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

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

Spectrum of electrons in two dimensions in magnetic field B  Density of states on each Landau level: L H is the magnetic length Filling factor:

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

Aharonov-Bohm effect Composite fermions  e = fermion if k=2n boson if k=n  n is an integer The statistical phase can be interpreted as an Aharonov-Bohm effect: when charge is moving around the flux (    it acquires a phase  time space

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

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

Mean field electrodynamics of HLR state Ohm’s law for composite Fermions:  both  xx  and  xy  are not quantized !

Experiments supporting HLR theory

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

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

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 N s =N

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

Two types of conventional superconductors

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

a) Type 2 QH fluids where roughly  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: , (or E F >>  In this case the surface energy between is positive. Quasiparticles (vortices) agglomerate and form multi-particle bound states electronic microemulsions Two possible types of quantum Hall fluids

If  vortices agglomerate into big bubbles N c is the number of electrons in the bubble If N b   ~1 the system is in “electronic microemulsion phase” which can be visualized as a mixture of HLR and Pfaffian on mesoscopic scale. N b is the bubble concentration

Schematic phase diagram

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

 1.Numerical simulations: H. Lu, S. das Sarma, K. Park, cond-mat ; P. Rondson, A. E. Feiguin, C. Nayak, cond. mat ; G. Moller, A. Woijs, N. Cooper, cond-mat  e 2 /  L H E F ~10-30,  /E F ~1 2.a) Activation energy in transport experiments is approximately two orders of magnitude smaller than E F, 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 E F Do we know that in the Pfaffian state  An exapmple: superfluid 3 He:

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.

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, Edge states

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, (2010)

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 Pfaffian HLR Vivek Venkatachalam, Amir Yacoby, Loren Pfeiffer, Ken West, Nature 469, 185, 2011

Experiments on the activation energy of  xx 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

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.

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

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.