Transcript Theoretical Considerations and Experimental Probes of the

Theoretical Considerations and
Experimental Probes of the =5/2
Fractional Quantized Hall State
by Bertrand I. Halperin, Harvard University
talk given at the
Rutgers Statistical Mechanics Meeting,
May 11, 2008
in honor of
Edouard Brézin and Giorgio Parisi
E. Brézin and the quantum Hall effect
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[2].
=5/2 Quantized Hall State
In 1987, Willett et al. discovered a Fractional Quantized Hall
plateau at Landau-level filling fraction =5/2 , the first evendenominator QHE state observed in a single-layer system.
The nature of this state is still under debate.
Moore-Read “Pfaffian” State
Moore and Read (1991) proposed a novel trial wave
function, involving a Pfaffian, as a model for a quantized
Hall state in a half-filled Landau level. Further clarified by
Greiter, Wen and Wilczek; suggested as an explanation for
the quantized Hall plateau at  = 5/2 = 2 +1/2.
Elementary charged excitations, which have charge  e/4,
obey “non-abelian statistics”.
State is related mathematically to a superconductor of
spinless fermions, with px+ipy pairing. (Described as a
“px+ipy superconductor of composite fermions.”)
Non-abelian statistics for Moore-Read
5/2 state
Consider a system containing 2N localized quasiparticles, far
from each other and far from boundaries. Then there exist M=2N-1
orthogonal degenerate ground states, which cannot be
distinguished from each other by any local measurement.
Moving various quasiparticles around each other and returning
them to their original positions, or interchanging quasiparticles,
can lead to a nontrivial unitary transformation of the ground
states, which depends on the order in which the winding is
performed. ( Unitary matrix depends on the topology of the
braiding of the world lines of the quasiparticles. Matrices form a
representation of the braid group).
If two quasiparticles come close together, degeneracy is broken;
but energy splittings fall off exponentially with separation.
Topological quantum computation
Non-abelian quasiparticles may be useful for “topological
quantum computation”.
[ Kitaev, quant-ph/9707021; Freedman, Larson, Wang,
Commun. Math Phys (2002); Bonesteel, et al PRL (2005). ]
Manipulation of qubits would be carried out by moving
quasiparticles around each other, not bringing them close
together. Advantage: exponentially long decoherence times if
quasiparticles are sufficiently far apart.
Caveats:
1. Current materials are very far from this regime.
2. Moore-Read state is actually not rich enough for general
topological quantum computation. However, this may be
possible with some other states in the second Landau level.
What is the evidence that the  =5/2
Quantized Hall State is indeed of the
Moore-Read type ?
Evidence comes primarily from numerical calculations on finite
systems.(Morf & collaborators, 2002, 2003; Das Sarma et al. 2004;
Rezayi and Haldane, 2000).
Using parameters appropriate to the experimental situation, find a
spin-polarized ground state, which seems to have an energy gap,
and which has good overlap with Pfaffian wave function. But
evidence is not overwhelming, and is certainly open to question.
Existing experiments do not provide clear evidence on nature of
state. Recent improvements in quality of materials, and new
experimental techniques,give hope of resolving these questions.
What are the theoretical alternatives?
Anti-Pfaffian state: Topologically distinct from the Pfaffian
state, but has similar properties, e/4 quasiparticles, nonabelian statistics. (Would be equally interesting.)
Other kinds of paired states, including tightly bound pairs in
a fully spin-polarized system, or partially polarized or
unpolarized systems. Would have e/4 quasiparticles but not
non-abelian statistics. (Not so interesting.)
Other kinds of quantized Hall states we haven’t thought of?
Anti-Pfaffian State
The Pfaffian (Pf) state is not symmetric under particle-hole
conjugation. The anti-Pfaffian (APf) is its particle-hole
conjugate. Pf and APf have been shown to be topologically
distinct. (Rezayi and Haldane, 2000; Levin, Halperin, and
Rosenow, 2007; Lee, Ryu, Nayak and Fisher, 2007. )
If you vary the parameters in a system so that the ground state
changes from Pf to APf, there must be a phase transition
separating the two phases.
If the parameters of a system vary in space, so that one region
is Pf and one is APf, there must be a boundary separating them,
with gapless low-energy excitations. (Both states have an
energy gap in the bulk.)
Boundary between a Pf or APf =5/2 state
with a =2 Integer Quantized Hall state (filled
Landau level)
The simplest boundary between a Pf =5/2 state and a =2 state
should have two low-energy chiral modes: a bosonic phonon
mode and a neutral Majorana fermion mode, traveling in the
same direction. The edge has a thermal Hall conductance with
K = 1 +1/2 = 3/2.
The boundary between an APf =5/2 state and a =2 state has a
different structure, and has K=-1/2.
The thermal Hall conductance is a topological invariant, cannot
be altered by disorder or boundary reconstruction.
Q = T K  2kBT/3h
Edges of a Pf or APf state
=3
=3
K= 3/2
K= -1/2
Pf
APf
K= 3/2
K= -1/2
=2
Thermal Conductance K: 3/2 = 1/2 +1
=2
-1/2 = -1/2 - 1 + 1
Nonlinear electrical resistance
Experimentally, it is difficult to measure the thermal Hall
conductance. However, the different boundary structures of Pf
and APf with, say, a vacuum or a simple =2 state should lead to
different Luttinger-liquid-type properties, which should give rise
to different forms of non-linear electrical resistance at a narrow
constriction, which has been studied experimentally.
Recent experimental results seem to favor APf. (Marcus lab)
The complications expected in a real system have not been
completely sorted out.
Competition between the Pfaffian and
Anti-Pfaffian State
If one has only two-body interactions, and one ignores Landaulevel mixing, the Hamiltonian of the half-full Landau level is
particle-hole symmetric. Since the Pf and APf states are
particle -hole conjugates of each other, they must have
identical energies in this model.
Degeneracy can be broken by inter-Landau level mixing,
effects of impurities, sample boundaries, and deviations from
half filling. Sample boundaries may be particularly important
in a narrow constriction. Pf and APf may coexist, with a
boundary between them.
Note: The Pf and APf trial wave functions are exact ground
states of models with three-body interactions, which break
particle-hole symmetry explicitly.
Proposed experiments to look for non-abelian
statistics,
or at least test whether =5/2 state is Pf, APf
or something else.
The most direct demonstration of non-abelian statistics wold
require the ability to move one quasiparticle around another in a
controlled way. Possible in principle, but we are far from being
able to accomplish this technologically. We seek other
experiments to examine the =5/2 state to see if it is of the
Moore-Read type.
Actual recent experiments
*Measurements of the non-linear resistance of a narrow
constriction at =5/2 can give important information about the
state. Interpretation seems to be complicated
*Measurements of the quasiparticle charge. Moore-Read
quasiparticles have charge e/4 . Recent measurements of shot
noise from a quantum point contact at =5/2 support this result.
(Heiblum group).
Quasiparticles with charge e/4 are necessary, but not sufficient:
could also result from other states without non-abelian
statistics.
Proposed experiments (1)
.
*Interference-type experiments directly related to nonabelian statistics.
Proposed Interference Experiments at
=5/2
Discussed by: Ady Stern and B. I. Halperin (PRL 2006)
Other theoretical papers discussing interference
experiments with non-abelian quasiparticles include:
Das Sarma, Freedman and Nayak, (PRL 2005)
Bonderson, Kitaev, and Shtengel, (PRL 2006)
Fradkin et al., Nucl Phys B 1998
Bonderson, Shtengel and Slingerland, cond-mat/0601242:
Discuss consequences for Read-Rezayi parafermion states,
possibly applicable to =12/5 .
Fix gate voltage at point contacts. Vary area A by varying
voltage on side gate. Measure resistance V12/I. Expect
oscillations in the resistance as a function of A
1
I
2
+
= 5/2
+
t1
t2
+
+
+ = quasihole
Side Gate
= 5/2
I
If =5//2 state is non-abelian Pfaffian or Anti-Pfaffian state:
the period of resistance oscillations should depend on
whether the number of localized quasiholes encircled by the
path is even or odd.
Weak back-scattering: V12t1 + t2 ei2 , with
=A B/40 , only if the qh number is even.
1
I
2
+
= 5/2
+
t1
t2
+
= 5/2
I
+
If interference path contains an odd number of localized
quasiholes, quasiparticle path tunneling at point t2 changes the
state of enclosed zero-energy modes, and cannot interfere with
path tunneling at t1.
Will these experiments actually work
and show non-abelian statstics?
We don’t know.
Real systems can be pretty complicated.
Acknowledgments
Co-authors: Ady Stern, Bernd Rosenow, Michael Levin,
Steve Simon, Chetan Nayak, Ivalo Dimov.
Discussions with experimentalists, too numerous to name.
Financial support: NSF, Microsoft Corporation, US-Israel
Binational Science Foundation.
Proposed experiments (2).
:
*Measurements of spin polarization. Moore-Read has complete
polarization in second Landau Level. Measurement should be
possible, but difficult at very low temperatures. (Would be a
consistency check, because some of the alternatives to MooreRead are not fully polarized, but not a definitive test.)
Existence of a quantized Hall state at
=5/2
If central region contains an odd number of localized
quasiparticles, this interference term is absent. Then leading
interference term varies as
Re [t1* t2 e2i ]2 . (Period corresponds to an area containing
two flux quanta, rather than four.)
Zero-energy modes
Specifically, in a px+ipy superconductor, an isolated vortex, at
point Ri , has a zero energy mode, with Majorana fermion
operator gi : (from solution of the Bogoliubov-de Gennes equations)
gi = gi† ,
gi2 = 1 , {gi , gj} = 2ij
To form ordinary fermion creation or annihilation operator: need
pair of vortices: e.g.
c12 = (g1+ i g) /  ,
c12† = (g1- i g) / ,
obey usual fermion commutations rules
N1 = c12†c12 has eigenvalues = 0, 1. [N12,N34] = 0 , etc.
Constraint : Number of occupied pairs = Nelectrons (mod 2) .
-> 2N vortices gives 2N-1 independent states
Explicit relation between Majorana operator
and electron operators
gi =
 dr [ u(r) y(r) + v(r) y†(r) ]
with v(r) = u*(r), localized near vortex.
If vortices are far apart, so there is no overlap between the
wave functions of their zero-energy states, then these states
must have precisely zero energy.
This relates to the fact that solutions of the BdG equations
must occur in pairs with E1=-E2.
Braiding properties of vortices
Vortices at points
.
R1
R2
R3
R4
Braiding properties
Vortices at points
R1
R2
Move vortex 2 around vortex 3
~ g2 g3 .
R3
.
R4
Gives unitary transformation
Changes N12 -> (1-N12) , N34 -> (1-N34) .
Braiding properties
Vortices at points
R1
R2
R3
R4
Move vortex 2 around 3 and 4. Gives unitary transformation
~ g2g4 g2g3 = g3 g4 : leaves N12 and N34 unchanged.
Since vortices are indistinguishable, get other unitary
transformations by simply interchanging positions of two
vortices.
Order of interchanges matter: The unitary transformations do
not commute.