Conductivity close to antiferromagnetic criticality

0.7 MB·English

by  S. V. Syzranov

Conductivity close to antiferromagnetic criticality
S.V. Syzranov and J. Schmalian
Institute for Theoretical Condensed Matter Physics,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
We study the conductivity of a 3D disordered metal close to antiferromagnetic instability within
the framework of the spin-fermion model using the diagrammatic technique. We calculate the
interaction correction δσ(ω,T) to the conductivity, assuming that the latter is dominated by the
disorderscattering,andtheinteractionisweak. Althoughthefermionicscatteringrateshowscritical
behaviourontheentireFermisurface,theinteractioncorrectionisdominatedbytheprocessesnear
the hot spots, narrow regions of the Fermi-surface corresponding to the strongest spin-fermion
scattering. Exactly at the critical point δσ ∝ [min(ω,T)]3/2. At sufficiently large frequencies ω
the conductivity is independent of the temperature, and δσ ∝ (τ−1 −iω)−2, τ being the elastic
2 scattering time. In a certain intermediate frequency range δσ(ω)∝iω(τ−1−iω)−2.
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PACSnumbers: 71.27.+a,72.10.Di,75.50.Ee,75.30.-m
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l
u
Transport close to quantum criticality fascinates and residual resistivity ρ caused by impurities. The frac-
J 0
challenges researches in many fields of condensed mat- tionalpowerherewasexplainedbythewidthofthe“hot
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ter, ranging from physics of high-temperature super- lines” ∝ T1/2 [9]. It is however unclear why such a
1
conductors and heavy-fermion materials to the conduc- quasi-classicalBoltzmannapproachshouldbeapplicable,

] tion in graphene or granulated superconductors near the considering, in particular, the singular scattering rate
l
e superconductor-insulator transition[1, 2]. At low tem- and vanishing quasi-particle weight. Another interesting
- peratures, the interplay of disorder and interactions in- question,thatdeservesaseparateinvestigation,isrelated
r
t evitably plays an important role in transport. Efforts to to the notion of impurity-induced ”local criticality”[10]:
s
. investigateconductivitynearmagneticinstabilitiesoften the single-particle self-energy may become momentum
t
a relyonthespin-fermionmodel[3]: theconductivityisde- independent Σ(k,iω) (cid:39) Σ(iω) and show singular be-
m
terminedbylow-energyfermionicexcitations,interacting haviour on the entire Fermi surface away from the hot
- with collective bosonic spin modes that carry no charge spot, that would manifest itself in the resistivity ρ(T).
d
yet become soft modes at the magnetic critical point. As estimated in Ref. [11], such local scattering processes
n
o If the interaction corrections to the conductivity are lead to the quasiparticle scattering time τs−p1(T)∝T3/2.
c small,theycanbeanalysedintheframeworkofthespin- This momentum-independent scattering rate implies a
[ fermion model using perturbation theory. For instance, similar temperature dependence of ∆ρ. The emergence

1 the interaction contributions to the conductivity of a 2D of an impurities-induced local single-particle scattering
v metalnearaferromagnetic(FM)instabilitycanbefound rateisparticularlyinterestinggiventheexperimentalin-
4 microscopically[4]usingthediagrammatictechniquesim- dicationsinfavouroflocal, i.e. momentumindependent,
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ilar to the electron-electron interaction corrections[5, 6] criticality of electrons near the antiferromagnetic quan-
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3 to the conductivity of a disordered metal. A separate tum critical points[2].
. analysis is needed in the case of an antiferromagnetic InthisLetterwestudymicroscopicallytransportinthe
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0 (AFM) instability, characteristic of pnictide supercon- spin-fermion model in presence of impurities using the
2 ductors,certainheavy-fermionmaterialsandpossiblythe diagrammatic technique. We demonstrate that at suffi-
1 cuprate systems and organic charge transfer salts. Near ciently strong disorder and weak spin-fermion coupling
:
v the AFM transition, the momenta of the lowest-energy the interaction corrections to the resistivity are domi-
i spinfluctuationsarelargeandclosetothereciprocalvec- natedbytheprocessesnearthehotspots. Thequasipar-
X
tors of the spin superlattice in the AFM phase. As a ticle self-energies in the “cold regions” also show critical
r
a result, electrons strongly interact with spin fluctuations behaviour,thatleads,however,toasmallercontribution
only close to narrow regions of the Fermi surface, the so- to the resistivity correction. We find the conductivity
called “hot spots”, and can be scattered from one such dependency on temperature and frequency. In the zero-
region to another. frequency low-temperature limit we recover the results
Inasystemwithoutdisorderquasiparticlelifetimeand previously known from the kinetic-equation approach.
weightvanishatthehotspotsatlowenergies. Theanal- Model. We consider for simplicity a spherical Fermi
ysis of the Boltzmann transport equation concluded[7], surface. A fermion can absorb or emit a boson only if
however, that the conductivity is dominated by weakly the momentum of the latter is close to a certain large
scattering“coldspots”. Theroleofimpuritieswithinthe vector Q (|Q|∼k ), determined by the geometry of the
F
kinetic-equation approach was investigated by Ueda[8] AFM spin superlattice in the AFM phase. In order to
and Rosch[9] who found a correction ∆ρ ∝ T3/2 to the address the experimentally relevant case of an isotropic
2
conduction we assume that there are three such (incom- other hand, to address small interaction corrections to
mensurate) vectors, Q , Q , and Q , of equal length theconductivityofadisorderedmetal,observedinexper-
1 2 3
anddirectedrespectivelyalongdifferentcoordinateaxes. iments, we assume that the dimensionless coupling con-
The fermions strongly interact with bosons only close to stant, α(cid:39)g2/v (cid:28)1, between the fermions and bosons
F
the “hot-spots”, points on the Fermi surface separated istheverysmallestparameterinthetheory,whichallows
by the vectors Q , Fig. 1. In our case the hot spots are us to treat the interactions perturbatively. In principle,
n
three pairs of circles; an electron near each circle can be the renormalized bosonic propagator D (iΩ,q) depends
n
inelastically scattered to the corresponding circle in the on the strength of disorder and the coupling g. How-
opposite hemisphere. ever,weassumeinwhatfollowsthatthebosondynamics
is characterised by phenomenological energy scales inde-
pendent of g and the elastic scattering time τ.
The assumption of small α implies, in particular, that
the conductivity is dominated by disorder scattering.
For simplicity, the impurity scattering is assumed to be
isotropic, and the disorder potential – weak and Gaus-
sian;(cid:104)U(r)U(r(cid:48))(cid:105)=(2πρFτ)−1δ(r−r(cid:48)),whereρF = 2kπF22v
isthedensityofstatesontheFermisurface. Theparam-
eter (ε τ)−1 (cid:28) 1 is assumed to be the second smallest
FIG. 1: (Colour online) The Fermi surface. F
in the problem, ε being the Fermi energy.
F
Perturbation theory. Under the assumptions that
The action of the spin-fermion model reads
the bosonic modes are correlated on a short scale yet
(cid:90) the spin-fermion coupling is weak, one can conveniently
(cid:88)
S = c†σ(x)[∂τ +ε(−i∇)+U(r)]cσ(x) calculate the interaction corrections to the conductiv-
σ x ity perturbatively. To the 0th order in the interac-
−1(cid:88)(cid:90) S (x)D−1(x−x(cid:48))S (x(cid:48)) tion, the conductivity is given by the Drude contribu-
2 n x,x(cid:48) n n n tion σ0(ω)=(2/3)v2ρF (cid:0)τ−1−iω(cid:1)−1, together with the
(cid:88) (cid:90) weak-localization and the electron-electron interaction
+g c† (x)σ c (x)S (x), (1)
σ σσ(cid:48) σ(cid:48) n correctionstoit. Nextwecalculatetheleadingcorrection
σσ(cid:48)n x σ ∝g2 in the spin-fermion coupling g.
2
Letusconsiderfirsttherenormalizationofthefermion-
where c† (x) and c (x) are the Grassman fields for an
σ σ boson interaction vertex by disorder. The first non-
electron (fermion) with spin σ, x = (r,τ); S (x) is the
n
vanishing correction to the coupling g, Fig. 2, reads
bosonic field of the collective spin excitations; n=1,2,3
labels a pair of hot spot circles separated by the vector g (cid:90) dk
Q , g is the coupling constant between the bosons and δg(q,iω,iε)= G(k,iε)G(k+q,iε+iω),
n 2πρ τ (2π)3
the fermions, ξ is the electron spectrum, D (x−x(cid:48)) F
k n (2)
is the propagator of the bosonic modes; the disorder
where iω and q (cid:39) Q are respectively the frequency
is represented by the random potential U(r), that acts
and the momentum of the spin excitation at the ver-
(cid:82)
(cid:82)ond3trhe(cid:82)βfedrτm·io··nsanondlys.etWee=us(cid:126)e=th1e.aAbblsroe,vibaetlioown wxe··su·p=- taevxe,raGg(eidε,fker)m=io[niεp−roξpka+gait/o(r2[τ1)2]s.ignBεec]−a1usies tthheerdeisaorredenro-
0
press the bosonic index n, if the respective expression excitations with energies larger than Λ, the integration
does not depend on it, and use conventions k = (k,iω) isconfinedtonarrowregionsaroundthehot-spotcircles,
and (cid:82) ···=T (cid:80) (cid:82) dk . such that |ξ ,ξ | < Λ. This allows one to integrate
k ω (2π)3 k k+q
The energies of all excitations in the spin-fermion separatelythetwopropagatorsnearthetwodifferenthot-
model are limited by some energy scale Λ, which is sig- spot circles, arriving at a very small renormalization of
nificantly exceeded by the microscopic bandwidth W of the coupling δg/g ∼(ε τ)−1sign(ω)sign(ω+ε), which,
F
the tight-binding Hamiltonian of the underlying lattice. as we show below, may be neglected when calculating
The shape of the Fermi surface and the excitation spec- the conductivity. Due to the large momentum transfer
traareassumedrenormalizeduponhavingintegratedout by the AFM bosonic fluctuations, |q| ∼ |Q| ∼ k , the
F
all the modes with higher energies Λ < ε (cid:46) W [3]. The renormalizationisfullyperturbative, i.e. maybeconsid-
relative smallness of the cutoff Λ allows one to linearise ered only to the leading order in the disorder potential,
the fermionic spectrum ξ with respect to k−k . and does not involve the summation of the whole diffu-
k F
We consider a system sufficiently close to the criti- sion ladder, unlike the case of electron-electron or FM
cal point, so that the collective spin excitations are the interactions, when the momentum scattering at the ver-
most important bosonic excitations in the model, so one tex is small[5, 6].
can disregard the other types of interaction. On the Let us notice also that the Hartree contributions to
3
theconductivityandtothequasiparticlescatteringrates, δσ(iω)O(cid:2)(ε τ)−1(cid:3) is suppressed by the small parame-
F
thatcontainonebosonpropagator,(e.g.,inFig.2b)van- ter(ε τ)−1 compared tothe diagrams 1-3, which canbe
F
ish due to the symmetry of the spin-fermion interaction understood as follows. The extra impurity line in dia-
vertex. Indeed, such diagrams contain an independent gram 4 adds to the respective integral one more momen-
summation of one interaction vertex with respect to the tum integration and two propagators, whose momenta
(cid:80)
spin polarizations ∝ σ =0. are shifted by a large constant vector of the order of k
σ σσ(cid:48) F
withrespecttoeachother. Thisextraintegrationresults
in the relative smallness (ε τ)−1 of the diagram. A sim-
F
ilar argument proves the smallness of the diagrams with
crossing impurity lines in the usual disorder averaging-
diagrammatic technique[12]. Let us emphasize that this
FIG.2: a)Renormalizationofthespin-fermionineractionver- smallness is specific of the case of the large momen-
tex. b)Hartree contributions to the conductivity and to the
tum scattering by the bosonic modes close to the AFM
fermion scattering rate.
criticality. If the electron dynamics is strongly affected
by collective excitations with small momenta, e.g., FM
fluctuations[4] or electron-electron interactions[5], then
taking into account the whole diffusion ladder is neces-
saryinplaceoftheimpuritylinesforthediagrams4and
5, as well as in the renormalization of the spin-fermion
interaction vertex.
Similarly, the contribution of the diagram 5,
FIG.3: Interactioncorrectionstotheconductivity. Diagrams (cid:90)
1-5 mimic processes close to the hot spots, while 6 is the δσ (iω)∼ατ−1(τ−1+ω)2 D(iΩ,q) (5)
5
contribution of electrons on the whole Fermi surface. q
is also suppressed by the small parameter (ε τ)−1, com-
To the lowest order in interactions and in disorder F
pared to the diagrams 1-3. However, Eq. (5) involves
strength the conductivity correction is given by the di-
the summation of the boson propagator over all Mat-
agrams 1-3 in Fig. 3 (thin solid lines correspond to the
subara frequencies, unlike Eq. (3), where the frequency
disorder-averagedelectronpropagators). Theircontribu-
summation is restricted to |Ω|<|ω|. This difference can
tiontotheconductivityisgivenbytheanalyticcontinua-
makediagram5importantatsufficientlylowfrequencies
tionfromtheMatsubaratorealfrequencies,iω →ω+i0,
ω. However, the temperature dependencies of both dia-
of the quantity
grams are determined by a few terms with Ω∼T in the
αv k (cid:82) (ω−|Ω|)θ(ω−|Ω|)D(iΩ,q) sumsinEqs.(5)and(3),andwecanneglectthetempera-
δσ(iω)= (cid:107) F q , (3) turedependencycontributionofthediagram5solongas
πω(τ−1+ω)2
ε τ (cid:29) 1. δσ (ω = 0,T = 0) amounts to a contribution
F 5
where q = (q,iΩ) and v if the component of velocity to the residual conductivity due to the spin fluctuations
(cid:107)
at the hot spot parallel to the respective vector Q. The and may be disregarded in what follows. Similarly, one
analytic continuation reads can neglect the diagrams for the conductivity that cor-
respond to the renormalisation of the interaction vertex
2iαv k (cid:82) DR(ε,q)A(ω,ε)dεd3q since εFτ (cid:29)1.
δσ(ω)=− (cid:107) F (2π)4 (4) Let us proceed to the diagram 6 in Fig. 3. This in-
πω(τ−1−iω)2
teractioncorrectioncomesfromelectronsnearthewhole
Fermisurface, unlikethepreviouslyconsidereddiagrams
where DR(ε,q) is the retarded boson propagator,
1-5, that correspond to processes near the hot spots. In-
A(ω,ε) = (ω − ε)[n(ε)−n(ε−ω)] with the Bose dis-
deed, anywhere on the Fermi surface an electron can be
tribution function n(ε).
scattered into a hot spot, where its dynamics is impeded
Diagrams 4-6 in Fig. 3 contain one extra impurity line
by the spin fluctuations, cf. diagram 3 in Fig. 4. The
and represent the next-order corrections to the conduc-
value of the diagram 6 in Fig. 3
tivity in the disorder strength, which are not accounted
for by the renormalization of the interaction vertex. We δσ (iω)∼(Λτ)−1δσ(iω) (6)
6
show in what immediately follows that, because of the
large bosonic momenta, adding more impurity lines to is small compared to the value δσ(iω) of the diagrams
the diagrams 1-3 results in small corrections, that are 1-3 due to the small parameter (Λτ)−1 (cid:28) 1, which can
perturbative in the disorder strength. be understood as follows. On one hand, extra smallness,
The straightforward evaluation of the diagram 4 ∼ (ε τ)−1, comes from the extra impurity line. On the
F
shows that the corresponding contribution δσ (iω) ∼ other hand, the diagram contains an extra largeness in
4
4
it, ∼εF/Λ,– the ratio of the area of the Fermi surface to where C = 9π32ζ√(3/2) = 23.14.., which up to a numerical
the characteristic size of the hot spots. The combination 4 2
coefficientmatchestheresultobtainedwithinthekinetic-
of the two factors leads to Eq. (6).
equation approach.
High frequencies. At larger ω the frequency depen-
dency of the conductivity can be found regardless of the
particularformofthebosonpropagator. Ifthefrequency
is very large, the cutoff Λ of the excitation energies in
FIG. 4: Interaction corrections to the scattering rate. Dia- the spin-fermion model should be chosen larger than ω
grams1and2describescatteringclosetothehotspots,while
to ensure the existence of quasiparticles that can absorb
3 is an isotropic contribution to the scattering on the whole
a quantum ω.
Fermi surface.
Letusassumefirstthatthebosonicdynamicsisdeter-
Thus,wehaveshownthattotheleadingorderindisor- mined by certain energy scales independent of the cutoff
derandinteractionstheconductivityisgivenbyEq.(3), Λ. Then one can neglect the bosonic frequencies Ω in
corresponding to the diagrams 1-3 in Fig. 3. To the first comparison with the large ω in Eq. (3). Making the an-
order in α the conductivity reads alytic continuation and introducing the average value of
the spin fluctuations (cid:104)S2(cid:105)=−(cid:82) D(iΩ,q), we arrive at
σ(ω)= 2v2ρ (cid:2)τ−1+δτ−1−iω(1+λ)(cid:3)−1, (7) q
3 F
αv k
δσ(ω)=− (cid:107) F (cid:104)S2(cid:105). (14)
where we introduced
π(τ−1−iω)2
6παv (cid:90) dεdq
δτ−1 =− (cid:107) ImDR(ε,q)A(ω,ε) , (8)
k ω v (2π)4 The result (14) has a simple physical interpretation.
F
6πα v (cid:90) dεdq At very high frequencies the spin fluctuations may be
λ=− (cid:107) ReDR(ε,q)A(ω,ε) . (9) considered frozen and equivalent to static disorder. The
k ω2 v (2π)4
F appropriate modification of the elastic scattering time,
The limit of low frequencies and temperatures. To averaged with respect to angles, estimates δ(cid:104)1/τ(cid:105) ∼
make further progress at sufficiently small ω and T we α(cid:104)S2(cid:105)/k and causes correction δσ ∼ (δτ−1)∂σ /∂τ−1
F 0
consider the most general form of the overdamped bo- to the Drude conductivity σ . As necessary, this correc-
0
son propagator that follows from the effective Ginzburg- tion matches Eq. (14).
Landau-type description: Intermediate frequencies. Let us proceed to the inter-
DR(ε,q)=−(cid:88)(cid:2)ξ−2+(q±Q)2−iεk2γ−1(cid:3)−1. (10) mediate frequencies; ω exceeds the temperature T, but
n F is smaller than the smallest characteristic energy of the
± boson dynamics, e.g., the bosonic scattering rate ε2γ−1.
F
at |ε| (cid:28) ε2Fγ−1. For |ε| (cid:29) ε2Fγ−1, DnR ∝ ε−2. Here γ is Then at |Ω|<ω the integral (cid:82) D(iΩ,q)(2dπq)3 is nearly
thephenomenologicalscalethatcharacteriseselasticand
independent of Ω, because the value of Ω affects the bo-
inelastic scattering of bosons.
son propagator only on a sufficiently small momentum
Substituting Eq. (10) into (8) yields δτ =
interval, while the rest of the integral is accumulated on
3α4ζ√(32/π2)vv(cid:107)Tγ13//22 for (kFξ)2 (cid:29) γT−1 and ω = 0. While at a greater interval. (cid:82)
(k ξ)2 (cid:28) γT−1 and ω = 0 it holds δτ = παv(cid:107)T2kFξ. Thus, in Eq. (3) we may set D(iΩ,q)(dq) ≈
F 2 v γ (cid:82) D(0,q)(dq). ThenthefrequencysummationinEq.(3)
Similarly, for T = 0 and finite frequencies δτ =
α√2v(cid:107)ω3/2. The crossover between these regimes is de- and the analytic continuation to real frequencies yield
5π v γ1/2
scribed by the expression δσ(ω)∝−iω(τ−1−iω)−2 (15)
δτ−1 =αv(cid:107)T32Φ(cid:18) γ , ω(cid:19) (11) independently of the form of the bosonic propagator.
v γ21 (kFξ)2T T Discussion. Spin-fermion interactions modify the
with the scaling function quasiparticleself-energypartonthewholeFermisurface.
(cid:16) (cid:17) Away from the hot spots the modification
3 (cid:90) x(z−x)t2 ex1−1 − ex−1z−1
Φ(y,z)= dxdt. (cid:90)
π2z (y+t2)2+x2 Σ (iε)∼−iα(k Λτ)−1 θ(|ε|−|Ω|)D(iΩ,q)signε.
lc F
(12) q
In particular, in the dc limit the correction to the resis- (16)
tivity due to the spin fluctuations at the AFM critical is given by diagram 3 in Fig. 4. It is momentum-
point reads independent and “locally critical”, corresponding to the
scattering rate τ−1 ∝ const+T3/2. Let us notice that
v(cid:107)T32 Eqs. (8), (9), andlc(16) hold for an arbitrary boson prop-
δρ=αC , (13)
v2kF2τ2γ12 agator and can be used to analyse transport in, e.g., a
5
system with a two-dimensional spin dynamics[2]. In ar- [2] H.v.Lo¨hneysen,A.Rosch,M.Vojta,andP.Wo¨lfle,Rev.
bitrary dimensions d holds τ−1 ∝const+Td/2. Mod. Phys. 79, 1015 (2007).
lc
We demonastrated, however, that the interaction cor- [3] A. Abanov, A. V. Chubukov, and J. Schmalian, Adv.
Phys. 52, 119 (2002).
rection to the conductivity near the AFM instability is
[4] I. Paul, C. Pepin, B. N. Narozhny, and D. L. Maslov,
dominated by the processes near the hot spots, corre-
Phys. Rev. Lett. 95, 017206 (2005).
sponding to diagrams 1-3 in Fig. 3. We have found the
[5] B. L. Altshuler and A. G. Aronov, in Electron-electron
dependency of conductivity on frequency ω and temper- interactions in disordered systems, edited by A. L. Efros
ature T. In the limit ω = 0 we recover the temperature and M. Pollak (North-Holland, Amsterdam, 1985).
dependencies of the interaction correction to conductiv- [6] G. Zala, B. Narozhny, and I. Aleiner, Phys. Rev. B 64,
ity previously known from the kinetic equation analysis; 214204 (2001).
δσ ∝ T3/2 and δσ ∝ T2 at the critical point and away [7] R.HlubinaandT.M.Rice,Phys.Rev.B51,9253(1995).
[8] K. Ueda, J. Phys. Soc. Japan 43, 1497 (1977).
from it respectively. At T = 0 we find δσ ∝ ω3/2. At
[9] A. Rosch, Phys. Rev. Lett. 82, 4280 (1999).
sufficientlyhighfrequenciesthecorrectionisindependent
[10] Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature
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gator. Atveryhighandmoderatefrequenciesthedepen- [11] P.Wo¨lfleandE.Abrahams,Phys.Rev.B84,041101(R)
dency is given by Eqs. (14) and (15) respectively. (2011).
Acknowledgements. We appreciate useful discussions [12] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,
Methods of Quantum Field Theory in Statistical Physics
with B.N. Narozhny and P. W¨olfle.
(Dover, New York, 1975).
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