arXiv:0901.0238v1 [cond-mat.supr-con] 2 Jan 2009
Non-analytic spin susceptibility of a nested Fermi liquid: the case of Fe-based
M.M. Korshunov,1,2, ∗I. Eremin,1,3D.V. Efremov,4D.L. Maslov,5and A.V. Chubukov6
1Max-Planck-Institut f¨ ur Physik komplexer Systeme, D-01187 Dresden, Germany
2L. V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036 Krasnoyarsk, Russia
3Institute f¨ ur Mathematische und Theoretische Physik,
TU Braunschweig, 38106 Braunschweig, Germany
4Institut f¨ ur Theoretische Physik, TU Dresden, 01062 Dresden, Germany
5Department of Physics, University of Florida, Gainesville, Florida 32611, USA
6Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
(Dated: January 2, 2009)
We propose an explanation of the peculiar linear temperature dependence of the uniform spin
susceptibility χ(T) in ferropnictides. We argue that the linear in T term appears due to non-analytic
temperature dependence of χ(T) in a two-dimensional Fermi liquid. We show that the prefactor of
the T term is expressed via the square of the spin-density-wave (SDW) amplitude connecting nested
hole and electron pockets. Due to an incipient SDW instability, this amplitude is large, which, along
with a small value of the Fermi energy, makes the T dependence of χ(T) strong. We demonstrate
that this mechanism is in quantitative agreement with the experiment.
PACS numbers: 71.10.Ay, 75.30.Cr, 74.25.Ha, 74.25.Jb
uid (FL) theory is that a system of strongly interact-
ing fermions can be considered effectively as a gas of
weakly interacting quasiparticles. In the absence of resid-
ual interaction between quasiparticles, the static uniform
spin susceptibility, χ(T), and the specific heat coefficient,
γ(T), are finite at T = 0 and obey quadratic dependen-
cies on T at low temperatures. The effect of residual
interactions on χ(T) and γ(T) has been studied inten-
sively in recent years , with the key result that in two
dimensions (2D) both γ(T) and χ(T) are linear rather
than quadratic in T [2, 3, 4].
The key hypothesis of the Fermi liq-
Theory predicts that the behavior of γ(T) is univer-
sal in a sense that the (negative) slope is given by the
square of the backscattering amplitude . This linear
decrease has been observed in monolayers of3He . On
the contrary, a linear in T term in the spin susceptibility
is not universal and can be of either sign [3, 7], causing
uncertainty in the interpretation of the experiments on
semiconductor heterostructures .
Recently, a pronounced linear temperature dependence
of the uniform susceptibility has been observed in high-
Tc superconductors with iron-based layered structure
[9, 10, 11, 12]. It extends from temperatures above either
SDW or superconducting transitions up to 500-700Kwith
almost doping-independent slope. The T dependence is
quite strong: χ(T) increases roughly by a factor of 2 be-
tween 200K and 700K. An explanation of this behavior
based on the J1− J2 model of localized spins has been
proposed in Ref. 13; however, given that the linear T de-
pendence persists up to large dopings, where local probes,
such as nuclear magnetic resonance (NMR) and µSR, do
not see localized moments [14, 15], this explanation is
The itinerant character of Fe-pnictides is suggested by
the agreement between the band structure obtained in
ab initio calculations  and observed in de Haas-van
Alphen and angle-resolved photoemission spectroscopy
(ARPES) experiments [17, 18, 19, 20, 21]. It is firmly
established by now that the Fermi surface (FS) of Fe-
pnictides consists of two small hole pockets near (0,0)
and two electron pockets near (π,π) points of the folded
Brillouin zone. In such a system, an obvious origin of
the SDW order is a logarithmic divergence of a particle-
hole vertex involving states on nested parts of the FS
[16, 22, 23, 24, 25, 26].
In this Letter, we propose an explanation of the ex-
perimental T-dependence of spin susceptibility based on
the itinerant picture. We argue that the origin of the
linear increase of χ(T) with temperature in ferropnic-
tides is the same as in a 2D FL. Furthermore, we show
that this behavior is universal for FLs with strong (π,π)
SDW fluctuations, namely, the slope of the linear in T-
dependence is determined by the square of the SDW am-
plitude with nesting momentum Q = (π,π). This am-
plitude is large which, along with a small value of the
Fermi energy εF, amplifies the T dependence of χ(T).
Choosing the SDW coupling to reproduce the observed
SDW transition temperature TNat zero doping, we find a
good agreement between calculated and measured slopes
of χ(T). The doping dependence of the slope also agrees
with the data. We view this agreement as a strong in-
dication that the linear temperature dependence of χ(T)
in pnictides [9, 10, 11, 12] is in fact the first unambigu-
ous observation of a non-analytic behavior of the 2D spin
susceptibility. Besides being fundamentally important on
its own right, this observation also strengthens the case
for the itinerant scenario for Fe-pnictides.
u5 + (2×) u1
u3 + (2×) u2
FIG. 1: Second-order diagrams for the thermodynamic poten-
tial. Solid and dashed lines correspond to f-fermions (elec-
trons) and to c-fermions (holes), respectively.
Theory. We consider a two-band model of interacting
fermions occupying the electron and hole FSs:
Hint = u1c†
Here, ckσ (fkσ) is the annihilation operator for a hole
(electron) with momentum k and spin σ (for an electron,
k is measured from the (π,π)-point), εc
2µ represent single-particle dispersions, and µ measures a
deviation from perfect nesting. Models of this type were
considered in the past in the context of an “excitonic
We assume that each of the electron and hole FSs is
doubly degenerate. The terms with u4and u5are intra-
band interactions, the terms with u1 and u2 are inter-
band interactions with momentum transfer 0 and Q, re-
spectively, and the term with u3 is the inter-band pair
hopping. All couplings flow from their initial values at
energies of order of the bandwidth to renormalized values
at εF [24, 25]. We assume that this renormalization is
already included into Eq. (1) and analyze the behavior of
the system at energies below εF, where the actual values
of momenta become relevant.
We first obtain the linear-in-T contribution to the spin
susceptibility, δχ(T), to second order in the interaction
and then show that the prefactor of the T term in δχ(T)
is expressed via the SDW vertex to all orders in the in-
Fig. 1 depicts all topologically inequivalent second-
order diagramsfor the
Φ(T,H), describing both intra- and inter-band processes.
Each of the diagrams contains the object
ϕ(T,H) = T3?
f fermion with momentum k and spin-σ in a magnetic
field H. As in an ordinary FL, the non-analytic H2T
term in ϕ(T,H) comes from a dynamic Kohn anomaly,
i.e. from diagrams with momentum 2kF carried by the
interaction lines [3, 4].It is, however, more conve-
nient to re-express the result via the polarization bub-
bles with small rather than 2kF momenta. The Green
functions in Eq. (2) can be combined in two different
ways: either as Πσσ
the cc, ff, or cf type with small momenta q ≪ kF. A
bubble depends on the magnetic field if the Zeeman en-
ergies of two fermions add up. One can readily show that
the cc and ff bubbles depend on the field via Π↑↓
the field enters the cf bubble through Π↑↑
terms. Evaluating individual diagrams, we find that each
of them can be expressed as a product of two dynamic
spin up/down bubbles:
k,ωis a Matsubara Green’s function of either c or
q,Ω, or as Πσσ′
k+q,ω+Ωis a polarization bubble of
q,Ω, where Πσσ′
?(Ω − 2iµBH − 2iµ∗)2+ (vFq)2,
where µ∗= 0 for intra-band scattering and µ∗= µ
for inter-band scattering. The rest of the calculations
proceeds in the same way as for an ordinary FL [3, 4].
In short, one integrates Π2
q,Ωover q first, replaces the
Matsubara sum by a contour integral, differentiates the
result twice with respect to H, sets H = 0, and ob-
tains the O(T) terms in χ. The first two diagrams in
Fig. 1 describe intra-band processes and give the same
results for χ(T) as in Refs. [3, 4]: δχ1,2(T) = λu2
where λ = 4χ0N2
is the Pauli susceptibility per one sheet of the FS, and
NF = m/2π is the density of states. A factor of 4 in
λ results from the double degeneracy of electron and
hole bands.The third through fifth diagrams involve
inter-band scattering and yield δχ3(T) = 2λu2
δχ4(T) = λu2
spectively. Here, η(x) = 2xcothx−x2/sinh2x−2x. The
third and forth diagrams give finite contributions if σ =
σ′, while the fifth diagram contributes if σ = −σ′. The
sixth and seventh diagrams do not contribute to δχ(T),
and the remaining two give δχ8(T) = −2λu1u2Tη(µ/T)
and δχ9(T) = −λu2
F/(2εF), εF = vFkF/2, χ0 = 2µ2
3Tη(µ/T), and δχ5(T) = 2λu2
Combining all diagrams, we obtain
δχ(T) = λT?u2
The first two terms are the contributions from intra-
band processes – they are the same as in an ordinary
FL. The rest of the terms correspond to inter-band pro-
cesses, specific to the electronic structure of nested fer-
ropnictides. For T ≫ µ (perfect nesting), η(µ/T) ≈ 1
and both intra- and inter-band processes contribute to
the linear term in the spin susceptibility. In the opposite
limit of T ≪ µ (poor nesting), η(µ/T) ∼ exp(−2µ/T)
and the inter-band contribution to χ(T) is suppressed
exponentially. In the intermediate regime T ∼ µ, η(µ/T)
is non-monotonic. Note that the pair-hopping term u3,
which gives rise to an attraction in the extended s-wave
(s±) pairing channel, does not contribute to the T term
The coupling constants in Eq. (4) are interactions at
the scale of the Fermi energy. These couplings are al-
ready renormalized from their bare values at the scale
of the bandwidth  by parquet renormalization group
(RG), in such a way that u4, u5, and u1flow to the same
value, while u2flows to a smaller value. The renormal-
ized coupling at the scale of εF depend only weakly on
the incoming and transferred momenta. Further renor-
malization below εF differentiate between uiwith differ-
ent transformed and incoming momentum. Such renor-
malization is particularly relevant for our system as the
coupling u1, which corresponds to the scattering process,
u1(k,p + q;p,k + q)c†
set of the SDW instability for q = 0 (we recall that mo-
menta for f-fermions are measured from Q, so that the
momentum transfer between c and f fermions is actu-
ally q = Q). The singular vertex is u1(k,p;p,k), which
means that electrons and holes swap their respective mo-
menta. Note that the divergence occurs for any angle
between k and p. At weak coupling, the enhancement of
u1is confined to q ≪ kF, while for a generic q ∼ kF the
coupling u1retains its bare value.
We now show that the u2
same coupling that diverges at the SDW instability. To
see this, we first note that the fermionic momenta in
diagrams for Φ(T,H) are constrained by two require-
ments: i) the momentum transfers are near 2kF and ii)
all four momenta are near the FS. For the third diagram
in Fig. 1, this implies that p ≈ −k, |k| ≈ kF, while q is
small (∼ T/vF). Therefore, the vertex in this diagram is
u1(k,−k;−k,k). This is an analog of the backscattering
amplitude in a 2D FL. The vertex u1(k,−k;−k,k) is a
special case of the SDW vertex u1(k,p;p,k) for p = −k.
Next, we consider higher-order diagrams. They can
be separated into two classes. In diagrams of the first
class, one obtains the non-analyticity by keeping only
two dynamic bubbles Πq,Ω and lumping the rest of the
diagram into renormalization of the static scattering am-
k+qfp+qck, diverges at the on-
1term in χ(T) ∝ T is the
FIG. 2: (a) T dependence of the spin susceptibility χ(T), as
given by Eq. (6), for perfect nesting (µ = 0) and for a range
of couplings uNF, as indicated in the plot. In a calculation,
max(T,µ) in Eq. (6) is approximated by
Calculated slope dχ(T)/dT as a function of uNF at T = 300K
(chosen to match Ref. 11). (b) Calculated χ(T) for uNF = 0.5
for a range of µ, as shown in the plot. Inset: Same curves as
in (a) normalized by χ(0)extr, obtained by extrapolating χ(T)
down to T = 0. (c) χ(T) in BaFe2−xCoxAs2 from Ref. 12.
Inset: same data as in the main panel normalized by χ(0). (d)
Calculated slope dχ(T = 300K)/dT as function of µ. Inset:
measured slope as a function of doping (from Ref. 11).
T2+ µ2. Inset:
u1(k,−k;−k,k) is renormalized into an effective cou-
1, which diverges at the SDW instability. Sin-
gular renormalizations of u1form ladder series which is
In particular, the backscattering amplitude
1 − u1NFln
The second class of higher-order diagrams contain three
and more dynamic bubbles. Terms of order of u3
not expressed in terms of backscattering amplitude but
rather contain u1with typical q of order kF. In this range
of momenta, u1is not enhanced by SDW fluctuations and
remains small at weak coupling.
Neglecting u2and setting u1= u4= u5≡ u > 0, we
1 − uNFln
We see that the full result for δχ(T) is obtained from the
second-order expression by replacing u1= u by the exact
SDW amplitude ueff
1, given by Eq. (5). This is similar to
the result for the specific heat , but differs from the
result for δχ(T) in an ordinary 2D FL, where the full
δχ(T) is not expressed via the backscattering amplitude
[3, 7]. This difference can be traced down to the symme-
try between the particle-particle (Cooper) channel in an
ordinary FL and the particle-hole channel in a nested FL.
Indeed, the backscattering contribution to both γ(T) and
δχ(T) for the ordinary case undergoes logarithmic renor-
malization in the Cooper channel. However, the Cooper
ladder for the ordinary case is identical to the particle-
hole ladder for the nested case, except for that the sign
of the interaction is reversed, i.e. the SDW instability
for u > 0 for the nested case is related to the Cooper
instability for u < 0 for the ordinary case. In both cases,
there are also non-backscattering contributions δχ(T).
For the ordinary case, the backscattering term is reduced
by Cooper renormalization, and non-backscattering con-
tributions play the dominant role. For the nested case,
SDW renormalization enhances the backscattering term,
and other contributions can be ignored. On the other
hand, the Cooper channel composed of electrons and
holes for the nested case is equivalent to the particle-
hole channel for an ordinary case and, therefore, is not
Comparison with experiments. We now apply Eq. (6)
to ferropnictides. The experimental results for χ(T) in
BaFe2−xCoxAs2  are shown in panel (c) of Fig. 2.
From the data, we estimate the slope of the T dependence
as (χ(700K)/χ(0))exp≈ 2, where χ(0) is obtained by ex-
trapolating χ(T) to T = 0 (theoretically, χ(0) ≈ 4χ0).
Taking vF = 0.45eV·˚ A and kF ≈ 0.16˚ A−1from the
ARPES data , we obtain εF ∼ 0.04 eV . The
only unknown parameter of the theory – the dimen-
sionless coupling constant uNF – is fixed by requiring
that the SDW vertex ueff
increases upon approaching
TN (TN = 140K at zero doping). As Eq. (5) is an ap-
proximate one-loop formula, we set a criterium that ueff
increases by a factor of 2 at TN. This yields uNF≈ 0.5.
We then find (χ(T = 700K)/χ(0))theor≈ 1.7, which is
quite close to the experimental (χ(700K)/χ(0))exp≈ 2.
A more detailed comparison between the experiments
and Eq. (6) is presented in Fig. 2, where we also show
the dependencies of the slope on uNF and µ. We find
quite a good agreement with the experimental data.
To summarize, we analyzed a non-
analytic, linear in T term in the spin susceptibility of
a 2D Fermi liquid with nested electron and hole pockets
of the Fermi surface. We found that the prefactor of the
T-term contains the same inter-band coupling ueff
is enhanced by SDW fluctuations. These results describe
quantitatively a strong temperature dependence of the
spin susceptibility in ferropnictides, observed in a num-
ber of recent experiments. An immediate consequence of
the proposed mechanism is that χ should exhibit equally
strong linear dependencies on the magnetic field and on
the wave number . We suggest to perform these mea-
surements as a crucial test for the origin of the observed
We thank R. Klingeler for useful discussions. M.M.K.
acknowledges support from RFBR 07-02-00226, OFN
RAS program on “Strong electronic correlations”, and
RAS program on “Low temperature quantum phe-
nomena”.I.E. acknowledges support from Asian-
Pacific Center for Theoretical Physics.
knowledges support from Laboratoire de Physique des
Solides, Universit´ e Paris-Sud (France) and RTRA Trian-
gle de la Physique. A.V.C. acknowledges support from
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 For a generic dispersion, the Fermi energy, εF, defined in
this way does not correspond to the energy scale separat-
ing the regimes of Fermi and Boltzmann statistics. The
latter occurs at T ∼ T∗≡?
that higher than linear terms in δχ(T) become impor-
tant also at T ∼ T∗. We assume that T∗in pnictides
is high enough so that temperatures of interest for this
study correspond to Fermi statistics and linear scaling of
F. It can be shown