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Bethe ansatz solutions of the τ 2-model with arbitrary boundary fields
Received: August
Bethe ansatz solutions of the
Xiaotian Xu 1 4 6 7 8
Kun Hao 1 4 6 7 8
Tao Yang 1 4 6 7 8
Junpeng Cao 1 2 3 5 7 8
Wen-Li Yang 0 1 4 6 7 8
Kangjie Shi 1 4 6 7 8
Open Access, c The Authors.
0 Beijing Center for Mathematics and Information Interdisciplinary Sciences
1 Institute of Physics, Chinese Academy of Sciences
2 Collaborative Innovation Center of Quantum Matter
3 School of Physical Sciences, University of Chinese Academy of Sciences
4 Shaanxi Key Laboratory for Theoretical Physics Frontiers
5 Beijing National Laboratory for Condensed Matter Physics
6 Institute of Modern Physics, Northwest University
7 Beijing , 100048 , China
8 Beijing 100190 , China
The quantum 2-model with generic site-dependent inhomogeneity and arbitrary boundary elds is studied via the o -diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.
Bethe Ansatz; Lattice Integrable Models
-
-model with arbitrary
boundary
Contents
1 Introduction 2 3 4
Transfer matrix
Properties of the transfer matrix
Asymptotic behaviors and average values
Fusion hierarchy
Truncation identity
Eigenvalues of the fundamental transfer matrix
Functional relations of eigenvalues
T-Q relation
Generic case
Degenerate case
A Speci c cases of the fused K-matrices
B Explicit expression of the average value functions
Introduction
simple Q-operator solution in terms of Baxter's T
Q relation. In fact, the Q-operator is
Q relation
Bethe states.
in the conventional T
Q relation (i.e., the inhomogeneous T
Q relation), we obtain the
special case
appendices A and B.
Transfer matrix
Let us x an odd integer p such that p
3, and let V be a p-dimensional vector space
matrices X and Z which act on the basis as
Xjmi = qmjmi;
Zjmi = jm + 1i;
where q
; N g denote the generators of the ultra-local Weyl algebra:
XnZm = q nm ZmXn;
Xnp = Znp = 1;
8n; m 2 f1;
Ln(u) =
eud(n+)Xn + e ud(n )Xn 1 (gn(+)Xn 1 + gn( )
Xn)Zn
(h(n+)Xn 1 + hn Xn)Zn 1 eufn(+)Xn 1 + e ufn( )
( )
An(u) Bn(u)
Cn(u) Dn(u)
n = 1; : : : ; N;
R(u) = BBB
0 sinh(u + )
sinh u sinh
sinh(u + )
with the crossing parameter
taking the special values
= 2i =p;
p = 2l + 1; l = 1; 2;
Antisymmetric-fusion conditions : R(
) =
Diag(cosh ; 1; 1; cosh ) P (+);
the one-row monodromy matrix T (u)
T (u) =
A(u) B(u)
C(u) D(u)
= LN (u) LN 1(u)
Baxter algebra
T (v)) = (1
T (v))(T (u)
where d(n+), d(n ), gn(+), gn( ), h(n+), h(n ), fn(+) and fn
each site. These parameters are subjected to two constraints,
gn( )h( ) = fn( )d(n+);
n
gn(+)h(+) = fn(+)d(n );
n
n = 1;
v)(Ln(u)
Ln(v)) = (1
Ln(v))(Ln(u)
n = 1; : : : ; N: (2.5)
The associated R-matrix R(u) 2 End(C2
C2) is the well-known six-vertex R-matrix
and the latter satis es the dual RE
R12(u1
u2)K1 (u1)R21(u1 + u2)K2 (u2)
R12(u2
u1)K1+(u1)R21( u1
= K2+(u2)R12( u1
2 )K1+(u1)R21(u2
L^n(u) =
e u fn(+)Xn 1 + eu+ fn( )Xn
(gn(+)Xn 1 + gn( )Xn)Zn
(h(n+)Xn 1 + h(n )Xn)Zn 1 e u d(n+)Xn + eu+ d(n )Xn 1
It is easy to check that Ln(u) enjoys the crossing property
Ln(u) = yL^tn( u
) y; n = 1; : : : ; N;
and the inverse relation
Ln(u) L^n( u) = DetqfLn(u)g
id; n = 1; : : : ; N;
let us introduce another one-row monodromy matrix T^(u) (cf., (2.10))
T^(u) =
A^ (u) B^ (u) !
C^ (u) D^ (u)
= L^1(u) L^2(u)
L^N (u):
with open boundaries can be constructed as [31]
t(u) = trfK+(u)T (u)K (u)T^(u)g;
of the conserved quantities.
refs. [33, 34], which is in the form of
K (u) =
K21(u) K22(u)
K11(u) = 2[sinh(
K22(u) = 2[sinh(
) cosh( ) cosh(u) + cosh( ) sinh( ) sinh(u)];
) cosh( ) cosh(u) cosh(
) sinh( ) sinh(u)];
K12(u) = e sinh(2u);
K21(u) = e
are three free boundary parameters. The most generic non-diagonal
K-matrix K+(u) is given by
K+(u) = K ( u
)j( ; ; )!( +; +; +):
We note that the two K-matrices possess the following properties
K (u + i ) =
K (0) =
tr K (0)
Properties of the transfer matrix
Asymptotic behaviors and average values
lim t(u) =
e f(2N+4)u+(N+2) g e +
id; (3.1)
( ) =
F ( ) =
with the help of the relations (2.22), namely,
t(0) =
sinh + cosh + cosh
23 cosh
cosh + sinh + cosh
d(n+)f (+) + e d(n )f ( )
n n
d(n+)f (+) + e d(n )f ( ) + e gn(+)h( ) + e
n n n
property of the transfer matrix t(u)
Ln(u + i ) =
L^n(u + i ) =
t(u + i ) = t(u):
as a Laurent polynomial of the form
where ftnjn = N +2; N +1;
tN+2 and t (N+2) are given by
; (N +2)g form the 2N +5 conserved charges. In particular,
tN+2 =
t (N+2) =
where the constants D( ) and F ( ) are determined by (3.2).
sponding transfer matrix t(u) satis es the following crossing relation
) = t(u):
averaging procedure [38]:
O(u) =
; N g. It was shown
in ref. [38] that
and the average values of each L-operator and L^-operator are given by
T (u) =
T^ (u) =
C(u) D(u)
A^(u) B^(u) !
C^(u) D^(u)
= LN (u) LN 1(u)
= L^1(u) L^2(u)
Ln(u) =
L^n(u) (...truncated)