, and these results will be discussed. \\[4pt] [1] Moore, G. and Read, N., Nucl. Phys. B360, 362 (1991). \\[0pt] [2] Stern, A. and Halperin, B. I., Phys. Rev. Lett. 96, 016802-016805 (2006). \\[0pt] [3] Bonderson, P., Kitaev, A. and Shtengel, K., Phys. Rev. Lett. 96, 016803-016806 (2006). \\[0pt] [4] Willett, R.L., Pfeiffer, L.N., West, K.W., PNAS 106: 8853-8858 (2009). \\[0pt] [5] Willett, R.L., Pfeiffer, L.N., West, K.W., PRB 82: 205301 (2010).   

Observation of Braiding Statistics in the Fractional Quantum Hall Effect

It has been postulated that the quasiparticles excitations of fractional quantum Hall (FQH) states are Abelian anyons with fractional statistical angles. More interestingly, non-Abelian anyons have been predicted for certain FQH states such as that found at filling factors $nu = 5/2$ and 12/5. To date experimental detection of anyons and their braiding statistics in quantum interference experiments has remained controversial. In this talk I will present results from the study of Abelian and non-Abelian braiding statistics of anyons in the fractional quantum Hall (FQH) systems through Fabry-Perot interferometry. In the $\nu = 7/3$ FQHE state we confirm the anyonic braiding statistics by detecting the postulated statistical phase angle of $2\pi/3$. This result is consistent with a change of the anyon number by one. In the interference study of the $\nu = 5/2$ FQH state we observe phase slips $5\pi/4, \pi,$ and $\pi/4$. These observed statistical phase slips agree with a theoretical model of braiding of Majorana modes of the $\nu = 5/2$ non-Abelian state that is strongly coupled to each other and to the edge modes of interferometer in presence of Coulomb interaction. Our results provide compelling support for the existence of non-Abelian anyons.   

Quantum Hall interferometry in abelian and non-abelian states

This talk deals with the theory of quantum Hall interferometers. I will first consider an idealized model of such interferometers and review how they may potentially identify anyonic quasi-particles, both abelian and non-abelian. I will then describe lessons learned from experiments regarding this idealized model, and how it should be amended to take into account effects of coupling between the bulk and the edge. In particular, I will describe how electron-electron interaction affects the outcome of interferometry experiments and distinguishes between ``Aharonov-Bohm'' and ``Coulomb-dominated'' regimes. These two regimes are characterized by different magnetic field periodicities and different directions of the equi-phase lines in the plane of magnetic field and voltage applied to a side gate. I will discuss how the contributions of the Aharonov-Bohm effect, the Coulomb interaction and the fractional statistics (abelian and non-abelian) may be distinguished from one another in the two regimes. Finally, I will address issues unique to non-abelian states, particularly the effect of bulk-edge tunnel coupling on the interference, and comment on experimental observations of interference in the $\nu=5/2$ state.   

Plasma Analogy and non-Abelian Braiding Statistics in Ising-type Quantum Hall States

Quantum Hall systems are effectively two-dimensional and thus allow for quasiparticle excitations with exotic exchange statistics based on the braid group. These may be ``anyons'' with fractional statistics that lie somewhere between that of bosons and fermions, or, even more exotically, ``non-Abelian anyons'' with quasiparticle exchange represented by matrices. The quasiparticle statistics of candidate quantum Hall states (universality classes) is usually conjectured, but in some instances can be deduced from their representative trial wavefunctions. I will explain how this can be done for wavefunctions with a ``plasma analogy'' similar to Laughlin's. Then I will construct a plasma analogy for the non-Abelian Ising-type states, e.g. the Moore-Read Pfaffian state. This provides the first complete proof of the non-Abelian braiding statistics of quasiparticles in these states, which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction $\nu=5/2$.   

Insights into the Nature of the Exotic Fractional Quantum Hall States from Ultra-low Temperature Transport

It is believed that the $\nu=5/2$ and 12/5 fractional quantum Hall states (FQHS) in the second Landau level of a two-dimensional electron gas support excitations with non-Abelian braiding statistics. Two outstanding questions concerning the nature of the both the odd and even denominator FQHS of the second Landau level as probed by transport measurements at temperatures as low as 5mK will be addressed. We report the discovery of a new odd denominator FQHS state at $\nu=2+6/13$. The energy gaps of this and other states at $\nu=2+1/3$, 2+2/3, and 2+2/5 reveal a markedly different dependence on the effective magnetic field as compared to that of the corresponding lowest Landau level states at 6/13, 1/3, 2/3, and 2/5. If, in addition, we assume a Landau level-independent effective mass, we find that the 7/3 and 8/3 states are consistent, whereas the 2+2/5 and the 2+6/13 states show a strong deviation from the predictions of the model of free composite fermions. For the even denominator states at $\nu=5/2$ and 7/2 we extended measurements to the new regime of very low densities and to samples grown in two MBE chambers: one at Princeton and one at Purdue. Comparisons found in the literature of the experimentally measured intrinsic gaps at $\nu=5/2$ with numerical results are confusing: three methods find a large difference whereas a fourth method finds a good agreement. Our data suggests that the former three methods have deficiencies and therefore cannot be used. Using the fourth and a new method we introduce we find an excellent agreement of the experimental and numerical intrinsic gaps at $\nu=5/2$. These findings lend a strong support to the Pfaffian description of the $\nu=5/2$ fractional state. Work done in collaboration with A. Kumar, N. Samkharadze, N. Deng, J. Watson, G. Gardner, M. Manfra, L. Pfeiffer, and K. West. G.A.C. has been supported by the NSF DMR-0907172 and DOE DE-SC0006671 grants.