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On the semi-classical limit of scalar products of the XXZ spin chain
Received: November
On the semi-classical limit of scalar products of the XXZ spin chain
Yunfeng Jiang 0 1
Joren Brunekreef 0 1
chain. The 0 1
Open Access 0 1
c The Authors. 0 1
0 Wolfgang Pauli Strasse 27 , CH-8093 Zurich , Switzerland
1 Institut fur Theoretische Physik, ETH Zurich
We study the scalar products between Bethe states in the XXZ spin chain with anisotropy j j > 1 in the semi-classical limit where the length of the spin chain and the number of magnons tend to in nity with their ratio kept nite and is a natural yet non-trivial generalization of similar methods developed for the XXX spin nal result can be written in a compact form as a contour integral in terms of Faddeev's quantum dilogarithm function, which in the isotropic limit reduces to the classical dilogarithm function.
Bethe Ansatz; Integrable Field Theories; Lattice Integrable Models
Contents
1 Introduction 2 3 4
The q-deformed A -functional
Algebraic Bethe ansatz of XXZ spin chain
Scalar products of XXZ spin chain
Slavnov determinant as an A -functional
Factorizing the Slavnov determinant
A symmetric representation
Semi-classical limit of XXZ spin chain
XXZ root distributions
Semi-classical limit of scalar products
The A -functional as a grand partition function
Semi-classical limit of the A -functional
Semi-classical limit of Slavnov determinant
Conclusions and outlook
A Commutation relations of operators in the XXZ model
B Large rapidity expansion
Gauge transformation
B.2 Limiting behavior
B.3 Commutation relations
Numerical solution of the XXZ Bethe equations
D Dilogarithm and quantum dilogarithm
Introduction
integrability for decades, see for example [2{5] and references therein.
xed and
nite. In this limit, the solutions of
representation [32] called the Slavnov determinant.
the XXZ spin chain, due to the presence of an anisotropy parameter
, the structure of
loghvjuiXXX
loghvjuiXXZ
of the logarithm in the trigonometric case. The q-analog is de ned as
The functions g(u) of the two cases are given by
x) =
n=1 [n]q
[n]q =
gXXX(u) =
gXXZ(u) =
tanh u
+ GuXXX(u) + GvXXX(u);
+ GuXXZ(u) + GvXXZ(u):
where the resolvents Gu(u) for the two cases are
GuXXX(u) =
GuXXZ(u) =
of Bethe roots on the cut. The full expression can be found in section 5.
actually be written in terms of Faddeev's quantum dilogarithm function
b(z)1 [35] as
loghvjuiXXZ
log p (gXXZ(u) + )
where the anisotropy
= i ,
> 0. The de nition of
b(z) and its relation to the
dilogarithm function are given in appendix D.
1We thank Ivan Kostov for pointing out this fact to us.
quantum dilogarithm function and its relation to the classical dilogarithm.
Note added.
At the nishing stage of this paper, we became aware that the same
prob
The q-deformed A -functional
products of more general states.
Ansatz. This will also serve to set up our notations and conventions.
Algebraic Bethe ansatz of XXZ spin chain
The Hamiltonian of the XXZ spin chain is given by
= 1,
H = J X
is the anisotropy. We impose a periodic boundary condition: L+1
1. For we recover the Hamiltonian of the XXX spin chain. The XXZ spin chain can be considered as a q-deformation of the XXX spin chain. To see this, we de ne the parameters q and :
q = ei :
We then obtain the isotropic case
q-deformed Lax operator takes the form
L^n;a(u) =
sinh (u + iSnz )
sinh (u
written as
L~n;a(x) =
and a hat for the additive parameters.
where we use the q-deformed operators
The Lax operator satis es the following RLL relation:
where the R-matrix takes the form
q Snz =
Ra;b(x; y) = BBB
0 a(x; y)
a(x; y) = q
b(x; y) =
c(x; y) = q
with the functions
de ned by:
Ta(x) = La;1(x)La;2(x)
La;L(x) =
The monodromy matrix satis es the following RT T -relation:
b(x; y) c(x; y)
c(x; y) b(x; y)
A(x) B(x)
C(x) D(x)
= YN sinh (uj
k6=j sinh (uj
j = 1;
de ne the transfer matrix t(x)
Tra Ta(x) which generates all the conserved charges of
the system.
j"Li, with all spins pointing up. This is an eigenstate of the operators A and D:
A(x) j i = a(x) j i
a(x) =
D(x) j i = d(x) j i
d(x) =
reference state:
jxi = Y
The scalar products we will compute are of the form
A~N (x)
Sq =
and provide a list of the resulting relations.
scalar product A~N (x) takes the following form:
[N ]q! = [N ]q
[N ]q =
K~N (x)
A~N (x) = K~N (x)
[ =x
eiLp~(x) =
[ =x
We now group parts of this expression together:
we refer to [38].
In terms of additive spectral parameters ui, we have
K^N (u)
A^N (u) = K^N (u)
[ =u
sinh (ui
A -functional then reads
= i . Our nal de nition of the
Auq [ ] =
[ =u
The q-deformed A -functional can be written equivalently as
u j=1
Auq [ ] =
(uj )e @=@uj
u =
Y sinh(uj
u is the trigonometric
Vandermonde determinant
representation similar to the one in [31].
Slavnov determinant as an A -functional
being replaced by their trigonometric counterparts.
The scalar product is then given by
hvjui = Y a(vi)d(ui)Su;v;
where Su;v is the Slavnov determinant,
The function Tu(v) is the eigenvalue of the transfer matrix:
where Qu(v) is the B (...truncated)