The elementary excitations of the BCS model in the canonical ensemble

11 Pages·0.28 MB·English

by  G. Sierra

The elementary excitations of the BCS model in the canonical ensemble
Germ´an Sierra,1∗ Jos´e Mar´ıa Roma´n1 and Jorge Dukelsky2
1Instituto de F´ısica Te´orica, CSIC/UAM, Madrid, Spain.
2Instituto de Estructura de la Materia, CSIC, Madrid, Spain
(Dated: February 2, 2008)
3
0 We summarize previous works on the exact ground state and the elementary excitations of the
0 exactly solvable BCS model in the canonical ensemble. The BCS model is solved by Richardson
2 equations,and,inthelargecouplinglimit,byGaudinequations. Therelationshipbetweenthistwo
kindsof solutions are used toclassiffy theexcitations.
n
a
J
I. INTRODUCTION
2
2
Oneofthe mostfundamentalproblemsinamanybody systemis toknowwhicharethe groundstate(GS)andthe

] lowenergyexcitedstates,whichdeterminethethermodynamicpropertiesandtheresponsetoexternalfields. Inmost
n
o cases the excited states can be understood in terms of a collection of elementary excitations characterized by their
c statistics,discretequantumnumbersasspin,chargeanddispersionrelation. Inthepairingmodelofsuperconductivity,
- proposedbyBardeen,CooperandSchriefferin1957,thisproblemwassolvedlongagointhegrandcanonicalensemble
r
p for a large number of particles [1, 2]. However recent studies, motivated by the fabrication of ultrasmall metallic
u grains,show that the grandcanonical BCS solution deviates strongly from the exact numerical or analytical solution
s forsystemswithafixedandsmallnumberofparticles(forareviewsee[3]). Mostofthepreviousstudieshavefocused
.
t on the ground state of the BCS system, and some of its excitations. In this contribution we shall review further
a
m progress concerning the understanding of the full excitation spectrum [4].
-
d
n II. THE GRAND CANONICAL BCS ANSATZ
o
c
The BCS model of superconductivity is characterized by the energy levels of the electrons and the scattering
[
potential between them. When the latter is a constant, the BCS model is exactly solvable `a la Bethe [5] and

1 integrable[6]. This is the modelwe shallusedto study the GSandexcitations. The Hamiltonianofthe reducedBCS
v
is defined by [3]
7
1
1
14 HBCS = 2j,Xσ=±εjσc†jσcjσ −GXj,j′ c†j+c†j−cj′−cj′+ , (1)
0
3 where c (resp. c† ) is an electron annihilation (resp. creation) operator in the time-reversed states j, with
0 j,± j,± | ±i
/ energies εj/2, and G is the BCS dimensionful coupling constant. The sums in (1) run over a set of N doubly
t
a degenerate energy levels εj/2(j =1,...,N). We adopt Gaudin’s notation, according to which εj denotes the energy
m of a pair occupying the level j [7]. To fix ideas we shall use the so called equally space or picket fence model,
that is the one employed in the study of ultrasmall superconducting grains [3]. This model is given by the choice
-
d ε = d(2j N 1), where d = ω/N is the single particle energy level spacing and ω/2 is the Debye energy. The
j
n − −
coupling G can be written as G=gd, where g is dimensionless.
o
The BCS ansatz for the GS of this Hamiltonian in the grand canonical ensemble is given by [1, 2]
c
:
v BCS = (u +v c† c† ) 0 , (2)
i | i j j j,+ j,− | i
X Yj
r
a where 0 is the Fock vacuum of the electron operators and u ,v are the BCS variational parameters given by
j j
| i
1 ξ 1 ξ
u2 = 1+ j , v2 = 1 j , (3)
j 2(cid:18) E (cid:19) j 2(cid:18) − E (cid:19)
j j
ξ =ε ε G, E = ξ2+∆2. (4)
j j − 0− j q j
∗ Talkpresentedatthe6thInternational workshoponConformalFieldTheoryandIntegrableModels,Chernologka,Russia,Sept. 2002.
2
In these eqs. ε and ∆ are twice the chemical potential and the BCS gap, which are found by solving the following
0
equations:
1 1 1 ξ
j
= , M = 1 , (5)
G E 2 (cid:18) − E (cid:19)
Xj j Xj j
where M is the number of electron pairs. The most studied case in the literature corresponds to the half-filled
situation, where the number of electrons, N = 2M, equals the number of levels N [3]. In this case the solution of
e
eqs. (5) in the large N limit is given by ∆=ω/sinh(1/g) and ε =0.
0
TheexcitedstatesinthegrandcanonicalensemblecanbeobtainedactingontheGSansatz(2)withtheBogoliubov
operators γ (σ = )
j,σ
±
γ ...γ BCS , γ =u c v c† , (6)
j1,σ1 jn,σn| i j,± j j±∓ j j∓
and have an energy 1(E +...+E ). Recall that in our conventions E is twice the quasiparticle energy, thus the
2 j1 j2 j
factor 1/2 in the energy of the states (6).
III. PAIR-HOLE REPRESENTATION OF BCS MODEL
Every energy level j =1,...,N has four possible states given by
0 : empty,
| i
c† 0 : singly occupied, (7)
jσ| i
c† c† 0 : doubly occupied.
j+ j−| i
Animportantpropertyofthe Hamiltonian(1)isthe“blocking”oflevelswhicharesinglyoccupied,whichmeansthat
these levels decouple from the rest of the system. This is a consequence of the equation
ε
H c† ψ = j c† ψ +c† H ψ , (8)
BCS jσ| i 2 jσ | i jσ BCS| i
where the state ψ does not contain the operators c† . Thus the singly occupied levels only contribute with their
| i jσ
kinetic energy and one can study the dynamics on those levels which are either empty or doubly occupied.
Let us call the Hilbert space of states of M pairs distributed among N different energy levels. To describe
N,M
H
let us define the hard-core boson operators
N,M
H
b =c c , b† =c† c† , N =b†b , (9)
j j,− j,+ j j,+ j,− j j j
which satisfy the commutation relations
[bj,b†j′]=δj,j′ (1−2Nj). (10)
The Hamiltonian (1) restricted to the Hilbert space can then be written as
N,M
H
HBCS = εjb†jbj −G b†jbj′ . (11)
Xj Xj,j′
A basis of states of is given by
N,M
H
I = b† 0 , (12)
| i j | i
jY∈I
where I denotes a set of M different integers ranging from 1 to N. Thus the dimension of is given by the
N,M
combinatorial number CN. A convenient pictorial representation of the singly and occupied stHates is given by [4]
M
0 , b† 0 . (13)
◦↔| i •↔ j| i
AtG=0thegroundstateof(11)istheFermiseaobtainedfillingallthelevelsbelowtheFermienergyε . ThesetI
F 0
is givenin this case by I = 1,2,...,M (see fig. 1). Similarly the lowest energy excited state at G=0 corresponds
0
{ }
to the choice I = 1,2,...,M 1,M +1 (see fig. 2).
1
{ − }
3
ε
8
GROUND STATE FOR N=8 LEVELS AND M=4 PAIRS
ε
7
ε
6
ε EF
5
EF
ε
4
ε
3 I 0 = {1, 2, 3, 4 }
ε
2
ε1
FIG. 1: Pair-hole representation of the ground state at G=0 for N =2M =8.
ε
8
LOWEST EXCITED STATE FOR N=8 LEVELS AND M=4 PAIRS
ε
7
ε
6
ε EF
5
EF
ε
4
ε
3
I 1 = { 1, 2, 3, 5 }
ε
2
ε1
FIG. 2: Pair-hole representation of thelowest energy excited state at G=0 for N =2M =8.
IV. EXACT SOLUTION OF THE BCS MODEL IN THE CANONICAL ENSEMBLE
In 1963 Richardson showed that the eigenstates of the Hamiltonian (11) with M pairs have the (unnormalized)
product form [5]
M N
1
M = B† vac , B† = b† , (14)
| i ν| i ν ε E j
νY=1 Xj=1 j − ν
where the parameters E (ν =1,...,M) are, in general, complex solutions of the M coupled algebraic equations
ν
N M
1 1 2
= , ν =1,...,M, (15)
G ε E − E E
Xj=1 j − ν µ=X1(6=ν) µ− ν
whichplaythe roleofBethe ansatzequationsforthis problem[8, 9, 10,11, 12,13, 14,15]. The energyofthese states
is given by the sum of the auxiliary parameters E , i.e.
ν
M
E(M)= E . (16)
ν
νX=1
ThegroundstateofH isgivenbythesolutionofeqs.(15)whichgivesthelowestvalueofE(M). The(normalized)
BCS
states (14) can also be written as [16]
C
M = ψ(j ,...,j )b† b† vac , (17)
| i √M! 1 M j1··· jM| i
j1,X···,jM
4
0 0.2 0.4 0.6 0.8 1 1.2
12
8
4
µ
E
m 0
I -4
-8
-12
16
14
12
µ 10
E
e 8
R
6
4
2
0
0 0.2 0.4 0.6 0.8 1 1.2
g
FIG.3: EvolutionoftherealandimaginarypartsofEµ(g),inunitsofd=ω/N,fortheequallyspacedmodelwithM =N/2=8,
asafunctionofthecouplingconstantg=G/d [17,18]. Forconveniencetheenergylevelsarechoosen inthisfigureasεj =2j.
where the sum excludes double occupancy of pair states and the wave function ψ takes the form
M
1
ψ(j , ,j )= . (18)
1 M
··· ε E
XP kY=1 jk − Pk
The sumin(18)runsoverallthe permutations, ,of1, ,M. The constantC in(17) guaranteesthe normalization
P ···
of the state [16] (i.e. M M =1).
h | i
The number ofsolutions ofthe Richardson’seqs.(15) is equalto the dimensionofthe Hilbertspace , namely
N,M
CN [7]. For finite andsmallvalues of N, M the solutions E (G) M haveto be found numerically. THhese solutions
M { µ }µ=1
and the corresponding eigenstates can be classified according to the values taken by E (G) at G=0, namely [17]
µ
lim E (G)=ε , j I, (19)
µ j
G→0 ∈
whereI is the setI thatlabelthe basis(12)of . This meansthatthe spectrumofthe BCSHamiltonianfollows
N,M
H
an adiabatic evolution as a function of G [17, 18] For very small values of G all the roots E (G) of eq. (15) are real
µ
and close to their G=0 value given in eq. (19). This happens for all the states labelled by I.
Ground State solution
Inparticularforthe GS,i.e.I ,asweincreaseGtherealrootsE andE ,nearesttothe Fermilevel,approach
0 M M−1
fromaboveand below the energyε , andbecome equalto it at some criticalvalue G=G . For G>G the two
M−1 c1 c1
roots E and E become a complex conjugate pair. IncreasingG further one encountersa similar phenomena for
M M−1
the roots E and E , and so on, until all the roots become complex (see fig. 3).
M−2 M−3
Inthe limitwhere N is large,while M/N andω =dN remainfinite the complexrootsE formanopenarcΓ with
µ
end points ε i ∆ (see fig. 4) [7, 18, 19]. This gives an interesting geometrical meaning to the chemical potential
0
±
(ε /2) and the BCS gap (∆/2) for the exactly solvable BCS model. There are also roots which stay real staying in
0
the segment ( ω,ε ) where ε is the intersection point of Γ with the real axis.
1 1
−
5
g = 1.5
1 g = 1.0
g = 0.5
µ
E
m 0
I
-1
-1 -0.5 0
ReE
µ
FIG.4: PlotoftherootsEµ fortheequallyspacedmodelinthecomplexplane[18]. Thediscretesymbolsdenotethenumerical
values for M =100. Thecontinuous lines are the analytical curvesobtained in [18]. All theenergies are in unitsof ω.
V. EXCITATIONS OF THE CANONICAL BCS MODEL
In the canonical ensemble, where the number of electrons is fixed, the excited states can be obtained from the GS
in two ways: by breaking Cooper pairs or by pair-hole excitations [4].
Breaking Cooper pairs
In fig. 5 we show at G=0 the excited state obtained by breaking the electron pair nearest to the Fermi level. The
spinupelectronremainsinthesameenergylevel,whilethespindowngoesonelevelup,producingtwoblockedlevels.
As shown in fig. 5 the problem becomes identical to that of two decoupled levels and a system with two less levels
active for pairing interactions. Hence for G> 0 the energy of this excitation is the sum of the single particle energy
of the blocked levels plus the GS energy of the system with these levels being removed. This situation is similar to
whathappens whenthe systemhas an oddnumber of electronsor whenit is under the effectof anexternalmagnetic
field [3]. In both cases there are singly occupied levels, which only contribute with their non interacting energy. This
sort of excitations are the ones that have been mainly studied in the literature.
FIG. 5: Excited state obtained by breaking a Cooper pair at G = 0. On the rhs the singly occupied levels are blocked and
decouple from the rest of thesystem.
6
Pair-hole excitations
The pair-hole excitations consist in promoting a pair below the Fermi level to a level above it. At G = 0 the
resulting state can be viewed as a pair-hole excitation of the Fermi sea (see fig. 6). This sort of excitations belong to
the Hilbert space and were first consider in reference [4]. It is clear that a general excitation consists in the
N,M
H
combination of broken Cooper pairs and pair hole-excitations. The rest of this review will focuse on the latter ones.
ε
8
ε
7
ε
6
ε
5
ε
4
ε
3
ε
2
ε
1
FIG. 6: Two possible excited states obtained by moving one and two pairs above theFermi level.
In fig. 7 we plot the real part of E (g) M for three states, labelled I ,I ,I and a system with N = 40 energy
{ µ }µ=1 0 1 2
levels and M =20 pairs as a function of the BCS dimensionless coupling g from0 to 1.5. The states are givenby the
choices
I = 1,...,18,19,20 , I = 1,...,18,19,21 , I = 1,...,18,20,22 . (20)
0 1 2
{ } { } { }
The state I is the GS of the system and the pattern that follows ReE (g) is the same as in fig. 3. Notice that as
0 µ
g all the roots become complex and escape to infinity. The state I is the lowest energy state of the system,
1
→ ∞
and as we see in fig. 7, in the limit g , the root E (g) stays real and finite while the remaining M 1 = 19
20
→ ∞ −
rootsescape to . Finally, the stateI has two rootsE (g) andE (g)that stay realandfinite inthe g limit.
2 20 19
∞ →∞
This is a general feature of the numerical solutions of the Richardson’s equations (15), namely in the limit g
→ ∞
there are N roots E (g) that remain finite, not necessarily real, while the M N remaining ones go to . As we
G µ G
− ∞
shall see below the number N of finite roots can be interpreted as the number of elementary excitations of a given
G
excited state.
Ground State (a) 1 excitation (b) 2 excitations (c)
4
0
-4
-8
-12
d
/ -16

µ
E -20

e -24
R
-28
-32
-36
-40
-44
0 0.5 1 0 0.5 g 1 0 0.5 1 1.5
FIG. 7: Real part of Eµ for theequally spaced model with M =N/2=20 pairs and NG =0,1,2 excitations [4].
7
68
1 excitation
64
2 excitations
60
3 excitations
56
52
48
44
d
40

/
36
c
ex32
E
28
24
20
16
12
8
4
0
0 0.5 1 1.5
g
FIG. 8: Excitation energies Eexc = E −EGS ≤ 14d for M = 20 pairs at half filling [4]. There are 44 =13+26+5 states
corresponding toNG =1,2,3 respectively. The particle-hole symmetry reducesthese numbersto 25 = 7+15+3.
Excitation energies
To support this conjecture we have computed the excitation energy E = E E of some low lying excited
exc GS
−
states. Fig.8 shows the energies E for a system with M = N/2 = 20 as a function of g. At g = 0 E is simply
exc exc
givenby the energy needed to lift the pairs fromthe Fermi sea up to the unoccupied energy levels, but as g increases
the excitation energies quickly converge to asymptotas whose slope is given essentially by N ,
G
lim E =N ∆, ∆ gω. (21)
exc G
g→∞ ∼
MoreoverusingthemethodsofRefs.7,18onecanshowinthelargeN limitthattheexcitationenergyofaRichardson
state is given by
NG
E = (E ε )2+∆2 , (22)
exc α 0
−
αX=1p
where ε is twice the chemical potential, and the energies E NG are the ones that remain finite in the g
0 { α}α=1 → ∞
limit. In the latter limit one has ∆ gω and eq. (22) becomes eq. (21). The excitation energy given by eq. (22) fits
∼
quite well the excitation energies of our prototype example (N =40, M =20), as shown in fig. 8.
In order to compare our results with the BCS standard solution let us consider the excitation energy of a real
Cooper pair, γ† γ† in the Bogoliubov approach, which is given by ε2+∆2 (notice that ∆ 2∆ ). The
j+ j− j ≡ BCS
q
standard Bogoliubovquasiparticle with an energy 1 ε2+∆2 would have to be comparedwith excitations involving
2 j
q
broken Cooper pairs. Since E in eq. (22) lies between two energy levels, with ε ε = 2d 1/N (e.g. in fig. 7b
α j+1 j
− ∼
E ( )=0withε <E <ε ),E =ε +O(1/N). Therefore,ourtheoryisconsistentwithinO(1/N)corrections,
20 20 20 21 α j
∞
as it is well known from the existing relation between the canonical and grand canonical ensembles.
VI. COUNTING THE NUMBER OF STATES WITH NG ELEMENTARY EXCITATIONS
We said above that the dimension of the Hilbert space of the states with M pairs distributed into N
N,M
different energy levels is given by the combinatorial numberHCN, which is also equal to the number of solutions of
M
the Richardson’s equations (15). The next question is: how many of these solutions contain N roots that remain
G
finite in the g limit? Let us call d that number which must obviously satisfy CN = M d .
→ ∞ N,M,NG M NG=0 N,M,NG
P
8
In reference [7] Gaudin made the following conjecture
CN CN 0 N M
dN,M,NG =(cid:26) 0NG − NG−1 N≤G >GM≤ . (23)
In table 1 we give some examples of this formula. Since d = 1, there is only one state where all the energies E go
0 µ
to infinity as g , which is nothing but the GS of the system.
→∞
No of solutions d d d =N 1 d =1
M M−1 1 0
··· −
E finite M M 1 1 0
µ
− ···
E infinite 0 1 M 1 M
µ
··· −
Table 1.- Classification of roots in the g limit. Here d stands for d .
→∞ NG N,M,NG
Thisconjecturecanbe motivatedasfollows. Supposethatinthe limitg there isaset E ( ) NG ofN roots
→∞ { α ∞ }α=1 G
thatremainfinite whilethe remainingM N onesescapetoinfinity. Thenfromeqs.(15)onederivesthatthe roots
G
−
E =E ( ) satisfy
α α
∞
N 1 NG 2
0= , α=1,...,N . (24)
G
ε E − E E
Xj=1 j − α βX6=α β − α
These equations are due to Gaudin and they appear in the diagonalization of a spin chain model known as Gaudin
magnets [20]. The number of solutions of eqs. (24) was shown by Gaudin to be given by (23) [7]. In reference [4] we
checked numerically this Gaudin’s conjecture for small systems. Furthermore we were able to show, in the large N
limit, that for finite g the roots E =E (g) satisfy an “effective” Gaudin equations
α α
N 1 NG 2
0= , (25)
R(ε )(ε E ) − R(E )(E E )
Xj=1 j j − α βX6=α β β − α
with R(E)= (E ε )2+∆2. As g one has ∆ gω and eqs. (25) become eqs. (24).
0
− →∞ ∼
p
VII. A CLASSIFICATION PROBLEM
Gaudin’s conjecture (23) gives the number d of states with a given value of N , but says nothing of how
N,M,NG G
to find them. Recall that the solutions of Richardson’s eqs. (15) are obtained by starting from a given initial state
at g = 0, labelled by the set I, and then increasing the value of g. Hence the problem is to find for each state I
which is the number of roots N , which remains finite in the g limit. This defines the function N (I) which,
G G
→ ∞
in physicalterms, gives the number of elementary excitations present in the state. The total number of states I with
N =N (I) is nothing but d .
G G N,M,NG
Finding N (I) is a highly non trivial problem because it connects the two extreme cases g =0 and g = . From
G
∞
a numerical point of view it is also a challenging problem due to the existence of several sorts of singularities in the
merging and splitting of roots with no obvious a priori pattern, except for the GS. Despite of these difficulties it is
possible to give a simple algorithm yielding N (I), which has an interesting combinatorial interpretation in terms of
G
Young diagrams [4]. Before we give this algorithm we need some preliminary definitions.
Path representation of states and Young diagrams
ThefirstideaistoassociatetoeverystateI,intheHilbertspace ,apathγ inthelattice 2. Thisisdone,in
N,M I
thepair-holerepresentationofI,byassociatingtoeveryemptyleveHl, ,ahorizontallinkin 2,anZdtoeveryoccupied
◦ Z
level, ,averticallink,startingfromthelowestenergystate. Placingtheinitialpointofγ at(0,0),thenitsfinalpoint
I
is at(•M,N M). The number ofup andrightgoingpaths from(0,0)and(N M,M)is givenby CN,whichequals
− − M
the dimension of . This means that the path representation is faithful. In fig. 9 we show the paths associated
N,M
H
tothe groundstateI andanexcitedstateI atg =0. The pathγ isgivenby(0,0) (0,M) (N M,M),while
0 I0 → → −
the pathassociatedto anyexcitedstateI lies alwaysbelowit. This factmakespossibleto associateaYoungdiagram
Y to every state I, whose boundary is given by the paths γ and γ with their common links being removed (see
I I I0
9
FIG. 9: Path representation of the ground state I0 and an excited state I. Combining these two paths one gets the Young
diagram YI. The numberof elementary excitations of thestate I is given by NG =3, according to therule (26).
fig. 9). The GS is associated to the empty Young diagram, i.e. Y = Ø. Using these definitions, the funtion N (I)
I0 G
is given by the following algorithm [4]:
N (I)=numberof boxesonthelongestSW NEdiagonalof Y . (26)
G I
−
In the example shown in fig.9 we get N = 3. The algorithm (26) was proposed in reference [4] to explain a large
G
amount of numerical simulations. As explained in [4] the physical mechanism underlying eq. (26) is the collective
behaviorof the holes and pairsthat occupy the closestlevels to the Fermi level. There arealsocombinatorialreasons
to support it (see below), however an analytical proof is needed.
(a) (b)
4 (1)
F
0
-4
(2)
-8
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) d -12
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) / -16
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) Eµ -20
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) e
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) R -24
F -28
-32
-36
-40
-44
0 0.5 1 1.5
g
FIG. 10: a) The path and Youngdiagram of an excited state with NG =3. The dotted line denotes the position of the Fermi
level. b) Real part of Eµ. For g large enough there is a real root (1) and a complex root (2) in agreement with the result
NG =3.
The non-triviality of (26) can be seen in fig. 10, which shows the evolution of the real part of the roots E as a
µ
function of g. Only for sufficiently large values of g, and after a complicated pattern of fusion and splitting of roots
10
(2 real roots 1 complex root) one observes that, indeed, N = 3. It seems very difficult to explain this result on
G
↔
the basis of Bogoliubov’s quasiparticle picture which only works properly for large number of particles in the grand
canonical ensemble.
We have yet no proof of eq. (26), however there is the following consistency check based on Gaudin’s conjecture
(23): d must be equal to the number of Young diagrams, Y , associated to the paths γ going from (0,0) to
N,M,NG I I
(M,N M), which have N boxes on their longest SW-NE diagonal. The proof of this result uses the methods of
G
−
Ref. 21. This resultcanalsobe provedusing the RSOS counting formulas. (We thanks A. Berkovichfor pointing out
this fact). As an example we show in fig. 11 how the paths corresponding to the case N = 6,M = N = 2 can be
G
mapped into RSOS paths.
We would like finally to mention the close relationship between the path representation and the associated Young
diagrams of the BCS states with the crystal basis used in the Fock space representation of the affine quantum group
U (Sl(2))[22,23,24]. Atthemomentofwrittingthispaperwedonotknowifthereexistsaquantumgroupstructure
q
underlying the exactly solvable BCS model. This may indeed be the case given that the integrability of the BCS
d
model can be explained from an inhomogenous XXZ spin chain with boundary operators [11, 12, 13, 14].
N=6, M=N = 2
G
1) ALL PATHS (0,0) −−> (4,2) 2) ADDING TWO BOXES
(4,2)
(0,0)
3) PATHS WITH N G = 2 4) MAPPING TO RSOS PATHS
FIG. 11: 1) Shows the C26 = 9 paths from (0,0) to (4,2), 2) The paths corresponding to NG = 2 are those below the shaded
boxes, 3) Weselect only those pathswith NG =2 and 4) after arotation theNG =2 pathsbecome RSOSpathson aBratelli
diagram.
VIII. CONCLUSIONS
We have shownin reference[4] that the pair-holeexcited states of the exactly solvable BCSmodel in the canonical
ensemblecanbeinterpretedintermsofelementaryexcitationswithpeculiarcountingpropertiesrelatedtotheGaudin
model, which have no counterpart in the BCS-Bogoliubov’s picture of quasiparticles.
The combinatorics involved in the counting of elementary excitations is similar to that of RSOS models, which
suggests that these excitations are in fact solitons which were called “gaudinos” in [4].
Some open problems are 1) the relation between these gaudinos and the Bogoliubov quasiparticles, 2) analytic
proof of the algorithm for computing the number of gaudinos N (I) for each BCS state, 3) working out the physical
G
consequences of these excitations in the canonical ensemble for small systems, 4) generalization of these results to
higher spin representations and other Lie groups and supergroups.