Integrability of the spin-1/2 fermions with charge pairing and Hubbard interaction

In this paper we study the exact solution of a one-dimensional model of spin-$\frac{1}{2}$ electrons composed by a nearest-neighbor triplet pairing term and the on-site Hubbard interaction. We argue that this model admits a Bethe anstaz solution through a mapping to a Hubbard chain with imaginary kinetic hopping terms. The Bethe equations are similar to that found by Lieb and Wu \cite{LW} but with additional twist phases which are dependent on the ring size. We have studied the spectrum of the model with repulsive interaction by exact diagonalization and through the Bethe equations for large lattice sizes. One feature of the model is that it is possible to define the charge gap for even and odd lattice sites and both converge to the same value in the infinite size limit. We analyze the finite-size corrections to the low-lying spin excitations and argue that they are equivalent to that of the spin-$\frac{1}{2}$ isotropic Heisenberg model with a boundary twist depending on the lattice parity. We present the classical statistical mechanics model whose transfer matrix commutes with the model Hamiltonian. To this end we have used the construction employed by Shastry \cite{SHA1,SHA2} for the Hubbard model. In our case, however, the building block is a free-fermion eight-vertex model with a particular null weight.


The Model Hamiltonian
In general, correlations among fermions in one-dimension give rise to complex phase diagram with charge and spin ordering. One of the simplest lattice system model that describes the effect of such correlations is the Hubbard model [1,2]. This model encodes the basics physics concerning the competition between electron kinetic energy and the on-site Coulomb interaction denoted here by U. The Hamiltonian of this model on a ring of size L with a electron-hole symmetric interaction is given by, where c † α (j) and c α (j) creates and annihilates fermions on site j with spin α and n α (j) = c † α (j)c α (j) is the occupation number operator. Here we apply periodic boundary conditions by identifying the sites L + 1 ≡ 1.
In 1968 Lieb and Wu showed that the Hamiltonian (1) can be diagonalized by an extension of the Bethe ansatz technique [3]. They used this solution to argue that the Hubbard model at half-filling is an insulator for positive values of U and undergoes a Mott transition at U = 0. The literature exploring the solution by Lieb and Wu is nowadays vast and for a collection of reprints and an extensive review on this subject see for instance [4,5]. In the context of this paper we mention the progresses made by Shastry towards the understanding of the algebraic structure associated to the integrability of the one-dimensional Hubbard model [6,7]. In particular, this author discovered a two-dimensional vertex model of classical statistical mechanics whose transfer matrix commutes among themselves and with the Hubbard Hamiltonian.
The purpose of this work is to introduce a variant of the Hubbard model and to discuss its solution by the Bethe anstaz as well as to uncover the underlying covering vertex model. The model is defined replacing the hopping term of the Hubbard chain by a nearest-neighbor charge pairing potential. The corresponding model Hamiltonian is, where periodic boundary conditions is assumed.
We observe that the first term of Hamiltonian (2) causes charges to be created or annihilated in pairs being similar to a triplet pairing in the p-wave theory of superconductivity. Here are we considering the situation in which the pairing energy is the same for both spin up and down channels. The interaction term is the same as that of the Hubbard model which is taken symmetric under the electron-hole transformation c α (j) ↔ c † α (j). The charge pair model (2) enjoys of translation invariance, the symmetry under spin flips and the invariance under two Z 2 symmetries represented by the unitary transformation V σ H c V † α with V α = e iπ L j=1 nα(j) . Besides that we have other global invariance with respect to specific rotations associated to the spin space. In order to describe that we first recall the structure of the isomorphic SU(2) algebras which can be constructed out of the possible six non-vanishing on-site combinations of spin-1 2 fermionic operators. The on-site generators of the standard spin SU(2) algebra is known to be given by, Yet another basis can be obtained by applying for instance the electron-hole transformation on the spin down fermionic operators defined by (3). This gives rises to the so-called pseudo-spin or charge SU(2) algebra whose on-site generators are, [n ↑ (j) + n ↓ (j) − 1]. (4) For arbitrary values of L the Hubbard model (1) is invariant by the full non-Abelian spin SU(2) symmetry (3). However, this is not the case of the Hamiltonian (2) which is invariant only when this symmetry is broken down to rotations around the y axis. The same observation applies to the pseudo-spin SU(2) algebra (4) since the charge pair model is invariant by such symmetry when it is restricted to rotations around the x axis. More precisely, for arbitrary values of L we have the following conservation laws 1 In next section we shall explore the commutations (5) in order to determine the eigenspectrum of the model by the coordinate nested Bethe ansatz method. These charges can be made equivalent to the conservation of particle numbers by means of a redefinition of the original electrons operators. In this new fermionic basis, the Hamiltonian (2) is mapped to the form of the Hubbard model but with pure imaginary and asymmetric hopping terms. We find that the corresponding Bethe equations for L even are distinct from that with L odd through suitable boundary twists.
In section 3 we investigate the properties of the spectrum of the model for repulsive interactions.
We argue that for both L even and odd we can define a lattice charge gap with respect to the ground state which in the thermodynamic limit converges to the value computed by Lieb and Wu for the Hubbard model at half-filling [3]. We have used the Bethe equations to study the finite-size corrections associated to the excitations due to the spin degrees of freedom. We find that they are equivalent to that of the isotropic spin-1 2 Heisenberg model with periodic boundary for L even and with a twisted toroidal boundary when L is odd. In section 4 we describe the lattice vertex model whose transfer matrix commutes with the Hamiltonian (2). This is done by using a construction due to Shastry devised to couple two symmetric six-vertex models satisfying the free-fermion condition [6,7]. However, in our case the building block has the form of an eightvertex model in which one of the weights is zero. The fact that Shastry's formulation also works for such special Z 2 invariant vertex model seems to have been unnoticed in the literature. Our concluding remarks are given in section 5 and in Appendix A we present the technical details on the underlying Yang-Baxter algebra.

The Energy Hamiltonian Spectrum
The space of states of spin-1 2 fermions associated with every lattice is four-dimensional and they can be represented as, where |0 denotes the vacuum state defined by the condition c α (j) |0 = 0.
In the above canonical basis the local conserved charges S y j and L x j are viewed as anti-diagonal matrices. However, they can be both diagonalized by on-site unitary transformation with the following similarity matrix, We can use this transformation to define new on-site fermionic operators, while the one associated to the pseudo-spin algebra is, By the same token the Hamiltonian (2) of the charge pair model can be expressed as follows, which has the typical form of the Hubbard Hamiltonian however with imaginary and asymmetric hopping terms.
The conserved charges (10,11) imply that the Hilbert space of the Hamiltonian (12) can be separated into block disjoint sectors labeled by the total number N α = L j=1 d † α (j)d α (j) of fermions of spin α. This means that eigenvalue problem can be formally written as, The range of the quantum numbers can be constrained observing that Hamiltonian (12) is invariant under the particle-hole symmetry d α (j) ↔ d † α (j). In fact, taking into account this invariance we obtain the spectral identity, Here we remark that such spectral relation is valid for arbitrary L in the case of the transformed charge pair model Hamiltonian (12). Therefore, unlike the Hubbard model no restriction to bipartite lattices is necessary in order to relate the energies of different sectors with the same coupling U [3]. The spectral identity (14) together with spin flip invariance tell us that we may restrict our considerations to states, for even and odd values of L. We emphasize that in the case of the Hubbard model (1) the spectral relation (14) and constraint (15) only works when L is even [3].
We now can determine the eigenspectrum of the Hamiltonian (12) by adapting the nested Bethe ansatz approach employed Lieb and Wu [3] in the presence of hopping phases. In a given sector with total number of fermions N = N ↑ + N ↓ the wave function may be represented as linear combination of N-particle states, where the reference state 0 is taken such that d α (j) 0 = 0 for any site j and spin α. The exponentials terms in (16) are able to pull the hopping bulk phases up to the boundary terms. In our case this happens when we choose the twists to be fixed as φ ↑ = π 2 and φ ↓ = − π 2 . In the Bethe ansatz approach one assumes that the N-particle amplitudes have a plane wave form [3], where it is assumed the ordering denotes the N! permutations of fermions with positions x Q 1 , x Q 2 , . . . , x Q N and spins α 1 , α 2 , . . . , α N while P = {P 1 , P 2 , . . . , P N } refers to similar permutations on the fermions momenta k P 1 , k P 2 , . . . , k P N .
The coefficients associated to these permutation are denoted by A(Q|P ).
In this formulation the hopping phases are all removed excepted those associated to fermions hopping among the boundary sites j = 1 and j = L. At this point the situation becomes equivalent to that of generalized diagonal boundary conditions discussed for the Hubbard model in [8,9]. This fact is taken into account by requiring that A(Q, P ) satisfy the condition, From now on the procedure is analog to that already exposed by Lieb and Wu [3] and we shall present only the main results. The spectrum of the Hamiltonian (12) is parametrized in terms of a set of variables {k j , µ j } which fulfill the following nested Bethe equations, while the eigenvalue of the transformed Hamiltonian (12) associated with the state specified by the rapidities {k j , µ j } is given by, We would like to close this section with the following comments. We first observe that the fermionic chain for L = 2 is somehow special since the charge pairing terms are canceled and the Hamiltonian (2) becomes a diagonal operator. From the Bethe solution point of view this peculiarity is associated with the presence of the minus sign factor in the first level Bethe equation (19). We have checked this fact by solving the two sites Bethe equations (19,20) for roots configurations satisfying the restriction (15). These solutions indeed reproduce the expected Hamiltonian energies and our findings have been summarized in Table 1.  (2,12) for L = 2 where the sectors (N ↑ , N ↓ ) satisfy (15).
The eigenvalues are obtained substituting the Bethe roots into the relation (21).
We next note that for arbitrary L the Bethe equations (19,20) are similar to that of the Hubbard to be rather different because of the canonical transformation (9). We can see that considering examples of simple states whose energy per site is independent of the size L. From the Hubbard model perspective we already know that there exists two such eigenvalues associated to the trivial ferromagnetic and anti-ferromagnetic states. We find out that these energies also belong to the spectrum of the charge pair model Hamiltonian (2) but with distinct wave-function structure.
The form of the wave-function on the canonical basis can be uncovered with the help of the transformation (9). The final results for such states have been summarized on Table 2. Finally, we remark that for a bipartite lattice the rotation invariance (5) of the charge pair Hamiltonian around specific axes are enlarged to the invariance under two SU(2) symmetries.
This is similar to the case of the Hubbard model (1) which for L even has besides the spin SU (2) symmetry (3) another distinct SU(2) invariance named the η-pairing symmetry [10,11].
In fact, for an even number of lattice sites the invariance of the Hamiltonian (2) under the rotation around the y-axis of the spin algebra (3) extends to a full "spin" SU(2) symmetry, namely where now the extra on-site generatorsS x j andS z j alternate among the lattice sites, The same happens to the rotation around the x-axis of the charge algebra (4). For L even it is enlarged to the following "charge" SU(2) symmetry, where the expression for the additional staggered on-site generatorsR y j andR z j are given by, 3 The Spectrum Properties for U > 0 As far as the energy spectrum is concerned the difference among the charge pair model (2) and the Hubbard chain (1) is the presence of size dependent twists in the Bethe equations. However, these fluxes are not expected to affect the value of ground state energy per site in the thermodynamic limit. The value should be same as that of the Hubbard model in the half-filled case determined long ago by Lieb and Wu [3]. Denoting this energy by e ∞ we have, where J 0 (x) and J 1 (x) are Bessel functions.
The other basic feature of the half-filled Hubbard model is the presence of energy gap in the charge excitation sector. For L even this mass gap was defined by Lieb and Wu [3] as the energy ∆(L) of a particle or a hole excitation with respect to the half-filled state. In the thermodynamic limit its value was computed to be [3], We shall argue that for the charge pair model (2) it is possible to define the energy gap for both even and odd lattice sites. It turns out that the phase factors for L odd compensate the effects of frustration due to the lattice parity and the energy gap of either a hole or a particle excitation over the double degenerated ground state is the same. We shall present numerical evidences that the value of the gap for even and odd sites converges in the thermodynamic limit to the result (27).
The other known feature of the Hubbard model at half-filling is that the spin excitations are gapless in the repulsive regime. The phases twists for the charge pair model will not change this behaviour but the conformal data will be dependent on the parity of the lattice size. In what follows we will also study the finite-size effects for some of the gapless states of the charge pair Hamiltonian (2).

Finite-size effects for L even
From the Bethe solution we concluded that the energy spectrum of the charge pair model and the Hubbard model coincides for L/2 even. However, when L/2 is odd the energy spectrum of these two models are not the same due to the presence of a minus sign in the first Bethe equation (19). In what follows we shall therefore restrict our analysis of the spectrum for lattice sites not multiple of four.
From the exact diagonalization of the Hamiltonian (2) we conclude that the ground state is a singlet and lies in the sector ( L 2 , L 2 ). The situation is similar to that of the Hubbard model at half-filling. In Figure 1 we exhibit the low-lying energies per site for L = 6 in which the states are label using the quantum numbers associated with the Bethe ansatz solution of the transformed Hamiltonian (12).
We find out that the ground states as well as many of the low-lying excitations can be described by real roots of the Bethe ansatz equations (19,20). Based on this observation we can take the logarithm of the Bethe equations to obtain, where the numbers Q (1,2) j define the many possible branches of the logarithm.
In Table (3) we give the numbers Q (1,2) j for the ground state and lowest states associated to the charge and spin sectors. We remark that such sequence of numbers are not the same as that of  (19,20).
the corresponding states of the Hubbard model 2 because of the extra sign on the Bethe equation (19).
We now start to report on the numerical analysis about the eigenstates described in Table (3).
In what follows we shall denote by E j (N ↑ , N ↓ , U) the j-th energy level in a given sector (N ↑ , N ↓ ).
The energy gap of one hole excitation over the singlet ground state can then be defined as, 2 For the Hubbard models such Bethe numbers depend on whether L/2 is even or odd see for instance [12].  (27).  We now turn to the analysis of the finite-size corrections to the spin degrees of freedom. As remarked these excitations should be gapless since the spectrum of the charge pair model (2) and the Hubbard model (1) is the same for lattice sizes multiple of four. For the Hubbard model such spin excitations is known to show the same critical behaviour of the isotropic spin-1 2 Heisenberg model with periodic boundary conditions [13]. It is therefore expected similar critical behaviour for charge pair model (2) in the case of even number of lattice sizes. In particular, the finite-size dependence of the ground state energy should be governed by a conformal theory with central charge c = 1. More precisely, following the results obtained by Woynarovich and Eckle [13] one expects, where ξ = 2I 1 (2π/U)/I 0 (2π/U) is the sound velocity of the spin excitation. The functions I 0 (x) and I 1 (x) are modified Bessel functions.
We have checked the above result by numerically computing the estimators, for lattice sizes up to L = 1038. In Table (5) we have presented these estimates and we observe the rapid converge to the expected value c = 1.
From the above informations we conclude that for L even the leading behaviour of the finitesize corrections of charge pair model should be same of that discussed Woynarovich and Eckle [13] for the Hubbard model. As far as finite-size effects are concerned the difference among these two models for L/2 odd appears to be associated with the subleading corrections. The amplitudes of subleading terms are probably affected by presence of distinct phase factors in the first level Bethe equation.

Finite-size effects for L odd
We have performed numerical diagonalization of the charge pair Hamiltonian (2) for small values of odd lattice sites. In Figure (2) we present the low-lying energies in the spectrum of the charge pair model for L = 7. Considering this analysis we conclude that the ground state sits in the sectors ( L+1 2 , L−1 2 ) and ( L−1 2 , L+1 2 ). We find that these states have zero momenta and  consequently the energy of the ground state is double degenerated. For sake of comparison we note that these same states for the Hubbard model carries non-zero momenta and the respective energy is therefore four-fold degenerated.
We find that the low-lying states are well described by real Bethe roots and as before we can take the the logarithm of the Bethe equations (19,20). Considering the presence of the phase factors we obtain, where in Table (  The energies are labeled by the quantum numbers (N ↑ , N ↓ ) of the Bethe equations (19,20).
odd number of sites in analogy to what has been done for L even. In fact, from Figure (2) we observe that either a hole or a particle excitation has the same energy with respect to the ground state. This leads us to define the following charge gap for odd number of sites, We have computed the gap estimates (33) by numerically solving the Bethe equations (32) for the respective energies up to L = 1025. The results are exhibited in Table (7) and we see that the extrapolated estimators are very close to the exact values (27).
Let us now discuss the behaviour of the finite-size corrections to the ground state energy. The for the ground state and two low-lying excitations.
computation of these corrections can be done within the root density formalism [14,15] since the Bethe equations are solved by real roots. At this point we recall that this approach has already been applied to the Hubbard model with even number of sites [13]. By adapting the computations of [13] to tackle the Bethe equations (32) we find that the leading behaviour of the finite-size corrections for the ground state is, and from the predictions of conformal field theory [16] we conclude that this state has conformal dimension X 0 = 1 8 . To support the above result for the scaling dimension we compute the following finite size two-step estimators for large sizes, The strong logarithmic correction in the finite-size estimators is considered as follows. For each two consecutive values of lattice sites we eliminate the logarithmic amplitude A 0 (L) and calculate the respective scaling dimension X 0 (L). In Table (8) we present the results for X 0 (L) together with the extrapolated value for large L. We observe that the data approach the value predicted by the root density method X 0 = 1 8 with reasonable precision. We observe that in analogy to the Hubbard model with L even the scaling dimension lacks of dependence on the coupling U [13]. This fact suggests that further insights about the finite-size corrections may be easily obtained by exploring the strong coupling limit of the Bethe equations (19,20). In what follows we will pursue this analysis for the sector with total number of fermions  N ↑ + N ↓ = L and spin N ↑ − N ↓ = 2n where n takes values on half-integers for L odd. Formally, this limit may be performed by scaling the spin rapidities as µ j = U 2 λ j and afterwards taking the limit U → ∞. For real momenta sin(k j ) is always bounded and through lowest order in 1/U the two-level Bethe equations for the momenta and spin variables decouple. The first Bethe equation (19) turn into a momenta condition for free-fermions while the second level one (20) becomes equivalent to that of the isotropic spin-1 2 model with twisted boundary condition. More precisely, the equation for the renormalized spin rapidities becomes, The critical exponents associated to the spin degrees of freedom can therefore be inferred from previous analytical and numerical works for the spin-1 2 Heisenberg, see for instance [17][18][19]. Here we have to combine the frustrated character of the ground state of the Heisenberg chain with the presence of boundary twist. Following [19] and performing the adaption to our situation we find where the number m indicates the vorticity of the state.
We note that the ground state scaling dimension X 0 = 1 8 coincides with the lowest dimensioñ X(1/2, 1/2) of the twisted Heisenberg chain. Thus it is plausible to believe that the conformal dimensions (37) should be present in the finite-size corrections of the charge pair model for L odd.
In order to give further support to this conjecture we now consider the first excitation in the sector ( L+1 2 , L−1 2 ) of the charge pair model. This state has momenta being double degenerated and the respective logarithmic branch numbers are given in the third line of Table (6). By applying the root density method to this state we obtain, whose corresponding conformal dimension is X 1 = 5 8 . In Table (9) we provide numerical support for the above analytical computation. The extrapolated value is in reasonable accordance with the analytical prediction.  Once again we note the dimension X 1 = 5 8 can be obtained either fromX(1/2, −1/2) or X(1/2, 3/2) in agreement with fact we are dealing with a momenta state. We think that the above arguments strongly suggests that the contributions of the spin degrees of freedom to the finite-size corrections of the charge pair model with L odd are indeed governed by the conformal dimensions (37).
We conclude with the following comments. We expect that the finite-size behaviour of the Hubbard model for L odd will be different from that described above for the charge pair model.
First we remark that the gap definition (33) does not apply for the Hubbard model because its eigenvalues do not satisfy the spectral property (14). Besides that exact diagonalization of the Hubbard Hamiltonian (1) reveal us that there exits level crossing among the first two lowest energies states for some finite value of U. For L = 5 we find that the level crossing is among the ground states lying in the sectors (3,2) and (3,3). For L = 7 the crossing occurs for states in the sectors (3,4) and (3,3) rather than among the energies in sectors (3,4) and (4,4). Therefore, the nature of such crossings seems to depend on the parity of the number (L − 1)/2 and the understanding of the large L behaviour of the low-lying states requires further investigation. We plan to expand on this preliminary analysis and present it elsewhere since most of finite-size results for the Hubbard model appears to be concentrated on even number of sites.

The Covering Vertex Model
Here we argue that the fermionic Hamiltonian (2) can be derived in the context of the commuting transfer matrix approach [20]. We start by recalling the Boltzmann weights structure of the symmetric free-fermion eight-vertex model. The model has four weights a, b, c, d and the its Lax operator can be represented as, where the indices 0 and j refer to the horizontal and vertical spaces of states of the vertex model, respectively. It is assumed the free-fermion condition among the weights, For d=0 Shastry devised a way to couple two free-fermion six-vertex models by a particular diagonal vertex interaction. As a result was obtained a new integrable vertex model of statistical mechanics with non-additive R-matrix [6,7]. In addition, Shastry showed that the transfer matrix of such model commutes with the Hamiltonian of an equivalent spin chain derived from that of the Hubbard model by means of the Jordan-Wigner transformation, where {σ ± j , σ z j } and {τ ± j , τ z j } are two commuting sets of Pauli matrices acting the j-th lattice site. In what follows we shall point out that Shastry's approach also works in the subspace of weights with b = 0. Before that we recall that Shastry's construction has been shown to be applicable when we couple certain special vertex models invariant under the gl(n|m) superalgebra [21][22][23].
We emphasize such generalizations lead to Hamiltonian models with higher number of states per site than that of the charge pair model (2) introduced here. The fact that Shastry's method also works using an eight-vertex model with b = 0 appears to have been overlooked so far. For b = 0 free-fermion condition (40) is a circle in the affine plane and it can be parametrized as, where λ is the spectral parameter. Note that at λ = 0 the Lax operator (39) becomes a twodimensional permutator.
The construction of coupled vertex models for b = 0 is fairly parallel to that devised by Shastry and in what follows we shall summarize only the main results. The Lax operator of the coupled model has the standard Shastry's form, where I denotes the four-dimensional identity matrix and h(λ) characterizes the strength of the coupling.
In our case, however, the operators L As usual the transfer matrix of the respective vertex model on the square lattice can be written as the trace of an ordered product of Lax operators (43) on the horizontal space, which gives rise to a family of commuting of transfer matrices provide the coupling h(λ) satisfies the Shastry's spectral constraint, At this point we remark that the condition (40) with b = 0 and the constraint (46) can be translated into a single algebraic relation after a suitable definition of the ring variables. Indeed, following [24] it is possible to define new affine variables, such that the spectral curve assuring the integrability of the model is the following genus one quartic curve, Now the spin Hamiltonian H s associated with this vertex model is obtained by expanding the logarithm of the transfer matrix (45) around the regular point λ = 0. Apart from an additive constant its expression is given by, with periodic boundary conditions imposed.
With the help of the Jordan-Wigner transformation (41) the fermionic Hamiltonian (2) can be rewritten in terms of Pauli operators. It turns out that this transformation is able to reproduce only bulk part of the coupled spin chain (49), since the boundary terms are clearly distinct from that of the coupled spin model (49).
In order to match the boundary term we can exploit the fact that integrability is still preserved by performing certain suitable twist transformations on the Lax operators [25]. Besides that we have to consider that the local states of the fermionic Hamiltonian (2) are constituted by a graded space with two bosonic and two fermionic degrees of freedom. We expect that the respective Lax operator at the regular point should be proportional to the graded permutation operator, where e jk are the standard Weyl matrices. We choose the Grassmann parities p j according to the basis ordering (6) and therefore we set p 1 = 0, p 2 = 1, p 3 = 1, p 4 = 0.
Combining the procedures mentioned above we find that the suitable fermionic Lax operator is obtained by the following twist transformation, where the twists M and M are the following diagonal matrices, It turns out that the explicit matrix representation of the Lax operator (52) in terms of the spectral variables x and y is given by, where the weights w 1 , . . . , w 4 dependence on the spectral variables are, We now show that this fermionic Lax operator is able to produce the two-body part of the charge pair Hamiltonian (2) through its expansion around the regular point x = 1 and y = 0. It turns out that the first order expansion of the spectral variables constrained by the curve (48) is given by, where ǫ is the expansion parameter.
Now considering the expansion of the Lax operator (54) we obtain, where the operator H j,j+1 is given by which coincides with the the two-body term of Hamiltonian (2) apart from a trivial additive factor.
We close this section mentioning that both Lax operators (43,54) fulfill the Yang-Baxter relation. This factorization condition together with corresponding R-matrices has been summarized in Appendix A.

Concluding Remarks
In this paper we have introduced a variant of the Hubbard model whose next-neighbor term plays the role of a triplet charge pair potential. For arbitrary lattice sizes the model has two conserved charges which can be added to the Hamiltonian without affecting its integrability.
Besides that gauge fluxes can be attached to both the pair potential and the conserved charges and an extended charge pair Hamiltonian can be written, where θ ↑ , θ ↓ are flux phases and h 1 , h 2 are the chemical potentials associated to the conserved charges.
The fluxes can be removed from the Hamiltonian (59) by means of the canonical transformation c α (j) → e −iθα c α (j) and c † α (j) → e iθα c † α (j). The Bethe ansatz solution for extended Hamiltonian (59) follows that given in section 3 and the respective Bethe equations are given by the same relations (19,20). The basic change is in the expression for the eigenenergies which now is, We have argued that the exact integrability of the charge pair model (2) can be established by using a construction devised by Shastry for the Hubbard model [6,7]. This procedure gives rise to an equivalent spin chain (49) which can be seen as two coupled special XY models. We now show that such spin chain can be mapped into two coupled XX models where the boundary conditions depend on if we have an even or odd number of sites. To this end we define the following transformation acting on the even sites of the lattice, For L even the form of the transformed Hamiltonian (49) is, which is exactly the same spin chain associated to integrability of the Hubbard model [6,7]. The corresponding Bethe equations have been discussed before [27,28] and for sake of completeness we also present them here, where now the phase factors depend on the combined parities of the quantum numbers of the model. The eigenvalues are once again determined by the expression (21).
On the other hand when L is odd the transformed Hamiltonian (49) is given by, Now we see that the boundary term in (64) breaks explicitly the two U(1) symmetries present in the bulk part of the Hamiltonian. Despite of this fact we found out that the transformed model (64) still preserves the property of having factorized reference states associated with the exact eigenvalues E = ± LU 4 . The situation is similar to what we have found for the charge pair model as shown in Table (2). The structure of such eigenstates for the spin model (64) are however a bit different since it contain alternating phases in the tensor product. The form of these reference states are summarized in Table ( In addition to that we have been able to built few low-lying states on top of the reference state given in Table ( (19,20). This strongly suggests that the eigenenergies of the charge pair model (2) and the coupled spin chain (49) are the same for L odd. We have indeed confirmed this fact by comparing the spectrum of these models with the help of exact diagonalization for L = 3, 5, 7 sites. The eigenfunctions structure of such two models should be related but a more concrete relationship among them has eluded us so far.
Lastly, one characteristic of the one-dimensional charge pair model is that the thermodynamic limit properties do not depend on fact that the lattice is bipartite. It was argued that the charge gap can be defined for even and odd number of sites both converging to the same value in the infinite size limit. This should be contrasted to the case of the Hubbard model in which the lattice bipartiteness plays important role to stablish certain exact results for the repulsive interaction in any lattice dimension [29]. It seems interesting to investigate whether or not the methods used to obtain significant informations for the Hubbard model in all dimensions can also be adapted to the case of the charge pair model. In particular, if one can state concrete informations for the charge pair model in higher dimension without the need of a bipartite lattice assumption.
Appendix A: The Yang-Baxter algebra.
We finally recall that it is possible to rewrite the Yang-Baxter relation in an alternative form which is insensitive to the grading of the spaces [26].