Statistics of skyrmions in Quantum Hall systems

7 Pages·0.14 MB·

by  Jacek Dziarmaga

Statistics of skyrmions in quantum Hall systems
Jacek Dziarmaga
Department of Mathematical Sciences,
University of Durham, South Road, Durham, DH1 3LE,
United Kingdom.
E-mail address: [email protected] ∗
6 December 11, 1995
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n Abstract
a
J We analyze statistical interactions of skyrmions in the quantum Hall system near a critical filling fraction in the
5 frameworkoftheGinzburg-Landaumodel.Thephasepickedupbythewave-functionduringanexchangeoftwoskyrmions
1 close to ν = 1/(2n+1) is π[S+1/2(2n+1)], where S is the skyrmion’s spin. In the same setting an exchange of two
fully polarized vortices gives rise to the phase π/(2n+1). Skyrmions with odd and even numbers of reversed spins have

2 different quantum statistics. Condensation of skyrmions with an even number of reversed spins leads to filling fractions
v with odd denominators, while condensation of those with an odd number of reversed spins gives rise to filling fractions
7 with even denominators.
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5DTP-95/79
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Recentlyskyrmionexcitationsnearthefilling factorν =1havedrawnremarkableattention[1,2,3].Therearetwokinds
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dof topological excitations in single-layer quantum Hall systems. When the system is fully polarized, the relevant charged
nquasiparticlesaretopologicalvortices.Atν =1/(2n+1)thechargeofsuchavortexise/(2n+1)andthespinis1/2(2n+1).
o
It has been suggested [1] that for a weak Zeeman coupling the lowest energy quasiparticle is a skyrmion or a slowly varying
c
topological texture. Its electric charge is still e/(2n+1) but its total spin S can be substantially larger. Such large spin but
:
vmoderatechargeexcitationscanexplaintheobserveddepolarizationasthefillingfactorslightlydeviatesfromν =1/(2n+1)
i
X[4].
Ithasbeenproposedtodescribeskyrmionexcitationsintermsofaneffectiveferromagneticmodel[1].Themodelexplains
r
athe dependence of the skyrmion size on relative strenght of Coulomb and Zeeman interaction. However, it does not predict
correctlyskyrmionenergies.Theenergyofskyrmionandantiskyrmionseemtobethesame.Microscopiccalculations[2]show
that it is not the case. The effective model has been derived [1, 5] from the Ginzburg-Landau model of the quantum Hall
system [6, 7]. In the derivation, among other things, the charge density was assumed to be constant. This leaves polarized
vortices outside the reach of the effective model. In terms of the Ginzburg-Landau model the difference between a vortex
and a skyrmion is that a vortex is fully polarized while inside the skyrmion core there is some number of electrons with
reversedspins. Sucha depolarizationhelps to minimize the total Coulombenergy.It costs some Zeemanenergy but it is not
an obstacle when the Zeeman coupling is sufficiently weak.
InthispaperweconsidersometopologicalpropertiesofskyrmionsandvorticesintheframeworkoftheGinzburg-Landau
model.WeshowthattheMagnusforceactingonvorticesandskyrmionsisthesame.Wealsoconsiderstatisticalinteractions
of skyrmions. We will show that during a clockwise exchange of two skyrmions the wave-function picks up the geometrical
phase
1 t2 d
Γ=[S+ ] Arg[ξ (t) ξ (t)] , (1)
2(2n+1) Z dt 1 − 2
t1
where S is the skyrmion’sspin.This formulaapplies also to vorticeswhenwe put S =S = 1 .Thus the Ginzburg-
vortex (2n+1)
Landau treatement shows that the statistical interaction is much stronger for skyrmions than for vortices. It is proportional
to the number of reversedspins.
∗OnleaveofabsencefromInstituteofPhysics,JagiellonianUniversity,Krako´w,Poland
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1 The Ginzburg-Landau model and topological defects
We wish to investigate the statistical interaction of skyrmions in the quantum Hall effect. We will describe this system by
the Zhang-Hansson-Kivelsonmodel [6] generalized by Kane and Lee [7] to describe unpolarized quantum Hall systems.
The Lagrangianof the model is
1 1 1
L=iφ†(∂ +ia )φ (∂ +i(a +eA ))φ 2 + εµνσa ∂ a λ[φ†φ(~x) ρ ]2+γ[φ⋆φ φ⋆φ ]+µ[φ⋆φ +φ⋆φ ] . (2)
t 0 −2m | k k k | 4Θ µ ν σ−2 − 0 ↓ ↓− ↑ ↑ ↓ ↓ ↑ ↑
Greek indices denote 0,1,2, while Latin indices take values 1,2. When they are repeated, summation is understood. We
use the signature (+, , ). φ is a two-component complex scalar field, φ = (φ ,φ ). a is a statistical gauge field while
↓ ↑ µ
− −
A = 1Bε xl is a gauge potential of the external uniform magnetic field B directed down the z-axis. eρ is the positive
k −2 kl 0
background charge density, related to the external magnetic field by eB = 2Θρ . For the boson field to represent fermions
0
the parameter Θ must take one of the values Θ = (2n+1)π, where n is a nonnegative integer. m is the effective electronic
mass and γ is the effective Zeeman coupling. µ is a chemical potential, chosen so that the ground state of the system (2)
is, up to a gauge transformation, the solution φ↓ = √ρ0,φ↑ = 0 with the statistical gauge field ”screening” the external
magnetic field, a +eA = 0. The chemical potential has to be µ = γ. In this ground state the system is fully polarized.
k k
−
The Lagrangian(2) should be supplemented by the Coulomb interaction term
1 e2/ε
d2x′[φ†φ(~x) ρ ] [φ†φ(x~′) ρ ] , (3)
− 2Z − 0 ~x x~′ − 0
| − |
where ε is a dielectric constant of the host material.
There are two types of relevant topological excitations in the model. One of them is simply a vortex or a fully polarized
quasihole. This configuration is given by the Ansatz
φ =f (r)e−iθ ,
↓ v
φ =0 ,
↑
eB
a = r+a (r) ,
θ v
2
a =b (r) . (4)
0 v
The modulus fv(r) has to interpolate between fv(0)=0 and fv(∞)=√ρ0. The asymptote of av(r) at infinity is av(r)≈ r1.
The vortex is by definition restricted to the lower spin component. This restriction is consistent with field equations of
the model (2) for any value of the Zeeman coupling γ. Such a solution indeed exists as is well known from the studies on
vortices in the Ginzburg-Landau model of the fully polarized quantum Hall effect [8]. The total charge is proportional to
the statistical magnetic flux and equals to e/(2n+1). The distribution of the electronic density is such that it vanishes in
the very center of the vortex and tends to eρ ouside the vortex core. This is not the most favourable charge distribution
0
−
fromthe pointofviewofboth the Coulombinteractionandthe quarticself-interactionterm.While the totalchargeisfixed,
the Coulomb interaction tends to minimize the integral of the charge density squared. In other words it tends to make the
chargedistribution asdiluted aspossible.When werestricttothe spin-downcomponent,the deviationofthe chargedensity
from eρ mustequalto eρ atthe centerofthe vortex.This is atopologicalobstruction.Atthe same time the totalcharge
0 0
−
is quantized so there is not much freedom left to make the charge distribution more dilute. However, for sufficiently small
Zeemancoupling the configurationmay find a wayto minimize its Coulombenergyby nucleationof the spin-up component.
A more general configuration, with more freedom to minimize the Coulomb energy, is
φ =f (r)e−iθ ,
↓ s
φ =g(r) ,
↑
eB
a = r+a (r) ,
0 s
2
a =b (r) . (5)
θ s
Now we admit a nonzero spin-up component. We want it to be nonzero in the vortex core, g(0)=0, and to cost a minimal
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gradient energy so the phase of the spin-up component has to be constant. For a finite energy configuration g( ) = 0.
∞
Excitation of the upper component costs some Zeeman energy so it is not energetically favourable except for small effective
Zeeman couplings γ, where the gain in Coulomb energy can prevail the loss in Zeeman energy. Note that with the Ansatz
(5) the average spin points up in the middle of the soliton while it points down outside the core. This configuration is a
skyrmion but in the language of the untruncated model (2). As compared to a uniform background with charge density ρ ,
0
the component adds a negative charge Q ,
↑
↑
∞
Q = 2πe rdr g2(r) , (6)
↑ − Z
0
2
while the component contributes a positive electronic charge deficit
↓
∞
Q = 2πe rdr [f2(r) ρ ] . (7)
↓ − Z − 0
0
The total charge is the same as for the fully polarized vortex; the two contributions add up to Q +Q = e .
↑ ↓ (2n+1)
Thus there is an interplay of two factors, namely the Coulomb and the Zeeman energy. For strong Zeeman coupling
vortices are the relevantquasiparticles.At weak Zeeman coupling vorticesare still solutions of field equations but they have
higher energy than skyrmions. In this limit skyrmions become the relevant quasiparticles.
2 Magnus force acting on skyrmions and vortices
We apply the adiabatic approximation to investigate the dynamics of vortices and skyrmions. The only terms in the La-
grangian (2), which can contribute to the terms in the effective mechanical Lagrangianwhich are linear in velocity, are
1
L(1) = d2x[iφ†∂ φ εkla ∂ a ] . (8)
eff Z t − 4Θ k t l
The prescription for the adiabatic approximationis as follows. Take the static vortex or skyrmion solution described by the
fields φ(~x ξ~),a (~x ξ~) and located at an arbitraryposition ξ~. Promote the parameter ξ~to the role of a time-dependent
µ
{ − − }
collective coordinate ξ~(t). In this way the static fields become time-dependent. The final step is to substitute such a time-
dependent field configurations to the Lagrangian and integrate out their spatial dependence. After the spatial integration
one should be left with a purely mechanical Lagrangianbeing a functional of the trajectory ξ~(t).
The first term in (8) does not contribute to the effective Lagrangian.The static field is in the Coulomb gauge,∂ a =0,
k k
so that the gauge field can be expressed as a = ε ∂ U, where U is an auxillary potential. In the adiabatic approximation
k kl l
we replace U(~x) U[~x ξ~(t)]. The second term in (8) becomes
→ −
1 1
d2x[ εklε ∂ U∂ (ε ∂ U)]= d2x[ ε ∂ U∂ ∂ U] . (9)
Z −4Θ km m t ln n Z −4Θ mn m t n
It can be shown, by an integration by parts, that this term does not contribute to the effective Lagrangian. Thus the only
contribution to the part of the effective mechanical Lagrangianwhich is linear in velocity comes from
L(1) = d2xiφ†∂ φ . (10)
eff Z t
When we introduce moduli and phases of the scalar fields, φA = √ρAeiχA with A = , , the effective Lagrangian will split
↑ ↓
into contributions from spin-up and spin-down components
L(1) = d2x[ρ ∂ χ +ρ ∂ χ ] . (11)
eff −Z ↑ t ↑ ↓ t ↓
Letus applythe procedureto asinglevortexorskyrmion.Thestaticconfigurationis describedby(5).Inthe vortexcase
we can put formally g(r) = 0. Let us take the trajectory ξ~(t) which passes through the origin at t = 0. For a very small ξ~
the fields in (11) can be expanded as
dρ
ρ (~x ξ~)=ρ (r) (ξ1cosθ+ξ2sinθ) A(r)+O( ξ 2) ,
A A
− − dr | |
sinθ cosθ
∂ χ (~x ξ~)= n ( ξ˙1 +ξ˙2 )+O( ξ ) . (12)
t A A
− − − r r | |
With this expansion the effective Lagrangianbecomes
L(1) = πε ξkξ˙l n [ρ ( ) ρ (0)]+O( ξ 2) . (13)
eff − kl A A ∞ − A | |
X
A
Thesystemwithjustonequasiparticleistranslationallyinvariant.Thetermproportionaltoε ξkξ˙l isthemostgeneralterm
kl
linear in velocity which is, up to a total time derivative, translationally invariant. This proves that the term O( ξ 2) in
| |
Eq.(13) vanishes identically.
There are two contributions to the effective Lagrangian.The contribution from the spin-up component vanishes because
ρ ( )=n ρ (0)=0. Thus the only contribution comes from the lower component (ρ (0)=0, ρ ( )=ρ ) and amounts
↑ ↑ ↑ ↓ ↓ 0
∞ ∞
to
eB
L(1) =πρ ε ξkξ˙l ε ξkξ˙l. (14)
eff 0 kl ≡ 2(2n+1) kl
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The Magnusforce is exactly the same for skyrmionsandfor vortices.It equalsthe Lorenzforce acting ona particle with the
charge e . As discussed in [5] this is not a mere coincidence. A quasiparticle is not an independent object but a defect
2n+1
composed out ofelectrons. As the locationof defect movesit act with a Magnusforce onthe surroundingelectrons.They in
turn respond with a currentwhich interacts through the Lorentz force with the external magnetic field. Thus the two forces
combine to be just one.
If the inertial mass were zero then the Magnus force would prevent a quasiparticle from moving with respect to the
condesate. Note that in our derivation we have been slowly varying the location of a quasiparticle without moving the
condesate so that the ξ’s in Eq.(14) are quasiparticle coordinates with respect to the condesate’s frame. The Lorentz force
acts on the condensate as a whole. Namely, if the fields φ(t,~x),a (t,~x) are solutions of field equations of the model (2),
µ
{ }
then there are also the following boosted solutions
φ˜(t,~x)=φ[t,~x R~(t)]eiχB ,
−
a˜ (t,~x)=a [t,~x R~(t)] R˙k(t)a [t,~x R~(t)] ,
0 0 k
− − −
a˜ (t,~x)=a [t,~x R~(t)] ,
k k
−
1 t1
χ =mR˙kxk m[R˙kR˙k]t e dτ R˙k(τ)A [~x R~(τ)] ,
B − 2 − Z k −
t0
R~(t )=0 ,
0
eB
R¨k = εklR˙l ,
− m
B
A (~x)= ε xl . (15)
k kl
−2
Any solution can be boosted to move along an electronic cyclotron orbit. Even the uniform condesate feels the Lorentz
force in spite of the fact that the external magnetic field seems to be screened by the statistical gauge field. If there is a
quasiparticle in the original solution then after the boost it follows the cyclotron orbit but it does not move with respect to
the condensate. The Magnus force remains zero.
3 Statistical interactions of quasiparticles
WehaveshownthattheMagnusforceisthesameforbothskyrmionsandvortices.Nowweproceedtotheirmutualstatistical
interactions. We will show that the statistical interaction of skyrmions depends on the total number of inversed spins they
carry.
Once again we will apply the formula (11) but this time to the system of two distant antiskyrmions. The total phase of
the lower component reads
χ (~x)= Arg(~x ξ~ ) Arg(~x ξ~ ) (16)
↓ 1 2
− − − −
whiletheuppercomponentphaseisconstant.Thus,onceagain,onlythe componentcontributestotheeffectiveLagrangian.
↓
The formula (11) simplifies to
L(1) = d2x[δρ ∂ χ ] , (17)
eff −Z ↓ t ↓
where δρ = ρ ρ is a deviation of the down component’s density from the uniform background. When the distance
↓ ↓ 0
−
between the skyrmions is large as compared to their widths the deviation can be approximated by the sum
δρ (t,~x) δρ [~x ξ~ (t)]+δρ [~x ξ~ (t)] (18)
↓ ↓ 1 ↓ 2
≈ − −
of two nonoverlapping rotationally-symmetric deviations. When the formulas (16) and (18) are substituted to (17), there
appear four terms to be integrated out. Two of them are ”self-interaction” terms which lead to Magnus forces as discussed
in the previous section but in addition there are two mutual-interaction terms
L(1) = d2x δρ [~x ξ~ (t)]∂ Arg[~x ξ~ (t)]+δρ [~x ξ~ (t)]∂ Arg[~x ξ~ (t)] . (19)
eff Z { ↓ − 1 t − 2 ↓ − 2 t − 1 }
For very distant skyrmions the density deviations can be approximated by δρ [~x ξ~ (t)] Q↓δ(2)[~x ξ~ (t)], where Q is
↓ − 1 ≈− e − 1 ↓
the charge deficit of the spin-down component, compare with (7). In this approximationthe integral (19) becomes
Q Q d
L(1) = ↓ d2x δ(2)[~x ξ~ (t)]∂ Arg[~x ξ~ (t)]+δ(2)[~x ξ~ (t)]∂ Arg[~x ξ~ (t)] ↓ Arg[ξ (t) ξ (t)] , (20)
eff − e Z { − 1 t − 2 − 2 t − 1 }≡− e dt 1 − 2
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whichisjustthestatisticalinteractionwewerelookingfor.TheprefactorQ /eisthetotalnumberofelectronsmissinginthe
↓
lower component as compared to the uniform ground state. As the number of electrons in the upper component is Q /e,
↑
the total spin of the skyrmionas comparedto the uniform backgroundis S = 1(Q Q )/e. Q = Q e , so t−hat the
2 ↑− ↓ ↑ − ↓− 2n+1
statistical interaction becomes
1 d
L(1) = [S+ ] Arg[ξ (t) ξ (t)] , (21)
eff − 2(2n+1) dt 1 − 2
where S is the total spin of the skyrmion with respect to the uniform polarized background. In the case of the vortex
S =S =1/2(2n+1).
vortex
4 Dual formulation of the model
To get further insight into statistical interactions of vortices we will perform Hubbard-Stratanovich transformation on the
model (2). The phase gradients squared in the Lagrangian(2) can be rewritten as
ρ23 expi d2x[ ρA∂ χ ∂ χ ]= [DIA]expi d2x[IkAIkA IkA∂kχA] , (22)
A Z −2m k A k A Z k Z 2mρ − m
A
wheretheauxillaryfieldsIA havebeenintroduced.Inthenextstepthephasescanbesplitintotwopartsχ =χ0 +η .χ0’s
k A A A
aremultivaluedphasesduetovortices/skyrmionswhileη’saresinglevaluedcomponents.AtthepresentstagetheLagrangian
is linearized in phase gradients. Functional integration over η’s leads to the conservation laws
IA+ρ (aA+eA )
ρ˙ +∂ [ k A k k ]=0 . (23)
A k
m
These conservation laws will be identically satisfied when we introduce a pair of dual gauge fields
ρ , 1 [IA+ρ (aA+eA )] =εµνσ∂ BA d=ef 1εµνσHA . (24)
{ A m k A k k } ν σ 2 νσ
Thisdefinitiontogetherwiththedefinitionofthetopologicalcurrents,2πKµ =εµνσ∂ ∂ χ ,leads,aftersomerearrangement
A ν σ A
and integration by parts, to the following dual Lagrangian
1
L =L + εµνσa ∂ a [εµνσ(a +eA )∂ BA+2πBAKµ] . (25)
D B 4Θ µ ν σ− µ µ ν σ µ A
X
A
L is the Lagrangianof the dual gauge field
B
mHAHA ∂ HA∂ HA λ
L = ok ok k 12 k 12 [( HA) ρ ]2 2γH↑ (26)
B { 2HA − 8mHA }− 2 12 − 0 − 12
XA 12 12 XA
Let us concentrate now on the minimal coupling between the topological current and the dual gauge field. The topological
current due to the antivortex/antiskyrmionmoving along the trajectory ξ~(t) is
K0 = δ δ2[~x ξ~(t)] , Kk = δ ξ˙k(t)δ2[~x ξ~(t)] . (27)
A − ↓A − A − ↓A −
Only the topological current of the component is nonzero. It is like a current of a negatively chargedpoint particle. What
↓
is the dual gauge potential which couples to the topological current? We know that the relation between the dual magnetic
field and the original density is ρ = HA. Far from any topological defects the dual magnetic field is just HA = δ ρ
A − 21 21 − ↓A 0
and is directed down the z-axis. A single vortex/skyrmion moves in a uniform dual magnetic field. This is the origin of the
Magnus force. Thus Magnus force is in fact a dual Lorentz force. Topological defects distort the uniform background. Their
contribution to the dual gauge potential is
Q xl ξl
B↓(~x)= ↓ ε − v , (28)
k 2πeXv kl|~x−ξ~v |2
where v runs over topological defects. With this form of the gauge potential we are able to work out the mutual statistical
interaction
Q xl ξl Q d
2π d2xKµBA = ↓ d2xε − v ξ˙kδ(2)(~x ξ~ )= ↓ Arg[ξ~ ξ~ ] (29)
− Z A µ e vX6=wZ kl|~x−ξ~v |2 w − w − e vX<w dt v − w
which is the same as Eq.(21).
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Even if the inertial mass of quasiparticles is zero there still remains some possibility of motion provided that there are
long range potential interactions between them. For nonzero Zeeman coupling the quasiparticles are exponentially localized
field configurations. There is thus no long-range potential interaction through the matter fields. However there is still the
long range Coulomb interaction (3) which gives rise to the following effective Lagrangianfor diluted quasiholes
e2 1 1 d
L(1) =πρ ε ξkξ˙l [S+ ] Arg[ξ~ ξ~ ] (30)
eff 0Xv kl v v− ε(2n+1)2 vX<w |ξ~v−ξ~w | −vX<w 2(2n+1) dt v − w
The last term is a total time derivative and can be neglected in classical considerations. According to the classical part of
this Lagrangianthe particles movealong the trajectoriesof constanttotalpotential energy.For example a pair of quasiholes
performs a strictly circular clockwise motion.
5 Remarks
We havederivedstatisticalinteractionofskyrmionsintheframeworkofthe time-dependentGinzburg-Landaumodel.These
interactions can be much stronger than interactions of fully polarized vortices. This conclusion could not be obtained in
the effective Heisenberg model. This example shows that the Heisenberg model is not relevant not only to description of
quasiparticles’ energies but it also fails to give a correct answer to more basic topological questions.
We did not make any quantitative predictions as to what is the skyrmion’s spin S which determines the strenght of the
statistical interaction. Such calculations are possible in the Ginzburg-Landau model but, there exist already microscopic
Hartree-Fock techniques to establish the value of S, see [2].
Fromourcalculationsitturnsoutthatthatthegeometricalphasepickedupbyelectronicwave-functionduringaclockwise
exchange of two skyrmions is
1 1
Γ=π[S+ ] π[N + ] , (31)
↑
2(2n+1) ≡ (2n+1)
whereN isthenumberofelectronswithreversedspinstrappedinsidetheskyrmioncore.Thisphasedeterminesthequantum
↑
statistics ofskyrmions.There aretwo types ofanyons.For N eventhe phaseis π upto anevenmultipilicity ofπ. This
↑ (2n+1)
is the same kind of statistics as for fully polarized Laughlin quasiparticles (or quasiholes). Condensation of such skyrmions
leads to hierarchy states with odd denominator filling fractions. For skyrmions with N odd the exchange phase (31) is
↑
π2n+2 up to unrelevant even multiplicities of π. This type of anyons differs from the Lauglin quasiparticles by an addition
2n+1
of one flux quantum. Their condensation gives rise to even-denominator filling fractions. To summarize the condesation of
skyrmions of the primary state with ν = 1 gives rise to the following filling fractions labelled by p
2n+1
1
ν = , p=1,2,3.... , (32)
(2n+1) α
− p
where α = +1 or 1 dependent on whether skyrmions or antiskyrmions condense. Note that in the above formula p is an
−
arbitrary integer so that the filling fractions can have both even and odd denominators.
Aknowledgement. I would like to thank Wojtek Zakrzewski and Jens Gladikowski for useful discussions and inspiring
atmosphere. This research was supported in part by the KBN grant 2P03B08508 and in part by Foundation for Polish
Science.
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