Physics
Physics Research Publications
Purdue University Year
Theory of electron nematic order in
LaFeAsO
C. Fang H. Yao W. F. Tsai
J. P. Hu S. A. Kivelson
This paper is posted at Purdue e-Pubs.
http://docs.lib.purdue.edu/physics articles/743
Theory of electron nematic order in LaFeAsO
Chen Fang,
1
Hong Yao,
2
Wei-Feng Tsai,
2,3
JiangPing Hu,
1
and Steven A. Kivelson
2
1
Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
2
Department of Physics, Stanford University, Stanford, California 94305, USA
3
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
共Received 26 April 2008; published 20 June 2008
兲
We study a spin S quantum Heisenberg model on the Fe lattice of the rare-earth oxypnictide superconduct-
ors. Using both large S and large N methods, we show that this model exhibits a sequence of two phase
transitions: from a high-temperature symmetric phase to a narrow region of intermediate “nematic” phase, and
then to a low-temperature spin ordered phase. Identifying phases by their broken symmetries, these phases
correspond precisely to the sequence of structural 共tetragonal to monoclinic兲 and magnetic transitions that have
been recently revealed in neutron-scattering studies of LaFeAsO. The structural transition can thus be identified
with the existence of incipient 共“fluctuating”兲 magnetic order.
DOI: 10.1103/PhysRevB.77.224509 PACS number共s兲: 71.27.⫹a, 71.10.⫺w, 74.25.Ha
I. CONTEXT
Of course, the big issue of the day is whether the physics
of high-temperature superconductivity
1–6
in the rare-earth
oxypnictides is related to that in the cuprates. In favor of this
association is the observation that both are “bad metals,”
7
and so presumably not well described by Fermi-liquid theory
in their normal states. Preliminary evidence
8,9
suggests that
the superconducting state in the oxypnictides, like that in the
cuprates, has gapless nodal quasiparticle excitations, and
hence, probably, an unconventional pairing symmetry. Fi-
nally, there is tantalizing evidence that competing ordered
states, and possibly an associated quantum critical point,
may play a role in both cases.
5,10,11
In the case of the cuprates, superconductivity is produced
by doping a commensurate spin ordered, insulating parent
“Neel” state. Probably, Neel order does not coexist with su-
perconductivity; however, other ordered states, including
spin and charge stripe 共unidirectional density wave兲 ordered
states,
12,13
an Ising nematic state
13–15
共about which, more
later兲 and a form of time-reversal symmetry-breaking
order
16,17
共whose character is still being debated兲 seem to
coexist 共in some cases, at least兲 with superconductivity, and
possibly to vanish at quantum critical points somewhere un-
der the superconducting dome.
The oxypnictides in question have chemical makeup
RO
1−x
F
x
FeAs, where R is a rare earth, and x is the dopant
concentration; the behavior 共including the maximum super-
conducting T
c
兲 depends systematically on the particular
choice of R. The situation with competing orders in the ox-
ypnictides is only beginning to be explored. Undoped RO-
FeAs is not cleanly insulating, but its resistivity is strikingly
large 共e.g.,
⬃7m⍀-cm in SmOFeAs at room
temperature
10,18
兲 for a metal; it does, however, exhibit 共in
neutron-scattering experiments on LaFeAsO
11,19
兲 commensu-
rate spin order below T
SDW
=135 K. Moreover, a closely as-
sociated structural transition, which we wish to identify as
the transition to an “electron nematic phase,”
20
occurs at the
slightly higher temperature, T
N
=150 K.
11
The evolution of
these orders as a function of doping, x, has not yet been
directly probed with neutrons. However, the sharp onset of
an anomalous drop in the resistivity occurs at T
=T
N
in the
undoped x=0 material. The onset temperature for the resis-
tance drop has been tracked in resistivity measurements for
different x, and found to extrapolate to 0 at a critical value of
x close to the point at which the superconducting T
c
first
reaches its maximum value max关T
c
兴=55 K in
SmO
1−x
F
x
FeAs.
10
Assuming that the association between T
and T
N
persists, this means that there is an electron nematic
to isotropic quantum phase transition in the superconducting
dome in at least some members of both families of high-
temperature superconductors, a suggestive evidence both of
a common thread in the behavior of both materials, and of
the conjecture that nematic order is, in some way, a crucial
part of the physics.
II. INTRODUCTION
The purpose of this paper is to propose a unified perspec-
tive on the occurrence of both the magnetic and the structural
phase transitions in undoped LaFeAsO. To the extent that
this proposal is correct, it justifies the identification of the
observed structural transition with the occurrence of an elec-
tron nematic phase.
Following the lead of two insightful recent papers by
Yildirim
21
and by Si and Abrahams,
22
we will treat undoped
ROFeAs as if it were a magnetic 共“Mott”兲 insulator,
23
and
hence we consider a simple model of localized spins on the
iron sites interacting with neighboring spins by an antiferro-
magnetic superexchange interaction mediated through the in-
tervening As atoms 共Eq. 共3.1兲, below兲. Some microscopic
justification for this approach is contained in those two ear-
lier papers. In addition, the fact that the magnetic order
11
in
ROFeAs is commensurate is, a priori, indicative of strong-
coupling physics. Moreover, although the bare susceptibility
computed
24–27
from the LDA band structure is peaked at the
appropriate ordering vector, Q, the Fermi surface is not well
nested and this peak is neither pronounced nor sharp.
However, it is important to acknowledge from the start
that a model of localized spins cannot be taken as a realistic
representation of the electronic structure of ROFeAs. The
PHYSICAL REVIEW B 77, 224509 共2008兲
1098-0121/2008/77共22兲/224509共6兲 ©2008 The American Physical Society224509-1
most obvious point is that ROFeAs is not an insulator! At
best, it is our hope that the magnetic and structural properties
of this material can be qualitatively understood on the basis
of the present model. Even there, we will show that the small
magnitude of the ordered moment, m =0.35
B
, at low tem-
perature is inconsistent with the predictions of the present
model. Indeed, as pointed out by Si and Abrahams, it is not
even clear whether we should be considering a spin S=2 or
S=1 model, in this context. Still, the model is sufficiently
simple that its behavior can be cleanly derived. As we shall
see, it inevitably exhibits two ordered phases 共see Fig. 2兲 of
precisely the character of those seen in experiment. More-
over, numerous spectroscopic and a few thermodynamic pre-
dictions 共see remarks in Sec. VI兲 can be made on the basis of
this model that are sufficiently robust, which we may hope
they transcend the deficiencies of the model.
III. THE MODEL
In the tetragonal phase, the iron sites form square planar
arrays, such that the sites of adjacent planes lie above one
another. Because the superexchange is mediated through off
plane but plaquette centered As atoms, the first- and second-
neighbor antiferromagnetic exchange couplings, J
1
and J
2
,
are expected to be of roughly the same magnitude. However,
the coupling between spins on neighboring planes, J
z
, while
still antiferromagnetic, is expected to be much smaller than
the in-plane couplings. 共See Fig. 1兲 Estimates from previous
work
21,28,29
are J
1
⬇0.5J
2
⬇400–700 K. J
z
is several orders
of magnitude smaller than J
2
. The resulting minimal Hamil-
tonian is
H =
兺
n,R,
␦
1
关J
1
S
ជ
R,n
· S
ជ
R+
␦
1
,n
− K共S
ជ
R,n
· S
ជ
R+
␦
1
,n
兲
2
兴
+ J
2
兺
n,R,
␦
2
S
ជ
R,n
· S
ជ
R+
␦
2
,n
+ J
z
兺
n,R
S
ជ
R,n
· S
ជ
R,n+1
, 共3.1兲
where S
R,n
is a spin S operator on site R in plane n and,
␦
1
and
␦
2
are, respectively, the first and second-nearest-
neighbor lattice vectors in square plane. We set the lattice
spacing between nearest neighboring iron sites to be 1. The
biquadratic interaction term, K, is small for well localized
spins, but even if we were to omit this term in the bare
Hamiltonian, it would rise from quantum or thermal fluctua-
tion through the so-called “order out of disorder”
mechanism
28,30–33
in the long-wavelength limit. Note that
this Hamiltonian has the C
4
lattice rotational symmetry of
the high-temperature tetragonal phase.
In the broken-symmetry “nematic” phase, the spin nem-
atic order parameter
N ⬅具
兺
␦
1
F
d
共
␦
1
兲S
ជ
R,n
· S
ជ
R+
␦
1
,n
典共3.2兲
is nonzero, where F
d
is a “d-wave” form factor, F
d
共⫾x
ˆ
兲=
−F
d
共⫾y
ˆ
兲=1. Since a structural distortion of appropriate sym-
metry is linearly coupled to the spin nematic order param-
eter, the magnitude of the structural distortion u will be pro-
portional to N in the presence of weak electron-lattice
coupling. It is a central conclusion of our work that the struc-
tural distortion is a response to a purely electronic pattern of
symmetry breaking, so we will typically take u=0, although
in some places, we will consider the effects of a small per-
turbation
H
⬘
= J
N
兺
n,R,
␦
1
关F
d
共
␦
1
兲S
ជ
R,n
· S
ជ
R+
␦
1
,n
兴共3.3兲
where J
N
⬀u.
There is a subtlety of the crystal structure that is not ap-
parent in the model presented in Eq. 共3.1兲: Because of the
presence of a glide plane, the spins on the Fe sites are all
equivalent, so in the model, there appears to be only one
atom per unit cell. Thus, the distortion that produces a non-
zero u looks to be an orthorhombic distortion, in which the
elementary square plaquette become slightly rectangular.
However, because of the three dimensional placement of the
As atoms out of the Fe plane, there are actually two Fe sites
per unit cell, and consequently the correct classification of
the low-symmetry phase is monoclinic. Because it simplifies
the discussion, we will henceforth ignore this subtlety, and
phrase our discussion on an idealized crystal structure with a
single Fe atom per unit cell, as is appropriate to the magnetic
Hamiltonian in Eq. 共3.1兲.
We will consider this model in the limit J
2
⬎J
1
/ 2ⰇJ
z
,
K⬎0. We shall derive T=0 properties of this model to low-
est order in a spin-wave 共1/ S兲 expansion. To treat the finite
temperature properties of the model, we consider S
ជ
to be an
N dimensional unit vector 关SO共N兲 spin兴 and obtain a system-
atic solution to the problem in the large N limit. It is gener-
ally found that the spin-wave theory is accurate
34
at T=0,
even in the limit S =1/ 2, so it should be quite reliable in the
present case. Similarly, the physical N =3 typically is well
approximated by large N.
IV. ZERO TEMPERATURE (LARGE S)
We can compute the properties of the system described in
Eq. 共3.1兲 using the standard 共Holstein–Primakov兲 spin-wave
a
b
c
z
J
1
J
2
J
y
x
z
FIG. 1. 共Color online兲 Schematic graph for the proposed model
with nearest-neighbor coupling J
1
, next-nearest-neighbor coupling
J
2
and interlayer coupling J
z
. The orientation of the spins in the
low-temperature phase are drawn according to Ref. 11. Note that we
use coordinate system with axis x, y, and z in the current study,
which is 45° rotated along the c =z direction from the realistic crys-
tal axis a, b, and c.
FANG et al. PHYSICAL REVIEW B 77, 224509 共2008兲
224509-2
theory. We compute the classical expression and the leading
corrections to order 1/ S.
Since at T=0, the small interplane coupling J
z
does not
qualitatively effect the magnetic properties of the system, we
will simplify expressions for various quantities by evaluating
them in the limit J
z
→ 0. For J
1
⬎2J
2
ⰇK⬎0, the classical
共S → ⬁兲 ground state is the Neel state with ordering wave
vector Q=共
,
兲. For J
2
⬎J
1
/ 2ⰇK⬎0, the regime of inter-
est in the present study, the classical ground state is the
“striped” phase with ordering vector Q=共
,0兲, as shown in
Fig. 1,or共0,
兲. Note that, in addition to breaking spin-
rotational symmetry and time-reversal symmetry, this state
共even in the absence of any spin-orbit coupling兲 spontane-
ously breaks the lattice symmetry. Specifically, we compute
the antiferromagnetic and the nematic order parameters,
m = e
iQ·R
具S
ជ
R,n
典 = S −
␣
+ ...
N = 具
兺
␦
1
F
d
共
␦
1
兲S
ជ
R,n
· S
ជ
R+
␦
ˆ
1
,n
典 =−4关S
2
−

S + ...兴共4.1兲
where
␣
and

are dimensionless functions of J
2
/ J
1
that we
compute below. Here, we have taken the ordering vector Q
to be 共
,0兲 as in shown Fig. 1, which means that the nearest-
neighbor bonds along the x direction are satisfied, but the
y-directed bonds are ferromagnetic, and “frustrated.”
The spin-wave spectrum to leading order in S is given by
k
2
=4S
2
关共J
y
cos k
y
+2J
2
+ J
x
− J
y
兲
2
− 共J
x
+2J
2
cos k
y
兲
2
cos
2
k
x
兴, 共4.2兲
where J
y
⬅J
1
−2J
N
−2KS
2
and J
x
⬅J
1
+2J
N
+2KS
2
. As ex-
pected, the spectrum is gapless at k =共0,0兲 and k =Q
=共
,0兲. These are the Goldstone modes, which have an an-
isotropic linear dispersion,
k
⬇ 2S
冑
共2J
2
+ J
x
兲关共2J
2
+ J
x
兲q
x
2
+ 共2J
2
− J
y
兲q
y
2
兴, 共4.3兲
in which q is the small deviation from the gapless points.
The spectrum is also almost gapless at k=Q
⬘
=共0,
兲 and k
=Q
⬙
=共
,
兲, where the gap,
⌬ =4S
冑
共J
x
− J
y
兲共2J
2
− J
y
兲共4.4兲
is determined by the small terms in the Hamiltonian. The
ordered state can be thought of as two interpenetrating Neel
states, which at the classical level do not lock to each other.
It is the small terms, K and J
N
, that lock them together in a
collinear state, and gap what would otherwise be two inde-
pendent sets of Goldstone modes. Notice that this gap also
vanishes at the critical coupling J
2
→ J
1
/ 2. This is somewhat
surprising, as the striped to Neel transition might otherwise
be expected to be first order. 共It is an open question, which
we will not address at present, whether there is an interest-
ing, unconventional
35
quantum critical point, here, or possi-
bly some additional intermediate zero-temperature phases
stabilized by quantum fluctuations.兲
At the same level of approximation, we can compute the
leading-order quantum corrections to the sublattice magneti-
zation and nematic order parameters,
␣
and

in Eq. 共4.1兲.To
simplify matters, we compute both quantities in the limit K
and J
N
→ 0, since these small couplings make only negligible
differences in the results. Then
␣
=
1
2
冋
冕
d
2
k
共2
兲
2
2共J
1
cos k
y
+2J
2
兲
k
/S
−1
册
, 共4.5兲
and

=2
␣
−
冕
d
2
k
共2
兲
2
J
1
共cos
2
k
x
+ cos
2
k
y
兲
k
/S
−
冕
d
2
k
共2
兲
2
2J
2
cos k
y
共1 + cos
2
k
x
兲
k
/S
. 共4.6兲
These integrals are readily evaluated numerically. For in-
stances, for J
2
/ J
1
=2.0,
␣
=0.20,

=0.30, and for J
2
/ J
1
=1.0,
␣
=0.22,

=0.21. Both
␣
and

diverge as J
2
→ J
1
/ 2, but
only logarithmically,
␣
,

⬃共2
兲
−1
ln关J
1
/ 共2J
2
−J
1
兲兴. Thus,
except extraordinarily close to the quantum critical point,
quantum fluctuations do not significantly reduce the ordered
moment.
The dynamic structure can also be readily computed. The
transverse piece has the form
S
⬜
共k,
兲 = A共k兲
␦
共
−
k
兲, 共4.7兲
where A共k兲=4
S
2
关J
x
共1−cos k
x
−J
y
共1−cos k
y
兲+2J
2
共1
−cos k
x
cos k
y
兲兴/
k
兴. Note that interesting behavior is ob-
served near k=Q
⬘
and k=Q
⬙
: A共Q
⬘
兲
=2
S
冑
共2J
2
−J
y
兲/ 共J
x
−J
y
兲, which is large; however, A共Q
⬙
兲
=2
S
冑
共J
x
−J
y
兲/ 共2J
2
−J
y
兲, which is small. The longitudinal
structure factor has the form of a two spin-wave continuum,
and can also be computed explicitly.
36
V. FINITE T (LARGE N) SOLUTION
To study the phase diagram at finite temperature, and in
particular to gain insight into the regime of fluctuating mag-
netic order above the spin ordering transition temperature, it
is sufficient to treat the problem classically, as the effects of
quantum fluctuations simply produce small renormalizations
of the effective parameters, as above. Since we are interested
in the region J
2
⬎J
1
/ 2, we break the system up into two
interpenetrating square lattices on which J
2
is the nearest-
neighbor coupling. On each sublattice, we define the stag-
gered magnetization
ជ
n,
␣
for plane n and sublattice
␣
=1 or
2. To make our calculations analytically tractable, we take
the continuum limit and for convenience we write the model
with respect to the real crystal axis, i.e., x , y are equivalent to
a,b crystal directions, respectively, in the following model.
So
H
c
=
冕
d
2
r
兺
n,
␣
冋
1
2
J
˜
2
兩ⵜ
ជ
n,
␣
共r兲兩
2
− J
˜
z
ជ
n,
␣
共r兲 ·
ជ
n+1,
␣
共r兲
册
− K
˜
兺
n
关
ជ
n,1
共r兲 ·
ជ
n,2
共r兲兴
2
+ J
˜
1
兺
n
ជ
n,1
共r兲
x
y
ជ
n,2
共r兲,
共5.1兲
where we have used the same symbols with a tilde for the
THEORY OF ELECTRON NEMATIC ORDER IN LaFeAsO PHYSICAL REVIEW B 77, 224509 共2008兲
224509-3
couplings as in Eq. 共3.1兲, although the present quantities
should include renormalizations due to both quantum effects
and high energy thermal fluctuations. J
˜
i
⬃J
i
S
2
,共i =1,2兲 and
J
˜
z
⬃J
z
S
2
. Specially, as we pointed out before, the K
˜
term can
be shown
28
to rise through fluctuations in the following form
K
˜
⬃0.13J
˜
1
2
S
2
/ 4J
˜
2
, which is about 0.03J
˜
2
for S =1 and J
2
=2J
1
, given approximately in Ref. 21 and 29.
We generalize the order parameter to N components, take
the large N limit, and solve the problem. Note that the tran-
sition temperatures are obtained for the physical N=3. The
self-consistency equations can be derived using the same
method employed in Ref. 37. Define the nematic order
=K
˜
具
ជ
n,1
共r兲·
ជ
n,2
共r兲典/ 共NT兲 and
n,
␣
共r兲,
␣
=1,2 are the La-
grangian multiplies for
ជ
n,
␣
共r兲. The saddle point of above
Lagrangian is determined by the following self-consistent
equations where we take
n,
␣
共r兲= ,
␣
=1,2:
=
K
˜
共2
兲
3
·
冕
−⌳
⌳
dk
x
冕
−⌳
⌳
dk
y
冕
0
2
dk
z
G
12
共k
ជ
兲共5.2兲
1=
NT
共2
兲
3
·
冕
−⌳
⌳
dk
x
冕
−⌳
⌳
dk
y
冕
0
2
dk
z
G
11
共k
ជ
兲共5.3兲
where ⌳⬃O共1兲 is momentum cutoff, and
G共k
ជ
兲
−1
=
冉
A共k
ជ
兲
− B共k
ជ
兲
− B共k
ជ
兲
A共k
ជ
兲
冊
, 共5.4兲
where A共k
ជ
兲=J
˜
2
k
2
−2J
˜
z
cos k
z
+2NT, B共k
ជ
兲=2NT
+J
˜
1
k
x
k
y
with k
ជ
=共k ,k
z
兲 and k
2
=k
x
2
+k
y
2
. From these self-consistent
equations, we can determine the transitions temperatures. For
K
˜
,J
˜
z
ⰆJ
˜
2
, the nematic transition temperature T
N
is deter-
mined by the following equation
4
J
˜
2
NT
N
=ln
J
˜
2
/共NT
N
兲
冑
共
K
˜
4
J
˜
2
兲
2
+
共
J
˜
z
NT
N
兲
2
+
K
˜
4
J
˜
2
共5.5兲
The spin-density wave transition temperature T
SDW
takes
place when =
SDW
+J
˜
z
/ 共NT兲. It is determined by the fol-
lowing equations
SDW
K
˜
=
1
8
J
˜
2
ln
2
SDW
+
J
˜
z
NT
SDW
+2
冑
SDW
2
+
SDW
J
˜
z
NT
SDW
J
˜
z
/共NT
SDW
兲
,
SDW
K
˜
+
1
NT
SDW
=
1
4
J
˜
2
ln
J
˜
2
J
˜
z
. 共5.6兲
By solving these equations, we find that the above model has
two second-order phase transitions. The nematic transition
temperature T
N
is always larger than the SDW transition
temperature T
SDW
. In Fig. 2, we show the transition tempera-
tures T
N
and T
SDW
as the function of J
˜
z
for a fixed K
˜
=0.0075J
˜
2
. T
N
is largely insensitive to J
˜
z
so long as it is
small. In Fig. 3, we plot the difference 共T
N
−T
SDW
兲/ T
SDW
as
the function of J
˜
z
. If we compare the experimental result in
LaFeAsO where 共T
N
−T
SDW
兲/ T
SDW
⬃11%, our results sug-
gest J
˜
z
⬃10
−4
J
˜
2
. From the suggested value J
˜
z
⬃10
−4
J
2
,we
derive that T
N
⬃0.61J
˜
2
and T
SDW
⬃0.55J
˜
2
, as shown in Fig.
2. In this parameter region, by increasing J
˜
z
, which can be
achieved by applying external pressure along z axis, the dif-
ference between T
N
and T
SDW
can be reduced.
VI. FINAL REMARKS
The localized spin model we have solved produces results
that have enough in common with the observed ordered
phases of ROFeAs that, we believe, it is clear that it captures
some of the correct physics, as was first proposed by
Yildirim
21
and Si and Abrahams.
22
Both the stripelike pattern
of magnetic order and the existence of a monoclinic 共nem-
atic兲 lattice distortion are found to be inevitable conse-
quences of the magnetic interactions. By analyzing the
model carefully, we have reached some additional conclu-
sions.
共1兲 Most importantly, we find that there is inevitably a
narrow range of temperatures above the magnetic ordering
)10(
3
FIG. 2. 共Color online兲 T
N
and T
SDW
as the function of J
˜
z
for
J
˜
2
=2J
˜
1
, N =3, and S=1.
)10(
3
FIG. 3.
T
N
−T
SDW
T
SDW
as the function of J
˜
z
for J
˜
2
=2J
˜
1
, N=3, and S
=1.
FANG et al. PHYSICAL REVIEW B 77, 224509 共2008兲
224509-4
