Quantal molecular description and universal aspects of the spectra of bosons and fermions in the lowest Landau level

1 MB·English

by  Constantine Yannouleas

Quantal molecular description and universal aspects of the spectra of bosons and
fermions in the lowest Landau level
Constantine Yannouleas∗ and Uzi Landman†
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430
(Dated: 29 November2009)
Through the introduction of a class of trial wave functions portraying combined rotations and
vibrations of molecules formed through particle localization in concentric polygonal rings, a cor-
related basis is constructed that spans the translationally invariant part of the lower-Landau-level
0
(LLL) spectra. These trial functions, referred to as rovibrational molecular (RVM) functions, gen-
1
0 eralize our previous work which focused exclusively on electronic cusp states, describing them as
2 pure vibrationless rotations. From a computational viewpoint, the RVM correlated basis enables
controlled and systematic improvements of the original strongly-correlated variational wave func-
n
tion. Conceptually, it providesthebasis for thedevelopmentof aquantalmolecular description for
a
the full LLL spectra. This quantal molecular description is universal, being valid for both bosons
J
and fermions, for both the yrast and excited states of the LLL spectra, and for both low and high
7
angular momenta. Furthermore, it follows that all other translationally invariant trial functions

(e.g.,theJastrow-Laughlin, compactcomposite-fermion, orMoore-Read functions)arereducibleto
]
s a description in terms of an excited rotating/vibrating quantalmolecule.
a
g PACSnumbers: 03.75.Hh,03.75.Lm,73.43.-f,73.21.La
-
t
n
a I. INTRODUCTION ticipatedthatsmall(and/ormesoscopic)assembliesoful-
u tracoldbosonicatomswillbecometechnicallyavailablein
q
the near future [32–35] andthat they will providean ex-
. A. Motivation
t cellentvehicle[27,28,31–35]forexperimentallyreaching
a
exoticphasesandfortesting the richvarietyofproposed
m
Following the discovery [1] of the fractional quantum
LLL trial wave functions.
- Halleffect(FQHE)intwo-dimensional(2D)semiconduc-
d Despite the rich literature and unabated theoretical
tor heterostructures under high magnetic fields (B) in
n interest, a unifying physical (as well as mathematical)
o the 1980’s, the description of strongly correlated elec-
description of the full LLL spectra (including both yrast
c trons in the lowest Landau level (LLL) developed into
[36] and all excited states), however, is still missing. In
[ a major branch of theoretical condensed matter physics
this paper, a universal theory for the LLL spectra of a
[2–19]. Earlyon,it wasrealizedthat the essentialmany-
1 finite number of particles valid for both statistics (i.e.,
v body physicsinthe LLLcouldbe capturedthroughtrial
for both bosons and fermions) is introduced. The LLL
0 wave functions. Prominent examples are the Jastrow-
spectraareshowntobeassociatedwithfullyquantal [37]
9 type Laughlin (JL) [2], composite fermion (CF) [6], and
0 MooreandRead’s(MR)[7]Pfaffianfunctions,represent- and strongly correlated ro-vibrational molecular (RVM)
1 states, i.e., with (analytic) trial functions describing vi-
ing quantum-liquid states [2]. In the last ten years, the
. brational excitations relative to the set of the special
1 fieldofsemiconductorquantumdots [15]helped tofocus
yrast states known as cusp states.
0 attention on finite systems with a small number (N) of
0 electrons. Theoretical investigations of such finite sys- The cusp states exhibit enhanced stability and magic
1
tems led to the introduction of “crystalline”-type LLL angular momenta (see below), and as such they have at-
:
v trial functions referred to as rotating electron molecules tracted considerable attention. However,the cusp states
i [12, 15] (REMs). In particular in their intrinsic frame representonlyasmallfractionoftheLLLspectrum. The
X
of reference, the REMs describe electrons localized at moleculartrialfunctionsassociatedwiththemarepurely
r
a the apexes of concentric polygonal-ring configurations rotational(i.e.,vibrationless)andwereintroducedforthe
(n ,n ,...,n ), where r n = N and r is the num- case of electrons in Ref. [12] under the name rotating
1 2 r q=1 q
ber of concentric rings. electron molecules (REMs). The corresponding purely
Morerecently,theemPergingfieldofgraphenequantum rotationalbosonic analytic trialfunctions for cuspstates
dots [20, 21], and the burgeoning field of rapidly rotat- [calledrotatingbosonmolecules(RBMs)] areintroduced
ing trapped ultracold neutral gases [22–34] have gener- inthispaper;seeSectionIIA. Moreimportantly,thispa-
atedsignificantinterestpertainingto stronglycorrelated per shows that the quantal molecular description can be
states in the lowest Landau level. Furthermore, it is an- extendedtoallotherLLLstates(beyondthespecialcusp
states)by introducing (see Section IID) analytic expres-
sions for trial functions representing ro-vibrational exci-
tations of both REMs and RBMs. These ro-vibrational
∗Electronicaddress: [email protected] trial functions include the REM or RBM expressions as
†Electronicaddress: [email protected] a special case, and they will be referred to in general as
2
RVM trial functions.
1.5
It is remarkable that the numerical results of the
present theory were found in all tested cases to be
amenable(if sodesired)to anagreementwithin machine
precision with exact-diagonalization (EXD) results, in-
cluding energies, wave functions, and overlaps. This nu- N=3 bosons
merical behavior points toward a deeper mathematical
finding,i.e.,thattheRVMtrialfunctionsforbothstatis-
1
tics provide a complete and correlated basis (see below)
that spans the translationally invariant (TI) subspace y
g
[5] of the LLL spectrum. An uncorrelated basis, with-
r
out physical meaning, built out of products of elemen- e
n
tary symmetric polynomials is also known to span the
E
(bosonic) TI subspace [38].
For the sake of clarity, we comment here on the use of
0.5
the terms “correlated functions” and/or “correlated ba-
sis.” Indeed, the exact many-body eigenstates are cus-
tomarilycalledcorrelatedwheninteractionsplayadomi-
nantrole. Consequently,abasisiscalledcorrelatedwhen
its members incorporate/anticipate effects of the strong
two-body interaction a priori (before the explicit use of
thetwo-bodyinteractioninanexactdiagonalization). In
this respect, Jastrow-type basis wavefunctions (e.g., the 0
Feenberg-Clarkmethodofcorrelated-basisfunctions[39– 0 1 2 3 4 5 7
6
41] and/or the composite-fermion basis [18, 19, 42, 43])
are described as correlated, since the Jastrow factors in- L
corporate the effect of a strong two-body repulsion in
keeping the interacting particles apart on the average.
OurRVMbasisisreferredtoascorrelatedsince,inaddi- FIG. 1: (Color online) LLL spectra for N = 3 scalar bosons
calculatedusingexactdiagonalization. OnlytheHamiltonian
tion to keeping the interacting particles away from each
term containing the two-body repulsive contact interaction,
other, the RVM functions incorporate the strong-two-
gδ(zi −zj), [see Eq. (3)] was considered in the exact diag-
body-repulsioneffectofparticlelocalizationinconcentric
onalization. The gray solid dots (marked by arrows; green
polygonal rings and formation of Wigner molecules; this
online) denote the translationally invariant states. The dark
localization effect has been repeatedly demonstrated via solid dots are thespurious states (see text). The dashed line
EXDcalculationsinthepastdecade(see,e.g.,thereview denotes the yrast band, while the cusp states are marked by
in Ref. [15] and references therein). In this spirit, we de- acircle. Energiesinunitsofg/(πΛ2). Thenumberoftransla-
scribe the basis of elementary symmetric polynomials as tionallyinvariantstatesismuchsmallerthanthetotalnumber
“uncorrelated,” since the elementary symmetric polyno- of LLL states.
mials do not incorporate/anticipate this dominant effect
of a strong two-body repulsion, i.e., that of keeping the
interacting particles apart.
TI subspace is of importance in the following two ways:
We are unawareof any other strongly-correlatedfunc- (1) From a practical (and calculational) viewpoint, one
tions which span the TI subspace. Indeed, although canperformcontrolledandsystematicstepwiseimprove-
the Jastrow-Laughlinfunction (used for describing yrast mentsoftheoriginalstrongly-correlatedvariationalwave
states) is translationally invariant, its quasi-hole and function, e.g., the pure REM or RBM. (For detailed il-
quasi-electronexcitationsarenot[5]. Similarly,thecom- lustrative examples of the rapid-convergence properties
pactcomposite-fermiontrialfunctionsaretranslationally of the RVM basis, see the Appendix.) This calculational
invariant[30],buttheCFexcitationswhichareneededto viewpoint was also the motivation behind the introduc-
completetheCFbasisarenot[18,19,42,44]. Theshort- tionofother correlatedbasesin many-bodyphysics; see,
coming of the above well known correlated LLL theo- e.g., the treatment of quantum liquids and nuclear mat-
riestosatisfyfundamentalsymmetriesofthemany-body terinRefs.[39–41]andthecomposite-fermioncorrelated
Hamiltonianrepresentsanunsatisfactorystateofaffairs, basis in Refs. [18, 19, 42, 43]. (2) Conceptually, it guar-
and the present paper provides a remedy to this effect. antees that the properties of the RVM functions, and
In this context, we note that although the Moore-Read in particular the molecular point-group symmetries, are
functions [7] are also translationally invariant, they ad- irrevocably incorporated in the properties of the exact
dress only certain specific LLL states and they do not LLL wave functions. Furthermore, it follows that all
form a basis spanning the TI subspace. other translationally invariant trial functions (e.g., the
Our introduction of a correlated basis that spans the JL,compactCF,orMoore-Readfunctions),arereducible
3
2.3 a givenmagnetic field B. This specialformtakes advan-
tage of the simplifications at the limit of large B, i.e.,
when the relevant Hilbert space can be restricted to the
lowest Landau level, given that ¯hω << ¯hω /2; the fre-
0 c
quency ω specifies the external harmonic confinement
0
and ω =eB/(mc) is the cyclotron frequency. Then the
2.1 c
many-body hamiltonian reduces to
He,global =
LLL
y ¯hω N e2
g N c +¯h( ω2+ω2/4 ω /2)L+ , (1)
er 1.9 2 0 c − c κzi zj
n q Xi<j | − |
E
where L = N l is the total angular momentum and
i=1 i
z =x+iy.
P
InthecaseofN rapidlyrotatingbosons(withω spec-
0
1.7 ifying the external confinement of the two-dimensional
harmonictrapandΩ denotingtherotationalfrequency),
the corresponding Hamiltonian [24, 25, 29, 45] (in the
N=4 fermions
limit Ω/ω 1) is written as [46]
0
→
Hb,global =
1.5 LLL
N
6 8 10 12 14
N¯hω +¯h(ω Ω)L+ gδ(z z ). (2)
0 0 i j
− −
L Xi<j
Since we will consider many-body energy eigenstates
FIG. 2: (Color online) LLL spectra for N =4 spin-polarized that are eigenstates of the total angular momentum as
electrons calculated using exact diagonalization. Only the well, it follows that only the interaction terms are non-
Hamiltonian term containingthetwo-bodyCoulomb interac- trivial in both Hamiltonians (1) and (2). As a result,
tion, e2/(κ|zi −zj|), [see Eq. (3)] was considered in the ex- wewill henceforthfollowthe practice offocussingonthe
act diagonalization. The gray solid dots (marked by arrows; simpler interaction-only LLL Hamiltonian
greenonline)denotethetranslationally invariantstates. The
darksoliddotsarethespuriousstates(seetext). Thedashed N
linedenotestheyrastband,whilethecuspstatesaremarked H = v(z z ), (3)
LLL i j
by a circle. Energies in units of e2/(κlB). The number of i<j −
translationallyinvariantstatesismuchsmallerthanthetotal X
numberof LLL states. where v(z z ) denotes the two-body interaction
i j
−
(Coulomb for electrons and repulsive contact potential
for bosons). The “ground states” of the Hamiltonian in
toadescriptionintermsofanexcitedrotating/vibrating
Eq. (3) coincide with the “yrast” band [36].
quantalmolecule. Specificexamplesofthereducibilityof
We remind the reader that the EXD method is based
the JL and Moore-Read states to the molecular descrip-
on the fact that the full LLL Hilbert space at a given
tion introduced in this paper are provided in Sections
total angular momentum L is spanned by the set of all
IIIC and IIID. This is a surprising result, since these
possible uncorrelated permanents (for bosons) or Slater
Jastrowbasedtrialfunctions arewidely described in the
determinants (for electrons) made out from the Darwin-
previousliteratureasbeingliquid-likeinanessentialway.
Fock zero-node single-particle levels (referred to also as
orbitals)
B. Characteristic properties of the zli
ψ (z)= exp( zz∗/2), (4)
lowest-Landau-level spectra li √πl ! −
i
For completeness and clarity in the presentation, we with li 0. The position variable z is given in units of
≥
briefly provide in this section a graphical illustration of Λ = ¯h/(mω0) in the case of a rotating harmonic trap
some of the main characteristicsof the LLL spectra,cal- (withplateral confinement frequency ω0) or lB√2 in the
culated via the exact-diagonalizationapproach. caseofanappliedmagneticfieldB,withl = ¯h/(mω )
B c
First we describe here the special form [11, 15, 30] beingthemagneticlengthandω thecyclotronfrequency
c
p
of the many-body Hamiltonian used for calculating the [46]. (For details concerning the EXD method, see, e.g.,
global ground state of a finite number N of electrons at Refs. [15, 29].) In the following, we use the convention
4
thatanuncorrelatedstateisdescribedbyasingleperma- for any arbitrary constant complex number c.
nent (or Slater determinant) made out from the orbitals The LLL states belonging to the TI subspaces are de-
inEq.(4),whicharecharacterizedbygoodsingle-particle noted by gray solid dots (marked by an arrow; green
angular momenta l . online) in Figs. 1 and 2. The dimension DTI(L) of the
i
A small part (sufficient for our purposes here) of the translationalinvariantsubspaceismuchsmallerthanthe
EXD LLL spectra (as a function of L) are plotted in dimension DEXD(L) of the exact-diagonalization(EXD)
Fig. 1 for N = 3 scalar bosons and in Fig. 2 for N = 4 [15]space(whichisspannedbyuncorrelatedpermanents
spin-polarized electrons. As is usually done in the LLL, or Slater determinants as discussedabove). The remain-
the one-body terms of the Hamiltonian (i.e., confining ingDEXD(L) DTI(L)statesarespurious center-of-mass
−
potential and kinetic-energy) were omitted [11, 15, 30], excitations, generated by multiplying the TI states with
andtheexactdiagonalizationinvolvedonlythetwo-body the operator zm, m=0,1,2,..., where
c
interaction [see Eq. (3)].
In Figs. 1 and 2, the yrast bands [36] are denoted by 1 N
z = z , (8)
a dashed line. Along the yrast bands there appear spe- c i
N
cial cusp states denoted by a circle. The cusp states are Xi=1
important because they exhibit enhanced stability when
is the coordinate of the center of mass [49].
theone-bodytermsoftheHamiltonian(i.e.,externalcon-
The energies of these spurious states coincide with
finementandkinetic energy)areadded[seeEqs. (1)and
thoseappearingatalltheothersmallerangularmomenta
(2)],andthustheydetermine[47]theglobalgroundstates
[5]. Thus (see TABLES I, II, and IV)
[11, 15, 24, 29, 48] as a function of the applied magnetic
field B or the rotationalfrequency Ω of the trap(for the DTI(L)=DEXD(L) DEXD(L 1). (9)
− −
correspondence between B and Ω, see Ref. [46]). In all
studied cases [11, 15, 24, 28, 29, 31, 48] (including both We further note that for N particles (bosons or
electronsandbosonsuptoN =9particles),thetotalan- fermions)
gular momenta of the globalgroundstates belong to the
DTI(L)=DTI L+N(N 1)/2 , (10)
set of magic angular momenta given by Eq. (30) below. b f −
We note that the emergence of these magic angular mo-
where the subscripts b a(cid:0)(cid:0)(cid:0)nd f stand for(cid:1)(cid:1)(cid:1)bosons and
menta are a direct signature of the molecular nature of
fermions (electrons), respectively. N(N 1)/2 is the
the cuspstates,afactthatfurthermotivatesourpresent −
smallest value of the total angular momentum for spin
investigationsconcerningthemoleculardescriptionofthe
polarized fermions in the LLL.
full LLL spectra beyond the electronic [12, 13, 15] cusp
states.
In the LLL, all many-body wave functions have the
C. Plan of the paper
general form[5, 25, 30]
W(z ,z ,...,z )0 , (5) The paper is organized as follows:
1 2 N
| i
The analytic trial functions associated with pure ro-
where W(z ,z ,...,z ) is an homogeneous polynomial
1 2 N tations of bosonic molecules (i.e., the RBMs) are intro-
of degree L (being antisymmetric for fermions and sym-
duced in Section IIA, followed by a description of the
metric for bosons).
purely rotationalelectronic molecularfunctions (i.e., the
InEq.(5),thesymbol 0 standsforaproductofGaus-
| i REMs) in Section IIB.
sians [see Eq. (4)], i.e.,
Properties of the RBMs and REMs are discussed in
N Section IIC.
0 =exp( z z∗/2). (6) Section IID introduces the generalro-vibrationaltrial
| i − i i
functions (i.e., the RVMs).
i=1
X
Case studies of the quantal molecular description of
To simplify the notation, this common factor will be
the LLL spectra are presented in Section III. In par-
omitted henceforth in the algebraic expressions and ma-
ticular, Section IIIA discusses the case of N = 3 LLL
nipulations [except in Eqs. (18) and (33)], where it is
scalar bosons, while Section IIIB discusses the case of
repeated for clarity]. Its contribution, however,is neces-
N = 3 spin-polarized LLL electrons. The case of N = 4
sary when numerical results are calculated.
LLL electrons is presented in Section IIIC, along with
A central property of the LLL spectra is the existence
an analysis of the Jastrow-Laughlin state (for fractional
ofatranslationallyinvariant(TI)subspace[5,25,30]for
filling ν =1/3)from the viewpoint of the present molec-
a given L. This subspace is associated with a special
ular theory. Section IIID investigates the case of N =5
subset ofthe generalwavefunctions in Eq.(5), i.e., with
LLL bosons, along with an analysis of the Moore-Read
wave functions having translationally invariant polyno-
state according to the molecular picture.
mials W(z ,z ,...,z ). Specifically, the TI polynomials
1 2 N Section IV offers a summary and discussion.
obey the relationship
Finally, the Appendix discusses the rapid-convergence
W(z +c,z +c,...,z +c)=W(z ,z ,...,z ), (7) properties of the RVM basis.
1 2 N 1 2 N
5
We note that, going from N = 3 to N = 5 particles, IntheLLL,onecanspecificallyconsiderthelimitwhen
the molecular description requires consideration of suc- theconfiningpotentialcanbeneglectedcomparedtothe
cessivelylargernumbersofisomericmolecularstructures effect induced by the gauge field. The localized u(z,Z)
as elaborated in Section III. In particular, for N = 3 single-particlestates(referredtoalsoasorbitals)arethen
onlyonemolecularisomerisneeded,whilethreedifferent takento be displaced zero-nodeDarwin-Fockstates with
molecular isomers are needed for N = 5. It is remark- appropriate Peierls phases due to the presence of a per-
able that these isomers are independent of the statistics pendicular magnetic field [see Eq. (1) in Ref. 12], or due
(bosons or fermions). to the rotation of the trap with angular frequency Ω.
Then, assuming a symmetric gauge, the orbitals can be
represented [12, 13, 53] by displaced Gaussian analytic
II. MOLECULAR TRIAL FUNCTIONS functions, centered at different positions Z X +Y
j j j
≡
accordingto the equilibrium configurationof N classical
The molecular trial functions introduced in this paper point charges[54, 55] arranged at the vertices of nested
are derived with the help of a first-principles methodol- regularpolygons(eachGaussianrepresentinga localized
ogyof hierarchicalsuccessiveapproximationswhichcon- particle). Such displaced Gaussians are written as
vergetotheexactsolutionofthemany-bodySchr¨odinger
equation [15]. Specifically, this methodology is based on u(z , Zj)=(1/√π)
the theory of symmetry breaking at the mean-field level exp[ z Z 2/2]exp[ i(xY yX )], (14)
j j j
× −| − | − −
and of subsequent symmetry restoration via projection
techniques [15]. In this Section, we present (and/or re- where the phase factor is due to the gauge invariance.
view where appropriate) this derivation in some detail. z x+iy,andalllengthsareinunits ofΛinthe caseof
≡
arotatingtraporl √2inthecaseofanappliedmagnetic
B
field; see Section IB.
A. Purely rotational bosonic trial functions The localized orbital u(z,Z) can be expanded in a
(RBMs) series over the complete set of zero-node single-particle
wave functions in Eq. (4). One gets [see Appendix A in
RBM analytical wave functions in the LLL for N Ref. [56]]
bosons in two-dimensional rotating traps can be derived
∞
followingearlieranalogousderivationsforthecaseofelec-
u(z,Z)= C (Z)ψ (z), (15)
trons [12]. Our approach consists of two steps: l l
(I)Atthefirststep,oneconstructsapermanent(Slater Xl=0
determinant for fermions) ΨN(z ,...,z ) out of dis-
1 N with
placed single-particle states u(z ,Z ), j = 1,...,N that
j j
represent scalar bosons localized at the positions Zj, Cl(Z)=(Z∗)lexp( ZZ∗/2)/√l! (16)
with (omitting the particle indices) z = x+ iy = and −
Z =X +iY =Reiφ. for Z =0. Naturally, C (0)=1 and C (0)=0.
0 l>0
6
For an N-particle system, the bosons are situated at
ΨN[z]=perm( N[z]), (11)
the apexes of r concentric regular polygons. The ensu-
M
ingmulti-ringstructureisdenotedby(n ,n ,...,n )with
with the matrix MN[z] being rq=1nq = N. The position of the j-th1ele2ctronron the
q-th ring is given by
u z ,Z ) ... u z ,Z )
( 1 1 ( N 1 P
MN[z]= ... ... ... . (12) Zjq =Zqexp[i2π(1−j)/nq], 1≤j ≤nq. (17)
u(z ,Z ) ... u(z ,Z )
1 N N N
 
  The singleepermanent ΨN[z] represents a static boson
For the permanent of a matrix, we follow here the def- molecule. Using Eq. (15), one finds the following expan-
inition in Ref. [51], that is, the permanent is an analog sion (within a proportionality constant):
of a determinant where all the signs in the expansion by
minors are taken as positive. This definition provides ∞ C (Z )C (Z ) C (Z )
an unnormalized expressionfor the permanent. If a nor- ΨN[z]= l1 1 l2 2 ··· lN N
√l !l ! l !
malized expressionis needed, one has to multiply with a l1=0X,...,lN=0 1 2 ··· N
normalization constant P(l ,l ,...,l )0 , (18)
1 2 N
× | i
1
= , (13) where P(l ,l ,...,l ) perm[zl1,zl2,...,zlN]; the el-
N √N!p !p !...p ! 1 2 N ≡ 1 2 N
1 2 M ements of the permanent are the functions zlj , with
i
where p ,p ,...,p denote the occupations (multi- zl1,zl2,...,zlN being the diagonal elements. The Z ’s
plicities{) o1f t2he orbiMta}ls, assuming that there are M dis- (w1ith21 kN N) in Eq. (18) are the Zq’s of Eq. (17k),
≤ ≤ j
tinct orbitals in a given permanent (M N) [52]. but relabeled.
≤
6
InEq.(18),thecommonfactor 0 representstheprod- as (within a proportionality constant)
| i
uct of Gaussians defined in Eq. (6). To simplify the no-
tation, this common factor is usually omitted. l1+···+lN=LP(l ,...,l )
(II)Secondstep: Inthefollowing,wewillcontinuewith ΦN[z]= 1 N ei(φ1l1+···+φNlN), (21)
L l !...l !
thedetailsofthecompletederivationinthesimplestcase l1X,...,lN 1 N
ofa single (0,N)ring. Thus we considerthe specialcase
with φ =2π(j 1)/N.
j
Zj =Re2πi(1−j)/N, 1 j N, (19) We further ob−serve that it is advantageous to rewrite
≤ ≤
Eq.(21) by restricting the summation to the orderedar-
where R is the radius of the single ring.
rangements l l ... l , in which case we get
The Slater permanent ΨN[z] breaks the rotational 1 ≤ 2 ≤ ≤ N
symmetry and thus it is not an eigenstate of the total
angular momentum ¯hLˆ = h¯ Nj=1ˆlj. However, one can ΦNL[z] = l1+l2+···+lN=LP(ll1!,......l,lN! )
restore[12,15]therotationalsymmetrybyapplyingonto 1 N
P 0≤l1≤Xl2≤···≤lN
ΨN[z] the projection operator
perm[eiφ1l1,eiφ2l2,...,eiφNlN]
. (22)
1 2π × p1!p2!...pM!
dγeiγ(Lˆ−L), (20)
L
P ≡ 2π
Z0 The second permanent in Eq. (22) can be shown [57] to
be equal (within a proportionality constant) to a sum of
where ¯hL are the eigenvalues of the total angular mo-
cosine terms times a phase factor
mentum.
When applied onto ΨN[z], the projection operator
L
acts as a Kronecker delta: from the unrestricted sumPin eiπ(N−1)L/N, (23)
Eq.(18)itpicksuponlythosetermshavingagiventotal
angular momentum L (henceforth we drop the constant which is independent of the individual lj’s.
prefactorh¯ whenreferringtoangularmomenta). Asare- The final result for the (0,N) RBM wave function is
sult the projected wave function ΦN = ΨN is written (within a proportionality constant):
L PL
l1+l2+...+lN=L
ΦRBM(0,N)[z]= C (l ,l ,...,l )perm[zl1,zl2,...,zlN], (24)
L b 1 2 N 1 2 N
0≤l1≤Xl2≤...≤lN
where the coefficients are given by different expressions (i) For even N, one has
for even or odd number of bosons N.
−1 −1
N M
C (l ,l ,...,l )= l ! p !
b 1 2 N i k
! !
i=1 k=1
Y Y
π
cos [(N 1)l +(N 3)l +...+l l ... (N 3)l (N 1)l ] , (25)
× − σ1 − σ2 σ(N/2) − σ(N/2+1) − − − σ(N−1) − − σN N
Xσ n o
where the summation runs overthe permutations of (ii) With the notation K = N 1, the corresponding
σ −
the set of N indices 1,2,...,N . coefficients for odd N are:
{P }
7
−1 −1
N M
C (l ,l ,...,l )= l ! p !
b 1 2 N i k
! !
i=1 k=1
Y Y
π
cos [Kl +(K 2)l +...+2l 2l ... (K 2)l Kl ] , (26)
× σ1 − σ2 σ(K/2) − σ(K/2+1) − − − σ(K−1) − σK N
σX{K} n o
where runsoverallpermutationsofN 1indices one has M =4 and p =p =p =p =1.
σ{K} − 1 2 3 4
selected out from the set 1,2,...,N (of N indices).
P { } We further note that for both Eqs. (25) and (26) the
In both Eqs. (25) and (26), the index M (with 1
≤ total number of distinct cosine terms is N!/2 [58], with
M N denotes the number of different single-particle
≤ thedivisionby2followingfromthesymmetryproperties
angularmomental ’s(j =1,2,...,M)intheorderedlist
j of cosine, i.e., from cos( x)=cos(x).
l ,l ,...,l and the p ’s are the multiplicities of each −
1 2 N k
{ }
oneofthedifferentl ’s[occupationsofthecorresponding TheRBMexpresionforan(n ,n )two-ringconfigura-
j 1 2
single-particle orbitals ψ (z)]. For example, for N = 4 tion(with N =n +n ) canbe derivedfollowingsimilar
lj 1 2
and l = 2,l = 2,l = 2,l = 5 , one has M = 2 and stepsasinthederivationoftheexpressionsforthemulti-
1 2 3 4
{ }
p = 3, p = 1; for l = 0,l = 0,l = 0,l = 0 , one ring REMs [12]. If L +L =L, the final two-ringRBM
1 2 1 2 3 4 1 2
{ }
hasM =1andp =4;for l =1,l =2,l =3,l =9 , expression is
1 1 2 3 4
{ }
ΦRBM(n ,n )[z] =
L 1 2
l1+l2+...+ln1=L1ln1+1+ln1+2+...+lN=L2
C (l ,l ,...,l )C (l ,l ,...,l )perm[zl1,zl2,...,zlN],
b 1 2 n1 b n1+1 n1+2 N 1 2 N
0≤l1≤lX2≤...≤ln1 0≤ln1+1≤Xln2+2≤...≤lN
(27)
where C (l ,l ,...,l ) and C (l ,l ,...,l ) are Refs. [12, 13] following similar steps as those in Section
b 1 2 n1 b n1+1 n1+2 N
calculatedbyapplyingthesingle-ringexpressionsofEqs. IIA. A determinant needs to be used, however, instead
(25) and (26). ofa permanent,to conformwith the antisymmetrization
Generalizations of expression (27) to structures with properties of the electronic (fermionic) many-body wave
a larger number r of concentric rings involve for each function. IfL +L =L,thefinaltwo-ringREMexpres-
1 2
q-th ring (1 q r): (I) Consideration of a separate sion is
≤ ≤
factor C (l ,l ,...,l ); (II) A restric-
b nq−1+1 nq−1+2 nq−1+nq
tion on the summation of the associated n angular mo-
q
menta, i.e., l +l +...+l =L , with
nq−1+1 nq−1+2 nq−1+nq q
r L =L.
q=1 q
P
B. Purely rotational fermionic trial functions
(REMs)
TheREMexpresionforany(n ,n ,...,n )multi-ring
1 2 r
configuration(with N = r n ) was derivedearlierin
q=1 q
P
8
ΦREM(n ,n )[z] =
L 1 2
l1+l2+...+ln1=L1,ln1+1+ln1+2+...+lN=L2
C (l ,l ,...,l )C (l ,l ,...,l )det[zl1,zl2,...,zlN],
f 1 2 n1 f n1+1 n1+2 N 1 2 N
0≤l1<l2<...<lXn1<ln1+1<...<lN
(28)
where the fermionic coefficients C (l ,l ,...,l ) and configurations. Indeed under condition (32) the C (...)
f 1 2 n1 b
C (l ,l ,...,l ) are calculated by applying to andC (...)coefficientsareidenticallyzero,ascanbeeas-
f n1+1 n1+2 N f
each one of them the single-ring [(0,N)] expression ilycheckedusingMATHEMATICA[57]. Inotherwords,
purely rotational states are allowed only for certain an-
Cf(l1,l2,...,lN) = gular momenta that do not conflict with the intrinsic
−1 molecular point-group symmetries.
N
π
l ! sin (l l ) . (29) The yraststates correspondingto magic ’s [Eq.(30)]
i !  N i− j  areassociatedwiththe specialcuspstatesdLescribedpre-
iY=1 1≤iY<j≤N h i
viouslyinFigs.1and2andinSectionIB. Furthermore,
 
It is straighforward to generalize the two-ring REM the enhanced stability associated with the cusp states
expression in Eq. (28) to more complicated or simpler (see Section IB) is obviously due to the selection rule
[i.e., (0,N) and (1,N 1)] configurations by (I) consid- described by Eqs. (31) and (32).
−
eringa separatefactorC (l ,l ,...,l ) An important property of the REM and RBM trial
f nq−1+1 nq−1+2 nq−1+nq
for each qth ring; (II) restricting the summation of the functions is their translational invariance (in the sense
associatedn angular momenta, i.e., l +l + described in Section IB).
q nq−1+1 nq−1+2
...+l =L , with r L =L.
nq−1+nq q q=1 q
Apart from the permanent being replaced by a deter-
minant, we note two otherPmain differences between the D. General ro-vibrational trial functions (RVMs)
REM and RBM expressions. That is, for electrons (1)
M = N in all instances (single occupancy) and (2) a The RVM functions that account for the general exci-
product of sine terms in the coefficients C (...) replaces tationsofthe rotatingmoleculeshavethe form(within a
f
the sum of cosine terms in the coefficients C (...). normalization constant):
b
ΦRXM(n ,n ,...,n )Qm 0 , (33)
L 1 2 r λ| i
C. Properties of RBMs and REMs
The index RXM stands for either REM, i.e., a ro-
tating electron molecule, or RBM, i.e., a rotating
The analytic expressions for ΦRXM(n ,n ,...,n )[z]
L 1 2 r boson molecule. The purely rotational functions
(the index RXM standing for either RBM or REM) de-
ΦRXM(n ,n ,...,n ) have been described in detail in
scribepuremolecularrotationsassociatedwithmagican- L 1 2 r
earlier sections. The product in Eq. (33) combines ro-
gular momenta
tations with vibrational excitations, the latter being de-
r noted by Qm, with λ being an angular momentum; the
λ
L=L0+ nqkq, (30) superscript denotes raising to a power m. Both ΦRLXM
q=1 and Qm are homogeneous polynomials of the complex
X λ
particle coordinates z ,z ,...,z , of order and λm,
1 2 N
with k , q = 1,...,r being nonnegative integers; = L
q L0 respectively. The total angularmomentum L= +λm.
N(ANc−en1tr)a/l2pfroorpeelretcytroofntshaesnedtLri0al=fu0ncfotironbsosisontsh.atiden- Qmλ isalwayssymmetricinthesevariables;ΦRLXMLisanti-
symmetric (symmetric) for fermions (bosons). 0 is the
tically | i
product of Gaussians defined in Eq. (6); this product of
Gaussians is usually omitted.
ΦRXM(n ,n ,...,n )[z]=0 (31)
L 1 2 r The vibrational excitations Q are given by the same
λ
expressionforboth bosonsandelectrons,namely, by the
for both bosons and electrons when
symmetric polynomials:
r
= 0+ nqkq (32) N
L6 L
Xq=1 Qλ = (zi−zc)λ, (34)
i=1
This selection rule follows directly from the point group X
symmetries of the (n ,n ,...,n ) multi-ring polygonal where z is the coordinate of the center of mass defined
1 2 r c
9
TABLE I: LLL spectra of three spinless bosons interacting via a repulsive contact interaction gδ(zi − zj). 2nd column:
DimensionsoftheEXDandthenonspuriousTI(inparenthesis)spaces(theEXDspaceisspannedbyuncorrelatedpermanents
ofDarwin-Fockorbitals). 4thto6thcolumns: Matrixelements[inunitsofg/(πΛ2),Λ=p¯h/(mω0)]ofthecontactinteraction
between the correlated RVM states {k,m} [see Eq. (36)]. The total angular momentum L = 3k+2m. Last three columns:
EnergyeigenvaluesfromtheRVMdiagonalization oftheassociatedmatrixofdimensionDTI(L). Thereisnononspuriousstate
withL=1. ThefullEXDspectrumatagivenLisconstructedbyincluding,inadditiontothelistedTIeigenvalues[DTI(L)in
number],all theenergies associated with angular momentasmaller than L. An integerin squarebracketsindicates theenergy
orderinginthefullEXDspectrum(includingbothspuriousandTIstates). Sevendecimal digitsaredisplayed,buttheenergy
eigenvalues from theRVM diagonalization agree with thecorresponding EXDTI ones within machine precision.
L DEXD(DTI) {k,m} Matrix elements Energy eigenvalues (RVMdiag. or EXDTI)
0 1(1) {0,0} 1.5000000 1.5000000[1]
2 2(1) {0,1} 0.7500000 0.7500000[1]
3 3(1) {1,0} 0.3750000 0.3750000[1]
4 4(1) {0,2} 0.5625000 0.5625000[2]
5 5(1) {1,1} 0.4687500 0.4687500[2]
6 7(2) {2,0} 0.0468750 0.1482318
{0,3} 0.1482318 0.4687500 0.0000000[1] 0.5156250[4]
7 8(1) {1,2} 0.4921875 0.4921875[4]
8 10(2) {2,1} 0.0937500 0.1960922
{0,4} 0.1960922 0.4101562 0.0000000 0.5039062[6]
12 19(3) {4,0} 7.3242187×10−4 1.0863572×10−2 1.5742811×10−2
{2,3} 1.0863572×10−2 0.1611328 0.2335036
{0,6} 1.5742811×10−2 0.2335036 0.3383789 0.0000000 0.0000000 0.5002441[13]
in Eq. (8) and λ > 1 is an integer positive number. Vi- functions
brational excitations of a similar form, i.e.,
ΦRBM(0,3)Qm k,m , (36)
3k 2 ⇒{ }
N
withk,m=0,1,2,...,andL=3k+2m;thesestatesare
Q˜ = zλ (35)
λ i always orthogonal.
i=1
X FollowingEq.(24), asimplifiedanalyticexpressionfor
the (0,3) RBM can be derived, i.e.,
(and certain other variants), have been used earlier to
approximate part of the LLL spectra. Such earlier en-
l1+l2+l3=L
deavors provided valuable insights, but overall they re- ΦRBM(0,3)= C (l ,l ,l )Perm[zl1,zl2,zl3],
L b 1 2 3 1 2 3
mained inconclusive; for electrons over the maximum
density droplet [with magic = 0], see Refs. [8] and 0≤l1X≤l2≤l3 (37)
L L
[9]; for electrons over the ν = 1/3 (ν = L0/L) Jastrow- where the coefficients Cb(...) are given by:
Laughlin trial function [with magic = 3 ], see Ref.
0
[10]; and for bosons in the range 0 LL LN, see Refs. 3 −1 M −1
≤ ≤
[22, 25, 26]. C (l ,l ,l ) = l ! p !
b 1 2 3 i k
TheadvantageofQ [50](comparedtoQ˜ )isthatitis i=1 ! k=1 !
λ λ Y Y
translationally invariant (TI) [5, 25], a property shared
2π(l l )
with both ΦRBM and ΦREM. In the following, we will cos i− j , (38)
L L ×  3 
discuss illustrative cases, which will show that the RVM 1≤i<j≤3 (cid:20) (cid:21)
X
functionsofEq.(33)provideacorrelatedbasis(RVMba-  
sis)thatspanstheTIsubspace[5,25,30]ofnonspurious where1 M 3denotes the number ofdifferentsingle-
≤ ≤
states in the LLL spectra. particle angular momenta in the triad (l1,l2,l3) and the
p ’s are the multiplicities of each one of these different
k
angular momenta.
TABLE I provides the systematics of the molecular
III. MOLECULAR DESCRIPTION OF LLL
description for the beginning (0 L 12) of the LLL
SPECTRA ≤ ≤
spectrum. There are severalcases when the TI subspace
hasdimensiononeandtheexactsolutionΦexactcoincides
A. Three spinless bosons with a single k,m state. For L= 0 the exact solution
coincides with{ΦRB}M =1 (Q0 =1); this is the only case
0 λ
Onlythe(0,3)molecularconfigurationandthedipolar whenanLLLstate hasa Gross-Pitaevskiiform,i.e., itis
λ = 2 vibrations are at play (as checked numerically), a single (normalized) permanent [see Eq. (33)] given by
i.e.,thefullTIspectraatanyLarespannedbythewave 0 as defined in Eq. (6).
| i
10
For L = 2, we found Φexact Q , with the index [i] N=4e L=18
[1] ∝ 2
indicating the energy orderingin the full EXD spectrum
[1] [2] [4]
(including both spurious and TI states). Since [see Eq.
TI
(34)] D -
X 6 6
E 4 4 4
Q2 ∝(z1−zc)(z2−zc)+(z1−zc)(z3−zc)+(z2−zc)(z3−(z3c9)), -6 -4 -2 0 2 4 6-6-4-2 0 2 -6 -4 -2 0 2 4 6 -6-4-2 0 2 -6 -4 -2 0 2 4 6-6-4-2 0 2
this result agrees with the findings of Refs. [25, 59] con-
cerning groundstates ofbosons in the range0 L N. 1)] |1> |5> |2>
For L = 3, one finds Φexact ΦRBM. Since≤[see≤Eq. q. (
[1] ∝ 3 E
(37)] s [ 4 4 6 4
ΦR3BM ∝(z1−zc)(z2−zc)(z3−zc), (40) RVM -4 0 4 -4 0 -6 -4 -2 0 2 4 6-6-4-2 0 2 -6 -4 -2 0 2 4 6-6-4-2 0 2
pure (0,4) REM(cid:13) pure (1,3) REM(cid:13) (0,4) REM ( =14) +(cid:13)
thisresultagreesagainwiththefindingsofRefs.[25,59]. ( =18) ( =18) ( =2, mL=2)
L L l
For L = 5, the single nonspurious state is an excited
FIG. 3: (Color online) CPDs for N = 4 LLL electrons with
one, Φexact ΦRBMQ .
[2] ∝ 3 2 L = 18 (ν = 1/3). Top row: The three lowest-in-energy
For L=6 (ν =1/2), the ground-state is found to be EXDTI states (see TABLE II). Bottom row: The RVM trial
functions associated with the largest expansion coefficients
Φexact 160ΦRBM+ 1Q3 (underlined, see TABLE III) of these three EXDTI states in
[1] ∝ − 9 6 4 2 thecorrelated RVMbasis. Seethetext for details. The solid
= (z z )2(z z )2(z z )2, (41) dot denotes thefixed point r0. Distances in nm.
1 2 1 3 2 3
− − −
i.e.,thebosonicJastrow-Laughlinfunctionforν =1/2is
equivalenttoanRBMstatethatincorporatesvibrational Expression(42)ispreciselyofthe formΦR3kEMQm2 ,ascan
correlations. be checked after transforming back to Cartesian coordi-
For L N(N 1) (i.e., ν 1/2), the EXD yrast nates z1, z2, and z3. Thus the wave functions k,m of
energies e≥qual zero−, and with in≤creasing L the degener- Ref. [60] describe both pure molecular rotations|, as wiell
acy of the zero-energy states for a given L increases. It as vibrational excitations, and they cover the transla-
is important that this nontrivial behavior is reproduced tionally invariant LLL subspace. We note that the pairs
faithfully by the present method (see TABLE I). of indices k,m are universal and independent of the
{ }
statistics, i.e., the same for both bosons [Eq. (36)] and
electrons [Eq. (42)], as can be explicitly seen through a
comparison of TABLE I here and TABLE I in Ref. [60].
B. Three electrons
WefurthernotethatLaughlindidnotpresentmolecu-
lartrialfunctionsforelectronswithN >3,orforbosons
Although unrecognized,the solutionofthe problemof
for any N. This is done in the present paper.
three spin-polarized electrons in the LLL using molecu-
lartrialfunctionswaspresentedbyLaughlininRef.[60].
Indeed, the main result ofRef. [60][see Eq.(18)therein]
were the following wave functions (we display the poly- C. Four electrons
nomial part only)
For N = 4 spin-polarized electrons, one needs to con-
k,m sidertwodistinctmolecularconfigurations,i.e.,(0,4)and
| i ∝
(z +iz )3k (z iz )3k (1,3). Vibrationswithλ 2mustalsobeconsidered. In
a b 2−i a− b (za2+zb2)m, (42) this case the RVM states≥are not alwaysorthogonal,and
(cid:20) (cid:21) the Gram-Schmidt orthogonalizationis implemented.
where the three-particle Jacobi coordinates are Of particular interest is the L = 18 case (ν = 1/3)
which is considered [2] as the prototype of quantum-
z1+z2+z3 liquidstates. However,inthiscasewefound(seeTABLE
z = , (43)
c
3 II) that the exact TI solutions are linear superpositions
of the following seven RVM states [involving both the
(0,4) and (1,3) configurations]:
1/2
2 z +z
1 2
z = z , (44)
a 3 2 − 3 1 =ΦREM(0,4), 2 =ΦREM(0,4)Q2,
(cid:18) (cid:19) (cid:20) (cid:21) | i 18 | i 14 2
3 =ΦREM(0,4)Q4, 4 =ΦREM(0,4)Q6,
| i 10 2 | i 6 2
z = 1 (z z ). (45) |5i=ΦR18EM(1,3), |6i=ΦR12EM(1,3)Q32,
b √2 1− 2 |7i=ΦR15EM(1,3)Q3. (46)