Non-analytic spin susceptibility of a nested Fermi liquid: the case of Fe-based pnictides

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by  M. M. Korshunov

Non-analytic spin susceptibility of a nested Fermi liquid: the case of Fe-based
pnictides
M.M. Korshunov,1,2,∗ I. Eremin,1,3 D.V. Efremov,4 D.L. Maslov,5 and A.V. Chubukov6
1Max-Planck-Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
2L. V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036 Krasnoyarsk, Russia
3Institute fu¨r Mathematische und Theoretische Physik,
TU Braunschweig, 38106 Braunschweig, Germany
4Institut fu¨r Theoretische Physik, TU Dresden, 01062 Dresden, Germany
9 5Department of Physics, University of Florida, Gainesville, Florida 32611, USA
0 6Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
0 (Dated: January 2, 2009)
2
We propose an explanation of the peculiar linear temperature dependence of the uniform spin
n
susceptibilityχ(T)inferropnictides. WearguethatthelinearinT termappearsduetonon-analytic
a
temperaturedependenceof χ(T)in a two-dimensional Fermiliquid. Weshow that theprefactor of
J
theT termisexpressedviathesquareofthespin-density-wave(SDW)amplitudeconnectingnested
2
holeandelectronpockets. DuetoanincipientSDWinstability,thisamplitudeislarge,which,along

with a small value of the Fermi energy, makes the T dependence of χ(T) strong. We demonstrate
]
n that this mechanism is in quantitativeagreement with theexperiment.
o
c PACSnumbers: 71.10.Ay,75.30.Cr,74.25.Ha,74.25.Jb
-
r
p
u Introduction. The key hypothesis of the Fermi liq- The itinerantcharacterof Fe-pnictides is suggestedby
s uid (FL) theory is that a system of strongly interact- the agreement between the band structure obtained in
.
t ing fermions can be considered effectively as a gas of ab initio calculations [16] and observed in de Haas-van
a
weaklyinteractingquasiparticles. Intheabsenceofresid- Alphen and angle-resolved photoemission spectroscopy
m
ualinteractionbetweenquasiparticles,thestaticuniform (ARPES) experiments [17, 18, 19, 20, 21]. It is firmly
-
d spinsusceptibility,χ(T),andthespecificheatcoefficient, established by now that the Fermi surface (FS) of Fe-
n γ(T), are finite at T = 0 and obey quadratic dependen- pnictides consists of two small hole pockets near (0,0)
o cies on T at low temperatures. The effect of residual and two electron pockets near (π,π) points of the folded
c
interactions on χ(T) and γ(T) has been studied inten- Brillouin zone. In such a system, an obvious origin of
[
sively in recent years [1], with the key result that in two the SDW order is a logarithmic divergence of a particle-

1 dimensions (2D) both γ(T) and χ(T) are linear rather hole vertex involving states on nested parts of the FS
v
than quadratic in T [2, 3, 4]. [16, 22, 23, 24, 25, 26].
8
3 Theory predicts that the behavior of γ(T) is univer-
In this Letter, we propose an explanation of the ex-
2
sal in a sense that the (negative) slope is given by the
0 perimental T-dependence of spin susceptibility based on
square of the backscattering amplitude [5]. This linear
. the itinerant picture. We argue that the origin of the
1 decreasehas been observedin monolayersof 3He [6]. On
linear increase of χ(T) with temperature in ferropnic-
0
the contrary,a linearin T termin the spinsusceptibility
9 tides is the same as in a 2D FL. Furthermore, we show
is not universal and can be of either sign [3, 7], causing
0 that this behavior is universalfor FLs with strong (π,π)
v: uncertainty in the interpretation of the experiments on SDW fluctuations, namely, the slope of the linear in T-
semiconductor heterostructures [8].
i dependenceisdeterminedbythesquareoftheSDWam-
X
Recently,apronouncedlineartemperaturedependence plitude with nesting momentum Q = (π,π). This am-
r
of the uniform susceptibility has been observed in high- plitude is large which, along with a small value of the
a
T superconductors with iron-based layered structure Fermi energy ε , amplifies the T dependence of χ(T).
c F
[9,10,11,12]. Itextendsfromtemperaturesaboveeither Choosing the SDW coupling to reproduce the observed
SDWorsuperconductingtransitionsupto500-700Kwith SDWtransitiontemperatureT atzerodoping,wefinda
N
almost doping-independent slope. The T dependence is goodagreementbetweencalculatedandmeasuredslopes
quite strong: χ(T) increases roughly by a factor of 2 be- of χ(T). The doping dependence of the slope also agrees
tween 200K and 700K. An explanation of this behavior with the data. We view this agreement as a strong in-
based on the J −J model of localized spins has been dicationthatthe lineartemperaturedependence ofχ(T)
1 2
proposedinRef.13; however,giventhatthe linearT de- in pnictides [9, 10, 11, 12] is in fact the first unambigu-
pendencepersistsuptolargedopings,wherelocalprobes, ousobservationofanon-analyticbehaviorofthe2Dspin
suchas nuclearmagnetic resonance(NMR) andµSR, do susceptibility. Besidesbeingfundamentallyimportanton
not see localized moments [14, 15], this explanation is its own right, this observation also strengthens the case
questionable. for the itinerant scenario for Fe-pnictides.
2
k,ω,σ k,ω,σ k,ω,σ Fig. 1 depicts all topologically inequivalent second-
p,ω',σ p,ω',σ p,ω',σ order diagrams for the thermodynamic potential,
Φ(T,H),describingbothintra-andinter-bandprocesses.
u4 u4 + u5 u5 + (2× )u1 u1
p+q,ω'+Ω,σ' p+q,ω'+Ω,σ' -k+q,-ω-Ω,σ' Each of the diagrams contains the object
k+q,ω+Ω,σ' k+q,ω+Ω,σ' -p+q,-ω'-Ω,σ'
d2q d2kd2p
k,ω,σ k,ω,σ ϕ(T,H) = T3
p-q,ω'-Ω,σ p,ω',σ k,ω,σ k+q,ω+Ω,σ XΩ Z (2π)2 Xσ,σ′ωX,ω′Z (2π)4
u
+u3 u3 + (2× )u2 u2 + u 4 × Gσk,ωGσp,ω′Gσp′+q,ω′+ΩGσk′+q,ω+Ω, (2)
p,ω',σ' p+q,ω'+Ω,σ' 4
k+q,ω+Ω,σ' k+q,ω+Ω,σ' p-q,ω'-Ω,σ p,ω',σ whereGσ isaMatsubaraGreen’sfunctionofeithercor
k,ω
f fermion with momentum k and spin-σ in a magnetic
k,ω,σ k+q,ω+Ω,σ k,ω,σ p,ω',σ k,ω,σ k+q,ω+Ω,σ field H. As in an ordinary FL, the non-analytic H2T
u u u
5 1 3 term in ϕ(T,H) comes from a dynamic Kohn anomaly,
+ + (2×) +
u5 u2 u3 i.e. from diagrams with momentum 2kF carried by the
interaction lines [3, 4]. It is, however, more conve-
p-q,ω'-Ω,σ p,ω',σ -p+q,-ω'-Ω,σ -k+q,-ω-Ω,σ p-q,ω'-Ω,σ p,ω',σ
nient to re-express the result via the polarization bub-
FIG.1: Second-orderdiagramsforthethermodynamicpoten- bles with small rather than 2kF momenta. The Green
tial. Solid and dashed lines correspond to f-fermions (elec- functions in Eq. (2) can be combined in two different
trons) and to c-fermions (holes), respectively. ways: eitherasΠσσ Πσ′σ′,orasΠσσ′Πσσ′,whereΠσσ′ =
q,Ω q,Ω q,Ω q,Ω q,Ω
T d2k Gσ Gσ′ is a polarization bubble of
ω (2π)2 k,ω k+q,ω+Ω
thPe cc,Rff, or cf type with small momenta q ≪ k . A
Theory. We consider a two-band model of interacting F
bubble depends on the magnetic field if the Zeeman en-
fermions occupying the electron and hole FSs:
ergiesoftwofermionsaddup. Onecanreadilyshowthat
theccandff bubblesdependonthefieldviaΠ↑↓ ,while
H = εcc† c +εff† f + H , (1) q,Ω
Xk,σ h k kσ kσ k kσ kσi piX,σ,σ′ int the field enters the cf bubble through Π↑q↑,Ω and Π↓q↓,Ω
terms. Evaluatingindividualdiagrams,wefindthateach
Hint = u1c†p3σfp†4σ′fp2σ′cp1σ+u2fp†3σc†p4σ′fp2σ′cp1σ of them can be expressed as a product of two dynamic
u
+ 23 hfp†3σfp†4σ′cp2σ′cp1σ+H.ci spin up/down bubbles:
u u
+ 24fp†3σfp†4σ′fp2σ′fp1σ+ 25c†p3σc†p4σ′cp2σ′cp1σ. Π = m |Ω| , (3)
q,Ω
2π (Ω−2iµ H −2iµ∗)2+(v q)2
B F
Here, c (f ) is the annihilation operator for a hole p
kσ kσ
(electron)withmomentumkandspinσ (foranelectron, where µ∗ = 0 for intra-band scattering and µ∗ = µ
k ismeasuredfromthe (π,π)-point), εc andεf =−εc + for inter-band scattering. The rest of the calculations
k k k
2µrepresentsingle-particledispersions,andµmeasuresa proceeds in the same way as for an ordinary FL [3, 4].
deviation from perfect nesting. Models of this type were In short, one integrates Π2 over q first, replaces the
q,Ω
considered in the past in the context of an “excitonic Matsubara sum by a contour integral, differentiates the
insulator” [27]. result twice with respect to H, sets H = 0, and ob-
We assume that each of the electron and hole FSs is tains the O(T) terms in χ. The first two diagrams in
doubly degenerate. The terms with u and u are intra- Fig. 1 describe intra-band processes and give the same
4 5
band interactions, the terms with u and u are inter- results for χ(T) as in Refs. [3, 4]: δχ (T) = λu2 T,
1 2 1,2 4,5
band interactions with momentum transfer 0 and Q, re- where λ = 4χ N2/(2ε ), ε = v k /2, χ = 2µ2N
0 F F F F F 0 B F
spectively, and the term with u is the inter-band pair is the Pauli susceptibility per one sheet of the FS, and
3
hopping. All couplings flow from their initial values at N = m/2π is the density of states. A factor of 4 in
F
energiesoforderofthebandwidthtorenormalizedvalues λ results from the double degeneracy of electron and
at ε [24, 25]. We assume that this renormalization is hole bands. The third through fifth diagrams involve
F
alreadyincludedintoEq.(1)andanalyzethebehaviorof inter-band scattering and yield δχ (T)= 2λu2Tη(µ/T),
3 1
the systematenergiesbelowε , wherethe actualvalues δχ (T)=λu2Tη(µ/T), and δχ (T)=2λu2Tη(µ/T), re-
F 4 3 5 2
of momenta become relevant. spectively. Here,η(x)=2xcothx−x2/sinh2x−2x. The
Wefirstobtainthelinear-in-T contributiontothespin third and forth diagrams give finite contributions if σ =
susceptibility, δχ(T), to second order in the interaction σ′, while the fifth diagram contributes if σ = −σ′. The
andthen showthatthe prefactorof theT termin δχ(T) sixth and seventh diagrams do not contribute to δχ(T),
is expressed via the SDW vertex to all orders in the in- and the remaining two give δχ (T)= −2λu u Tη(µ/T)
8 1 2
teraction. and δχ (T)=−λu2Tη(µ/T).
9 3
3
Combining all diagrams, we obtain
δχ(T)=λT u2+u2+2 u2+u2−u u η(µ/T) .
4 5 1 2 1 2
(cid:2) (cid:0) (cid:1) (cid:3)(4)
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 linearterminthe spinsusceptibility. Inthe opposite
limit of T ≪ µ (poor nesting), η(µ/T) ∼ exp(−2µ/T)
and the inter-band contribution to χ(T) is suppressed
exponentially. IntheintermediateregimeT ∼µ,η(µ/T)
is non-monotonic. Note that the pair-hopping term u ,
3
which gives rise to an attraction in the extended s-wave
(s ) pairing channel, does not contribute to the T term
±
FIG. 2: (a) T dependence of the spin susceptibility χ(T), as
in χ(T).
given by Eq. (6), for perfect nesting (µ=0) and for a range
The coupling constants in Eq. (4) are interactions at
of couplings uNF, as indicated in the plot. In a calculation,
the scale of the Fermi energy. These couplings are al- max(T,µ) in Eq. (6) is approximated by T2+µ2. Inset:
ready renormalized from their bare values at the scale Calculatedslopedχ(T)/dT asafunctionofpuNF atT =300K
of the bandwidth [24] by parquet renormalization group (chosentomatchRef.11). (b)Calculatedχ(T)foruNF =0.5
(RG),insuchawaythatu ,u ,andu flowtothe same for a range of µ, as shown in theplot. Inset: Same curvesas
4 5 1
value, while u flows to a smaller value. The renormal- in(a)normalizedbyχ(0)extr,obtainedbyextrapolatingχ(T)
2
ized coupling at the scale of ε depend only weakly on down to T = 0. (c) χ(T) in BaFe2−xCoxAs2 from Ref. 12.
F Inset: samedataasinthemainpanelnormalizedbyχ(0). (d)
the incoming and transferred momenta. Further renor-
Calculated slope dχ(T = 300K)/dT as function of µ. Inset:
malization below ε differentiate between u with differ-
F i measured slope as a function of doping (from Ref. 11).
ent transformed and incoming momentum. Such renor-
malization is particularly relevant for our system as the
couplingu ,whichcorrespondstothescatteringprocess, plitude. In particular, the backscattering amplitude
1
u (k,p+q;p,k+q)c†f† f c , diverges at the on- u (k,−k;−k,k) is renormalized into an effective cou-
1 p k+q p+q k 1
set of the SDW instability for q=0 (we recall that mo- pling ueff, which diverges at the SDW instability. Sin-
1
menta for f-fermions are measured from Q, so that the gular renormalizations of u form ladder series which is
1
momentum transfer between c and f fermions is actu- summed into
ally q=Q). The singular vertex is u1(k,p;p,k), which ueff = u1 . (5)
meansthatelectronsandholesswaptheirrespectivemo- 1 1−u N ln εF
1 F max(T,µ)
menta. Note that the divergence occurs for any angle
betweenkandp. Atweakcoupling,the enhancementof The second class of higher-order diagrams contain three
u is confined to q ≪k , while for a generic q ∼k the andmoredynamicbubbles. Termsoforderofu3 etc. are
1 F F 1
coupling u retains its bare value. not expressed in terms of backscattering amplitude but
1
We now show that the u21 term in χ(T) ∝ T is the rathercontainu1withtypicalqoforderkF. Inthisrange
same coupling that diverges at the SDW instability. To ofmomenta,u1isnotenhancedbySDWfluctuationsand
see this, we first note that the fermionic momenta in remains small at weak coupling.
diagrams for Φ(T,H) are constrained by two require- Neglecting u2 and setting u1 = u4 = u5 ≡ u > 0, we
ments: i) the momentum transfers are near 2k and ii) finally obtain
F
allfour momenta are near the FS. For the third diagram
in Fig. 1, this implies that p ≈ −k, |k| ≈ kF, while q is 8(uNF)2χ0  η(µ/T) 
small(∼T/v ). Therefore,the vertexin this diagramis δχ(T)≈ T 1+ .
F v k 2
u1(k,−k;−k,k). Thisis ananalogofthe backscattering F F  (cid:16)1−uNF lnmaxε(FT,µ)(cid:17) 
amplitude in a 2D FL. The vertex u1(k,−k;−k,k) is a (6)
specialcaseoftheSDWvertexu1(k,p;p,k)forp=−k. Weseethatthe fullresultforδχ(T)isobtainedfromthe
Next, we consider higher-order diagrams. They can second-orderexpressionbyreplacingu =ubytheexact
1
be separated into two classes. In diagrams of the first SDW amplitude ueff, given by Eq. (5). This is similar to
1
class, one obtains the non-analyticity by keeping only the result for the specific heat [5], but differs from the
two dynamic bubbles Π and lumping the rest of the result for δχ(T) in an ordinary 2D FL, where the full
q,Ω
diagramintorenormalizationofthestaticscatteringam- δχ(T) is not expressed via the backscattering amplitude
4
[3,7]. This difference canbe traceddownto the symme- acknowledges support from RFBR 07-02-00226, OFN
try between the particle-particle (Cooper) channel in an RAS program on “Strong electronic correlations”, and
ordinaryFLandtheparticle-holechannelinanestedFL. RAS program on “Low temperature quantum phe-
Indeed,thebackscatteringcontributiontobothγ(T)and nomena”. I.E. acknowledges support from Asian-
δχ(T)fortheordinarycaseundergoeslogarithmicrenor- Pacific Center for Theoretical Physics. D.L.M. ac-
malization in the Cooper channel. However, the Cooper knowledges support from Laboratoire de Physique des
ladder for the ordinary case is identical to the particle- Solides,Universit´eParis-Sud(France)andRTRATrian-
hole ladder for the nested case, except for that the sign gle de la Physique. A.V.C. acknowledges support from
of the interaction is reversed, i.e. the SDW instability nsf-dmr 0604406.
for u > 0 for the nested case is related to the Cooper
instability foru<0for the ordinarycase. Inbothcases,
there are also non-backscattering contributions δχ(T).
Fortheordinarycase,thebackscatteringtermisreduced
∗
Electronic address: [email protected]
by Cooper renormalization,and non-backscatteringcon-
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Vojta, Rev.Mod.Phys.77,579 (2005); H.v.L¨ohneysen,
SDW renormalizationenhances the backscatteringterm, A.Rosch,M.Vojta,andP.W¨olfle,ibid 79,1015(2007).
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arXiv:0811.1732 (unpublished).
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to ferropnictides. The experimental results for χ(T) in
[5] A.V. Chubukov, D.L. Maslov, S. Gangadharaiah, and
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1
T (T = 140K at zero doping). As Eq. (5) is an ap- Phys. Rev. Lett. 90, 056805 (2003); T. Gokmen et al.,
N N
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F
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isenhancedbySDWfluctuations. Theseresults1describe [18] D.H. Lu et al., Nature455, 81 (2008).
[19] C. Liu et al.,arXiv:0806.2147 (unpublished).
quantitatively a strong temperature dependence of the
[20] C. Liu et al.,Phys.Rev. Lett.101, 177005 (2008).
spin susceptibility in ferropnictides, observed in a num-
[21] H. Ding et al.,arXiv:0812.0534 (unpublished).
ber ofrecentexperiments. Animmediate consequenceof [22] J. Dong et al.,Europhys.Lett. 83, 27006 (2008).
the proposedmechanismis thatχ shouldexhibitequally [23] V.CvetkovicandZ.Tesanovic,arXiv:0804.4678 (unpub-
strong linear dependencies on the magnetic field and on lished).
the wave number [3]. We suggest to perform these mea- [24] A.V. Chubukov, D.V. Efremov, and I. Eremin, Phys.
surements as a crucial test for the origin of the observed Rev. B 78, 134512 (2008).
[25] F. Wang et al., arXiv:0805.3343 (unpublished);
effect.
arXiv:0807.0498 (unpublished).
We thank R. Klingeler for useful discussions. M.M.K.
5
[26] M.M. Korshunov and I. Eremin, Phys. Rev. B 78, ing the regimes of Fermi and Boltzmann statistics. The
140509(R) (2008); Europhys.Lett. 83, 67003 (2008). latter occurs at T ∼ T∗ ≡ NF/NF′′. It can be shown
[27] L.V. Keldysh and Yu.V. Kopaev, Sov. Phys.–Sol. State that higher than linear termps in δχ(T) become impor-
Phys. 6, 2219 (1965); J. De Cloizeaux, J. Phys. Chem. tant also at T ∼ T∗. We assume that T∗ in pnictides
Sol. 26, 259 (1965); B.I. Halperin and M.T. Rice, Sol. is high enough so that temperatures of interest for this
StatePhys. 21, 125 (1968). studycorrespondtoFermistatisticsandlinearscalingof
[28] Foragenericdispersion,theFermienergy,εF,definedin δχ(T).
thiswaydoesnotcorrespondtotheenergyscaleseparat-