A Gaudin-like determinant for overlaps of N\'eel and XXZ Bethe states

0.16 MB·

by  Michael Brockmann

A Gaudin-like determinant for overlaps of N´eel and
XXZ Bethe states
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1 M. Brockmann1, J. De Nardis1, B. Wouters1, and J.-S. Caux1
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2 Institute for Theoretical Physics, University of Amsterdam, Science Park 904,
r Postbus 94485,1090 GL Amsterdam, The Netherlands
a
M E-mail: [email protected]

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Abstract. We derive a determinant expression for overlaps of Bethe states of
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the XXZ spin chain with the N´eel state, the ground state of the system in the

] antiferromagnetic Ising limit. Our formula, of determinant form, is valid for generic
h
c system size. Interestingly, it is remarkably similar to the well-known Gaudin formula
e forthenormofBethestates,andtoanotherrecently-derivedoverlapformulaappearing
m
in the Lieb-Liniger model.
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s
PACS numbers: 02.30.Ik, 05.30.-d
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1. Introduction
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In the last few decades one-dimensional quantum integrable models have proved to
o
c be immensely fertile theoretical laboratories for studying nonperturbative effects in
[
strongly-correlated systems. It however remains very arduous to obtain results going

2
v beyond formal expressions for eigenfunctions and basic spectral properties. Significant
7
further progress in the field has been obtained based on the fundamental breakthrough
7
8 computations of certain forms of scalar products [1] and of matrix elements of physical
2
operators [2] from the underlying algebraic structure of the models (most economically
.
1
expressed using the Algebraic Bethe Ansatz [3,4]).
0
4 A significant open problem, which has up to now received little attention, consists
1
: in asking how a generic but well-defined quantum state overlaps with eigenfunctions
v
i of a certain integrable model. For example, one might ask how a spin chain state
X
built from well-defined spin projections at each site projects onto eigenstates of a
r
a
Bethe Ansatz-solvable [5] Hamiltonian. Such overlaps form the basic building blocks
of a recently-proposed approach to address out-of-equilibrium dynamics in integrable
systems [6], but their calculation including the required knowledge of their scaling in
the thermodynamic limit poses significant challenges. For the Lieb-Liniger Bose gas [7],
an analytic expression for certain overlaps (namely of the condensate-like state with
eigenstates at finite interaction) has been recently found [8]. Similar results for the
XXZ model [9] are not yet known.
Overlap of the N´eel state with XXZ Bethe states 2
In this work we derive a practical determinant expression for the overlap of the N´eel
state with parity-invariant Bethe states of the XXZ model, starting from the original
determinant formula obtained in Refs. [10–12]. The advantage of our expression is that
it can be numerically evaluated for large system sizes, and that its thermodynamic
limit can be extracted analytically. Remarkably, it shows peculiar similarities with the
determinant expression for the overlaps in the Lieb-Liniger model [8]. Although we do
not fully understand the origin of these similarities at this stage, this coincidence is
most probably not fortuitous, and points to possible deep-rooted and ‘universal’ links
between quench situations in different models.
The paper is organized as follows. In Sec. 2 we review the standard Bethe Ansatz
solution of the XXZ chain and we present the most important formulas related to the
Bethe Ansatz that are needed in later chapters. In Sec. 3 we first present the final result
and then we show in its subsection 3.1 the proof of the formula. We also discuss the
limit to the isotropic spin chain in subsection 3.2.
2. The XXZ spin-1/2 chain
The one-dimensional spin-1/2 XXZ model is given by the Hamiltonian
N
H = J σxσx +σyσy +∆(σzσz 1) . (1)
j j+1 j j+1 j j+1 −
j=1
X(cid:0) (cid:1)
The coupling constant J > 0 fixes the energy scale and the parameter ∆ describes
the anisotropy of the nearest neighbour spin-spin coupling. The length of the chain is
given by N (which we choose to be even) and we impose periodic boundary conditions
σα = σα, α = x,y,z.
N+1 1
This Hamiltonian can be diagonalized by Bethe Ansatz [9]. We choose the
ferromagnetic state ... with all spins up as a reference state and construct
|↑↑ ↑i
interacting spin waves above this state. A state with M down spins reads
λ M = Ψ s M λ M σ− ...σ− ... (2a)
|{ j}j=1i {sj}Mj=1⊂{1,...,N} M { j}j=1|{ j}j=1 s1 sM| ↑↑ ↑i
X (cid:0) (cid:1)
with the explicit wave function in coordinate space
M M
i
Ψ s M λ M = ( 1)[Q]exp i s P(λ ) θ(λ λ ) .
M { j}j=1|{ j}j=1 − − a Qa − 2 Qb − Qa 
(cid:0) (cid:1) QX∈SM  Xa=1 aXb,b>=a1 
(2b)
 
 
Here, the coordinates s , j = 1,...,M, denote the positions of the down spins, and we
j
assume s < s for j < k. The set is the set of all permutations of integers 1,...,M,
j k M
S
and ( 1)[Q] denotes the parity of the permutation Q. Further, P(λ) is the momentum
−
associated to the rapidity λ,
sin(λ+iη/2)
P(λ) = iln , (3a)
− sin(λ iη/2)
(cid:20) − (cid:21)
Overlap of the N´eel state with XXZ Bethe states 3
and θ(λ) is the scattering phase shift given by
sin(λ+iη)
θ(λ) = iln . (3b)
sin(λ iη)
(cid:18) − (cid:19)
The parameter η is determined by the anisotropy parameter ∆ = cosh(η), and the set
of rapidities λ M in Eqs. (2) specifies the state. The latter is an eigenstate of the
{ j}j=1
Hamiltonian (1), and it is called Bethe state if the rapidities λ , j = 1,...,M, satisfy
j
the Bethe equations
N M
sin(λ +iη/2) sin(λ λ +iη)
j j k
= − , j = 1,...,M . (4)
sin(λ iη/2) − sin(λ λ iη)
(cid:18) j − (cid:19) k=1 j − k −
Y
In this case the rapidities λ , j = 1,...,M, are roots of the function
j
N M
sin(λ+iη/2) sin(λ λ iη)
A(λ) = 1+a(λ) with a(λ) = − k − , (5)
sin(λ iη/2) sin(λ λ +iη)
(cid:18) − (cid:19) k=1 − k
Y
and they are called Bethe roots. The norm of a Bethe state is given by [13]
λ M = λ M λ M , (6a)
k{ j}j=1k h{ j}j=1|{ j}j=1i
q M
sin(λ λ +iη)
λ M λ M = sinhM(η) j − k det (G ), (6b)
h{ j}j=1|{ j}j=1i sin(λ λ ) M jk
j k
j,k=1 −
Y
j6=k
M
G = δ NK (λ ) K (λ λ ) +K (λ λ ), (6c)
jk jk η/2 j η j l η j k
− − −
!
l=1
X
where K (λ) = sinh(2η) is the derivative of the scattering phase shift θ(λ).
η sin(λ+iη)sin(λ−iη)
In the following we will call states of the form (2) Bethe states or “on-shell” if the
parameters λ M fulfill the Bethe equations (4). If they are arbitrary the state is
{ j}j=1
M/2
called “off-shell”. We call a state parity invariant, λ , if the rapidities fulfill
|{± j}j=1i
the symmetry λ M = λ M = λ M/2 λ M/2 λ M/2.
{ j}j=1 {− j}j=1 { j}j=1 ∪{− j}j=1 ≡ {± j}j=1
3. Determinant expression for the overlaps with the N´eel state
We are interested in the overlap of the zero-momentum N´eel state given by
1
Ψ = ... + ... (7)
0
| i √2 |↑↓↑↓ i |↓↑↓↑ i
(cid:0) (cid:1)
with the XXZ Bethe states of the form (2). Therefore we only consider Bethe states
with M = N/2 flipped spins since the N´eel state lies in this sector of the XXZ chain.
For the sake of simplicity we choose N divisible by four such that M is even. We
Overlap of the N´eel state with XXZ Bethe states 4
M/2
consider parity-invariant Bethe states λ which have non-vanishing overlap
|{± j}j=1i
M/2
Ψ λ . For the total momentum of these states we find
h 0|{± j}j=1i
N
sin(λ +iη/2)
j
i ln = 0, (8a)
− sin(λ iη/2)
j=1 (cid:20) j − (cid:21)
X
and all other odd conserved charges Qˆ [14] evaluate to zero as well,
2n+1
M
ˆ M/2 M/2
Q λ = P (λ ) λ = 0, (8b)
2n+1|{± j}j=1i 2n+1 j |{± j}j=1i
j=1
X
because P is an odd function, P (λ) = i ∂2n ln sin(λ−µ+iη/2) . That we are
2n+1 2n+1 ∂µ2n sin(λ−µ−iη/2)
µ→0
only interested in the overlap with parity-invariant Behthe states isimotivated by the
fact that the odd conserved charges, evaluated on non-parity-invariant Bethe states,
are in general non-zero whereas their expectation value on the N´eel state vanishes [14].
Most-recently it was proven that the overlap of the N´eel state with non-parity-invariant
Bethe states is indeed zero [17].
For clarity, let us here quote the main result of our paper (whose derivation will be
presented below): the normalized overlap of the zero-momentum N´eel state Ψ with a
0
| i
M/2
parity-invariant Bethe state λ , which reads as follows:
|{± j}j=1i
Ψ λ M/2 M/2 tan(λ +iη/2)tan(λ iη/2) det (G+)
h 0|{± j}j=1i = √2 j j − M/2 jk (9a)
k{±λj}Mj=/12k Yj=1 p 2sin(2λj) sdetM/2(G−jk)
 
where
M/2
G± = δ NK (λ ) K+(λ ,λ ) +K±(λ ,λ ), j,k = 1,...,M/2, (9b)
jk jk η/2 j − η j l  η j k
l=1
X
 
K±(λ,µ) = K (λ µ) K (λ+µ), and K (λ) = sinh(2η) . Note that here Bethe
η η − ± η η sin(λ+iη)sin(λ−iη)
roots can be complex numbers (string solutions). The XXZ overlap formula (9) for
the N´eel state looks very similar to the Lieb-Liniger overlap formula for a state which
describes a Bose-Einstein condensate of one-dimensional free Bosons [8].
3.1. Proof of an off-shell formula
In order to prove overlap formula (9) we start with an expression for the overlap of
the N´eel state with an unnormalized off-shell state λ˜ M of the form (2), which was
|{ j}j=1i
proven inRef. [11,12] (see Eqs. (2.26) – (2.27)in Ref. [11] with ξ =: iη/2). Introducing
−
the short-hand notation s = sin(α+iβ) these equations from Ref. [11] can be written
α,β
Overlap of the N´eel state with XXZ Bethe states 5
as (note that N = 2M)
M s sM M s
Ψ λ˜ M = √2 λ˜j,+η/2 λ˜j,−η/2 λ˜j+λ˜k,η det (δ +U ), (10a)
h 0|{ j}j=1i s sM s M jk jk
"Yj=1 2λ˜j,0 λ˜j,+η/2#"j>Yk=1 λ˜j+λ˜k,0#
s s M s M s s 2M
U = 2λ˜k,η 2λ˜k,0 λ˜k+λ˜l,0 λ˜k−λ˜l,−η λ˜k,+η/2 . (10b)
jk  
s s s s s
λ˜j+λ˜k,0 λ˜j−λ˜k,η l=1 λ˜k−λ˜l,0 "l=1 λ˜k+λ˜l,+η# λ˜k,−η/2!
Yl6=k  Y
 
This expression is unhandy to perform the thermodynamic limit as well as for parity-
M/2
invariant states λ due to zeroes of the determinant and singularities in
|{± j}j=1i
the prefactor. To perform the limit to parity-invariant states (not necessarily Bethe
˜ ˜
states) we set λ = λ + ǫ for j = 1,...,M/2 and λ = λ + ǫ for
j j j j j−M/2 j−M/2
−
j = M/2+1,...,M. Here, the parameters λ , j = 1,...,M/2, are arbitrary complex
j
numbers. We shall see that the main ingredients to the derivation of formula (9)
are the limits ǫ 0, j = 1,...,M/2, and the pseudo parity invariance of the set
j
→
λ˜ M = λ +ǫ M/2 λ +ǫ M/2. We derive then an off-shell version of Eq. (9),
{ j}j=1 { j j}j=1 ∪{− j j}j=1
which has the same form up to corrections which are zero when the rapidities satisfy
the Bethe equations (4).
If we multiply the prefactor in (10a) with α = M/2 s2ǫj,0 and the determinant
reg j=1 s0,η
by its inverse α−1 we get regular expressions with well-d(cid:16)efined(cid:17)limits ǫ 0, j =
reg Q j →
1,...,M/2, as well as a well-defined XXX scaling limit λ ηλ and η 0 afterwards.
→ →
M/2
Assuminganappropriateorderofrapiditiesthelowestorderin ǫ oftheregularized
{ j}j=1
prefactor and of the determinant read
M/2 M/2
s s s s
γ = √2 λj,+η/2 λj,−η/2 λj+σλk,+η λj+σλk,−η , (11a)
 s2  s2 
j=1 2λj,0 j>k=1 λj+σλk,0
Y Y
 σ=± 
 
 
M/2
s
0,η
det = lim det (δ +U ) . (11b)
reg {ǫj→0}jM=/12Yj=1 s2ǫj,0 M jk jk 
The task is now to calculate the limits ǫ 0, j = 1,...,M/2, in (11b). For this
j
→
purpose we reorder the rows and columns of the matrix 1+U under the determinant in
such a way that the pair (λ +ǫ , λ +ǫ ) belongs to the two rows and two columns
k k k k
−
with indices 2k 1 and 2k.
−
We consider the resulting M M matrix as a M/2 M/2 block matrix built of
× ×
2 2 blocks. Collecting all terms up to first order in all ǫ , j = 1,...,M/2, we obtain
j
×
on the diagonal the 2 2 blocks
×
s s
1+U = 1+δ 2λk,+ηa , U = 2λk,−ηa−1 +δ b− ,
2k−1,2k−1 k s k 2k−1,2k −s k k k
2λk,0 2λk,+η
s s
U = 2λk,+ηa +δ b+ , 1+U = 1+δ 2λk,−ηa−1 , (12a)
2k,2k−1 −s k k k 2k,2k k s k
2λk,−η 2λk,0
Overlap of the N´eel state with XXZ Bethe states 6
and for the off-diagonal blocks (1 j,k M/2, j = k)
≤ ≤ 6
s s s s
U = δ 2λk,+η 0,η a , U = δ 2λk,−η 0,η a−1 ,
2j−1,2k−1 ks s k 2j−1,2k ks s k
λj+λk,0 λj−λk,+η λk−λj,0 λj+λk,+η
s s s s
U = δ 2λk,+η 0,η a , U = δ 2λk,−η 0,η a−1 , (12b)
2j,2k−1 ks s k 2j,2k ks s k
λj−λk,0 λj+λk,−η λj+λk,0 λk−λj,+η
where we defined the abbreviations δ = s /s for k = 1,...,M/2 and
k 2ǫk,0 0,η
M/2 s s 2M
a = a˜(λ ) = λk−σλl,−η λk,+η/2 . (13)
k k  
s s
l=1 λk−σλl,+η (cid:18) λk,−η/2(cid:19)
 Yσ=± 
 
Note that the function a˜ is different from the function a of Eq. (5) because in the
definition of a the parameters λ M are Bethe roots whereas here in Eq. (13) they
{ k}k=1
are arbitrary. The symbols b+ and b− in Eq. (12a) denote the first order corrections of
k k
the elements U and U , respectively. After a short calculation we obtain up
2k,2k−1 2k−1,2k
to zeroth order in δ
k
s s s2M M/2 s
2λk,−ηa−1b++ 2λk,+ηa b− = 2cosh(η) s ∂ ln λk,+η/2 λk+σλl,−η . (14)
s k k s k k − 0,η λk s2M s
2λk,+η 2λk,−η n λk,−η/2 Yll=6=1j σY=± λk+σλl,+ηo
We further define α = s2λk,+ηa and multiply the M M matrix 1+U from the
k −s2λk,−η k ×
left and from the right rqespectively with the matrices
diag α ,α−1,...,α ,α−1 , diag α−1,α ,...,α−1,α . (15)
M 1 1 M M M 1 1 M M
(cid:0) (cid:1) (cid:0) (cid:1)
Since the determinants of these diagonal matrices are equal to one this transformation
does not change the value of the determinant det (1+U). We see that the structure
M
of the matrix becomes
1−δ1ss22λλ11,−,0ηα12 1+δ1b−1α12 δ a12 b12 ...
1+δ b+α−2 1 δ s2λ1,+ηα−2 2 c d
 " 1 1 1 − 1 s2λ1,0 1 # " 12 12 # 
 δ1 ac21 db21 11−+δδ2bss2+2λλα22,−,−0η2α22 11+δδ2bs2−2λα2,+22ηα−2  ,
 " 21 21 # " 2 2 2 − 2 s2λ2,0 2 # 
 ... ... 
 
 (16)
where the elements a , b , c , and d , j,k = 1,...M/2, of the off-diagonal blocks
jk jk jk jk
can be calculated by multiplying the 2 2 block (12b) with diag (α ,α−1) from the left
× 2 j j
and with diag (α−1,α ) from the right.
2 k k
Under the determinant the matrix (16) can be further simplified by replacing
column 2k 1 by the difference of columns 2k 1 and 2k for all k = 1,...,M/2
− −
and afterwards by replacing row 2j 1 by the difference of rows 2j 1 and 2j for all
− −
Overlap of the N´eel state with XXZ Bethe states 7
j = 1,...,M/2. Up to first order in each δ ,δ ,...,δ the determinant of 1 + U
1 2 M/2
becomes
δ D 0 δ e 0 δ e 0
1 1 2 12 3 13
...
0 1 0 0 0 0
 " #" #" # 
D e e ...
 δ e 0 δ D 0 δ e 0  1 12 13
 1 21 2 2 3 23  M/2 e D e
detM  " 0 0 #" 0 1 #" 0 0 #  = δk detM/2 21 2 23 
  e e D
  31 32 3
 δ1e31 0 δ2e32 0 δ3D3 0  Yj=k   ... ... 
   
 " 0 0 #" 0 0 #" 0 1 #   
 
 ... ... 
 
  (17)
where e = a b c +d . The diagonal elements D , k = 1,...,M/2, are given
jk jk jk jk jk k
− −
by
s s
D = lim 2λk,−ηα2 2λk,+ηα−2 b+α−2 b−α2
k {δk→0}kM=/12 (cid:18)− s2λk,0 k − s2λk,0 k − k k − k k(cid:19)
s s s2M M/2 s
= 2λk,+ηa + 2λk,−ηa−1 +2cosh(η) s ∂ ln λk,+η/2 λk+σλl,−η
s k s k − 0,η λk s2M s
2λk,0 2λk,0 n λk,−η/2 Yll6==k1 σY=± λk+σλl,+ηo
M/2
s s
= 2λk,+ηA + 2λk,−ηA¯ +2Ms K (λ ) s K+(λ ,λ ), (18)
s k s k 0,η η/2 k − 0,η η k l
2λk,0 2λk,0 l=1
Xl6=k
where we defined K+(λ,µ) = K (λ µ)+K (λ+µ), K (λ) = s0,2η and A = 1+a ,
η η − η η sλ,+ηsλ,−η k k
A¯ = 1+a−1. Using these definitions the off-diagonal elements can be simplified to
k k
s s a
e = 2λk,+η 2λk,−η j K+(λ ,λ )+f , (19)
jk s s a η j k jk
r 2λj,+η 2λj,−η k
(cid:0) (cid:1)
where the symbols f are specified below (see Eq. (20c)). We can forget about the
jk
M/2
square roots as they cancel each other under the determinant. The factor δ
k=1 k
cancels exactly the factor α−1 = M/2 s2ǫj,0 = M/2δ−1 in Eq. (11b).
reg j=1 s0,η k=1 k Q
Hence, wereducedtheM dimensional determinant ofoverlaps withdeviatedparity-
Q Q
invariant states in the limit of vanishing deviations to an M/2 dimensional determinant
which depends on M/2 independent parameters. The result can be eventually summed
up as follows:
M/2 M/2 M/2
Ψ λ = Ψ λ +ǫ λ +ǫ
h 0|{± j}j=1i h 0|{ j j}j=1 ∪{− j j}j=1i {ǫj→0}Mj=/12
(cid:12)
(cid:12)
= γdet (G+)+ ǫ M/2 (cid:12) = γdet (G+), (20a)
M/2 jk O { j}j=1 {ǫj→0}Mj=/12 M/2 jk
h (cid:16) (cid:17)i
Overlap of the N´eel state with XXZ Bethe states 8
where the prefactor γ is given by Eq. (11a) and the matrix G+ reads
jk
M/2
G+ = δ Ns K (λ ) s K+(λ ,λ ) +s K+(λ ,λ )
jk jk 0,η η/2 j − 0,η η j l  0,η η j k
l=1
X
 s A +s A¯ 
+δ 2λj,+η j 2λj,−η j +(1 δ )f , j,k = 1,...,M/2 (20b)
jk jk jk
s −
2λj,0
s s s s s s
f = A 2λj,+η 0,η 2λj,−η 0,η +A A¯ 2λj,−η 0,η
jk k k j
s s − s s s s
(cid:18) λj+λk,0 λj−λk,+η λj−λk,0 λj+λk,−η(cid:19) λj−λk,0 λj+λk,−η
s s s s
A¯ 2λj,−η 0,η + 2λj,−η 0,η . (20c)
j
− s s s s
(cid:18) λj−λk,0 λj+λk,−η λj+λk,0 λj−λk,−η(cid:19)
Note that here the matrix elements G+ contain an additional factor s = isinh(η)
jk 0,η
compared to the earlier definition of G+ in Eq. (9b), which is convenient for taking
jk
the XXX limit η 0 which we shall do in Sec. 3.2. Formula (20) holds for arbitrary
→
complex numbers λ , j = 1,...,M/2.
j
M/2
If we are on-shell in Eq. (20), i.e. if the set λ satisfies the Bethe equations
{± j}j=1
(5), and if we are not in the XXX limit with rapidities at infinity, A and A¯ just vanish.
j j
Together with the norm of Bethe states and using the symmetry of the Gaudin matrix
(6c) as well as the relation
A B
det = det (A+B)det (A B) (21)
M M/2 M/2
B A −
!
for block matrices we finally gain the overlap formula (9), where the additional factors
s in G+, 1 j,k M/2, cancel against the prefactor sM in the norm formula (6).
0,η jk ≤ ≤ 0,η
q
3.2. The XXX limit including rapidities at infinity
At the isotropic point ∆ = 1, there are Bethe states with rapidities at infinity that need
to be specially treated. In order to obtain an appropriate overlap formula for the XXX
case including such rapidities at infinity we use the off-shell formula (20). As long as
we are off-shell we can scale all rapidities by η, send η to zero and can afterwards send
some of the rapidities to infinity. Within this scaling Eq. (20a) reduces to
Ψ λ M/2 = γ˜det (G˜+), (22a)
h 0|{± j}j=1i M/2 jk
Overlap of the N´eel state with XXZ Bethe states 9
where now
M/2 λ2 + 1 M/2 (λ +σλ )2 +1
γ˜ = √2 j 4 j k (22b)
 4λ2  (λ +σλ )2 
j=1 j j>k=1σ=± j k
Y Y Y
  
M/2
G˜+ = δ 2MK˜ (λ ) K˜+(λ ,λ ) +K˜+(λ ,λ ) [j,k = 1,...,M/2]
jk jk 1/2 j − 1 j l  1 j k
l=1
X
 (2λ +i)A +(2λ i)A¯
j j j j ˜
+δ − +(1 δ )f , (22c)
jk jk jk
2λ −
j
K˜+(λ,µ) = K˜ (λ µ)+K˜ (λ+µ), and K˜ (λ) = 2α . The explicit form of f˜ is not
α α − α α λ2+α2 jk
˜
interesting since either the f vanish due to Bethe equations or they give subleading
jk
corrections if one of the parameters λ goes to infinity. The term a reads now
j j
M/2 2M
λ σλ i λ +i/2
a = j − k − j , (23)
j
 λ σλ +i λ i/2
k=1σ=± j − k (cid:18) j − (cid:19)
Y Y
 
and we have again A = 1+a = 0 if the set λ M/2 satisfies the Bethe equations of
j j {± j}j=1
the XXX model.
Let us consider the case when n pairs of Bethe roots are at in such a way that
±∞
the difference and the sum of two Bethe roots which do not belong to the same pair is
also infinity. Let us denote the corresponding parameters by µ , j = 1,...,n. We have
j
m = M/2 n finite pairs (λ , λ ) that fulfill the Bethe equations
j j
− −
m 2M
λ σλ i λ +i/2
a = j − k − j = 1, j = 1,...,m. (24)
j
λ σλ +i λ i/2 −
"k=1σ=± j − k #(cid:18) j − (cid:19)
Y Y
We used the fact that all factors including one of the parameters µ , k = 1,...,n, are
k
equal to one. The Bethe equations for all infinite Bethe roots are trivial. The prefactor
γ˜ has no divergencies in µ and just reads
j
√2 m λ2 +1/4 m (λ +σλ )2 +1
j j k
γ˜ = . (25)
4n 4λ2 (λ +σλ )2
"j=1 j #"j>k=1σ=± j k #
Y Y Y
Now, wetakethelimitsµ ,j = 1,...,n, inEq.(22c)andtreatalltermscontaining
j
→ ∞
A or A¯ carefully. The determinant simplifies to
j j
det (G˜+)
lim M/2 jk = 4n(2n)!det (Gˆ+) (26)
{µj→∞}nj=1 N({µj}nj=1)! m jk
where Gˆ+ is a reduced m m version of the M/2 M/2 matrix G˜+ in Eq. (22c), and
jk × × jk
where we used a factor ( µ n ) = n µ−2 to get a non-vanishing result.
N { j}j=1 j=1 j
We need the same factor ( µ n ) to correctly renormalize the norm formula (6).
N { j}j=Q1
We introduce the abbreviations Λ2m = λ m , 2n = µ n , and ΛM =
± {± j}j=1 M± {± j}j=1 ±
Overlap of the N´eel state with XXZ Bethe states 10
Λ2m 2n where the subscripts and 2m, 2n are reminiscent to the parity invariance
± ∪M± ±
of the state and the 2m flipped spins, 2n pairs at infinity, respectively. The square of
the norm of the state can be written as
ΛM ΛM
h ±| ±i = Λ2m S+ 2n S− 2n Λ2m = (4n)! Λ2m Λ2m . (27)
N2({µj}nj=1)(cid:12){µj→∞}nj=1 h ± | | ± i h ± | ± i
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:12)
The norm square (cid:12)of the Bethe state Λ2m is given by the XXX limit of the norm
| ± i
formula (6b) and reads
m λ2 +1/4 m ((λ +σλ )2 +1)2
Λ2m Λ2m = j j k det (Gˆ ), (28)
h ± | ± i λ2 (λ +σλ )4 2m jk
"j=1 j #"j>k=1σ=± j k #
Y Y Y
where Gˆ is a reduced 2m 2m version of the M M Gaudin matrix G˜ .
jk jk
× ×
All together we obtain for the overlap of the N´eel state (7) with a normalized,
(m,n)
parity-invariant XXX Bethe state Λ , where m pairs (λ , λ ) are finite and all
| ± i j − j
other rapidities µ , j = 1,...,n, are at (N = 2n, 2m+2n = M = N/2):
j ∞
± ±∞
Ψ Λ(m,n) = hΨ0|ΛM±i = √2N∞! m λ2j +1/4 detmGˆ+jk , (29a)
h 0| ± i kΛM±k (cid:12)(cid:12){µj→∞}nj=1 (2N∞)! Yj=1 q 4λj vuudetmGˆ−jk
(cid:12) m p  t
(cid:12)
Gˆ± = δ 2MK˜ (λ ) K˜+(λ ,λ ) +K˜±(λ ,λ ), j,k = 1,...,m (29b)
jk jk 1/2 j − 1 j l 1 j k
!
l=1
X
with K˜±(λ,µ) = K˜ (λ µ) K˜ (λ + µ) and K˜ (λ) = 2α . Note again that in
α α − ± α α λ2+α2
Eqs. (29) Bethe roots can be complex numbers (string solutions).
4. Summary
In this work we obtained an exact expression for the overlap of the zero-momentum N´eel
state with parity-invariant Bethe states of the XXZ model for any value of the aniso-
tropy ∆. Generalizations to other simple classes of initial states (dimer and q-dimer
states) as in Ref. [12] are straightforward. Our result has a remarkable “Gaudin-like”
form, which has also recently been found in the Lieb-Liniger case [8], and suggests
a potentially deep link between such overlaps in integrable systems. On one hand,
it allows for a rigorous proof of the Lieb-Liniger overlap formula [15]. On the other
hand, the fact that our formula applies in the thermodynamic limit N opens the
→ ∞
possibility of using the method of Ref. [6] to investigate out-of-equilibrium dynamics in
spin chains, paralleling the results of Ref. [8]. Other applications include the comparison
with asymptotic results for the dynamical free energy [16], and finally taking the infinite
Trotter number limit for the surface free energy of an open spin chain as in Ref. [11].
We will investigate these topics in future publications.