Universality of Many-Body States in Rotating Bose and Fermi Systems

0.49 MB·English

by  M. Borgh

Universality of Many-Body States in Rotating Bose and Fermi Systems
M. Borgh1, M. Koskinen2, J. Christensson1, M. Manninen2, and S. M. Reimann1
1Mathematical Physics, LTH, Lund University, 22100 Lund, Sweden
2NanoScience Center, 40014 University of Jyv¨askyl¨a, Finland
(Dated: February 2, 2008)
Weproposeauniversaltransformationfromamany-bosonstatetoacorrespondingmany-fermion
state in the lowest Landau level approximation of rotating many-body systems, inspired by the
Laughlin wave function and by the Jain composite-fermion construction. We employ the exact-
8 diagonalization technique for finding the many-body states. The overlap between the transformed
0 boson ground state and thetruefermion ground state is calculated in ordertomeasure thequality
0 of the transformation. For very small and high angular momenta, the overlap is typically above
2 90%. For intermediate angular momenta, mixing between states complicates the picture and leads
tosmall ground-state overlaps at some angular momenta.
n
a
J PACSnumbers: 60.10.-j,70.10.Pm,68.65.-k,03.75.Kk,32.80.Pj

7

The properties of rotating many-body systems have netically trapped ions at ultra-low temperatures. In this
]
l been the topic of intense study, theoretical as well as workweconcentrateonharmonicallyconfined,Coulomb-
l
a experimental [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, interacting particles in two dimensions.
h 14, 15, 16, 17, 18, 19, 20]. In particular, Bose-Einstein This paper is organized as follows: The next section
-
s condensates have been of great interest since the advent gives an introduction to the mathematics of the pro-
e of the laser-cooling technique [21]. It is by now a well- posed boson–fermion universality of many-body systems
m
established fact that quantum many-body systems un- in the lowest Landau level. This section also introduces
. derrotationformquantizedvortices,apropertythathas the concepts used throughout the paper, and defines
t
a beenknowninthe contexts ofsuperconductivityandsu- a mathematical transformation of a many-boson wave
m perfluidity[22,23]. Inearlierworks,theformationofvor- function into a corresponding many-fermion wave func-
- tices in few-body systems was studied quite extensively, tion. Section II gives a brief description of the exact-
d
including both boson and fermion systems [8, 15, 24], as diagonalization method in the lowest Landau level, and
n
well as their two-component generalizations [24, 25, 26]. of the implementation of the boson–fermion transforma-
o
InRef.[8]wenotedthatsomepropertiesofvortexforma- tion. The reader who is familiar with exact diagonaliza-
c
[ tion in few-body systems of repulsively interacting par- tion may skip directly to section III, where our results

ticles in the lowest Landau level are universal, not only are presented and discussed. Some concluding remarks
1
with respect to the details of the interaction (the boson are given in section IV.
v
system was studied both with the long-range Coulomb
6
6 interaction and with a short-range, contact interaction),
9 but also with respect to the statistics of the constituent I. INTRODUCTION TO BOSON–FERMION
0 particles. We found remarkable similarities between the UNIVERSALITY
1. yrast spectra of the boson and fermion systems, and we
0 also noted that vortices enter at very specific, and cor- In order to formulate a mathematical expression for
8 responding, angular momenta in the boson and fermion thedirectcomparisonbetweenbosonandfermionmany-
0 systems, respectively. bodywavefunctions,weusecomplexcoordinatesforthe
:
v two-dimensional plane, z = x +iy. The general wave
In the present paper, we formalize this boson–fermion
i function in the lowest Landau level of one particle in a
X universalitybymeansofatransformationfromabosonic harmonic-oscillatorpotentialisψ (z)∝zℓe−|z|2,whereℓ
r many-body wave function to a corresponding fermionic ℓ
is the angular momentum. (Atomic units and one fixed
a wave function. The transformation is inspired by the
frequency ω = 1 of the harmonic confinement are used
Laughlin wave function [27], and forms a direct paral-
throughout the paper.) In the boson system at zero an-
lel to the Chern–Simons transformation in the theory
gular momentum, all particles reside in the ℓ = 0 state,
of the fractional quantum Hall effect [28, 29] and Jain’s
and the many-body wave function is
composite-fermion picture [30]. (The latter was recently
successfully applied to the problem of small quantum ΨB ∝e−Pk|zk|2. (1)
0
dotsinmagneticfields,andothersmallquantumsystems
athighangularmomentum[9,10,11,12,13,16,18,19].) As the system is set rotating, particles are lifted from
This allows us to investigate quantitatively to what ex- the ℓ = 0 single-particle state in order to carry angular
tent this universality holds in a general comparison be- momentum. A single vortex is formed at the center of
tweenfew-body bosonandfermionsystemsinthelowest mass when the total angular momentum of the system
Landau level. The particles may be electrons in a quan- equalsthenumberofparticles,inwhichcaseallparticles
tum dot (in the case of fermions) or optically or mag- are in the ℓ = 1 state. As angular momentum increases
2
further, more vortices successively enter the system. As The overlap between the true wave function and the
detailed in Ref. [8], a vortex-generating state carrying transformed boson wave function can the be calculated:
n vortices can be obtained from Eq. (1) by successive
O= ΨF DFΨB 2. (6)
multiplications with symmetric polynomials: LMDD+LB LB
(cid:10) (cid:11)
The argument given above for the boson–fermion cor-
N N
respondenceisentirelyheuristic: itisbasedonobserving
Ψ = (z −aeiα1)×···× (z −aeiαn)ΨB
n j1 jn 0 the common features of fermion and boson wave func-
Yj1 Yjn tions in the lowest Landau level. The correspondence
N formalized in Eq. (5) is ultimately justified by the ex-
= (zn−an)ΨB. (2)
j 0 plicit demonstration of its performance in section III.
Yj However,it is reasonableto anticipate the existence of a
boson–fermion transformation of this type from boson-
This wave function describes a state with n vortices
Chern–Simons theory [28, 29] and composite-fermion
evenly spaced on a ring with radius a centered at the
theory [29, 30]. A fermionization of repulsively interact-
origin, and may be used to obtain a trial many-body
ingbosonsinthelowestLandaulevelhasbeendescribed
wave function [8].
by several authors [7, 11, 12, 13, 16, 17]. In particular,
Thesamelineofreasoningcanberepeatedcompletely
Cazalillaet al. [12], Chang et al. [11]as wellasRegnault
analogouslyforfermions. However,inthelowestLandau
et al. [16] explicitly map interacting bosons onto non-
level the angular momentum of a many-fermion system
interacting, spinless fermions by means of a composite-
cannotbe zero. Insteadthe smallestpossible totalangu-
fermionconstruction. Itisknownthatthemean-fieldap-
larmomentumisachievedbyputtingonefermionineach
proximation of non-interacting composite fermions loses
of the N lowestsingle-particle states with single-particle
accuracy as angular momentum increases, and effects of
angular momenta ranging from 0 to N −1. This state
residualinteractionsbecomeimportant[9,11,16,18]. In
is calledthe maximum density droplet (MDD) andis the
thecaseofaharmonicinteraction,thetransformation(5)
fermionequivalentofthe zero-angular-momentumstates
betweeninteracting bosonsand interactingfermions was
for bosons. The wave function of the MDD is given by
rigorouslyderivedrecentlybyRuuskaandManninen[32].
the Laughlin wave function with filling factor one [31]:
Inthe Chern–Simonsapproachto the fractionalquan-
tum Hall effect, the many-body wave function Ψ of the
N e
ΨFMDD ∝ (zi−zj)e−Pk|zk|2. (3) electrons of the two-dimensional electron liquid is trans-
Yi<j formed into a wave function ΨCS of particles moving in
aneffectivemagneticfield. Thetransformationisdefined
Intheconstructionofthe vortex-generatingstateforthe as follows [29]:
fermion system, the boson condensate is replacedby the p
z −z
MDD. Ψe = i j ΨCS. (7)
(cid:20) z −z (cid:21)
Now we compare Eq. (1) with Eq. (3) and note that Yi<j i j
the latter can be obtained from the former simply by
The prefactor determines the number of flux quanta as-
multiplication with the polynomial
sociated with each particle, and it is symmetric or anti-
N symmetric with respect to particle interchange depend-
DF = (z −z ). (4) ing on whether p is chosen even or odd. In the sim-
i j
Yi<j plest possible model, we choose p=1. Since the original
electron wave function is fermionic, this means that the
Makingthe assumptionthatthetransformationbetween Chern–Simons wave function describes a corresponding
the lowest angular momentum states, ΨFMDD = DFΨB0, boson system. The transformation between the boson
holds whenever ΨB0 is replaced by ΨFMDD, we arrive at systemandthe fermionsystemis the givenby the factor
a very general transformation from any bosonic many- [(z −z )/z −z ].
i<j i j i j
body wave function to a corresponding fermionic wave QJain took the Chern–Simons approach further to a
function,shiftedinangularmomentumbyexactlyLMDD: more sophisticated ansatz in the lowest Landau level by
introducing his composite-fermion theory [29, 30]. In
ΨFLMDD+LB =DFΨBLB. (5) this approach, the power p of the Chern–Simons factor
is explicitly taken to be an even integer, thereby mak-
Thedegreeofuniversalityofthestructureofthemany-
ing the composite particles fermions. Also, the prefactor
body wave function between boson and fermion systems is simply taken to be the Jastrow factor (z −z )p,
will be reflected in how well the transformation (5) re- i<j i j
whichattachesvorticesratherthanfluxtuQbestothepar-
produces the true fermion wave function. The straight-
ticles [29]. The wave function of the original fermions is
forward way of determining this is to calculate the wave
retained by projection to the lowest Landau level:
functionofagivenmany-fermionsystembothdirectly,in
ordertoobtainthetruewavefunction,andbytransform- Ψe =P (zi−zj)pΨCF, (8)
ing the corresponding boson state according to Eq. (5). Yi<j
3
whereP istheprojectionoperatorintothelowestLandau Boson wave function Transform
level.
0.685 04000i - 0.895 01111000i
Jain’s construction and the bosonic Chern–Simons (cid:17)(cid:17)3
wavefunctionimmediatelysuggestamapping betweena −0.168 30001i (cid:17)(cid:17)(cid:8)(cid:8)(cid:8)*−0.400 10110100i
botohseornmsyasgtenmetiacnfidealdn(eoqru,iveaqlueinvtafleenrmtlyio,nansygsutleamr matosmoemne- ++00..139941 2210021000ii(cid:17)(cid:8)(cid:24)(cid:17)(cid:8)(cid:24)(cid:17)(cid:8)(cid:24)(cid:8)(cid:24)(cid:24)(cid:24)(cid:24):++00..113188 1111000110100100ii
tum) of the form described by Eq. (5). For a harmonic
−0.559 12100i −0.0775 11100001i
particle–particle interaction, this relation is exact, and
can be derived analytically by constructing a spectrum-
generating algebra, and applying the corresponding lad- FIG. 1: As an example, we show the transformation of the
deroperatorstotheground-statewavefunctionsΨ =1 LB = 4 boson wave function into a fermionic wave function
B with LF = 10. One term in the boson wave function may
and Ψ = (z −z ) respectively [32].
F i<j i j contribute to the weight of several terms in the transformed
The numQerical study presented in this paper shows wave function. Correspondingly, terms in the transformed
thatthetransformationisverygeneralandyieldsagood
wavefunction may receivecontributions from more than one
approximation for the fermion wave function in the low- term in theoriginal wave function.
est Landau level, also for Coulomb-interacting particles.
the interaction part of the Hamiltonian. Working in the
II. THE EXACT-DIAGONALIZATION
lowest Landau level already guarantees that we have a
METHOD AND THE BOSON–FERMION
finite number of possible Fock states for any given to-
TRANSFORMATION
tal angular momentum, and in this study we have used
all of these in the basis. Thus no further truncation of
In order to calculate the overlap, Eq. (6), the many-
the Hilbert space is done. The resulting matrix is di-
bodywavefunctionsmustbeobtainedwithsufficientac-
agonalized numerically, yielding the interaction-energy
curacy. In this study we take as our model system har-
spectrum and the many-body eigenstates. The diago-
monically confined, Coulomb-interacting particles, and
nalizationisperformedusingtheLanczosalgorithm[33],
we solve both the fermion and the boson problems with
exceptat the smallestangularmomenta,where a full di-
theexact-diagonalizationmethod. Thebosonwavefunc-
agonalizationis needed due to the small matrix sizes.
tionistransformedintoacorrespondingfermionicmany-
The exact-diagonalization method outlined above is
body wave function by means of Eq. (5), which we have
applicabletobothfermionsandbosons;onlythecommu-
implemented numerically to operate on a bosonic wave
tationrelationsofthecreationandannihilationoperators
function in the Fock–Darwin basis.
differ,andthemany-bodyFock–Darwinbasisstatesmust
Since the model system used in this study is that
begiventheappropriatesymmetryproperties(Slaterde-
of a harmonic confining potential, a natural choice of
terminants for fermions, product states for bosons).
single-particle basis is the set of eigenfunctions of the
The numerical implementation of the transformation,
two-dimensional harmonic oscillator in the lowest Lan-
Eq. (5), requires some care. A general bosonic many-
dau level:
body wave function consists of a linear combination of a
ψ (z)=A zℓe−|z|2. (9) (possiblyverylarge)numberofFockstates. Uponmulti-
ℓ ℓ
plication by the determinant DF (Eq. (4)), each bosonic
The many-body wave function is then built in the Fock– Fock state is transformed into a fermionic many-body
Darwin basis formed from these single-particle states, wavefunction,whichisingeneralnotabasisstateinthe
and the Hamiltonian (up to an additive constant) is fermion Fock–Darwin basis, but instead must be repre-
straightforwardlywritten as sented as a linear combination of fermionic basis states.
Thus, each Fock state in the boson wave function will
1
H = ℓ aˆ†aˆ + U aˆ†aˆ†aˆ aˆ , (10) yieldseveralFockstatesinthefinalfermionicwavefunc-
Xi i i i 2i,Xj,k,l i,j,k,l i j k l tion, and correspondingly, the weight of each Fock state
in the final fermionic wave function may receive contri-
where the interaction matrix element is butions from several Fock states in the original bosonic
wave function.
U = ij Uˆ(r,r′) kl
i,j,k,l For example, the four-particle bosonic state 202i
D E
(with angular momentum L = 4) transforms into the
= ψ∗(r)ψ∗(r′)U(r,r′)ψ (r)ψ (r′)drdr′. B
i j k l fermionic wave function 0.79 110011i−0.56 101101i+
ZZ
(11) 0.25 011110i, a linear combination of three states from
the Fock basis (total angularmomentum L =10). The
F
Since we are working with many-body systems at given true wave function for the four-bosonsystem with angu-
total angular momentum L, the one-body term yields lar momentum L = 4 is a linear combination of five
B
the same energycontribution for all allowedFock states. basis states, as depicted in Fig. 1. The state in our
Therefore we may drop this term, and diagonalize only toy example appears in the fourth term. Under appli-
4
cation of the transformation, Eq. (5), this wave func- tozero. Thesecaseswillbediscussedinsomedetaillater
tion transforms into the fermionic wave function shown in this section
intherightcolumnofFig.1. Asindicatedbyarrows,the Away from the extremes of very small and very large
bosonicstate 202icontributestothefirstthreetermsof angular momenta, the overlap drops and begins to fluc-
the transformedwave function. Contributions come also tuate. In the case of five particles (top panel of Fig. 3)
from other terms in the original wave function. For ex- this is seenasa generaldecreaseofthe overlapfor angu-
ample,contributionstothefirsttermcomefromallterms lar momenta in the range L =14–35together with two
F
of the bosonic wave function. The contribution from the drops to zero at L = 16 and L = 46. This pattern is
F F
boson basis state 04000i is also indicated in the figure repeated in a much more complicated picture in the six-
(in fact, this particular term in the boson wave func- particlecase,showninFig.3,bottompanel. Again,there
tioncontributesonlytothe firsttermofthe transformed is a general decrease in the overlap as the total angular
wave function). This example shows that it is necessary momentumincreasesbeyondthesmallangularmomenta
to carefully collect the weights of the basis states of the wheretheoverlapislargeduetothesmallHilbertspace.
fermionic wave function resulting from the transforma- As the angular momentum increases beyond L ≈ 40
F
tion. there is a region where the overlap tends to be either
large or vanishing. At the very largestangular momenta
studied, the overlapsettles at over 90%.
III. RESULTS The behavior described for the five- and six-particle
systems is echoed in smaller and larger systems as well.
In the four-particle system (Fig. 2) the overlap is well
We perform calculations of energies, wave functions
above 90% for all angular momenta studied, L =6–82.
and ground-state overlaps systematically over a wide F
However, for angular momenta in the range L = 10–
range of angular momenta for systems containing be- F
40 the overlap oscillates between 96% and 99%, and for
tween four and eight Coulomb-interacting particles.
L = 24 dropping to 93%, before returning to values
Fig. 2 shows the overlapof the groundstate of a four- F
very close to 100% for larger angular momenta. In sys-
fermion system with the transformed ground state of a
tems with seven and eight particles (Fig. 4) the overlap
corresponding four-boson system, for total fermion an-
oscillates wildly over most of the studied region of an-
gular momenta in the range L = 18–62. In the small
F gular momentum, where overlaps at or above 80% are
four-particle system, the overlap is very good, close to
interspersedwith dropsto about50%orbelow. Thereis
100%forallangularmomenta. Thetopandmiddle pan-
a tendency towards better overlaps in the high angular
els of the figure show the yrast spectra of the five lowest
momentum limit of the data.
states for the boson and fermion systems respectively.
We may conclude from these observations that the
The similarity of the yrastlines is readily apparentfrom
transformation defined in Eq. (5) from a bosonic many-
this figure. Both systems show a four-fold periodicity,
body wave function to a fermionic wave function for
and the kinks in the yrast line—where there is also a
the same number of particles and angular momentum
relatively large gap between the ground state and the
firstexcitedstate—appearatcorrespondingangularmo- LF =LB+LMDD performs well in the low and high an-
gularmomentumlimits,wheregoodperformancemaybe
menta, LF = LB +LMDD. The overlap oscillates with expected due to the restrictedHilbert spaceand particle
thesamefour-foldperiodforhighangularmomenta,with
localization respectively. For intermediate values of the
the overlaphaving a local maximum whenever there is a
angular momentum, the performance of the transforma-
kink in the yrast line.
tion needs a deeper study.
Looking at a system with more particles, the picture
The poor performance of Eq. (5) for certain combi-
becomes more complicated. Fig. 3 shows the ground-
nations of particle number and angular momentum is a
state overlap in the five- and six-particle systems over
result of a restructuring of levels in the yrast spectrum
theentirerangeoftotalangularmomentastudied,L =
F between the boson and fermion systems. The result is
LMDD = 10 to LF = 85 for five particles and LF = thatthebosongroundstatewillhaveadifferentstructure
LMDD = 15 to LF = 89 for six. At the smallest angular than the fermion state, and therefore so will the trans-
momenta,theoverlapisalways100%duetothefactthat
formed boson wave function. In order to study the par-
the Hilbert space is so small that only one or a few wave
ticle configurations correspondingto the different many-
functions are possible. It is interesting to note that also
bodywavefunctions,welookatthepair-correlationfunc-
in the limit of high angular momentum, the overlap be-
tion
tweenthetruefermiongroundstateandthetransformed
bosongroundstatetendstowards100%. Thisshowsthat g(r′,r)= Ψ ψˆ†(r′)ψˆ†(r)ψˆ(r′)ψˆ(r) Ψ , (12)
the performance of Eq. (5) is not a consequence of the D E
Laughlin wave function being a good approximation. In whichgivestheprobabilityoffindingaparticleatr,given
fact,theoverlapbetweentheLaughlinwavefunctionand that there is a particle at r′.
the exact wave function decreases with decreasing filling We may compare the pair-correlation functions ob-
factor [34]. tainedfromthe truefermionwavefunctionandfromthe
Fig. 3 also shows some cases where the overlap drops transformed boson wave function. Six particles at total
5
FIG. 2: Yrast spectra and overlaps for a four-particle system. (A) Yrast spectrum of four harmonically confined bosons. The
five lowest states for each angular momentum are included. (B) The corresponding yrast spectrum for four fermions. (C)
Overlap of thetransformed boson ground state with the fermion ground state, calculated from Eq. (6).
fermionic angular momentum L =57 and L =58 are the six particles of the system are equidistantly situated
F F
two examples from the intermediate angular momentum on a circle. This is one of two classically (meta-)stable
range where the transformed boson ground state some- configurations for six coulomb-interacting particles in
timesreproducesthe fermiongroundstateverywell,but a harmonic confining potential [35]. This structure is
sometimes fails spectacularly. The former is the case at also seen in the the bosonic wave function, and the ef-
L = 57, while the latter is true for L = 58. Fig. 5 fect of multiplication with DF, apart from changing the
F F
shows the pair-correlation functions of the boson wave particle-exchangesymmetry,istopushtheparticlesout-
function (top panel), the transformed boson wave func- ward.
tion(middle panel),andthe fermionwavefunction(bot-
The other classically stable configuration of six equal
tompanel)forL =57(leftcolumn)andL =58(right
F F electricalchargesis a ring of five particles with the sixth
column). The reference particle in each plot is indicated
particle sitting precisely at the center of the ring. In
by a black dot in the figure. Its position is chosen such
the classical system, this configuration actually has a
thatitsitsatthedistancermax fromtheoriginwherethe slightly lower energy than the ring of six particles [35].
particle density has its maximum.
These two configurations, (6,0) and (5,1), are known to
At L = 57 the particle structure is extremely well compete in quantum-mechanical six-body systems with
F
reproducedbythe transformationfromthebosonicstate repulsive coulomb interaction [36, 37]. In an electron
withLB =LF−LMDD =42,andthetwopair-correlation system where the spin degree of freedom is not frozen
functions from the true fermion wave function and the out, the (6,0) configuration tends to be favored (unless
transformed boson wave function are almost indistin- the system is very dilute), because of frustration in the
guishable. The true fermionic many-body wave function (5,1) configuration [36, 37, 38]. In the present study, we
displaysaninternalstructureoflocalizedparticles,where dealwithsystemsofspinlessbosonsandofspin-polarized
6
FIG. 3: Overlap between the true fermion ground state and FIG. 4: Overlap between the true fermion ground state and
the transformed boson ground state as a function of total the transformed boson ground state as a function of total
angularmomentumforfive(toppanel)andsix(bottompanel) angular momentum for seven (top panel) and eight (bottom
particles. panel) particles.
fermions. Therefore, spin frustration does not come into
play,andthetwoclassicalconfigurationscompete. When
the angular momentum is increased just by one unit to
L =58,thefermiongroundstateisthe(5,1)configura-
F
tion,asshowninthebottomrightpanelofFig.5. Inthe
boson system, however, another change takes place with
the corresponding increase of one unit of angular mo-
mentum: the particles remain seated on one ring of six
particles,butthedegreeoflocalizationdecreases. Corre-
spondingly,themultiplicationwithDF expandsthestate
toaringofsixslightlysmearedoutmaxima,astatethat
is orthogonal to the true ground state.
Indeed the orthogonality of the transformed state to
FIG. 5: Pair-correlation functions for the six-particle sys-
thetruefermiongroundstateisnocoincidence,asinfact
tem at total angular momentum LF = 57, corresponding to
the transformedbosonicgroundstate correspondsto the
LB = 42 in the boson system (top row), and at LF = 58
firstexcitedstateofthefermionsystem,andcorrespond-
(bottom row). The panels in each row show, from left to
ingly the fermion ground state is well reproduced by ap-
right,thepair-correlationfunctionofthebosongroundstate,
plyingthe transformationtothe firstexcitedstateofthe thetransformed boson groundstate, and thefermion ground
boson system. Table I shows the overlap of the fermion state,respectively. Theblackdotindicatesthepositionofthe
ground-statewavefunctionwitheachofthe transformed fixedreference particle
wavefunctions of the five lowestbosonstates. The same
table also shows the overlap of the transformed boson
ground-state wave function with each of the five lowest pair-correlationfunctions of the first excited state in the
states of the fermion system. As can be seen from the boson system and the transformed version of this state,
table, the transformed bosonic ground state almost per- and the first excited fermionic state. These plots may
fectly reproduces the first excited state of the fermion be compared with the corresponding ground-state pair
system. Plotting the pair-correlation functions, as done correlations in Fig. 5.
in Fig. 6, confirms this picture. This figure shows the N =6,L =58discussedaboveisaparticularlyclean
F
7
TABLEI:Overlapofthefermiongroundstatewiththetrans- TABLE II: Overlap of the fermion ground state with the
formed wave function of each of the five lowest states of the transformed wavefunction ofeach of thefivelowest statesof
boson system (middle column), in the system with N = 6 thebosonsystem(middlecolumn),inthesystemwithN =6
particles and fermionic angular momentum LF = 58. The particles and fermionic angular momentum LF = 53. The
table also shows the overlap between the transformed boson table also shows the overlap between the transformed boson
groundstatewitheachofthefivelowestfermionstatesinthe groundstatewitheachofthefivelowestfermionstatesinthe
same system (right column) same system (right column)
s ˙ΨFg.s. ΨBs¸ 2 [%] ˙ΨFs ΨBg.s.¸ 2 [%] s ˙ΨFg.s. ΨBs¸ 2 [%] ˙ΨFs ΨBg.s.¸ 2 [%]
g.s. 0.00 0.00 g.s. 68.32 68.32
1st 96.94 99.31 1st 25.74 20.71
2nd 0.46 0.00 2nd 4.67 9.29
3rd 0.00 0.00 3rd 0.00 0.00
4th 0.00 0.00 4th 0.00 0.00
FIG.6: Pair-correlationfunctionsforthefirstexcitedstatein FIG. 7: Pair-correlation functions for the six-particle system
thesix-particle system at total angular momentum LF =58, attotalangularmomentumLF =53,correspondingtoLB =
corresponding to LB = 43 in the boson system. The panels 38inthebosonsystem. Thepanelsshowthepair-correlation
showthepair-correlationfunctionofthebosonwavefunction function of the boson ground state (left), the transformed
(left), thetransformed boson wavefunction (center),andthe boson ground state (center), and the fermion ground state
fermion wave function (right), respectively. The black dot (right), respectively. The black dot indicates the position of
indicates the position of thefixed reference particle. thefixed reference particle.
example where the many-body configurations exchange boson state. However, the transformed bosonic ground
placesbetweenthe fermionandthe bosonspectra. More statedoesnothaveanyappreciableoverlap(1.94%)with
generally, states may mix, such that one fermionic state the first excited fermion state. Instead it corresponds
isreproducedbysomelinearcombinationoftransformed morecloselytothe secondexcitedstate(withanoverlap
boson states. Such mixing of states between the bosonic of 89.91%).
and fermionic systems accounts for the cases where the We haveseenthat when the particlenumber increases
overlap between the ground states is neither close to the overlapbetween the fermion and boson states is still
100% nor vanishing. One example is N = 6 particles at good, but in many cases the lowest energy state of the
angularmomentumLF =53,where the overlapbetween fermion system corresponds to an excited state of the
the ground states is 68.32%. In this case the fermionic boson system and vice versa. With increasing particle
ground state has a considerable overlap with all of the numbertheexchangeoftheorderoflevelsbecomesmore
three lowest bosonic states. This is shown in Table II. important. Thisisalreadyknowninconnectionwiththe
The pair-correlation functions are shown in Fig. 7. The study of vortices in boson and fermion systems [15]. For
transformed boson ground state displays a smeared-out example, in the boson case the single vortex reaches the
(5,1) configuration, while the fermion ground state is a origin,whileinthefermioncasethestatewithonevortex
(6,0) configuration distorted such that there is also an at the origin becomes an excited state when the particle
increased particle density at the origin. numberisabout10orgreater. Thisisalsotrueforstates
Thedescribedmixingoftheorderofthestatesbetween with multiple vortices: the same vortex states are there
bosonic and fermionic system is seen also in the other in both systems, but they appear as the lowest state at
cases were the overlapbetween the groundstates is bad, different (relative) angular momenta. As an example,
alsoforotherparticlenumbersthansix. Therearemany we may study three vortices in a rotating system of 20
examples at all particle numbers larger than four. One particles. For bosons the angular momentum is 48 and
particularly drastic example for eight particles is L = the state is the lowest-energystate, while for fermions it
F
57, where the fermion ground state has 91.20% overlap is the fourth state at angular momentum 238. Figure 8
with the tranformed wave function of the first excited shows the hole-hole correlation [39] (i.e. pair correlation
8
of vortices) which can be calculated for the fermionic pect large overlapsregardlessof the type of repulsive in-
wave functions (for the boson system, the transforma- teraction, merely due to the small Hilbert space. It is
tion, Eq. 5, is first applied to the wave function). With interesting to note, that not only are the overlaps large
this large number of particles we were not able to per- (over75%),but the overlapsobtainedfromδ-interacting
form the computation with the full Fock-Darwin basis. bosons follow those obtained from Coulomb-interacting
The resulting overlap between the converted boson and bosons closely.
the fermion wave functions is only 88%, but yet the re-
sulting pair-correlationfunctions in Fig. 8 are nearly in-
distinguishable. 100
80
%]
p [ 60
a
erl
v 40
O
20
0
15 20 25 30 35 40 45
Fermionic angular momentum
20 FERMIONS, L = 238 20 BOSONS, L = 48
FIG. 9: Overlap between the ground state of six Coulomb-
FIG. 8: Hole–hole pair-correlation function describing the
interacting fermions and the transformed ground state of six
similarvortexlocalizationinbosonandfermionsystems. The
bosons with δ-interaction, plotted as a function of total an-
boson wave function was first converted to a fermion wave
gular momentum in thefermion system.
function as described in the text.
One may speculate that the large overlaps between These results should be expected from what we know
fermionic wave function and transformed bosonic wave about the Laughlin wave function. The Laughlin wave
function occur when the wave function is particularly function is a good approximation to the exact many-
simple. One possible measure of the complexity of the body state as long as the angular momentum is not too
fermionwavefunctionwithinthegivensingle-particleba- large. Laughlin’s construction makes no explicit refer-
sis (here the harmonic-oscillatoreigenstates)is the num- ence to the details of the repulsive interaction between
ber of Fock states needed to make up 50% of its norm. the constituent particles [27].
For six particles, this number is five or below for angu- For angular momenta LF >∼ 30 the overlap drops to
lar momenta up to L = 30, and the overlap is 70% zeroexceptforcertainangularmomenta,wheretheover-
F
and above, but we see very large overlaps also when the lap is very good (95% and above), and very close to the
number of Fock states needed for 50% of the norm of overlap obtained when Coulomb interaction is used. In
the fermion wave function is larger than 40. This shows many cases the zero overlap is again simply the cause of
that the good overlapsbetween fermion state and trans- different orderings of the states in the the two systems,
formed boson state are not merely the result of simple but in some cases alsoa substantialmixing of states was
wave functions, but that the transformation defined by found.
Eq. (5) can handle also complex many body states.
The results detailed this far may to some extent be
robust against changes in the details of the repulsive in- IV. CONCLUSION
teractionbetweentheconstituentparticlesofthesystem.
This is suggestedby results where we calculate the over- We conclude that for particles in the lowest Landau
lapbetweentheground-statewavefunctionforCoulomb- level there is a far-reaching universality between bosons
interacting fermions and the transformed ground-state and fermions in the properties of the rotating systems.
wave function of bosons with a repulsive δ-type interac- Thisuniversalitymaybeformulatedmathematicallyasa
tion. This is done for the case of six particles at an- transformation(Eq.(5))fromabosonicmany-bodywave
gular momenta between L = 15 and L = 45. The function to a fermionic one. These two wave functions
F F
overlap is plotted as a function of LF in Fig. 9. This will differ in total angular momentum exactly by LMDD,
figure should be comparedwith the same range of angu- the smallest possible angular momentum in the fermion
lar momentum in the bottom panel of Fig. 3. For the system. The transformationproduces a verygoodcorre-
lower half of the range of angular momenta, the overlap spondence (as measured by calculating the overlap inte-
where δ-interaction is used for the bosons does not de- gral)between the bosonic and fermionic states when the
viate substantially from the overlap where both systems number of particles is small, when the angular momen-
have Coulomb interaction. For these angular momenta, tum is very small (due to the restricted Hilbert space),
the greater part of the many-body wave function comes and when the angular momentum is large (due to local-
from only a few Fock states and therefore we may ex- izationinto states welldescribedby Laughlinwavefunc-
9
tions). Away from these extremes, the correspondence to the fermion ground states.
between boson and fermion states is more complicated.
Apart from a general decrease of the the overlaps due
to the difficulty of reproducing the details of the more
complicated wave functions as the particle number in-
Acknowledgments
creases,there is also fluctuations betweenlarge and very
small overlapsbetween the groundstates. This is due to
a reshuffling of the levels in the spectrum between the This work was financially supported by the Swedish
boson and fermion systems. In these cases it is not the Research Council, the Swedish Foundation for Strategic
groundstatebutalow-lyingexcitedstatethattransforms Research, the Academy of Finland, and NordForsk.
[1] F.Dalfovo,S.Giorgini,L.P.Pitaevskii,andS.Stringari, 76, 085308 (2007).
Rev.Mod. Phys. 71, 463 (1999). [21] C.N.Cohen-Tannoudji,Rev.Mod.Phys.70,707(1998).
[2] D.A.ButtsandD.S.Rokhsar,Nature397,327 (1999). [22] P.G. deGennes, Superconductivity of Metals and Alloys
[3] B. Mottelson, Phys. Rev.Lett. 83, 2695 (1999). (Westview Press, Boulder, 1999).
[4] G. F. Bertsch and T. Papenbrock, Phys. Rev. Lett. 83, [23] A. A.Abrikosov, Sov.Phys. JETP 5, 1174 (1957).
5412 (1999). [24] S. Bargi, G. M. Kavoulakis, and S. M. Reimann, Phys.
[5] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Rev. A 73, 033613 (2006).
Hall,C.E.Wieman,andE.A.Cornell, Phys.Rev.Lett. [25] S. Bargi, J. Christensson, G. M. Kavoulakis, and S. M.
83, 2498 (1999). Reimann, Phys. Rev.Lett. 98, 130403 (2007).
[6] G.M.Kavoulakis,B.Mottelson,andC.J.Pethick,Phys. [26] M. Koskinen, S. M. Reimann, J.-P. Nikkarila, and
Rev.A 62, 063605 (2000). M. Manninen, J. Phys.: Cond. Mat. 19, 076211 (2007).
[7] I. Romanovsky, C. Yannouleas, and U. Landman, Phys. [27] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983).
Rev.Lett. 93, 230405 (2004). [28] S.C.Zhang,T.H.Hansson,andS.Kivelson,Phys.Rev.
[8] M.Toreblad,M.Borgh,M.Koskinen,M.Manninen,and Lett. 62, 82 (1989).
S.M. Reimann,Phys. Rev.Lett. 93, 090407 (2004). [29] G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101
[9] G.S.Jeon,C.-C.Chang,andJ.K.Jain,J.Phys.: Cond. (2003).
Mat. 16, L271 (2004). [30] J. K.Jain, Phys. Rev.Lett. 63, 199 (1989).
[10] C.-C.Chang,G.S.Jeon,andJ.K.Jain,Phys.Rev.Lett. [31] A. H. MacDonald, S. R. E. Yang, and M. D. Johnson,
94, 016809 (2005). Aust. J. Phys. 46, 345 (1993), ISSN0004-9506.
[11] C.-C. Chang, N. Regnault, T. Jolicoeur, and J. K. Jain, [32] V. Ruuska and M. Manninen, Phys. Rev. B 72, 153309
Phys.Rev.A 72, 013611 (2005). (2005).
[12] M. A. Cazalilla, N. Barber´an, and N. R. Cooper, Phys. [33] R. B. Lehoucq, D. C. Sorensen, and Y. Yang, Arpack
Rev.B 71, 121303 (2005). user’s guide: Solution to large scale eigenvalue problems
[13] G. S. Jeon, A. D. Gu¨¸clu¨, C. J. Umrigar, and J. K. Jain, with implicitly restarted arnoldi methods, FORTRAN
Phys.Rev.B 72, 245312 (2005). code http://www.caam.rice.edu/software/ARPACK.
[14] N. Barber´an, M. Lewenstein, K. Osterloh, and [34] C.YannouleasandU.Landman,Phys.Rev.B68,035326
D.Dagnino, Phys.Rev.A 73, 063623 (2006). (2003).
[15] S.M.Reimann,M. Koskinen,Y.Yu,andM. Manninen, [35] F. Bolton and U. Rossler, Superlattices and Microstruc-
Phys.Rev.A 74, 043603 (2006). tures 13, 139 (1993).
[16] N. Regnault, C.-C. Chang, T. Jolicoeur, and J. K. Jain, [36] S. M. Reimann, M. Koskinen, and M. Manninen, Phys.
J. Phys. B: At.Mol. Opt.Phys. 39, S89 (2006). Rev. B 62, 8108 (2000).
[17] C. Yannouleas and U. Landman, PNAS 103, 10600 [37] B. Reusch, W. H¨ausler, and H. Grabert, Phys. Rev. B
(2006). 63, 113313 (2001).
[18] G. S. Jeon, C.-C. Chang, and J. K. Jain, Eur. Phys. J. [38] A. Ghosal, A. D. Gu¨¸clu¨, C. J. Umrigar, D. Ullmo, and
B 55, 271 (2007). H. U.Baranger, Phys.Rev. B 76, 085341 (2007).
[19] C. Shi, G. S. Jeon, and J. K. Jain, Phys. Rev. B 75, [39] M. Manninen,S.M. Reimann,M.Koskinen,Y.Yu,and
165302 (2007). M. Toreblad, Phys. Rev.Lett. 94, 106405 (2005).
[20] Z. Dai, J.-L. Zhu, N. Yang, and Y. Wang, Phys. Rev. B