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arXiv:0901.0238v1 [cond-mat.supr-con] 2 Jan 2009

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

pnictides

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

Introduction.

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 [1], 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 [5]. This linear

decrease has been observed in monolayers of3He [6]. 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 [8].

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

questionable.

The itinerant character of Fe-pnictides is suggested by

the agreement between the band structure obtained in

ab initio calculations [16] 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.

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2

u4

u4

+u5

u5 + (2×) u1

u1

+ u3

u3 + (2×) u2

u2

+

u4

u4

+

u5

u5

+ (2×)

u2

u1

+

u3

u3

k,ω,σ

p,ω',σ

p+q,ω'+Ω,σ'

k+q,ω+Ω,σ'

k,ω,σ

p,ω',σ

p+q,ω'+Ω,σ'

k+q,ω+Ω,σ'

k,ω,σ

p,ω',σ

-k+q,-ω-Ω,σ'

-p+q,-ω'-Ω,σ'

k,ω,σ

p-q,ω'-Ω,σ

p,ω',σ'

k+q,ω+Ω,σ'

k,ω,σ

p,ω',σ

p+q,ω'+Ω,σ'

k+q,ω+Ω,σ'

k,ω,σ

k+q,ω+Ω,σ

p-q,ω'-Ω,σ

p,ω',σ

k,ω,σ

p,ω',σ

-p+q,-ω'-Ω,σ

-k+q,-ω-Ω,σ

k,ω,σ

k+q,ω+Ω,σ

p-q,ω'-Ω,σ

p,ω',σ

k,ω,σ

k+q,ω+Ω,σ

p-q,ω'-Ω,σ

p,ω',σ

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:

H =

?

k,σ

?

εc

kc†

kσckσ+ εf

kf†

kσfkσ

?

+

?

pi,σ,σ′

Hint,(1)

Hint = u1c†

p3σf†

?

2f†

p4σ′fp2σ′cp1σ+ u2f†

p3σc†

?

2c†

p4σ′fp2σ′cp1σ

+

u3

2

u4

f†

p3σf†

p4σ′cp2σ′cp1σ+ H.c

p4σ′fp2σ′fp1σ+u5

+

p3σf†

p3σc†

p4σ′cp2σ′cp1σ.

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

insulator” [27].

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-

teraction.

kand εf

k= −εc

k+

Fig. 1 depicts all topologically inequivalent second-

orderdiagrams forthe

Φ(T,H), describing both intra- and inter-band processes.

Each of the diagrams contains the object

thermodynamic potential,

ϕ(T,H) = T3?

Ω

?

d2q

(2π)2

?

σ,σ′

?

ω,ω′

?

d2kd2p

(2π)4

× Gσ

k,ωGσ

p,ω′Gσ′

p+q,ω′+ΩGσ′

k+q,ω+Ω,(2)

where Gσ

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

T?

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,ΩΠσ′σ′

k,ωGσ′

q,Ω, or as Πσσ′

k+q,ω+Ωis a polarization bubble of

q,ΩΠσσ′

q,Ω, where Πσσ′

q,Ω=

ω

?

d2k

(2π)2Gσ

q,Ω, while

q,Ωand Π↓↓

q,Ω

Πq,Ω=m

2π

|Ω|

?(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

(3)

4,5T,

BNF

F/(2εF), εF = vFkF/2, χ0 = 2µ2

1Tη(µ/T),

2Tη(µ/T), re-

3Tη(µ/T), and δχ5(T) = 2λu2

3Tη(µ/T).

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3

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

in χ(T).

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 [24] 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-

4+ u2

5+ 2?u2

1+ u2

2− u1u2

?η(µ/T)?.

(4)

pf†

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:

plitude.

u1(k,−k;−k,k) is renormalized into an effective cou-

pling ueff

1, which diverges at the SDW instability. Sin-

gular renormalizations of u1form ladder series which is

summed into

In particular, the backscattering amplitude

ueff

1 =

u1

1 − u1NFln

εF

max(T,µ)

.(5)

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

finally obtain

1etc. are

δχ(T) ≈8(uNF)2χ0

vFkF

T

1 +

η(µ/T)

?

1 − uNFln

εF

max(T,µ)

?2

(6)

.

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 [5], 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

Page 4

4

[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

logarithmically divergent.

Comparison with experiments. We now apply Eq. (6)

to ferropnictides. The experimental results for χ(T) in

BaFe2−xCoxAs2 [12] 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 [21], we obtain εF ∼ 0.04 eV [28]. The

only unknown parameter of the theory – the dimen-

sionless coupling constant uNF – is fixed by requiring

that the SDW vertex ueff

1

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.

Conclusion.

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 [3]. We suggest to perform these mea-

surements as a crucial test for the origin of the observed

effect.

We thank R. Klingeler for useful discussions. M.M.K.

1, which

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

nsf-dmr 0604406.

D.L.M. ac-

∗Electronic address: maxim@mpipks-dresden.mpg.de

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[28] 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

δχ(T).

NF/N′′

F. It can be shown