Bethe states for the two-site Bose-Hubbard model: a binomial approach

We calculate explicitly the Bethe vectors states by the algebraic Bethe ansatz method with the $gl(2)$-invariant $R$-matrix for the two-site Bose-Hubbard model. Using a binomial expansion of the n-th power of a sum of two operators we get and solve a recursion equation. We calculate the scalar product and the norm of the Bethe vectors states. The form factors of the imbalance current operator are also computed.


Introduction
The first experimental verification of the Bose-Einstein condensation (BEC) [1,2,3] occurred after a gap of more than seven decades following its theoretical prediction [4,5]. After its realization a great deal of progress has taken place both in the theoretical and experimental study of this physical phenomenon [6,7,8,9,10,11,12,13]. A particularly fruitful instrument in relation to ultracold physics are many-body atomic models related to BEC. In this direction the quantum inverse scattering method (QISM) [14,15,16,17,18] has been used to solve and study some prototypical many-body models that contribute to describe phenomena associated to BEC [19,20,21]. Some of these models, despite their simplicity, display a rich structure showing quantum phase transitions and interesting semi-classical behaviour that have been studied in [22,23,24,25,26,27], and explored in different areas such as nuclear physics, condensed matter and atomic-molecular physics. To keep things as simple as possible we shall consider here the two-site Bose-Hubbard, also known in special cases as the canonical Josephson Hamiltonian [7]. This model may be viewed as a particular case of the bosonic multi-state two-well model studied in [21], and can be used to describe a quadrupolar nuclei system in nuclear magnetic resonance [28] by a N/2 Schwinger pseudo-spin realization of the Hamiltonian. Conversely, there is also a link with the Schwinger bosonic realization of the Lipkin-Meshkov-Glick model [29,30] used to study closed shells in a nuclei model.
In spite, of course, from being a two-site specialization of the Bose-Hubbard model it is a very useful model in various realms as the understanding of tunnelling phenomena using two BEC [31,32,33,34,35,36,37], as well as quantum phase transitions using tools of quantum computation and quantum information. The model is described by the Hamiltonian where,â † 1 ,â † 2 , denote the single-particle creation operators in each site and,N 1 =â † 1â 1 ,N 2 = a † 2â 2 , are the corresponding boson number operators. The total boson particles number operator,N =N 1 +N 2 , is a conserved quantity, [Ĥ,N] = 0. The coupling K provides the interaction strength between the bosons and is proportional to the s-wave scattering length, ∆µ is the external potential and E J is the amplitude of tunnelling.
A very important problem in the algebraic Bethe ansatz method is the construction of the Bethe vectors states (BVS) [38,46,47] using the correspondent creation operator applied to the pseudo-vacuum. Employing this form of the BVS it is possible [47] to calculate their scalar product and then use it to calculate important physical quantities as the form factors. Form factors are defined as the matrix entries of operators in the base of the eigenvectors of the Hamiltonian. Another important application is in the calculation of the average values of the operators as for example correlation operators. Applying this method, some physical quantities for the Hamiltonian (1.1) were obtained in [38]. Recently, BVS have shown to be useful in fundamental issues of planar N = 4 super Yang-Mills (SYM) theory [48,49] in the context of the integrability in the AdS/CFT correspondence [50].
In the present work, we develop and use a new method to explicitly calculate the BVS and obtain the scalar product of two BVS for the two-site Bose-Hubbard model. Although we concentrate on this model the procedure is of general applicability. We use a Lax operator to construct a realization of the monodromy matrix and get an algebraic identity between the associated C-operator and the D-operator of the monodromy matrix (see next section) needed to calculate the BVS. We then use the binomial expansion for the n-th power of the sum of two operators: in a first step we will show that the binomial expansion can be written as a sum of permutations of the product of that operators or as a standard binomial expansion as in a commutative algebra plus a function of the commutator of these two operators; in a second step we write a recursion equation and give its solution. Next, we calculate the scalar product between one on-shell and one generic off-shell BVS, as well as the norm. As an application we obtain the form factors (non normalized) for the imbalance current operator.
It is easy to check that R(u) satisfies the Yang-Baxter equation where R jk (u) denotes the matrix acting non-trivially on the j-th and the k-th spaces and as the identity on the remaining space.
Next we define the monodromy matrixT (u), such that the Yang-Baxter algebra is satisfied To obtain a solution for the two-site Bose-Hubbard model (1.1) we need to choose a realization for the monodromy matrix π(T (u)) =L(u). In this construction, the Lax operator L(u) has to satisfy the algebra where we use the standard notation.
We are using the well known [19] Lax operator, solution of the equation (2.6), for the boson operatorsâ † i ,â i , andN i . These operators obey the standard canonical boson commutation rules.
The parameters of the Hamiltonian (1.1) are all real numbers, K, ∆µ, E J ∈ R. The parameters in the operators (2.10,2.11,2.12,2.13) can be complex numbers, u, η, ω ∈ C, but in this case the transfer matrix is not Hermitian. We will only consider the Hermitian case.

Bethe vectors states
In the algebraic Bethe ansatz method, the BVS are constructed by the application of thê C-operator to the pseudo-vacuum |0 , where the {v j } N 1 are solutions of the BAE (2.14).
Using theD-operator (2.13) we can write theĈ-operator (2.12) as, Now we can write the BVS in the product form as 19) or in the summation form as with the identification To explicitly write the BVS (3.19) or (3.20) we need to expand the powers of theDoperator (3.18). We thus consider the following n-th power binomial expansion for any two operatorsX andŶ , with n ≥ 2, proved by induction, If the operators commute, f([X,Ŷ ]) = 0, we clearly get the standard commutative binomial formula. In this case the n-th power of theD-operator (2.13) is given bŷ The n-th power of theD-operator (3.26) is acting in the pseudo-vacuum as D n |0 = η −2n |0 .

(3.27)
Redefining theD-operator (3.18) asD =X +Ŷ , we get the binomial expansion of theD-operator (3.18) for the n-th power, Applying the n-th power of theD-operator (3.30) to the pseudo-vacuum we get where the coefficients C n,j satisfies the recursion equation with the condition C j,j = 0.
The solution of the recursion equation (3.32) is Finally, using the binomial expansion of theD-operator (3.31) we can write the BVS (3.19) as The scalar product between one on-shell and one generic off-shell BVS (3.34), |Ψ and |Ψ , with the set of solutions of the BAE (2.14) {v j } N 1 and the generic set {ṽ j } N 1 [52], is From the scalar product (3.35) we can write the norm of the BVS (3.34), Because the total number of atoms is a conserved quantity all the BVS (3.34) are eigenfunctions of theN -operator with the same eigenvalue N, and so they are degenerate states for this operator. AsN 1 andN 2 are not conserved quantities the BVS are not eigenfunction of these operators but we can still write the form factors of these operators using the BVS (3.34). For instance, the non normalized form factors of the imbalance current between the two BEC is written as n j=0F * r F n × C r,r−n+j C n,j 1 − 2(n − j) N × (N − n + j)! j!(r − n + j)! . (3.37)

Summary
We have explicitly written the Bethe vectors states (BVS) by the algebraic Bethe ansatz method using the gl(2)-invariant R-matrix and an algebraic relation between theĈ and thê D operators. We use a binomial expansion of the n-th power of the sum of two operators obtained from that algebraic relation to write a recursion equation and solve it. The binomial expansion of the n-th power of the sum of two operators can be written as a commutative binomial expansion plus a function of the commutator of the operators. We calculate the scalar product and the norm of that BVS. The BVS are degenerate eigenfunctions of the total number of particlesN -operator with eigenvalue N. As an example of application of the BVS we calculate the form factors for the imbalance current operator.