Surface energy of the one-dimensional supersymmetric $t-J$ model with unparallel boundary fields

We investigate the thermodynamic limit of the exact solution, which is given by an inhomogeneous $T-Q$ relation, of the one-dimensional supersymmetric $t-J$ model with unparallel boundary magnetic fields. It is shown that the contribution of the inhomogeneous term at the ground state satisfies the $L^{-1}$ scaling law, where $L$ is the system-size. This fact enables us to calculate the surface (or boundary) energy of the system. The method used in this paper can be generalized to study the thermodynamic limit and surface energy of other models related to rational R-matrices.


Introduction
The t − J model is the strongly repulsive limit of the well-known Hubbard model [1,2,3], which has played a fundamental and important role in strongly correlated electronic systems.
The model is also one of the cornerstone models in the study of high-T c superconductivity [4,5,6,7]. In general, the Hamiltonian includes nearest-neighbor hopping (t) and nearestneighbor spin exchange and charge interactions (J) (see below (1.1)) for the periodic case [8].
For the open case, the Hamiltonian also includes the boundary chemical potentials χ 1 , χ L and the boundary fields h 1 , h L [9,10], i.e., S j · S j+1 − 1 4 n j n j+1 +χ 1 n 1 + 2h 1 · S 1 + χ L n L + 2h L · S L , (1.1) where L is the total number of lattice sites and the coupling constants χ 1 , χ L and h 1 , h L are given by (2.17) below. The operators c j,α and c † j,α are the annihilation and creation operators of the electron with spin α = ±1 on the lattice site j, which satisfies anticommutation relations, i.e., c † i,α , c j,τ = δ i,j δ α,τ . There are only three possible states at the lattice site i due to the factor P = (1 − n j,−α ) projects out double occupancies. The operator n j = α=±1 n j,α means the total number operator on site j and n j,α = c † j,α c j,α , and the total number operator of electronsN = L j=1 n j . The spin operators S = L j=1 S j , S † = L j=1 S † j and S z = L j=1 S z j with the local operators: S j = c † j,1 c j,−1 , S † j = c † j,−1 c j,1 , S z j = 1 2 (n j,1 − n j,−1 ) form the su(2) algebra. At the supersymmetric points J = ±2t, the Hamiltonian in one spatial dimension is supersymmetric and integrable [11,12,13,14,15,16,17]. One can obtain the exact solution of the one-dimensional supersymmetric t − J model with periodic boundary condition or parallel boundary fields by the nested Bethe ansatz method [8,9] or the off-shell Bethe ansatz [18,19]. Based on the exact solution, the properties of the t − J models, for example, surface energy, the elementary excitation, the correlation functions and the thermodynamics have attracted a great attention [20,21,22,23]. Compared with the periodic case and parallel boundary fields case, the one-dimensional supersymmetric t−J model with unparallel boundary fields is the most general integrable case. With the help of the exact solution of the one-dimensional supersymmetric t−J model with unparallel boundary fields [24,25,26], the thermodynamic limit and surface energy of the model is a fascinating question [27,28,29].
In this paper, our goals are to study the thermodynamic limit and boundary effects of the supersymmetric t − J model with unparallel boundary fields. Based on former works [30], one can not direct employ the thermodynamic Bethe ansatz (TBA) method to approach the thermodynamic limit of t − J model due to the inhomogeneous term in the T -Q relation.
Therefore, the first thing should be addressed is the contribution of the inhomogeneous term. In this paper, we choose the region of χ 1 > 1 and χ L < 1 as an example. Through the analysis of the finite-lattice systems, it is shown that the contribution of the inhomogeneous term in the associated T − Q relation to the ground state energy satisfies the scaling law where L is the system-size. Based on this fact, by using the standard thermodynamic Bethe ansatz method and taking the limit of temperature tending to zero, we find that all the Bethe roots are real at the ground state in the region of χ 1 > 1 and 0 ≤ χ L < 1. While in region of χ 1 > 1 and χ L < 0, besides the real Bethe roots, there exists the boundary bound state and the boundary bound state should be stable. Furthermore, the surface energy of the system is calculated. Comparison of the surface energy from the analytic expressions with that from the Hamiltonian by the extrapolation method, we show that they coincide with each other very well.
The plan of the paper is as follows. We briefly review the Bethe ansatz solutions of the one-dimensional supersymmetric t − J model with unparallel boundary fields in Section 2.
In Section 3, we focus on the contribution of the inhomogeneous term to the ground state energy. In Section 4, with the help of the Bethe ansatz solution for the finite-size system, we study the thermodynamic limit and surface energy of the model. We summarize our results and give some discussions in Section 5.

Bethe ansatz solutions
In this paper we consider J = 2t and t = −1, which corresponds to the supersymmetric and integrable point [8]. Let V (m|n) = V m ⊕ V n denotes a graded linear space with an orthonormal basis {|i , i = 1, · · · , m + n} having the Grassmann parity (denoted by ǫ i ): ǫ i = 0 for i = 1, · · · , m and ǫ i = 1 for i = m + 1, · · · , m + n, which endows the fundamental representation of su(m|n) algebra [31]. For the supersymmetric t − J model, we have m = 1 and n = 2 [8]. The integrability of the model is associated with the R-matrix where u is the spectral parameter and η is the crossing parameter, and Π i,j is the Z 2 -graded permutation operator The R-matrix satisfies the graded Yang-Baxter equation and possesses the properties: Initial condition: Unitarity relation: Crossing Unitarity relation: Here and below we adopt the standard notation: for any matrix A ∈ End(V (m|n) ), A j is an super embedding operator in the graded tensor product space V (m|n) ⊗ V (m|n) ⊗ · · · , which acts as A on the j-th space and as an identity on the other factor spaces; R ij (u) is an super embedding operator of R-matrix in the graded tensor product space, which acts as an identity on the factor spaces except for the i-th and j-th ones.
In this paper we consider the most general reflection matrices 3 : which satisfy the reflection equation (RE) which satisfies the dual RE respectively The above parameters in (2.7) and (2.9) have to satisfy the restrictions [26] to make sure that the associated K-matrices satisfy the RE (2.8) and its dual (2.10).
Let us introduce the one-row monodromy matrices (2.13) and the double-row monodromy matrix The transfer matrix is given by where str 0 denotes the supertrace carried out in auxiliary space [8,9].
With the same procedure introduced in [11], one can show that [t(u), t(v)] = 0, which ensures the integrability of the model described by the Hamiltonian (1.1). The first order derivative of the logarithm of the transfer matrix t(u) yields the Hamiltonian (1.1) where the coupling constants in the Hamiltonian are expressed in terms of the parameters in the corresponding K-matrices given in (2.7), (2.9) and (2.11) as follows: It is remarked that the total number operatorN is still a conserved charge for the model described by the Hamiltonian (1.1), i.e., [H,N ] = 0.
By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz [26], the eigenvalues Λ(u) of the transfer matrix t(u) is given by an inhomogeneous T − Q relation For simplicity, we introduce the new parameters θ and ϕ which satisfy and other two new parameters θ ′ and ϕ ′ which satisfy The above parameterizations make the constraints (2.11) fulfilled automatically. We further assume the parameters η, ξ, θ, φ, ξ ′ , θ ′ , φ ′ being real numbers to ensure the hermitian of the Hamiltonian (1.1). For ε = 1 case, the possible taking values of the parameters ξ and ξ ′ are constrained in the region of ξ < 0 and ξ ′ < 1 or ξ > 0 and ξ ′ > 1 , respectively. While for ε = −1 case, the possible taking values of the parameters ξ and ξ ′ are constrained in the region of ξ < 0 and ξ ′ > 1 or ξ > 0 and ξ ′ < 1, respectively. In this paper, we choose the region of ξ < 0 and ξ ′ < 1 as an example. It is straightforward to extend the analysis below to other ranges of the fields. 4 Table 1: Solutions of BAEs (2.24)-(2.25) for the case of L = 2, η = 1, ξ = 0.6, θ = π/5, φ = π/3, ξ ′ = 1.5, θ ′ = 2π/3, φ ′ = π/4. The symbol n indicates the number of the eigenvalues, and E n is the corresponding eigenenergy. The energy E n calculated from (2.26) is the same as that from the exact diagonalization of the Hamiltonian (1.1).
To ensure Λ(u) to be a polynomial, the residues of Λ(u) at the poles v j and λ j must vanish, i.e., the 2M parameters {v j |j = 1, · · · , M} and {λ j |j = 1, · · · , M} must satisfy the nested Bethe ansatz equations (BAEs) From the relation (2.16), we have the eigenvalue of the Hamiltonian (1.1) in terms of the Bethe roots, which is given by

Finite-size effects
In order to study the contribution of the inhomogeneous term (the last term in (2.18)) to the ground state energy, we first consider the T − Q relation without the inhomogeneous The singular property of the T − Q relation (3.1) gives rise to the associated BAEs and where we have put v j = µ j − η 2 . Assume that µ j → µ j i, λ j → λ j i and η = 1, we obtain and The corresponding eigenvalue reads Now, we consider the contribution of the inhomogeneous term in Eq. (2.18) to the ground state energy of the system. In order to this, we should analyze the distribution of Bethe roots in the BAEs (3.4) and (3.5). For ξ < 0 and ξ ′ < 1 (equivalent to χ 1 > 1 and χ L < 1), by using the standard thermodynamic Bethe ansatz method and taking the limit of temperature tending to zero, we find that all the Bethe roots are real at the ground state in the region of ξ < 0 and ξ ′ ≤ 1/2 (equivalent to χ 1 > 1 and 0 ≤ χ L < 1). While in region of ξ < 0 and 1/2 < ξ ′ < 1 (equivalent to χ 1 > 1 and χ L < 0), besides the real Bethe roots, there exists an imaginary Bethe root which corresponds to a boundary bound state. Let us discuss them separately.

Region of ξ < 0 and ξ ′ ≤ 1/2
Firstly, we consider the case of ξ < 0 and ξ ′ ≤ 1/2 [21,25], in which all the Bethe roots are real at the ground state. Taking the logarithm of BAEs (3.4)-(3.5), we obtain 2 arctan(λ j − λ l ) + 2 arctan(λ j + λ l ), (3.8) where I j and J j are both quantum numbers which determine the eigenenergy and the corresponding eigenstates. It is well-known that the size of the system L, with either even or odd value, gives the same physics properties in the thermodynamic limit. Therefore, for simplicity, we set L as an even number.
The values of E inh , the contribution of the inhomogeneous term to the ground state energy, versus the system size L are shown in Figure 2. From the fitting, we find the power law relation between E inh and L, i.e., E inh = γL β . Due to the fact that β ≈ −1, the value of E inh tends to zero when the size of the system tends to infinity, which means that the 3.2 Region of ξ < 0 and 1/2 < ξ ′ < 1 In the region of ξ < 0 and 1/2 < ξ ′ < 1, one of the Bethe roots at the ground state goes to ( 1 2 − ξ ′ )i when the system-size L tends to infinity [25,35,36,37]. We note the value of this Bethe root is related with the boundary parameter ξ ′ . Without losing generality, we assume following BAEs = −1, j = 1, 2, · · · , M − 1. (3.11) Taking the logarithm of Eq.(3.11), we have 2πI j = 2 arctan µ j ζ + 4L arctan(2µ j ), j = 1, · · · , M − 1, (3.12) where the quantum numbers {I j } are chosen as {1, 2, · · · , L/2 − 1}. The corresponding energy reads The values of E inh versus the system size L are shown in Figure 3. From the fitted curves in Figure 3, we see that the E inh also satisfies the scaling law L −1 . 6 Therefore, the contribution of the inhomogeneous term to the ground state energy in the thermodynamic limit is zero and we have E hom = E ture ≡ E. In addition, the results indicating that the boundary bound state should be stable. The surface energy will compute in the next section.

Surface energy
In order to analyze the influence of the boundary fields, now we calculate the surface energy [38,39,40] of the system.
where Ξ n (x) = 2 arctan(2x/n). It turns to be a continuous function in the thermodynamic limit as the distribution of Bethe roots is continuous, i.e., Z(µ j ) → Z(u). In the thermodynamic limit, the density distributions are determined by Taking the derivative of Z(u) with respect to u, we obtain the density of states as where a n (u) = 1 2π (4.4) The ground state energy is equal to 5) and the energy density of the ground state is The energy density e g is equal to −2/π in the thermodynamic limit, which is the same with that of the periodic case [41]. The surface energy then can be given by By using the relation (4.7), one can calculate the surface energy of the one-dimensional supersymmetric t−J model with unparallel boundary fields. The results are shown in Figure   4, where the blue solid lines are the surface energy calculated by using the relation (4.7) and the red points and green stars are data obtained by employing the BST algorithms [42] to solve the surface energy of the Hamiltonian (1.1) in the thermodynamic limit. Specifically, for one of the red points or green stars, we first calculate the ground state energy E 0 (L) with L = 4, 8, · · · , 48 by the DMRG. Then, the large-L extrapolation of the surface energy was performed using BST algorithms from the sequence Note that e ∞ g = −2/π. From the Figure 4, we can see that the analytical and numerical results agree with each other very well for all tunable parameters. The surface energy increased with the increase of ζ. Taking the ζ → 0 limit of Eq.(4.7), we have E b (ζ → 0) = 0. Taking the ζ → ∞ limit of Eq.(4.7), we have E b (ζ → ∞) = 1.
4.2 Region of ξ < 0 and 1/2 < ξ ′ < 1 In this region, the ground state energy of the system in the thermodynamic limit reads  The results are shown in Figure 5. Again, we see that the analytical results and the numerical ones agree with each other very well.

Conclusions
In this paper, we have studied the thermodynamic limit of the one-dimensional supersymmetric t−J with unparallel boundary fields. It is shown that the contribution of the inhomogeneous term to the ground state energy is inversely proportional with L, i.e., E inh ∝ L −1 .
This fact enables us to calculate the surface energy (4.7) and (4.10), which is same as that for the case of parallel boundary fields [21]. Moreover, it implies that the inhomogeneous term in (2.18) surely gives some contributions to the other physical qualities such as the boundary conformal charge, which are related to the coefficients in the expansion of energy E in terms of the powers of L −1 (namely, the coefficient of L −1 corresponds to the conformal charge [36]).
The method used in this paper can be generalized to study the thermodynamic limit and surface energy of other models related to rational R-matrices, such as the spin-s XXX chain or the su(n) spin chain with unparallel boundary fields. These results may be applied to the theory of ultra-cold atom systems, asymmetric simple exclusion process.