Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: spin-independent interactions and spin-dependent external fields

0.22 MB·English

by  Vladimir A. Yurovsky

Sum rules for spin-1/2 quantum gases in states with well-defined spins:
spin-independent interactions and spin-dependent external fields.
Vladimir A. Yurovsky
School of Chemistry, Tel Aviv University, 6997801 Tel Aviv, Israel
(Dated: May 4, 2015)
Analytical expressions are derived for sums of matrix elements and their squared moduli over
5 many-body states with given total spin — the states built from spin and spatial wavefunctions
1 belonging to multidimensional irreducible representations of the symmetric group, unless the total
0 spin has the maximal allowed value. For spin-dependent one-body interactions with external fields
2
andspin-independenttwo-bodyonesbetweentheparticles, thesumdependenceonthemany-body

y statesisgivenbyuniversalfactors,whichareindependentoftheinteractiondetailsandHamiltonians
a of non-interacting particles. The sum rules are applied to perturbativeanalysis of energy spectra.
M
PACSnumbers: 67.85.Fg,67.85.Lm,02.20.-a,03.65.Fd

1

] INTRODUCTION latedtothetotalmany-bodyspin. Ifthetotalspinisless
s thanthe maximalallowedone(N/2forN particles),the
a
g wavefunctions belong to multidimensional, non-Abelian,
Calculationsofquantum-mechanicalsystemproperties
- irreducible representations of the symmetric group (see
t require matrix elements between its states. For complex
n [4–7]), beyond the conventional paradigm of symmetric-
systems, even a calculation of the matrix elements can
a antisymmetricfunctions. Thesymmetricorantisymmet-
constitute a complicated problem. However, in certain
u rictotalwavefunction—suchfunctions only areallowed
q cases, sum rules can be derived from general principles,
by the Pauli exclusion principle — is represented as a
. providing analytical expressions for sums of matrix ele-
t sum of products of the spin and spatial wavefunctions
a ments or their products. The Thomas-Reiche-Kuhn and
[see Eq. (2) in Sec. I below].
m
theBethesumruleswereobtainedatearlyyearsofquan-
In the case of non-interacting particles in spin-
- tum mechanics. These and similar rules (see [1]) are for-
d independent potentials, all states with the given set of
mulated for weighted sums of oscillator strengths, which
n spatial quantum numbers are energy-degenerateand the
are proportional to squared moduli of transition matrix
o two kinds of wavefunctions are applicable, related by a
elements,overcertainsets ofeigenstates. The ruleswere
c lineartransformation. The effect ofspin-independentin-
[ applied to radiative transitions and scattering problems.
teractions between particles was analyzed by Heitler [8]
Sum rules fordynamic structurefactors (see [2]) are em-
3 using the theory of the symmetric groupirreducible rep-
ployed to obtain information on collective behavior of
v resentations. That work demonstrates that the average
2 many-body systems. Various sum rules are also used in energyofstateswithingivenirreduciblerepresentationis
8 nuclear and solid-state physics, as well as in quantum
proportionalto acertainsumofthe representationchar-
1 field theory.
acters. The character dependence on the representation
6
0 Thepresentworkderivessumrulesformany-bodysys- lifts the degeneracy of states related to different repre-
. tems of indistinguishable spinor particles. The particles sentations, and the wavefunctions with defined individ-
1 can be composite, e.g., atoms or molecules, and the spin ual spin projections become inapplicable. It is a gener-
0
can be either a real angular momentum of the particle alization of the well-known energy splitting between the
5
1 or a formalspin, whose projectionsare attributed to the singlet and triplet states in two-electron problems.
: particle’sinternalstates(e.g. hyperfine states ofatoms). Although the derivation [8], being done at early years
iv In the latter case, the particle spin 21 means that only of quantum mechanics, did not take into account spin
X two internal states are present in the system. This for- degrees of freedom and supposed that total wavefunc-
r mal spin is not related to the real, physical, spin of the tions can have arbitrary permutation symmetry, the re-
a particles, which can be either bosons or fermions. sults remain valid for symmetric or antisymmetric to-
Many-body states of spinor particles can be described tal wavefunctions, composed from spin and spatial func-
in two ways (see [3]). In the first one, each particle is tions of arbitrary symmetry. Matrix elements of spin-
characterized by its spin projection and coordinate, and independent Hamiltonians between the latter wavefunc-
the total wavefunction is symmetrized for bosons or an- tions can be reduced to the matrix elements between
tisymmetrized for fermions over permutations of all par- spatial wavefunctions due to orthogonality of the spin
ticles [see Eq. (19) in Sec. I below]. The second ap- wavefunctions (see Sec. III). Besides justification of the
proach is based on collective spin and spatial wavefunc- Heitler results,this reductionprovidesbasisfor spin-free
tions. These wavefunctions depend on spins or coordi- quantum chemistry (see [6, 7]) — the method of cal-
nates, respectively, of all particles and form representa- culations of energies and other properties of atoms and
tions of the symmetric group (see [4–7]). In the case of molecules.
spin-1 particles, the representationis unambiguously re- Spinor quantum gases are intensively studied start-
2
2
ing from the first experimental [9, 10] and theoretical I. THE HAMILTONIAN AND
[11, 12] works (see book [2], reviews [13, 14] and ref- WAVEFUNCTIONS
erences therein). The collective spin and spatial wave-
functions wereusedin derivationofexactquantumsolu- ConsiderasystemofN particleswiththeHamiltonian
tionsforone-dimensionalhomogeneousgas[15,16]andin
analysesofselectionrulesandcorrelations[17]. SU(M)- Hˆ =Hˆ +Hˆ (1)
spat spin
symmetricgases,introducedinRefs. [18–20]andrecently
observed in Refs. [21, 22], are described in similar way being a sum of the spin-independent Hˆ and
spat
[19], where the total wavefunction is composed of spin coordinate-independent Hˆ . Each of Hˆ and Hˆ
spin spat spin
and electronic functions. is permutation-invariant.
The total wavefunction is expressed in the form
Otherformsofmany-bodywavefunctionswithdefined
total spin have been employed as well. The Lieb-Mattis Ψ(nSl) =fS−1/2 Φ(tnS)Ξ(tlS). (2)
theoremfororderingofenergylevelsinfermionicsystems t
X
has been derived in Ref. [23]. One-dimensional gas of
spin-1 fermions in arbitrarypotential has been analyzed Here spatial Φ(tnS) and spin Ξ(tlS) functions form bases of
2 irreduciblerepresentationsofthesymmetricgroup of
for hardcore zero-range interactions in Ref. [24], where N
S
N-symbolpermutations[4–7]. This means that a permu-
anexactsolutionwasderived,andforzero-rangeinterac-
tation of the particles transforms each function to a
tionsofarbitrarystrengthinRef. [25],wherequalitative
P
linear combination of functions in the same representa-
properties of energy spectra are presented. An exact so-
tion,
lution for one-dimensional hardcore Bose-Fermi mixture
was derived in Ref. [26]. Intersystem degeneracies in
Φ(S) =sgn( ) D[λ]( )Φ(S)
spin-1 Fermi gases and energy spectra for certain few- P tn P t′t P t′n
body2systems have been obtained in Ref. [27]. Symme- Xt′
triesoftrappedandinteractingbosonsandfermionsand Ξ(S) = D[λ]( )Ξ(S)
P tl t′t P t′l
qualitativebehavioroftheenergyspectraatintermediate t′
X
interaction strengths were analyzed in Refs. [28, 29].
Here the factor sgn( ) is the permutation parity for
P
fermionsandsgn( ) 1forbosons. Thisfactorprovides
P ≡
The sum of matrix elements of spin-independent in- theproperpermutationsymmetryofthe totalwavefunc-
terparticle interactions directly follows from the Heitler tion
results [8]. The present paper provides the sums of
(S) (S)
squaredmoduliofthesematrixelements,aswellassums Ψ =sgn( )Ψ (3)
P nl P nl
of matrix elements and their squared moduli for spin-
dependent external fields. Such fields can be used for The matrices of the Young orthogonal representation
transferofpopulationbetweenstateswithdifferenttotal [4–7] D[λ]( ) of the symmetric group are associated
t′t P SN
spins, as described in [17]. Besides, spin-changing ma- withthetwo-rowYoungdiagramsλ=[N/2+S,N/2 S],
−
trix elements can provide an estimate of stability of the whichareunambiguouslyrelatedtothetotalspinS. Dif-
well-defined-spin states. ferent representations, associated with the same Young
diagram, are labeled by multi-indices n and l for the
spatial and spin functions, respectively. The represen-
Section I sets the analyzed problem and provides rep-
tationbasisfunctions arelabeledbythe standardYoung
resentations of spin, spatial, and total wavefunctions for
tableaux t and t′ of the shape λ. The dimension of
separablespinandspatialdegreesoffreedomandfornon-
the representation is equal to the number of different
interacting particles. Wavefunctions with defined par-
tableaux,
ticle spin projections are discussed in this section too.
Section II contains derivation of the sum rules. Matrix N!(2S+1)
elementsofspin-dependentexternalfieldsfordifferentto- fS = . (4)
(N/2+S+1)!(N/2 S)!
tal spin projections are related using the Wigner-Eckart −
theorem. Then sums of these matrix elements and their If S =N/2, f =1, D[λ]( )=1, and the functions Φ(S)
squared moduli are calculated for the maximal allowed S t′t P tn
spinprojections. Sumrulesforspin-independentinterac- and Ξ(tlS) remain unchanged on permutations of parti-
tionsbetweenparticlesareprovidedinSec. III. Thesum cles or change their sign (Φ(S) for fermions). Otherwise,
tn
rulesareappliedtodescriptionoftheshiftsandsplittings the functions belong to multidimensional, non-Abelian
of energy levels in Sec. IV. The quantitative properties irreducible representations of the symmetric group. For
of energy spectra are provided for arbitrary number of example, the states of N =3 particles with S =1/2 are
particlesinthe regimeofweakinteractionsusingpertur- associated with the Young diagram [2,1] and there are
bation theory. Appendix contains calculation of sums, f = 2 standard Young tableaux with the Yamanouchi
S
used in Sec. II. symbols (see [4, 5]) (2,1,1) and (1,2,1).
3
TheYoungorthogonalmatricesobeytheorthogonality Therepresentationisdeterminedbythesetofthespatial
relation[6, 7] quantum numbers n and the Young tableau r, which
{ }
can take one of f values. Then the multi-index n is
S
Dt[λ′r′′](P)Dt[λr](P)= NfS!δtt′δrr′δλλ′, (5) tshpeecsifietcalnly cahroesesnupapsorse{dn}t.oAbleldqiuffaenrteunmt. nTuhmisbseirtsuantjioinn
XP { }
takesplaceinnon-degenerategases,whenprobabilitiesof
the general relation for representation matrices
multipleoccupationofspatialstatesarenegligiblysmall,
although the multiple occupation is not forbidden by it-
D[λ]( )D[λ]( )=D[λ]( ), (6)
r′t P tr Q r′r PQ self. Another example is an optical lattice in the unit-
Xt filling regime [17].
and the relation for orthogonalmatrices The functions (11) satisfy the Schr¨odinger equation
D[λ]( −1)=D[λ]( ). (7) N N
tr P rt P Hˆ (j)Φ˜(S) = ε Φ˜(S)
0 tr{n} nj tr{n}
Additional relations can be obtained for elements of j=1 j=1
X X
the first column D[λ]( ) of the Young orthogonal ma-
t[0] P Their eigenenergies are independent of r. Therefore,
trices. Here [0] is the first Young tableau, in which the
there are f degenerate states of non-interacting parti-
S
symbols are arrangedby rows in the sequence of natural
cles for each set n . Tilde denotes wavefunctions cor-
numbers. For example, the Young tableaux [0] have Ya- { }
responding to the spatial Hamiltonian without interac-
manouchi symbols (2,1,1), (2,2,1,1), and (2,1,1,1) for
tions between particles. Then Eq. (2) gives us the total
the Young diagrams [2,1], [22], and [3,1], respectively.
wavefunctions of particles with no coordinate-dependent
Each permutation involving symbols between j and
min interactions
j can be written as a product of elementary trans-
max
positions with j j < j (see [5–7]). Ac- Ψ˜(S) =f−1/2 Φ˜(S) Ξ(S). (13)
Pjj+1 min ≤ max r{n}l S tr{n} tl
cordingtotheYoungorthogonalmatrixcalculationrules t
X
(see [5–7]), Dr[λt](Pjj+1) = δrt if j and j +1 are in the In the absence of interactions between spins, the spin
same row of the Young tableau t. Then Eq. (6) leads to wavefunction are eigenfunctions of the total spin projec-
D[λ]( ) = δ if the permutation involves the sym- tion operator Sˆ and can be expressed as
t[0] P t[0] P z
bols in one row only and can be, therefore, written as a
N/2+Sz N
product of elementary transpositions of symbols in the Ξ(S) =C D[λ]( ) ( j) ( j) .
samerow. Let ′ and ′′ be, respectively,arbitraryper- tSz SSz t[0] P |↑ P i |↓ P i
mutations of thPe symbPols in the first and in the second XP jY=1 j=N/Y2+Sz+1
(14)
rowoftheYoungtableau[0],whichdonotpermutesym-
Here the multi-index l is specifically chosen as the total
bols between the rows. Then we get, using Eq. (6),
spin projection S . In the case of the spin wavefunction,
z
Dt[λ[0]](PP′P′′)= Dt[λr](P)Dr[λ[0′]](P′P′′)=Dt[λ[0]](P). beaychseovfertawlopsaprtiniclsetsa,tiefs,N|↑>i a2n.dH|e↓nic,ehtahsetonobremoacliczuaptiioedn
r
X (8) factor [30]
The spatial and spin wavefunctions form orthonormal
basis sets,
1 (2S+1)(S+S )!
z
C =
hΦ(t′Sn′′)|Φ(tnS)i=δS′Sδt′tδn′n (9) SSz (N/2+Sz)!(N/2−S)!s(N/2+S+1)(2S)!(S−Sz)!
hΞt(′Sl′′)|Ξ(tlS)i=δS′Sδt′tδl′l (10) differsfromtheoneinthespatialwavefunction(11).(B15e)-
sides, the Young tableau r can take now only the value
The spatial functions of non-interacting particles are
of [0]. As a result, only one representation is associated
expressed as [6, 7]
with given total spin S and its projection S . The total
z
f 1/2 N wavefunction with the defined Sz is then expressed as
Φ˜(S) = S sgn( )D[λ]( ) ϕ (r )
tr{n} N! P tr P nj Pj
(cid:18) (cid:19) XP jY=1 (11) Ψ(nSS)z =fS−1/2 Φ(tnS)Ξ(tSSz). (16)
in terms of the spatial orbitals — the eigenfunctions t
X
ϕn(r) of the one-body Hamiltonian of non-interacting Incombinationwiththespatialwavefunction(11),the
particle Hˆ0(j), spin wavefunctions lead to the total wavefunctions of
non-interacting particles,
Hˆ (j)ϕ (r )=ε ϕ (r ), (12)
0 n j n n j
where rj is the D-dimensional coordinate of jth parti- Ψ˜(S) =f−1/2 Φ˜(S) Ξ(S) (17)
cle (D can be either 1, 2, or 3 in real physical systems). r{n}Sz S tr{n} tSz
t
X
4
(again, tilde denotes that the wavefunctions involve spa-
tial orbitals ϕ (r) of non-interacting particles). There TABLE I.Coefficients X(S,S′,1) in Eq. (23)
n Szk
are fS wavefunctions, labeled by the Young tableau r, k S−S′
having the total spin S and the set of spatial quantum 0 1
wnuitmhbtehresg{inv}e.n tTohtaenl stphine ptortoajlecntuiomnbSer wofilwl baveefunctions -1 √(S−Sz√+21S)(S+Sz) q(S+S2Sz(−21S)(S1+)Sz)
z 0 SSz −qSS(22S−−Sz12)
N/2 fS =N!/[(N/2+Sz)!(N/2−Sz)!]. (18) 1 −√(S−Sz√)2(SS+Sz+1) q(S−S2Sz(−21S)−(−S1−)Sz)
SX=Sz
In the alternative approach, mentioned in Introduc- are the spin raising and lowering operators for jth par-
tion, each particle has a given spin projection and the ticle. The interaction Uˆ0 conserves the z-projection of
total many-body wavefunctionis represented as (see [3]) the total many-body spin, while Uˆ±1 raises or lowers it.
The interactionof the spin-up or spin-downstate can be
N expressed in terms of Uˆ0 and the scalar Uˆ,
Ψ˜ =(N!)−1/2 sgn( ) ϕ (r )σ ( j) ,
{n}{σ} P nj Pj | j P i 1
XP jY=1 Uˆ↑ U(rj) (j) (j) =Uˆ0+ Uˆ
(19) ≡ |↑ ih↑ | 2
j
where the spin projection σ can be either or and X (22)
given total spin projectionjS , the set σ↑ con↓tains Uˆ U(r ) (j) (j) = Uˆ + 1Uˆ.
z { } ↓ ≡ j |↓ ih↓ | − 0 2
N/2+S spins andN/2 S spins . Forafixedsetof
z z j
↑ − ↓ X
spatialquantumnumbers n ,thenumberofsuchstates
{ }
is the number of distinct choices of N/2+S particles Consider matrix elements of the spherical vector and
z
withspinup,andisthenequaltothenumber(18)ofthe scalar interactions between eigenfunctions (16) of Sˆz.
states (17). Then the sets of degenerate states Ψ˜(S) Their dependence on Sz follows from the Wigner-Eckart
r{n}Sz theorem (see [32]). The matrix elements of the spherical
and Ψ˜ can be related by an unitarily transforma-
{n}{σ} scalar (20) are diagonal in spins and independent of the
tion. For interacting particles, the energy degeneracy of
spin projection,
statesΨ˜(S) islifted,asshownbyHeitler[8]andwillbe
dimispcuossssiebdrl{eni.n}SSzec. IV, and such transformation becomes hΨ(nS′S′)z′|Uˆ|Ψ(nSS)zi=δSS′δSzSz′hΨ(nS′S)|Uˆ|Ψ(nSS)i.
According to the Wigner-Eckart theorem, the matrix el-
ements of the spherical vector components (21) can be
II. SUM RULES FOR ONE-BODY factorized into the 3j-Wigner symbols and the reduced
INTERACTIONS matrix elements
A. The spin-projection dependence hΨ(nS′S′)z′|Uˆk|Ψ(nSS)zi=(−1)S′−Sz′ (cid:18)−SS′z′ k1 SSz (cid:19)hn′,S′||Uˆ||n,Si.
Permutation-invariantinteractionsofparticleswithex-
Thenthereducedmatrixelementsareexpressedinterms
ternal fields can be expressed in terms of the spherical ofthematrixelementsofUˆ forthemaximalallowedspin
k
scalar
projection
Uˆ =Xj U(rj) (20) hn′,S′||Uˆ||n,Si= SS′′ S′1 S SS −1hΨ(nS′S′)′|UˆS′−S|Ψ(nSS)i,
(cid:18)− − (cid:19)
and three spherical vector components
and the matrix elements with arbitrary spin projections
1 can be expressed as
Uˆ = U(r )sˆ (j), Uˆ = U(r )sˆ (j)
0 j z ±1 j ±
∓√2
Xj Xj (21) hΨ(nS′S′)z′|Uˆk|Ψ(nSS)zi=δSz′Sz+kXS(Sz,kS′,1)hΨ(nS′S′)′|UˆS′−S|Ψ(nSS)i
(23)
(see [31]). Here
with the factors
1
sˆz(j)= 2(|↑(j)ih↑(j)|−|↓(j)ih↓(j)|) X(S,S′,q) =( 1)S′−Sz−k S S′ q
Szk − Sz Sz k k
(cid:18) − − (cid:19)
is the z-component of the spin and S S′ q −1
sˆ (j)= (j) (j), sˆ (j)= (j) (j) × S S′ S′ S
+ − (cid:18) − − (cid:19)
|↑ ih↓ | |↓ ih↑ |
5
Here S′ S and, according to the properties of the the identity permutation, one finally gets
≤
3j-Wigner symbols, the matrix elements (23) vanish if
|VSal−ueSs′o|f>no1n-(vinanaisghrienegmceonetffitocietnhtes,seclaelcctuiloanterdulwesith[17th])e. hΨ˜(rS′{′n)′}l′|Uˆ|Ψ˜(rS{n)}li=δSS′δll′δr′r N hn′j|U|nji δn′j′,nj′,
3j-Wigner symbols [3, 32], are presented in Tab. I. Her- Xj=1 jY′6=j
(26)
mitian conjugate of Eq. (23), together with relations
It is a special case of the matrix elements obtained by
Uˆ = Uˆ† and Uˆ =Uˆ†, gives us the matrix elements
fo+r1S′ =−S−+11. 0 0 Heitler [8] and Kaplan [6].
For the spherical vector interactions (21), the matrix
Thus, each matrix element of a spin-dependent one-
elementscannotberepresentedinsosimpleaform. How-
body interaction with an external field is related to ma-
ever, rather simple expressions can be derived for sums
trix elements for the maximal allowed spin projections,
and sums of squared moduli of the matrix elements be-
which will be evaluated in the next section. tween eigenfunctions of Sˆ . It is enough to consider ma-
z
trix elements of Uˆ and the spin-up state interaction
−1
Uˆ for the maximal allowed spin projection, S′ = S′,
↑ z
B. Matrix elements for non-interacting particles S =S,asEq.(22)andtheWigner-Eckarttheorem(23)
z
relate to them each matrix element of each interaction.
Matrix elements of the spherical scalar (20) can be Inthebasisofthenon-interactingparticlewavefunctions
evaluatedexactlyforgeneralspinwavefunctions. Due to (17), the matrix elements of Uˆ↑ can be decomposed into
theorthogonalityofthespinwavefunctions(10),thema- the spatial and spin parts,
trix elements arediagonalinspin quantumnumbers and
can be reduced to the matrix elements between spatial hΨ˜(rS′{′n)′}S′|Uˆ↑|Ψ˜(rS{n)}Si=(fSfS′)−1/2
wavefunctions,
Φ˜(S′) U(r )Φ˜(S) Ξ(S′) (i) (i)Ξ(S) .
× h t′r′{n′}| i | tr{n}ih t′S′|↑ ih↑ | tS i
hΨ˜(rS′{′n)′}l′|Uˆ|Ψ˜r(S{n)}li=δSS′δll′f1 tX,t′,i (27)
S
Φ˜(S) U(r )Φ˜(S) . (24) The spatial matrix elements are given by Eq. (25). The
× h tr′{n′}| i | tr{n}i
spinmatrixelementsincludeprojectionsofthespinwave-
t i
XX
functions (14)
Letuscalculatethespatialmatrixelementforthegeneral
case,S =S′,havinginmindfurtheranalysisofspherical λ1
vectors.6 Equations (11) and (7) lead to (i)Ξ(S) =C D[λ]( ) δ
h↑ | tS i SS t[0] P iPl
P l=1
X X
hΦ˜t(′Sr′′){n′}|U(ri)|Φ˜(trS{)n}i= √fNSf!S′ sgn(Q)Dr[λ′t′′](Q) × λ1 |↑(Pj)i N |↓(Pj)i.
RX,Q Yj6=l j=Yλ1+1
×sgn(R)Dr[λt](R)hϕn′Qi|U(ri)|ϕnRiii′6=iδn′Qi′,nRi′. Substituting P =QPlλ1we get
Y
λ1
TheKroneckerδ-symbolsappearheredueto theorthog- (i)Ξ(S) =C D[λ]( )δ
onalityofthespatialorbitalsϕ andtheabsenceofequal h↑ | tS i SS t[0] QPlλ1 iQλ1
n Q l=1
quantum numbers in eachof the sets n and n′ . Due XX
{ } { } λ1−1 N
to the δ-symbols, all but one spatial quantum numbers
( j) ( j) .
remain unchanging. Supposing that the unchanged nj′ × |↑ Q i |↓ Q i
are in the same positions in the sets n and n′ , one jY=1 j=Yλ1+1
{ } { }
can see that the Kroneckersymbols lead to = , and, The permutation permute symbols in the first row
therefore, Q R of the Young tabPlelaλu1 [0]. Therefore, D[λ]( ) =
t[0] QPlλ1
D[λ]( ) [see Eq. (8)] , the summand in the equation
hΦ˜t(′Sr′′){n′}|U(ri)|Φ˜(trS{)n}i= √fNSf!S′ Dr[λ′t′′](R)Dr[λt](R) abto[0v]eQis independent of l, and the projection can be ex-
R pressed as
X
×hn′Ri|U|nRiij′Y6=Riδn′j′,nj′, (25) h↑(i)|Ξ(tSS)i=λ1CSS Dt[λ[0]](Q)δiQλ1
Q
X
where hn′|U|ni = dDrϕ∗n′(r)U(r)ϕn(r). Then, substi- λ1−1 N
tuting this expression into (24), using (6), (7), and the ( j) ( j) .
R × |↑ Q i |↓ Q i
property of representations Dr[λ′r](E) = δr′r, where E is jY=1 j=Yλ1+1
6
The projection involved into matrix elements of Uˆ is whereλ′ =[λ 1,λ +1],is calculatedinasimilarway.
−1 1 2
−
evaluated in the same way,
The explicit expressions (28)and (29) are rather com-
plicated as they include Young orthogonal matrices and
(i)Ξ(S) =λ C D[λ]( )δ
h↓ | tS i 2 SS t[0] Q iQ(λ1+1) summation over all permutations. The next section pro-
XQ vides expressions for sums of the matrix elements and
λ1 N their squared moduli, which are much simpler.
( j) ( j) .
C. Sum rules
× |↑ Q i |↓ Q i
jY=1 j=Yλ1+2
In the spin matrix elements of Uˆ↑, The sum of diagonal in total spin S and r matrix ele-
ments can be written out as
hΞt(′SS′′)|↑(i)ih↑(i)|Ξ(tSS)i=δSS′[λ1CSS]2 Dt[λ[0]](Q)δiQλ1
Q
X
f
× Dt[λ′[]0](R)δiRλ1 δR,QP′P′′ hΨ˜(rS′{)n′}S|Uˆa|Ψ˜(rS{n)}Si=Y(S)[Uˆa]NS
R P′,P′′ r
X X X
N
tthioensperm′ uoftatthioenfisrRstaλnd Q1csaynmbbeoldsiffaenrdent′b′yofpethrmeulatast- × hn′j|U|nji δn′j′,nj′ (30a)
λ onPes. As the perm1u−tations ′ and P′′ do not per- Xj=1 jY′6=j
2
P P
mute symbolsbetweenrowsinthe Young tableau[0],we
have Dt[λ′[]0](R) = Dt[λ′[]0](Q) [see Eq. (8)]. Since the num- The universal factors Y(S) are independent of the
rbeesrpseoctfivpeelrym, tuhteatsiponinsmPa′tarinxdePlem′′ eanrtes(tλa1ke−th1)e!faonrmd,λ2!, Ym(aSt)r[iUxˆ ]ecleamnebnetsdehrniv′je|Ud|nfrjoim. (28F)oursinUˆg↑,thetheequafalicttioesr
↑
D[λ] ( )D[λ] ( )= D[λ] ( ) = 1 [obtained with (6)
r [0]r P [0]r P [0][0] E
hΞt(′SS)|↑(i)ih↑(i)|Ξ(tSS)i=(λ1−1)!λ2!λ21CS2S aPnd (7) ] and Pδλ1Pj =(N −1)!, as
D[λ]( )D[λ] ( )δ . P
× t[0] Q t′[0] Q iQλ1
XQ Y(S)[Uˆ ]= N +S. (30b)
↑
2
Let us substitute this equation and (25) into (27), per-
form the summation over t and t′ using Eq. (6), and
substitute = −1 −1, j = i. Then the Kronecker
symbol leadPs toQj =R −1i=λR, and we get It is equal to the number of the spin-up atoms. For the
P Q 1 sphericalvectorcomponentUˆ ,thefactorY(S)[Uˆ ]isob-
0 0
tained using Eq. (22),
Ψ˜(S) Uˆ Ψ˜(S) =λ !λ !λ C2 D[λ] ( )D[λ] ( )
h r′{n′}S| ↑| r{n}Si 1 2 1 SS [0]r′ P [0]r P
P
X Y(S)[Uˆ ]=S. (30c)
N 0
× δλ1Pjhn′j|U|nji δn′j′,nj′. (28)
j=1 j′6=j
X Y
Equation (26) leads to
The matrix element
1
hΨ˜(rS′{−n1′})S−1|Uˆ−1|Ψ˜(rS{n)}Si= √2λ1!λ2!(λ2+1)CSSCS−1S−1 Y(S)[Uˆ]=N. (30d)
N
× D[[0λ]′r]′(P)D[[0λ]]r(P) δλ1Pjhn′j|U|nji δn′j′,nj′,
XP Xj=1 jY′6=j Thesumofsquaredmoduliofthematrixelements(28)
(29)
and (29) can be expressed, using Eqs. (4) and (15), as
2
λ f
|hΨ˜(rS′{)n′}S|Uˆ↑|Ψ˜(rS{n)}Si|2 = N1 !S Σj(jS′,S)hn′j|U|njihnj′|U|n′j′i δn′j′′,nj′′ δn′j′′′,nj′′′ (31)
r,r′ (cid:18) (cid:19) jj′ j′′6=j j′′′6=j′
X X Y Y
7
λ (λ +1)f f
|hΨ˜(rS′{−n1′})S−1|Uˆ−1|Ψ˜(rS{n)}Si|2 = 1 22(N!)2S S−1 Σj(jS′−1,S)hn′j|U|njihnj′|U|n′j′i δn′j′′,nj′′ δn′j′′′,nj′′′,
r,r′ jj′ j′′6=j j′′′6=j′
X X Y Y
(32)
where
Σj(jS′′,S) = D[[0λ]′r]′(P)D[[0λ]]r(P)δλ1Pj D[[0λ]′r]′(Q)D[[0λ]]r(Q)δλ1Qj′. (33)
r,r′ P Q
XX X
Thesesums arecalculatedinAppendix. Itisshownthat For transitions conserving the spatial quantum num-
bers, n′ = n and the Kroneckersymbols in (31) and
N!(N 1)! λ { } { }
Σ(S,S) = − λ 2 (34) (32) are equal to one for any j and j′. Then sums of
jj fSλ21 (cid:20) 1− λ1−λ2+2(cid:21) squaredmoduliofthematrixelementscanberepresented
N!(N 1)! as
Σ(S−1,S) = − (35)
jj f λ
are independent of j,San1d |hΨ˜r(S′{′n)}S′|Uˆa|Ψ˜(rS{n)}Si|2 =fS′ Y0(S,0)[Uˆa,Uˆa]hUi2
Xr,r′ h
+Y(S,0)[Uˆ ,Uˆ ] ∆U 2 (38a)
Σj(jS′′,S) = N!(Nf −2)!δSS′ − N1 1Σj(jS′,S) (36) 1 a a h i i
S − where
for any j′ =j.
If the se6ts of spatial quantum numbers n and n′ 1 N
U = n U n
{ } { } j j
are different, the product of Kronecker symbols in (31) h i N h | | i
j=1
and (32) does not vanish only if j = j′. Then the sum X
of squaredmoduli of the matrix elements can be written is the average matrix element and
out as
1/2
N
|hΨ˜(rS′{′n)′}S|Uˆa|Ψ˜(rS{n)}Si|2 =Y(S,1)[Uˆa,Uˆa]fNS′ h∆Ui=N1 (hnj|U|nji−hUi)2
Xr,r′ Xj=1
N  
× |hn′j|U|nji|2 δn′j′,nj′, (37a) TishteheuanviveerrasgaeldfaevctiaotrisonYo(Sf,0t)h[eUˆm,aUˆtr]ixaenldemYen(St,s0)o[UfˆU,(Uˆr).]
j=1 j′6=j 0 a a 1 a a
X Y
areindependentofthematrixelements n U n . Equa-
j j
where S′ S and the difference S S′ is unambigu- tions (34), (35), and (36) lead to h | | i
ously deter≤mined by the operator Uˆ .−Each term in the
a
sum herechangesone spatialquantum number,conserv- Y(S,0)[Uˆ ,Uˆ ]= Y(S)[Uˆ ] 2 (38b)
ing other ones. If U(r) = const, the sums vanish since 0 a a a
hfaϕcnt′o|Urs|ϕYn(Si,1=)[UUˆh,ϕUˆn′]|,ϕwnihi=ch0afroerinnd6=epenn′.deTntheofutnhiveemrsaa-l (where defined Y(S)[Uˆ−1] = 0,(cid:16)in additio(cid:17)n to Eq. (30)),
a a
trixelements n′ U n ,areexpressedintermsofΣ(S′,S). and
Then Eqs. (3h4)ja|nd| (j3i5) lead to jj Y(S,0)[Uˆ ,Uˆ ]= N(N −2S+2) (38c)
1 −1 −1 4(N 1)
N N 2S −
Y(S,1)[Uˆ↑,Uˆ↑]= +S − (37b) Y(S,0)[Uˆ ,Uˆ ]=Y(S,0)[Uˆ ,Uˆ ]
2 − 4(S+1) 1 ↑ ↑ 1 0 0
N 2S+2 S(N 2S)(N +2S+2)
Y(S,1)[Uˆ ,Uˆ ]= − , (37c) = − . (38d)
−1 −1 4 4(S+1)(N 1)
−
and Eq. (26) gives If U(r) = const, ∆U = 0, and, therefore,
Y(S,1)[Uˆ,Uˆ]=N. r,r′|hΨ˜(rS′{−n1})S−1|Uˆ−1|Ψ˜(rS{n)}Si|2 = 0. Indeed, in this
case, the spatial matrix elements (25) are equal to zero
P
The factor Y(S,1)[Uˆ ,Uˆ ] for the sphericalvectorcompo- duetotheorthogonalityofthespatialwavefunctionswith
0 0
nentUˆ isobtainedusing(22). Sincethematrixelements different spins.
0
of Uˆ are diagonal in r [see Eq. (26)], one gets Thus,sumsofmatrixelementsandtheirsquaredmod-
uli are expressed in terms of universal factors, which are
S(N +2) independent of the spatial orbitals and details of the ex-
Y(S,1)[Uˆ ,Uˆ ]= . (37d)
0 0 4(S+1) ternal fields, and sums of one-body matrix elements (or
8
their squared moduli), which are independent of many- III. SUM RULES FOR TWO-BODY
body spins. The sum rules, combined with the spin- SPIN-INDEPENDENT INTERACTIONS
projectiondependence (23), provideinformationoneach
matrix element for an one-body spin-dependent interac- The permutation-invariant interaction between parti-
tion with an external field. cles is given by
Vˆ = V(rj rj′). (39)
−
j6=j′
X
Without loss of generality, we can restrict consideration
to even potential functions, V(r) = V( r), since their
−
odd parts are canceled. Matrix elements of this inter-
action can be evaluated for general spin wavefunctions.
Due to the orthogonality of the spin wavefunctions (10),
the matrix elements are diagonal in spin quantum num-
bers andcanbe reduced to the matrix elements between
spatial wavefunctions,
hΨ˜(rS′{′n)′}l′|Vˆ|Ψ˜(rS{n)}li=δSS′δll′f2 hΦ˜(trS′){n′}|V(ri−ri′)|Φ˜(trS{)n}i. (40)
S t i<i′
XX
(this reduction is used in spin-free quantum chemistry [6, 7]). Then, using (11), (39), and the property (7) of the
Young orthogonal matrices, the spatial matrix elements can be expressed as
f
hΦ˜t(rS′){n′}|V(ri−ri′)|Φ˜(trS{)n}i= NS! sgn(Q)Dr[λ′t](Q)sgn(R)Dr[λt](R)
R,Q
X
× dDridDri′ϕ∗n′Qi(ri)ϕ∗n′Qi′(ri′)V(ri−ri′)ϕnRi(ri)ϕnRi′(ri′) δn′Qi′′,nRi′′. (41)
Z i′6=i′′6=i
Y
The Kronecker δ-symbols appear here due to the orthogonality of the spatial orbitals ϕ and the absence of equal
n
quantum numbers in each of the sets n and n′ . Due to the δ-symbols, all but two spatial quantum numbers
{ } { }
remain unchanging. Supposing that the unchanged ni′′ are in the same positions in the sets n and n′ , one can
{ } { }
see that the Kronecker symbols allow only = or = ii′. Then substitution of (41) into (40), using (6) and
Q R Q RP
(7), leads to
hΨ˜(rS′{′n)′}l′|Vˆ|Ψ˜r(S{n)}li=2δSS′δll′N1! δn′j′′,nj′′
R i<i′Ri′6=j′′6=Ri
XX Y
× δr′rhn′Rin′Ri′|V|nRinRi′i+sgn(Pii′)Dr[λ′r](RPii′R−1)hn′Ri′n′Ri|V|nRinRi′i , (42)
h i
where n′n′ V n n = dDr dDr ϕ∗ (r )ϕ∗ (r )V(r r )ϕ (r )ϕ (r ).
h 1 2| | 1 2i 1 2 n′1 1 n′2 2 1− 2 n1 1 n2 2
Taking into account that
R
ii′ −1 = PiPi′ (43)
PP P P
(see [7]) and substituting i=j, one finally gets
R
hΨ˜(rS′{′n)′}l′|Vˆ|Ψ˜r(S{n)}li = 2δSS′δll′ δn′j′′,nj′′ δr′rhn′jn′j′|V|njnj′i+sgn(Pjj′)Dr[λ′r](Pjj′)hn′j′n′j|V|njnj′i .
jX<j′j′6=Yj′′6=j h i
(44)
It is a special case of the matrix elements obtained by Heitler [8] and Kaplan [6].
The sum of diagonal elements of the representation matrix, the character
χ ( ) D[λ]( ),
S C ≡ rr P
r
X
9
TABLE II.Characters χS( ) of theclasses of conjugate elements of the symmetric group N of permutations of N symbols
C C S
intheirreduciblerepresentations,correspondingtothespinS. ThecharactersarecalculatedwiththeFrobeniusformula[7,34]
and scaled to therepresentation dimension fS.
χS( )/fS
{C2} 4S2+2NN(2CN+4S1)−4N
{3} 12S2+4NN2(N+1−21S)−10N
{4} N4−24N3+4N2(6S2+6S+8N29()N−161N)((N10−S22)(+N10S3+)9)+16S(S+1)(S2+S+12)
{22} N4−12N3+8N2(S2+S4+N7()N+8−N1)((1N0S−22+)(1N0S−+39))+16S(S+1)(S2+S+6)
− − −
isthesameforallpermutations ,whichformtheclassofconjugateelements [4–7]. TableIIpresentsthecharacters
P C
for the classes appearing here. (Supplemental material for [17] contains a code based on the explicit expressions [33]
forthecharacters.) Theconjugatedclassesofthesymmetricgroup arecharacterizedbythe cyclicstructureofthe
N
S
permutations. All permutations in the class = NνN ...2ν2 have νl cycles of length l. This class notation omits
lνl if νl = 0 and the number of cycles of theCleng{th one, i.e. }the number of symbols which are not affected by the
permutations in the class. This number is determined by the condition N lν =N. Permutations of two symbols
l=1 l
form the class 2 . This leads to the sum of diagonal in r matrix elements
{ } P
hΨ˜(rS{n)′}l|Vˆ|Ψ˜(rS{n)}li=2 fShn′jn′j′|V|njnj′i±χS({2})hn′j′n′j|V|njnj′i δn′j′′,nj′′, (45a)
r j<j′ j′6=j′′6=j
X X(cid:2) (cid:3) Y
where the sign + or is taken for bosons or fermions, respectively. Similar expressions have been obtained for the
−
total energy [8] and arbitrary observables [17]. If n′ = n , the Kronecker symbols are equal to one for any j and
j′ and the sum can be transformed to the form { } { }
χ ( 2 )
Ψ˜(S) Vˆ Ψ˜(S) =N(N 1)f V S { } V . (45b)
h r{n}l| | r{n}li − S h idir± f h iex
r (cid:18) S (cid:19)
X
Here and above, the dependence on many-body states is given by universal functions f and χ ( 2 ), which are
S S
independent of the matrix elements n′n′ V n n , while the average matrix elements { }
h 1 2| | 1 2i
2 2
V dir = njnj′ V njnj′ , V ex = nj′nj V njnj′ (46)
h i N(N 1) h | | i h i N(N 1) h | | i
− j<j′ − j<j′
X X
of the direct and exchange interactions, respectively, are independent of the many-body states.
Calculating the sum of squaredmoduli of the matrix elements (44), one can see that if the sets of spatial quantum
numbers n and n′ are different by two elements, the product of Kronecker symbols in the product of the matrix
elements{doe}snot{van}ishonlyifthe pairj, j′ isthe sameinbothmatrix elements. Thenthe sumcanbe expressedas
|hΨ˜r(S′{)n′}l|Vˆ|Ψ˜(rS{n)}li|2 =4fS δn′j′′,nj′′ |hn′jn′j′|V|njnj′i|2+|hn′j′n′j|V|njnj′i|2
r,r′ j<j′j′6=j′′6=j (cid:20)
X X Y
χ ( 2 )
±2 Sf{ } Re hn′jn′j′|V|njnj′ihn′j′n′j|V|njnj′i∗ . (47a)
S (cid:21)
(cid:0) (cid:1)
Here the equality rr′Dr[λ′r](Pjj′)Dr[λ′r](Pjj′) = rDr[λr](E) = fS was used. Each term in the sum above changes
two of the spatial quantum numbers, conserving other ones. The case of a single changed quantum number will be
P P
considered elsewhere.
For transitions conserving the spatial quantum numbers, n′ = n and the Kronecker symbols in (44) are equal
to one for any j and j′. Then { } { }
Ψ˜(S) Vˆ Ψ˜(S) 2 = f N2(N 1)2 V 2 2χ ( 2 )N2(N 1)2 V V
|h r′{n}l| | r{n}li| S − h idir± S { } − h idirh iex
Xr,r′ h
+ Dr[λr](Pj1j1′Pj2j2′)hnj1′nj1|V|nj1nj1′ihnj2′nj2|V|nj2nj2′i
jX16=j1′ jX26=j2′ Xr i
10
The trace of the Young matrix can be transformed in the following way (since j =j′ and j =j′)
1 6 1 2 6 2
Dr[λr](Pj1j1′Pj2j2′)=χS({22})+(δj1j2 +δj1j2′ +δj1′j2 +δj1′j2′)(χS({3})−χS({22}))
r
X
+(δj1j2δj1′j2′ +δj1j2′δj1′j2)(fS −2χS({3})+χS({22})),
since Pj1j1′Pj1j2′ ∈{3} for j1′ 6=j2′, Pj1j1′Pj1j1′ =E, and χS(E)=fS. Here and in what follows, χS({3}) and χS({22})
have to be equated to zero at N < 3 and N < 4, respectively, when the corresponding permutations do not exist.
Usingtheidentity2f +4(N 2)χ ( 3 )+(N 2)(N 3)χ ( 22 )=N(N 1)χ2( 2 )/f (itcanbedirectlyproved
S − S { } − − S { } − S { } S
with the characters in Table II), the sum of squared moduli of the matrix elements can be represented as
2
1
Ψ˜(S) Vˆ Ψ˜(S) 2 =f Y(S,0)[Vˆ,Vˆ] ∆ V 2+Y(S,0)[Vˆ,Vˆ] ∆ V 2 + Ψ˜(S) Vˆ Ψ˜(S) (47b)
Xr,r′ |h r′{n}l| | r{n}li| S(cid:16) 1 h 1 i 2 h 2 i (cid:17) fS Xr h r{n}l| | r{n}li!
with the universal factors
χ ( 3 ) χ ( 22 ) 2χ ( 3 ) χ ( 22 )
Y(S,0)[Vˆ,Vˆ]=4N(N 1)2 S { } − S { } , Y(S,0)[Vˆ,Vˆ]=2N(N 1) 1 S { } − S { } . (47c)
1 − f 2 − − f
S (cid:18) S (cid:19)
Here tion theory [3], the eigenenergies E (counted from the
Sn
2 multiplet-independentenergyofnon-interactingparticles
1 N 1 N ε ) are determined by the secular equation
∆1V 2 = nj′nj V njnj′ V ex j=1 nj
h i N N 1 h | | i−h i 
Xj=1 − jX′6=j P V(S)A(S) =E A(S), (50)
2   rr′ nr′ Sn nr
∆2V 2 = ( nj′nj V njnj′ V ex)2 Xr′
h i N(N 1) h | | i−h i
− jX<j′ whereA(nSr)aretheexpansioncoefficientsofthewavefunc-
(48)
tion(16)intermsofthewavefunctionsofnon-interacting
measure the average deviation of the exchange matrix particles (17),
elements.
Ψ(S) = A(S)Ψ˜(S) (51)
Thus,sumsofmatrixelementsandtheirsquaredmod- nSz nr r{n}Sz
uli are expressed in terms of universalfactors, which are Xr
independent of the spatial orbitals and interaction po- and the matrix elements of the spin-independent two-
tentials,andsums oftwo-bodymatrix elements (ortheir body interaction (44)
squared moduli), which are independent of the many-
V(S) = Ψ˜(S) Vˆ Ψ˜(S)
body spins. The universalfactors are expressedin terms rr′ h r′{n}Sz| | r{n}Szi
of characters of irreducible representations of the sym-
do not couple states with different spins.
metric group. The characters are functions of the total
Consider at first the case when the matrix elements
spin and the number of particles. V = n n Vˆ n n and V = n n Vˆ n n are in-
dir 1 2 1 2 ex 1 2 2 1
h | | i h | | i
dependent of the spatial quantum numbers. E.g., this
can take place in the case of zero-range interactions
IV. MULTIPLET ENERGIES FOR
V(r)=Vδ(r), ifthespatialorbitalshaveaformofplane
WEAKLY-INTERACTING GASES
waves. In this case, the summation over in the ma-
trix element (42) for n = n′ can be perRformed using
As an example of applications of the sum rules, con- { } { }
Eqs. (6), (7), and the orthogonality relation (5) in the
sider splitting of degenerate energy levels due to weak
following way [5]
two-body spin-independent interactions. The Hamilto-
nHˆia0(nj)ofofthneons-yinstteemracitsinagspuamrticolfesonaen-dbotwdyo-bHoadmyilitnotneiraancs- Dr[λ′r](RPii′R−1)= Dt[λ′t](Pii′) Dr[λ′t]′(R)Dr[λt](R)
R t,t′ R
tions (39), X X X
N!
N = δr′rχS( 2 ).
f { }
Hˆ = Hˆ (j)+Vˆ (49) S
spat 0
Then the matrix elements become diagonal in r,
j=1
X
T(1h3e).intIenractthieonzsesrpo-liotrdeenrerogfiesthoef tdheegedneegreanteerapteertsutrabteas- Vr(rS′) =δrr′N(N −1) Vdir± χSf({2})Vex ,
(cid:18) S (cid:19)