Bethe vectors for XXX-spin chain

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.

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 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 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) = (λ + 1 2 )I a,i + 3 α=1 σ α a S α i (4) where σ α a are usual Pauli matrices acting in V a , S α i = 1 2 σ α 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 L a,i (λ) = λ Its matrix elements form an associative algebra of local operators acting in the quantum space h i . Introducing permutation operator P P = 1 2 (here I denotes 2 × 2 unit matrix), we can rewrite it as L a,i (λ) = λI a,i + P a,i .
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 Relation (8) describes the so called fundamental commutation relation. The explicit expression where I a,b resp. P a,b is identity resp. permutation operator in V a ⊗ V b . In matrix form The operator R ab (λ − µ) is called R-matrix and satisfies Yang-Baxter equation (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,1 (λ)L a,2 (λ) . . . L a,L (λ) provide generators of Yang-Baxter algebra. Equation (8) provides the following relation for monodromy matrix which describes commutation relations for generators of Yang-Baxter algebra A(λ), B(λ), C(λ) and D(λ). We call it global fundamental commutation relation. Equation (14) implies commutativity of transfer matrices where transfer matrix τ (λ) is defined as the trace of monodromy matrix over auxiliary space V a Obviously, transfer matrix τ (λ) is a polynomial of degree L in λ Due to commutativity (15) of transfer matrices, we see that operators Q k mutually commute Hamiltonian of the chain is expresed in terms of Pauli matrices resp. permutation operators where we impose cyclic condition S L+1 = S 1 resp. P L,L+1 = P L,1 . It can be expressed as a function of transfer matrix This is the reason why we can say that transfer matrix τ (λ) is a generating function for commuting conserved charges.

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 τ (λ). There is a pseudovacuum vector |0 ∈ H such that C(λ) |0 = 0 which is an eigenvector of operators A(λ) and D(λ) It is a tensor product of local pseudovacua where |0 k = ( 1 0 ). It can be easily seen that α(λ) = (λ + 1) L , d(λ) = λ L . Other eigenstates of transfer matrix (16) are of the form with eigenvalue Explicit form of Bethe equations (35) is

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 Commutation relations for the fermions (37) are of the form It is a straightforward task to check the following identities 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+1 permuting just the neighboring vector spaces h k and h k+1 . Due to identities (39)-(42) and definition of permutation operator (6), it is straightforward to check that Permutation operator P j,k in non-neighboring vector spaces h j , h k where j < k − 1, 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 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 (12) 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 (11) we can define the matrixR ab (λ) = R ab (λ)P ab which satisfieŝ Substituting L a,i (λ) =R a,i (λ)P a,i in monodromy matrix (12), we obtain very convenient expression It contains operatorsR k,k+1 resp. P k,k+1 acting only in the neighboring spaces (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 X(λ) acts nontrivially only in the quantum spaces H = h 1 ⊗ · · · ⊗ h L and is a scalar in the auxiliary space V a . Moreover, we know due to equations (43) and (49) how to express X(λ) in the terms of fermions. Permutation matrix (6) can be rewritten as where we have used (37), (41), and N 1 =ψ 1 ψ 1 . Hence, we get R a,1 (λ) = I a,i + λP a,1 = (λ + 1)I − λN 1 λψ 1 λψ 1 λN 1 + I . where

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 B(λ) is convenient. The operator X(λ) = R 12 (λ) . . .R L−1,L (λ)P L−1,L . . . P 12 can be written in terms of fermions due to equations (43) and (49). It can be easily seen that for all k = 1, . . . , L.
If we are able to write B(λ) 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 B(λ) 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 For higher magnons, we need alsô

2-magnon
Using (67) and (68) we obtain We get for 2-magnon state using (71) The finite sum in (72) can be calculated explicitly by means of geometric progression

M-magnon
From results (69), (74) and (78) we can conjecture that general M -magnon state is of the form where σ λ is a permutation of the parameters {λ 1 , . . . , λ M } and 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.
Each of these monodromy matrices satisfies exactly the same commutation relations (14) as original undivided monodromy matrix (12). Moreover, we have and operators corresponding to different components mutually commute. From construction, we see that The full monodromy matrix T (λ) for the complete chain [1, . . . , L] is and the M -magnon state is represented in the form 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 1 (λ k ) and D 2 (λ k ) to the right with the help of (26) and (27) where f (λ k 1 , λ k 2 ) is defined in (30) and the summation is performed over all divisions of index set I into two disjoint subsets I 1 and I 2 where I = I 1 ∪ 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 k∈I B(λ k ) |0 = where summation is performed over all divisions of the set I into its N mutually disjoint subsets I 1 , 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.

Bethe vectors explicitly
By assumption we have a chain of length L. Let us divide it in L components, i.e. into L 1-chains. Using proposition 2 we get for M -magnon (Bethe vector) with M ≤ L: It holdsd for 1-chain, which is a chain with Hilbert space h = C 2 , that Therefore, the sum over all divisions of {1, . . . , M } into L subsets contains just divisions into subsets containing at most one element, i.e. |I j | = 0, 1. Moreover, only M of them is nonempty, let us denote them I n 1 , I n 2 , . . . , I n M . We have to sum over all possible combinations of such sets, i.e. over all M -tuples n 1 < n 2 < · · · < n M ; and then, to sum over all distributions of parameters λ 1 , λ 2 , . . . , λ M into the sets I n 1 , . . . , I n M . After all, we get Moreover, it holds for 1-chain that B(λ) = B is parameter independent and eigenvalues α i (λ) = a(λ), δ i (λ) = d(λ) are still the same for all components i = 1, . . . , L, where a(λ) = λ + 1 and d(λ) = λ. We get Again, the reader is refered for more details to [3]. Let us return to (79). To prove its validity, it is sufficient to realize thatψ k 1ψ k 2 . . .ψ k M |0 = B k 1 B k 2 . . . B k M |0 for k 1 < k 2 < · · · < k M .
To get explicit formula for Bethe vectors we have to divide the chain into L components of length 1 as we did in the last section. We get for M -magnon where, again, B-operators B ξn j n j (λ) = B n j are parameter independent for 1-chains. For more details, see [3].

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.