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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)
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