b.spivak university of washington with s. kivelson, s. sondhi, s. parameswaran
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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. Integer quantum Hall effect. Fractional quantum Hall effect. I will discuss the cases m/n=1/2, 5/2, …. - PowerPoint PPT PresentationTRANSCRIPT
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
integerarenm,,2
xy e
n
m
Integer quantum Hall effect
integerisn,2
xy en
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
c
mc
eBn cc ;12
Density of states on each Landau level:LH is the magnetic length
2
1
2 Hc Lc
eB
Filling factor:c
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
quantumfluxtheis2 00 e
cdl
Φ
Φ
Φ
Composite fermions
0k
e = fermion if k=2nboson 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)
)ln(Im,
,
,
0
kjjkkj
jkj
ii
zze
ck
k
c
ei
c
ei
a
ABrrarb
aAA
B and b are the magnetic field and statistical magnetic field A and a are the vector potential and statistical vector potential
0k
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:
2
222 1,,
2 H
HF
F
HpotF
Le
mLp
m
p
L
eEE
What are the effective mass and the Fermi energyof composite fermions?
at the mean field level the system is in a Fermi liquid state in a zero effective magnetic field!
B + b =0
jze c
k 0
Mean field electrodynamics of HLR state
eEj CFOhm’s law for composite Fermions:
,fieldeffectivethein
movefermionscompositeif
2/10
2/1
kbB
bothxxandxyare not quantized !
Experiments supporting HLR theory
bB
bB
cc
as diverge fermions composite of r radius cyclotron
Fv
cr
mc
ec
|| bBbB
0bB
J.P. Eisenstein, R.L. Willet, H.L. Stormer, L.N. Pffiffer, K.W. West
Superconductivity of composite fermions
),(')()( r'r,rrr'rr'r, ΔΠΔ g
Chern-Simons Superconducting order parameter
222Fyx
yx
ppp
ipp
pΔ
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
)(
2
2,
0
0
eEv
aAvvj
jez
ab
s
et
m
c
e
meN
k
Nk
s
ss
z is a unit vector perpendicular to the plane, at T=0 Ns=N
42
1||
.......,2
5,
2
1
,2
0)(
0
00
2*
22
0
e
k
ed
kdee
ee
et
m
xy
s
rar
ezj
eEeEv
Ψ
Correspondence between the perfect conductivity of the superconductors and the quantization of the Hall conductance:
Meissner effect incompressibility
Quantized vortices fractionally charged quasiparticles
gaptheisE
constantstructurefinetheis
ionconcentratelectrontheis
radiusBohrtheis
||
1,1
1
F
3/1
2/1
2/1
2
2
Δ
Δ
e
B
F
eLs
BeeL
n
a
v
nrif
anen
mc
Two types of conventional superconductors
jBc
curl4
lengthmagnetictheis
field"magneticlstatisticathe"oflengthnPenetratio.2
||lengthCoherence.1
H
H
F
L
L
v
Δ
Two characteristic lengths in the Pfaffian state at T=0
Two characteristic energy scales
2
2
energyFermiThe
||gapThe
HF mLE
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 EF >> 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
1||
1
vorticestwoofmergingwithassociatedenergyCoulomb
)1
(votexaofenergyonCondensati
2/1
2/12/1
2
2
22
Δ
ΔΔ
e
EN
Ee
E
Fc
F
F
If vortices agglomerate into big bubbles
Nc is the number of electrons in the bubble
If Nb ~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
Schematic phase diagram
Bosonic Chern-Simons theory.At Bogomolni’s point vortexes do not interact
!interactnotdovortexes
point)s(Bogomolnilinearisequationthe,If
D
casestatic
rr
02
'2
0||2/2
,0||22
||,2
1,
2
0
20
2
22
4int
0int0
m
D
m/kΦλ
mkm
D
iDDD
AaiDm
ddtSaak
ddtSSSSS
yx
kjiijkcscs
Φ
Φ
1.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 e2/LHEF ~10-30, /EF ~12.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
Do we know that in the Pfaffian state
An exapmple: superfluid 3He:
310~;40~
FF
pot
EE
U Δ
Existing experiments on measuring the effective vortex charge near 5/2filling 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.
KEF 100
Willet’s experiments measure the totalnumber 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
c
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)
The Yacoby group’s experiments are based on the fact that samples are disordered and there are puddles of HLR states embeddedinto the Pfaffian state. The charge of big HLR puddles grows in steps e/4 as a functionof 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 Jij have random sign, near the critical point the system is Pfaffian (p-wave superconducting) glass
An effective model of Joshepson junctions
energykineticquantumcos
jiji
ijJ Σ
Conclusion:Weakly coupled Pfaffian state is equivalent toType 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 QHfractions can be type 1 as well.