Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic fields

The one-dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly via the Bethe ansatz methods. With the coordinate Bethe ansatz in the charge sector, the second eigenvalue problem associated with the spin sector is constructed. It is shown that the second eigenvalue problem can be transformed into that of the inhomogeneous XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.


Introduction
The Hubbard model is one of the essential models in condensed matter physics.An interesting issue is that the model is exactly solvable in one dimension [1], which provides an important benchmark for understanding the Mott insulators.After Lieb and Wu's pioneering work, a lot of attentions have been paid to the integrability, symmetry [2,3,4,5] and physical properties of this model.A remarkable result was obtained by Shastry [6] and by Olmedilla et al [7,8] who constructed the corresponding R-matrix of the one-dimensional Hubbard model and therefore demonstrated its complete integrability in the framework of Yang-Baxter equation [9,10].Subsequently, the model was resolved [11] via the algebraic Bethe ansatz method based on the result of Shastry.Another interesting issue about this model is the open-boundary problem, which is tightly related to the impurity problem in a Luttinger liquid [12].The exact solution of the open Hubbard chain was firstly obtained by Shulz [13].Subsequently, the exact solution of the model with boundary potentials was obtained [14,15].The integrability of the one-dimensional Hubbard model with diagonal open boundary was demonstrated in [16] by constructing the Lax representation and solved by algebraic Bethe ansatz method [17].The generic integrable boundary conditions were obtained in [18] by solving the reflection equation [19,20,21].It was found [18] that in the spin sector magnetic fields applied on the two end sites do not break the integrability of this model.Although the integrability has been known for long time, the exact solutions (or diagonalization of the Hamiltonian) of the model with arbitrary boundary magnetic fields are still lacking.
In this paper, we study the open Hubbard chain with arbitrary boundary magnetic fields.
The Hamiltonian of the model is where c † j,α and c j,α are the creation and annihilation operators of electrons on site j with spin component α =↑, ↓; t and U are the hopping constant and the on-site repulsion constant as usual; n j,α are particle number operators, respectively; indicate the boundary fields and h ± j = h x j ± ih y j for j = 1, N. We shall show in the following that the model can be exactly solved by combining the coordinate Bethe ansatz and the off-diagonal Bethe ansatz proposed recently in [22,23,24] for arbitrary h 1 and h N .We remark that the unparallel boundary fields break the U(1) symmetry in spin sector and make the total spin no longer a conserved charge.
The paper is organized as follows.In Section 2, we use the coordinate Bethe ansatz method to derive the eigenvalue equation in the spin sector as that in the periodic case [9].
In Section 3, we transform this eigenvalue problem into that of the inhomogeneous XXX spin chain with boundary fields, which allows us to apply the recently proposed off-diagonal Bethe ansatz method [22,23,24] to solve it.The exact spectrum of the Hamiltonian and the Bethe ansatz equations are thus obtained.Section 4 is attributed to the reduction to the parallel or anti-parallel boundary case.Concluding remarks are given in Section 5.

Coordinate Bethe ansatz
Though the U(1) symmetry in the spin sector is broken by the unparallel boundary fields, the U(1) symmetry in the charge sector is still reserved.The conserved charge corresponding to this reserved symmetry is the total number operator of electrons, namely, N = N j=1 {n j,↑ + n j,↓ } , [H, N ] = 0. (2.1) The symmetry allows us to construct the eigenstate of the Hamiltonian (1.1) with a fixed number of electrons as follows: where M is the number of electrons and {α} = (α 1 , . . ., α M ).The eigenvalue equation of the Hamiltonian then reads (2.3) This eigenvalue equation can be rewritten as where {α} j means α j is replaced by β j in the set {α} and σ = (σ x , σ y , σ z ) with σ x , σ y and σ z being the Pauli matrices.The wave function takes the following Bethe ansatz form [9]: where P = (P 1 , . . ., P M ) and Q = (Q 1 , . . ., Q M ) are the permutations of (1, . . ., M); r = (r 1 , . . ., r M ) with r j = ± and θ(x 1 < . . .< x M ) is the generalized step function.For all x j = 1, N and x j = x l case, (2.4) is automatically satisfied and the corresponding eigenvalue is For two electrons occupy the same site case, we should consider the continuity of the wave function Ψ {α} (x 1 , . . ., x M ).Considering the sector I: (2.7) For convenience, we omit the superscript {α} and treat A r P (Q) as a column vector in the spin space.Then the continuity condition (2.7) of the wave function implies where Q ′ = (. . ., Q j+1 , Q j , . ..), P ′ = (. . ., P j+1 , P j , . ..) and r ′ = (. . ., r j+1 , r j , . ..).For (2.9) Substituting (2.8) into (2.9),we have Now, we define the coordinate permutation operator Pi,j , Due to the fact that the wave function of fermion is completely antisymmetric under exchanging both the coordinates and spins of two particles, if we denote P i,j as the spin permutation operator, we have P i,j Pi,j = −1, P 2 i,j = P 2 i,j = 1.
(2.12) Thus, we have the following relation: Substituting this relation into (2.10),we readily have with the S-matrix given by (2.15) Now let us turn to the case of In this case, the eigenvalue equation (2.4) becomes This induces which gives With the help of the identity we have

.19)
Similarly, for the case of namely, which induces with (2.23) the situation becomes a little bit subtle.We have to check the self-consistence of the ansatz.For the case of and i, j = 1) and x Q M = N, the eigenvalue equation (2.4) becomes And in this case, the S-matrix makes the following equation hold: Combining Eq.(2.24) and Eq.(2.25), we have the following relation need to be confirmed: For the case of we have the relation (2.17).For Obviously (2.17) and (2.27) make (2.26) hold.With the same procedure we can demonstrate that the ansatz is also satisfied when two electrons both occupy the site N. Now let us consider the following process.The j-th particle moves from the l-th site to the left end by scattering with all the other particles to their left, and then is reflected by the left boundary.After scattering with all the other particles, it is reflected by the right boundary and then moves back to its original position.This process can be described by the following relations: Consequently, this gives rise to the following eigenvalue problem: τ (k j )A (...,+,...) = e −2ik j N A (...,+,...) , (2.28) with the resulting operators (2.29) Let V denotes a two-dimensional linear space.Throughout the paper we adopt the standard notations: for any matrix A ∈ End(V), A j is an embedding operator in the tensor space V ⊗ V ⊗ • • •, which acts as A on the j-th space and as identity on the other factor spaces; R ij (u) is an embedding operator of R-matrix in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
In the next section we shall show that τ (k j ) is proportional to the transfer matrix of the inhomogeneous XXX spin chain with arbitrary boundary fields and thus the eigenvalue problem (2.28) can be further solved by the off-diagonal Bethe ansatz method.

Off-diagonal Bethe ansatz
Before going further, let us introduce the following R-matrix and K-matrices: where 2t .
The R-matrix possesses the following properties: Unitarity relation : Crossing relation : The following Yang-Baxter equation, the reflection equation and its dual also hold: Now let us define the inhomogeneous double-row monodromy matrix2 [19,21], and the transfer matrices with different spectrum parameters commute with each other, Putting u = − sin k j , we readily have Noticing that we have the following important identification between the operators {τ ( The eigenvalue problem (2.28) is thus equivalent to that of diagonalizing the transfer matrix of the inhomogeneous open XXX chain model with boundary fields.Here we naturally have the "inhomogeneous" parameters θ j = sin k j and the crossing parameter η = −i U 2t .Thanks to the works [22,23,24], the transfer matrix (3.11) of the open XXX chain with arbitrary boundary fields which is specified by the K-matrices K ± (u) (3.2) and (3.3) can be exactly diagonalized by off-diagonal Bethe ansatz method.In the following, we shall use the method in [24] to the eigenvalue problem (2.28) of the Hubbard model with arbitrary boundary fields.
For this purpose, we introduce some functions at first:

Even M case
Following [24], we construct the following ansatz of the eigenvalue of the transfer matrix τ (u) for an even M: in which the functions Q 1 (u) and Q 2 (u) are parameterized by M different from each other parameters {µ j |j = 1, . . ., M} for a generic non-vanishing c as follows: It has been shown [24] that Λ(u) becomes the eigenvalue of the transfer matrix τ (u) given by (3.11) if the M parameters {µ j |j = 1, . . ., M} satisfies the following Bethe ansatz equations: where the parameter c is expressed in terms of the boundary fields (3.23).Numerical checks of the completeness of the above solutions for small size of M (the results for the odd M see the next subsection) was given in [25,26] (see also [27]).A beautiful expression for the corresponding eigenvectors was proposed recently in [28].
Based on the expressions (3.24) of Λ(u) for the eigenvalue of the transfer matrix (3.11) and the relation (3.19) between the operator τ (k j ) and the transfer matrix at special point τ (− sin k j ), the eigenvalue problem (2.28) gives rise to the following constraints on the quasimomentum {k j }: Noticing that d(− sin k j ) = A(sin k j − η) = 0, the above Bethe ansatz equations become

Odd M case
Following [24], we construct the following ansatz of the eigenvalue of the transfer matrix τ (u) for an odd M: {λ j } and M quasi-momentum {k j } satisfy the following Bethe ansatz equations:

. 10 )
and the transfer matrix τ (u),τ (u) = tr 0 K + 0 (u)T 0 (u) .(3.11)From the Yang-Baxter equation and the reflection equation and its dual one may derive[19] k j )} (2.29) appeared in the eigenvalue problem of the open-boundary Hubbard model and the transfer matrix of the open XXX spin chain with boundary fields:

1 ,
. . ., M. (3.29) Then from the solutions of the Bethe ansatz equations (3.27) and (3.29), one can reconstruct the exact wave functions (2.5) with even number of electrons for the Hubbard model with boundary fields, the corresponding eigenvalues are given by (2.6).
one-dimensional Hubbard model with arbitrary boundary magnetic fields described by the Hamiltonian (1.1) is studied by combining the coordinate Bethe ansatz and off-diagonal Bethe ansatz methods.With the coordinate Bethe ansatz, eigen-functions of the Hamiltonian of the model are given in terms of some quasi-momentum {k j } as (2.5).The constraints (2.28) on these quasi-momentum is transformed into the eigenvalues problem of the resulting transfer matrix of the associated open XXX spin chain with arbitrary boundary fields.The second eigenvalue problem is then solved via the off-diagonal Bethe ansatz method.The corresponding Bethe ansatz equations (3.27) and (3.29) for the even number of electrons case, (3.33) and (3.34) for the odd number of electrons are constructed respectively when two boundary fields are unparallel, which corresponds to the case of the U(1) symmetry in the spin sector being broken.When the two boundary fields h 1 and h N are parallel or antiparallel, the U(1) symmetry in spin sector is recovered, the resulting Bethe ansatz equations become (4.3) and (4.4) which are labeled by a discrete number m = 0, . . ., M.