Heisenberg model and Rigged Configurations

0.55 MB·English

by  Pulak Ranjan Giri

Heisenberg Model and Rigged Configurations
Pulak Ranjan Giri∗ and Tetsuo Deguchi†
Department of Physics, Graduate School of Humanities and Sciences,
Ochanomizu University, Ohtsuka 2-1-1, Bunkyo-ku, Tokyo, 112-8610, Japan
(Dated: February 2, 2015)
Weshowacorrespondenceofallthesolutionsofthespin-1/2isotropicHeisenbergmodelforN =
12 to the rigged configurations based on the comparison of the set of Takahashi quantum numbers
in lexicographical order with the set of riggings of the rigged configurations in co-lexicographical
order.
PACSnumbers: 71.10.Jm,02.30Ik,03.65Fd
5
1
0
2 I. INTRODUCTION

n
a The study of the isotropic spin-1/2 Heisenberg model still remains to be very much important in the researchfield
J of the integrable models due to its complex structure of solutions, which have not been completely understood till to

0 date. Although the eigenvalue equation was solved several decades ago [1], the completeness issue of the spectrum
3
still remains an open problem. For a deeper understanding of the isotropic spin-1/2 Heisenberg model, its variants

andforvariousapproachestosolvethesemodelsseerefs. [2–16]. Gettingthesolutionstothe Betheansatzequations,
]
h which produce eigenvalues and their corresponding Bethe eigenstates, are difficult because of their high degree of
t nonlinearity and multi-variate nature. Numerical method is the only way,where severalattempts have been made to
-
p obtain the solutions to the Bethe ansatz equations. First complete solutions for N =8,10 spin-1/2 XXX chains were
e obtained in ref. [17], where the Bethe ansatz equations are solved by an iteration method. On the other hand in ref.
h
[18] (see supplemental material) solutions up to N = 14 are obtained by making use of the homotopy continuation
[
method.

1 The solutions to the Bethe ansatz equations have a rich and diverse structure. As known to Bethe himself, the
v complex solutions come in complex conjugate pairs and typically arrangethemselves in a set ofstring like structures.
1 This apparent distribution of the rapidities in the form of strings leads to the string hypothesis [19]. The strings
0
together with their deformations are very much useful for the study of the finite length spin chains. Although it is
8
known that the string hypothesis does not work for some solutions of relatively large yet finite spin chains due to
7
0 the collapse of certain numbers to string solutions, it still works well in practical purpose for most of the solutions
. of the finite length chains. The complex solutions, which are responsible for the formation of bound states, present
1
more challenges in the numerical methods comparedto the real solutions. Nonetheless the knowledge of the complex
0
solutions are essential for the understanding of the dynamics of the model and for the evaluations of the correlation
5
1 functions[20,21]andotherphysicalquantities. Recentlyithasbeenshown[22]thatsomeofthesolutionsofN =12,
: which have some odd-length strings with complex centers, do not fit into the standard form of the string solutions.
v
Theexistenceofthecomplexcenteredodd-lengthstringsmakethosestringsnon-self-conjugateindividually,although
i
X thesolutionsasawholeremainself-conjugate. Anotherinterestingsolutionsarethesingularsolutions,which,because
r of their two constituent rapidities of the form ±i, make the eigenvalues and their corresponding Bethe eigenstates
a 2
ill-defined. A suitable regularization scheme [18, 23–25] is necessary to obtain well-defined eigenvalues and their
corresponding Bethe eigenstates for these type of solutions.
The different classes of solutions make the counting process, known as the completeness of the spectrum, very
difficult. Although string hypothesis provides a proof for the completeness, its certain assumptions are not legitimate
[26–30] for the finite length chains. One way out is to check the completeness by obtaining all the solutions of the
Bethe ansatz equations. On the other hand, there is a nice one-to-one and onto mapping between the solutions of
the Bethe ansatz equations and a combinatorialobject knownas the riggedconfiguration[31–34], which is helpful to
understand the completeness problem. However, there are certain solutions for the large length chains, such as the
solutions corresponding to the collapse of the 2-strings, which do not have straight forward mapping to the rigged
configurations. Recent conjecture [35] on the rigged configurations has predicted the numbers of singular solutions
presentinaspinchain. Although,thecorrespondencebetweenafewsolutionsofthe Betheansatzequationsforsome
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
lengths of the spin chain to the riggedconfigurations are known, the complete correspondences of all the solutions of
a spin chain has never been shown.
The purpose of this present paper is to provide a complete correspondence between all the solutions of the Bethe
ansatz equations of an isotropic spin-1/2 Heisenberg chain for N = 12 and the rigged configurations. The reason
for choosing N = 12 case is the following: For an even-length spin chain, N = 12 is the lowest length which has
some intriguing features in the solutions of the Bethe ansatz equations. One is, as mentioned above, there are some
roots, called non-self-conjugate string solutions, which do not fit in the standard framework of the string hypothesis.
The other being, which we will show in this paper, there are some solutions which seem to be of the form λ±0.5i.
However, numerical evaluations with higher working precision show that there are small imaginary deviation ǫNi
to these solutions, which are usually overlooked in the numerical methods. Note that N = 12 can be directly
diagonalized to obtain the eigenvalues. Recently it has been completely solved by homotopy continuation method
[18],wheretheabovementionedtwotypesofsolutionshavenotbegivenproperattention. Wealternativelystudythe
N =12 spin chain by an iteration method making use of the deformed strings structures of the solutions and obtain
a correspondence among the solutions to the Bethe quantum numbers, Takahashi quantum numbers and the rigged
configurations. OneadvantageofthismethodisthatitenablesustoobtaintheBethequantumnumbersandalsothe
Takahashi quantum numbers, which are compared to the riggings of a rigged configuration to obtain the mapping.
We organize this paper in the following fashion: In section II, we review the the algebraic Bethe ansatz method
for the spin-1/2 isotropic Heisenberg model. In section III we discuss the string solutions which include non self-
conjugate string solutions, almost singular string solutions and singular string solutions. In section IV we study the
correspondenceoftheriggedconfigurationstotheN =12spin-1/2isotropicHeisenbergmodelbasedonacomparison
of the Takahashi quantum numbers with the riggings and finally we conclude.
II. ALGEBRAIC BETHE ANSATZ METHOD
In the algebraic Bethe ansatz method the Lax operator for the spin-1/2 isotropic Heisenberg model is given by
λ−iSz −iS−
L (λ)= γ γ , (1)
γ −iS+ λ+iSz
(cid:18) γ γ (cid:19)
where S± =Sx±iSy, Sj(j =x,y,z)is the spin-1/2 operatoratthe i-thlattice site and in j-direction. Eachelement
γ γ γ i
of L (λ) is a matrix of dimension 2N ×2N, which acts nontrivially on the γ-th lattice site. The monodromy matrix,
γ
T(λ), is then given by the direct product of the Lax matrices at each site
A(λ) B(λ)
T(λ)=L (λ)L (λ)···L (λ)= . (2)
N N−1 1 C(λ) D(λ)
(cid:18) (cid:19)
The transfer matrix is obtained from the monodromy matrix (2) as
t(λ)=A(λ)+D(λ). (3)
The Hamiltonianofthe spin-1/2isotropicHeisenbergmodelonaone-dimensionalperiodiclattice oflengthN is then
obtained by taking the logarithm of t(λ) at λ= i as
2
N
J d 1
H = i logt(λ) −N =J SxSx +SySy +SzSz − . (4)
2 dλ i i+1 i i+1 i i+1 4
(cid:20) (cid:21)λ=2i ! Xi=1(cid:18) (cid:19)
The Bethe states for M down-spins are given by
M
|λ ,λ ,··· ,λ i= B(λ )|Ωi, (5)
1 2 M α
α=1
Y
where|Ωiisthe referenceeigenstatewithallup-spins, B(λ )isanelementofthe monodromymatrixT(λ )obtained
α α
from eq. (2) and λ are the rapidities. More explicitly the Bethe state (5) can be expressed as [36]
α
M B(λ )|Ωi= (−2i)M M λj −λk+i M (λj − 2i)N ×
α=1 α j<k λj −λk j=1 λj + 2i
Y Y Y
N M! M λPj −λPk−i H(j−k) M λPj + 2i xj M S−|Ωi, (6)
1≤x1<xX2···<xM≤NPX∈SMPjY<Pk(cid:18)λPj −λPk+i(cid:19) jY=1 λPj − 2i! jY=1 xj
3
where S is the permutation group of M numbers with elements P and the Heaviside step function is defined as
M
H(x)=1 for x>0 and H(x)=0 for x≤0. Acting the transfer matrix t(λ) from the left on the Bethe state (5) we
obtain
M M M M
t(λ) B(λ )|Ωi=Λ(λ,{λ }) B(λ )|Ωi+ Λ (λ,{λ })B(λ) B(λ )|Ωi, (7)
α α α k α α
α=1 α=1 k=1 α6=k
Y Y X Y
where eigenvalue of the transfer matrix is given by
N M N M
i λ−λ −i i λ−λ +i
α α
Λ(λ,{λ })= λ+ + λ− , (8)
α
2 λ−λ 2 λ−λ
(cid:18) (cid:19) α=1 α (cid:18) (cid:19) α=1 α
Y Y
and the unwanted terms are given by
N M N M
i i λ −λ −i i λ −λ +i
k α k α
Λ (λ,{λ })= λ + + λ − , k=1,2,··· ,M. (9)
k α λ−λ  k 2 λ −λ k 2 λ −λ 
k (cid:18) (cid:19) α=1 k α (cid:18) (cid:19) α=1 k α
 αY6=k αY6=k 
 
Eq. (7) reduces to the eigenvalue equation, if the unwanted terms (9) vanish, which enables us to obtain the Bethe
ansatz equations
λα− 2i N = M λα−λβ −i, α=1,2,··· ,M. (10)
λα+ 2i ! β=1 λα−λβ +i
Y
β6=α
Once the solutions λ of the Bethe ansatz equations (10), knownas the Bethe roots, are obtained, the eigenvalues of
α
the Hamiltonian H for the M down-spin sector are expressed as
M
J d 1 1
E = i logΛ(λ,{λ }) −N =−J . (11)
2 (cid:20)dλ α (cid:21)λ=2i ! 2αX=1 λ2α+ 41
The logarithmic form of eq. (10) is expressed as (cid:0) (cid:1)
M
2π 2
2arctan(2λ )=J + arctan(λ −λ ), α=1,2,··· ,M, mod 2π, (12)
α α α β
N N
β=1
X
β6=α
where {J ,α = 1,2,··· ,M} are the Bethe quantum numbers, which take integral (half integral) values if N −M
α
is odd (even) respectively. Since the Bethe quantum numbers are repetitive, they are not useful for the process of
counting the total number of states. Therefore a strictly non-repetitive quantum numbers are necessary, which are
obtained as follows.
In string hypothesis, the rapidities of a solution to the Bethe ansatz equations are arranged in form of strings of
different lengths as
i
λj =λj + (j+1−2a)+∆j , a=1,2,··· ,j, α=1,2,..,M , (13)
αa α 2 αa j
where the j-string of length j has the realcenter λj with α being the index to distinguish all the M strings of same
α j
length and ∆j are the string deviations. In the limit that these deviations vanish, ∆j →0, equations (10) reduce
αa αa
to the convenient form
2λj Ij 1 Ns Mk
arctan α = π α + Θ λj −λk , mod π,
j N N jk α β
k=1 β
XX (cid:0) (cid:1)
2λ 2λ 2λ 2λ
Θ (λ) = (1−δ )arctan +2arctan +···+2arctan +arctan , (14)
jk jk
|j−k| |j−k|+2 j+k−2 j+k
where M down-spins are partitioned into M k-stings with the length of the largest string being N such that
k s
NskM = M. We now obtain the strictly non-repetitive quantum numbers Ij, known as the Takahashi quantum
k k α
numbers, from eq. (14), which have the following bounds
P
1
|Ij |≤ N −1− [2min(j,k)−δ ]M . (15)
α 2 j,k k
!
k=1
X
4
III. STRING SOLUTIONS
The solutions of the Bethe ansatz equations are usually arranged in a set of strings. In string hypothesis language
the real solutions are 1-string solutions. Other solutions are composed of strings of different lengths as given in eq.
(13). In the numerical iteration method first the centers of the strings λj are evaluated, which are used as the initial
α
input for the next iteration method to calculate the deviations ∆j and the centers. For finite but not so large spin
αa
chains, the deviations of the strings are of the form
∆j =ǫj +iδj , (16)
αa αa αa
where the real ǫj and δj decreases exponentially with respect the length N of the spin chain. While implementing
αa αa
the iteration scheme, the complex conjugacy condition of the solutions are exploited to simplify the forms of these
deviations. One useful way to implement it is to consider the strings to be individually self-conjugate, which implies
[17]
∆j =(∆j )∗. (17)
αa αj+1−a
The centre for the odd-length strings with these conditions are inevitably real. Since each of the stings in a given
solution is self-conjugate, they are not related to each other. For the even-length spin chains up to N = 10 all the
solutions obey the conditions (17). Even most of the solutions for N = 12 given in Tables 1−6 also obey these
conditions. However a small number of solutions for N = 12 do not obey the self-conjugacy conditions (17), which
we discuss in the next subsection.
A. Non Self-conjugate String solutions
There are a few number of solutions for N = 12 spin-1/2 chain, which do not fall in the standard form of the
sting solutions discussed above. Note that in a scheme of small deviations, the self-conjugacy conditions (17) are too
restrictive to be valid for all solutions. It is therefor necessary to impose the self conjugacy condition on whole set of
the rapidities in a solution
{λj }={(λj )∗}. (18)
αa αa
Note that (18) implies (17), however the reverse is not true always. Relaxing the self-conjugacy condition as (18)
allows the centre of the odd-length strings to be complex with small imaginary part in such a way that the solutions
remainself-conjugate. Thesetypeofcomplexcenteredodd-stringsolutions,wecallnonself-conjugatestringsolutions,
have been observed to appear for even-length spin chains for N ≥ 12. Here we remark that it is essential to keep
the deviations small, so that the ∆j → 0 limit lead to the Takahashi equations (14) and the Takahashi quantum
αa
numbers (15) can be obtained.
The first two examples of such non self-conjugate string solutions appear in a set of solutions with two 1-strings
and one 3-string in M = 5 down-spins sector of the N = 12 spin chain. The self-conjugacy conditions (17) allow us
to write the rapidities of the solutions in a simple form as
{λ },{λ },{λ+ǫ+(1+δ)i,λ,λ+ǫ−(1+δ)i}, (19)
1 2
where the real parameters λ ,λ ,λ are the centers and ǫ,δ are small deviations. The curly brackets in the above
1 2
expressionsmeanasetofrapiditiesofagivensting. InTable 5.3allsolutionsexpecttwoareobtainedmakinguseof
(19)intheiterationprocess. Forthetwonon self-conjugate stringsolutionswemakeuseoftherelaxedself-conjugacy
condition (18), which allows us to write the rapidities as
{λ },{λ+iδ },{λ+ǫ+(1+δ )i,λ−iδ ,λ+ǫ−(1+δ )i}, (20)
1 1 2 1 2
where the real parameters λ ,λ are the centers and ǫ,δ ,δ are the deviations. The solutions 171 and 172 in Table
1 1 2
5.3 are obtained by making use of (20) in the iteration process. Notice that because of the imaginary term iδ the
1
1-string {λ+iδ } and the 3-string {λ+ǫ+(1+δ )i,λ−iδ ,λ+ǫ−(1+δ )i} are individually non self-conjugate
1 2 1 2
although together remain self-conjugate.
The next three examples of such non self-conjugate string solutions appear in a set of solutions with one 1-string,
one 2-string and one 3-string in M = 6 down-spins sector of the N = 12 spin chain. The self-conjugacy conditions
(17) allow us to write the rapidities of the solutions in a simple form as
i i
{λ },{λ + (1+2δ),λ − (1+2δ)},{λ+ǫ+(1+δ )i,λ,λ+ǫ−(1+δ )i}, (21)
2 1 1 1 1
2 2
5
where the real parameters λ ,λ ,λ are the centers and the real parameters δ,δ ,ǫ are small deviations. In Table 6.8
1 2 1
all solutions expect three are obtained making use of (21) in the iteration process. For the three non self-conjugate
string solutions we make use of the relaxed self-conjugacy condition (18), which allows us to write the rapidities as
i i
{λ+iδ },{λ + (1+2δ),λ − (1+2δ)},{λ+ǫ+(1+δ )i,λ−iδ ,λ+ǫ−(1+δ )i}, (22)
1 1 1 2 1 2
2 2
where the realparametersλ,λ ,δ are the centers and the realparametersδ,δ ,ǫ are smalldeviations. The solutions
1 2 1
105,118 and 119 in Table 6.8 are obtained by making use of (22) in the iteration process. The presence of the
imaginary term iδ makes the 1-string {λ+iδ } and the 3-string {λ+ǫ+(1+δ )i,λ−iδ ,λ+ǫ−(1+δ )i} non
1 1 2 1 2
self-conjugate individually although together they remain self-conjugate.
B. Almost singular string solutions
Therearesomesolutions{λ ,λ ,λ ,···λ }forN =12spin-1/2chain,inwhichapairofrapiditiesareofthe form
1 2 3 M
i i
λ =aǫ+ 1+2ǫN , λ =aǫ− 1+2ǫN , (23)
1 2
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
where a and ǫ are small numbers such that aǫ is real in the case of N = 12. ǫN is so small that in numerical
calculation the contribution of ǫNi does not usually appear unless the working precision is increased significantly.
Most calculations wrongly produce solutions, in which the above pair takes the form
i i
λ =aǫ+ , λ =aǫ− , (24)
1 2
2 2
as obtained in the supplements of ref. [18]. However,analytically also it can be argued that there does not exist any
solution containing a pair of rapidities of the form (24). Because of the similarity with the two regularized rapidities
of the singular solutions with (23), we call such solutions almost singular. For M = 4 down-spins sector there are
two such solutions 257,258 with two 2-strings shown in Table 4.4. For M = 5 down-spins sector there are six such
solutions, two solutions 48,49 with three 1-strings and one 2-string, which are shown in Table 5.2, two solutions
212,213 with one 1-string and one 4-string shown in Table 5.4 and the last two solutions 242,243 with one 1-string
and two 2-strings shown in Table 5.6. For M = 6 there are two such solutions 55,56 with two 1-strings and one
4-string shown in Table 6.4. Note that almost singular solutions obtained for N = 12 are all self-conjugate string
solutions.
C. Singular string solutions
There are some solutions, known as the singular string solutions, of the form
i i
λ = ,λ =− ,λ ,λ ,··· ,λ , (25)
1 2 3 4 M
2 2
n o
which produce ill-defined eigenvalues and the Bethe eigenstates due to the presence of the two rapidities of the form
λ = i,λ = −i. To obtain finite and well-defined eigenvalues and the Bethe eigenstates for these solutions it is
1 2 2 2
imperative to exploit a suitable regularization scheme [10, 23–25, 37, 38]. For the even length spin chains it has
been observed that the rapidities and the corresponding Takahashi quantum numbers are distributed symmetrically
making the sum of the rapidities to vanish
M
λ =0. (26)
α
α=1
X
Note that the condition (26) is not only satisfied by singular string solutions, but also by other solutions, which also
are invariant under the negation of their rapidities. The condition (26), noted in [17], forms a part of a conjecture in
ref. [35].
The simplest singular string solution, found in M =2 down-spins sector, is of the form
i i
,− . (27)
2 2
n o
6
From condition (26) it clear that there is one singular solution in this sector, which is given as solution 46 with one
2-string in Table 2.2. In M =3 down-spins sector also condition (26) gives one solution of the form
i i
,− ,0 . (28)
2 2
n o
In Table 3.2 such solutionis given in 85 with one 1-string andone 2-string. In M =4 down-spins sector two class of
solutions, compatible with the condition (26), are given by
i i
,− ,a ,−a for a ∈R, (29)
1 1 1
2 2
ni i o
,− ,ia ,−ia for a ∈R. (30)
1 1 1
2 2
n o
The solutions 71,76,91,98in Table 4.2 with two 1-stringsand one 2-stringare of the form(29) andthe solution 271
in Table 4.5 with one 4-string is of the form (30). In M =5 down-spins sector there are two classes of singular sting
solutions of the form
i i
,− ,0,a ,−a for a ∈R, (31)
1 1 1
2 2
ni i o
,− ,0,ia ,−ia for a ∈R. (32)
1 1 1
2 2
n o
The solutions 22,29,50 in Table 5.2 with three 1-strings and one 2-string are of the form of (31). Whereas the
solution 211 in Table 5.4 with one 1-string and one 4-string and the solution 283 in Table 5.7 with one 2-stringand
one 3-string are of the form of (32). In M =6 down-spins sector there are four classes of solutions of the form
i i
,− ,a ,−a ,a ,−a for a ,a ∈R, (33)
1 1 2 2 1 2
2 2
ni i o
,− ,a ,−a ,ia ,−ia for a ,a ∈R, (34)
1 1 2 2 1 2
2 2
ni i o
,− ,ia ,−ia ,ia ,−ia for a ,a ∈R, (35)
1 1 2 2 1 2
2 2
ni i o
,− ,a ±ia ,−a ±ia for a ,a ∈R. (36)
1 2 1 2 1 2
2 2
n o
Thesolutions2,11,16inTable6.2withfour1-stringsandone2-stringareoftheform(33). Thesolutions52,57,72,79
in Table 6.4with two 1-stringsandone 4-stringareof the form(34). The solution 89in Table 6.6with one 6-string
and the solution 105 in Table 6.8 with one 1-string, one 2-string and one 3-string are of the form (35). And finally
the solution 126 in Table 6.9 with three 2-strings are of the form (36).
IV. RIGGED CONFIGURATIONS
For finite length spin chains with no collapsing strings there is a correspondence between the solutions of the
Bethe ansatz equations and a combinatorial object, known as the rigged configurations [35, 37]. Although, this
correspondencehas been shownfor a few solutions of some given length spin-1/2chains, the complete bijection of all
the solutionsof the Bethe ansatzequations to the riggedconfigurationshaveneverbeen shown. Since this knowledge
of correspondence is very much important in the completeness of the solutions, we in this section discuss the the
bijection based on the comparison of the Takahashi quantum numbers with the riggings of the rigged configurations.
Before we study a correspondencebetween the solutions of the Bethe ansatz equations to the riggedconfigurations
let us briefly formulate the riggedconfigurationsfirst. These are the Young Tableau like objects, which havetwo sets
of integers to label them. The numbers on the left hand sides of the boxes are known as the vacancy numbers P (ν)
k
andthe numbersonthe righthandsides ofthe boxesareknownasthe riggingsJ . Inordertohavethese numbers,
k,α
first the total numbers of down-spins M in a given solution is divided into s parts as ν ={ν ,ν ,··· ,ν }, such that
1 2 s
the parts ν ’s are positive and satisfy s ν = M. For a spin-1/2 chain the vacancy number P (ν) for a string of
i i=1 i k
length k in a specific partition ν is given by
P
s
P (ν)=N −2 min(k,ν ) , (37)
k i
i=1
X
7
whereallthevacancynumbersarenon-negative. GivenavacancynumberthecorrespondingriggingsJ aredefined
k,α
as
0≤J ≤J ≤···≤J ≤P (ν), (38)
k,1 k,2 k,Mk k
The flip map κ among the riggings in a rigged configuration is defined as
κ(J )=P (ν)−J , (39)
k,α k k,Mk−α+1
whichplaysthesameroleasthetransformationofalltherapiditiesinasolutionundernegation. Foracorrespondence
between the solutions of the Bethe ansaz equations and the rigged configurations we need a rule such that a rigged
configuration can be assigned to a particular solution. Such assignment can be done by comparing the riggings J
with the real parts of the rapidities of the solutions of the Bethe ansatz equations. The higher value of the real part
of the solutions to the Bethe equations is assignedto the higher value of the riggings[35]. For sucha scheme to work
one needs to actually solve the Bethe ansatz equations to obtain the rapidities. Moreover sometimes the rapidities
of the solutions of the Bethe ansatz equations are very close to each other, which can make the scheme ineffective.
We therefore consider another approach in which the comparison between the riggings with the Takahashi quantum
numbers are done.
A. Rigged configurations and Bethe roots correspondence
The mapping f starts from the largest strings to the lowest strings in decreasing order. For example, for M = 6
down-spin sectors with one 3-string, one 2-string and one 1-string, ν = {3,2,1}, first the rigging {J } of 3-string
3,1
is compared with the Takahashi quantum number {I1}, then the rigging {J } of 2-string is compared with the
3 2,1
Takahashi quantum number {I1} and finally the rigging {J } of 1-string is compared with the Takahashi quantum
2 1,1
number {I1}. In this example a specific length of string appear once. There are solutions where a specific length of
1
string may appear more than once. Let us consider such an example in M = 6 down-spin sector with one 3-string
and three 1-strings, ν = {3,1,1,1}. In this case first the rigging {J } of 3-string is compared with the Takahashi
3,1
quantum number {I1}, then the the riggings{J ,J ,J } of the three 1-strings are compared with the Takahashi
3 1,1 1,2 1,3
quantum numbers {I1,I2,I3}. In a general perspective we have the following map
1 1 1
f :{J ,J ,··· ,J }→{I1,I2,··· ,IMk}. (40)
k,1 k,2 k,Mk k k k
When there is one string of a particular length in a partition, the set of rigging {J } is arrangedin increasing order
k,1
{J }={J1 ,J2 ,··· ,J2|Ik|+1}, (41)
k,1 k,1 k,1 k,1
where Ji <Ji+1. Similarly the set of Takahashi quantum numbers {I1} is also arranged in increasing order
k,1 k,1 k
{I1}={I1,1,I1,2,··· ,I1,2|Ik|+1}, (42)
k k k k
where I1,i <I1,i+1. We then easily obtain a mapping from {J } to {I1} as
k k k,1 k
f :Ji →I1,i i=1,2,··· ,2|I |+1. (43)
k,1 k k
When there are more than one k-strings in a partition, M 6=1, then arrangethe set of riggings in co-lexicographical
k
order [39] as
J1 ,J1 ,··· ,J1 , J2 ,J2 ,··· ,J2 ,··· , J2|Ik|+1CMk,J2|Ik|+1CMk,··· ,J2|Ik|+1CMk . (44)
k,1 k,2 k,Mk k,1 k,2 k,Mk k,1 k,2 k,Mk
n(cid:0) (cid:1) (cid:0) (cid:1) (cid:18) (cid:19)o
where any element Jki,1,Jki,2,··· ,Jki,Mk ,i = 1,2,··· ,2|Ik|+1CMk of the set (44) is arranged in weakly increasing
order defined in (38)(cid:16). In co-lexicographic(cid:17)alorder the elements i,j(i<j) of (44) satisfy the following relation
Ji ,Ji ,··· ,Ji < Jj ,Jj ,··· ,Jj , (45)
k,1 k,2 k,Mk k,1 k,2 k,Mk
(cid:0) (cid:1) (cid:16) (cid:17)
8
if and only if the m elements from the right satisfy Ji = Jj , l = M ,M −1,··· ,M −m+1 and Ji <
k,l k,l k k k k,Mk−m
Jj . The set of Takahashi quantum numbers are arrangedin lexicographicalorder [39]
k,Mk−m
I1,1,I2,1,··· ,IMk,1 , I1,2,I2,2,··· ,IMk,2 ,··· , I1,2|Ik|+1CMk,I2,2|Ik|+1CMk,··· ,IMk,2|Ik|+1CMk . (46)
k k k k k k k k k
n(cid:16) (cid:17) (cid:16) (cid:17) (cid:18) (cid:19)o
where any element is of the form Ik1,i <Jk2,i <···<JkMk,i ,i = 1,2,··· ,2|Ik|+1CMk. In lexicographical order the
i,j(i<j) elements of (46) satisfy t(cid:16)he following relation (cid:17)
I1,i,I2,i,··· ,IMk,i < I1,j,I2,j,··· ,IMk,j , (47)
k k k k k k
(cid:16) (cid:17) (cid:16) (cid:17)
if and only if the m elements from the left satisfy Il,i = Jl,j, l = 0,1,2,··· ,m−1 and Im+1,i < Im+1,j. Now the
k k k k
mapping between the elements of (44) and (46) are given by
f : Jki,1,Jki,2,··· ,Jki,Mk → Ik1,i,Ik2,i,··· ,IkMk,i , i=1,2,··· ,2|Ik|+1CMk,. (48)
On the right most co(cid:0)lumn of the Tables 1(cid:1)−6(cid:16)we have shown the(cid:17)rigged configurations for each of the solutions.
V. CONCLUSIONS
There is a natural correspondence between the solutions of the Bethe ansatz equations for the spin-1/2 isotropic
Heisenberg model to the riggedconfigurations. We in this paper study this correspondence by a comparisonbetween
the sets of the rigged configurations in the co-lexicographical order and the corresponding sets of the Takahashi
quantum numbers in the lexicographicalorder. As an example we consider the N =12 case, because the solutions in
this case starts to exhibit several interesting structures which are not present for any even length chain for N < 12.
For example, there are solutions in which some of the individual odd-length strings are not self-conjugate. These
solutions, known non self-conjugate string solutions, have been observed in [22] recently. Another type of solutions,
we call almost singular string solutions, also start to appear from N = 12, in which one pair of the rapidities takes
the form of eq. (23) with ǫN 6= 0. The existence of ǫN 6= 0 is very much important theoretically in the calculations
of the form factors, scalar product of the Bethe states. We obtained the Bethe quantum numbers and the Takahashi
quantum numbers, which are essential for the purpose of obtaining a correspondence between the solutions of the
Bethe equations and the rigged configurations.
9
TABLE 1: Bethe roots for N =12, M =1. There are 11 solutions with one 1-string
J In λ E Rigged Configuration
1. 0 01 0 −2 10 5
2. −1 −11 −0.1339745962155614 −1.8660254037844388 10 4
3. 1 11 0.1339745962155614 −1.8660254037844388 10 6
4. −2 −21 −0.2886751345948129 −1.4999999999999998 10 3
5. 2 21 0.2886751345948129 −1.4999999999999998 10 7
6. −3 −31 −0.5 −1 10 2
7. 3 31 0.5 −1 10 8
8. −4 −41 −0.8660254037844386 −0.5000000000000001 10 1
9. 4 41 0.8660254037844386 −0.5000000000000001 10 9
10. −5 −51 −1.8660254037844386 −0.13397459621556135 10 0
11. 5 51 1.8660254037844386 −0.13397459621556135 10 10
10
TABLE 2.1: Bethe roots for N =12, M =2. There are 45 solutions with two 1-strings
J In λ E Rigged Configuration
1. −1/2 −1/21 −0.07188914699748664 −3.918985947228995 8 4
1/2 1/2 0.07188914699748664
1 8 4
2. −1/2 −1/21 −0.05930327105788041 −3.660965635573501 8 3
−3/2 −3/2 −0.21467167951993998
1 8 3
3. 1/2 1/21 0.05930327105788041 −3.660965635573501 8 5
3/2 3/2 0.21467167951993998
1 8 5
4. −1/2 −1/21 −0.078210204956388 −3.6242082443384067 8 4
3/2 3/2 0.22146708985545924
1 8 5
5. −13//22 −13//221 0−.007.282211406270048995865358485924 −3.6242082443384067 8 3
1 8 4
6. −3/2 −3/21 −0.22834234895177763 −3.3097214678905704 8 3
3/2 3/2 0.22834234895177763
1 8 5
7. −−15//22 −−15//221 −−00..04501647072847409639154248471929935 −3.182542835560352 8 2
1 8 3
8. 1/2 1/21 0.051402449314441295 −3.182542835560352 8 5
5/2 5/2 0.40677870695287993
1 8 6
9. −1/2 −1/21 −0.08562408262485187 −3.126368759905621 8 4
5/2 5/2 0.4153665837561704
1 8 6
10. 1/2 1/21 0.08562408262485187 −3.126368759905621 8 2
−5/2 −5/2 −0.4153665837561704
1 8 4
11. −−53//22 −−53//221 −−00..3199678556285302859414272198459 −2.956923031168023 8 2
1 8 2
12. 3/2 3/21 0.197568325447185 −2.956923031168023 8 6
5/2 5/2 0.3968525089122949
1 8 6
13. −3/2 −3/21 −0.2361932593092894 −2.7989518211725315 8 3
5/2 5/2 0.4238121234847884
1 8 6
14. 3/2 3/21 0.2361932593092894 −2.7989518211725315 8 2
−5/2 −5/2 −0.4238121234847884
1 8 5
15. −1/2 −1/21 −0.04056536687024037 −2.6330645003205206 8 1
−7/2 −7/2 −0.7237559965997632
1 8 3
16. 1/2 1/21 0.04056536687024037 −2.6330645003205206 8 5
7/2 7/2 0.7237559965997632
1 8 7
17. −1/2 −1/21 −0.09548959777704773 −2.5611034015776353 8 4
7/2 7/2 0.7360621562567807
1 8 7
18. −17//22 −17//221 0−.009.7534680965291757672750647787037 −2.5611034015776353 8 1
1 8 4
19. −3/2 −3/21 −0.18387644015227955 −2.425995851844449 8 1
−7/2 −7/2 −0.7090278224426931
1 8 2
20. 3/2 3/21 0.18387644015227955 −2.425995851844449 8 6
7/2 7/2 0.7090278224426931
1 8 7
21. −5/2 −5/21 −0.43325246627153147 −2.28462967654657 8 2
5/2 5/2 0.43325246627153147
1 8 6