B.Spivak University of Washington with S. Kivelson, S. Sondhi, S. Parameswaran

1. A typology of quantum Hall liquids. Weakly coupled Pfaffian state as a type 1 quantum Hall fluid B.Spivak University of Washington with S. Kivelson, S. Sondhi, S. Parameswaran

2. Integer quantum Hall effect Fractional quantum Hall effect I will discuss the cases m/n=1/2, 5/2, ….

3. Spectrum of electrons in two dimensions in magnetic field B e Density of states on each Landau level: LH is the magnetic length Filling factor:

4. J. Jain, R. Laughlin, S. Girvin, A. McDonnald, S. Kivelson, S.C. Zheng, E. Fradkin, F. Wilczek, P. Lee, N. Read, G. Moore, B. Halperin, D. Haldane

5. Aharonov-Bohm effect time F space Composite fermions fermion if k=2n boson if k=n + e = n is an integer The statistical phase can be interpreted as an Aharonov-Bohm effect: when charge is moving around the flux (kf0) it acquires a phase kp

6. Chern-Simons theory of the quantum Hall effect (Fermion version k=2) B and b are the magnetic field and statistical magnetic field A and a are the vector potential and statistical vector potential + e = composite fermion

7. Halperin- Lee-Read (HLR) state: “Fermi liquid” of composite Fermions, k=2 At the filling factor n=½ the statistical and external magnetic fields cancel each other: B + b =0 at the mean field level the system is in a Fermi liquid state in a zero effective magnetic field! What are the effective mass and the Fermi energy of composite fermions?

8. Mean field electrodynamics of HLR state Ohm’s law for composite Fermions: both sxxand sxyare not quantized !

9. Experiments supporting HLR theory

10. Superconductivity of composite fermions J.P. Eisenstein, R.L. Willet, H.L. Stormer, L.N. Pffiffer, K.W. West

11. Chern-Simons Superconducting order parameter P-wave (triplet) order parameter the system has an isotropic gap

12. Moore-Read Pfaffian 5/2 QH state, weakly coupled (BCS) p-wave superconductivity of composite fermions z is a unit vector perpendicular to the plane, at T=0 Ns=N

13. Correspondence between the perfect conductivity of the superconductors and the quantization of the Hall conductance: Meissner effect incompressibility Quantized vortices fractionally charged quasiparticles

14. Two types of conventional superconductors

15. Two characteristic lengths in the Pfaffian state at T=0 Two characteristic energy scales

16. Two possible types of quantum Hall fluids a) Type 2 QH fluids where roughly x~l . In this case the surface energy between HLR and Pfaffian states is negative. Consequently density deviations are accommodated by the introduction of single quasiparticles/vortices b) Type I QH state: x>>l , (or EF >> D) In this case the surface energy between is positive. Quasiparticles (vortices) agglomerate and form multi-particle bound states electronic microemulsions

17. If x>>l vortices agglomerate into big bubbles Nc is the number of electrons in the bubble If Nbx2 ~1 the system is in “electronic microemulsion phase” which can be visualized as a mixture of HLR and Pfaffian on mesoscopic scale. Nb is the bubble concentration

18. Schematic phase diagram

19. Bosonic Chern-Simons theory. At Bogomolni’s point vortexes do not interact

20. Do we know that in the Pfaffian state x>>l ? • Numerical simulations: • H. Lu, S. das Sarma, K. Park, cond-mat. 1008.1587; • P. Rondson, A. E. Feiguin, C. Nayak, cond. mat. 1008.4173; • G. Moller, A. Woijs, N. Cooper, cond-mat. 1009.4956 • x/l~10-30, e2/eLHEF ~10-30, D/EF ~1 • a) Activation energy in transport experiments is approximately two orders of magnitude smaller than EF , and sometimes decreases further as a function of gate voltage and parallel magnetic field. • b) the characteristic temperature where the 5/2 plateau of QHE disappears is much smaller then EF An exapmple: superfluid 3He:

21. Existing experiments on measuring the effective vortex charge near 5/2 filling fraction cannot distinguish between the first and second type of quantum Hall states. They only prove that the elementary building blocks for any charged structure (either vortices, or bubbles, or more complex objects) have charge e/4.

23. Willet’s experiments measure the total number of vortices of charge e/4 in a sample R. L. Willett, L. N. Pfeiffer, and K. W. West, Phys. Rev.B 82, 205301 2010 Edge states

24. In Heiblum’s group experiments the edge state carrier charge is inferred from shot noise measurements. Edge states exist even exactly at 5/2 filling fraction. J. Nuebler, V. Umansky, R. Morf, M. Heiblum, K. von Klitzing, and J. Smet, Phys. Rev. B 81, 035316 (2010)

25. The Yacoby group’s experiments are based on the fact that samples are disordered and there are puddles of HLR states embedded into the Pfaffian state. The charge of big HLR puddles grows in steps e/4 as a function of the gate voltage Vivek Venkatachalam, Amir Yacoby, Loren Pfeiffer, Ken West, Nature 469, 185, 2011 Pfaffian HLR

26. Experiments on the activation energy of sxx The longitudinal resistance exists due to motion of vortices. The activation energy is determined by the pinning of vortices. Thus these experiments do not provide direct information about the value of the gap

27. In pure samples the value of the “critical temperature” is directly related to the value of the gap. However in disordered samples the value of the “critical temperature” may be determined by weak links between superconducting droplets. The situation is quite similar to that in granular superconductors.

28. An effective model of Joshepson junctions Since the Jij have random sign, near the critical point the system is Pfaffian (p-wave superconducting) glass T Pfaffian glass Pfaffian HLR disorder

29. Conclusion: Weakly coupled Pfaffian state is equivalent to Type 1 p+ip superconducting state. In this state vortices attract each other and agglomerate into big bubbles. There is a quantum phase transition between HLR and Pfaffian states as a function of disorder Depending on interaction, conventional QH fractions can be type 1 as well.