Composite fermion basis for two-component Bose gases

6 MB·

by  Ola Liabøtrø

Composite fermion basis for two-component Bose gases
O. Liabøtrø and M. L. Meyer
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
(Dated: January 17, 2017)
Despite its success, the composite fermion (CF) construction possesses some mathematical fea-
turesthathave,untilrecently,notbeenfullyunderstood. Inparticular,itisknowntoproducewave
functions that are not necessarily orthogonal, or even linearly independent, after projection to the
lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime,
we have previously shown that it presents a technical challenge for rotating Bose gases with low
7 angular momentum. These are systems where the CF approach yields surprisingly good approx-
1 imations to the exact eigenstates of weak short-range interactions, and so solving the problem of
0 linearly dependent wave functions is of interest. It can also be useful for studying higher bands of
2 fermionicquantumHallstates. Herewepresentseveralwaysofconstructingabasisforthespaceof
so-called“simple”CFstatesfortwo-componentrotatingBosegasesinthelowestLandaulevel,and
n
provethattheyallgivesetsoflinearlyindependentwavefunctionsthatspanthespace. Usingthis
a
J basis,westudythestructureofthelowest-lyingstateusingso-calledrestrictedwavefunctions. We
also examine the scaling of the overlap between the exact and CF wave functions at the maximal
6
possible angular momentum for simple states.
1

]
s I. INTRODUCTION modelsandexperiments,calledpseudospin[18,19]. Typ-
a ically this means multicomponent mixtures where the
g
different components are different internal states of the
- Almost 20 years ago, the connection between the
t atoms. Varying inter- and intraspecies interactions in-
n physics of charged particles moving in two dimensions in
dependently allows for novel behaviours [20]. This pseu-
a a strong magnetic field, and dilute cold atoms rotating
u dospin degree of freedom has been incorporated in the
rapidly in a harmonic trap, was noticed [1]. Since then,
q composite fermion scheme used for cold atom systems,
a large body of theoretical and experimental work has
. and has been used in the quantum Hall regime of high
t accumulated that explore the various aspects of rapidly
a angular momenta [21, 22]. On the other hand, near the
m rotating atomic gases and the associated quantum phe-
lowerendoftheangularmomentumscale, wheretheCF
nomena; for reviews, see e.g. [2–4]. In particular, one
- description is not a priori expected to work, it has been
d expectsstronglycorrelatedphasessimilartothosefound
shown that it actually works surprisingly well, both in
n in the quantum Hall effect when the atoms experience
o strongsyntheticmagneticfields. Theeffectofasynthetic scalar and two-component cases [23, 24].
c magnetic field can be generated by simply rotating the A property of the CF method of constructing wave
[
cloud, or by more advanced techniques [5–9]. functions is that one typically needs to do a projection

1 For electron systems, one prominent way of theoreti- into the lowest Landau level (LLL) in order to either
v callystudyingthequantumHalleffectinvolvesconstruct- compare different CF wave functions, or to compare a
8 ing (classes of) explicit trial wave functions that approx- CF to an eigenstate found by numerical diagonalization
7
imate the true low-energy eigenstates of the interacting oftheinteractingHamiltonian[25]. Thisprojectionleads
2
4 system. At least in the case of Coulomb interaction, the tonon-orthogonal,andoftenalsolinearlydependent,CF
0 true many-body eigenstates are extremely complicated. wave functions. This has been known in the context of
. Still, many successful trial wave functions exist, most fa- electrons in the quantum Hall regime [26–28], but the
1
0 mously the Laughlin wave functions [10], the family of issueismuchmoreprominentforbosonswithlowangular
7 composite fermion (CF) states [11], and trial wave func- momentum: we have previously observed [29] one or two
1 tions addressing non-Abelian quantum Hall states [12– orders of magnitude difference between the number of
: 14]. The success of these trial wave functions in explain- seeminglydistinctCFcandidatesandtheactualnumber
v
i ing various phenomena is linked to the way they capture of linearly independent wave functions. In some extreme
X the important topological properties of the phases they cases the number of seemingly distinct CF candidates
r describe. one can write down is even larger than the dimension of
a
therelevantsectorofHilbertspace,meaningtheycannot
Manyofthemethodsmentionedabovehavebeenmod-
possibly be independent. Until very recently, little was
ified to be applicable to weakly interacting cold atom
understood about the mechanisms responsible for these
systems [15, 16]. The hope is to be able to experimen-
lineardependenciesafterprojection. Inapreviouspaper
tally study strongly correlated states in a cold atom set-
[29],wediscussedthreetypesofrelationsbetweencertain
ting, where parameters like density, disorder and scat-
types of low-lying CF candidates, but were only able to
tering lengths may be tuned much more finely than in
giveexamplesdemonstratinghowtheserelationsseemto
the semiconductor systems traditionally used in quan-
explain all the linear dependencies.
tum Hall experiments [17]. More recently, an additional
degree of freedom is often taken into consideration in In this paper, we present sets of CF candidates for
2
two-component systems that we rigorously prove are ba- mayworkwithmany-bodywavefunctionsthatareeigen-
sissetsforthesubspacethatminimizestheCFcyclotron states of L. These are homogeneous polynomials of de-
energy for low angular momenta. We use these states greeL, symmetricinthecoordinatesofeachspeciessep-
to study the real-space distribution of particles and vor- arately. As is common [2, 24] we will focus on transla-
ticesofthelowest-lyingwavefunctionswhenwevarythe tionallyinvariantstates,i.e. polynomialsinvariantunder
angular momentum, and also give additional attention a constant shift η0 of all coordinates,
to certain special cases, comparing to known behaviour
from scalar gases. Ψ({ηi+η0})=Ψ({ηi}). (4)
As mentioned in the introduction, one may adapt the
CF approach to produce wave functions for 2D bosons,
II. TWO-COMPONENT ROTATING BOSE
bothinthescalarandmulti-componentcases. ACFtrial
GASES
wavefunctionforthebosonictwo-speciessystemisofthe
form [25]
We first summarize the model for two-species Bose
gases in the lowest Landau level, including their descrip- Ψ =P (Φ Φ J(z,w)q) (5)
CF LLL Z W
tion in terms of composite fermions. A more detailed
introduction can be found in [24]. The two species of where q is an odd number, q = 1,3,5,.... ΦZ, ΦW are
bosons experience a two-dimensional harmonic trap po- Slater determinants for each species of CFs, treated as
tential of strength ω, and are rotating about the mini- non-interacting to lowest order. They consist of CF or-
mum of the potential at frequency Ω. The Hamiltonian bitals
is: (cid:18) (cid:19)
ηη¯
ψ (η)=N ηmLm , m≥−n, (6)
NX+M(cid:18)p2 1 (cid:19) n,m n,m n 2
H = i + mω2r2−Ωl
2m 2 i i where Lm is the associated Laguerre polynomial, and
i=1 n
(1) N is a normalization factor. The interpretation of
N+M N+M n,m
X X theseorbitalsisthattheyarecompositefermionsoccupy-
+ 2πg δ(r −r ).
i,j i j ingLandau-likelevels,labeledbyn(oftencalledΛ-levels)
i=1 j=i+1
inareducedeffectivemagneticfield. J isaJastrowfactor
involving both species,
Here N denotes the number of particles of the minority
species, and M ≥ N denotes the number of particles of Y
J(z,w)= (η −η )
the majority species, all with the same mass m. l is i j
i
theangularmomentumofparticlei. Thestrengthofthe i<j (7)
Y Y Y
contact interaction gi,j depends only on the species of = (zi−zj) (wk−wl) (zi−wk).
particles i,j. In the species-independent case g = g =
i,j i<j k<l i,k
constant, the system posesses a pseudospin-1/2 symme-
try, which we will assume here. For sufficiently dilute J(z,w)q has q units of angular momentum per pair of
gases, i.e. in the weak interaction limit, this reduces to particles, in total LJ = q(N +M)(N +M −1)/2. We
the well known lowest Landau level problem [2, 3] in the will be considering low total angular momentum, hence
effective magnetic field 2mω, we will choose the smallest possibility q = 1. PLLL de-
notes projection to the lowest Landau level. We use the
H =X(ω−Ω)l +2πgXδ(η −η ). (2) projection of Girvin and Jach [30] (called Method I in
i i j
[25]), which amounts to first moving the conjugate vari-
i i<j
ables η to the left of the η , and then replacing η by
i i i
In the ideal limit (ω−Ω)→0 the Landau levels become ∂ηi.
flat, meaning that the many-body eigenstates are solely Inthispaper,wefocusonlowangularmomenta,specif-
determinedbytheinteraction. Hereη =x +iy arethe icallyL≤MN. Weconsiderthesetoftranslationallyin-
j j j
dimensionless complex positions of the particles in units variantCFwavefunctionsthatminimizethetotalCFcy-
of the “magnetic” length p(cid:126)/(2mω). We name the co- clotronenergyEc ∝Pini inthisLrange. Thesumruns
ordinates of the two components z =η , 1≤i≤N and overtheCForbitalsoccupiedinapairofSlaterdetermi-
i i
wi = ηN+i, 1 ≤ i ≤ M. Working in symmetric gauge, nants. SinceLP≤MN <1/2(N+M)(N+M−1)=LJ,
the single particle eigenstates in the lowest Landau level we see that imi must be negative. We have previ-
with angular momentum l are ously shown that this set is spanned by CF candidates
whereonlytheorbitalsψ areoccupied,andthatlin-
n,−n
ψ (η)=N ηlexp(−ηη¯/4) l≥0 (3) earcombinationsofcandidatesinthissetgivegoodover-
0,l l
laps with the very lowest-lying states in the exact yrast
TheGaussianfactorsareubiquitous,sowesuppressthem spectrum [24]. After projection to the LLL, the orbitals
for simplicity from now on. Since the Hamiltonian com- ψ (η) become ∂n. This simple form of the Slater de-
n,−n η
P
mutes with the total angular momentum L = l , we terminants has led these CF wave functions to be called
i i
3
simple states. Since the differential operators commute,
we may perform the projection to the LLL on the Slater
determinants individually, as long as we keep them to
the left of the Jastrow factor. This is understood in the
following sections.
III. COMPOSITE FERMION BASIS
Wenowpresentsomesetsofstatesthatweprovetobe
basesforthespaceofsimplestateswithN+M particles
and angular momentum L. First, we need some defini-
tions. The simple states are on the form (5), where the
Slater determinants after projection are
N
X Y
Φ (a)= (−1)|ρ|∂ai (8)
Z zρi
ρ∈SNi=1
and
FIG.1: Youngdiagramscorrespondingtopartitionsof6into
M a 5×3 box.
X Y
Φ (b)= (−1)|ρ|∂bi (9)
W wρi
ρ∈SMi=1
We have that
with a ,b <N +M −1. In the following, we will focus
i i N N
oWn,wΦ,ZM, b↔utZw,ezw,NillinseavlewrauysseptohsastibNle.≤ M, so replacing ∆Z(p)ΦZ(a)= X (−1)|ρ|YY∂zpσii∂zaρii
σ,ρ∈SN i=1j=1
We define
N N
XN = X (−1)|ρ|YY∂zpρσii∂zaρii (14)
P ={p∈ZN | M ≥p ≥...≥p ≥0, p =L}. σ,ρ∈SN i=1j=1
N,M,L 1 N i
N
i=1 X X
(10) = ΦZ(a+ pσkek)
This is the setof partitionsof L into an M×N box. We σ∈SN k=1
orderthepartitionslexicographically. Thepartitionscan
and
be visualized using Young diagrams. Such a diagram is
obtained by coloring L cells of an M×N box compactly N
X
from the lower left. The number of colored cells in the ∆ Φ (a)= Φ (a+ne ). (15)
zn Z Z k
lowest row is p , the next row corresponds to p , and
1 2 k=1
so on. As an example, we show all the Young diagrams
The maximal L = MN state occurs when the Slater
corresponding to the set P in FIG. 1.
3,5,6
determinants contain minimal differentiation operators,
There is a one to one correspondence between P
N,M,Λ i.e.
and {Φ (a)|PN a +1−i=Λ} given by
Z i=1 i
Ψ =Φ (α)Φ (β)J, (16)
(L=MN) Z W
p↔Φ (a(p)), a (p)=i−1+p . (11)
Z i N+1−i
where α ,β =i−1. We will continue to use the vectors
i i
We define the following differentiation operators: α and β as they are useful not only for L = MN but
also for general L. We can now state our main theorem:
N M
∆ ≡X∂n, ∆ ≡X∂n , (12) Theorem 1. The sets
zn zi wn wi
i=1 i=1 B ={Φ (a(p))Φ (β)J | p∈P }
Z,L Z W N,M,MN−L
D ={∆ (p)Φ (α)Φ (β )J | p∈P }
and Z,L Z Z W N,M,MN−L
(17)
N are bases for the set of simple CF states with N + M
X Y
∆Z(p∈PN,M,Λ)≡ ∂zpρii. (13) pDarticleasreaanldsoanbagsuelas.r momentum L. The sets BW,L and
ρ∈SNi=1 W,L
4
In fact, B = B for MN −L even. Otherwise, �+�=��
Z,L W,L ��
thesetscontain(−1)timesthevectorsoftheother. This
�=�
can be seen as a consequence of reflection symmetry, as
�� �=�
introduced in [29].
�=�
��
Proof. We first show that D spans B ; we already
Z,L Z,L � �=�
know from Eq. (14) that BZ,L spans DZ,L. We then ��� �� �=�
show that BZ,L spans the set of simple CF states, and ���
finally that B is a linearly independent set. This �� �=�
Z,L ��
must also hold for D since |D |=|B |. We refer
Z,L Z,L Z,L
to lemmas that we prove in the appendix.
��
Lemma 1 states that D spans B :
Z,L Z,L �
� � �� �� �� �� �� ��
X
ΦZ(a)= cp∆Z(p)ΦZ(α) (18) ����������������
p∈PN,M,Λ
FIG.2: ThesizeofthesimpleCFbasisasafunctionofLfor
for coefficients c ∈ Q|PN,M,Λ|, where Λ = MN −L. The a total of N +M =12 particles.
lemma also shows that we can write a general simple CF
state as
X
Ψ=φ (a)φ (b)J = c ∆ (p)φ (α)φ (b)J.
Z W p Z Z W
p∈PN,M,Λ
(19)
Lemma 2 states that we can write
N
X Y
∆Z(p)= dpp˜ ∆zp˜i, (20) (a) (b)
p˜∈PN,Λ,Λ i=1
with coefficients d ∈Q. We can therefore write
pp˜
N
X X Y
Ψ= c d ∆ φ (α)φ (b)J.
p pp˜ zp˜i Z W
p∈PN,M,Λp˜∈PN,Λ,Λ i=1
(21)
Next, we can apply generalized translation invariance
(Lemma 3),
(∆ +∆ )Ψ=0, (22)
zn wn
(c) (d)
to replace the ∆ operators with −∆ , giving
zp˜i wp˜i
FIG. 3: Young diagrams illustrating 1-1 correspondences.
Theleft-rightcorrespondenceisbetweenpartitionsofP
N,M,L
N and P , explaining the symmetry of the number of
X X Y N,M,NM−L
Ψ= s(p˜)c d ∆ φ (α)φ (b)J, statesdisplayedinFIG.2. Thetop-bottomcorrespondenceis
p pp˜ wp˜i Z W
between P and P giving bijections B ↔ B
p∈PN,M,Λp˜∈PN,Λ,Λ i=1 N,M,L M,N,L Z,L W,L
(23) and DZ,L ↔DW,L.
where s(p˜) is (−1) if there is an odd number of p˜ that
i
are non-zero, and 1 otherwise.
We can see from eqs. (14, 15) that this gives terms
that are all on the form Φ (α)Φ (b0)J for varying b0. states, we can always fix ΦW (ΦZ) to ΦW(β) (ΦZ(α))
Z Z
and then vary the Z (W) determinant. The number of
Wehavenowexpressedageneralsimplestateasalinear
ways to do that, i.e. the size of the basis, is simply
combination of elements of B , and the same can be
W,L
|P |. WeplotthisasfunctionofLforN+M =
doneforB . AllthatremainsistoshowthatB isa N,M,MN−L
Z,L Z,L
12 in FIG. 2. The symmetry of each curve about its
linearlyindependentset, andthisresultisLemma4.
midpointL=MN/2thatwasoriginallyobservedin[29]
To summarize, we have now shown that either of the is now easily understood because the number of ways to
setsB ,B ,D ,D (Eq. (17))arebasissetsfor color L cells in a Young diagram leaves MN −L cells
Z,L W,L Z,L W,L
the simple CF wave functions at angular momentum L. uncolored,meaningthattheLdiagramsare1-1withthe
In other words, when constructing basis sets for simple MN −L diagrams. We have illustrated this in FIG. 3.
5
IV. STRUCTURE OF THE LOWEST-LYING
WAVE FUNCTIONS
Using the simple CF basis sets presented in the pre-
vious section, we can now diagonalize the interaction
Hamiltonian in the simple CF subspace, to produce ap-
proximations to the lowest-lying wave functions of the
two-component rotating gas. Because the size of the
simple CF basis is so much smaller than the size of the
Hilbert space, this is often a huge computational simpli-
fication. We now take advantage of this to study some
aspects of these low-lying states.
In order to increase our physical understanding of the
structure of these states, one option is to study density
and pair correlation functions. Another approach that
is particularly suitable to visualizing vortex structure is
tocomputetheso-calledrestrictedwavefunction(RWF)
ψ (r)[31, 32]. To find the RWF of a given many-body
r
wave function Ψ, one first calculates a set of particle co-
ordinates {r∗}N+M that maximises |Ψ|2. The restricted
i i=1
wave function is defined as
Ψ(r,r∗,r∗,...)
ψ (r)= 2 3 (24)
r Ψ(r∗,r∗,r∗,...)
1 2 3
FIG. 4: Plot of the restricted wave function for the single
We see that this function varies as one of the particles
vortex state of 8 particles.
is allowed to move from its maximizing position. The
amplitudeofψ givestherelativeamplitudeofthemany-
r
body wave function compared to the maximum, and the
sees a vortex coinciding with the lump of majority parti-
argument gives the change in phase. The vortices can
cles already at L=1, a so-called coreless vortex [20, 34].
be identified from plots of ψ where the nodes of the
r
This is consistent with findings using full diagonaliza-
amplitude meet lines where the phase jumps.
tion [35]. The multiplicity of this vortex increases with
An example of an RWF plot is shown in FIG. 4. The
L, and the vortex and majority particles move together,
wavefunctionΨinthisexampleistheexactgroundstate
away from the lone minority particle. This displays an
for a single-species gas with 8 particles at L = M = 8.
exampleofhowthe“perspectives”ofthetwospeciescan
For a single component, the cases L = M are known as
be very different for a given state Ψ. A similar pattern
singlevortexstates[23,33]. Thetriangularplotmarkers
show the optimal positions {r∗}. The number on each is also observed for (N,M)=(2,6),(3,5) for L<M.
i
plot marker specifies how many particles share that po- For L≥M, a variety of configurations are realized as
sition. The plot marker with black filling corresponds thosemaximizing|Ψ|2. Thecommonfeaturesarethat,as
to the particle whose position r is varied in the plot. L increases and the particles spread out more, particles
The contour lines show lines of constant amplitude of ofthesamespeciesstilltendtostayclosetoorontopof
ψ and the color shows the phase change, where black some of the other ones, rather than all of them spread-
r
corresponds to −π and white to π. In this case, the con- ing out. The vortices are exclusively located near the
figuration of highest |Ψ|2 is a ring of particles, with a particles of the opposite species: no same-species nodes
vortex clearly visible close to the center of the ring. are observed for the L we consider. Finally, the mean
In our two-component case, we can define an RWF for number of phase jump lines per particle of the opposite
each species, ψ and ψ for minority and majority speciesincreaseswithL. Amoreorlesstypicalsituation
rZ rW
species respectively. The difference is simply the species is shown in FIG. 6. The majority species has split into
of the particle whose position we vary. We will use tri- three groups, but three of them are still close together.
angles pointing down as plot markers for the minority The minority particles are all in the same position. In
component particles, and triangles pointing up for ma- both (b) and (c), there are two nodes close to the lump
jority particles (like ∇ and ∆, respectively). of three minority particles. Since the only difference be-
In FIG. 5 we see that there is a substantial difference tween(b)and(c)iswhichmajorityparticlecoordinatewe
betweentheRWFplotsψ andψ . Inthetoprowwe choose to vary, this demonstrates that the vortex config-
rZ rW
see a vortex approaching the cloud from the side where uration that is seen is not sensitive to that choice, which
the lone minority particle is located, starting far outside is what we would expect of a physical vortex state.
thecloudandcomingcloserandclosertothecenterasL Finally we discuss the maximally rotating simple
increases. On the other hand, the lone minority particle states, namely the states in Eq. (16). For a given N,M
6
(a) (b) (c)
(d) (e) (f)
FIG. 5: Restricted wave functions for very low angular momenta for 1 minority and 7 majority particles. (a) - (c) show ψ ,
rW
(d) - (f) show ψ .
rZ
this is the only possible simple state at L = MN, i.e. seem to indicate that the vortex of winding N appears
no CF diagonalization is necessary. The restricted wave at L = MN, but that the vortex is a coreless vortex
functions of some L = MN states are shown in FIG. 7. in the majority component. On the other hand, (d) -
Again we see a very clear distinction between minority (f) show how ψ evolves from a more or less Gaussian
rZ
and majority species particles. The majority particles distribution in (d) to a situation where one node radi-
are positioned on the vertices of a regular M-gon, with ally approaches each majority particle from outside the
alltheminorityparticlesinthecenter. In(a)-(c)wesee cloud.
what looks like a single, double and triple vortex struc-
ture, respectively, but filled by the minority component.
The qualitative behaviour of the amplitude contours is V. OVERLAPS
largely the same in the three plots. In particular, 7(a)
looks remarkably similar to FIG. 4 except for the mi- When working with trial wave functions like CF wave
nority particle in the middle. We will come back to this functions, especially in a context for which CF was not
point in SEC. V. originallyintended,liketheslowlyrotatingBosesystems
Itcanbenotedthat,whilethesinglevortexforascalar we are discussing, one should carefully compare the re-
gas appears at L = N, the general multiply quantized sults obtained with ones obtained by other means. In
vortex of winding k appears at L /N < k; for 20 par- the lowest Landau level, we are fortunate because the
k
ticles, the double and triple vortices appear at L/N 1.8 Hilbert space of each L sector is finite. Given enough
and 2.85 respectively [33]. The results presented here computer resources, one can therefore in principle com-
7
(a) (b) (c)
FIG. 6: Three visualizations of the RWF for (N,M,L)=(3,5,9). (a) shows ψ , while (b) and (c) show ψ for two choices
rZ rW
of the particle whose position we vary.
(a) (b) (c)
(d) (e) (f)
FIG. 7: Restricted wave functions for L=MN. (a) - (c) show ψ , (d) - (f) show ψ .
rW rZ
8
pute the exact spectrum (at least to machine precision) VI. CONCLUSIONS AND OUTLOOK
for a given L, and compare the CF results to this. In
practice, desktop computers can handle two-component
The main results of this paper are the identification of
systems with up to a total of around 15-20 particles for
basis sets of simple CF states, and the proof that these
the L considered in this paper. This method has been
sets are in fact spanning the simple state subspace and
usedtoverifytheapplicabilityoftheCFconstructionto
are linearly independent. We have used these basis sets
scalar [15, 23] and two-component systems both below
to revisit the spatial structure of particles and vortices
and in the quantum Hall regime [21, 22, 24].
for the rotating two-component Bose gas in the LLL at
Inparticular,[23,36]showedthattheoverlapbetween lowangularmomenta,andhavepaidspecialattentionto
the exact and CF state for the scalar case N = L (the the unique simple CF candidates at angular momentum
single vortex) increases to unity in the N → ∞ limit. L=MN. We find that for N =1 minority particle, the
As mentioned in the previous section, this is the state system mimics the CF candidate for the single vortex
plotted in FIG. 4 for M =8. It strikingly resembles the state of the scalar rotating gas. This includes an overlap
restricted wave function plot of the L state with a with the exact wave function that converges to 1 in the
max
single minority particle, FIG. 7. This resemblance, and N →∞limit. ForN >1,weobserveacorelessvortexof
thefactthattheL stateisuniqueforgivenN,M,led winding N in the majority species at exactly L = MN.
max
ustocomputetheoverlapbetweentheexactlowest-lying From the plots in FIG. 8, however, we see that the exact
state and the CF L state as function of M for given wavefunctionmusthavecontributionsfromCFstatesin
max
N =1,2,3. higher E bands. To answer whether or not this makes
c
In FIG. 8, the squared overlaps |hΨ |Ψ i|2 be- anyqualitativedifferencesfromtheresultspresentedhere
CF exact
tween the maximally rotating simple CF wave functions would require further investigation.
at L=MN and the exact lowest-lying states are plottet As presented in [29], there are still linear dependence
as functions of M for N =1,2,3 in (a) - (c) respectively. relationsbetweenCFcandidatesinhigherEc bandsthat
In (a), we see exactly the same convergence to unity as are not understood. However, now that we have a good
wasreportedin[23,36]foronecomponent. TheCFcan- understanding of the relations for simple states, it might
didate in the one-component case is not simple (simple be possible to make further progress for these so-called
statesdon’texistinthescalarcase),butitisuniqueand “compact states” [25]. They are the relevant candidates
minimizes the CF cyclotron energy. The exact ground forL>MN inthesystemstudiedhere,buttheyarealso
state of the single vortex, on the other hand, is known, relevant for electrons in strong magnetic fields, confined
and its polynomial part is simply proportional to to quantum dots [38].
For other projection methods [39] and/or geometries
[40], we expect some linear dependence relations similar
Ψ =e (z˜) (25)
s.v. M to the ones in this paper. The reason is that, as we have
seen, it is in fact the Jastrow factor that is responsible
P
where z˜i = zi −1/M jzj are the particle coordinates for the equations that relate different Λ-level configura-
relative to the center of mass, and ek is the elementary tion patterns. Certainly some rules will need to be mod-
symmetricpolynomialofdegreek. Infact,forN =1,the ified, but the principle of translation invariance should
exactlowest-lyingL=M stateisknownalsointhetwo- still hold.
component case: it was given in [37] and its polynomial
part is proportional to
Acknowledgement
Ψ =z˜e (w˜)−Me (w˜) (26)
N=1,L=M M−1 M
We would like to thank Susanne Viefers for helpful com-
P
Here η˜ =η −1/(N +M) η are the particle coordi- mentsonthemanuscript. Thisworkwasfinanciallysup-
i i i i
nates relative to the center of mass for all particles. ported by the Research Council of Norway.
In(b)and(c)however,weclearlyseethatthesquared
overlap decreases with system size, as is usually the case
with most trial wave function approaches. It should Appendix A: Mathematical results
be stressed that the dimension of the relevant sector of
Hilbert space grows very rapidly with system size, so
Lemma 1. There exists c∈Q|PN,M,Λ| such that
given that Ψ is a unique state in this space, it
L=MN
is still surprising that the overlap with the true lowest- X
Φ (a)= c ∆ (p)Φ (α),
lying state is as large as it is. We should also remember Z p Z Z (A1)
that we have restricted ourselves to simple states in this p∈PN,M,Λ
analysis: the fact that the overlap decreases with system
size therefore tells us that higher bands of CF cyclotron where α =i−1. Λ=PN a −α .
i i=1 i i
energy E contribute significantly for larger systems.
c
9
1.00
1.000
0.95
rloveap 0.995 overlap 0.95 overlap
rSquaed 0.990 Squared 0.90 Squared 0.90
0.85
0.85
0.985
0 5 10 15 20 0 2 4 6 8 10 12 14 3 4 5 6 7 8
Numberofmajorityparticles Numberofmajorityparticles Numberofmajorityparticles
(a) (b) (c)
FIG. 8: The squared overlap |hΨ |Ψ i|2 between the numerically exact lowest-lying states, and the CF states (16), at
CF exact
L=MN, for (a) N =1, (b) N =2, and (c) N =3, as function of M.
Proof. Weusetheonetoonecorrespondencebetweenthe Eq. (A7) says that the Lemma holds for Φ (a) =
Z
elements of P and Φ (a) given by Φ (a(p ))andeq. (A8)saysthatitholdsforΦ (a)=
N,M,Λ Z Z min Z
Φ (a(p)) if it holds for all p0 <p. It must therefore hold
Z
p↔Φ (a(p)), a (p)=i−1+p . (A2)
Z i N+1−i for all p.
and induce an ordering on the Slater determinants such Lemma 2. For all p ∈ P , there exists d ∈
N,M,Λ p
that Q|PN,Λ,Λ| such that
p<p0 ⇔Φ (a(p))<Φ (a(p0)). (A3) N
Z Z X Y
∆Z(p)= dpp˜ ∆zp˜i, (A9)
We have that
p˜∈PN,Λ,Λ i=1
N where we define ∆ =N.
X X z0
∆ (p)Φ (α)= Φ (α+ p e ). (A4)
Z Z Z σk k Proof. Our proof is based on induction. We first define
σ∈Sn k=1 the subset
The non-zero terms are all Slater determinants. The
P ={p∈P | p =0 ⇔ i>K} (A10)
greatest determinant with respect to the ordering occurs N,M,Λ|K N,M,Λ i
when σ orders the partition non-decreasingly. This par- and note that trivially, for all p ∈ P ⊃ P ,
N,Λ,Λ|1 N,M,Λ|1
ticular determinant is ΦZ(a(p)) and we may therefore there exists dp ∈ Q|PN,Λ,Λ| such that eq. (A9) holds.
write There is only one such p and ∆ (p) = ∆ . Now, we
Z zΛ
assume as induction hypothesis that (A9) holds for all
X p∈PN,Λ,Λ|k when k ≤K.
∆ (p)Φ (α)= n Φ (a(p))
Z Z pp0 Z (A5) Let p∈P and consider the polynomial
N,Λ,Λ|K+1
p0≤p
N
Y
for some integers npp0. It is important that ∆zpi. (A11)
i=1
n 6=0. (A6)
pp
It is a product of N factors. Each factor is a sum of
Itisinfactpositive,sinceeverypermutationσthatgives N differentiation operators with respect to N different
this determinant leaves the elements of α+PN p e variables. The resulting terms with the highest number
k=1 σk k
in increasing order. It follows that for the smallest par- ofnon-zeroexponentsarethosethatdonotmultiplydif-
tition, p =min(P ), we have ferentiation operators for the same variable from several
min N,M,Λ
differentfactors∆ . ThisisexceptforthelastN−K−1
zpi
1 ∆ as ∂0 =∂0 and any of the NN−K−1 combinations
ΦZ(a(pmin))= npminpmin∆Z(p)ΦZ(α) (A7) givzpeis thezsiame.zTj his restricted part of QNi=1∆zpi can be
and in general described by permutations as
Φ (a(p))= 1 ∆ (p)Φ (α)− X npp0Φ (a(p)). NN−K−1 X KY+1∂pi = NN−K−1 ∆ (p),
Z n Z Z n Z ρi (N −K−1)! Z
pp p0<p pp ρ∈SK+1 i=1
(A8) (A12)
10
and this means that that this set is linearly independent and therefore B
Z,L
is as well.
N
(N −K−1)!Y Thew-independenttermsofΨ arisewhenthe∂ op-
∆Z(p)= NN−K−1 ∆zpi +D (A13) eratorsactontheM lowestorderpvariablesintheJawsitrow
i=1
factor. We can therefore write
where D is a polynomial of differentiation operators
where each term has at most K non-zero exponents. By
tfohremin(dAu9c)t.ioTnhhiyspiomtphleiseiss,ththaetstehceanhyapllotbheeswisritisteanlsoontrthuee Ψ¯p =(YM k!) X (−1)NM+|σ|+|τ|YN ∂zaσii(p)YN zτMj−1+j
for all k ≤K+1 and completes the proof. k=0 σ,τ∈SN i=1 j=1
M N
As was introduced in [29], the simple CF states obey =(Yk!) X (−1)NM+|σ|Y∂zaτσii(p)zτMi−1+i.
Lemma 3 (Generalized translation invariance). k=0 σ,τ∈SN i=1
(A18)
We are interested in the particular symmetrized term
(∆ +∆ )Ψ=0 (A14)
zn wn
that occurs when σ is the identity operator. We name
for all integers n>0. this term tp, and it can be written as
Ordinarytranslationinvarianceiscapturedinthecase N
X Y
n=1. t =K xM−pi, (A19)
p p τi
Proof. The operator (∆ + ∆ ) commutes with the τ∈SNi=1
zn wn
Slater determinants, so it is enough to show that (∆ +
zn where K is an integer that results from differentiation
∆ )J = 0. This result is independent of the splitting p
wn andpossiblyapermutationsign. Thetermhastheprop-
of particles into Z and W type. We can therefore use
erty that the smallest exponent is as great as possible
variables {ηi}Ni=+1M. We have that among terms in Ψ¯ . Given that, the second smallest ex-
p
ponent is as great as possible and so on. We use this to
N N+M
∆ J =X∂n X (−1)|ρ| Y ηj−1 define an ordering on the tp terms, saying that
ηn ηi ρj
i=1 ρ∈SN+M j=1 t <t (A20)
p p0
N N+M
=X X (−1)|ρ| (ρi−1)! Y ηρj−1−nδj,i.
(ρ −n−1)! j iff the k’th least exponent of t is greater than the k’th
i p
i=1ρ| ρi>n j=1 least exponent of t and their k−1 least exponents are
(A15) p0
pairwise equal. This is equivalent to p < p0, and if we
Now, for each pair (i,ρ), there is a unique pair (i0,ρ0)
pickanothernon-zerotermt0 ofΨ¯ bychoosingaσ that
such that ρ −n = ρ and ρ0 −n = ρ0 and ρ = ρ0 for p p
i i0 i0 i j j is not the identity, then we have t < t0. We index the
all j 6= i,i0. Since ρ and ρ0 only differ by a permutation p p
oftwoelements,(−1)|ρ|+|ρ0| =−1andthecorresponding partitions such that
terms in eq. (A15) cancel out. Since this happens for all
p <p <...<p , (A21)
(i,ρ), it follows that ∆ηnJ =0. 1 2 PN,M,NM−L
Lemma 4 (Linear independence). Now, if i < j, then t < t < t0 . This means that Ψ¯
pi pj pj i
contains a polynomial term that is not contained in any
BZ,L ={ΦZ(a(p))ΦW(β)J | p∈PN,M,MN−L} (A16) Ψ¯ for all j > i. This means that if there exists a linear
j
dependence relation
is a linearly independent set.
Proof. If c Ψ¯ +...+c Ψ¯ =0, (A22)
1 p1 |PN,M,NM−L| p|PN,M,NM−L|
Ψ =Φ (a(p))Φ (β)J ∈B , (A17)
p Z W Z,L then the leftmost non-zero coefficient of this relation
mustbe0,andtherelationmustthereforebetrivial.
then we denote the projection onto w = 0 ∀w by Ψ¯ ,
i i p
and the set of projected states by B¯ . We will show
Z,L
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