On the Bethe states of the onedimensional supersymmetric t − J model with generic open boundaries
HJE
On the Bethe states of the onedimensional 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
GuangLiang Li 0 1 2 8 9 10 11
Tao Yang 1 2 5 7 8 9 10 11
WenLi 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 wellknown that the onedimensional 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
nondiagonal 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 onedimensional 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 wellde 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 Rmatrix and
corresponding generic integral nondiagonal 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 Rmatrix
R(u) given by
R12(u) = BB
0 u +
B
B
B
B
B
B
B
B
B
B
B
B
CC :
The Rmatrix 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 ith 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 jth 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 ith and jth 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 Rmatrix is an even operator (i.e., the parities of the nonzero matrix elements
Rbadc of the Rmatrix satis es p(a) + p(b) + p(c) + p(d) = 0) and satis es the graded quantum
YangBaxter 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 nondiagonal Kmatrices 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 nondiagonal 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 \rowtorow"
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 onerow 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 doublerow 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 YangBaxter 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 wellde 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 spin1=2 XXX chain of length M with nondiagonal 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 rmatrix possesses the properties
Initial condition:
Unitary relation:
Crossing Unitary relation: r1t12( ) r2t11(
PTsymmetry:
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 rmatrix
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 Kmatrix 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 smallsite 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 smallsite 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 onedimensional 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
nondiagonal Kmatrices 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 wellde 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. 2017ZDJC32), 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 (CCBY 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
highTc copper compounds, Phys. Rev. Lett. 61 (1988) 1415.
[4] M.S. Hybertsen, M.S. Schluter and N.E. Christensen, Calculation of Coulombinteraction
parameters for La2CuO4 using a constraineddensityfunctional approach, Phys. Rev. B 39
(1989) 9028.
[5] M.S. Hybertsen, E.B. Stechel, M. Schluter and D.R. Jennison, Renormalization from
densityfunctional theory to strongcoupling models for electronic states in CuO 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. GonzalezRuiz, Integrable open boundary conditions for the supersymmetric t
J model.
The Quantum group invariant case, Nucl. Phys. B 424 (1994) 468 [hepth/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
OneDimensional 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 nondiagonal 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
IzerginKorepin model with general nondiagonal 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 onedimensional
supersymmetric 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 openboundary conditions
for the qdeformed 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, Eightstate supersymmetric U model of strongly
correlated fermions, Phys. Rev. B 57 (1998) 9498 [condmat/9709129].
[34] J. Cao, W.L. Yang, K. Shi and Y. Wang, O diagonal Bethe ansatz solution of the XXX
spinchain 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].