On the Bethe states of the one-dimensional supersymmetric t − J model with generic open boundaries

Journal of High Energy Physics, Jul 2017

By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the supersymmetric t − J model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T − Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe states which have welldefined homogeneous limit. This exact solution provides basis for further analyzing the thermodynamic properties and correlation functions of the model.

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On the Bethe states of the one-dimensional supersymmetric t − J model with generic open boundaries

HJE On the Bethe states of the one-dimensional supersymmetric t Pei Sun 1 2 5 7 8 9 10 11 Fakai Wen 1 2 5 7 8 9 10 11 Kun Hao 1 2 5 7 8 9 10 11 Junpeng Cao 1 2 3 4 6 8 9 10 11 Guang-Liang Li 0 1 2 8 9 10 11 Tao Yang 1 2 5 7 8 9 10 11 Wen-Li Yang 1 2 5 7 8 9 10 11 Kangjie Shi 1 2 5 7 8 9 10 11 Beijing 1 2 8 9 10 11 China 1 2 8 9 10 11 Beijing 1 2 8 9 10 11 China 1 2 8 9 10 11 0 Department of Applied Physics, Xian Jiaotong University 1 Institute of Physics, Chinese Academy of Sciences 2 229 Taibai Beilu , Xian 710069 , China 3 School of Physical Sciences, University of Chinese Academy of Sciences 4 Collaborative Innovation Center of Quantum Matter 5 Shaanxi Key Laboratory for Theoretical Physics Frontiers 6 Beijing National Laboratory for Condensed Matter Physics 7 Institute of Modern Physics, Northwest University 8 Q relation, and the 9 Beijing , 100048 , China 10 28 Xianning West Road , Xian 710049 , China 11 8 3rd South Street, Zhongguancun, Beijing 100190 , China By combining the algebraic Bethe ansatz and the o -diagonal Bethe ansatz, we investigate the supersymmetric t J model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T corresponding eigenstates are expressed in terms of nested Bethe states which have wellde ned homogeneous limit. This exact solution provides basis for further analyzing the thermodynamic properties and correlation functions of the model. Bethe Ansatz; Lattice Integrable Models - J 1 Introduction 2 Integrability of the model 3 Nested algebraic Bethe ansatz 4 Reduced spectrum problem 5 Nested inhomogeneous T 6 Concluding remarks model have played essential roles in theoretical study of strongly correlated copperoxide based materials [6]. In general, the Hamiltonian of the supersymmetric t J model with the general boundary interaction terms is given by L 1 ;j=1 H = t X P hcj+; cj+1; + cj++1; cj; i P + J X Sk Sk+1 L 1 k=1 1 4 nknk+1 + X nl + nl+1 L 1 l=1 N^ + 1n1 +2hz1S1z +2h1 S1 +2h1+S1+ + LnL +2hzLSLz +2hL SL +2hL+SL+; (1.1) where t is the nearest neighbor hopping of electrons and J is the antiferomagetic exchange; + L is the total number of lattice sites; the operators cj; and cj; are the annihilation and creation operators of the electron with spin = 1 on the lattice site j, which satis es anticommutation relations, i.e., fci+; ; cj; g = i;j ; . There are only three possible states at the lattice site i due to the factor P = (1 nj; operator nj = P = nj; means the total number operator on site j and nj; = cj+; cj; ; ) ruled out double occupancies; the is the chemical potential and N^ = PL j=1 nj; 1;L are the boundary chemical potentials; z h1;L and h1;L are the boundary elds; the spin operators S and Sz = PjL=1 Sjz, form the su(2) algebra and can be expressed by = PjL=1 Sj , S+ = PjL=1 Sj+ Sj = cj+;1cj; 1; Sj+ = cj+; 1cj; (nj;1 nj; 1): (1.2) Sjz = 1 2 It is well-known that the one-dimensional t J model is integrable at the supersymmetric point J = 2t [7{9], and the model with the periodic boundary condition or the diagonal { 1 { boundaries has been studied by employing many Bethe ansatz methods [10{20]. For the non-diagonal boundary case, the nested algebraic Bethe ansatz method doesn't work since the U(1) symmetry is broken. With the help of the o -diagonal Bethe ansatz [21{27], the exact energy spectrum of the one-dimensional supersymmetric t J model with unparallel boundary elds has been obtained [28]. However, the eigenstates (or Bethe states) which have played important roles in applications of the model are still missing. In this paper, we study the supersymmetric t J model with generic integrable boundary conditions in grading: bosonic, fermionic and fermionic (BFF). By combining the graded nested algebraic Bethe ansatz and o -diagonal Bethe ansatz, we obtain the Bethe states which have well-de ned homogeneous limit and the corresponding eigenvalues of the transfer matrix of the model. Numerical results for the small size systems suggest that the spectrum obtained by the nested Bethe ansatz equations (BAEs) is complete. The paper is organized as follows. In section 2, the associated graded R-matrix and corresponding generic integral non-diagonal boundary re ection matrices are introduced. In section 3, by using the graded algebraic Bethe ansatz, we derive the eigenvalues of the transfer matrix of the system which related with the eigenvalues of the nested transfer matrix. In section 4, the eigenvalues of the nested transfer matrix are derived by o diagonal Bethe ansatz, and the Bethe states are also be given. In section 5, we construct the nested inhomogeneous T Q relation and the nested Bethe ansatz equations of the J model. Section 6 contains our results and give some discussions. 2 Integrability of the model In this paper we consider J = 2t = 2 which corresponds to the supersymmetric and integrable point [29]. The integrability of the model is associated with the rational R-matrix R(u) given by R12(u) = BB 0 u + B B B B B B B B B B B B CC : The R-matrix R(u) possesses the following properties Initial condition: Unitarity relation: Crossing Unitarity relation: R12(0) = P12; R12(u)R21( u) = 1(u) R1st21 ( u + ) R2st11 (u) = 2(u) id; id: Here P12 is the graded permutation operator with the de nition P 11 22 = ( 1)p( 1)p( 2) p( i) is the Grassmann parities which is one for fermions and zero for bosons. Here, we choose BFF grading which means p(1) = 0; p(2) = p(3) = 1 and R21(u) = P12R12(u)P12, sti denotes the super transposition in the i-th space (Ast)ij = Aji( 1)p(i)[p(i)+p(j)] and isti denotes the inverse super transposition. The functions 1(u) and 2(u) are given by 1(u) = (u )(u + ); 2(u) = u(u ): (2.6) Here and below we adopt the standard notations: for any matrix A 2 End(V), Aj is an super embedding operator in the Z2 graded tensor space V V , which acts as A on the j-th space and as identity on the other factor spaces. For R 2 End(V V), Rij is an super embedding operator of R in the Z2 graded tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones. The super tensor product of two = ( 1)[p( )+p( )]p( )A B . (For further details operators are de ned through (A B) we refer the reader to [30]). The R-matrix is an even operator (i.e., the parities of the non-zero matrix elements Rbadc of the R-matrix satis es p(a) + p(b) + p(c) + p(d) = 0) and satis es the graded quantum Yang-Baxter equation (QYBE) R12(u v) R13(u) R23(v) = R23(v) R13(u) R12(u v): In terms of the matrix entries, it reads R( satis es the graded re ection equation (RE) [31] R12(u1 u2)K1 (u1)R21(u1 + u2)K2 (u2) and the latter satis es the dual RE which take the form [32] t R12(u2 u1)K1+(u1)R21( u1 u2)ist1;st2K2+(u2) = K2+(u2)R~12( u1 u2)ist1;st2K1+(u1)R21(u2 u1); t R21(u)ist1;st2 = R~12(u)ist1;st2 = fR211(u)gist2 1 st2 fR121(u)gst1 1 ist1 ; : R12(u2 u1)K1+(u1)R21( u1 u2 + )K2+(u2) For our case, the dual re ection equation (2.10) reduces to In this paper we consider the generic non-diagonal K-matrices K (u) 0 + (2c 1)u 0 0 0 2c2u 1 0 u 2c1u CA + u 0 k11 0 Here the four boundary parameters c, c1, c2 and are not independent with each other, and satisfy a constraint The dual non-diagonal re ection matrix K+(u) is given by K+(u) = K ( u + =2) ( ;c;c1;c2)!( 0;c0;c01;c02) with the constraint 0 k32 k33 In order to show the integrability of the system, we rst introduce the \row-to-row" monodromy matrices T0(u) and T^0(u) T0(u) = R0L(u L)R0 L 1(u T^0(u) = R10(u + 1)R20(u + 2) L 1) R01(u 1); RL0(u + L); where f j ; j = 1 Lg are the inhomogeneous parameters and L is the number of sites. The one-row monodromy matrices are the 3 3 matrices in the auxillary space 0 and their elements act on the quantum space V L . The tensor product is in the graded space, so we can write n [T (u)]abo 1::: L 1::: L = R0N (u)caL LL : : : R0j (u)ccjj+j1 j : : : R01(u)bc2 1 1 ( 1)PjL=2(p( j)+p( j)) Pij=11 p( i): For the system with open boundaries, we need to de ne the double-row monodromy matrix T0(u) = T0(u)K0 (u)T^0(u); which satis es the similar relation as (2.9), in terms of matrix entries, they are Then the transfer matrix of the system is constructed as R(u )ba11ba22 T(u)bc11 R(u + )bc22cd11 T( )cd22 ( 1)(p(b1)+p(c1))p(b2) = T( )ba22 R(u + )ba11cb22 T(u)bc11 R(u )cd22cd11 ( 1)(p(b1)+p(c1))p(c2): t(u) = str0fK0+(u)T0(u)g = X( 1)p( ) K0+(u)T0(u) : (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) By using the (2.8), (2.9) and (2.10), we can prove the commutativity of t(u). (For further details about the commuting transfer matrix with boundaries for graded case, we refer the reader to [14, 32, 33]). The Hamiltonian (1.1) can be constructed by taking the derivative of the logarithm of the transfer matrix t(u) of the system with the parameters chosen as follows: H = 2 1 = hzL = 2 (1 2c); =2 (c0 1=2) u=0;f j=0g hz1 = + 2 0 ; h L = (c0 1=2) (2c 2 1) + (c0 0 1=2) ; h Then the transfer matrix can be expressed by From the relations (2.21), (3.1) and (3.3), the elements of matrix T0(u) acting on the reference state j 0i give rise to A(u) j 0i = k11(u)a0(u) j 0i ; D11(u) j 0i = D22(u) j 0i = 2u + 2u + k11(u)a0(u) + k22(u) k11(u)a0(u) + k33(u) 2u + 2u + k11(u) b0(u) j 0i ; k11(u) b0(u) j 0i ; D12(u) j 0i = k23(u)b0(u) j 0i ; D21(u) j 0i = k32(u)b0(u) j 0i ; Bi(u) j 0i 6= 0; Ci(u) j 0i = 0; i = 1; 2; { 5 { T0(u) = B C1(u) D11(u) D12(u) C : 0 A(u) B1(u) B2(u) 1 C2(u) D21(u) D22(u) A 2 3 t(u) = 4k1+1(u)A(u) X ki+1;j+1(u)Dji(u)5 ; + 2 i;j=1 where kij is the K matrix element in the ith row and jth column. Now we use the graded version of the nested algebraic Bethe ansatz method to obtain the eigenvalues of the transfer matrix (3.2). For this purpose, we rst de ne the reference state j 0i as j 0i = j0ij ; j0ij = B 0 C : 0 (3.1) (3.2) (3.3) (3.4) where The operators B1(u) and B2(u) acting on the reference state give nonzero values, and can be regarded as the creation operators of the eigenstates of the system. Following the procedure of the nested algebraic Bethe ansatz, the eigenstates of the transfer matrix can be constructed as j)(u + j); a0(u) = b0(u + ): (3.5) ju1; : : : ; uM ; F i = Ba1(u1)Ba2(u2) : : : BaM (uM )F a1a2:::aM j 0i ; (3.6) where we have used the convention that the repeated indices indict the sum over the values 1,2, and F a1:::an is a function of the spectral parameters uj. Moreover, the coe cients F a1:::an are actually the vector components of the nested Bethe state (see below (4.27)). As the transfer matrix (3.2) acting on the assumed states (3.6), we should exchange the positions of the operators A(u), Dij(u) and the operators Baj (uj). With the help of the re ection equation (2.20) and the Yang-Baxter equation (2.8), we can derive commutation 2u + b0(u) Y M i=2 i=1 (u 1 ui)(u + ui + ) ^ (u; fujg); { 6 { relations Bi(u)Bj(v) = A(u)Bj(v) = Bk(v)Bl(u) rlikj (u v) ; u v + 2v (u v (u + v + )(u )(u + v) v) Bj(v)A(u) + (u v)(2v + ) Bj(u)A(v); u + v + Bi(u)D~ij(v) D~ij(u)Bk(v) = reidf (u + v + )rkfjg(u (u + v + )(u v) v) Bd(v)D~eg(u) + reidj(2u + ) (2u + )(u v) Bd(u)D~ek(v) 2v rkidj(2u + ) 2u + (2v + )(u + v + ) Bd(u)A(v); where rij = u + Pij, P 11 22 = ( 1)p( 1)p( 2) 1 2 1 2 with the grading p(1) = p(2) = 1, and D~ij(u) = Dij(u) ij 2u + A(u): Acting the transfer matrix t(u) on the state j i and repeatedly using the commutation relations (3.8) and (3.9), we obtain t(u) ju1; : : : ; uM ; F i = (u) ju1; : : : ; uM ; F i + unwanted terms; where the corresponding eigenvalue (u) is (u) = " 3 # X ki+i (u) + k1+1(u) k11(u)a0(u) YM (u i=1 ui ) (u + ui) (u ui) (u + ui + ) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) K+(u) = K (u) = 2u 2u + 2u + 2u and ^ (u; fuj g) is the eigenvalue of the nested transfer matrix t^(u; fuj g) given by t^(u; fuj g) = g allow us to reconstruct the associated Bethe state (3.6), while the eigenvalue ^ (u; fuj g) gives rise to the associated eigenvalue (3.12) of the transfer matrix t(u) of the model. We shall determine the eigenvalue ^ (u; fuj g) and the corresponding eigenstate jF i in the next section. The condition that the unwanted terms should be zero gives rise to that the M Bethe roots must satisfy the associated Bethe ansatz equations (BAEs) 1 = K(1)(uk)a0(uk)Q(1)(uk (2uk + )b0(uk) ^ (uk; fuj g) ) ; k = 1 : : : M; (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) In the previous section, we have reduced searching eigenstates of the original transfer matrix t(u) (2.21) into the spectrum problem (3.16) of the nested transfer matrix t^(u; fuj g) given by (3.13). Now, we are in the position to calculate the eigenvalue ^ (u; fuj g) and the corresponding eigenstate jF i of the nested transfer matrix t^(u; fuj g) which allows us to reconstruct the Bethe state (3.6) of the supersymmetric t J model. Because the re ection matrices (3.14) and (3.15) have the o -diagonal elements. The traditional algebraic Bethe ansatz is invalid [22] due to the fact that the system doesnot have the obvious reference { 7 { where t 4 M i=1 Q(1)(u) = Y(u ui)(u + ui + ); K(1)(u) = (2 4c0)u2 + 2 0u 0 2 1 2 + 2c0 ( + (2c 1)u) : Some remarks are in order. It is easy to check that the nested Bethe state ju1; : : : ; uM ; F i given by (3.6) and the eigenvalue (u) given by (3.12) both have well-de ned homogeneous limit (i.e., j ! 0). This implies that in the homogeneous limit, the resulting Bethe states and the eigenvalue give rise to the eigenstate and the corresponding eigenvalue of the super J model described by the Hamiltonian (1.1). Reduced spectrum problem state. Thanks to the works [34{37], we can solve the spectrum problem (3.16) as follows. For simplicity, let and j = uj + 12 . We recognize the t^(u; fuj g) as the transfer matrix of the open spin-1=2 XXX chain of length M with non-diagonal boundary terms. Following the procedure in [34] t^( ; f j g) = = 2 2 2 2 t ( ; f j g) Tr0 n K+( ) T0( ; f j g) K ( ) Tb0( ; f j g) ; 0 0 o where and and that (4.3) satis es the dual one. The r-matrix possesses the properties Initial condition: Unitary relation: Crossing Unitary relation: r1t12( ) r2t11( PT-symmetry: 1 2 1 2 . The functions 1( ) and 2( ) are given by 1( ) = ( )( + ); 2( ) = ( 2 ): From the de nition (4.1), we know that the eigenvalue ( ) of the transfer matrix t ( ; f j g) is a polynomial of and satis es the relations: Crossing symmetry : Asymptotic behavior : ( ) = ( ) ( ( 2 + ); 4c1c02 4c01c2)u2M+2; ! 1; We have checked that (4.4) is the solution of the normal RE of the following form ( j ) ( j + ) = 4q( j ) ( 2 j )( + 2 j ) ; j = 1; ; M; { 8 { (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) HJEP07(21)5 ( ) (2 (2 2 2 2 ) K(2)( )a( ) Q(2)( + ) Q(2)( ) + K(3)( )d( ) Q(2)( ) Q(2)( ) + (2 )(2 2 )a( )a( h + ) Q(2)( ) ; = M j=1 M j=1 1 2 M j=1 (0) = Y ( ) = Y c ); + ) + + 1=2 c ); + 1=2 ) + 0) q ) d=ef Y M j=1 { 9 { Some special points can also be calculated directly by using the properties of the r-matrix and the re ection matrices K( )(u) as: It is remarked that the above relations were derived independently by the Separation of Variables [38]. These conditions (4.11){(4.16) allow us to construct the eigenvalue ^ ( ) in terms of an inhomogeneous T Q relation as [34, 35] )( j j + )( + j + ): (4.14) 1( l)trfK+(0)gK (0) id; 2( l + )trfK ( )gK+( ) id: where a( ) = Y( + j )( j ); d( ) = Y( j )( + j ); K(2)( ) = ( K(3)( ) = (p1 + 4(c02 p1 + 4(c02 (p1 + 4(c2 (p1 + 4(c2 Such parametrization obviously satis es the crossing symmetry (4.11), asymptotic behavior (4.12), production identity (4.13) and the values of the special points (4.15) and (4.16). To ensure ^ ( ) to be a polynomial, the residues of ^ ( ) at the poles wj must vanish, i.e., the M Bethe roots must satisfy the BAEs (2wj 2 )K(2)(wj )a(wj )Q(2)(wj + ) + 2wj K(3)(wj )d(wj )Q(2)(wj ) ideas in [36, 37], we rst introduce two transformation matrices g( ) Now, we construct the eigenstates jF i of the nested transfer matrix t^( ). Following the 2c01c2)p1+4c1c2+2c1c02+2c01c2 and n = 4c1c2 (2c01c2 2c1c02)p1+4c1c2+2c01c2+2c1c02 . The gauge matrices diagonalize the nested K-matrix K ( ) given by (4.3) and the matrix g( )K+( ) fg( ) g 1 respectively, namely, g(+)fg( )K+( )fg( )g 1gfg(+)g 1 ing vector components fF a1a2:::aM given by (3.6) of the original system.1 provided that the parameters fwj jj = 1; : : : ; M g satisfy the BAEs (4.24). The correspondg allow us to reconstruct the eigenstates ju1; : : : ; uM ; F i 1We have numerically checked, for small-site cases (such as L = 2; 3), that the states constructed by (3.6) with vector components fFa1a2:::aM g given by (4.27) give rise to the complete set of eigenstates the BAEs (5.2){(5.3). With the gauge transformation, we can introduce the gauged monodromy matrix U( ) U( ) = g(+) T ( ) ng( )K ( )fg( )g 1o ^ T ( ) fg(+)g 1 = A( ) B( ) C( ) D( ) ! : Then it was shown in [36, 37] that the eigenstate jF i in (3.16) can be expressed as where the reference state j0i is M O j=1 jF i = fg(j) g ( ) 1 Y B(wj )j0i = F a1a2:::aM ja1; : : : ; aM i; X ai=1;2 M j=1 M O j=1 j0i = j1ij ; j1ij = 1 0 ! ; Nested inhomogeneous T Now we are ready to write out the eigenvalues (u) of the transfer matrices t(u) in terms of some inhomogeneous T Q relation with the help of (3.12) and (4.17) as2 (u) = 1 (2u + ) (2u where the 2M Bethe roots must satisfy the BAEs (3.17) and (4.24), namely, In the homogeneous limit, the corresponding T Q relation and associated BAEs become (5.1) and (5.2){(5.3) by setting j = 0; j = 1; : : : ; N . Therefore the energy of the Hamiltonian (1.1) reads ju=0;f jg=0 + (2c 2 1) + (c0 0 1=2) 0 M + L 1 ) 2 E = = 2 M X where the 2M parameters fuj jj = 1; : : : ; M g and fvj jj = 1; : : : ; M g satisfy the resulting BAEs (5.2) and (5.3). Here we present the results for the L = 2 and L = 3 cases: the numerical solutions of the BAEs are shown in table 1 and table 2, which indicated that the eigenvalues are identical with the results we get from the exact diagonalization of the Hamiltonian (1.1). Numerical results for the small-site cases suggest that the spectrum obtained by the nested BAEs (5.2){(5.3) is complete. 2Although the inhomogeneous T Q relation given by (5.1) is di erent from that obtained in [28], each of them gives rise to the complete set of eigenvalues of the transfer matrix. The T Q relation (5.1) takes problem (3.16). 0:1000 1:6602i n 5:312156 1 corresponding eigenenergy. The energy En calculated from (5.4) is the same as that from the exact diagonalization of the Hamiltonian (1.1). 6 Concluding remarks In this paper, we have studied the one-dimensional supersymmetric t J model with the most generic integrable boundary condition, which is described by the Hamiltonian (1.1) and the corresponding integrable boundary terms are associated with the most generic non-diagonal K-matrices given by (2.14){(2.15). By combining the algebraic Bethe ansatz and the o -diagonal Bethe ansatz, we construct the eigenstates of the transfer matrix in terms of the nested Bethe states given by (3.6) and (4.27), which have well-de ned homogeneous limit. The corresponding eigenvalues are given in terms of the inhomogeneous T Q relation (5.1) and the associated BAEs (5.2){(5.3). The exact solution of this paper provides basis for further analyzing the thermodynamic properties and correlation functions of the model. These are under investigation and results will be reported elsewhere. Acknowledgments We would like to thank Prof. Y. Wang for his valuable discussions and continuous encouragements. The nancial supports from the National Program for Basic Research of MOST (Grant No. 2016YFA0300600 and 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11434013, 11425522 and 11547045), the Major Basic Research Program of Natural Science of Shaanxi Province (Grant No. 2017ZDJC-32), BCMIIS and the Strategic Priority Research Program of the Chinese Academy of Sciences are gratefully acknowledged. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Rev. B 37 (1988) 3759. model, Nucl. Phys. B 546 (1999) 691 [INSPIRE]. [1] F.C. Zhang and T.M. Rice, E ective Hamiltonian for the superconducting Cuoxides, Phys. [2] Z.-N. Hu and F.-C. Pu, Two magnetic impurities with arbitrary spins in open boundary t J HJEP07(21)5 [3] H. Eskes and G.A. Sawatzky, Tendency towards local spin compensation of holes in the high-Tc copper compounds, Phys. Rev. Lett. 61 (1988) 1415. [4] M.S. Hybertsen, M.S. Schluter and N.E. Christensen, Calculation of Coulomb-interaction parameters for La2CuO4 using a constrained-density-functional approach, Phys. Rev. B 39 (1989) 9028. [5] M.S. Hybertsen, E.B. Stechel, M. Schluter and D.R. Jennison, Renormalization from density-functional theory to strong-coupling models for electronic states in Cu-O materials, Phys. Rev. B 41 (1990) 11068. [6] S. Reja, J.V.D. Brink and S. Nishimoto, Strongly Enhanced Superconductivity in Coupled t J Segments, Phys. Rev. Lett. 116 (2016) 067002 [arXiv:1509.04117]. [7] C.K. Lai, Lattice gas with nearest neighbor interaction in one dimension with arbitrary statistics, J. Math. Phys. 15 (1974) 1675. [8] B. Sutherland, Model for a multicomponent quantum system, Phys. Rev. B 12 (1975) 3795. [9] S. SARKAR, The supersymmetric t J model in one dimension, J. Phys. A 24 (1991) 1137 [11] A. Gonzalez-Ruiz, Integrable open boundary conditions for the supersymmetric t J model. The Quantum group invariant case, Nucl. Phys. B 424 (1994) 468 [hep-th/9401118] [12] F.H. Essler, The supersymmetric t J model with a boundary, J. Phys. A 29 (1996) 6183. [13] Y. Wang, J. Dai, Z. Hu and F.C. Pu, Exact Results for a Kondo Problem in a One-Dimensional t J Model, Phys. Rev. Lett. 79 (1997) 1901. [14] H. Fan, B.-y. Hou and K.-j. Shi, Algebraic Bethe ansatz for the supersymmetric t J model with re ecting boundary conditions, Nucl. Phys. B 541 (1999) 483 [INSPIRE]. [15] Y.K. Zhou and M.T. Batchelor, Spin excitations in the integrable open quantum group invariant supersymmetric t J model, Nucl. Phys. B 490 (1997) 576 [INSPIRE]. [16] H. Fan and M. Wadati, Integrable boundary impurities in the t J model with di erent gradings, Nucl. Phys. B 599 (2001) 561 [INSPIRE]. t J model, J. Phys. A 33 (2000) 6187. [17] H. Fan, M. Wadati and R.H. Yue, Boundary impurities in the generalized supersymmetric 31 (1998) 5241. Nucl. Phys. B 777 (2007) 352 [INSPIRE]. [19] Z.H. Hu, F.C. Pu and Y. Wang, Integrabilities of the t J model with impurities, J. Phys. A [20] W. Galleas, Spectrum of the supersymmetric t J model with non-diagonal open boundaries, [21] J. Cao, W. Yang, K. Shi and Y. Wang, O -diagonal Bethe ansatz and exact solution of a topological spin ring, Phys. Rev. Lett. 111 (2013) 137201 [arXiv:1305.7328] [INSPIRE]. [22] Y. Wang, W.L. Yang, J. Cao and K. Shi, O -diagonal Bethe ansatz for exactly solvable [26] J. Cao, W.-L. Yang, K. Shi and Y. Wang, Nested o -diagonal Bethe ansatz and exact solutions of the SU(N ) spin chain with generic integrable boundaries, JHEP 04 (2014) 143 [arXiv:1312.4770] [INSPIRE]. [27] K. Hao, J. Cao, G.-L. Li, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms, JHEP 06 (2014) 128 [arXiv:1403.7915] [INSPIRE]. [28] X. Zhang, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the one-dimensional super-symmetric t J model with unparallel boundary elds, J. Stat. Mech. 1404 (2014) P04031 [arXiv:1312.0376] [INSPIRE]. [29] F.H. Essler and V.E. Korepin, Higher conservation laws and algebraic Bethe Ansatze for the [30] A.M. Grabinski and H. Frahm, Truncation identities for the small polaron fusion hierarchy, supersymmetric t J model, Phys. Rev. B 46 (1992) 9147. New J. Phys. 15 (2013) 043026 [arXiv:1211.6328] [INSPIRE]. t J model with boundaries, Phys. Rev. B 61 (2000) 3450. [31] H. Fan, M. Wadati and X.M. Wang, Exact diagonalization of the generalized supersymmetric [32] A.J. Bracken, X.-Y. Ge, Y.-Z. Zhang and H.-Q. Zhou, Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons, Nucl. Phys. B 516 (1998) 588 [INSPIRE]. [33] M.D. Gould, Y.Z. Zhang and H.Q. Zhou, Eight-state supersymmetric U model of strongly correlated fermions, Phys. Rev. B 57 (1998) 9498 [cond-mat/9709129]. [34] J. Cao, W.-L. Yang, K. Shi and Y. Wang, O -diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions, Nucl. Phys. B 875 (2013) 152 [arXiv:1306.1742] [INSPIRE]. Q equation for the open XXX chain with general [arXiv:1307.5049] [INSPIRE]. from Algebraic Bethe Ansatz, SIGMA 9 (2013) 072 [arXiv:1309.6165] [INSPIRE]. HJEP07(21)5 [18] G. Bedu rftig and H. Frahm , Open t J chain with boundary impurities , J. Phys. A 32 [35] R.I. Nepomechie , An inhomogeneous T boundary terms: completeness and arbitrary spin , J. Phys. A 46 ( 2013 ) 442002 [36] S. Belliard and N. Crampe , Heisenberg XXX Model with General Boundaries: Eigenvectors [37] X. Zhang , Y.- Y. Li , J. Cao , W.-L. Yang , K. Shi and Y. Wang , Retrieve the Bethe states of quantum integrable models solved via o -diagonal Bethe Ansatz , J. Stat . Mech. 1505 ( 2015 ) P05014 [arXiv: 1407 .5294] [INSPIRE]. [38] H. Frahm , A. Seel and T. Wirth , Separation of Variables in the open XXX chain , Nucl. Phys. B 802 ( 2008 ) 351 [arXiv: 0803 .1776] [INSPIRE].


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Pei Sun, Fakai Wen, Kun Hao, Junpeng Cao, Guang-Liang Li, Tao Yang, Wen-Li Yang, Kangjie Shi. On the Bethe states of the one-dimensional supersymmetric t − J model with generic open boundaries, Journal of High Energy Physics, 2017, 1-17, DOI: 10.1007/JHEP07(2017)051