Completeness of Bethe s states for generalized XXZ model, II.

Transcription

1 Copleteness of Bethe s states for generalized XXZ odel, II Anatol N Kirillov and Nadejda A Liskova Steklov Matheatical Institute, Fontanka 27, StPetersburg, 90, Russia Abstract For any rational nuber 2 we prove an identity of Rogers- Raanujan s type Bijection between the space of states for XXZ odel and that of XXX odel is constructed The ain goal of our paper is to study a cobinatorial relationship between the space of states for generalized XXZ odel and that for XXX one In our previous paper [KL] we gave a cobinatorial description of states for generalized XXZ odel in ters of the so-called rigged sl(2) XXZ configurations On the other hand it is well-known that when the anisotropy paraeter of XXZ odel goes to infinity then the XXZ odel under consideration transfors to the XXX one We are going to describe this transforation fro cobinatorial point of view in the case when is an integer A cobinatorial copleteness of Bethe s states for generalized XXXodel was proven in [K] and appears to be a starting point for nuerous applications to cobinatorics of Young tableaux and representation theory of syetric and general linear groups, see eg [K2] Here we ention only a ferionic forula for the Kostka-Foulkes polynoials, see eg [K2], and the relationship of the last with ŝl(2)- branching functions bkλ 0 λ (), see

2 eg [K3] We will show in, Theore 2, that -counting of the nuber of XXZ states using Bethe s ansatz approach [TS], [KR], gives rise to the Rogers-Raanujan type forula for any rational nuber > 2 It sees an interesting proble to find a polynoial version of the Rogers- Raanujan type identity fro our Theore 2 Another uestion which we are interested in is to understand a cobinatorial nature of the liit XXZ XXX + In 2 we will describe a cobinatorial rule which shows how the XXZconfigurations fall to the XXX pieces For siplicity we consider in our paper only the case > s General case will be considered elsewhere Rogers-Raanujan s type identity This paper is a continuation of our previous work [KL] Let us reind the ain definitions, notations and results fro [KL] For fixed R, 2 let us define (cf [TS]) a seuence of real nubers p i and seuences of integer nubers ν i, i,y i : :=,p =, ν i = [ pi p i+ ],p i+ = p i ν i p i,i=,2, () y =0, y 0 =, y = ν 0,y i+ = y i + ν i y i,i=0,,2, (2) 0 =0, =ν 0, i+ = i + ν i,i=0,,2, (3) r(j) =i, if i j< i+,j=0,,2, (4) It is clear that integer nubers ν i define the decoposition of into a continuous fraction =[ν 0,ν,ν 2,]=ν 0 + ν + ν 2 + Let us define (see Fig ) a piecewise linear function n j, j 0, n j := y i +(j i )y i, if i j< i+ (5) 2

3 It is clear that for any integer n> there exists the uniue rational nuber t such that n = n t y i n j y i+ y i+ y i y i +y i y i y i y i +y i 2 y i y i 2 6 * i i + i i i + i+ i+ - j Fig Iage of piecewise linear function n j in the interval [ i, i+ ] Let us introduce additionally the following functions (see [KL]) j =( ) i (p i (j i )p i+ ), if i j< i+, (6) ( k k n χ ), if n k > 2s, 2p Φ k,2s = 0 ( k χ n k )+ ( )r(k), if n k 2s, 2 2 where 2s = n χ 3

4 In order to forulate our ain result of part I about the nuber of Bethe s states for generalized XXZ odel, let us consider the following syetric atrix Θ =(c ij ) i,j α+ : i) c ij = c ji and c ij =0, if i j 2 ii) c j,j =( ) i, if i j< i+ { 2( ) iii) c ij = i, if i j< i+, i α, ( ) i, if j = α+ Exaple For =4+ 5 one can find Θ = Further, let us consider a atrix E =(e jk ) j,k α+,where e jk =( ) r(k) (δ j,k δ j,α+ δ k,α+ + δ j,α+ δ k,α+ ) Then one can check that (cf [KL], (39)) P j (λ)+λ j =((E 2Θ) λ t + b t ) j, where b =(b,,b α+ )and ( { } b j =( ) r(j) 2s N 2l n j 2Φ j,2s N ) 4

5 Theore ([KL]) The nuber of Bethe s states of generalized XXZ odel, Z XXZ (N, s l), iseualto ( ) ((E B) λ t + b t ) j, (7) λ j where suation is taken over all configurations λ = {λ k } such that α+ k= n k λ k = l, λ k 0; λ j λ =( λ, λ α+ ), λ j =( ) r(j) λ j, B =2Θ One of the ain goal of the present paper is to consider a natural analog for (7) Naely, let us define the following analog of the su (7) [ ] 2 λb λt ((E B) λ t + b t ) j, (8) λ j where ɛ j =( ) r(j) [ ] M Let us reind that is the Gaussian binoial coefficient: N (M) [ ] M, if 0 N M, (N) = (M N) N 0 otherwise Reark In our previous paper [KL], see (5) and (52), we had considered another -analog of (7) It turned out however that the -analog (5) fro [KL], probably, does not possess the good cobinatorial properties One of the ain results of Part II is the following Theore 2 (Rogers-Raanujan s type identity) Assue that be a rational nuber, 2,and λ j ɛ j V l () = λ λb λt, (9) (; ɛ(j) ) λj j 5

6 where suation in (9) is taken over all configurations λ = {λ k } such that l = k n k λ k, λ k 0 Then we have k 0 ( ) k k 2 + k(k ) 2 ( + k )= l 0 l2 V l () A proof is a -version of that given in [KL], Theore 4 2 XXZ XXX bijection In this section we are going to describe a bijection between the space of states for XXZ-odel and that of XXX-odel Let us forulate the corresponding cobinatorial proble ore exactly First of all as it follows fro the results of our previous paper, the cobinatorial copleteness of Bethe s states for XXZ odel is euivalent to the following identity N (2s +) N = Z XXZ (N, s l), (0) l=0 where N = 2s N and Z XXZ (N, s l) is given by (7) On the other hand it follows fro the cobinatorial copleteness of Bethe s states for XXX odel (see [K]) that 2 N (2s +) N = (N 2l +)Z XXX (N, s l), () l=0 where Z XXX (N, s l) is the ultiplicity of ( N 2 l) -spin irreducible representation of sl(2) in the tensor product V N s V N s LetusrearkthatbothZ XXZ (N, s l) andz XXX (N, s l) aditsa cobinatorial interpretation in ters of rigged configurations The difference between the space of states of XXX odel and that of XXZ odel is the 6

7 availability of the so-called -configurations (or string) in the space of states for the last odel The presence of -strings in the space of states for XXZ-odel is a conseuence of broken sl(2)-syetry of the XXZodel Our goal in this section is to understand fro a cobinatorial point of view how the anisotropy of XXZ odel breaks the syetry of the XXX chain More exactly, we suppose to describe a bijection between XXZ-rigged configurations and XXX-rigged configurations Let us start with reinding a definition of rigged configurations We consider at first the case of sl(2) XXX-agnet Given a coposition µ =(µ,µ 2,) and a natural integer l by definition a sl(2)-configuration of type (l, µ) is a partition ν l such that all vacancy nubers P n (ν; µ) := k in(n, µ k ) 2 k n ν k (2) are nonnegative Here ν is the conjugate partition A rigged configuration of type (l, µ) is a configuration ν of type (l, µ) together with a collection of integer nubers {J α } n(ν) α= such that 0 J J 2 J n(ν) P n (ν;µ) Here n (ν) is eual to the nuber of parts eual to n of the partition ν It is clear that total nuber of rigged configurations of type (l, µ) iseualto Z(l µ):= ( ) Pn (ν;µ)+ n (ν) ν ln n (ν) The following result was proven in [K] Theore 3 Multiplicity of (N 2l +)-diensional irreducible representation of sl(2) in the tensor product V N s Vs N is eual to Z l 2s,,2s }{{,,2s },,2s }{{} N N 7

8 Exaple 2 One can check that V 5 =6V 0 +5V +5V 2 +0V 3 +4V 4 +V 5 In our case we have µ =(2 5 ) Let us consider l = 5 It turns out that there exists three configurations of type (3, (2 5 )), naely ( ) Hence Z(3 (2 5 ))=+2+3=6=Mult V0 V 5 Now let us give a definition of sl(2)-xxz configuration We consider in our paper only the case when the anisotropy paraeter is an integer, Z 2 Under this assuption the forulae (5) and (6) take the following for: n j = j, if j<,v j =+; n p0 =, v p0 = ; 2Φ k,2s = 2sk in(k, 2s), if k<, 2s+< ; 2Φ p0,2s = 2s, if 2s +< ; b kj = k j, if j k< ; b kp0 =, if k< ; a j := a j (l µ) = [ ] µ 2l in(j, µ ) 2l j, if j< ; [ ] a p0 (l µ)= µ 2l Definition A sl(2)-xxz-configuration of type (l, µ) is a pair (λ, λ p0 ), where λ is a coposition with all parts strictly less than, j< jλ j +λ p0 = l, and such that all vacancy nubers P j (λ µ) are nonnegative 8

9 Let us reind ([KL]) that P j (λ µ) :=a j (l µ)+2 (k j)λ k + λ p0, if j< ; (3) j<k< P p0 (λ µ) :=a p0 (l µ)+λ p0 ; P p0 (λ µ):=a p0 (l µ)+λ p0 Exaple 3 Let us consider =6,s= 3,N=5,l= 5 The total 2 nuber of type (5, (3 5 )) sl(2) XXZ configurations is eual to The total nuber of type (5, (3 5 )) rigged configurations is eual to Z XXZ (5 (3 5 ))=0= Here we used a sybol to ark a strings Now we are ready to describe a ap fro the space of states for XXZ odel to that of XXX one More exactly we are going to describe a rule how a XXZ-configuration fall to the XXX-pieces At first we describe this rule scheatically: k λ λ λ λ λ π k 9

10 This decoposition corresponds to the well-known identity [ ] [ ] + k k + j = k j j j=0 In what follows we will assue that > s Theore 4 The ap π is well-defined and gives rise to a bijection between the space of states of XXZ-odel and that of XXX one Proof Let us start with rewriting the forulae (3) for the XXZ-vacancy nubers in ore convenient for, naely, Pj XXZ ( ν µ)= in(j, 2s ) 2 [ ] ν k 2s 2l j, p k j 0 if j< ; P XXZ ( ν µ)= P XXZ ( ν µ)= { } [ 2s 2l + [ ] 2s 2l + p0 (ν) 2s 2l ] +λ p0 ; (4) Here µ =(2s,,2s )and νis a pair ν =(ν, λ p0 ), where ν is a partition such that l(ν), ν +λ p0 = l Relationship between λ fro Definition and ν is the following j (ν) =λ j, ie ν =( λ 2 λ 2 ( ) λ ) Now let us consider an integer l s and let ν l be a XXXconfiguration Let λ p0 be integer such that 2 s 2l <λ p0 s l and consider the pair ν =(ν, λ p0 ) It is easy to check that Pj XXZ ( ν µ)= if j< ; in(j, 2s ) 2 k j ν k = P XXX j (ν µ) 0, Pp XXZ 0 ( ν µ)= 2s 2l+λ p0 0; Pp XXZ 0 ( ν µ)=λ p0 0 0

11 Thus the pair ν =(ν, λ p0 )isaxxz-configuration Furtherore it follows fro our assuptions (naely, s <, λ p0 > 0) that λ p0 = 0 and both -strings and ( )-strings do not give a contribution to the space of XXZ-states Thus we see that both XXX-configuration ν and XXZ-configuration ν =(ν, λ p0 ) defines the sae nuber of states Now, if ν =(ν, µ) isaxxz-configuration then ν is a XXX configuration as well This is clear because (see (4)) Pj XXX (ν µ) Pj XXZ ( ν µ), j By the siilar reasons if ( ν,λ p0 ) is a XXZ-configuration then for any 0 k λ p0 the pair ( ν,λ p0 k) isalsoxxz-configuration It follows fro what we say above that π is the well-defined ap Furtherore there exists one to one correspondence between the space of XXX-configurations and that of XXZ-configurations, naely, ν ν =(ν, λ p0 ), where λ p0 =[ s ν ] All others XXZ-configurations (ν, k) with0 k< s ν give a contribution to the space of descendants for ν ν Acknowledgents We are pleased to thank for hospitality our colleagues fro Tokyo University, where this work was copleted References [K] AN Kirillov, Cobinatorial identities and copleteness of states for the generalized Heisenberg agnet, Zap Nauch Se LOMI, 984, v3, p88-05 [K2] AN Kirillov, On the Kostka-Green-Foulkes polynoials and Clebsh-Gordan nubers, Journ Geo and Phys, 988, v5, n3, p [K3] AN Kirillov, Dilogarith identities, Progress of Theor Phys Suppl, 995, n8, p6-42

12 [KL] AN Kirillov and NA Liskova, Copleteness of Bethe s states for generalized XXZ odel, Preprint UTMS-94-47, 994, 20p [TS] M Takahashi and M Suzuki, One diensional anisotropic Heisenberg odel at finite teperature, Progr of Theor Phys, 972, v48, N6B, p [KR] AN Kirillov, NYu Reshetikhin, Exact solution of the integrable XXZ Heisenberg odel with arbitrary spin, J Phys A: Math Gen, 987, v 20, p