Emergence of fractional statistics for tracer particles in a Laughlin liquid

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by  Douglas Lundholm

Emergence of fractional statistics for tracer particles in a Laughlin liquid
Douglas Lundholm1 and Nicolas Rougerie2
1KTH Royal Institute of Technology, Department of Mathematics, SE-100 44 Stockholm, Sweden∗
2Universit´e Grenoble 1 & CNRS, LPMMC (UMR 5493), B.P. 166, F-38 042 Grenoble, France†
(Dated: March, 2016)
We consider a thought experiment where two distinct species of 2D particles in a perpendicular
magneticfieldinteractviarepulsivepotentials. Ifthemagneticfieldandtheinteractionsarestrong
enough, one type of particles forms a Laughlin state and the other ones couple to Laughlin quasi-
holes. We show that in this situation, the motion of the second type of particles is described by
6
an effective Hamiltonian, corresponding to the magnetic gauge picture for non-interacting anyons.
1
Theargumentisinaccordwith,butdistinctfrom,theBerryphasecalculation ofArovas-Schrieffer-
0
Wilczek. It suggests possibilities to observe the influence of effective anyon statistics in fractional
2
quantumHall systems.

r
p
PACSnumbers: 05.30.Pr,03.75.Hh
A

1
The basic explanationfor the fractionalquantum Hall mersedinanatomicgasformingaquantumHalldroplet.

] effect[1–5]istheoccurenceofstronglycorrelatedfluidsas Itthusseemstimelytorevisittheemergenceoffractional
l
e the ground states of 2D electron gases under strong per- statistics in this context.
- pendicular magnetic fields. The elementary excitations In this letter we present a derivation of the anyonic
r
t (quasi-particles) of these peculiar states of matter carry nature of the Laughlin quasi-holes that does not appeal
s
. a fraction of an electron’s charge, leading to the quan- to the Berry phase concept, and suggest a mechanism
t
a tization of the Hall conductance in certain fractions of bywhichthe statisticalmechanicsinfluenceofthe anyon
m e2/h. Even more fascinating is the possibility that these statistics could be directly ascertained. Our main result
- quasi-particles may have fractional statistics, i.e. a be- istoderiveexplicitlyaneffective,emergent,Hamiltonian
d
havior under continuous exchanges that interpolates be- for the test particles, see Equation (10) below. We then
n
o tweenthatofbosonsandthatoffermions[6–9]. Recently, propose an ansatz (19) for its ground state in a specific
c it has been proposed [10–14] that this physics could be experimental regime, and compute the associated den-
[ emulatedinultra-coldatomicgasessubjectedtoartificial sity (20), a measurable quantity one could use to probe

3 magnetic fields. the emergence of fractional statistics.
v The main evidence for the emergence of fractional
8 statistics concerns the quasi-hole excitations of the We consider,as in the proposals [19, 20], two different
0
Laughlin wave functions [15], which occur when the fill- species of interacting 2D particles. The Hilbert space is
5
ing factor ν of the 2D electron gas is the inverse of an
2
0 oddinteger. BasedonaBerryphasecalculation,Arovas, M+N =L2 (R2M) L2 (R2N), (1)
. SchriefferandWilczek[16]arguedthatacontinuous,adi- H sym ⊗ sym
1
abatic, exchange of two such quasi-holes leads the elec-
0 where M is the number of particles of the first type
6 trons’ wave function to pick up a phase factor e−iπν.
and N the number of particles of the second type. For
1 This suggeststhatif the quasi-holesaretobe considered
definiteness we assume that the two types of particles
v: as genuine quantum particles, they should be treated as be bosons, whence the imposed symmetry in the above
i anyons with statistics parameter ν.
X − Hilbert spaces. The following however applies to any
While this constitutes a powerful argument, a more
choice of statistics for both types of particles.
r
a direct derivation of the emergence of fractional statistics
We write the Hamiltonian for the full system as (spin
seems desirable (see e.g. the discussion in [17]). It is
is neglected)
indeed not entirely obvious that one should identify the
change in the phase of the electrons’ wave function with
M N
the statistics parameter attributed to the quasi-holes. H =H 1+1 H + W (x y ),
M+N M N 12 j k
Furthermore,themoststrikingconsequencesofquantum ⊗ ⊗ −
j=1k=1
statistics—thepresenceorabsenceofanexclusionprin- XX
ciple (cf. [18]), of condensationor a degeneracy pressure
where
— are statistical mechanics effects that cannot be ob-
served by measuring Berry phases.
M
1
It has recently been proposed [19, 20] that the frac- H = (p +eA(x ))2 + W (x x )
M 2m xj j 11 i− j
tionalstatistics ofquantumHallquasi-particlescouldbe j=1(cid:18) (cid:19) 1≤i<j≤M
X X
probed by observing the behavior of test particles im- (2)
2
is the Hamiltonian for the first type of particles and W has zerorangeandn islargeenough[21–23]. Inthe
22
context of (1), n should be even, n = 2 being the most
N
1 relevant case. For 2D electron gases, the second Hilbert
H = (p +A(y ))2 + W (y y )
N 2 yk k 22 i− j spaceshouldbeantisymmetricandnshouldbeodd. The
k=1(cid:18) (cid:19) 1≤i<j≤N
X X filling factor is then ν =1/n.
(3)
Next we consider the situation where the inter-species
that for the second type of particles. We shall denote
interaction potential W is strong enough to force the
X = (x ,...,x ) and Y = (y ,...,y ) the coordi- 12
M 1 M N 1 N
jointwavefunctionofthesystemtovanishwheneverpar-
nates of the two types of particles and choose units so
that ~ = c = 1, and the mass and charge of the second ticles of different species meet, i.e. Ψ(xj = yk) = 0 for
allj,k. SinceΨmuststaywithinthelowestLandaulevel
type of particles are respectively 1 and 1. We keep the
− of the bath particles, this forces the form
freedom that the first type of particles might have a dif-
ferent mass m and a different charge e < 0. The first M N
typeofparticlesshouldbethoughtofas−tracersimmersed Ψ(XM,YN)=Φ(XM,YN) (ζj zk)ΨLau(ZN),
−
in a large sea of the second type. We shall accordingly j=1k=1
Y Y
use the terms “tracer particles” and “bath particles” in where we identify ζ C with x R2. We assume
j j
the sequel. that choosing such an∈ansatz cance∈ls the inter-species
We have also introduced: interaction,i.e. thatthe latterissufficientlyshortrange.
Then, all terms of the total Hamiltonian acting on the
theusualmomentap = i andp = i .
• xj − ∇xj yk − ∇yk bath particles are frozen and thus Φ depends only on
a uniform magnetic field of strength B > 0. Our x ,...,x . This leads us to our final ansatz
1 M
•
convention is that it points downwards:
Ψ(X ,Y )=Φ(X )c (X )Ψqh(X ,Y ) (6)
M N M qh M M N
B B
A(x):= x⊥ = ( x ,x ). where Ψqh describes a Laughlinstate of the N bathpar-
2 1
−2 −2 − ticles, coupled to M quasi-holes at the locations of the
tracer particles:
intra-species interaction potentials, W and W .
11 22
•
M N
• an inter-species interaction potential W12. Ψqh(XM,YN)= (ζj zk)
−
The splitting between Landau levels of the one-body j=1k=1
Y Y
Hamiltonian appearing in HN is proportional to B, and (zi zj)ne−BPNj=1|zj|2/4. (7)
we assume that it is largeenough to force all bath parti- −
1≤i<j≤N
Y
cles to live in the lowest Landau level (LLL)
Here we choose c (X )>0 to enforce
qh M
H= ψ(x)=f(z)e−B|z|2/4, f holomorphic . (4)
c (X )2 Ψqh(X ,Y )2dY =1 (8)
qh M M N N
n o ZR2N | |
Note that the splitting between Landau levels for the
for any X . We thus ensure normalization of the full
tracer particles is rather proportional to eB/m so if M
wave function Ψ by demanding that
m > e it is reasonable to allow that they occupy sev-
eral Landau levels. Φ(X )2dX =1.
M M
If in addition the bath particles’ interaction potential ZR2M | |
W22 is sufficiently repulsive, we are led to an ansatz of Wenextarguethat,forawavefunctionoftheform(6)
the form and if N M, N 1,
≫ ≫
Ψ(XM,YN)=ΨLau(z1,...,zN)Φ(XM,YN) (5) Ψ,H Ψ Φ,HeffΦ + BN (9)
h M+N i≃ M 2
forthejointwavefunctionofthefullsystem,whereΨLau for an effective Hamiltonian(cid:10)Heff, so(cid:11)that the physics is
is a Laughlin wave function: M
completely reducedto the motionof the tracerparticles.
ΨLau(z1,...,zN)=cLau (zi zj)ne−BPNj=1|zj|2/4 Inthisdescription,theinteractionwiththebathparticles
− boils down to the emergence of effective magnetic fields
1≤i<j≤N
Y in Heff:
M
with the coordinates Z = (z ,...,z ) CN identified
swpiathceyo1n,.e.m.,uysNt i∈mRpo2.seTNtohsattayΦ1winith(i5n)Ntbhee∈halolloowmeodrHphilibceirnt HMeff = M 21m pxj +eA(xj)+Ar(xj)+Aa(xj) 2
j=1
y ,...,y . We assume that tuning the integer n allows X (cid:0) (cid:1)
1 N
to completely cancel the interaction W22 between par- + W11(xi−xj). (10)
ticles of the second type. It is for example the case if 1≤i<j≤M
X
3
Here the original potentials A and W are inherited We turn to vindicating our claim (9). Clearly, it is
11
from (2), while A and A emerge from the interaction sufficient to show that, for any j = 1...M we have es-
r a
withthe bathparticles. Thesubscripts standfor “renor- sentially
malizing”and“anyon”vectorpotentialsrespectively. We
p Ψ c Ψqh p +A (x )+A (x ) Φ. (15)
have the expressions xj ≃ qh xj r j a j
Indeed,insertingthis(cid:0)intheexpressionforthe(cid:1)energyand
B (x y)⊥
Ar(x)= 2πnZR2 |x−−y|2 1D(0,R)(y)dy (11) rdeecdaullcieng(9()8.)wTehemcaoynisnttaengtrainte(fi9r)stisinjutshtetYhNe LvaLrLiaebnleesrgtyo
of the N bath particles.
for some R √N to be defined below, and
∝ We need to recall a few facts about the Laughlin and
quasi-holes wave functions. Let us denote ̺ the one-
M B (x y)⊥ qh
A (x )= j − 1 (y)dy particle density of the quasi-holes ansatz,
a j −k=1,k6=j 2πnZR2 |xj −y|2 D(xk,lB)
X
(12) ̺ (y)=N c 2 Ψqh(X ,y,y ,...,y )2
qh qh M 2 N
with the (conveniently scaled) magnetic length | | ZR2(N−1)| |
dy ...dy , (16)
2 N
l = 2/B.
B
which implicitly depends on X . Laughlin’s plasma
M
Everywhere in the paper wpe denote D(x,R) the disk of analogy (see the appendix for details) provides reason-
center x and radius R, and 1D(x,R) the corresponding able approximations for ̺qh and cqh in terms of an aux-
indicator function (equal to 1 in the disk and 0 outside). iliary classical mean-field problem, whose density we de-
Note that A correspondsto a constantmagnetic field note ̺MF. Explicitly, assuming N 1 and N M, we
r ≫ ≫
supported in a large disk, approximate
curlAr(x)= Bn1D(0,R)(x), N−1̺qh ≃̺MF = 2πBnN 1D(0,R)− M 1D(xj,lB)
j=1
X
while Aa is generated by Aharonov-Bohm-like units of  (17)
flux attached to the tracer particles’ locations: withR2 =2(nN+M)/B.Wealsoassumethatthe disks
D(x ,l ) in (17) do not overlap, e.g. because the tracer
j B
M
B particles feel a hard core preventing them from getting
curlA (x )= 1 (x ).
a j − n D(xk,lB) j tooclosetooneanother. Onemightstillproceedwithout
k=1,k6=j
X theseassumptions,butweshallsticktothesimplestcase
Inthisconvention,thewavefunctionΦstillhasthesym- inthisletter. ApplyingtheFeynman-Hellmannprinciple
metry imposed at the outset, and the above thus de- to the effective plasma also leads to a useful approxima-
scribes anyons in the magnetic gauge picture [6–9]. tionforthederivativesofcqh withrespecttothelocation
We remark that the Aharonov-Bohm magnetic fluxes of the tracer particles:
are naturally smeared over a certain length scale in the c y x
above (extended anyons model [24–27]). However, the ∇xj qh N − j ̺MF(y)dy. (18)
derivationstrictlyspeakingrequiresthatthedisksin(12) cqh ≃ ZR2 |y−xj|2
do not overlap, e.g. that the interaction potential W With these approximationsat handwe canproceedto
11
containsasufficientlyextendedhardcore. Thus,byNew- the derivation of (15) where, from now on and without
ton’s theorem one may replace loss of generality, we take j =1. Clearly,
M 1(x x )⊥ ∇x1Ψ=(∇x1Φ)cqhΨqh+Φ cqh∇x1Ψqh+Ψqh∇x1cqh .
A (x )= j − k . (13)
a j − n x x 2 A straightforwardcalculation(cid:0) shows that (cid:1)
j k
k=1,k6=j | − |
X
Ψqh =V(x )Ψqh,
Finally, we shall see below that for M N, R √N ∇x1 1
≪ ∝ N 1
is typically very large,so that one mightwant to further V(x )= (e +ie )
1 1 2
approximate ζ1 zk!
k=1 −
X
B N x y N (x y )⊥
A (x)= x⊥. (14) = 1− k +i 1− k
r 2n x y 2 x y 2
1 k 1 k
k=1| − | k=1 | − |
X X
The effect of this field is thus to reduce the value of the x y (x y)⊥ N
external one. This charge renormalization is due to the = x1−y2 +i x1− y2 δy=yk dy
fractional charge associated with Laughlin quasi-holes. ZR2(cid:18)| 1− | | 1− | (cid:19) k=1 !
X
4
where e and e arethe basis vectorsin R2. In the state The associated free Hamiltonian (case W = 0) is then
1 2 11
Ψ, the bath particles y ,...,y are distributed accord- essentially exactly soluble (see [8, 29] and references
1 N
ingtothedensity̺ ,sothatwemaysafelyapproximate therein). InthecaseofaweakinteractionpotentialW ,
qh 11
it is reasonable to expect that the ground state is also
N
dictated by the free Hamiltonian, namely
δ ̺ (y) N̺MF(y)
Xk=1 y=yk ≃ qh ≃ Φ(XM)=cΦ ζi ζj 1/ne−m2ωtPMj=1|ζj|2, (19)
| − |
forthe purposeofcomputing(9). Insertingintheabove, 1≤iY<j≤M
recalling (18), we observethat the realpartofV cancels see [8, Equations(6)and (36)to (38)]. We havedenoted
with c , leading to
∇x1 qh
eB∗ 2
p Ψ=c Ψqh(p +ImV(x ))Φ ω = ω2+
x1 qh x1 1 t s 2m
(cid:18) (cid:19)
where
an effective trapping frequency and c a normalization
Φ
ImV(x1)≃NZR2 (|xx11−−yy)|2⊥̺MF(y)dy. ceooxfnpasetrcaetnpttuh.lastiNvteohtpiesotathennasttaitaszliniWscest(i.l1l9r)eavsaonnisahbeles ianttζhie=preζsje,nwcee
11
Inserting the expression (17) of ̺MF yields As a possible probe of the situation just discussed, we
note that if the state of the full system is given by (6)-
ImV(x )= B (x1−y)⊥1 (y)dy (8)-(19), the one-particle density of the tracer particles
1 2πnZR2 |x1−y|2 D(0,R) can be approximated, for large M, as
− 2BπnZR2 (|xx11−−yy)|⊥2 1D(x1,lB)(y)dy ρtracer ≃ mωπtn1D(0,R′), (20)
B M (x y)⊥ whereR′ = M/(mω n)isfixedbynormalization. This
1− 1 (y)dy. t
− 2πn j=2ZR2 |x1−y|2 D(xj,lB) follows frompthe same kind of considerations that lead
X to(17),seetheappendixfordetails. Boththedistinctive
The second term is clearly 0, while the first and third flat profile and the (length and density) scales involved
termsgivethecontributionsof (11)and(12)respectively. are signatures of the effective anyon Hamiltonian, and
This closes the argument establishing (15), and (9) fol- thusitsemergencecanbedirectlyseeninameasurement
lows because of (8). ofthedensityprofileofthetracerparticles. Indeed,both
the gaussian profile one would get for free bosons in the
We now discuss possible measurable consequences of
LLL,andtheThomas-Fermi-likeprofileinthepresenceof
the above derivation. The effective Hamiltonian (10)
weak pair interactions (see e.g. [30–32]), differ markedly
is notoriously hard to solve, even at the ground state
from (20).
level(see[6–9]forreviews). Onecanhoweverusecertain
known results for comparisons with experiments. For il- Conclusions. Assuming anansatz ofthe form(6) for
lustration,weshallconsideracasewherethecomparison the joint wave function of the system, we have demon-
seems the mostsimple anddirect. We haveacoldatoms stratedthatthetracerparticlesfeelaneffectivemagnetic
systeminmind,withbothtypesofparticlesbeingbosons Hamiltonian (10). The latter contains long-range mag-
held by a harmonic confinement neticinteractionswhoseformisidenticaltothoseappear-
inginthe“magneticgaugepicture”descriptionofanyons
1
V(r)= ω2r2. withstatisticsparameter 1/n. Onemaythus(formally)
2 extract from Φ a phase fa−ctor to gauge A away (singu-
a
lar gauge transformation). The effective Hamiltonian is
The (artificial) magnetic field can be imposed by rotat-
thena usualmagnetic Laplacian,with reducedmagnetic
ing the trap or by more elaborate means [28]. The pre-
field, but the effective wave function (formally) picks up
vious discussion is unchanged if the energy scale asso-
a factor of e−iπ/n upon exchanging two particles, hence
ciated with the trap is smaller than those entering the
describes anyons.
derivation, thus the effective Hamiltonian for the tracer
The assumptions we made are consistent with those
particles is (10), supplemented by m times the trapping
of the usual Berry phase calculation [16]. Namely, the
term. Aspreviouslydiscussedweusetheexpressions(13)
energy scales associated to the effective Hamiltonian we
and (14) for the effective fields. If eB/m is large com-
derivedshouldbe smallerthan the energygapabovethe
pared to the trapping energy, it makes sense to project
ground state ansatz (6).
thisHamiltonianontotheLLLoftheassociatedeffective
The reasoning we proposed suggests a way to probe
magnetic field, of strength
theanyonstatistics,byadirectobservationofthecollec-
eB∗ =(e n−1)B. tivebehaviorofthetracerparticles. Iftheyareoriginally
−
5
bosons, they will acquire some form of exclusion princi- which can be regarded as the Hamiltonian for N 2D
ple [18, 29, 33–35] whose influence could be observed. classical charged particles in a quadratic external po-
We have discussed a possible set-up where simple cal- tential, interacting via repulsive Coulomb forces. The
culations show measurable effects of the emergent frac- quasi-holes/tracerparticlesappearinthis representation
tional statistics. The consideration of more generalsitu- asfixedrepulsivepointchargeswhoseinfluence is feltby
ations will demand further studies of the trapped anyon the bath particles.
gas (see however [27] and references therein for the dis- Thus, µ is characterized as the probability measure
qh
cussion of approximate models). This remains a topic minimizing the free energy
for future investigation, as does a more mathematically
rigorous derivation of the phenomenon. [µ]= Hplas(Y )µ(Y )dY + µlogµ (23)
qh N N N
As for possible generalizations,non-Abelian Quantum F ZR2N ZR2N
Hallstatesasconsideredin[36]arenotcurrentlycovered
and the minimum free energy F satisfies
qh
by our methods. However, if one is willing to take for
granted, or argue for, appropriate replacements to (17)- F = logZ =2logc . (24)
qh qh qh
−
(18), the approach applies to the case where the bath
particlesformanotherAbelianQuantumHallstate,such Without further approximations, this rewriting is not
as a composite fermions state as discussed in [19]. particularly useful, but the point is that we may use the
good scaling properties of Ψqh, inherited from the fact
Acknowledgments: This work is supported by the that it is of the form polynomial gaussian. Indeed,
×
ANR (Project Mathostaq ANR-13-JS01-0005-01) and scaling length units by a factor √N transformsthe limit
the Swedish Research Council (grant no. 2013-4734). N for the effective plasma problem into a mean-
→ ∞
ManythankstoThierryChampel,MicheleCorreggi,Ste- field/small temperature regime. That is, it gets mapped
fano Forte, St´ephane Ouvry, Gianluca Panati, Franc¸ois toyetanotherequilibriumstatisticalmechanicsproblem,
D. Parmentier, Jan Philip Solovej and Jakob Yngvason butthistime withacouplingconstant N−1 andanef-
∝
for useful and stimulating discussions. fective temperature N−1.
∝
Since the minimization of (23) can be mapped to a
mean-field/smalltemperatureregimebyasimplechange
Appendix : The plasma analogy
of scale, it is reasonable to use an ansatz of the form
µ = ρ⊗N and neglect the entropy to perform the min-
Hereweprovidesomesupportforourclaimsaboutthe imization. We refer to [23, 38, 39] where this approx-
approximationof̺qh andcqh inEquations(17)and(18) imation is rigorously justified for related models, and
respectively. As mentioned previously, we argue these to [40, 41] for numerical confirmation. Making this ma-
are reasonable provided N 1, N M and nipulation leads to a classical mean-field energy func-
≫ ≫
D(x ,l ) D(x ,l )=∅ for all j =k tional
j B k B
∩ 6
D(x ,l ) D(0,R) for all j =1...M. (21)
j B ⊂ B M
MF[ρ]= x2 2 log x x ρ(x)dx
Pclhesyssihcoaulllyd,tbheetsheoausgsuhmtopftiaosnassmmeaalnlnthuamtbtehreotfraimceprupraitriteis- E ZR22| | − Xj=1 | − j|
 
immersed in a large sea of the bath particles. That the
nN ρ(x)log x yρ(y)dxdy, (25)
disks around xj and xk do not overlap assumes a hard- − ZZR2×R2 | − |
core condition, that can be provided by the interaction
and we should expect
potential W .
11
Laughlin’splasmaanalogy,originatingin[15,37],con-
̺ N̺MF and F NEMF
sists in writing qh N→≃∞ qh N→≃∞
c2 Ψqh 2 =µ = 1 exp Hplas(Y ) with EMF and ̺MF respectively the minimum and min-
qh| | qh Zqh − N imizer of (25). Furthermore, the Feynman-Hellmann
(cid:0) (cid:1) principle tells us that the derivatives of EMF with re-
as a Gibbs state for a classical electrostatic Hamiltonian
Hplas. In this convention, the partition function is Z spect to xj are given by integrating the derivative of the
qh
energy functional against the minimizer:
and the temperature is 1. Explicitly we have
x x
Hplas(YN)= N B2|yj|2−2 M log|yj −xk|! ∇xjEMF =2ZR2 |x−−xjj|2̺MF(x)dx. (26)
j=1 k=1
X X At this stage, combining with (24) we have thus jus-
2n log yi yj , (22) tified (or, at least, explained) the approximations (17)
− | − |
1≤i<j≤N and (18). What remains to be discussed is the explicit
X
6
expressionfor̺MF (right-handsideof (17))thatwehave [11] I.Bloch,J.Dalibard, andW.Zwerger,Rev.Mod.Phys.
used in the main text. Note that the density profile (20) 80, 885 (2008).
for the tracer particles is obtained in the same way, by a [12] A. Morris and D. Feder, Phys. Rev. Lett. 99, 240401
(2007).
simple change of units.
[13] M. Roncaglia, M. Rizzi, and J. Dalibard,
Recall that the expression is clearly physically sound:
www.nature.com, ScientificReports 1 (2011).
the B x2/2 term in the energy functional (25) corre-
[14] S. Viefers, J. Phys. C 12, 123202 (2008).
| |
sponds to the potential generated by a constant back- [15] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983).
ground of charge density 2B (in somewhat arbitrary [16] S.Arovas,J.Schrieffer, andF.Wilczek,Phys.Rev.Lett.
−
units). Each tracer particle/quasi-hole corresponds to a 53, 722 (1984).
fixed point with charge 4π, and the bath particles have [17] S. Forte, Mod. Phys.Lett. A 6, 3152 (1991).
[18] D.LundholmandJ.P.Solovej,Phys.Rev.A88,062106
charge4πnN inthis representation. Eachtracerparticle
(2013).
thusscreensthebackgroundchargeinadiskofradiusl
B [19] Y.Zhang,G.J.Sreejith,N.D.Gemelke, andJ.K.Jain,
around it, and the bath particles distribute in order to
Phys. Rev.Lett. 113, 160404 (2014).
screen the remaining background density. [20] N. R. Cooper and S. H. Simon, Phys. Rev. Lett. 114,
Foramorerigorousderivation,notethattheminimizer 106802 (2015).
̺MF for (25) must satisfy the variational inequality [21] S. Trugman and S. Kivelson, Phys. Rev. B 31, 5280
(1985).
B M [22] T. Papenbrock and G. F. Bertsch,
x2 2 log x xj 2nN̺MF log . λMF (27) Phys. Rev.A 63, 023616 (2001).
2| | − | − |− ∗ | |≥
j=1 [23] N.Rougerie,S.Serfaty, andJ.Yngvason,J.Stat.Phys.
X
154, 2 (2014).
with equality on the support of ̺MF. Here,
[24] S. Mashkevich, Phys.Rev.D 54, 6537 (1996).
[25] C. Trugenberger, Phys. Lett. B 288, 121 (1992).
λMF =EMF nN ̺MF(x)log x y̺MF(y)dxdy [26] M. Y. Choi, C. Lee, and J. Lee, Phys. Rev. B 46, 1489
− ZZR2×R2 | − | (1992).
is a Lagrange multiplier. A useful result is the following [27] D. Lundholm and N. Rougerie, J. Stat. Phys. 161, 1236
(2015).
(see [42, Theorem 1.2]): let ρ be a probability measure
that satisfies, for some λ R, [28] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg,
∈ Rev. Mod. Phys. 83, 1523 (2011).
M [29] A.Dasni`eresdeVeigyandS.Ouvry,Phys.Rev.Lett.72,
B
x2 2 log x x 2nNρ log . λ ( λ) 600 (1994).
j
2| | − | − |− ∗ | |≤ ≥ [30] A. Aftalion, X. Blanc, and J. Dalibard,
j=1
X Phys. Rev.A 71, 023611 (2005).
where ρ > 0 (respectively ρ = 0). Then it must be that [31] A. Aftalion, X. Blanc, and F. Nier, Phys. Rev. A 73,
λ = λMF and ρ = ̺MF. It is easy to see that the right- 011601(R) (2006).
handsideof (17)satisfiestheabove,provided(21)holds. [32] A.Aftalion,X.Blanc, andF.Nier,J.Funct.Anal.241,
661 (2006).
Indeed, the potential ρ log . can in this case be easily
∗ | | [33] Y.-S. Wu,Phys.Rev.Lett. 73, 922 (1994).
computed using Newton’s theorem [43, Theorem 9.7].
[34] D.LundholmandJ.P.Solovej,Comm.Math.Phys.322,
883 (2013).
[35] D.LundholmandJ.P.Solovej,Ann.HenriPoincar´e15,
1061 (2014).
[36] Y. Zhang, G. J. Sreejith, and J. K. Jain, Phys. Rev.B.
∗ [email protected] 92, 075116 (2015).
† [email protected] [37] R. B. Laughlin, in The quantum Hall effect, edited by
[1] S.Girvin, S´eminaire Poincar´e 2, 54 (2004). R. E. Prange and S. E. Girvin (Springer, Heidelberg,
[2] M. O. Goerbig, (2009), arXiv:0909.1998. 1987).
[3] H. St¨ormer, D. Tsui, and A. Gossard, Rev.Mod. Phys. [38] N. Rougerie, S.Serfaty, and J. Yngvason, Phys.Rev.A
71, S298 (1999). 87, 023618 (2013).
[4] R.B. Laughlin, Rev.Mod. Phys. 71, 863 (1999). [39] N.Rougerie and J.Yngvason,Comm. Math.Phys.336,
[5] J. K. Jain, Composite fermions (Cambridge University 1109 (2015).
Press, 2007). [40] J.M.Caillol, D.Levesque,J.J.Weis, andJ.P.Hansen,
[6] A.Khare,FractionalStatisticsandQuantumTheory,2nd J. Stat. Phys.28, 325 (1982).
ed.(World Scientific, Singapore, 2005). [41] J. M. Caillol and D. Levesque,
[7] J. Myrheim, in Topological aspects of low dimensional Phys. Rev.B 33, 499 (1986).
systems, Les Houches - Ecole d’Ete de Physique Theo- [42] D.Chafa¨ı, N.Gozlan, andP.-A.Zitt,AnnalsofApplied
rique,Vol. 69, edited by A. Comtet, T. Jolicœur, S.Ou- Probability 24, 2371 (2014).
vry, and F. David (1999) pp.265–413. [43] E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate
[8] S.Ouvry,S´eminaire Poincar´e 11, 77 (2007). Studies in Mathematics, Vol. 14 (American Mathemati-
[9] F. Wilczek, Fractional Statistics and Anyon Supercon- cal Society, Providence, RI,2001) pp.xxii+346.
ductivity (World Scientific,Singapore, 1990).
[10] N.R. Cooper, Advancesin Physics (2008).