Adjacent Spin Operator Dynamical Structure Factor of the S=1/2 Heisenberg Chain

Abstract

Considering the adjacent spin operators $S^z_jS^z_{j+1}$ and $S^-_jS^-_{j+1}$
in the S=1/2 Heisenberg chain, we give a determinant representation of their
form factors. The dynamical structure factors of the respective operators are
computed over the whole Brillouin zone in several magnetic fields and the
resulting signal is analyzed in terms of excitation types. Among other results,
we find that the $S^z_jS^z_{j+1}$ dynamical structure factor carries a large
weight of the 4-spinon excitations which are distinguishable from the 2-spinon
signal because they are located outside the 2-spinon spectrum.

Full text

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Adjacent Spin Operator Dynamical Structure Factor of the S = 1/2 Heisenberg Chain
Antoine Klauser1,2, Jorn Mossel2, Jean-S´ ebastien Caux2
1Instituut-Lorentz, Universiteit Leiden, P. O. Box 9506, 2300 RA Leiden, The Netherlands and
2Institute for Theoretical Physics, Universiteit van Amsterdam,
P. O. Box 94485, 1090 GL Amsterdam, The Netherlands
(Dated: January 5, 2012)
Considering the adjacent spin operators Sz
we give a determinant representation of their form factors. The dynamical structure factors of
the respective operators are computed over the whole Brillouin zone in several magnetic fields and
the resulting signal is analyzed in terms of excitation types. Among other results, we find that
the Sz
distinguishable from the 2-spinon signal because they are located outside the 2-spinon spectrum.
jSz
j+1and S−
jS−
j+1in the S = 1/2 Heisenberg chain,
jSz
j+1dynamical structure factor carries a large weight of the 4-spinon excitations which are
Contents
I. Introduction1
II. Setup
A. Bethe Ansatz
B. String solutions
C. Spin of the eigenstates and infinite rapidity 3
2
2
2
III. Form factor determinant representations 3
A. Sz
j+1
1. Homogeneous limit
2. Fourier transform
B. S−
j+1
1. Reduction of determinant for string
solutions
2. Fourier transform
jSz
4
5
5
6
jS−
7
7
IV. Sz
jSz
Heisenberg spin chain
A. Spin sectors
B. Sum rules
1. Sz
j+1integrated intensity
2. Sz
j+1first frequency moment
j+1dynamical structure factor in the
7
8
9
9
9
jSz
jSz
V. S−
jS−
Heisenberg spin chain
A. Sum rules
1. S−
j+1integrated intensity
2. S−
j+1first frequency moment at zero
magnetic field
j+1dynamical structure factor in the
9
10
10
jS−
jS−
11
VI. Identification of the excitations in the
isotropic spin chain
A. Excitations at h = 0
1. String excitations
2. Spin raising and infinite rapidities
B. Excitations at h > 0
1. Particle-hole
2. Spin raising
3. String excitations
11
11
12
12
12
12
12
13
VII. DSF Evaluations
A. S−−++(q,ω)
13
13
B. S4z(q,ω)14
VIII. Conclusion15
References16
I.INTRODUCTION
One-dimensional (1D) systems are particularly prone
to quantum criticality as compared to higher-dimensional
ones.They offer a prolific playground for low-
temperature phenomenology [1, 2]. Realizations of 1D
critical quantum liquids are numerous and can take the
form of e.g. stripes in cuprate high-temperature super-
conductors, trapped ultracold atomic gases and carbon
nanotubes. One of the simplest strongly correlated mod-
els, which remains of great interest due to its underlying
richness, is the Heisenberg S = 1/2 antiferromagnetic
chain [3] with Hamiltonian given by
H = J
N
?
j=1
1
2
?S−
jS+
j+1+ S+
jS−
j+1
?+
?
Sz
jSz
j+1−1
4
?
−hSz
j
(1)
where N is the number of lattice sites, periodic boundary
conditions are understood. Several crystals are known
to realize 1D spin chain e.g. KCuF3, SrCuO3, CuPzN
and have been probed by neutron scattering [4–10]. For
a positive coupling constant J > 0 and zero external
magnetic field h = 0 the ground state is antiferromag-
netic without long-range order. Its basic excitations are
spinons, fractionalized spin excitations that emerge in the
critical state [11]. The Heisenberg model is integrable
with an exact solution given by Bethe Ansatz [12, 13].
The properties of its ground state have been extensively
studied in [14–16]. Moreover, in the case where a spin-
exchange anisotropy is introduced, one can obtain a com-
plete phase diagram in terms of the magnetic field and
the anisotropy parameter [17]. The isotropic Heisenberg
chain belongs to the universality class of Luttinger liq-
uids and the asymptotics of its correlation functions can
be calculated by effective field theory methods. Whereas
the computation of correlation functions was for a long
arXiv:1201.0867v1 [cond-mat.str-el] 4 Jan 2012

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2
time a limitation of the Bethe Ansatz method, significant
progresses have been made in the last decade [18, 19] al-
lowing the calculation of single spin operator dynamical
structure factors (DSF) in the S = 1/2 spin chain [20–
28].
Going one step further from what has been studied so
far, we consider the calculation of what we shall call the
adjacent spin operator DSF
Sab¯ a¯b(q,ω) =
1
N
N
?
j,j?=1
?∞
j(t)Sb
−∞
dte−iq(j−j?)+iωt
?Sa
j+1(t)S¯ a
j?(0)S¯b
j?+1(0)? . (2)
where a,b = z,−,+. Resuming the study of the spin-
exchange DSF in [29], our interest lies in the computation
of specific components of the latter correlation function.
In addition to the resonant inelastic x-ray scattering re-
sponse function studied in this previous paper, the adja-
cent spin operator form factors (FF) are related to the
study of transport and spin currents along the spin chain
[30–34] and to the fidelity in the Heisenberg spin chain
[35, 36].
This paper presents a non-perturbative calculation of
the adjacent spin operator DSF for (a,b) = (z,z),(−,−)
(in (2)) in the antiferromagnetic S = 1/2 Heisenberg
chain.
Low-energy effective field theories have already been
used to predict the asymptotics of the adjacent spin op-
erator correlation function in a few cases [37, 38].
The contents of this publication are the following. In
section II, we present the exact eigenstates of the system
provided by the Bethe Ansatz. Using a determinant rep-
resentation, the FFs of the operators Sz
are formulated in terms of eigenstates rapidities in section
III. We show in section IV and V how the S−−++(q,ω)
and S4z(q,ω) DSF are constructed using the FF expres-
sions. In section VI, a catalog of the excitations is given
by describing the quasiparticles which occur in presence
or absence of magnetic field. In section VII, we evalu-
ate numerically the expression for the two DSFs over the
entire Brillouin zone. The contribution of each family of
excitations is shortly discussed. The results are summa-
rized in the conclusion in section VIII.
jSz
j+1and S−
jS−
j+1
II. SETUP
A.Bethe Ansatz
Thanks to the integrability of the Heisenberg spin
chain, the exact eigenfunctions of (1) can be obtained
via the Bethe Ansatz [12]. Because the Hamiltonian con-
serves the total magnetization of the chain, the Hilbert
space separates into subspaces of fixed magnetization
M = N/2−??N
j=1Sz
j?. Defining a reference state with all
j=1|↑?, the Bethe Ansatzspins pointing upwards |0? =?N
yields, for eigenstates belonging to the M-subspace:
|{k}M? =
?
?
P∈πM
(−1)[P]exp
?

i
kPaja
2
?
?
1≤a<b≤M
ϕ(kPa,kPb)


(3)
·
j1,...,jM
expi
M
?
a=1
|j1,...,jM? .
{k} is a set of M quasi-momenta determining the eigen-
state and the scattering phase ϕ(ka,kb) is defined by
2cotϕ(ka,kb)
2
= cotka
of M localized down-flipped spins as
2− cotkb
2. We introduced the state
|j1,...,jM? = S−
j1...S−
jM|0? .(4)
It is convenient to express the quasi-momenta in terms
of rapidities λi
exp(iki) =λi+ i/2
λi− i/2.(5)
Imposing periodic boundary conditions leads to the
Bethe equations for the rapidities
arctan(2λi) =π
NIi+1
N
M
?
k=1
arctan(λi− λk),
∀i . (6)
Taking N to be even, the quantum numbers I1,...,IM
are a set of distinct integers for odd M and half-integers
for even M.Each of these sets of quantum numbers
specifies a set of rapidities and vice versa.
shown that the set of quantum numbers for the ground
state is {I0
of an eigenstate are obtained, the energy and momentum
of the state follow straightforwardly
It can be
i= i−M+1
2
},i = 1,...,M. Once the rapidities
E = −J
M
?
i=1
1/2
1/4 + λ2
i
− h(N
2− M)
P = πM −2π
N
M
?
i=1
Ii (mod 2π). (7)
The Bethe equations represented in (6) provide a sys-
tematic way of constructing eigenstates and the solutions
allow one to obtain dynamical correlation functions and
thermodynamic quantities with high precision.
B.String solutions
The solutions of (6) are not restricted to real rapidi-
ties only. As Bethe mentioned in his seminal work [12],
self-conjugate complex solutions, which can be inter-
preted as bound states of down spins, also satisfy the
Bethe equations. Bethe [12] made the conjecture, com-
pleted later by Takahashi et al.
plex rapidities form string structures which are λj,a
[39], that such com-
α
=

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λj
the string length and the string deviation δnj,a
is an index that runs from 1 to Mjwhere Mjis the num-
ber of njstrings (strings of length nj) and j = 1,...,Ns
where Nsis the total number of different possible lengths.
The string hypothesis is unfortunately not always cor-
rect, although it can be shown that generally the string
deviations vanish exponentially with system size. The
correct number of complex solutions is moreover smaller
than the one constructed from the string hypothesis and
other kinds of complex solutions appear. The propor-
tion is of order 1/√N [40]. More extended discussions
about string structures can be found in [41]. The devi-
ated string solutions turn out to be unimportant for the
correlation functions presented in this paper.
In the presence of string solutions with vanishing de-
viations the Bethe equations (6) become undetermined.
The remedy is to rewrite the Bethe equations only in
terms of the real centers λnj
α of the strings to obtain the
Bethe-Takahashi equations [39]:
α+i
2(nj+ 1 − 2a) + iδj,a
α
with a = 1,...,nj, njbeing
α
. Here α
Nθnj(λj
α) −
Ns
?
k=1
Mk=1
?
β
Θnj,nk(λj
α− λk
β) = 2πIj
α
(8)
where
θnj(λ) = 2atan(2λ/nj)
Θnj,nk(λ) = (1 − δj,k)θ|nj−nk|+2(λ) + 2θnj−nk−2(λ) +
+ ... + 2θnj+nk−2(λ) + θnj+nk(λ)(9)
and Ij
strings of length nj.
αare the set of quantum numbers corresponding to
C.Spin of the eigenstates and infinite rapidity
In the isotropic spin chain the total spin operator
??
over, the eigenstates created with the Bethe Ansatz are
highest-weight with respect to the global su(2) algebra
i.e. the states constructed with M rapidities have the
spin eigenvalues Stot
z
= Stot=
der to access eigenstates with Stot
act on a Bethe state with the global spin lowering op-
erator S−
q=0=
are actually also Bethe solutions but they include in-
finite rapidities. Indeed the Bethe equations allow ra-
pidities to go to infinity and from eq. 3, one notices
that a state with M rapidities of which M?tend to
infinity can be written |{λ1,...,λM−M?,∞,...,∞
?S−
Stot
z
=N
jSj
?2commutes with the Hamiltonian and in con-
sequence its eigenvalues Stotare conserved.More-
N
2− M [42].
z
< Stot, one must
In or-
1
√N
?N
j=1S−
j. These descendant states
? ???
M?
}? =
q=0
?M?
|{λ1,...,λM−M?}?.This eigenstate is no
2− M + M?and
longer highest-weight but has Stot=N
2− M.
The norm of a state which contains one or two infinite
rapidities can easily be calculated by commutation of spin
operators and are
N({λ,∞}M) =
N − 2M + 2
N
2(N − 2M + 4)(N − 2M + 3)
N2
· N({λ}M−2) .
N({λ}M−1) ,(10)
N({λ,∞,∞}M) =
(11)
III. FORM FACTOR DETERMINANT
REPRESENTATIONS
The natural language for expressing normalized eigen-
states and form factors (FF) is the algebraic Bethe
Ansatz [19, 21, 22]. In the algebraic Bethe Ansatz for-
malism, one introduces the monodromy matrix T(λ) that
acts on the space C2⊗HNwhere HNis the Hilbert space
of the N-sites chain. The operator is represented in the
auxiliary space C2as
?A(λ) B(λ)
where λ is called the spectral parameter.
monodromy matrix is constructed as a product of
L−operators
T(λ) = LN(λ;ξN)...L1(λ;ξ1)
T(λ) =
C(λ) D(λ)
?
(12)
The
(13)
with Lj(λ;ξj) = R0j(λ−ξj), where 0 refers to the auxil-
iary space and j to the j−th site of the spin chain. The
R−matrix Rij must be a solution of the Yang-Baxter
equation. The ξjare inhomogeneity parameters and the
homogeneous spin chain (1) corresponds to setting all
ξj → i/2. However, it is more convenient to take this
limit at the end of the computation, as we will do in
this work. One can now define the transfer matrix as
T (λ) = A(λ) + D(λ), from which the Hamiltonian can
be recovered
H =i
2
d
dλlnT (λ)
????
λ=i/2
. (14)
Hence the transfer matrix and the Hamiltonian have com-
mon eigenfunctions and can be constructed as
B(λ1)...B(λM)|0?
(15)
and the dual state as
?0|C(λ1)...C(λM)(16)
where the rapidities λj have to satisfy the Bethe equa-
tions (6). The norm of eigenstates can now be expressed
as a determinant [42–44]. Another very import result is
a determinant expression for the scalar product [44]
?0|C(µ1)...C(µM)B(λ1)...B(λM)|0?
(17)

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where the µjcan be arbitrary and the λjhave to satisfy
the Bethe equations or vice versa. A last essential ingre-
dient is the inverse mapping that expresses the local spin
operators σz
jin terms of the non-local operators
A(λ),B(λ),C(λ),D(λ) [21]
j,σ+
j,σ−
σz
j= −2
j−1
?
i=1
[A + D](ξi)D(ξj)
N
?
k=j+1
[A + D](ξk) + 1,
σ−
j=
j−1
?
j−1
?
i=1
[A + D](ξi)B(ξj)
N
?
N
?
k=j+1
[A + D](ξk),
σ+
j=
i=1
[A + D](ξi)C(ξj)
k=j+1
[A + D](ξk) .(18)
Using these expressions and the scalar product (17), we
expressed in terms of determinants the adjacent spin op-
erator form factors which are defined as
?GS|Sa
jSb
j+1|α?
(19)
with |GS?, |α? respectively the groundstate and an eigen-
state of the system. The form factors involving a single
spin operator were first derived in [21], the general case
was treated in [22]. These expressions are not always
suitable for a numerical evaluation, therefore we re-derive
below the form factors of interest for this paper.
A.Sz
jSz
j+1
With ?{λ}| and |{µ}? two eigenstates of the system satisfying the Bethe equations, the form factor of the operator
Sz
j+1=
two adjacent σz
joperators is
jSz
1
4σz
jσz
j+1reads
1
4?{λ}|σz
jσz
j+1|{µ}?. Using (18) and the properties of the transfer matrix, the product of
σz
jσz
j+1= 4
j−1
?
+σz
i=1
j+ σz
[A + D](ξi)D(ξj)D(ξj+1)
N
?
i=j+2
[A + D](ξi)
j+1− 1 .(20)
The form factor for a single σz
form factor is the term:
the D(λ) operator on a general state [19] twice, we can write the form factor as a double summation of Slavnov
determinants
joperator has been considered in [21, 25] and the remaining non-trivial part of σz
?j−1
jσz
j+1
i=1[A + D](ξi)D(ξj)D(ξj+1)?N
k=j+2[A + D](ξk). Using the expression for the action of
?{λ}|D(ξj)D(ξj+1)|{µ}? =
M
?
n=1
d(µn)
?
i?=n
φ(µn− µi+ η)
φ(ξj− µi)
M
?
m=1,m?=n
d(µm)
?
1
i?=m
φ(µm− µi+ η)
φ(ξj+1− µi)
·
φ2(η)
φ(µm− µn+ η)
detH({λi},{µi?=m,n,ξj,ξj+1})
φ(ξj− ξj+1)
1
b(µm,ξj+1)
?
l>kφ(λk− λl)?
l<kφ(µk− µl)
·
(21)
with
Hab({λ},{µ}) =
φ(η)
φ(λa− µb)
?

?
b(µ,λ) =
k?=a
φ(λk− µb+ η) − d(µb)
?
k?=a
φ(λk− µb− η)


d(λ) =
N
i=1
b(λ,ξi),
φ(µ − λ)
φ(µ − λ + η).(22)
The value of η and the definition of φ(λ) depend on the anisotropy parameter of the spin chain [21]. We keep their
general expressions for the rest of this section in order to cover any anisotropy. However, in the isotropic case they
take the values
φ(λ) = λ,η = i .(23)

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5
1.Homogeneous limit
The homogeneous spin chain corresponds to the case where ξi→ η/2
order to obtain a finite expression for the ratio detH({λ},{µi?=m,n,ξj,ξj+1})/φ(ξj+1−ξj). We use l’Hˆ opital’s rule to
have a well-defined homogeneous limit and the matrix elements become
∀i. This limit should be taken with care in
?{λ}|D(η/2)D(η/2)|{µ}? =
?
iφ2(λi+ η/2)φ−2(µi− η/2)
?
j>kφ(λk− λj)?
j<kφ(µk− µj)
M
?
n=1
An
M
?
m=1
detFnm.(24)
Here we defined
An= d(µn)φ(µn− η/2)
M
?
i=1
φ(µi− µn− η)
Bn
ab= (1 − δb,n)d(µb)φ(µb+ η/2)
?
i?=n
φ(µi− µb− η)
φ(η)
φ(λa− η/2)φ(λa+ η/2)
Ca =
φ(η)φ(2λa)
φ2(λa− η/2)φ2(λa+ η/2)
?
Bn
b = m
?
Ca,
Fnm
ab
=
Gn
ab,
ab,
b ?= m
Gn
ab=
φ(η)
φ(λa−µb)
??
k?=aφ(λk− µb+ η) − d(µb)?
k?=aφ(λk− µb− η)
?
,b ?= n
b = n
.(25)
Using Laplace’s determinant formula (see chapter 9 appendix in [19]) and the fact that Bnis a rank one matrix, we
write the summation over determinants as a single determinant:
M
?
m=1
detFnm= det(Gn+ Bn) − detGn
(26)
and the complete form factor is then
?{λ}|σz
jσz
j+1|{µ}?
=
?{λ}|σz
+ 4ϕj−1({λ})
ϕj−1({µ})
j|{µ}?+?{λ}|σz
?
j+1|{µ}?− ?{λ}|{µ}?
j>kφ(λk− λj)?
?
iφ2(λi+ η/2)φ−2(µi+ η/2)
j<kφ(µk− µj)
M
?
n=1
An(det(Gn+ Bn) − detGn) .(27)
Here we introduced
ϕj({λ}) =
M
?
l=1
?φ(λl− η/2)
φ(λl+ η/2)
?j
(28)
and if {λ} is an eigenstate, then we can write ϕj({λ}) = e−ijP{λ}with P{λ}the total momentum of the state.
2.Fourier transform
For the further computation of the dynamical structure factor, the Fourier transform of the form factor (27) is
necessary. With (1/√N)?N
j=1e−iqj?{λ}|Sz
jSz
j+1|{µ}?= (1/√N)?
peip?{λ}|Sz
q−pSz
p|{µ}?, we give here an explicit