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arXiv:0706.4327v3 [cond-mat.str-el] 10 Aug 2007
Dynamical structure factor at small q for the XXZ
spin-1/2 chain
R G Pereira1†, J Sirker2, J-S Caux3, R Hagemans3, J M
Maillet4, S R White5and I Affleck1
1Department of Physics and Astronomy, University of British Columbia, Vancouver,
British Columbia, Canada V6T 1Z1
2Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart,
Germany
3Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam,
The Netherlands
4Laboratoire de Physique,´Ecole Normale Sup´ erieure de Lyon et CNRS, 69364 Lyon
C´ edex 07, France
5Department of Physics and Astronomy, University of California, Irvine CA 92697,
USA
E-mail:†rpereira@phas.ubc.ca
Abstract.
longitudinal dynamical structure factor Szz(q,ω) for the anisotropic spin-1/2 chain
in the gapless regime. Using bosonization, we derive a low energy effective model,
including the leading irrelevant operators (band curvature terms) which account for
boson decay processes. The coupling constants of the effective model for finite
anisotropy and finite magnetic field are determined exactly by comparison with
corrections to thermodynamic quantities calculated by Bethe Ansatz. We show that a
good approximation for the shape of the on-shell peak of Szz(q,ω) in the interacting
case is obtained by rescaling the result for free fermions by certain coefficients extracted
from the effective Hamiltonian. In particular, the width of the on-shell peak is argued
to scale like δωq ∼ q2and this prediction is shown to agree with the width of the
two-particle continuum at finite fields calculated from the Bethe Ansatz equations.
An exception to the q2scaling is found at finite field and large anisotropy parameter
(near the isotropic point). We also present the calculation of the high-frequency tail
of Szz(q,ω) in the region δωq≪ ω −vq ≪ J using finite-order perturbation theory in
the band curvature terms. Both the width of the on-shell peak and the high-frequency
tail are compared with Szz(q,ω) calculated by Bethe Ansatz for finite chains using
determinant expressions for the form factors and excellent agreement is obtained.
Finally, the accuracy of the form factors is checked against the exact first moment
sum rule and the static structure factor calculated by Density Matrix Renormalization
Group (DMRG).
We combine Bethe Ansatz and field theory methods to study the
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Dynamical structure factor at small q for the XXZ spin-1/2 chain2
1. Introduction
The problem of a spin-1/2 chain with anisotropic antiferromagnetic exchange interaction
has been extensively studied [1] and constitutes one of best known examples of strongly
correlated one-dimensional systems [2]. The XXZ model is integrable and exactly
solvable by Bethe Ansatz [3, 4], which makes it possible to calculate exact ground state
properties as well as thermodynamic quantities. At the same time, it exhibits a critical
regime as a function of the anisotropy parameter, in which the system falls into the
universality class of the Luttinger liquids. The long distance asymptotics of correlation
functions can then be calculated by applying field theory methods. The combination
of field theory and Bethe Ansatz has proved quite successful in explaining low energy
properties of spin chain compounds such as Sr2CuO3and KCuF3[5].
Recently, most of the interest in the XXZ model has turned to the study of
dynamical correlation functions.The relevant quantities for spin chains are the
dynamical structure factors Sµµ(q,ω), µ = x,y,z, defined as the Fourier transform
of the spin-spin correlation functions [6]. These are directly probed by inelastic neutron
scattering experiments [7, 8]. They are also probed indirectly by nuclear magnetic
resonance [9], since the spin lattice relaxation rate is proportional to the integral of the
transverse structure factor over momentum [10, 11].
Even though one can use the Bethe Ansatz to construct the exact eigenstates, the
evaluation of matrix elements, which still need to be summed up in order to obtain the
correlation functions, turns out to be very complicated in general. In the last ten years
significant progress has been made with the help of quantum group methods [12]. It is
now possible to write down analytical expressions for the form factors for the class of two-
spinon excitations for the Heisenberg chain (the isotropic point) at zero field [13, 14, 15],
as well as for four-spinon ones [16, 17, 18]. No such expressions are available for general
anisotropy in the gapless regime or for finite magnetic field, but in those cases the form
factors can be expressed in terms of determinant formulas [19, 20, 21] which can then
be evaluated numerically for finite chains for two-particle states [22, 23, 24] or for the
general multiparticle contributions throughout the Brillouin zone [25, 21].
From a field theory standpoint, dynamical correlations can be calculated fairly
easily using bosonization [26]. However, this approach is only asymptotically exact in
the limit of very low energies and relies on the approximation of linear dispersion for the
elementary excitations. In some cases, the main features of a dynamical response depend
on more detailed information about the excitation spectrum of the system at finite
energies – namely the breaking of Lorenz invariance by band curvature effects. That
poses a problem to the standard bosonization approach, in which nonlinear dispersion
and interaction effects cannot be accommodated simultaneously. For that reason, a
lot of effort has been put into understanding 1D physics beyond the Luttinger model
[27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
In particular, using the bosonization prescription one can relate the longitudinal
dynamical structure factor Szz(q,ω) at small momentum q to the spectral function of
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Dynamical structure factor at small q for the XXZ spin-1/2 chain3
the bosonic modes of the Luttinger model. In the linear dispersion approximation, the
conventional answer is that Szz(q,ω) is a delta function peak at the energy carried by the
noninteracting bosons [2]. As in the higher-dimensional counterparts, the broadening
of the peak is a signature of a finite lifetime. The problem of calculating the actual
lineshape of Szz(q,ω) at small q is thus related to the fundamental question of the
decay of elementary excitations in 1D.
In the bosonization approach, interactions are included exactly, but band curvature
effects must be treated perturbatively. All the difficulties stem from the fact that
band curvature operators introduce interactions between the bosons and ruin the exact
solvability of the Luttinger model. To make things worse, perturbation theory in those
operators breaks down near the mass shell of the bosonic excitations [37] and no proper
resummation scheme is known to date. The best alternative seems to be guided by
the fermionic approach, which treats band curvature exactly but applies perturbation
theory in the interaction [28].
In this paper we address this question using both bosonization and Bethe Ansatz.
Our goal is to make predictions about Szz(q,ω) that are nonperturbative in the
interaction (i.e., anisotropy) parameter and are therefore valid in the entire gapless
regime of the XXZ model (including the Heisenberg point). We focus on the finite field
case, which in the bosonization approach is described by a simpler class of irrelevant
operators. To go beyond the weakly interacting regime we can resort to the Bethe
Ansatz equations in the thermodynamic limit to calculate the exact coupling constants
of the low energy effective model. Our analysis is supported by another type of Bethe
Ansatz based method, which calculates the exact form factors for finite chains. This
provides a nontrivial consistency check of our results.
The outline of the paper is as follows. In section 2, we introduce the longitudinal
dynamical structure factor for the XXZ model in a finite magnetic field and review the
exact solution for the XX model. In section 3 we describe the effective bosonic model and
explain how to fix the coupling constants of the irrelevant operators. Section 4 provides
a short description of the Bethe Ansatz framework which is relevant for our analysis.
In section 5, we show how to obtain the broadening of Szz(q,ω) in a finite magnetic
field both from field theory and Bethe Ansatz and compare our formula with the exact
form factors for finite chains. In section 6 we present a more detailed derivation of the
high-frequency tail of Szz(q,ω) reported in [32]. The zero field case is briefly addressed
in section 7. Finally, we check the sum rules and discuss the finite size scaling of the
form factors in section 8.
2. XXZ model
We consider the XXZ spin-1/2 chain in a magnetic field
H = J
N
?
j=1
?Sx
jSx
j+1+ Sy
jSy
j+1+ ∆Sz
jSz
j+1− hSz
j
?. (2.1)
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Dynamical structure factor at small q for the XXZ spin-1/2 chain4
Here, J is the exchange coupling, ∆ is the anisotropy parameter, h is the magnetic
field in units of J and N is the number of sites in the chain with periodic boundary
conditions. We focus on the critical regime (given by −1 < ∆ ≤ 1 for h = 0). We are
interested in the longitudinal dynamical structure factor at zero temperature
Szz(q,ω) =1
N
N
?
j,j′=1
e−iq(j−j′)
?+∞
−∞
dteiωt?Sz
j(t)Sz
j′ (0)?, (2.2)
where q takes the discrete values q = 2πn/N, n∈ Z. It is instructive to write down the
Lehmann representation for Szz(q,ω)
Szz(q,ω) =2π
N
α
where Sz
state energy. The matrix elements
?0??Sz
Szz(q,ω) is a sum of delta function peaks at the energies of the eigenstates with fixed
momentum q. In this sense, Szz(q,ω) provides direct information about the excitation
spectrum of the spin chain. In the thermodynamic limit N → ∞, the spectrum is
continuous and Szz(q,ω) becomes a smooth function of q and ω. Equation (2.3) also
implies that Szz(q,ω) is real and positive and can be expressed as a spectral function
?
???0??Sz
q
??α???2δ (ω − Eα+ EGS),
??α?
(2.3)
q=?
jSz
je−iqj, |α? is an eigenstate with energy Eα and EGS is the ground
q
are called form factors. We denote by
??α???2the transition probabilities that appear in (2.3). For a finite system,
F2≡???0??Sz
q
Szz(q,ω) = −2Imχret(q,ω),
for ω > 0. χret(q,ω) is the retarded spin-spin correlation function and can be obtained
from the Matsubara correlation function
(2.4)
χ(q,iωn) = −1
N
N
?
j,j′=1
e−iq(j−j′)
?β
0
dτ eiωnτ?Sz
j(τ)Sz
j′ (0)?,(2.5)
where β is the inverse temperature, by the analytical continuation iωn→ ω + iε.
It is well known that the one-dimensional XXZ model is equivalent to interacting
spinless fermions on the lattice.The mapping is realized by the Wigner-Jordan
transformation
j→ nj−1
S+
S−
Sz
2,
j→ (−1)jc†
j→ (−1)jcje−iπφj,
jeiπφj,(2.6)
where cjis the annihilation operator for fermions at site j, nj= c†
In terms of fermionic operators, the Hamiltonian (2.1) is written as
jcjand φj=?j−1
ℓ=1nℓ.
H = J
N
?
+∆
j=1
?
−1
2
?
c†
jcj+1+ h.c.
?
− h
?
c†
jcj−1
2
?
?
nj−1
2
??
nj+1−1
2
??
.(2.7)
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Dynamical structure factor at small q for the XXZ spin-1/2 chain5
2.1. Exact solution for the XX model
One case of special interest is the XX point ∆ = 0, at which (2.7) reduces to a free
fermion model [38]. As the free fermion point will serve as a guide for the resummation of
the bosonic theory, we reproduce the solution in detail here. For ∆ = 0 the Hamiltonian
(2.7) can be easily diagonalized by introducing the operators in momentum space
cp=
1
√N
N
?
j=1
e−ipjcj. (2.8)
with p = 2πn/N, n∈ Z, for periodic boundary conditions. The free fermion Hamiltonian
is then
?
where ǫp = −J (cosp + h) is the fermion dispersion. In the fermionic language, the
dynamical structure factor reads
Szz(q,ω) =1
N
−∞
=2π
N
α
where nq=?
Fermi momentum kF. The latter is determined by the condition ǫkF= 0, which gives
?1
where σ ≡
excited states in terms of particle-hole excitations created on the Fermi sea. The only
nonvanishing form factors appearing in Szz(q,ω) are those for excited states with only
one particle-hole pair carrying total momentum q: |α? = c†
are simply
?0??Sz
choices for the hole momentum p below the Fermi surface. In the limit N → ∞, (2.10)
reduces to the integral
?π
=θ(ω − ωL(q))θ(ωU(q) − ω)
(dωpq/dp)|ωpq=ω
where ωpq= ǫp+q−ǫpis the energy of the particle-hole pair and ωL(q) and ωU(q) are the
lower and upper thresholds of the two-particle spectrum, respectively. For the cosine
dispersion, we have
?
H0=
p
ǫpc†
pcp,(2.9)
?+∞
?
pc†
dteiωt?nq(t)n−q(0)?
|?0|nq|α?|2δ (ω − Eα+ EGS), (2.10)
je−iqjnj=?
pcp+q.
We construct the ground state |0? by filling all the single-particle states up to the
kF= arccos(−h) = π
?Sz
2+ σ
?
, (2.11)
j
?
= ?nj? −1
2is the magnetization per site. We can also describe the
p+qcp|0?. The form factors
q
??α?= θ(kF− |p|)θ(|p + q| − kF).(2.12)
For a finite system there are qN/2π states with form factor 1, corresponding to different
Szz(q,ω) =
−π
dpθ(kF− |p|)θ(|p + q| − kF)δ (ω − ǫp+q+ ǫp)
, (2.13)
ωpq= 2J sinp +q
2
?
sinq
2. (2.14)
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Dynamical structure factor at small q for the XXZ spin-1/2 chain6
0.0560.0580.06
ω/J
0.0620.064
0
10
20
30
40
50
60
J Szz(q = π/50, ω)
σ = - 0.1
∆ = 0
ωU(q)
ωL(q)
∆Szz
Figure 1. Exact dynamical structure factor Szz(q,ω) for the free fermion point ∆ = 0.
For this graph we set σ = −0.1 (kF = 2π/5) and q = π/50.
The expressions for the lower and upper thresholds depend on the proximity to half-
filling (zero magnetic field). Here we shall restrict ourselves to finite field and small
momentum |q| ≪ kF. More precisely, we impose the condition
|q| < |2kF− π| = 2π|σ|.
For kF< π/2 (σ < 0), we have
(2.15)
ωL(q) = 2J sin|q|
2sin
?
?
kF−|q|
kF+|q|
2
?
?
, (2.16)
ωU(q) = 2J sin|q|
2sin2
. (2.17)
If kF > π/2, the above expressions for ωL(q) and ωU(q) are exchanged. Hereafter we
take kF< π/2 and q > 0. It follows from (2.16) and (2.17) that Szz(q,ω) for fixed q is
finite within an energy interval of width
δωq= ωU(q) − ωL(q) = 4J coskFsin2?q
for small q. In fact, we can calculate Szz(q,ω) explicitly using (2.13). The result is
2
?
≈ (J coskF)q2
(2.18)
Szz(q,ω) =θ(ω − ωL(q))θ(ωU(q) − ω)
??2J sinq
which is illustrated in figure 1. Note that, although the form factors are constant,
Szz(q,ω) is peaked at the upper threshold because of the larger density of states. The
values of Szz(q,ω) at the lower and upper thresholds are both finite
2J sinq
2
?2− ω2
,(2.19)
Szz(q,ω → ωL,U(q)) =
?
2cos
?
kF∓q
2
??−1
.(2.20)
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Dynamical structure factor at small q for the XXZ spin-1/2 chain7
In the small-q limit, only excitations created around the Fermi surface contribute
to Szz(q,ω). For this reason, a simplifying approach would be to expand the fermion
dispersion around the Fermi points
≈ ±vFk +k2
6
where k ≡ p ∓ kF for right (R) or left (L) movers, vF= J sinkF is the Fermi velocity,
m = (J coskF)−1is the effective mass at the Fermi level and γ = J sinkF. The free
fermion Hamiltonian is then approximated by
ǫR,L
k
2m∓γk3
+ ..., (2.21)
H0=
∞
?
k=−∞
?
ǫR
k: c†
kRckR: +ǫL
k: c†
kLckL:
?
, (2.22)
where ckR,Lare the annhilation operators for fermions with momentum around ±kF,
respectively, and : : denotes normal ordering with respect to the ground state. If we
retain only the linear term in the expansion, ωkqturns out to be independent of k. This
means that all particle-hole excitations are degenerate, and Szz(q,ω) is given by a single
delta function peak at the corresponding energy ω = vFq
Szz(q,ω) = qδ(ω − vFq). (2.23)
This is a direct consequence of the Lorentz invariance of the model with linear dispersion.
In order to get the broadening of Szz(q,ω), we must account for the nonlinearity of the
dispersion, i.e., band curvature at the Fermi level. If we keep the next (quadratic) term
in ǫR,L
k
, we find
Szz(q,ω) =m
qθ
?q2
2m− |ω − vFq|
?
.(2.24)
We note that this flat distribution of spectral weight is a good approximation to the
result in (2.18) and (2.19) in the limit q ≪ cotkF, in the sense that the difference
between the values of Szz(q,ω) at the lower and upper thresholds is small compared
to the average height of the peak (see figure 1). This difference stems from the energy
dependence of the density of states factor 1/(dωpq/dp)|ωpq=ω, which is recovered if we
keep the k3term in the dispersion. It is easy to verify that for q ≪ cotkF (γmq ≪ 1)
∆Szz≡ Szz(q,ωU(q)) − Szz(q,ωL(q)) ≈ γm2.
∆Szzis q-independent, therefore ∆Szz/(m/q) ∼ q vanishes as q → 0. This means that
if we compare Szz(q,ω) for different values of q – taking into account that δωq∼ q2and
Szz(q,ω) ∼ 1/q inside the peak and rescaling the functions accordingly – the rescaled
function becomes flatter as q → 0. On the other hand, the slope ∂Szz/∂ω near the
center of the peak diverges as q → 0.
The thresholds for the two-particle continuum,
ωU,L(q) ≈ vFq ±q2
are easy to interpret. For kF< π/2, the lower threshold corresponds to creating a hole
at the state with momentum q below kF (a “deep hole”) and placing the particle right
(2.25)
2m,
(2.26)
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Dynamical structure factor at small q for the XXZ spin-1/2 chain8
above the Fermi surface, whereas the upper one corresponds to the excitation composed
of a “high-energy particle” at kF+ q and a hole right at the Fermi surface [28].
Alternatively, we could have calculated the density-density correlation function,
which for ∆ = 0 is given by the fermionic bubble
χ(q,iω) =
?
dk
2π
θ(−k)θ(k + q)
iω − ǫk+q+ ǫk
− (ω → −ω). (2.27)
Using the quadratic dispersion ǫk≈ vFk + k2/2m, we find
m
2πqlog
χ(q,iω) =
?iω − vFq + q2/2m
iω − vFq − q2/2m
?
− (ω → −ω). (2.28)
The result (2.24) is then obtained by taking the imaginary part of χret(q,ω) according
to (2.4).
3. Low energy effective Hamiltonian
3.1. The free boson Hamiltonian
For a general anisotropy ∆ ?= 0, the Hamiltonian (2.7) describes interacting spinless
fermions. The standard approach to study the low-energy (long-wavelength) limit of
correlation functions of interacting one-dimensional systems is to use bosonization to
map the problem to a free boson model – the Luttinger model [1]. This approach has the
advantage of treating interactions exactly. As a first step, one introduces the fermionic
field operators ψR,L(x)
cj→ ψ (x) = eikFxψR(x) + e−ikFxψL(x), (3.1)
ψR,L(x) =
1
√L
+Λ
?
k=−Λ
ckR,Le±ikx, (3.2)
where L = N is the system size (we set the lattice spacing to 1) and Λ < π is a
momentum cutoff. In the continuum limit, the kinetic energy part of the Hamiltonian
in (2.22) can be written as
H0=
?L
0
dx
?
?
: ψ†
R
?
vF(−i∂x) +(−i∂x)2
2m
+ ...
?
?
ψR:
+ : ψ†
L
vF(−i∂x) +(−i∂x)2
2m
+ ...
?
ψL:. (3.3)
The 1/m term is usually dropped using the argument that it has a higher dimension
and is irrelevant in the sense of the renormalization group. However, it introduces
corrections to the Luttinger liquid fixed point which are associated with band curvature
effects. Similarly, if we write the interaction term in (2.7) in the continuum limit, we
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Dynamical structure factor at small q for the XXZ spin-1/2 chain9
get (following [2])
Hint= ∆J
?L
?L
0
dx : ψ†(x)ψ (x) :: ψ†(x + 1)ψ (x + 1) :
= ∆J
0
dx {ρR(x)ρR(x + 1) + ρL(x)ρL(x + 1)
+ ρR(x)ρL(x + 1) + ρL(x)ρR(x + 1)
?
+
?
+ei2kFψ†
R(x)ψL(x)ψ†
L(x + 1)ψR(x + 1) + h.c.
?
e−i2kF(2x+1)ψ†
R(x)ψL(x)ψ†
R(x + 1)ψL(x + 1) + h.c.
??
(3.4)
where ρR,L ≡: ψ†
oscillating except at half-filling (where 4kF = 2π). We will neglect that term for the
finite field case, but will restore it in section 7 when we discuss the zero field case.
We now use Abelian bosonization and write the fermion fields as
1
√2παe−i√2πφR,L(x),
where α ∼ k−1
a bosonic field˜φ and its dual field˜θ
˜φ =φL− φR
√2
˜θ =φL+ φR
√2
which satisfy [˜φ(x),∂x′˜θ(x′)] = iδ(x − x′).
fermions can be shown to be related to the derivative of the bosonic fields
1
√2π∂xφR,L,
so that
n(x) ∼1
Here we are interested in the uniform (small q) part of the fluctuation of Sz
which is proportional to the derivative of the bosonic field˜φ. Bosonizing the linear term
in the kinetic energy (3.3), we find
R,LψR,L:. The last term corresponds to Umklapp scattering and is
ψR,L(x) ∼
(3.5)
F
is a short-distance cutoff and φR,Lare the right and left components of
,(3.6)
,(3.7)
The density of right- and left-moving
ρR,L∼ ∓
(3.8)
2+ σ +
1
√π∂x˜φ +
1
2παcos
?√4π˜φ − 2kFx
?
.(3.9)
j∼ n(x),
Hlin
0
=
?L
0
dxivF
?
: ψ†
R∂xψR: − : ψ†
L∂xψL:
?
=vF
2
?L
0
dx?(∂xφR)2+ (∂xφL)2?.(3.10)
The terms that appear in the interaction part are
ρR,L(x)ρR,L(x + 1) =
1
2π(∂xφR,L)2,
1
2π∂xφR∂xφL,ρR(x)ρL(x + 1)= −
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Dynamical structure factor at small q for the XXZ spin-1/2 chain 10
ψ†
−cos(2kF)
R(x)ψL(x)ψ†
L(x + 1)ψR(x + 1) =
(∂xφR− ∂xφL)2+sin(2kF)
2π
3√2π
(∂xφR− ∂xφL)3+ ..., (3.11)
where we have set α = 1 (equal to the level spacing; see [2]). If we keep only the
marginal operators (quadratic in ∂xφR,L), we get an exactly solvable model
HLL=vF
2
−g2
πvF
?
dx
??
1 +
g4
2πvF
?
??(∂xφR)2+ (∂xφL)2?
∂xφL∂xφR
, (3.12)
where g2 = g4 = 2J∆[1 − cos(2kF)] = 4J∆sin2kF. The Hamiltonian (3.12) can be
rewritten in the form
HLL=1
2
?
dx
?
vK
?
∂x˜θ
?2
+v
K
?
∂x˜φ
?2?
, (3.13)
where v (the renormalized velocity) and K (the Luttinger parameter) are given by
v = vF
??
1 +
1 +
g4
2πvF
?2
g2
2πvF
g2
2πvF
−
?
g2
2πvF
?2
≈ vF
?
1 +2∆
π
sinkF
?
, (3.14)
K =
?
g4
2πvF−
g4
2πvF+
1 +
≈ 1 −2∆
π
sinkF. (3.15)
Expressions (3.14) and (3.15) are approximations valid in the limit ∆ ≪ 1.
Luttinger model describes free bosons that propagate with velocity v and is the correct
low energy fixed point for the XXZ chain for any value of ∆ and h in the gapless regime.
However, the correct values of v and K for finite ∆ must be obtained by comparison
with the exact Bethe Ansatz (BA) solution. In the case h = 0, the BA equations can
be solved analytically and yield
√1 − ∆2
arccos∆,
K (∆,h = 0) =
2(π − arccos∆).
There are also analytical expressions for h ≈ 0 and h close to the critical field [39]. For
arbitrary fields, one has to solve the BA equations numerically in order to get the exact
v and K.
The Luttinger parameter in the Hamiltonian (3.13) can be absorbed by performing a
canonical transformation that rescales the fields in the form˜φ →√Kφ and˜θ → θ/√K.
HLLthen reads
HLL=v
2
We can also define the right and left components of these rescaled bosonic fields by
ϕR,L=θ ∓ φ
√2
The
v(∆,h = 0) =Jπ
2
(3.16)
π
(3.17)
?
dx?(∂xθ)2+ (∂xφ)2?. (3.18)
. (3.19)
Page 11
Dynamical structure factor at small q for the XXZ spin-1/2 chain11
These are related to φR,Lby a Bogoliubov transformation. An explicit mode expansion
(neglecting zero mode operators) is
1
√qL
ϕR,L(x,τ) =
?
q>0
?aR,L
q
e−q(vτ∓ix)+ aR,L†
q
eq(vτ∓ix)?,
] = δqq′ and q = 2πn/L, n =
(3.20)
where aR,L
1,2,..., for periodic boundary conditions. The Hamiltonian (3.18) is then diagonal in
the boson operators
q
are bosonic operators obeying [aR,L
q
,aR,L†
q′
HLL=
?
q>0
vq?aR†
qaR
q + aL†
qaL
q
?.(3.21)
We can calculate the propagators for the free fields ∂xϕR,Lfrom the mode expansion
in (3.20). In real space, for L → ∞ and zero temperature (β → ∞), the propagators
read
D(0)
R,L(x,τ) = ?∂xϕR,L(x,τ)∂xϕR,L(0,0)?0=
1
2π
1
(vτ ∓ ix)2. (3.22)
In momentum space,
D(0)
R,L(q,iωn) ≡ −
?L
±q
0
dxe−iqx
?β
0
dτ eiωnτD(0)
R,L(x,τ)
=
iωn∓ vq.
(3.23)
In order to calculate the dynamical structure factor defined in (2.2), we express the
fluctuation of the spin operator in terms of the bosonic field φ. From (2.6) and (3.9),
we have
?
In the continuum limit,
?L
=K
2πD(0)(q,iωn),
where D(0)(q,iωn) is the free boson propagator (for the ∂xφ field)
Sz
j∼
K
π∂xφ. (3.24)
χ(q,iωn) = −K
π
0
dxe−iqx
?β
0
dτ eiωnτ?∂xφ(x,τ)∂xφ(0,0)?0
(3.25)
D(0)(q,iω) ≡ D(0)
R(q,iω) + D(0)
L(q,iω) =
2vq2
(iω)2− (vq)2. (3.26)
It follows that the retarded correlation function is
χret(q,ω) =Kq
2π
?
1
ω − vq + iη−
1
ω + vq + iη
?
. (3.27)
Finally, using (2.4), the dynamical structure factor for the free boson model is (q > 0)
Szz(q,ω) = Kqδ (ω − vq). (3.28)
The result in (3.28) is analogous to (2.23). Since the Luttinger model exhibits Lorentz
invariance, Szz(q,ω) is a delta function peak at the energy carried by the single
Page 12
Dynamical structure factor at small q for the XXZ spin-1/2 chain12
boson with momentum q. This solution should be asymptotically exact in the limit
q → 0, which means that any corrections to it must be suppressed by higher powers
of momentum. However, the free boson result misses many of the features that the
complete solution must have. For example, the exact solution for the XX point suggests
a broadening of the delta peak with a width δωq∼ q2. Like in that case, it is necessary
to incorporate information about band curvature at the Fermi level by keeping the
quadratic term in the fermion dispersion in order to get a finite width for Szz(q,ω).
As we shall discuss in the next section, the problem is that such a term is mapped
via bosonization onto a boson-boson interaction term. Even though the interaction
term is irrelevant, finite-order perturbation theory in these operators leads to a singular
frequency dependence close to ω = vq. It turns out that broadening the delta function
peak within a field theory approach is a not an easy task. A complete solution that
recovers the scaling δωq ∼ q2requires summing an infinite series of diagrams, as we
will point out in section 5. Another feature expected for Szz(q,ω) when ∆ ?= 0 is a
high-frequency tail associated with multiple particle-hole excitations. This tail can be
calculated in the region δωq ≪ ω − vq ≪ J by lowest-order perturbation theory in
the fermionic interaction (∝ ∆) starting from a model of free fermions with quadratic
dispersion [27]. In section 6 we obtain this result by including fermionic interactions
exactly (finite ∆) and doing perturbation theory in the band curvature terms.
3.2. Irrelevant operators
In order to go beyond the Luttinger model, we need to treat the irrelevant operators
that break Lorenz invariance. There are two sources of such terms: band curvature
terms, which are quadratic in fermions but involve higher derivatives, and irrelevant
interaction terms [33]. The first type appeared in (3.3) and corresponds to the k2term
in the expansion of the fermion dispersion
1
2m
We derive the bosonized version of a general band curvature term in the following way
(see [40]). We define the operator
?
=
?
=
?
where ... is a total derivative. Organizing by powers of ǫ, we can write
∞
?
where
F(n)(x) = ψ†
δHbc= −
?
: ψ†
R∂2
xψR: + : ψ†
L∂2
xψL:
?
.(3.29)
F (x,ǫ) = ψ†
R
x +ǫ
2
?
?k
?n
ψR
?
x −ǫ
∞
?
2
?
1
l!
∞
k=0
∞
1
k!
?ǫ
2
∂k
xψ†
R
l=0
?
−ǫ
2
?l
1
∂l
xψR
n=0
?
−ǫ
2
ψ†
R∂n
xψR
n
?
k=0
k!(n − k)!+ ...,(3.30)
F (x,ǫ) =
n=0
(−1)n
n!
ǫnF(n)(x), (3.31)
R∂n
xψR. (3.32)
Page 13
Dynamical structure factor at small q for the XXZ spin-1/2 chain13
According to (3.5), we have
ψR∼
1
√2παe−i√2πφR∼
1
√Le−i√2πφ+
Re−i√2πφ−
R, (3.33)
where φ±
have used the identity eA+B= eAeBe−[A,B]/2with
Rare the creation and annihilation parts of φR(x) = φ+
R(x) + φ−
R(x) and we
?φ−
R(x),φ+
R(y)?≈ −1
2πlog
?
−2πi
L
(x − y + iα)
?
, (3.34)
for large L. Then we express F (x,ǫ) in terms of the bosonic fields
F (x,ǫ) =1
Lei√2πφ+
R(x+ǫ/2)ei√2πφ−
R(x+ǫ/2)e−i√2πφ+
R(x−ǫ/2)e−i√2πφ−
R(x−ǫ/2).(3.35)
After normal ordering the operators, we can do the expansion in ǫ (dropping the normal
ordering sign)
x +ǫ
22
1
2πiǫexp
∞
?
From (3.31) and the coefficient of the ǫnterm in (3.36), we have
?2√2πi?P
ψ†
R
??
ψR
?
x −ǫ
?
?
?
= −
?
i√2πφR
?
ℓ!
x +ǫ
2
?
− φR
?P′
?
x −ǫ
2
???
?∂j
= −
ℓ=0
?2√2πi?ℓ
2πiǫℓ!
{mj}
?
jmj!
?ǫ
2
jjmj
?
j=1,3,···
xφR
j!
?mj
. (3.36)
F(n)(x) =(−1)n+1n!
2n+12πi
?
{mj}
jmj
?
j(mj!)
?
j=1,3,···
?∂j
xφR
j!
?mj
,(3.37)
where the mj’s obey the constraint?
jjmj= n + 1. In particular, for n = 2 the sum
in (3.37) contains only two terms (either m1= 3, m3= 0 or m1= 0, m3= 1). We get
√2π
3
F(2)(x) = ψ†
R∂2
xψR=
(∂xφR)3−
1
12√2π∂3
xφR. (3.38)
The last term is a total derivative and can be omitted from the Hamiltonian. Similar
expressions for the left-moving field φL are obtained straightforwardly by using the
symmetry under the parity transformation x → −x, R → L. The bosonized version of
the band curvature terms in (3.29) is then
√2π
6m
We now rewrite δHbcin terms of the right and left components of the rescaled field
φ. Using (3.6) and (3.7),
√2π
6m
=
6m
0
δHbc=
?(∂xφL)3− (∂xφR)3?.(3.39)
δHbc=
?
∂x˜θ + ∂x˜φ
√2
?3
−
?
∂x˜θ − ∂x˜φ
√2
?3
?π/K
?L
dx?3(∂xθ)2∂xφ + K2(∂xφ)3?. (3.40)
Page 14
Dynamical structure factor at small q for the XXZ spin-1/2 chain14
Finally, using (3.19), we get (in accordance with [35])
δHbc=
?2π/K
6
+
3 + K2
4m
3(1 − K2)
4m
?(∂xϕL)3− (∂xϕR)3?
?(∂xϕL)2∂xϕR− (∂xϕR)2∂xϕL
?2π/K
6
?. (3.41)
Besides δHbc, we need to include the irrelevant operators which arise from the
expansion of the fermionic interaction in the lattice spacing, as we encountered in (3.11).
In terms of ϕR,L, this contribution reads
δHint=J∆K3/2
3√2π
−3?(∂xϕL)2∂xϕR− (∂xϕR)2∂xϕL
sin(2kF)??(∂xϕL)3− (∂xϕR)3?
??. (3.42)
Combining (3.41) and (3.42), we can write the irrelevant operators in the most general
form
√2π
6
+η+
?(∂xϕL)2∂xϕR− (∂xϕR)2∂xϕL
To first order in ∆, the coupling constants η±are given by
η−≈1
mπ
η+≈ −3∆
The perturbation δH in (3.43) might as well have been introduced phenomenolog-
ically in the effective Hamiltonian. In fact, the dimension-three operators (∂xϕR,L)3are
the leading irrelevant operators that are allowed by symmetry. They obey the parity
symmetry ϕL→ ϕR, x → −x, but not spin reversal (or particle-hole) ϕR,L→ −ϕR,L,
which is absent for h ?= 0. Such terms give rise to three-legged interaction vertices
which scale with powers of the momenta of the scattered bosons (figure 2). They are
responsible, for example, for corrections to the long distance asymptotics of the cor-
relation functions [40]. Note that as ∆ → 0 (K → 1), η−→ 1/m while η+vanishes
because there is no mixing between right and left movers at the free fermion point.
Moreover, the weak coupling expressions predict that both η− and η+ vanish in the
limit h → 0 (m → ∞), in which particle-hole symmetry is recovered. (See, however,
figure 15 below.) For h = 0 the leading irrelevant operators are the dimension-four
operators (∂xϕR,L)4,(∂xϕR)2(∂xϕL)2and the umklapp interaction cos(4√πKφ), which
becomes nonoscillating [32].
The condition that a general model of the form HLL+ δH be unitarily equivalent
to free fermions up to dimension-four operators [33] amounts to imposing that the
Bogoliubov transformation that diagonalizes HLLin the R/L basis also diagonalizes the
cubic operators in δH. In our notation, this condition is expressed as η+= 0. That
condition is not satisfied by the XXZ model except for the trivial case ∆ = 0. However,
δH =
?
dx?η−
?(∂xϕL)3− (∂xϕR)3?
??. (3.43)
?
πmsinkF.
1 +2∆
sinkF
?
,(3.44)
(3.45)
Page 15
Dynamical structure factor at small q for the XXZ spin-1/2 chain 15
η+
η−
R
R
R
R
R
L
Figure 2. Interaction vertices in the low energy effective Hamiltonian. The solid
(dashed) lines represent propagators for right- (left-) moving bosons D(0)
R(D(0)
L).
the contributions from this extra (i.e., not present for free fermions) dimension-three
operator to Szz(q,ω) are of O(η2
Similarly to what happens for v and K, (3.44) and (3.45) should be regarded as
weak-coupling expressions. Again we can use the fact that the XXZ model is integrable
and obtain the exact (renormalized) values of η± by comparison with Bethe Ansatz.
In section 3.3 we will discuss how to fix these coupling constants in order to obtain a
parameter-free theory.
+), as we will discuss in section 6.
3.3. Determination of the renormalized coupling constants
As mentioned in section 3.2, the renormalized parameters η± can be determined by
comparison with exact Bethe Ansatz results for infinite length. We will proceed by
analogy with the calculation for the zero-field case in [41]. One difficulty is that there
are no analytical solutions of the Bethe Ansatz equations for finite fields, so we must be
satisfied with a numerical evaluation of the parameters. In the following, we will relate
η±to the coefficients of the expansion of v and K as functions of the magnetic field, by
comparing the corrections to the free boson result for the free energy calculated in two
different ways.
Let us consider the response to a small variation in the magnetic field around a
finite value h0. In the limit δh = h − h0≪ 1, such response is well described by the
Luttinger model
H =
?
dx
?
v
2
?(∂xθ)2+ (∂xφ)2?− Jδh
?
K
π∂xφ
?
, (3.46)
where v(h) and K (h) are known exactly from the Bethe Ansatz equations. For h0= 0,
the cutoff-independent terms of the free energy density according to field theory read
f (h0= 0) ∼ −πT2
6v
where v and K are given by (3.16) and (3.17), respectively. The magnetic susceptibility
at zero temperature is χ = −J−2(∂2f/∂h2)|T=0= K/πv, which is the familiar free
boson result. For finite field h0?= 0, the free energy assumes some general form
f (h0?= 0) ∼ −πT2
−
K
2πv(Jδh)2, (3.47)
6v(h)− C (h), (3.48)
Page 16
Dynamical structure factor at small q for the XXZ spin-1/2 chain16
and the T = 0 susceptibility is obtained by
χ = −1
J2
?∂2f
∂h2
?????
h,T=0
= −1
J2
?
∂2C
∂ (δh)2
?????
h,T=0
=K(h)
πv(h), (3.49)
where the last identity holds for any Luttinger liquid.
We would like to calculate the corrections to f and χ that involve higher powers
of the perturbation δh. Our first approach is to assume that the field dependence is
already completely contained in the definitions of v(h) and K(h), so that we can employ
the expansion
v(h) = v(h0)?1 + aδh + O?δh2??,
K(h) = K(h0)?1 + bδh + O?δh2??,
where the coefficients a and b can be extracted from the exact v and K by linearizing
the field dependence around h = h0. Consequently, the lowest-order correction to the
free boson susceptibility around h = h0is
χ =K(h0)
πv(h0)
Likewise, the free energy at finite temperature must contain a term of the form
δf ∼ aπ δhT2
due to the field dependence of the velocity. Both a and b depend on h0and the anisotropy
∆.As an example, at the XX point, K = 1 for any value of the field, therefore
b(∆ = 0,h0) = 0. From (2.11), we have
vF= J sinkF= J√1 − h2≈ vF(h0) −
(3.50)
(3.51)
?1 − (a − b)δh + O?δh2??. (3.52)
6v(h0), (3.53)
J2h0
vF(h0)δh + O?δh2?, (3.54)
so that we get
a(∆ = 0,h0) = −J2h0
v2
F(h0)=coskF
sin2kF
.(3.55)
In our second approach, we take v = v(h0) and K = K(h0) to be fixed and assume
that the corrections to the free boson result are generated by the irrelevant operators.
We consider the effective Hamiltonian H = HLL+ δH, with δH defined in (3.43). An
equivalent Lagrangian formulation in imaginary time is
L = L0+ δL, (3.56)
L0=(∂τφ)2
δL = −A√π
2v
+v
2(∂xφ)2− Jδh
6v2(∂τφ)2∂xφ +B√π
?
K
π∂xφ,(3.57)
6
(∂xφ)3+ O?η2
±
?, (3.58)
Page 17
Dynamical structure factor at small q for the XXZ spin-1/2 chain17
where A = 3η−+η+and B = η−−η+. We shift the field by φ → φ+Jδh
the term linear in ∂xφ and get
L0=(∂τφ)2
2v
δL = −A√KJδh
6v3
+BK3/2(Jδh)3
6πv3
We then calculate the free energy density from the partition function
?β
f = −T
L
v
?
K
πx to absorb
+v
2(∂xφ)2+K (Jδh)2
(∂τφ)2+B√KJδh
2πv
, (3.59)
2v
(∂xφ)2
+ odd powers of φ. (3.60)
Z =
?
Dφ exp
?
−
0
dτ
?L
0
dx (L0+ δL)
?
,(3.61)
LlnZ ≈ f0+T
?β
0
dτ
?L
0
dx?δL?, (3.62)
where f0reproduces the free boson result
f0∼ −πT2
and ?δL? is the expectation value of δL calculated with the unperturbed Hamiltonian.
In order to compute ?δL?, we need the finite temperature propagators
?∂xφ(x + ǫ)∂xφ(x)? = −1
1
2π
6v
−
K
2πv(Jδh)2, (3.63)
v2?∂τφ(x + ǫ)∂τφ(x)?
(πT/v)2
sinh2(πTǫ/v).
= −
(3.64)
Now we use the expansion sinh−2(πTǫ/v) ≈ (v/πTǫ)2− 1/3 for ǫ → 0 and drop the
cutoff-dependent terms in δf. The reason is that the latter simply renormalize the
corresponding terms in f0and have already been accounted for in the renormalization
of v and K. The correction to the free energy to first order in A and B becomes
?β
∼ (A + 3B)π√KJδhT2
36v
The susceptibility obtained from f0+ δf is
∂2(f0+ δf)
∂ (δh)2
T=0
Comparing with the expression (3.52), we can identify
√KJ
v2
v2
Besides, from the δhT2term in (3.53) and (3.65), we have
√KJ
6v2(A + 3B) =
δf =T
L
0
dτ
?L
0
dx?δL?
+ BK3/2(Jδh)3
6πv3
.(3.65)
χ = −1
J2
????
=K
πv− BK3/2Jδh
πv3
+ O?δh2?.(3.66)
a − b =B =
√KJ
(η−− η+).(3.67)
a =
√KJ
3v2(3η−− η+). (3.68)
Page 18
Dynamical structure factor at small q for the XXZ spin-1/2 chain18
Finally, combining (3.67) and (3.68) and writing a = v−1∂v/∂h and b = K−1∂K/∂h,
we find the formulas first presented in [32]
Jη−=
v
K1/2
3v2
2K3/2
∂v
∂h+
∂K
∂h.
v2
2K3/2
∂K
∂h,
(3.69)
Jη+=
(3.70)
The above relations allow us to calculate the renormalized values of η±once we have
the field dependence of v and K. Notice that η+∝ ∂K/∂h and as expected vanishes
at the XX point. On the other hand, η−remains finite at ∆ = 0 because ∂v/∂h ?= 0
and we recover η−= (vF/J)∂vF/∂h = J coskF= m−1. It is also possible to check the
validity of (3.69) and (3.70) explicitly in the weak coupling limit, using the expressions
for v (∆ ≪ 1,h) and K (∆ ≪ 1,h) in (3.14) and (3.15) as well as the weak coupling
expressions for η±in (3.44) and (3.45).
4. Bethe Ansatz solution
Although the Bethe Ansatz is first and foremost a method for calculating the energy
levels of an exactly solvable model (readers who are unfamiliar with the subject
are invited to consult standard textbooks, for example [42, 39, 43]), recent progress
stemming from the Algebraic Bethe Ansatz means that we can now use it to make
many nontrivial statements about dynamical quantities. Assuming that certain specific
families of excited states carry the dominant part of the structure factor, we can delimit
the energy and momentum continua where we expect most of the correlation weight
to be found, and provide the specific lineshape of the structure factor both within this
interval, and further up within the higher-energy tail. We start here by introducing
the important aspects of the Bethe Ansatz which we will make use of later on when
studying the correspondence with field theory results.
4.1. Bethe Ansatz setup and fundamental equations
As is well-known, an eigenbasis for the XXZ chain (2.1) on N sites is obtained from
the Bethe Ansatz [3, 4],
ΨM(j1,...,jM) =
?
P
(−1)[P]eiPM
a=1kPaja−i
2
P
1≤a<b≤Mφ(kPa,kPb). (4.1)
Here, M ≤ N/2 represents the number of overturned spins, starting from the reference
state |0? = ⊗N
The total magnetization of the system along the ˆ z axis, Sz
by the Hamiltonian. P represents a permutation of the integers {1,...,M} and jiare
the lattice coordinates. The quasi-momenta k are parametrized in terms of rapidities λ,
i=1| ↑?i(i.e. the state with all spins pointing upwards in the ˆ z direction).
tot= Nσ =N
2−M is conserved
eik=sinh(λ + iζ/2)
sinh(λ − iζ/2), ∆ = cosζ,(4.2)
Page 19
Dynamical structure factor at small q for the XXZ spin-1/2 chain19
such that the two-particle scattering phase shift becomes a function of the rapidity
difference only, φ(ka,kb) = φ1(λa−λb) with φ1defined below. An individual eigenstate is
thus fully characterized by a set of rapidities {λ}, satisfying the quantization conditions
(Bethe equations) obtained by requiring periodicity of the Bethe wavefunction (4.1):
φ1(λj) −1
N
M
?
k=1
φ2(λj− λk) = 2πIj
N,
j = 1,...,M,(4.3)
in which Ijare half-odd integers for N −M even and integers for N−M odd, and where
we have defined the functions
?
The energy and momentum of an eigenstate are simple functions of its rapidities,
?
P = πM −
j
in which
1
2ππ
Each solution of the set of coupled nonlinear equations (4.3) for sets of non-
coincident rapidities represents an eigenstate (if two rapidities coincide, the Bethe
wavefunction (4.1) formally vanishes). The space of solutions is not restricted to
real rapidities:it has been known since Bethe’s original paper that there exist
solutions having complex rapidities (’string’ states), representing bound states of
magnons. In fact, obtaining all wavefunctions from solutions to the Bethe equations (or
degenerations thereof) remains to this day an open problem in the theory of integrable
models. It is however possible to construct the vast majority of eigenstates using
this procedure, allowing to obtain reliable results for thermodynamic quantities and
correlation functions. In all our considerations in the present paper, we can and will
restrict ourselves to real solutions to the Bethe equations.
φn(λ) = 2arctan
tanh(λ)
tan(nζ/2)
?
. (4.4)
E = −πJ sinζ
j
a1(λj) − hSz
tot,
?
φ1(λj) = πM −2π
N
?
j
Ij
(4.5)
an(λ) =
d
dλφn(λ) =1
sin(nζ)
cosh(2λ) − cos(nζ).(4.6)
4.2. Ground state and excitations
The simplest state to construct is the ground state, which is obtained by setting the
quantum numbers Ijto (we consider N even from now on for simplicity)
= −M + 1
2
The simplest excited states which can be constructed at finite magnetic field are
obtained by introducing particle-hole excitations on the ground-state quantum number
distributions, see figure (3). Since we limit ourselves to real solutions to the Bethe
equations, we require |λj| < ∞ and thus |I| < I∞, where, from (4.3),
I∞=N − M
2
IGS
j
+ j,j = 1,...,M. (4.7)
− (N
2− M)ζ
π.
(4.8)
Page 20
Dynamical structure factor at small q for the XXZ spin-1/2 chain20
Figure 3. Representation of various quantum number configurations: a black (empty)
circle represents an occupied (unoccupied) allowable quantum number (which here are
taken to be half-odd integers; the figure is centered on zero). The top set represents
the ground state configuration, whereas the second and third from top represent two-
particle excitations at different momenta, obtained by creating a particle-hole pair on
the ground-state configuration. The bottom set is for a four-particle state obtained
from two particle-hole pairs. The dotted line delimits the ground state interval, whereas
the solid lines delimit the quantum numbers for which real solutions to the Bethe
equations can be obtained in this illustrative case (see main text).
The momentum of an excited state is simply given by the left-displacement of
the quantum numbers with respect to those in the ground state, q =
δlI =?
difference fixed. Since the energy of these two-particle states at fixed momentum are non-
degenerate, this defines a two-particle continuum whose characteristics will be studied
later. Higher-particle states can be similarly constructed and counted.
The restriction to real rapidities and a single particle-hole pair therefore means
that our subsequent arguments will apply only to the region q < Min(2kF,k∞) where
kF = πM
number, k∞= 2πI∞−M/2
N
q < Min{π(1 − 2σ),2σ(π − ζ)}
in terms of the magnetization, noting in particular that the window of validity of our
arguments vanishes in the case of zero magnetic field.
For a finite chain with N sites and M overturned spins, the Hilbert space is finite,
and therefore so is the sum over intermediate states in the Lehmann representation for
the structure factor (2.3). Each intermediate state is obtained by solving the Bethe
equations, the space of states being reconstructed by spanning through the sets of
allowable quantum numbers. The form factor of a local spin operator between the
ground state and a particular excited state is obtained from the Algebraic Bethe Ansatz
as a determinant of a matrix depending only on the rapidities of the eigenstates involved
[19, 20] even in the case of string states with complex rapidities [21]. This enables to
obtain extremely accurate results on the full dynamical spin-spin correlation functions
2π
NδlI, where
j(IGS
j
−Ij). At a given fixed (small) momentum, we can thus construct qN/2π
two-particle states by shifting the particle and hole quantum numbers, leaving their
Nand k∞ is given by the maximal displacement of the outermost quantum
. We can thus write our restriction as
(4.9)
Page 21
Dynamical structure factor at small q for the XXZ spin-1/2 chain21
in integrable Heisenberg chains [25, 21]. We will make use of this method in what follows
to compare results from the Bethe Ansatz to field theory predictions for the structure
factor at small momentum.
5. Width of the on-shell peak
Linearizing the dispersion around the Fermi points is a key step for the bosonization
technique. By doing so all the particle-hole excitations with same momentum q ≪ kF
become exactly degenerate and one can associate a particular linear combination with a
single-boson state [40]. In this approximation, the single boson state |b? ≡ aR†
only state that couples to the ground state via Sz
is given by
??b???2=KqN
whose energies are given by (4.5), are nondegenerate. In fact, most of the above spectral
weight is shared by qN/2π two-particle states whose energies are spread around ω = vq.
This is reminiscent of the exact solution for the free fermion point in section 2.1. In
the bosonic picture, on the other hand, the broadening δωqis related to a finite lifetime
for the bosons of the Luttinger model. Once band curvature is introduced via the
irrelevant operators in (3.43), the single boson is allowed to decay and the coupling
to the multiboson states lifts the previous degeneracy. The fact that the irrelevant
operators have the same scaling dimension as in the noninteracting case suggests that
for ∆ ?= 0 the width should also vanish as q2in the limit q → 0. In this section we
argue in favor of a q2scaling for δωqfor all values of ∆ in the gapless regime, as long
as η− ?= 0, based on two different approaches. First, we explain how the expansion
of the bosonic diagrams in the interaction vertex η−, neglecting η+, coincides with the
expansion of the free fermion result (2.28) in powers of 1/m. η− is then interpreted
as a renormalized inverse mass, in the sense that the width of the peak for ∆ ?= 0 is
given by δωq= |η−|q2. Second, we derive from the Bethe Ansatz equations an analytical
expression for the width of the two-particle continuum at finite fields and show that it
coincides with the field theory prediction for the width of Szz(q,ω). Finally, we confirm
these results directly by analyzing the numerical form factors calculated for finite chains
of lengths up to 7000 sites.
q|0? is the
q. The associated weight in Szz(q,ω)
???0??Sz
q>0
2π
???0??aR
q
??b???2=KqN
2π
. (5.1)
However, as we will see in section 5.3, the exact eigenstates in the Bethe Ansatz solution,
5.1. Width from field theory
We saw that the width δωq is well defined for the free fermion point, in which case
Szz(q,ω) has sharp lower and upper thresholds ωL,U(q).
Szz(q,ω) still vanishes below some finite lower threshold ωL(q) at zero temperature
due to simple kinematic constraints. However, the on-shell peak has to match a high-
frequency tail somewhere around ωU(q), hence the meaning of an upper threshold is no
longer clear.
For the interacting case,
Page 22
Dynamical structure factor at small q for the XXZ spin-1/2 chain22
In their solution for weakly interacting spinless fermions, Pustilnik et al. [28] found
that ωUhas to be interpreted as the energy at which the peak joins the high frequency
tail by approaching a finite value with an infinite slope. Although it is actually possible
that the singularity at ωU(q) get smoothed out if one treats the decay of the “high-
energy electron” for a general model [36], the singularity may be protected in integrable
models such as the XXZ model.
Of course the situation is a lot simpler for models with no high-frequency tail, where
the dynamical structure factor is finite only within the interval ωL(q) < ω < ωU(q).
Such is the case for the Calogero-Sutherland model [44]. The absence of a tail for S (q,ω)
in the Calogero-Sutherland model can be attributed to the remarkable property that
the quasiparticles are all right movers [29]. As we will discuss in section 6, the η+term
that mixes R and L in our low energy effective Hamiltonian (figure 2) is responsible
for the high-frequency tail for h ?= 0 because it allows for intermediate states with two
bosons moving in opposite directions, thus carrying small momentum and high energy
ω ≫ vq.
In contrast, the η−interaction has matrix elements between multiboson states which
contain only right movers. All these states have ω ≈ vq. Therefore η−must be related
to the broadening of the on-shell peak. It has already been pointed out in [33] that
the model with η+= 0 is equivalent to free fermions up to irrelevant operators with
dimension four and higher. For this case one can write down an approximate expression
for the dynamical structure factor which misses more subtle features in the lineshape
(e.g., the power law singularities at the thresholds) but accounts for the renormalization
of the width due to interactions. Even for models with nonzero η+, such as the XXZ
model in the entire gapless regime, it is reasonable to expect that δωq, if well defined,
will depend primarily on the interaction between excitations created around the same
Fermi point. For that reason, we will neglect the η+interaction in an attempt to derive
an expression for the width of Szz(q,ω) from the bosonic Hamiltonian. In the following
we apply perturbation theory in η−up to fourth order and show that it recovers the
expansion of the logarithm for the density-density correlation function. This fact has
already been noticed in [34, 35] up to O(η2
such as (3.42) were neglected in [34, 35]. Such terms are crucial to obtain the correct
effective inverse mass, since the correction of first order in the fermionic interaction ∆
stems from (3.42).
For η+= 0, the Hamiltoninan HLL+ δH decouples into right and left movers. For
excitations with q > 0, we can consider only right movers and work with
√2π
6
The first attempt to broaden the delta function peak in Szz(q,ω) would be to calculate
the corrections to the propagator
?+∞
−). However, irrelevant interaction terms
HR=v
2(∂xϕR)2−η−(∂xϕR)3. (5.2)
χ(q,iω) = −K
2π
−∞
dxe−iqx
?β
0
dτ eiωτ?Tτ∂xϕR(x,τ)∂xϕR(0,0)?,(5.3)
Page 23
Dynamical structure factor at small q for the XXZ spin-1/2 chain 23
by using perturbation theory in the cubic term.
perturbation theory in η− breaks down near ω ≈ vq.
approximation, which sums an infinite series but not all the diagrams, one finds that
the self-energy to O(η2
surprising if we look at the exact solution for the free fermion point. Expanding the
positive-frequency part of (2.28) in powers of 1/m, we get
Unfortunately, any finite order
Even using the Born
−) is divergent: Im Σ(q,ω) ∼ δ (ω − vq) [37]. This is actually not
χ(q,iω) =
q
2πw
?
1 +1
3
?q2/m
2w
?2
+1
5
?q2/m
2w
?4
+ ...
?
, (5.4)
where w ≡ iω−vFq. Stricly speaking, such expansion is valid only for ω−vFq ≫ q2/2m.
For ω ≈ vFq, the expansion in band curvature produces increasingly singular terms that
need to be summed up to produce the finite result in (2.28).
In any case, it is legitimate to examine the expansion of bosonic diagrams and ask
whether it can at least reproduce the free fermion result. We use the bare propagator
D(0)
R(q,iω) =q
w,
to calculate the expansion of χR(q,iω) up to O?η4
χ(0)(q,ω) =
2πw.
The O(η2
R(x,τ) = ?Tτ∂xϕR(x,τ)∂xϕR(0,0)?0in (3.22), with Fourier transform
D(0)
(5.5)
−
?, as represented in figure 4. The
zeroth-order result is simply the same as in (3.27)
Kq
(5.6)
−) correction is
χ(2)(q,iω) = −K
2π
?
d2xe−iqx+iωτ
?
d2x1
?
d2x21
2
?√2π
6
η−
?2
×
×?Tτ∂xϕR(x)[∂xϕR(1)]3[∂xϕR(2)]3∂xϕR(0)?
2π
=K
?
D(0)
R(q,iω)
?2
ΠRR(q,iω), (5.7)
where ΠRR(q,iω) is the bubble with two right-moving bosons
ΠRR(q,iω) ≡ − πη2
−
?q
0
dk
2π
?+∞
−∞
dν
2πD(0)
R(k,iν)D(0)
R(q − k,iω − iν)
=η2
−q3
12w.
(5.8)
Note that ΠRRis singular at ω = vq, which prevents us from treating it as a self-energy.
The origin of the singularity is that the two right-moving bosons in the intermediate
state always carry energy ω = vq, no matter how the momentum is distributed between
the pair. Substituting (5.8) back into (5.7), we get
χ(2)(q,iω) =
Kq
2πw
1
12
?η−q2
w
?2
. (5.9)
Page 24
Dynamical structure factor at small q for the XXZ spin-1/2 chain24
χ =
+
++
+ + ...
BC
A
Figure 4. Perturbative diagrams up to fourth order in η−.
To O?η4
−
?, there are three topologically distinct diagrams (figure 4), which give the
χ(4)
2πw144w
?η−q2
χ(4)
2πw280w
The coefficients for each diagram are nontrivial and result from both combinatorial
factors and integration over internal momenta (recall that the interaction vertex is
momentum-dependent because of the derivatives in (3.43)). Remarkably, all the fourth-
order diagrams have the same q and ω dependence with comparable amplitudes. We
are not allowed to drop any of them and there is no justification for the use of a self-
consistent Born approximation, for example [37]. Putting all the terms together, we end
up with the expansion
?
32w
which is analogous to (5.4) with the replacements 1/m → η−, vF→ v and an extra factor
of K. This proves that the expansion of bosonic diagrams reproduces the expansion of
the free fermion result up to fourth order in 1/m. Since there is no simple way to predict
the prefactors of each diagram, all we can do is to check this correspondence order by
order in perturbation theory. However, if we believe that the bosonic theory reproduces
the free fermion result to all orders in η−, we must conclude that in the interacting case
the series in (5.11) sums up to give the result
?iω − vq + η−q2/2
from which we obtain
K
|η−|qθ
This result predicts that Szz(q,ω) is finite and flat within an interval of width
following contributions
A(q,iω) =
Kq
1
?η−q2
?4
?4
?4
,
χ(4)
B(q,iω) =Kq
2πw
1
504w
, (5.10)
C(q,iω) =Kq
1
?η−q2
.
χ(q,iω) =Kq
2πw
1 +1
?η−q2
?2
+1
5
?η−q2
2w
?4
+ ...
?
, (5.11)
χ(q,iω) =
K
2πη−qlog
iω − vq − η−q2/2
?
, (5.12)
Szz(q,ω) =
?|η−|q2
2
− |ω − vq|
?
. (5.13)
δωq= |η−|q2.(5.14)
Page 25
Dynamical structure factor at small q for the XXZ spin-1/2 chain 25
η−
η−
vq
ω
ωL
ωU
q2
q
K
Szz(q,ω)
Figure 5. Lineshape in the approximation with the η−interaction only (solid line).
The dotted line illustrates the expected true lineshape for small ∆ (see section 5.3 ).
This lineshape (illustrated in figure 5) is the exact one for the case of free fermions with
quadratic dispersion. The reason is simple: because the bosonization of the operator
∼ k2c†
Hamiltonian (5.2) to an effective free fermion model with inverse mass η−. In a more
general model, more irrelevant operator have to be added to the effective Hamiltonian to
reproduce details of the lineshape that are higher order in q. For example, we expect the
power-law singularities present at ωL,Ufor ∆ ?= 0 [28] to be associated with dimension-
four operators such as (∂2
This means that if we write
q
δωqf
δωq
the rescaled function f(q,x) approaches the flat distribution of figure 5 in the limit
q → 0. Finally, we note that this approximate solution yields the same sum rules as the
free boson result
?∞
?∞
and also the magnetic susceptibility
1
π
0
independent of the value of η−.
kckonly generates the η−term, one could invert the problem and refermionize the
xϕR)2and (∂xϕR)4(with corrections of O(η2
+), see section 7).
Szz(q,ω) ≡
?
q,ω − vq
?
, (5.15)
0
dω Szz(q,ω)= Kq,(5.16)
0
dωωSzz(q,ω) = vKq2, (5.17)
χ = χ(q = 0) = lim
q→0
?∞
dω
ωSzz(q,ω) =K
πv,
(5.18)
5.2. Width from Bethe Ansatz
The purpose of this section is to provide an analytical derivation of the quadratic width
formula (5.14), making use of standard methods associated to the thermodynamic Bethe
Page 26
Dynamical structure factor at small q for the XXZ spin-1/2 chain 26
Ansatz, and assuming that single particle-hole type excitations in the Bethe eigenstates
basis carry the most important part of the structure factor. We first set our notations
and underline certain characteristics of the ground state of the infinite chain in a field
which will prove to be useful for our purposes. We then discuss particle-hole excitations
in the thermodynamic limit, and obtain a relationship giving the width in terms of
solutions of integral equations, simplifying to the conjectured field theory result in the
small momentum limit.
Let us begin by taking the thermodynamic limit N → ∞ of the equations of Section
4. To do this, we first define particle and hole densities as functions of the continuous
variable x =
I
N,
ρ(x) =1
N
?
l∈{I}
δ(x −
l
N),ρh(x) =1
N
?
m/ ∈{I}
δ(x −m
N) (5.19)
in such a way that ρ(x) + ρh(x) → 1 as N → ∞. We can also write these in rapidity
space by using the transformation rule for δ functions, so that the Bethe equations
become
?∞
where we view x as an implicit function of λ. Taking the derivative of this with respect
to λ and using
dλ
= ρ(λ) + ρh(λ) yields
?∞
For the particular case of the ground state, the occupation density ρGS(λ) is non-
vanishing in a symmetric interval [−B,B], with ρh
interval, ρGSvanishes but not ρh
the rapidity distribution of the ground state, which is obtained by solving
?B
The magnetic field dependence is encoded in the constraint
?B
where σ is the field-dependent average magnetization per site along the z axis. These
two equations determine B and ρGS, and therefore also ρh
solution as follows. Let us define the inverse operator L(λ,λ′), λ,λ′∈ [−B,B], inverse
of the kernel in (5.22) in the sense that
?B
This operator is symmetric, L(λ,λ′) = L(λ′,λ), unique and analytic in its domain of
definition [45]. In particular, the definition implies the identity
?B
φ1(λ) −
−∞
dλ′φ2(λ − λ′)ρ(λ′) = 2πx(λ) (5.20)
dx(λ)
a1(λ) −
−∞
dλ′a2(λ − λ′)ρ(λ′) = ρ(λ) + ρh(λ), λ ∈ R.(5.21)
GS(λ) vanishing. Outside of this
GS. λ = ±B therefore represent the two Fermi points in
ρGS(λ) +
−B
dλ′a2(λ − λ′)ρGS(λ′) = a1(λ), λ ∈ [−B,B]. (5.22)
−B
dλρGS(λ) =M
N=1
2− σ(5.23)
GS. We can write a formal
−B
dλ′[δ(λ − λ′) + L(λ,λ′)][δ(λ′−¯λ) + a2(λ′−¯λ)] = δ(λ −¯λ).(5.24)
a2(λ −¯λ) + L(λ,¯λ) +
−B
dλ′L(λ,λ′)a2(λ′−¯λ) = 0, λ,¯λ ∈ [−B,B]. (5.25)
Page 27
Dynamical structure factor at small q for the XXZ spin-1/2 chain 27
In terms of this operator, we have the explicit solution of equation (5.22) for the ground
state distribution,
??B
Knowing ρGSthen yields ρh
?
The ground state can also be obtained from the thermodynamic Bethe Ansatz formalism
[45] in the following way. Given distributions ρ(λ) and ρh(λ), the free energy f =
(E − TS)/N is written to leading order in N as
f = −h
−∞
in which we have suppressed the λ functional arguments and defined the bare energy
ρGS(λ) =
−Bdλ′[δ(λ − λ′) + L(λ,λ′)]a1(λ′) |λ| ≤ B,
0|λ| > B.
(5.26)
GSfrom (5.21), namely
ρh
GS(λ) =
0|λ| ≤ B,
a1(λ) −?B
−Bdλ′a2(λ − λ′)ρGS(λ′) |λ| > B.
(5.27)
2+
?∞
dλ?ε0ρ − T(ρ + ρh)ln(ρ + ρh) + Tρlnρ + Tρhlnρh?
(5.28)
ε0(λ) = h − πJ sinζa1(λ). (5.29)
Introducing the quasi-energy ε(λ) = T lnρh(λ)
equilibrium δF = 0 under the constraint of the Bethe equations (5.21) then gives after
standard manipulations [45] (taking the limit T → 0, so here and in what follows, ε(λ)
is for the ground state configuration)
?B
In particular, we have that
ρ(λ), the condition of thermodynamic
ε(λ) +
−B
dλ′a2(λ − λ′)ε(λ′) = ε0(λ),λ ∈] − ∞,∞[. (5.30)
ε(±B) = 0,ε(λ) ≤ 0(> 0)forλ ∈ (/ ∈)[−B,B]. (5.31)
Similarly to (5.26), we can also solve for ε(λ) = ε−(λ)+ε+(λ) with ε±(λ) ≥ (≤)0 using
the inverse integral kernel:
??B
?
The free energy simplifies to
?B
The magnetic equilibrium condition∂F
ε−(λ) =
−Bdλ′[δ(λ − λ′) + L(λ,λ′)]ε0(λ′) |λ| ≤ B,
0|λ| > B,
|λ| ≤ B,
(5.32)
ε+(λ) =
0
ε0(λ) −?B
−Bdλ′a2(λ − λ′)ε−(λ′) |λ| > B.
(5.33)
f = −h
2+
−B
dλa1(λ)ε(λ).(5.34)
∂h= 0 then is
?B
−B
dλa1(λ)∂ε(λ)
∂h
=1
2.(5.35)
By defining the dressed charge Z(λ) as solution to
?B
Z(λ) +
−B
dλ′a2(λ − λ′)Z(λ′) = 1, (5.36)
Page 28
Dynamical structure factor at small q for the XXZ spin-1/2 chain 28
which we can solve as
Z(λ) = 1 +
?B
−B
dλ′L(λ,λ′), (5.37)
we have the identity Z(λ) =
parameter K is given by the square of the dressed charge at the Fermi boundary (see
e.g. [39]),
∂ε(λ)
∂h
by making use of (5.29) and (5.30). The Luttinger
K = Z2(−B). (5.38)
The magnetic field dependence of the Fermi boundary B can be obtained by taking
the h derivative of (5.30):
?B
Since ε(−B) = 0, we have
∂ε(λ)
∂λ∂B
and therefore
∂h
∂B=ε′(−B)
The magnetization is
?B
To get the susceptibility, we start from
?B
The integral equation for the dressed charge (5.36) gives
?B
× [Z(B)a2(λ′− B) + Z(−B)a2(λ′+ B)]
which yields after simple manipulations and use of symmetry
∂σ
∂B= −2ρGS(−B)Z(−B).
The susceptibility is therefore given by
∂B= −2ρGS(−B)Z2(−B)
This expression will be related to the Fermi velocity after discussing elementary
excitations (see equation (5.68)).
Finally, we will need the slope of the ground state rapidity distribution at the Fermi
boundary,
∂B
|−B. From the integral equation for ρGS, we can write
∂ρGS(λ)
∂B
−B
dλ′[δ(λ − λ′) + a2(λ − λ′)]∂ε(λ′)
∂B
=∂h
∂B.
(5.39)
|λ=−B=∂ε(λ)
|λ=−B
(5.40)
Z(−B). (5.41)
σ = −∂f
∂h=1
2−
−B
dλa1(λ)∂ε(λ)
∂h
=1
2−
?B
−B
dλa1(λ)Z(λ). (5.42)
∂σ
∂B= −
−B
dλa1(λ)∂Z(λ)
∂B
− a1(B)Z(B) − a1(−B)Z(−B). (5.43)
∂Z(λ)
∂B
= −
−B
dλ′[δ(λ − λ′) + L(λ,λ′)]
(5.44)
(5.45)
χ =∂σ
∂h=∂B
∂h
∂σ
ε′(−B)
. (5.46)
∂ρGS(λ)
= ρGS(−B)[L(λ,B) + L(λ,−B)].(5.47)
Page 29
Dynamical structure factor at small q for the XXZ spin-1/2 chain29
This can be related to the derivative of the dressed charge by using the representation
∂Z(λ)
∂B
= L(λ,B) + L(λ,−B) +
?B
−B
dλ′∂L(λ,λ′)
∂B
. (5.48)
From the definition of L(λ,λ′), we can show that
∂L(λ,λ′)
∂B
and therefore
∂Z(λ)
∂B
finally yielding
= L(λ,B)L(λ′,B) + L(λ,−B)L(λ′,−B) (5.49)
= [L(λ,B) + L(λ,−B)]Z(−B), (5.50)
∂ρGS(λ)
∂B
|−B=ρGS(−B)
Z(−B)
∂Z(λ)
∂B
|−B. (5.51)
We will make use of these identities later, while relating the width of the two-particle
continuum to field-dependent physical quantities.
Let’s now construct an excited state over the finite-field ground state by generating
a single particle-hole pair. That is, we select a quantum number Ip/ ∈ {IGS} associated
to a particle and Ih∈ {IGS} associated to a hole, and write the excited state densities
in x space as
ρ(x) = ρGS(x) +1
Nδ(x −Ip
Nδ(x −Ip
N) −1
N) +1
Nδ(x −Ih
Nδ(x −Ih
N),
ρh(x) = ρh
GS(x) −1
N), (5.52)
with once again ρ(x)+ρh(x) → 1 as N → ∞. We can again map to rapidity space, with
λp≤ −B and |λh| ≤ B. Upon creating such a particle-hole pair, the induced distribution
ρ(λ) will be only very slightly shifted (order 1/N) as compared to the ground state one
(for λ ?= λp,λh). We therefore define a backflow function K(λ;λp,λh) ∼ O(N0) as
ρ(λ) = ρGS(λ) +1
N[K(λ;λp,λh) + δ(λ − λp) − δ(λ − λh)].
By subtracting the equations for the ground state from those of the excited state, the
backflow function is shown to obey the constraint
?B
for λ ∈ [−B,B], with K = 0 outside of this domain. We can again formally solve for K
by applying the inverse integral operator 1 + L,
?B
In terms of this kernel, the energy of the excited state is
?∞
(5.53)
K(λ;λp,λh)+
−B
dλ′a2(λ−λ′)K(λ′;λp,λh) = −a2(λ−λp)+a2(λ−λh)(5.54)
K(λ;λp;λh) = −a2(λ − λp) −
−B
dλ′L(λ,λ′)a2(λ′− λp) − L(λ,λh).(5.55)
E−EGS=N
−∞
dλε0[ρ − ρGS] = ε0(λp) − ε0(λh) +
?B
−B
dλε0(λ)K(λ;λp,λh),(5.56)
Page 30
Dynamical structure factor at small q for the XXZ spin-1/2 chain30
which can be rewritten after basic manipulations as (|λp| > B and |λh| < B)
E − EGS= ε(λp) − ε(λh).
Similarly, the momentum of the excited state is
(5.57)
P − PGS= − φ1(λp) + φ1(λh) −
?B
−B
dλφ1(λ)K(λ;λp,λh). (5.58)
Single particle-hole pairs as described above constitute a set of two-particle
excitations labeled by the particle and hole rapidities λp and λh. This continuum is
well-defined and spanned by the intervals λp∈ ] − ∞,−B], λh∈ [−B,B]. Assuming
that the mapping from (λp,λh) to (ω,q) is one-to-one and onto and that the particle
dispersion curvature is greater than the hole one (this monotonicity assumption will be
discussed further in section 5.3), the highest energy state at a given fixed momentum q
will be given by the choice λp= λp(q), λh= −B, where λp(q) is solution to
?B
Similarly, the lowest energy state will correspond to the choice λp= −B, λh= λh(q),
where λh(q) is solution to
?B
As discussed in Section 4, this continuum is well-defined (i.e. finite real solutions to
both (5.59) and (5.60) can be found) as long as q ≤ Min(2kF,k∞), with 2kF= π(1−2σ)
and k∞= 2σ(π − ζ). This is illustrated in Figure (6). The width of the two-particle
continuum defined by these excitations will thus be given by the energy difference
between these two limiting configurations, namely
q = −φ1(λp(q)) + φ1(−B) +
−B
dλφ1(λ)K(λ;λp(q);−B). (5.59)
q = φ1(λh(q)) − φ1(−B) −
−B
dλφ1(λ)K(λ;−B;λh(q)). (5.60)
W(q) = ε(λp(q)) + ε(λh(q)) − 2ε(−B) = ε(λp(q)) + ε(λh(q))
where we have used ε(±B) = 0. These functions are exact in the thermodynamic limit,
in the sense that they allow at least in principle to obtain the exact function W(q) for
the momentum region where these excitations are defined. These coupled equations
unfortunately cannot be solved explicitly at nonzero magnetic field (where B is finite).
We can however obtain analytical results in the small momentum limit, where these
excitations always exist in a finite region at finite field.
At small momentum, we can expand the width at fixed magnetic field as
(5.61)
W = qW(1)+ q2W(2)+ O(q3)(5.62)
with coefficients given explicitly by
W(1)=
∂
∂q(ε(λp(q)) + ε(λh(q)))|q=0,
W(2)=1
2
(5.63)
∂2
∂q2(ε(λp(q)) + ε(λh(q)))|q=0. (5.64)
Page 31
Dynamical structure factor at small q for the XXZ spin-1/2 chain 31
B −B
h
h
p
p
λλ
λλ
B−B
Figure 6. Highest and lowest energy two-particle excited states at fixed momentum.
The straight line represents the interval λ ∈ [−B,B] within which the ground-state
rapidity ρGS(λ) is nonvanishing. λp and λh respectively represent the positions of
the particle and hole rapidities for the highest (top) and lowest (bottom) two-particle
excited states at a fixed value of momentum.
Let us treat the linear term first. Considering that (5.60) also defines a function q(λh),
we can rewrite the hole contribution to the coefficient as
∂ε(λh)
∂λh|λh=−B
∂q(λh)
∂λh|λh=−B
The denominator is obtained from (5.60) as
?B
where we have used (5.55), the symmetry of L and partial integration. This contribution
is by definition related to the field-dependent Fermi velocity, namely
∂
∂qε(λh)|q=0=
2π
λ→−B+
In particular, this allows us to relate the susceptibility to the Fermi velocity and the
dressed charge using relation (5.46),
Z2(−B) = πvχ.
For the particle contribution to the linear term, we find similarly that
−2πρh
−1
2πlimλ→−B−
ρh
momentum expansion (5.62) for the width, the linear term vanishes:
∂
∂q(ε(λp(q)) + ε(λh(q)))|q=0= 0.
The width therefore depends at least quadratically on momentum. To compute the
coefficient of the quadratic term, we first note that given a function λ(q) and its inverse
q(λ), the chain rule allows us to write
∂2
∂q2ε(λ(q))
q=0
∂λ
−B
∂
∂qε(λh(q))|q=0=
. (5.65)
∂q
∂λh
= 2πa1(λh) −
−B
dλφ1(λ)∂K(λ;−B;λh)
∂λh
= 2πρGS(λh) (5.66)
1
lim
ε−′(λ)
ρGS(λ)≡ −v.(5.67)
(5.68)
∂q
∂λp
=
GS(λp). Since limλ→−B− ρh
ε+′(λ)
GS(λ)
GS(λ) = limλ→−B+ ρGS(λ), we also have
= v since ε is smooth around this point.
∂
∂qε(λp)|q=0=
Therefore, in the
W(1)=
(5.69)
????
=
?∂q
????
?−2?∂2ε(λ)
∂λ2
????
−B
−∂2q
∂λ2
????
−B
∂ε(λ)
∂λ
????
−B
?
. (5.70)
Page 32
Dynamical structure factor at small q for the XXZ spin-1/2 chain 32
From (5.59) and (5.60), we have that the particle and hole parts are related through
∂q(λp)
∂λp
∂2q(λp)
∂λ2
????
λp=−B
????
= −∂q(λh)
= −∂2q(λh)
∂λh
????
λh=−B
????
, (5.71)
p
λp=−B
∂λ2
h
λh=−B
, (5.72)
so using (5.70) for λp(q) and λh(q), we obtain that the quadratic coefficient of the width
can be simplified to
W(2)=1
2
∂2
∂q2(ε(λp(q)) + ε(λh(q)))
????
q=0
=
?∂q
∂λ
????
−B
?−2∂2ε(λ)
∂λ2
????
−B
. (5.73)
While this expression for the width is an end in itself, it is much more enlightening
to relate it to more physical quantities by making use of the identities derived earlier.
Starting from
get
∂2ε(λ)
∂λ2
−B
Putting this in (5.73) and making use of (5.41), (5.67) and (5.68) again, this finally gives
∂2ε(λ)
∂λ2 |−B=
∂2ε(λ)
∂B2 |−Band using (5.67) together with (5.51) and (5.68), we
????
?v
= −2πρGS(−B)
?3
2
∂v
∂B+1
2
v
χ
∂χ
∂B
?
. (5.74)
W(2)=
πχ
?3
2
∂v
∂h+1
2
v
χ
∂χ
∂h
?
. (5.75)
Since we have the identity K = Z2(−B) = πvχ, this coincides with (3.69). It also
reduces to the formula derived in [32] for ∆ ≪ 1 by linearizing the Bethe Ansatz
equations. While our derivation was done for the anisotropic chain in the gapless regime,
the same calculation can be performed for the isotropic antiferromagnet by simply using
the appropriate scattering kernels in the Bethe equations. This result is however limited
to chains with finite magnetization, in view of the fact that the region of validity of the
excitations we have used to compute the width collapses to zero when the field vanishes.
5.3. Comparison with numerical form factors
In order to compare the field theory results with the dynamical structure factor for finite
chains, we first fix the parameters of the bosonic model introduced in section 3.1. We
do that by calculating v(∆,h) and K(∆,h) = πv(∆,h)χ(∆,h) numerically using the
Bethe Ansatz integral equations in the thermodynamic limit. η−and η+are obtained
by linearizing the field dependence of v and K around some fixed h0and using (3.69)
and (3.70). As examples, we consider three values of the anisotropy, ∆ = 0.25, ∆ = 0.75
and the Heisenberg point ∆ = 1, at a fixed magnetization per site σ = −0.1. Table 1
lists the values of the important parameters (we set J = 1). Note that b is negative for
σ < 0 (m > 0) because K decreases as we approach half-filling [2].
As mentioned in section 4.2, we are able to calculate the exact transition
probabilities F2(q,ω) ≡
Ansatz [19, 20, 21]. Figure 7 illustrates a typical result obtained for finite anisotropy
???0??Sz
q
??α???2for finite chains by means of the Algebraic Bethe
Page 33
Dynamical structure factor at small q for the XXZ spin-1/2 chain33
Table 1. Parameters for the low-energy effective model for ∆ = 0.25, ∆ = 0.75 and
∆ = 1 and finite magnetic field h0(in all cases the magnetization per site is σ = −0.1).
∆h0
vKabη−
η+
0.25
0.75
1
-0.414
-0.652
-0.791
1.087
1.313
1.399
0.871
0.699
0.639
0.306
0.271
0.256
-0.050
-0.145
-0.188
0.356
0.409
0.397
-0.095
-0.449
-0.690
6
8 1012 14
16
1820 22
ωN/2πv
10-8
10-6
10-4
10-2
100
F2(q = 2π/25, ω)
∆ = 0.25
σ = - 0.1
Figure 7. Numerical form factors squared (transition probabilities) for states with
momentum q = 2π/25, for a chain with N = 200 sites, anisotropy ∆ = 0.25,
magnetization per site σ = −0.1. The energies of the eigenstates are rescaled by
the level spacing of the bosonic states predicted by field theory. The on-shell states
are the ones at ωN/2πv = qN/2π = 8.
and finite magnetic field. In contrast with the free fermion case, we observe two main
differences when we turn on the fermion interaction ∆: First, the form factors for
the two-particle (on-shell) states become ω-dependent; second, the form factors for
multiparticle states are now finite and account for a finite spectral weight extending
up to high energies. For the four-particle states (two particle-hole pairs), we expect
?0??Sz
evolve smoothly from the XX point, except close to the lower and upper thresholds. If
that is the case, the two-particle states still carry most of the spectral weight. In the
thermodynamic limit, F2(q,ω) has to be combined with the density of states factor
D(q,ω) =2π
N
α
q
??α?
∼ O(∆), but this is not true near ω ≈ vq where perturbation theory in
the interaction diverges [28]. Figure 7 suggests that most of the exact form factors
?
δ(ω − Eα+ EGS),(5.76)
Page 34
Dynamical structure factor at small q for the XXZ spin-1/2 chain 34
0.920.94
0.96
0.98
ω/vq
1 1.021.04
0.6
0.7
0.8
0.9
1
1.1
F2(q = 2π/25,ω)
N = 600
N = 1000
ωmin(N = 1000)
ωmax(N = 1000)
Figure 8. Form factors squared for the two-particle states for two values of system
size N (we set q = 2π/25, ∆ = 0.25 and σ = −0.1). The points seem to collapse
on a single curve, showing very little size dependence. The minimum and maximum
energies converge to the thresholds of the two-particle continuum when N → ∞.
to define the lineshape of Szz(q,ω) (see (2.3)).
We can count the states at each energy level of the finite system in the Bethe
Ansatz the same way we count states for weakly interacting fermions. For example,
in figure 7 we see n ≡ qN/2π = 8 two-particle states with F2∼ O(1). One can also
verify that for n = 8 there are 14 states with two right-moving particle-hole pairs (of the
form c†
states with F2(q,ω) < 10−3in figure 7 are all four-particle states. Furthermore, for small
∆ the main contribution to the high-frequency tail (ℓ ≡ ωN/2πv > n) is due to states
containing two particle-hole excitations created around the two different Fermi points
[27]. If the momenta of the pairs at the right and left branches are q1= 2πn1/N > 0
and q2 = 2πn2/N < 0, such that n1 = (ℓ + n)/2 and n2 = −(ℓ − n)/2, then the
number of such states is given by |n1× n2| = (ℓ2−n2)/4. This is in agreement with the
counting of states in figure 7. We also find much smaller form factors for states with
three particle-hole pairs (not shown in the figure).
We now focus on the two-particle states inside the peak, with ω ≈ vq. If we seek
only these states with dominant form factors it is possible to reach much larger system
sizes (we go up to 7000 sites). The number of two-particle states is always n = qN/2π.
Figure 8 shows F2(q,ω) for a fixed value of q = 2π/25 and two different system sizes.
We extract δωq from the numerical form factors as follows. We see from figure
8 that the separation between energy levels inside the peak is of order δωq/N and
p1+q1,Rcp1,Rc†
p2+q2,Rcp2,R|0?) and no states with three or more pairs. The 14 on-shell
Page 35
Dynamical structure factor at small q for the XXZ spin-1/2 chain35
0
0.0005
0.001
1/N
0.0015
0.002
0.9495
0.95
0.9505
0.951
0.9515
ωmin/vq
Figure 9. Finite size scaling of the minimum energy for two-particle states with
q = 2π/25, ∆ = 0.25 and σ = −0.1.
Table 2. Effective inverse mass, defined as the coefficient of the q2scaling of the width
δωq. The data are for σ = −0.1 and anisotropy parameters ∆ = 0.25,0.75,1.
∆1/m∗
FIT
η−
m
?1 +2∆
0.25 0.3540.3560.356
0.75 0.4080.4090.449
10.3960.397 0.496
1
πsinkF
?
decreases from ωL(q) to ωU(q). As N increases, the maximum and minimum energies
ωmax,min(N) converge to fixed values which we identify as the thresholds of the two-
particle continuum. Figure 9 shows the finite size scaling of the minimum energy for
∆ = 0.25, σ = −0.1 and q = 2π/25. The same N−1dependence is observed for the
maximum energy. We use this scaling to determine the lower and upper thresholds
ωL,U(q) in the thermodynamic limit for several values of q. We then calculate the
width δωq = ωU(q) − ωL(q). As expected, we find that δωq = q2/m∗for small q
(figure 10). Table 2 compares the coefficients 1/m∗
the predicted values of η− taken from Table 1. The perturbative result in (3.44) is
also shown for comparison. The agreement supports our formula for the width in the
strongly interacting (finite ∆) regime. Note that η−is a nonmonotonic function of ∆.
In figure 11 we confirm that, despite the enhancement (suppression) near the lower
(upper) threshold, F2(q,ω) converges to the constant value F2(q,ω) = K in the limit
q → 0, as expected from the box-like shape shown in figure 5 (see however the subtleties
about the thermodynamic limit in section 8.2). This is in agreement with the fact that
FITobtained by fitting the data with
Page 36
Dynamical structure factor at small q for the XXZ spin-1/2 chain36
Figure 10. Width of the on-shell peak (based on the two-particle contribution) as a
function of momentum q for σ = −0.1 and two values of anisotropy: ∆ = 0.25 (blue
diamonds) and ∆ = 0.75 (red circles). The lines are the best fit to the data.
the exponents of the singularities at the edges are linear in q for h ?= 0 [28]. Notice that
F2(q,ω) (and therefore Szz(q,ω)) is not a scaling function of (ω − vq)/δωq.
If the density of states D(q,ω) for the two-particle states were constant, the
extrapolation of figure 8 to the thermodynamic limit would be representative of the
lineshape of Szz(q,ω). This would be exactly the case if the exact energies Eα− EGS
could be written as the sum of the energies of particles and holes with parabolic
dispersion (as in a Galilean-invariant system, e.g. the Calogero-Sutherland model [29]).
This is also the case considered in [28]. In our case D(q,ω) does vary inside the peak
because of the cubic terms in the dispersion of the particles in the Bethe Ansatz. For
large enough N we can include the density of states factor (5.76) if we rescale F2(q,ω)
by the separation between energy levels inside the peak
Szz(q,ω) = D(q,ω)F2(q,ω) ≈2π
N
F2(q,ω)
Ej+1− Ej, (5.77)
where Ejand Ej+1are the energies of the two-particles states, ordered in energy, with
Ej− EGS = ω. The approximate density of states calculated this way is illustrated
in figure 12. The resulting lineshape is shown in figure 13. This lineshape should
be contrasted with the free fermion result in figure 1. The exact boundaries of the
two-particle continuum (dotted line in figure 13) are actually shifted to lower energies
relatively to the prediction ωU,L(q) = vq±η−q2/2 (dashed lines) because of the cubic term
in the exact dispersion, which was neglected in the field theory approach. Notice that
there appears to be a peak at the exact lower threshold of the two-particle continuum.
Page 37
Dynamical structure factor at small q for the XXZ spin-1/2 chain 37
-0.4 -0.20 0.2 0.4
0.6
(ω-vq)/δωq
0.75
0.8
0.85
0.9
0.95
F2(q,ω)
qN/2π = 40
qn/2π = 80
qN/2π = 120
N = 6000
Figure 11. Frequency dependence of the form factors squared for N = 6000, ∆ = 0.25,
σ = −0.1, and three values of momentum. The dashed line represents the field theory
prediction F2(q,ω) = K ≈ 0.871, as in figure 5.
0.0895
0.09
0.0905
0.091
ω
0.0915
0.092
30
32
34
36
38
40
D(q,ω)
qN/2π = 80
0.04520.0454 0.04560.0458
64
68
72
qN/2π = 40
N = 6000
Figure 12. Density of states D(q,ω) for the two-particle states obtained using (5.77).
As in figure 11, we use N = 6000, ∆ = 0.25 and σ = −0.1. The main graph is for
qN/2π = 80. The inset shows the density of states for a smaller value of momentum,
qN/2π = 40. The solid lines are meant to illustrate the deviation of D(q,ω) from the
linear dependence in ω.
Page 38
Dynamical structure factor at small q for the XXZ spin-1/2 chain38
0.0895
0.09
0.0905
0.091
ω
0.0915
0.092
0.0925
27
28
29
30
31
32
Szz(q,ω)
∆ = 0.25
σ = - 0.1
Figure 13. Lineshape of Szz(q,ω) estimated from the two-particle states (S2
in the notation of section 8.2). For this graph ∆ = 0.25, σ = −0.1, N = 6000 and
qN/2π = 80. The dashed line is the flat distribution of figure 5. The dotted lines are
the exact boundaries of the two-particle continuum in the thermodynamic limit.
N(q,ω)
The result of Pustilnik et al. predicts that there is actually a power-law singularity at
ωL(q), which is related to the physics of the X-ray edge problem [28]. We do not attempt
to study the singularity in the form factors in this paper (see discussion in section 8.2).
Interestingly, however, the density of states competes with the energy dependence of the
form factors, leading to a minimum in Szz(q,ω) above ωL(q) and a rounded peak below
ωU(q). In the limit q ≪ cotkFwe can linearize the density of states for the two-particle
states
2π/N
Ej+1− Ej
where ˜ γ is a fitting parameter analogous to γ in (2.21) for the free fermion model. The
inset of figure 12 shows the density of states for a smaller value of q = 2π(40/6000).
We have checked that D(q,ω) becomes more linear and ˜ γ converges to a finite value as
q decreases. For ∆ = 0.25 and σ = −0.1 we estimate ˜ γ ≈ 1.11, which is larger than
the value for free fermions γ = sin(2π/5) ≈ 0.951. Combining this density of states
with the power-law singularity proposed in [28], the behavior near the lower threshold
is described by the function
D(q,ω) ≈
≈
1
η−q
?
1 +˜ γq
η−
ω − vq
δωq
?
, (5.78)
Szz(q,ω) ≈
K
η−q
?
1 −
˜ γq
2η−
+˜ γq
η−
ω − ωL(q)
δωq
??ω − ωL(q)
δωq
?−µq
, (5.79)
where µqis the exponent of the X-ray edge singularity. The position of the minimum is
Page 39
Dynamical structure factor at small q for the XXZ spin-1/2 chain 39
0.1150.116
0.117
ω
0.1180.119
18
18.5
19
19.5
20
20.5
21
Szz(q = 2π/75,ω)
∆ = 1
σ = - 0.1
Figure 14. Lineshape for the Heisenberg chain at finite field (∆ = 1, σ = −0.1,
N = 6000 and qN/2π = 80). Lines and symbols are represented as in figure 13.
then
ω∗− ωL(q)
δωq
≈η−µq
˜ γq
,(5.80)
for ˜ γq/η− ≪ 1 and µq ≪ 1. Since µq ∝ q for small q, the right-hand side of (5.80)
becomes constant in the limit q → 0. In this sense, the X-ray edge singularity and
the energy dependence of the density of states are effects of the same order in q. We
notice that the difference ∆Szz, defined between the maximum and the minimum of
Szz(q,ω), converges to a finite value as q → 0 (as it did for free fermions). The precise
value depends on both the density of states D(q,ω) and the frequency dependence of
F2(q,ω) (which is approximately linear with a negative slope for |ω − vq| ≪ δωq). As
a result, the slope of Szz(q,ω) near the center of the peak diverges as 1/q2as q → 0.
This is a rather singular dependence of the lineshape on ˜ γ and is potentially important
for systems in which the dispersion is not exactly parabolic (e.g. due to band mixing in
semiconductor quantum wires).
Figure 14 shows the lineshape for the isotropic point ∆ = 1 and the same values of
σ and q used in figure 13. In comparison with the weak coupling value ∆ = 0.25, there
is an enhancement of the singularities near the lower and upper thresholds. The shift of
the peak to lower energies (another “q3effect”) is also more pronounced, but the width
is very well described by the field theory formula (prefactor given in table 2).
Finally, let us comment on the validity of the q2scaling for the width as a function
of magnetic field. Figure 15 shows the dependence of the coupling constants of the
irrelevant operators on the magnetization σ for ∆ = 0.25 and ∆ = 1. From the field
Page 40
Dynamical structure factor at small q for the XXZ spin-1/2 chain 40
-0.5
-0.4-0.3 -0.2-0.10
σ
-0.2
0
0.2
0.4
0.6
0.8
1
-0.5
-0.4-0.3 -0.2-0.10
σ
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η+
η-
(a)(b)
η-
η+
σinv
Figure 15. Parameters η±for the low energy effective Hamiltonian as a function of
the magnetization σ < 0 for two values of anisotropy: (a) ∆ = 0.25; (b) ∆ = 1. For
σ > 0, we have η±(−σ) = −η±(σ).
theory standpoint, we expect that the q2scaling is valid as long as η−q2≪ vq (the peak
is narrow) and η±q2≫ ˜ γq3(the cubic terms yield the leading correction to the free
boson result and we can drop operators with dimension four and higher in the effective
Hamiltonian). For ∆ = 0.25, we see that η±follow the behavior predicted by the weak
coupling expressions (3.44) and (3.45), vanishing at σ = 0. In this case, η±are always
of O(1). The restrictions are similar to the ones for the approximation (2.24) for the
dynamical structure factor of the XX model, namely q ≪ kF and q ≪ cotkF (which
becomes q ≪ πσ for small σ). On the other hand, for ∆ = 1 we find that |η±| → ∞
as σ → 0. This is a direct consequence of formulas (3.69) and (3.70) in the strong
coupling regime. It is known that the magnetic susceptibility at small fields is given
by χ(h) ∼ const + C1h2+ C2h8K−4, where C1,2are constants [43, 46]. The exponent
8K − 4 is a manifestation of the Umklapp scattering term at zero magnetic field. As a
result, ∂χ/∂h diverges as h → 0 for K < 5/8 or ∆ > cos(π/5) ≈ 0.81. In other words,
the Luttinger parameter has an infinite slope at h = 0 (see [47] for the isotropic case).
Since ∂K/∂h and ∂v/∂h have opposite signs, η−goes through zero for a finite value of
σ. Therefore, we predict that δωqis a nonmonotonic function of σ for ∆ > cos(π/5) and
|σ| ≪ 1. The sign change in η−is reflected in the Bethe Ansatz data as the inversion of
the ordering of the energies of the two-particle states as a function of hole momentum.
For ∆ = 1, the “inversion point” where η−= 0 occurs at |σinv| ≈ 0.030. At this point
we observe that δωq∝ q3for q ≪ πσ. However, η+∼ O(1), so the lineshape defined
by the two-particle states must be different from the one at zero field, where there is
also a q3scaling (see section 7). We have also confirmed that the q2scaling holds in
the region where η−< 0 and q ≪ π|σ| (figure 16). In this regime we find that the sign
change of η−is accompanied by the inversion of the lineshape of Szz(q,ω): The form
Page 41
Dynamical structure factor at small q for the XXZ spin-1/2 chain41
00.010.020.03
q
0.04
0.05 0.06
0
0.0005
0.001
0.0015
0.002
δωq
Bethe Ansatz
δωq = 0.246 q2
∆ = 1
σ = - 0.01
Figure 16. Width δωqfor ∆ = 1 and σ = −0.01 (in the region where η−< 0). The
blue diamonds represent the width defined as the difference between the maximum
and minimum energies of the two-particle states calculated in the Bethe Ansatz,
extrapolated to the thermodynamic limit. For q ≪ π|σ|, we recover the behavior
δωq= |η−|q2, with |η−| ≈ 0.246 (dashed line).
factors appear to vanish at the lower threshold and are peaked near the upper threshold
(with a possible divergence at ωU) (figure 17). There is no maximum or minimum near
the edges in this case. In order to understand this result, we recall that a converging
X-ray edge is possible in strongly interacting systems. An important point is that the
exponents of the X-ray edge singularities calculated in [28], which predict a diverging
X-ray edge, are valid only to first order in the interactions. Second order corrections,
which usually have the opposite sign because of the orthogonality catastrophe, tend to
kill the singularity at the lower edge [48]. For even smaller values of σ, the divergence
of η−seems to be consistent with the Bethe Ansatz results. For |σ| < σinv, the width
increases as |σ| decreases at least down to σ = −0.001, the lowest magnetization we
were able to analyze. In the limit σ → 0 and |η±| ≫ 1, we expect for the isotropic point
(using the results of [49, 47])
η+
3
→ η−→
J
8√2σln|σ0/σ|, (5.81)
where σ0=?32/πe. According to the conditions ˜ γq2≪ η−q ≪ v, the field theory result
point σinvand for σ → 0. In the limit σ → 0, as mentioned in section 4.2, the set of
allowable quantum numbers for the single particle-hole excitations becomes empty, as
the I∞quantum number tends to N/2 (meaning that the particle part becomes trapped
which predicts the q2scaling for a small fixed q breaks down both near the inversion
Page 42
Dynamical structure factor at small q for the XXZ spin-1/2 chain 42
0.01949
0.01950.01951 0.019520.01953
ω
150
200
250
300
Szz(q = π/250, ω)
0.0195 0.019510.019520.01953
ω
0.5
0.55
0.6
F2(q, ω)
∆ = 1
σ = -0.01
Figure 17. Lineshape of Szz(q,ω) for magnetization below the inversion point, i.e.
|σ| < σinv (∆ = 1, σ = −0.01, N = 7000 and qN/2π = 14). This value of q is in the
domain where δωq∼ q2(see figure 16), but the lineshape is inverted. In contrast with
figure 8, the form factors (shown in the inset) are peaked at the upper threshold of the
two-particle continuum.
at the Fermi surface), and this family of excitations disappears. The vanishing field two-
particle continuum at nonvanishing momentum is then obtained from considering the
next simplest excitations, which are states having two holes (spinons) within the ground
state configuration together with a single negative parity one-string (or, for the XXX
chain, an infinite rapidity). At finite but small field, the contributions from these states
dominates Szz(q,ω) for q ≫ πσ and allows to smoothly recover the zero field behavior.
A full discussion of all the possible lineshapes as a function of ∆ and σ together with
the characterization of the dominant families of excitations is accessible from the results
of [21], but is beyond the scope of the present paper.
6. High-frequency tail
We now turn to the calculation of Szz(q,ω) in the frequency range γq≪ ω − vq ≪ J,
where finite order perturbation theory is expected to be valid. This off-shell spectral
weight is possible because the η+interaction allows for two-boson intermediate states
with total momentum q = q1+ q2but energy ω = v |q1| + v |q2| > v|q| if sign(q1) =
−sign(q2). In other words, the incoming boson can decay into one right-moving and
one left-moving boson, which together can carry small momentum but high energy
ω ≫ v|q|. In the limit ∆ ≪ 1, this is equivalent to a state with two particle-hole pairs
created around the two different Fermi points [27]. In this sense, our η+is analogous
Page 43
Dynamical structure factor at small q for the XXZ spin-1/2 chain43
δχ
=+ + 2
Figure 18. Diagrams at O(η2
+) for the calculation of the tail.
to the Uqinteraction in [28]. We should stress that, although the tail carries a small
fraction of the spectral weight of Szz(q,ω), it is important for response functions that
depend on the overlap of two spectral functions, e.g. the drag resistivity in the fermionic
version of the problem [27]. In our formalism the calculation of the tail provides a direct
quantitative check of the accuracy of the low energy effective model against the form
factors calculated by Bethe Ansatz.
6.1. Field theory prediction
The lowest-order correction to χ(q,iωn) due to the η+interaction is
?L
where δχ(x,τ) is the correlation function in real space given by
?√2π
×?∂xφ(x)?(∂xϕL(1))2∂xϕR(1) − (R ↔ L)?
δχ(q,iωn) = −
0
dxe−iqx
?β
0
dτ eiωτδχ(x,τ), (6.1)
δχ(x,τ) =K
π
1
26
η+
?2?
d2x1
?
d2x2
×?(∂xϕR(2))2∂xϕL(2) − (R ↔ L)?∂xφ(0).?
δχ(q,iω) =K
2π
where ΠRL(q,iω) is the bubble with right- and left-moving bosons
(6.2)
This corresponds to the diagrams in figure 18. δχ can be factored in the form
?
D(0)
R(q,iω) + D(0)
L(q,iω)
?2
ΠRL(q,iω),(6.3)
ΠRL(q,iω) = −2πη2
+
9
?+∞
?+∞
L(q − k,iω − iν).
−∞
dxe−iqx
?β
dν
2πD(0)
0
dτ eiωτD(0)
R(x,τ)D(0)
L(x,τ)
= −2πη2
+
9
× D(0)
−∞
dk
2π
?+∞
−∞
R(k,iν)
(6.4)
After integrating over frequency, we get
Πret
RL(q,ω) = −η2
+
9
??Λ
dk
0
dk
k(q + k)
ω + vq + 2vk + iη
k(q − k)
ω + vq − 2vk + iη
+
?Λ
q
?
, (6.5)
Page 44
Dynamical structure factor at small q for the XXZ spin-1/2 chain 44
where Λ ∼ kF is a momentum cutoff. Note that the real part of Πret
divergent, but the imaginary part is not. The integration over the internal momentum
yields
?Λ2
Finally, using Eqs. (6.3) and (2.4), we find that the high-frequency tail of Szz(q,ω)
is given by
δSzz(q,ω) =Kη2
18v
This is the same ω−2dependence obtained for weakly interacting fermions with parabolic
dispersion [27]. Since the small parameter is η+∼ ∆/m, we approach the perturbative
regime either by ∆ → 0 or m → ∞ (more precisely, q/mv → 0). In this limit, our result
(6.7) agrees with equation (19) of [27] if we use (3.45) and U (q) = (∆/2)cosq.
Since our model predicts that δSzz(q,ω ≫ vq)
consequence is that there will be no tail in S (q,ω) for models where the Luttinger
parameter K is independent of particle density, since then η+= 0 according to (3.70).
This is the case for the Calogero-Sutherland model, where K is a function of the
amplitude of the long-range interaction only [50].
The divergence of the high-frequency tail of δSzz(q,ω) as ω → vq confirms that
the on-shell region is not accessible by our standard perturbation theory in the band
curvature terms. The matching of the tail to the on-shell peak at ωU(q) is a complicated
problem that has only been addressed in the regime ∆ ≪ 1 (see [28]). The (ω − vq)−1
divergence in (6.7) comes from the frequency dependence of the external legs in the
diagrams of figure 18. It is easy to see that if the bosonic propagators are replaced
by the “dressed” propagator (all orders in η−) given by (5.12), the singularity at the
upper threshold ωU(q) becomes only logarithmic. This supports the picture that the
η+interaction only modifies the shape of the on-shell peak very close to the edges. We
expect that η+ will contribute to the exponent of the singularity at the edges, since
the exponent µqderived in [28] picks up corrections of second order in the interaction
between right and left movers, i.e. O(η2
η+does not affect the width to O(q2). Evidence for that is that the perturbation theory
in η+ (second order given by (6.6)) does not generate terms with the same q and ω
dependence as in (5.9) and (5.10). If the frequency dependence is regularized in the
peak region by summing the perturbation theory in η−, the diagrams involving η+are
always suppressed by higher powers of q because of simple kinematics. Inside the peak
the energy of the left moving boson that is put on shell when taking the imaginary
part of χ(q,ω) (as in the “unitarity condition” method used in [35]) has to be of order
δωq= η−q2or smaller, which constrains the phase space for the internal momenta.
RLis ultraviolet-
Πret
RL(q,ω) = −η2
+
92v−ω2− v2q2
8v3
log
?(vq)2− (ω + iη)2
4v2Λ2
??
.(6.6)
+q4
θ(ω − vq)
ω2− v2q2. (6.7)
∼O(η2
+), one interesting
+). As discussed in section 5.1, we believe that
Page 45
Dynamical structure factor at small q for the XXZ spin-1/2 chain 45
1214
16
1820
ωN/2πv
2224
26
2830
0
1×10-5
2×10-5
3×10-5
4×10-5
2v Szz(q,ω)
σ = - 0.1
∆ = 0.25
(a)
1214
16
1820
ωN/2πv
2224
26
28 30
0
1×10-4
2×10-4
3×10-4
4×10-4
5×10-4
2v Szz(q,ω)
σ = - 0.1
∆ = 0.75
(b)
Figure 19. Tail of Szz(q,ω) for q = 2π/50 and δωq ≪ ω − vq ≪ J. The red dots
represent the sum of the numerical F2(q,ω) identified with each energy level predicted
by field theory (c.f. figure 7). The solid line is the field theory result (6.10). The chain
length is N = 600. (a) σ = −0.1, ∆ = 0.25; (b) σ = −0.1, ∆ = 0.75.
6.2. Comparison with numerical form factors
For a finite system with size N, the result for δSzz(q,ω) must be expressed in terms
of the transition probabilities F2(q,ω). If the intermediate bosons carry momenta
q1,2= 2πn1,2/N, such that q1+q2= q ≡ 2πn/N, the energy levels are given by the sum
of their individual energies ω = v|q1| + v |q2|, i.e.,
ωℓ=2πvℓ
N
,ℓ = n + 2,n + 4,... . (6.8)
Page 46
Dynamical structure factor at small q for the XXZ spin-1/2 chain46
Thus field theory predicts a uniform level spacing 4πv/N above the mass shell. It is
easy to verify (by simply replacing the integrals by sums in momentum space) that
δSzz(q,ω) for the finite system can be written as
δSzz
?
q =2πn
N
,ω
?
=2π
N
?
ℓ
F2(q,ω)δ (ω − ωℓ), (6.9)
where F2(q,ω) =???0??Sz
q
??α???2, with |α? a two-boson intermediate state, is the transition
F2(q,ω) = 2v δSzz(q,ω) =4π2Kη2
9v2N2
We compare our field theory prediction with the form factors calculated numerically
for a chain with N = 600 sites. We take q = 2π/50 (n = 12) and the previous values
σ = −0.1 and ∆ = 0.25 or ∆ = 0.75 (for which the parameters are shown in table 1). As
we saw in figure 7, the energies of the eigenstates calculated by BA are actually scattered
around the values of ωℓpredicted in (6.8). The broadening becomes comparable with
the level spacing 4πv/N when ℓ ≈ 30 (ω ≈ 0.4J). Again the number of states agrees
with a picture of multiple particle-hole excitations based on perturbation theory in the
interaction. These features are not predicted by the bosonization approach. In order
to make the comparison with (6.10), we group the form factors that can be identified
with a given energy level ωℓand plot the total F2(q,ω) as a function of the integers
ℓ = ωN/2πv. We emphasize that for very large ℓ we expect deviations from the lowest-
order field theory result due to the effect of more irrelevant operators we have neglected.
The results are shown in figure 19.
probability for the state with energy ωℓand is given by
+
n4
ℓ2− n2. (6.10)
7. The zero field case
So far we have focused on the dynamical structure factor at finite magnetic field, which
is somewhat analogous to interacting fermions with parabolic dispersion. One may then
ask whether the field theory calculations can be applied to the case h = 0 (kF= π/2).
Let us first review what is known for the free fermion point ∆ = 0. In this case Szz(q,ω)
is still given by (2.19), but the thresholds of the two-particle (two-spinon in the Bethe
Ansatz solution) continuum are given by
ωL(q) = J sinq,
ωU(q) = 2J sinq
(7.1)
2. (7.2)
As a result, Szz(q,ω) develops a square root divergence at the upper threshold ωU(q).
The width now scales like q3for small q
δωq≈Jq3
8
A crossover from q2to q3is observed as we decrease the magnetic field (or, equivalently,
increase q ≪ kF) so as to violate (2.15) or (4.9). The result (7.3) is also obtained by
.(7.3)
Page 47
Dynamical structure factor at small q for the XXZ spin-1/2 chain47
keeping the leading correction to the linear dispersion around kF
?
where γ = vF= J. Bosonizing the band curvature term according to (3.37), we find
δHbc= −πγ
which can be rewritten as
δHbc= −πγ
as follows from the operator product expansion of (7.6).
In the interacting case we also have to keep track of the irrelevant interaction terms,
including the Umklapp interaction in (3.4). The general form for the leading irrelevant
operators for zero field is [41, 32]
ǫR,L
k
≈ ±vFk −γk3
6
+ ...
?
,(7.4)
12: (∂xφR)4: −γ
24:?∂2
xφR
?2: +(R → L), (7.5)
12: (∂xφR)2:: (∂xφR)2: +(R → L), (7.6)
δH =πζ−
12
+πζ+
?: (∂xϕR)2:: (∂xϕR)2: + : (∂xϕL)2:: (∂xϕL)2:?
2
: (∂xϕR)2:: (∂xϕL)2: +λ1
2πcos(4√πKφ) + ..., (7.7)
where the dots stand for higher dimensional local counterterms. The coupling constants
to first order in ∆ can be obtained from the bosonization of the band curvature term
and the irrelevant interaction terms. We find
?
The exact coupling constants for finite ∆ can be taken from [41]
ζ−≈ −J1 +∆
π
?
,ζ+≈ −∆J
π,
λ1≈∆J
π.
(7.8)
ζ−= −
v
4πK
Γ?
Γ?
6K
4K−2
3
4K−2
?
?Γ3?
?Γ3?
πK
2K − 1
?
1
4K−2
2K
4K−2
,
?
?, (7.9)
ζ+= −
v
2πtan
?
(7.10)
λ1= −
4vΓ(2K)
Γ(1 − 2K)
Γ?1 +
1
4K−2
?
2√πΓ?1 +
K
2K−1
?
?4K−2
,(7.11)
where v and K are given by (3.16) and (3.17), respectively.
One important point is that the other possible type of dimension-four operator
(∂xϕR)3∂xϕL+ R ↔ L is absent from the effective Hamiltonian for the XXZ model.
We see this directly when calculating the coupling constants to first order in ∆, but
we can also show that it remains true for finite ∆ by imposing the constraint that the
XXZ model is integrable [32]. Integrability implies the existence of nontrivial conserved
quantities, the simplest one of which is the energy current operator JE=?
JE= J2?
+∆(Sz
jjE
jgiven
by [51, 52]
j
?Sy
j−1Sx
j−1Sz
jSx
j+1− Sx
j−1Sz
jSy
j+1+ ∆(Sx
j−1Sy
jSz
j+1− Sz
j−1Sy
jSx
j+1)
jSy
j+1− Sy
j−1Sx
jSz
j+1)?.(7.12)
Page 48
Dynamical structure factor at small q for the XXZ spin-1/2 chain48
The latter is defined by the continuity equation of the energy density at zero field
jE
j+1− jE
jHjis the Hamiltonian (2.1) with h = 0. One can then verify that JEis
conserved in the sense that [JE,H] = 0.
Let us now look at the corresponding quantity in the low energy effective model.
In the general case, we consider the Hamiltonian density H = HLL+ δH + δH3, where
δH is given by (7.7) and we also add the interaction
?(∂xϕR)3∂xϕL+ (∂xϕL)3∂xϕR
We obtain the energy current operator from the continuity equation in the continuum
limit
?
The energy current operator for the Luttinger model (with ζ±,3= λ1 = 0) takes the
form
?
= − v2
This coincides with the spatial translation operator of the Gaussian model [53]. A
nontrivial consequence of the conservation law arises when we consider the corrections
to JEdue to the irrelevant operators. We keep corrections up to operators of dimension
four. Using (7.15), we find JE= JE
?ζ−
+2ζ3
?(∂xϕR)3∂xϕL− (∂xϕL)3∂xϕR
Note that there are no first-order corrections to JEassociated with the ζ+interaction
or the Umklapp scattering λ1. (The case of the Umklapp perturbation was discussed in
[54]). The conservation of JEup to dimension-four operators implies
j= −∂tHj= i[Hj,H], (7.13)
where H =?
δH3= πζ3
?. (7.14)
∂xjE(x) = −∂tH(x) = idy[H(x),H(y)]. (7.15)
JE
0=
dxjE
0(x) =v2
2
?
dx?(∂xϕR)2− (∂xϕL)2?
?
dx∂xφ∂xθ.(7.16)
0+ δJEwith [32]
δJE= πv
?
dx
3
?(∂xϕR)4− (∂xϕL)4?
??.(7.17)
[JE,H] = [JE
0,HLL] + [JE
0,δH] + [JE
0,δH3] + [δJE,HLL] = 0. (7.18)
Since JE
commutes with any local operator of the form
conditions [53]. As a result, [JE
with the condition that the commutator [δJE,HLL] vanishes. This is automatically
satisfied by the contribution from the ζ−term because it does not mix R and L and
?(∂xϕR)4− (∂xϕL)4,HLL
[δJE,HLL] = πv2ζ3
dx dx′×
?(∂xϕR)3∂xϕL− (∂xϕL)3∂xϕR,(∂x′ϕR)2+ (∂x′ϕL)2?
= 4πiv2ζ3
dx?(∂xϕR)3∂2
0 is conserved in the Luttinger model, we have [JE
0,HLL] = 0. In fact, JE
?dxO(x) under periodic boundary
0
0,δH] = [JE
0,δH3] = 0 as well. We are left
?is a total derivative. We then have
??
?
xϕL+ (∂xϕL)3∂2
xϕR
?.(7.19)
Page 49
Dynamical structure factor at small q for the XXZ spin-1/2 chain49
Therefore, [JE,H] = 0 ⇔ ζ3= 0.
This argument also applies to the finite field case. The model is still integrable for
h ?= 0. Although the relevant quantity for thermal transport is now a linear combination
of the energy current and the spin current operator (which is not conserved for the XXZ
model), the energy current operator given by (7.12) commutes with the Hamiltonian
(2.1) for all values of h [52, 55]. The corresponding conserved quantity in the low energy
theory is the current operator JEobtained from the effective Hamiltonian at zero field,
which has no dependence on the coupling constants η±. Clearly, [JE
for δH(h ?= 0) given by (3.43), so integrability poses no constraints on the coupling
constants η±.
We have checked that ζ3?= 0 for a nonintegrable model obtained by adding to the
XXZ model the following next-nearest neighbour interaction
δHnnn= J∆′?
which is mapped by bosonization onto
?
The first term in (7.21) is quadratic in the bosons and modifies the velocity and the
Luttinger parameter of the Luttinger model. The second term is the irrelevant operator.
To first order in ∆ and ∆′, we find that it gives rise to a ζ3term in the Hamiltonian,
which is given by
δHnnn∼ −29
This shows that, unlike the XXZ model, a low energy effective model describing a
nonintegrable model must in general contain the ζ3interaction.
This result establishes a connection between integrability and the field theory
approach, by means of a restriction on the coupling constant of a band curvature type
operator in the low energy effective model. More generally, if we keep more irrelevant
operators in the effective Hamiltonian, integrability should manifest itself as a fine tuning
of the coupling constants and the absence of certain perturbations. This connection may
be important for understanding the role of integrability in the transport properties of
one-dimensional systems [51].
With ζ3= 0, only ζ+and λ1mix right and left movers. We can apply second order
perturbation theory in these interactions to calculate two contributions to the high-
frequency tail in the frequency range δωq≪ ω −vq ≪ J [32]. For a finite chain with N
sites and fixed momentum q = 2πn/N, the ζ+operator gives rise to intermediate states
with discrete energies ωℓ= 2πvℓ/N, ℓ = n + 2,n + 4,.... In the thermodynamic limit,
the contribution to the tail is
ζ+(q,ω) =K(ζ+/v)2
192v
The states generated by the Umklapp operator have energies ωℓ = 2πv(ℓ + 4K)/N,
ℓ = n,n + 2,... . For 4πv/N ≫ δωq it is easy to separate this contribution from
0,δH(h ?= 0)] = 0
j
Sz
jSz
j+2,(7.20)
δHnnn= J∆′
dx
?
−3
π
?
∂x˜φ
?2
+16
3
?
∂x˜φ
?4
+ ...
?
. (7.21)
6J∆′?(∂xϕR)3∂xϕL+ (∂xϕL)3∂xϕR
?. (7.22)
δSzz
q2
?ω2− v2q2
v2
?
θ(ω − vq). (7.23)
Page 50
Dynamical structure factor at small q for the XXZ spin-1/2 chain 50
the ζ+one because of the shift in the energy levels by the noninteger factor 4K. The
corresponding contribution to the tail is
δSzz
λ1(q,ω) =2λ2
1K2
Γ2(4K)(2v)3−8Kq2?ω2− v2q2?4K−3θ(ω − vq). (7.24)
The derivation of equations (7.23) and (7.24), as well as the result for the finite system,
is presented in the appendix.
The result in (7.23) and (7.24) shows that the Umklapp operator (dimension
4K) yields the dominant contribution to the tail near ω ∼ vq. For 0 < ∆ < 1/2
(3/4 < K < 1), the next-leading contribution is given by ζ+ (dimension 4).
1/2 < ∆ < 1 (1/2 < K < 3/4) it is important to include in the effective Hamiltonian
the operator
δHλ2= λ2∂xθcos(4√πKφ),
which is a descendant of the Umklapp operator and is allowed by all symmetries. This
operator has dimension 4K + 1 and is less irrelevant than ζ+for K < 3/4. Another
reason to include λ2is that for K ≤ 3/4 the exact amplitude ζ+in (7.10) diverges at the
points K = 1/2 + 1/(4n), n ≥ 1. This divergence has been discussed in the context of
corrections to the bulk and boundary susceptibility of the open XXZ chain [46]. There
it was found that the susceptibility as a function of magnetic field or temperature has
a correction of first order in ζ+. However, the corrections at any given order of h or
T are always finite because the divergences of ζ+at the points K = 1/2 + 1/(4n) are
cancelled by the contribution from the Umklapp operator and the cancellation gives
rise to logarithm corrections. In our case the tail (7.23) is of order ζ2
cancelled by a more irrelevant operator with a K-dependent dimension. Note also that
the O(λ2
frequency tail for all values of 0 < ∆ < 1 it is necessary that the amplitude of the terms
generated by λ2also diverge (have poles) at the above values of K. By simple power
counting, we expect that the divergence of the ζ2
cancelled by the term of second order in λ2and 2(n − 1)-th order in λ1, which scales
like δSzz∼ λ2(n−1)
q and ω dependence as in (7.23) and the cancellation is thus possible. In principle it is
possible to determine the amplitude λ2from the Bethe Ansatz, but that would require
solving the Wiener-Hopf equations to higher orders than was done in [46].
Computing the broadening δωq for h = 0 from bosonization is much more
challenging. Since ζ− is the only vertex present at the free fermion point, the naive
expectation is that we could derive the renormalization of the width at zero field by
summing all orders of ζ−, as we did for η−in section 5.1. Although we now have to
deal with a four-legged vertex, which introduces three-boson intermediate states, the
calculation of the lowest order diagrams is not much harder than the finite field case.
However, the fundamental difference is that for h = 0 the broadening has to be produced
by dimension-four operators and is therefore of the same order of q as the changes in the
lineshape (i.e., the density of states factor and the singularities near the thresholds).
For
(7.25)
+, so it must be
1) term in (7.24) does not diverge. Therefore, in order to recover a finite high-
+term at K = 1/2 + 1/(4n) will be
1
λ2
2q2(ω2− v2q2)n(4K−2). For n(4K − 2) = 1, this term has the same
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