Bethe vectors for XXX-spin chain

Journal of Physics: Conference Series OPEN ACCESS Bethe vectors for XXX-spin chain To cite this article: estmír Burdík et al 204 J. Phys.: Conf. Ser. 563 020 View the article online for updates and enhancements. Related content - On correlation functions of integrable models associated with the six-vertex R- matrix N Kitanine, K Kozlowski, J M Maillet et al. - Functional relations and the Yang-Baxter algebra Wellington Galleas - Antiperiodic dynamical 6-vertex model by separation of variables II: functional equations and form factors D Levy-Bencheton, G Niccoli and V Terras This content was downloaded from IP address 48.25.232.83 on 09/04/208 at 23:45

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 Bethe vectors for XXX-spin chain Čestmír Burdík, Jan Fuksa,2, Alexei Isaev 2 Department of mathematics, Faculty of nuclear sciences and physical engineering, Czech Technical University in Prague, Trojanova 3, 20 00 Prague, Czech Republic 2 Bogoliubov Laboratory of Theoretical Physics, JINR, Joliot-Curie 6, 4980 Dubna, Moscow region, Russia E-mail: burdices@kmlinux.fjfi.cvut.cz, fuksajan@fjfi.cvut.cz, isaevap@theor.jinr.ru Abstract. The paper deals with algebraic Bethe ansatz for XXX-spin chain. Generators of Yang-Baxter algebra are expressed in basis of free fermions and used to calculate explicit form of Bethe vectors. Their relation to N-component models is used to prove conjecture about their form in general. Some remarks on inhomogeneous XXX-spin chain are included.. Introduction Algebraic Bethe ansatz has turned out as remarkably sufficient tool in the theory of quantum integrable systems. Its origins come up to the 80 s and are connected mainly with Leningrad shool. Since that time, it was used successfuly to solve an amount of models. In this text we are going to discuss some features of well-known model solved by this method, the so called XXX-spin chain. In section 2, we repeat some properties of XXX-spin chain in terms of algebraic Bethe ansatz. In sections 3, we introduce free fermions and use them in section 4 to express generators of Yang-Baxter algebra. The aim of this text is to calculate an explicit form of Bethe vectors for XXX-spin chain in fermionic basis which is discussed in sections 5. To prove our conjectures about their form, we are forced to use N-component model in section 6. At the end of this text, in section 7, we mention some results for inhomogeneous case. 2. Algebraic Bethe ansatz for XXX-spin chain Suppose we have a chain of L nodes. A local Hilbert space h j C 2 corresponds to the j-th node. The total Hilbert space of the chain is H L h j L C 2. Let A be an operator acting on h C 2. We use the following notation for operator A acting nontrivially only in the space h j H and trivially in others throughout the text A j I j A I L j. 2 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 The basic tool of algebraic Bethe ansatz is Lax operator which is an parameter depending object acting on the tensor product V a h i explicitly defined as L a,i λ : V a h i V a h i 3 L a,i λ λ + 2 I a,i + 3 σa α Si α 4 where σa α are usual Pauli matrices acting in V a, Si α 2 σα i are spin operators on the i-th node and I a,i is identity matrix in V a h i. L a,i λ can be expressed as a matrix in the auxiliary space V a λ + L a,iλ 2 + Sz i Si S i + λ + 2. 5 Sz i Its matrix elements form an associative algebra of local operators acting in the quantum space h i. Introducing permutation operator P P 3 I I + σ α σ α 6 2 α here I denotes 2 2 unit matrix, we can rewrite it as α L a,i λ λi a,i + P a,i. 7 Assume two Lax operators L a,i λ resp. L b,i µ in the same quantum space h i but in different auxiliary spaces V a resp. V b. The product of L a,i λ and L b,i µ makes sense in the tensor product V a V b h i. It turns out that there is an operator R ab λ µ acting nontrivially in V a V b which intertwines Lax operators in the following way R ab λ µl a,i λl b,i µ L b,i µl a,i λr ab λ µ. 8 Relation 8 describes the so called fundamental commutation relation. The explicit expression for R ab λ µ is R ab λ µ λ µi a,b + P a,b 9 where I a,b resp. P a,b is identity resp. permutation operator in V a V b. In matrix form λ µ + 0 0 0 R ab λ µ 0 λ µ 0 0 λ µ 0. 0 0 0 0 λ µ + The operator R ab λ µ is called R-matrix and satisfies Yang-Baxter equation R ab λ µr ac λr bc µ R bc µr ac λr ab λ µ in V a V b V c. Comparing 7 and 9, we see that Lax operator and R-matrix have exactly the same form. Yang-Baxter algebra of global operators acting on the Hilbert space H of the full chain is defined via a monodromy matrix T a λ L a, λl a,2 λ... L a,l λ 2 2

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 which is a product of Lax operators along the chain, i.e. over all quantum spaces h i. Matrix elements of T a λ in the auxiliary space V a Aλ Bλ T a λ 3 Cλ Dλ provide generators of Yang-Baxter algebra. Equation 8 provides the following relation for monodromy matrix R ab λ µt a λt b µ T b µt a λr ab λ µ 4 which describes commutation relations for generators of Yang-Baxter algebra Aλ, Bλ, Cλ and Dλ. We call it global fundamental commutation relation. Equation 4 implies commutativity of transfer matrices τλτµ τµτλ 5 where transfer matrix τλ is defined as the trace of monodromy matrix over auxiliary space V a τλ Tr a T a λ Aλ + Dλ. 6 Obviously, transfer matrix τλ is a polynomial of degree L in λ τλ 2 λ + L 2 L + λ k Q k. 7 2 Due to commutativity 5 of transfer matrices, we see that operators Q k mutually commute k0 [Q j, Q k ] 0. 8 Hamiltonian of the chain is expresed in terms of Pauli matrices resp. permutation operators L H k α 3 Sk α Sα k+ 2 L P k,k+ L 4 k 9 where we impose cyclic condition S L+ S resp. P L,L+ P L,. It can be expressed as a function of transfer matrix H d 2 dλ ln τλ λ /2 L 4. 20 This is the reason why we can say that transfer matrix τλ is a generating function for commuting conserved charges. 2.. Global fundamental commutation relations Let us express global commutation relations 4 for generators of Yang-Baxter algebra. After a simple factorization, R-matrix 0 can be written as R ab λ fλ 0 0 0 0 gλ 0 0 gλ 0 0 0 0 fλ 2 3

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 where fλ λ+ λ T a λ Aλ Cλ and gλ λ. The matrices T aλ resp. T b µ take the form Aλ Cλ Bλ Dλ Bλ Dλ resp. T bµ Aµ Cµ Bµ Dµ Aµ Cµ Bµ Dµ Comparing matrix elements of 4 on the positions,,, 4, 4,, 4, 4 we obtain Comparing 2, 3 resp. 2, 2. 22 [Aλ, Aµ] [Bλ, Bµ] [Cλ, Cµ] [Dλ, Dµ] 0. 23 [Bλ, Cµ] gλ, µ DµAλ DλAµ, 24 [Aλ, Dµ] gλ, µ CµBλ CλBµ. 25 From comparison of matrix elements, 3, 3, 4, 2, resp. 4, 3 we obtain where AµBλ fλ, µbλaµ + gµ, λbµaλ, 26 DµBλ fµ, λbλdµ + gλ, µbµdλ, 27 AµCλ fµ, λcλaµ + gλ, µcµaλ, 28 DµCλ fλ, µcλdµ + gµ, λcµdλ, 29 fλ, µ fλ µ λ µ +, gλ, µ gλ µ λ µ λ µ. 30 We see that gµ, λ gλ, µ. 2.2. Eigenstates of transfer matrix Commutation relations 23-30 for Yang-Baxter algebra together with an assumption that the Hilbert space H has structure of a Fock space are sufficient to encover spectrum of the transfer matrix τλ. There is a pseudovacuum vector 0 H such that Cλ 0 0 which is an eigenvector of operators Aλ and Dλ It is a tensor product of local pseudovacua Aλ 0 αλ 0, Dλ 0 δλ 0. 3 0 0 0 2 0 L 32 where 0 k 0. It can be easily seen that αλ λ + L, dλ λ L. Other eigenstates of transfer matrix 6 are of the form λ,..., λ M Bλ Bλ 2... Bλ M 0 33 with eigenvalue M M Λλ, {λ} αλ fλ i, λ + δλ fλ, λ i. 34 i i 4

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 They are called Bethe vectors. For M N, we refer to Bethe vector λ,..., λ M as M-magnon state. To get eigenstates of the transfer matrix, parameters {λ} {λ,..., λ M } have to satisfy Bethe equations for all i,..., M M M αλ i gλ, λ i fλ k, λ i + δλ i gλ i, λ fλ i, λ k 0. 35 k k i Explicit form of Bethe equations 35 is λi + L M λ i k k i k k i λ i λ k + λ i λ k. 36 3. Free Fermions Our first aim is to express Bethe vectors in fermionic basis. We start with definition of free fermions. For tensor product of L copies of C 2 we define free fermions as ψ k k σ z j σ + k, ψk k Commutation relations for the fermions 37 are of the form σ z j σ k. 37 [ ψ i, ψ j ] + δ ij I, [ ψ i, ψ j ] + 0, [ψ i, ψ j ] + 0. 38 It is a straightforward task to check the following identities ψ k+ ψ k + ψ k ψ k+ + ψ k ψk+ + ψ k+ ψ k σ x k σx k+, 39 ψ k+ ψ k + ψ k ψ k+ ψ k ψk+ ψ k+ ψ k σ y k σy k+, 40 [ψ k, ψ k ] σ z k, 4 2 ψ k ψ k 2 ψ k+ ψ k+ σ z k σz k+. 42 4. Fermionic realization of monodromy matrix Equation 7 provides us an easy expression for Lax operator. Identity operator I is a member of algebra of fermions. Therefore, it remains to know a fermionic realization only for permutation operator P a,i. Let us start with permutation operator P k,k+ permuting just the neighboring vector spaces h k and h k+. Due to identities 39-42 and definition of permutation operator 6, it is straightforward to check that P k,k+ I + ψ k+ ψ k + ψ k ψ k+ ψ k ψ k ψ k+ ψ k+ + 2 ψ k ψ k ψk+ ψ k+. 43 Permutation operator P j,k in non-neighboring vector spaces h j, h k where j < k, becomes a non-local in terms of fermions. Using properties of Pauli matrices, it can be rewritten as The first part is local even in the terms of fermions P j,k 2 I + σz j σk z + σ+ j σ k + σ j σ+ k. 44 2 I + σz j σ z k I ψ k ψ k ψ j ψ j + 2 ψ k ψ k ψj ψ j, 45 5

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 but the second part is nonlocal σ j + σ k + σ j σ+ k ψ j ψ k + ψ k j ψ k σl z ψ j ψk + ψ k j ψ k I 2 ψ l ψ l. 46 lj The nonlocality of P j,k resp. R j,k λ is a serious problem. There appear difficulties when we attempt to express monodromy matrix 2 in terms of such nonlocal operators. We need to avoid the nonlocality. Let us remind that L a,i λ R a,i λ. For R-matrix R ab λ satisfying Yang-Baxter equation we can define the matrix ˆR ab λ R ab λp ab which satisfies ˆR ab λ ˆR bc λ + µ ˆR ab µ ˆR ab µ ˆR bc λ + µ ˆR ab λ. 47 Substituting L a,i λ ˆR a,i λp a,i in monodromy matrix 2, we obtain very convenient expression T a λ L a, λl a,2 λ... L a,l λ ˆR a, λp a, ˆRa,2 λp a,2... ˆR a,l λp a,l ˆR a, λ ˆR,2 λ... ˆR L,L λp L,L... P,2 P a,. 48 It contains operators ˆR k,k+ resp. P k,k+ acting only in the neighboring spaces h k h k+. Fermionic realization of R-matrix ˆR k,k+ λ is ˆR k,k+ λ λp k,k+ +I λ+i+λ ψ k+ ψ k + ψ k ψ k+ ψ k ψ k ψ k+ ψ k+ +2 ψ k ψ k ψk+ ψ k+. 49 To get fermionic represenation of Yang-Baxter algebra means to express monodromy matrix 48 as 2 2 matrix in the auxiliary space V a C 2. For this purpose, we rewrite 48 as where the operator Xλ T a λ ˆR a, λxλp a, 50 Xλ ˆR,2 λ... ˆR L,L λp L,L... P,2 5 acts nontrivially only in the quantum spaces H h h L and is a scalar in the auxiliary space V a. Moreover, we know due to equations 43 and 49 how to express Xλ in the terms of fermions. Permutation matrix 6 can be rewritten as P a, I I + σ x σ x + σ y σ y + σ z σ z 2 [ ] I 0 0 σ x 0 iσ y σ z 0 + 2 0 I σ x + 0 iσ y + 0 0 σ z 2 I + σz 2 σx iσ y ψ ψ ψ I N ψ 2 σx + iσ y 2 I σz ψ ψ ψ ψ N where we have used 37, 4, and N ψ ψ. Hence, we get lj λ + I λn λ ˆR a, λ I a,i + λp a, ψ λψ λn + I 52. 53 6

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 Using 52 and 53, monodromy matrix 50 can be written in the following form λ + I λn λ T a λ ψ I N ψ Aλ Bλ Xλ λψ λn + I ψ N Cλ Dλ where 54 Aλ λ + λn Xλ N + λ ψ Xλψ, 55 Bλ λ + λn Xλ ψ + λ ψ XλN, 56 Cλ λψ Xλ N + λn + Xλψ, 57 Dλ λψ Xλ ψ + λn + XλN. 58 5. Fermionic realization of Bethe vectors The goal of our text is to find expression for Bethe vectors 33. For this purpose, fermionic realization 56 of the creation operator Bλ is convenient. The operator Xλ ˆR 2 λ... ˆR L,L λp L,L... P 2 can be written in terms of fermions due to equations 43 and 49. It can be easily seen that ψ k 0 0 59 for all k,..., L. If we are able to write Bλ in normal form our work would be simple. Unfortunately, it seems as a rather difficult task. Instead, we have to use the weak approach, i.e. to apply Bλ on the pseudovacuum 0 and try to commute the fermions ψ k to the right and see what remains. For our purposes, we need the following set of useful identities, which follow from equations 52 and 53 where N k ψ k ψ k, again. We can see that and For higher magnons, we need also and P k,k+ ψk ψ k+ ψ k+ N k + ψ k N k+, 60 ˆR k,k+ λ ψ k+ ψ k+ + λ ψ k + λ ψ k+ N k ψ k N k+ 6 P k,k+ ψk 0 ψ k+ 0, 62 ˆR k,k+ µ ψ k+ 0 ψ k+ 0 + µ ψ k 0, 63 ˆR k,k+ µ 0 µ + 0, 64 P k,k+ 0 0. 65 ˆR k,k+ λ ψ k ψk+ λ + ψ k ψk+ 66 ˆR l,l+ λ... ˆR k,k λ ψ k 0 λ k l ψl 0 + λk l k l λ + j ψl+j 0, 67 λ + λ ˆR l,l+ λ... ˆR k,k+ λ ψ k 0 ˆR l,l+ λ... ˆR k,k λ ψ k 0 + λλ + k l ψk+ 0. 68 7

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 5.. -magnon It is obvious that the second term in 56 annihilates the pseudovacuum state. Then, using 67, we get for -magnon state µ Bµ 0 µ + µn ˆR 2 µ... ˆR L,L µp L,L... P 2 ψ 0 µ + µn ˆR 2 µ... ˆR L,L µ ψ L 0 µ + µn [µ L ψ 0 + µl µ + µl µ + L µ + k ψk 0 nµ k where we introduce concise notation 5.2. 2-magnon Using 67 and 68 we obtain + λl 2 k 2 L k λ + m0 µ L µ + µ j ψj+ ] 0 L [µ] k ψk 0 69 k [µ] µ + µl, and nµ µ µ +. 70 Bλ ψ k 2 k 0 λ + λ L 2 [λ] m ψm+ ψk 0 + m0 [λ] j+m ψm+ ψk+j 0 + λl L k λ + [λ]k [λ] j ψk ψk+j 0. 7 We get for 2-magnon state using 7 [ L λ, µ BλBµ 0 nµ [µ] k Bλ ψ L k 2 k 0 nµnλ [µ] k [λ] m+2 ψm+ ψk + + λλ + nµnλ L k 2 k2 m0 nµnλ r<s L k k2 m0 ] L k L L k [µ] k [λ] j+m+ ψm+ ψk+j + [µ] k [λ] k+j ψk ψk+j 0 + λλ + [ { r<s L L s 2 s3 r kr+ k [[µ] s [λ] r+ + [µ] r [λ] s ] ψ r ψs 0 + s [µ] s [λ] r+ + [µ] r [λ] s + [µ] k [λ] s+r k ψr ψs 0 λλ + s kr+ } [µ] k [λ] s+r k ] ψ r ψs 0. 72 The finite sum in 72 can be calculated explicitly by means of geometric progression µ λλ µ [µ]r+ [λ] s s [µ] k [λ] s+r k λλ + kr+ [µ] s r [λ] µ [µ] s [λ] r [µ] r+ [λ] s. λλ µ 73 8

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 Substitution of 73 into 72 gives formula for 2-magnon BλBµ 0 nµnλ 5.3. 3-magnon For 3-magnon we need at first r<s L r<s L [ [µ] s [λ] r+ + [µ] r [λ] s + µ ] + [µ] s [λ] r [µ] r+ [λ] s ψr ψs 0 λλ µ [ ] nµnλ [λ] r [µ] s λ µ + λ µ + [µ]r [λ] s µ λ + ψ r ψs 0. 74 µ λ Bν ψ r ψs 0 ν + νn X 2...L ν ψ ψr ψs r 2 s r ν L 3 ν + 2 [ν] m ψm+ ψr ψs 0 + ν L 3 m0 L s +ν L 3 L s +ν L 3 ν + 2 m0 l r 2 [ν] l+m ψm+ ψr+l ψs 0 + m0 r 2 [ν] j+m ψm+ ψr ψs+j 0 + s r l r 2 [ν] j+l+m ψm+ ψr+l ψs+j 0 + m0 L s +ν L s+r ν + s r 2 r 2 [ν] j+m ψm+ ψs ψs+j 0 + m0 L s s r +ν L s+2 ν + s 3 [ν] j ψr ψs ψs+j 0 + ν L r ν + r [ν] l ψr ψr+l ψs 0 + L s +ν L r ν + r 3 s r l 3-magnon state is obtained from 2-magnon state 74 ν, µ, λ BνBµBλ 0 nµnλ where we denote for more comfort l [ν] j+l ψr ψr+l ψs+j 0. 75 r<s L K 2 s, rbν ψ r ψs 0 76 K 2 s, r [µ] s [λ] r λ µ + λ µ + [µ]r [λ] s µ λ + µ λ. 77 9

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 Using75, we get ν, µ, λ nνnµnλ q<r<s L + s r ν 2 [ν] j+q K 2 s j, r + + ν + 2 + ν + 2 σ q<r<s L σ S 3 r lq+ s lr+ where S 3 denotes symmetric group. [ [ν] q+2 K 2 s, r + r q ν 2 [ν] l+q K 2 s, r l+ ν 2 ν + 2 s r r q l l [ν] j+l+q K 2 s j, r l+ [ν] s l+q K 2 r, l + [ν] s 2 K 2 r, q + [ν] r K 2 s, q+ [ν] s+r l K 2 l, q [ν] q [µ] r [λ] s ν µ + ν µ ν λ + ν λ ] ψ q ψr ψs 0 nνnµnλ µ λ + ψq ψr ψs 0 78 µ λ 5.4. M-magnon From results 69, 74 and 78 we can conjecture that general M-magnon state is of the form λ,..., λ M Bλ Bλ M 0 M nλ i i k <...<k M L σ λ S M σ λ M i<j λ i λ j + λ i λ j where σ λ is a permutation of the parameters {λ,..., λ M } and M [λ i ] i k ψk ψ km 0 79 i σ λ S M is the sum over all such permutations. We are going to provide a proof of this conjecture in more general form below. 6. N-component model In the literature, cf. [2, 5], the so-called two-component model appears. It was introduced to avoid problems with computation of correlation functions for local operators attached to some node x of the chain in the Yang-Baxter algebra generated by 3. The chain [,..., L] is divided into two components [,..., x] and [x +,..., L]. Then, the Hilbert space is splitted into two parts H H H 2 where H h h x and H 2 h x+ h L. The pseudovacuum 0 H is of the form 0 0 0 2 where 0 H and 0 2 H 2. We define on V a H H 2 a monodromy matrix for each component A λ B T λ L a, λ L a,x λ λ, 80 C λ D λ resp. A2 λ B T 2 λ L a,x+ λ L a,l λ 2 λ. 8 C 2 λ D 2 λ Each of these monodromy matrices satisfies exactly the same commutation relations 4 as original undivided monodromy matrix 2. Moreover, we have A j λ 0 j α j λ 0 j, D j λ 0 j δ j λ 0 j, C j λ 0 j 0, 82 0

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 and operators corresponding to different components mutually commute. From construction, we see that αλ α λα 2 λ, δλ δ λδ 2 λ. 83 The full monodromy matrix T λ for the complete chain [,..., L] is A λa T λ T λ T 2 λ 2 λ + B λc 2 λ A λb 2 λ + B λd 2 λ, 84 C λa 2 λ + D λc 2 λ C λb 2 λ + D λd 2 λ and the M-magnon state is represented in the form M M λ,..., λ M Bλ k 0 A λ k B 2 λ k + B λ k D 2 λ k 0 0 2. 85 k k Izergin and Korepin [5] state that Bethe vectors of the full model can be expressed in terms of Bethe vectors of its components. To obtain this expression we should commute in 85 all operators A λ k and D 2 λ k to the right with the help of 26 and 27 and then use 82. Proposition. An arbitrary Bethe vector corresponding to the full system can be expressed in terms of the Bethe vectors of the first and second component. Let I {λ,..., λ M } be a finite set of spectral parameters. To concise notation below we will consider the set I as a finite set of indices I {,..., M}, then Bλ k 0 k I I I 2 k I δ 2 λ k B λ k k 2 I 2 α λ k2 B 2 λ k2 0 0 2 k I k 2 I 2 fλ k, λ k2 86 where fλ k, λ k2 is defined in 30 and the summation is performed over all divisions of index set I into two disjoint subsets I and I 2 where I I I 2. For detailed proof, please, see e.g. [2]. This result can be straightforwardly generalized to arbitrary number of components N L. Proposition 2. An arbitrary Bethe vector of the full system can be expressed in terms of the Bethe vectors of its components. For N L components the Bethe vector is of the form Bλ k 0 α i λ kj δ j λ ki fλ ki, λ kj k I k 2 I 2 i<j N I I 2 I N k I k N I N B λ k 0 B 2 λ k2 0 2 B N λ kn 0 N 87 where summation is performed over all divisions of the set I into its N mutually disjoint subsets I, I 2,..., I N. The proof is simply performed using 86 by induction on number of components N. More details will be included in [3] where more general inhomogeneous XXZ-chain is concerned. 6.. Bethe vectors explicitly By assumption we have a chain of length L. Let us divide it in L components, i.e. into L -chains. Using proposition 2 we get for M-magnon Bethe vector with M L: M Bλ k 0 k I I 2 I L k I k 2 I 2 k L I L i<j L α i λ kj δ j λ ki fλ ki, λ kj B λ k 0 B 2 λ k2 0 2 B L λ kl 0 L. 88

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 It holdsd for -chain, which is a chain with Hilbert space h C 2, that BλBµ 0 0. 89 Therefore, the sum over all divisions of {,..., M} into L subsets contains just divisions into subsets containing at most one element, i.e. I j 0,. Moreover, only M of them is nonempty, let us denote them I n, I n2,..., I nm. We have to sum over all possible combinations of such sets, i.e. over all M-tuples n < n 2 < < n M ; and then, to sum over all distributions of parameters λ, λ 2,..., λ M into the sets I n,..., I nm. After all, we get M Bλ k 0 k n <n 2 < <n M L B n λ B n2 λ 2 B nm λ M σ λ S M σ λ M n j i α i λ j L in j + j δ i λ j fλ i, λ j i 0 0 2 0 L. 90 Moreover, it holds for -chain that Bλ B is parameter independent and eigenvalues α i λ aλ, δ i λ dλ are still the same for all components i,..., L, where aλ λ + and dλ λ. We get M M j Bλ k 0 aλ j nj dλ j L n j fλ i, λ j k n <n 2 < <n M L n <n 2 < <n M L σ S M σ λ B n B n2 B nm 0 0 2 0 L M dλ j L M fλ i, λ j aλ j σ S M σ λ i<j M i aλj nj dλ j B n B n2 B nm 0 0 2 0 L. 9 Again, the reader is refered for more details to [3]. Let us return to 79. To prove its validity, it is sufficient to realize that ψ k ψk2... ψ km 0 B k B k2... B km 0 for k < k 2 < < k M. 7. Some remarks on inhomogeneous XXX-chain We start from inhomogeneous monodromy matrix T ξ a λ L a, λ + ξ L a,2 λ + ξ 2 L a,l λ + ξ L 92 where L a,j λ are Lax operators defined in 4 and ξ ξ..., ξ L is an inhomogenity vector. ξ Expressing T a λ in the auxiliary space V a, we get ξ T A ξ λ a λ C ξ λ B ξ λ D ξ λ where, again, operators A ξ λ, B ξ λ, C ξ λ and D ξ λ are acting in H h h L. Acting on pseudovacuum vector 0 H we get 93 A ξ λ 0 α ξ λ 0, 94 D ξ λ 0 δ ξ λ 0, 95 C ξ λ 0 0 96 2

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 where α ξ λ aλ + ξ aλ + ξ 2 aλ + ξ L, 97 δ ξ λ dλ + ξ dλ + ξ 2 dλ + ξ L. 98 Here, functions aλ λ + and dλ λ. For inhomogeneous version, we can introduce N-component model as well as for homogeneous Bethe ansatz. For 2-compotent model, for example, we have ξ T ξ a λ L a, λ + ξ L a,x λ + ξ x L a,x+ λ + ξ x+ L a,l λ + ξ L T ξ a λt 2 a λ 99 }{{}}{{} st component 2nd component where ξ ξ,..., ξ x resp. ξ 2 ξ x+,..., ξ L. We have A ξ ξ λ 0 α λα ξ 2 2 λ 0, D ξ ξ λ 0 δ λδ ξ 2 2 λ 0. 00 The very important property of inhomogeneous chain is, that its operators A ξ λ, B ξ λ, C ξ λ and D ξ λ satisfy the same fundamental commutation relations as homogeneous chain 23-30. Therefore, analogy of propositions and 2 can be easily formulated. Proposition 3. Let N L. An arbitrary Bethe vector of the full system can be expressed in terms of Bethe vectors of its N components B ξ λ k 0 k I I I N k I k N I N i<j N ξ α i i λ ξ k j δ j j λ k i fλ ki, λ kj B ξ λ k B ξ 2 2 λ k 2 B ξ N N λ kn 0. 0 To get explicit formula for Bethe vectors we have to divide the chain into L components of length as we did in the last section. We get for M-magnon M B ξ λ k 0 k n < <n M L σ λ S M σ λ M B ξn n λ B ξn M n M λ M n < <n M L M i L dλ j + ξ i σ λ S M σ λ B n B nm 0. M n j i 0 n j i α ξ i i λ j α ξ i i λ j σ λ n < <n M L σ λ S M L in j + L in j + M j i λ j fλ i, λ j δ ξ i δ ξ i i j i λ j fλ i, λ j B n B nm 0 aλ j + ξ nj i n j i j aλ j + ξ i dλ j + ξ i i fλ i, λ j 02 where, again, B-operators B ξn j n j λ B nj are parameter independent for -chains. For more details, see [3]. 3

XXII International Conference on Integrable Systems and Quantum Symmetries ISQS-22 IOP Publishing Journal of Physics: Conference Series 563 204 020 8. Final remarks We showed in this text explicit expressions for Bethe vectors of XXX-spin chain, both in fermionic and usual representation. We discussed also inhomogeneous version. We refer the reader for more details to [3] where we plan to discuss Bethe vectors for XXZ-chain in more detailed form. Acknowledgments JF thanks CTU for support via Project SGS2/98/OHK4/3T/4. authors API was supported by the grant RFBR 4-0-00474. The work of one of the [] Faddeev L D 996 How Algebraic Bethe Ansatz works for integrable model arxiv:hep-th/960587 [2] Slavnov N A 2007 Algebraic Bethe ansatz and quantum integrable models Uspekhi Fiz. Nauk 62 727 [3] Fuksa J, Isaev A P and Slavnov N A in preparation [4] Jordan P and Wigner E 928 Z. Phys. 47 63 [5] Izergin A G and Korepin V E 984 The quantum inverse scattering method approach to correlation functions Comm. Math. Phys. 94 No. 67-92 [6] Göhmann F and Korepin V E 2000 Solution of the quantum inverse problem J. Phys. A: Math. Gen. 33 99-220 4