Nambu-Eliashberg theory for multi-scale quantum criticality : Application to ferromagnetic quantum criticality in the surface of three dimensional topological insulators

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by  Ki-Seok Kim

Nambu-Eliashberg theory for multi-scale quantum criticality : Application to
ferromagnetic quantum criticality in the surface of three dimensional topological
insulators
Ki-Seok Kim and Tetsuya Takimoto
Asia Pacific Center for Theoretical Physics, Hogil Kim Memorial building 5th floor,
POSTECH, Hyoja-dong, Namgu, Pohang 790-784, Korea
Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk 790-784, Korea
(Dated: January 25, 2011)
We develop an Eliashberg theory for multi-scale quantum criticality, considering ferromagnetic
1
quantumcriticality inthesurfaceofthreedimensionaltopological insulators. Althoughananalysis
1
based on the random phase approximation has been performed for multi-scale quantum criticality,
0
an extension to an Eliashberg framework was claimed to be far from triviality in respect that the
2
self-energy correction beyond the random phase approximation, which originates from scattering
n withz=3longitudinalfluctuations,changesthedynamicalexponentz=2inthetransversemode,
a explicitlydemonstratedinnematicquantumcriticality. Anovelingredientofthepresentstudyisto
J introduce an anomalous self-energy associated with the spin-flip channel. Such an anomalous self-

4 energy turns out to be essential for self-consistency of the Eliashberg framework in the multi-scale
2 quantum critical point because this off diagonal self-energy cancels the normal self-energy exactly
in the low energy limit, preserving the dynamics of both z = 3 longitudinal and z = 2 transverse

] modes. Thismulti-scalequantumcriticalityisconsistentwiththatinaperturbativeanalysisforthe
l
e nematic quantum critical point, where a vertex correction in the fermion bubble diagram cancels
- a singular contribution due to the self-energy correction, maintaining the z = 2 transverse mode.
r
t Wealso claim that thisoff diagonal self-energy gives risetoan artificial electric field in theenergy-
s momentum space in addition to the Berry curvature. We discuss the role of such an anomalous
.
t self-energy in theanomalous Hall conductivity.
a
m
PACSnumbers:
-
d
n I. INTRODUCTION within this analysis, stability of the RPA dynamics is
o
not guaranteed beyond this approximation. In order to
c
describe scaling near quantum criticality, one should go
[ Investigationontheroleofthemomentum-spaceBerry
beyond the perturbative analysis, introducing quantum
2 curvature has driven an interesting direction of research corrections fully self-consistently. An Eliashberg theory
v inmoderncondensedmatterphysics,coveringweakanti-
is desirable, where self-energy corrections are taken into
1 localization,anomalouschargeandspin Halleffects, and
account fully self-consistently in the one loop level [7].
5 topological insulators [1–3]. A recent trend is to study
8 The scaling expression of the free energy can be derived
interplay betweenthe topologicalband structure andin-
4 within the Eliashberg approximation, where critical ex-
teraction,suggestingthat topologicalexcitationssuchas
. ponents are determined [8]. Unfortunately, an extension
1 skyrmions and vortices carry fermion quantum numbers
to an Eliashberg framework was claimed to be far from
1 of electric charge or spin via the associated topological
0 triviality in multi-scale quantum criticality because the
term [4].
1 self-energy correction, which originates from scattering
: Quantum criticality has also driven an important di- with z = 3 critical fluctuations, turns out to change the
v
rection of researchin modern condensed matter physics, dynamical exponent z = 2 in the transverse mode, ex-
i
X particulary focusing on the nature of non-Fermi liquid plicitly demonstrated in nematic quantum criticality [9].
r physics and mechanism of superconductivity out of the In this study we construct an Eliashberg theory for
a non-Fermi liquid state [5]. Combined with the topolog-
multi-scale quantum criticality, considering ferromag-
ical band structure, critical degrees of freedom given by
netic quantum criticality in the topological surface. A
topological excitations in the duality picture will carry
novelingredientistointroduceananomalousself-energy
fermion quantum numbers, thus a novel type of non-
associatedwiththespin-flipchannel. Suchananomalous
Fermi liquid physics may arise.
self-energyturnsouttobeessentialforself-consistencyof
Inthispaperwestudyferromagneticquantumcritical- the Eliashberg framework because this off diagonal self-
ityinthe surfaceofthreedimensionaltopologicalinsula- energy cancels the normal self-energy exactly in the low
tors fromthe side ofa disorderedphase. A recentanaly- energy limit, preserving the dynamics of both z =3 and
sisbasedontherandomphaseapproximation(RPA)has z = 2 critical modes. This multi-scale quantum critical-
shown a multi-scale quantum critical point, where longi- ity is consistent with that in a perturbative analysis of
tudinal spin fluctuations are described by the dynamical the nematic quantum critical point, where a vertex cor-
critical exponent z =3 while transverse modes are given rection in the fermion bubble diagramcancels a singular
by z = 2 [6]. Although scaling can be demonstrated contribution due to the self-energy correction, maintain-
2
ingz =2forthetransversemode[9]. Weclaimthatsuch will be induced by doped magnetic impurities. It was
an off-diagonal self-energy generates an artificial electric demonstrated that Ruderman-Kittel-Kasuya-Yosida in-
field in the energy-momentum space [10] in addition to teractions between doped magnetic impurities are ferro-
theBerrycurvature. Basedonthe semi-classicalapprox- magnetic [11]. Through this exchange mechanism, mag-
imation for the Hall conductivity, we find an anomalous netic atoms are expected to forma ferromagneticallyor-
divergentbehavioratzerotemperature. Wediscussphys- deredfilm, deposited uniformly on the surface ofa topo-
icalimplicationofthisphenomenologyinconnectionwith logical insulator. In this case a possible ferromagnetic
the topological nature of Dirac fermions. transition may be realized, controlling the distance of
doped impurities in the regular impurity pattern.
We see two important features in dynamics of elec-
II. HERTZ-MORIYA-MILLIS THEORY IN THE
trons on the topological surface state. The first is that
SURFACE OF THE TOPOLOGICAL INSULATOR
spin dynamics is quenched to the orbital motion of sur-
face electrons. As a result, one finds that the role of
A. Model an external magnetic field differs from that in graphene,
where the pseudo-spin quantum number is constrained
We start from an effective field theory for ferromag- to the orbital motion. In particular, one of the authors
netic quantumcriticality inthe surfaceof topologicalin- couldrevealthatthe KondoeffectandFriedeloscillation
sulators around the magnetic impurity should be modified in the
presenceofanexternalmagneticfield,comparedwiththe
Z = DψαDφie−R0βdτRd2rL, graphene structure [12]. The second aspect is more fun-
Z damental,that is,only anodd number ofDirac fermions
L=ψα†[(∂τ −µ)δαα′ +ivfzˆ·(σ×∇)αα′]ψα′ is allowed to appear in the topological surface state [3].
+|∂ φ |2+v2|∇φ |2+m2|φ |2+V(|φ |) As a result, quantum anomaly is realized, giving rise to
τ i b i b i i nontrivial topological properties of the system [13]. For
+gφiψα†σαiα′ψα′, (1) example, a vortex in the XY ordered state will carry an
electric charge, giving rise to an anomalous Hall effect,
ψ
where ψ = ↑ represents an electron field in the different from the ”conventional” anomalous Hall effect
(cid:18)ψ↓ (cid:19) inconventionalferromagnets[2]. Itshouldbenotedthat
Nambu-spinor representation and φ=φ xˆ+φ yˆcorre-
x y this phenomenology does not occur in usual condensed
sponds to a spin excitation. We consider the case of an
matter systems because the quantum anomaly is often
XY spinforsimplicity, whichcanbe generalizedinto the
cancelled by the so called fermion doubling effect [3].
case of O(3) spin. v is the Dirac velocity and µ is the
f
Recently, interaction effects have been investigated in
chemical potential for surface electrons. σ is the Pauli
the graphene structure, proposing an interesting phase
matrix, associated with the spin state. v is the spin-
b
fluctuation velocity and m2 is the inverse of correlation- diagram based on the quantum Monte Carlo simulation
b [14]. Immediately, such a phase diagramwas interpreted
length for spin fluctuations, corresponding to an XY or-
dered state when m2 < 0 and a quantum disordered invariousanalyticalscenarios,where the singlet-channel
phase when m2 > 0b. V(|φ |) is an effective potential interaction was emphasized [15]. On the other hand, we
b i focus on the role of spin-flip scattering, allowed by in-
for spin fluctuations, where an easy plane anisotropy al-
teractions in the spin-triplet channel. We introduce an
lows the XY spin dynamics only. g is an effective cou-
anomalous electron self-energy associated with the spin-
pling constant between electrons and spin excitations,
flipscattering,anddiscussitspossibleimplicationincon-
where tuning g leads the spin correlation-length to di-
nection with the topological nature of the surface state.
verge, resulting in quantum criticality. The quantum
critical point between the XY ordered and disordered
phases is defined by the vanishing effective mass for spin
fluctuations, tuned by the coupling parameter g.
B. Eliashberg approximation
This effective field theory would be realized when an
insulating ferromagnet lies on the surface of topologi-
cal insulators. In this case ferromagnetic spin fluctua- One can perform an Eliashberg approximation for the
tions denoted by φ describe those in the insulating fer- effective Lagrangian [Eq. (1)] at the quantum critical
i
romagnet. Then, the coupling parameter g represents point (m2 = 0), where both fermion and boson self-
b
the strength of couplings between spins in the ferro- energy corrections are determined fully self-consistently
magnet and itinerant electrons on the topological sur- in the one-loop level. Considering the cumulant expan-
face. Another similar setting is that ferromagnetism sionuptothesecondorder,wecanfindaneffectiveaction
3
SQC =Z d3xZ d3x′(cid:16)ψα†(x)[{(∂τ −µ)δαα′ +ivfzˆ·(σ×∇)αα′}δ(x−x′)+Σαα′(x−x′)]ψα′(x′)
−NσΣαα′(x−x′)Gα′α(x′−x)
(cid:17)
+ d3x d3x′ φ (x)[(−∂2−v2∇2)δ(x−x′)δ +Π (x−x′)]φ (x′)−Π (x−x′)D (x−x′)
Z Z (cid:16) i τ b ij ij j ij ij (cid:17)
g2
+ d3x d3x′D (x−x′)tr σ G(x−x′)σ G(x′−x) . (2)
ij i j
2 Z Z (cid:16) (cid:17)
Σαα′(x − x′) and Gα′α(x′ − x) are the electron self- called the Luttinger-Ward functional [16], which is the
energyandGreen’sfunction,respectively,whereαandα′ key ingredient for an interaction effect obtained in the
representspinstates,↑and↓. Π (x−x′)andD (x−x′) Eliashberg approximation.
ij ij
arethespin-fluctuationself-energyandGreen’sfunction, Integrating over electrons and spin fluctuations, we
respectively,whereiandj expressspindirections, xand obtain an effective free energy as a functional for self-
y. All repeated indices are summed. The last term is energies
N d2k
FLW[Σ(iω),Π(q,iΩ)]=− βσ Xiω Z (2π)2trαα′nln[−G−1(k,iω)]+Σ(k,iω)G(k,iω)o
1 d2q
+ tr ln[D−1(q,iΩ)]−Π(q,iΩ)D(q,iΩ)
β XiΩ Z (2π)2 ijn o
g2 1 d2q 1 d2k
+Nσ 2 β XiΩ Z (2π)2β Xiω Z (2π)2Dij(q,iΩ)trαα′(cid:16)σiG(k+q,iω+iΩ)σjG(k,iω)(cid:17), (3)
where is the bare propagator of electrons, where ǫ is the two
ij
dimensional antisymmetric tensor with i,j = x,y. N
−1 σ
G(k,iω)= g−1(k,iω)−Σ(k,iω) , representsthespindegeneracy,inourcaseNσ =2. trαα′
(cid:16) (cid:17) and tr mean trace for spin states and spin directions,
ij
−1
D(q,iΩ)= [Ω2+v2q2]I+Π(q,iΩ) (4) respectively.
b
(cid:16) (cid:17)
are fully renormalized propagators of electrons and spin
Minimizing the free energy functional with respect
fluctuations, respectively.
to both self-energy corrections, we obtain fully self-
consistent coupled Eliashberg equations for electron and
(iω+µ)I +v ǫ k σ
g(k,iω)= f ij i j boson self-energies
(iω+µ)2−v2k2
f
g2 1 d2k
Πij(q,iΩ)=Nσ 2 β Xiω Z (2π)2trαα′(cid:16)σiG(k+q,iω+iΩ)σjG(k,iω)(cid:17),
1 d2q
Σ(k,iω)=g2 D (q,iΩ)σ G(k+q,iω+iΩ)σ . (5)
β Z (2π)2 ij i j
XiΩ
As discussed before, the triplet interactionchannel gives diagonal term in the electron self-energy matrix. The
rise to the spin-flip scattering, corresponding to an off- normal Eliashberg self-energy is proposed to depend on
4
frequency only because the momentum dependence is C. Polarization function
regularandthe singularbehaviorcanbe shownfromthe
frequency dependence [7, 17]. On the other hand, the
Inserting the self-energy expression into the Green’s
momentumdependencefortheanomalousself-energyhas
function, we obtain
not been clarified yet. We propose the following ansatz
for the electron self-energy ˆ
[iω+µ−Σ(iω)]I +[v |k|+Φ(iω)]ǫ k σ
G(k,iω)= f ij i j.
Σ(k,iω)=Σ(iω)I+Φ(iω)ǫijkˆiσj, (6) [iω+µ−Σ(iω)]2−[vf|k|+Φ(iω)]2
where the momentum dependence for the off-diagonal Basedonthis expression,wecanevaluatethe bosonself-
self-energyis givenby the same dependence as the topo- energy, given by the fermion polarization bubble. A de-
logical band structure. In other words, the off-diagonal tailed procedure is shown in appendix A.
self-energyisaddedtothe dispersionrelationasthe nor- It turns out that spin dynamics at the XY ferromag-
ˆ
mal self-energy in the conventional system. k=k/|k| is netic quantum critical point is given by the following ef-
the unit vector. fective Lagrangian
|Ω| Ω2
L = C g2 +v2|q|2 φ (q,iΩ)P φ (−q,−iΩ)+ C g2 +v2|q|2 φ (q,iΩ)(δ −P )φ (−q,−iΩ()7,)
φ L v |q| L i ij j T v2|q|2 T i ij ij j
(cid:16) f (cid:17) (cid:16) f (cid:17)
where where D (q,iΩ) is the kernel of the longitudinal
L(T)
q q (transverse) mode. Based on this spin-fluctuation prop-
Pij = |qi|2j agator, we find an electron self-energy in the Eliashberg
approximation. An important point is the role of the
is the projectionoperatorto satisfy P P =P . C multi-scalequantumcriticalityintheelectronself-energy.
ik kj ij L(T)
is a constantof the orderof 1 andv is the renormal- It turns out that the electron self-energy is governed by
L(T)
izedvelocityforspinfluctuations,whereL(T)represents the longitudinal spin dynamics at low energies because
the longitudinal (transverse) mode. An important fea- the z = 3 critical dynamics gives rise to more singular
ture for spindynamics is multi-scalequantumcriticality, corrections than the z = 2 transverse mode. Actually,
where the longitudinalspin dynamics is givenby the dy- thisresultwasexpectedintheEliashbergapproximation.
namical exponent z = 3 while the transverse one is de- But, itwasclaimedthata morecomplicatedstructureis
scribedbyz =2. Themulti-scalequantumcriticalitycan hidden beyondthe Eliashbergapproximation[18], which
be interpreted as follows. The longitudinal spin dynam- will be discussed in the last section.
icsisnotinvolvedwiththespin-flipscattering,described
by the z = 3 ferromagnetic quantum criticality. On the
other hand, the transverse dynamics is associated with
thespin-flipprocess. Inthetopologicalsurfacestatethis
process appears with finite momentum transfer, similar
to the process with antiferromagnetic fluctuations. As
a result, the transverse spin dynamics is described by
z = 2. We note that the self-consistent Eliashberg ap-
proximation gives essentially the same result as the ran-
dom phase approximation[6].
D. Electron self-energy
The spin-fluctuation propagator is given by
P δ −P
D (q,iΩ)= ij + ij ij
ij CLg2v|fΩ|q|| +vL2|q|2 CTg2vf2Ω|q2|2 +vT2|q|2 Our novel point is that the anomalous self-energy cor-
≡D (q,iΩ)P +D (q,iΩ)(δ −P ), (8) rection is exactly the same as the normal one, given by
L ij T ij ij
5
g2 1 ∞ sgn(ω+Ω)
Σ(iω)=−Φ(iω)≈−i dqq D (q,iΩ)+D (q,iΩ) ∝|ω|2/3, (9)
L T
2 β XiΩ Z0 (cid:16) (cid:17) (ω+Ω)2+vfq2
p
wherethelastapproximateequalityisgivenbythez =3 approximation, neglecting vertex corrections for self-
longitudinal mode D (q,iΩ). This equivalence is at the energies. Recently, it has been clarified that two di-
L
heart of the self-consistency in the Nambu-Eliashberg mensionalFermisurfaceproblemsarestillstronglyinter-
framework. As discussed in the introduction, the self- acting even in the large-N limit [19–24], implying that
energycorrectioninthefermionbubblediagramchanges vertex corrections should be incorporated. An impor-
the z = 2 dynamical exponent in the transverse mode. tant question is whether these vertex corrections give
Actually, one can see this effect from Eqs. (A9) and rise to novel critical exponents beyond the Eliashberg
(A13) in appendix A. If the anomalous self-energy is theory. Several perturbative analysis demonstrated that
notintroduced,thenormalself-energycontributiondom- although ladder-type vertex corrections do not change
inates over the bare frequency, modifying the dynamical critical exponents of the Eliashberg theory, Aslamasov-
exponentofthetransversemodefromz =2toz =12/5. Larkincorrectionsresultinmodificationsforsuchcritical
However,introduction of the anomalous self-energy can- exponents[21,22]. Ifthisisageneralfeaturebeyondthis
cels the normal self-energy exactly in the low energy levelofapproximation,variousquantumcriticalphenom-
limit, recovering the RPA result [6]. ena[25]shouldbereconsideredbecausenovelanomalous
Itisinterestingtocomparethemechanismofcancella- exponentsinfermionself-energycorrectionsareexpected
tion with the perturbative analysis in nematic quantum to cause various novel critical exponents for thermody-
criticality, where next leading corrections given by elec- namics,transport,andetc. However,consideringour re-
tron self-energy and Maki-Thompson vertex corrections centexperiencesoncomparisonwithvariousexperiments
are introduced for dynamics of critical bosonic modes inheavyfermionquantumcriticality,wearesurprisedat
[9]. Since the fermion self-energy is driven by z =3 crit- the fact that critical exponents based on the Eliashberg
ical fluctuations, the self-energy correction in the trans- theory explain thermodynamics [26], both electrical and
verse mode, corresponding to the z =2 dynamics in the thermal transport coefficients [27], uniform spin suscep-
RPAlevel,givesrisetoanadditionalsingularcorrection, tibility [28], and Seebeck effect [29] quite well.
changing the dynamical critical exponent from z = 2
Recently, one of the authors investigated the role of
to z = 12/5. This weird result is cured by the Maki-
vertex corrections non-perturbatively, summing vertex
Thompson vertex correction, where the singular correc-
corrections up to an infinite order [19, 24]. It turns out
tion of the self-energy is cancelled by the singular vertex
that particular vertex corrections given by ladder dia-
correction exactly, which is a fundamental cancellation
grams do not change Eliashberg critical exponents at all
based on the Ward identity.
[24], consistent with the perturbative analysis. This was
Theself-consistencyoftheNambu-Eliashbergtheoryis
performed in a fully self-consistent way, resorting to the
farfromtrivialityformulti-scalequantumcriticality,and
Ward identity. In contrast with the previous perturba-
this effective field theory is certainly beyond the simple
tive analysis, Aslamasov-Larkin corrections were shown
Eliashberg description in respect that vertex corrections
nottomodifytheEliashbergdynamics[19]. Resortingto
areintroducedinsome sense. However,it turns outthat
the analogywithsuperconductivity,wherethe supercon-
critical exponents for thermodynamics is completely the
ducting instability described by the Aslamasov-Larkin
same as the conventional Eliashberg theory without an
vertex correction is reformulated by the anomalous self-
anomalous self-energy correction [19].
energyintheEliashbergframeworkoftheNambuspinor
representation [30], we claimed that the off-diagonal
self-energy associated with the 2k particle-hole chan-
F
E. Stability of the Eliashberg framework
nel incorporates the same (Aslamasov-Larkin) class of
quantum corrections in the Nambu spinor representa-
The Eliashberg theory is non-perturbative in respect tion. We evaluated an anomalous pairing self-energy
that quantum corrections are introduced fully self- in the Nambu-Eliashbergapproximation,which vanishes
consistently in the one-loop level, consistent with one- at zero energy but displays the same power law depen-
loop renormalization group analysis. Indeed, the scaling denceforfrequencyasthenormalEliashbergself-energy.
expressionofthefreeenergywasderivedexplicitly,based As a result, even the pairing self-energy correction does
on the Eliashberg approximation [8]. In this respect the notmodify the Eliashbergdynamics without the Nambu
Eliashbergtheoryforquantumcriticalityimpliesthethe- spinor representation, resulting in essentially the same
ory with critical exponents given by the Eliashberg ap- scaling physics for thermodynamics at quantum critical-
proximation, satisfying the scaling theory. ity. However, we admit that this issue should be investi-
One cautious person may criticize the Eliashberg gatedmoresincerely,resortingtoadirectsummationfor
6
such a class of quantum corrections. The Berrycurvature andartificialelectric fieldare ex-
pressed in terms of the pseudospin vector
1 ˆ ˆ ˆ
III. ARTIFICIAL ELECTRIC FIELD IN THE Bz = 2∂kxN ×∂kyN ·N,
ENERGY-MOMENTUM SPACE
1 ˆ ˆ ˆ
E = ∂ N ×∂ N ·N,
x 2 ky ω
An essential question is on the role of the anoma- 1 ˆ ˆ ˆ
lous self-energy in transport. Considering that the off- Ey = 2∂ωN ×∂kxN ·N, (13)
diagonalself-energyhasthesamemomentumdependence
as the kinetic-energy term, it is natural to examine its where Nˆ =N/|N| is the unit vector.
role in the Hall conductivity. One can understand this expression based on another
An effective equation of motion for quasi-particle dy- equivalent representation, where the field strength is ex-
namics was proposed as follows [10] pressed by the gauge field. Resorting to the CP1 repre-
sentation [31]
dR dk
=v+(B−E ×v)× ,
dt dt N ·σ =|N|Uσ3U†, (14)
dk dR
=−E+B× , (10) where U is an SU(2) matrix field to describe a direction
dt dt
of the pseudospin vector, we can define an SU(2) gauge
describingthewave-packetdynamicsinthesemi-classical connection
level. R represents the center coordinate of the wave-
packet and k expresses the Bloch wave vector. E and Aµ =−i[∂µU†]U. (15)
B are external electric and magnetic fields while E and
B are artificial electric and magnetic fields in the en- This naturally leads to both Berry curvature and artifi-
ergy and momentum space. B is usually referred as the cial electric field
Berry curvature. v = ∂Ek is the group velocity, where
∂k B =∂ A −∂ A ,
Ek is the quasi-particle energy dispersion. A key ingre- z kx y ky x
dient is that interactions give rise to an artificial elec- Ex =∂kyAω −∂ωAky,
tric field in the energy-momentum space, modifying the E =∂ A −∂ A , (16)
y ω kx kx ω
quasi-particle dynamics according to the Maxwell equa-
tion. As a result, the Hall conductivity should be cor- where the U(1) projectedcomponent ofthe SU(2) gauge
rected in the following way [10] field is given by A =A(3).
µ µ
InsertingthepseudospinvectorEq. (12)intoEq. (13),
e2 d2k weobtainbothBerrycurvatureandartificialelectricfield
σ = f(E ) B −[v E −v E ] ,(11)
xy ~ Z (2π)2 k (cid:16) z y x x y (cid:17)
1 [v +Φ(ω)/k]2[m+∆d (ω)]
f z
B ≈ ,
where the Berry curvature is shifted by E × v due to z 2 3/2
[v k+Φ(ω)]2+[m+∆d (ω)]2
interactions. f(E ) is the Fermi-Dirac distributionfunc- f z
k (cid:16) (cid:17)
tion. In this section we calculate this quantity with an 1 1
E ≈ k F, E ≈ k F,
introduction of the anomalous self-energy correction. x 2 x y 2 y
We consider the following effective Hamiltonian
H =N(k,ω)·σ,whichincorporatestheeffectofinter- where F is given by
eff
actions through the self-energy. The pseudospin vector
[∂ Φ(ω)](1/k)[v +Φ(ω)/k][m+∆d (ω)]
is given by F =− ω f z
3/2
[v k+Φ(ω)]2+[m+∆d (ω)]2
f z
N(k,ω)=d(k)+∆d(ω), (cid:16) (cid:17)
d(k)=−vfkyσˆx+vfkxσˆy+dz(k)σˆz, + [vf +Φ(ω)/k]2[∂ω∆dz(ω)] .
3/2
∆d(ω)=−Φ(ω)(k /k)σˆ +Φ(ω)(k /k)σˆ +∆d (ω)σˆ , [v k+Φ(ω)]2+[m+∆d (ω)]2
y x x y z z f z
(cid:16) (cid:17)
(12)
Considering the group velocity
where the d(k)vector is nothing but the bare dispersion
and ∆d(ω) is associated with the anomalous self-energy vy =kyV, vx =kxV,
correction. d (k) is introduced for time reversalsymme-
z
with
trybreakingasthez-directionalmagneticfield,settobe
d (k) → m. ∆d (ω) is also introduced for consistency.
z z 1 [v k+Φ(ω)]v
f f
V = ,
k = kx2+ky2 is the amplitude of the momentum. k [vfk+Φ(ω)]2+[m+∆dz(ω)]2
q
p
7
wefindthatthecontributionoftheelectricfieldE tothe ified in the presence of the self-energy correction. As a
shift of B in Eq. (11) vanishes identically result, we obtain the following expression for the Hall
conductivity
v E −v E =0.
y x x y
On the other hand, the Berry curvature should be mod-
1e2 ∞ 1 [v k+Φ(ω)]2[m+∆d (ω)] 1e2
f z
σ = dk = F[k →∞]−F[k →0] , (17)
xy 2 h Z0 k [v k+Φ(ω)]2+[m+∆d (ω)]2 3/2 2 h(cid:16) H L (cid:17)
f z
(cid:16) (cid:17)
where
[m+∆d (ω)]Φ(ω) [m+∆d (ω)]Φ2(ω) 1
z z
F[k →∞]= + ln ,
H [m+∆dz(ω)]2+Φ2(ω) Φ2(ω)+[m+∆dz(ω)]2 3/2 (cid:16)Φ(ω)+ Φ2(ω)+[m+∆dz(ω)]2(cid:17)
(cid:16) (cid:17) (cid:16) (cid:17) p
−[m+∆d (ω)]3+[m+∆d (ω)]Φ2(ω)
z z
F[k →0]=
L 3/2
Φ2(ω)+[m+∆d (ω)]2
z
(cid:16) (cid:17)
[m+∆d (ω)]Φ2(ω) k
z L
+ ln .
Φ2(ω)+[m+∆d (ω)]2 3/2 (cid:16)[m+∆dz(ω)]2+Φ2(ω)+ Φ2(ω)+[m+∆dz(ω)]2 Φ2(ω)+[m+∆dz(ω)]2(cid:17)
z
(cid:16) (cid:17) p p
It is straightforward to see that the noninteracting case IV. DISCUSSION AND SUMMARY
in Φ(ω) = 0 recovers the well known result, that is, the
half Hall conductivity [3], where F[k → ∞] = 0 and
H In this study we constructed an Eliashbergframework
F[k →0]=−sgn(m).
L for the Hertz-Moriya-Millis theory [7], describing ferro-
magneticquantumcriticalityinthesurfacestateofthree
An interesting point in this expressionis that the Hall
dimensional topological insulators. An idea is to intro-
conductivity diverges in the k →0 limit, seen from the
L duceananomalousself-energycorrectionbeyondthepre-
log term. Although we do not understand the reasonfor
vious study [6], resulting from spin-flip scattering. We
thisdivergence,itimpliesthatweshouldincorporatever-
couldshowthatouransatzfortheanomalousself-energy
tex corrections even for the intrinsic topologicaleffect in
allows a set of fully self-consistentsolutions for electrons
the Hall conductivity when interactions are introduced.
and spin fluctuations.
If we take into account vertex corrections, this log di-
The spin-fluctuation dynamics was shown to coincide
vergence may be re-summed to result in the common
withthesolutionbasedontherandomphaseapproxima-
power-lawbehavior with an exponent, which vanishes in
tion, where both z = 3 and z = 2 quantum critical dy-
the k →0 limit.
L
namicsarise[6]. The z =3quantumcriticalitydescribes
We suspect that the underlying mechanism for this thelongitudinalspindynamics,whereonlyforwardscat-
divergence may be charged vortices. As discussed be- tering contributions result in the z = 3 Landau damp-
fore, the parity anomaly assigns an electric charge to ing, while the z = 2 quantum criticality for transverse
a vortex excitation [13]. From the ordered side in the spin fluctuations originates from the spin-flip scattering,
duality picture, such vortex excitations are well defined involved with the finite-momentum transfer due to the
particles, thus taken into account with electrons on an spin-orbital quenching.
equal footing. An interplay between electrons and vor- This multi-scale quantum criticality was shown to al-
tices would be described by the mutual Chern-Simons lowtheself-consistentNambu-Eliashbergsolutionforthe
theory[3], wheretheir mutualstatistics isguaranteedby electron dynamics. The normal self-energy turns out to
themutualChern-Simonsterm. Itisnotclearatallhow followthez =3quantumcriticalitybecausesuchcritical
suchaninterplayaffectstransport,inparticular,theHall fluctuationsgiverisetomoresingularcontributionsthan
conductivity. It will be an interesting future direction to thez =2transversemode. Wefoundthattheanomalous
investigate this problem in the duality picture because self-energyshowsexactlythesamefrequencydependence
topologicalexcitationscarrynontrivialfermionquantum as the normalself-energy althoughthe underlying mech-
numbers in this situation. anism is not completely clear. The equivalence between
8
the normal and anomalous self-energies with a different tex corrections should be introduced up to an infinite
signis atthe heartofthe self-consistencyinthe Nambu- order. Basedonacomplicatedrenormalizationgroupar-
Eliashberg theory for multi-scale quantum criticality. In gument, the authors proposed an interesting expression
otherwords,thesimpleEliashbergtheorywithouttheoff for the electron Green’s function
diagonal self-energy cannot be self-consistent for multi-
scalequantumcriticality. Wearguedthatthemechanism G(k,iω)∝ D0−η .
of cancellation between the normal and anomalous self- [iω+Σ(iω)−ǫk]1−η
energiesinthefermionbubble diagrammayberootedin
the Wardidentity, comparingourresultwiththe pertur- D0 has a constant of an energy unit proportional to the
bativeanalysisfornematicquantumcriticality,wherethe Fermienergy,andΣ(iω)∝|ω|2/3 isthez =3self-energy.
singular self-energy correction is cancelled by the vertex ǫk is the band dispersion. It should be noted that the
correction based on the Ward identity [9]. anomalousexponentη resultsfromscatteringwithz =2
Wedemonstratedthatsuchanoff-diagonalself-energy critical fluctuations.
gives rise to an artificial electric field in the energy- In the present system a significant quantity is the off
momentumspacebeyondthestaticBerrycurvaturefrom diagonal self-energy. Although such an anomalous self-
the topological band structure. In particular, we calcu- energywillnotbeallowedinnematicquantumcriticality
lated the anomalous Hall conductivity in terms of both duetotheabsenceofthespin-orbitcoupling,itmayplay
Berry curvature and artificial electric field. Although an important role for the exponent η in ferromagnetic
the contribution from the artificial electric field vanishes quantum criticality of the topological surface. A more
identically, we could find that the Berry curvature is completeframeworkwouldbetointroducevertexcorrec-
modified due to the anomalous self-energy correction, tionsfullyself-consistentlywithself-energycorrectionsin
resulting in the log-divergence for the Hall conductiv- the Nambu-spinorrepresentation,where the off-diagonal
ity. We argued that the log-divergence may disappear self-energy contribution is also included.
if vertex corrections are re-summed up to an infinite or- In summary, quantum criticality with the topological
der,whichisexpectedtoturnthelog-divergenceintothe band structure is expected to open a novel direction of
power-law dependence, vanishing algebraically. We sug- research in respect that the interplay between interac-
gested an idea that this phenomenology may be related tion and topology may cause unexpected non-Fermi liq-
with charged vortices in the duality picture, where the uid physics. We found an anomalous behavior in the
electron quantum number of the vortex results from the intrinsic topological contribution of the Hall coefficient,
quantumanomaly,anessentialfeature ofthe topological the source of which is the energy-momentum-space field
surface state. strengthbeyondthe static Berry curvatureas a resultof
Before closing, we discuss the stability of the Eliash- theinterplaybetweenquantumcriticalityandtopological
berg approximation in another aspect, considering an structure.
interesting recent study for nematic quantum criticality
[18]. The nematic quantum critical point is well known
toshowmulti-scalequantumcriticality,wherecriticaldy- Acknowledgments
namicsofnematicfluctuationsisgovernedbybothz =3
and z = 2. An important notice is that although the We would like to thank S. Ryu for his eight-hours lec-
self-energy correction results from scattering with z = 3 ture on topologicalinsulators in APCTP.K.-S. Kimwas
critical fluctuations dominantly, scattering with z = 2 supportedbytheNationalResearchFoundationofKorea
critical modes gives rise to the logarithmic singularity (NRF) grant funded by the Korea government (MEST)
for the quasi-particleresidue [18], which means that ver- (No. 2010-0074542).
Appendix A: Polarization function
In appendix A we derive the self-energy correction for spin fluctuations, where the longitudinal spin dynamics is
givenby z =3 while the transverseone is describedby z =2. Insertingthe electronGreen’s function withthe ansatz
for the electron self-energy matrix into the Eliashberg equation for the boson self-energy, we obtain
g2 1 d2k [iω+iΩ+µ−Σ(iω+iΩ)]I+[v |k+q|+Φ(iω+iΩ)]ǫ (kˆ +qˆ )σ
Π (q,iΩ)= tr σ f nm n n m
ij 2 β Xiω Z (2π)2 n i [iω+iΩ+µ−Σ(iω+iΩ)]2−[vf|k+q|+Φ(iω+iΩ)]2
ˆ
σ [iω+µ−Σ(iω)]I+[vf|k|+Φ(iω)]ǫn′m′kn′σm′ , (A1)
j [iω+µ−Σ(iω)]2−[v |k|+Φ(iω)]2
f o
9
where the following approximationfor the momentum
kˆ +qˆ ≡ kn+qn ≈ kn + qn (A2)
n n |k+q| |k | |k |
F F
is utilized.
Resorting to identities for Pauli matrix products
σ σ =δ I+iǫ σ ,
i j ij ijl l
σ σ σ =δ σ +iǫ I −(δ δ −δ δ )σ ,
i m j im j imj ij ms is mj s
σiσmσjσm′ =δim δjm′I +iǫjm′l′σl′ +iǫimjσm′ −(δijδms−δisδmj) δsm′I+iǫsm′l′σl′ (A3)
(cid:16) (cid:17) (cid:16) (cid:17)
and performing the decomposition for the denominator, we can rewrite Eq. (A1) as follows
1 d2k 1 1
Π (q,iΩ)=N g2
ij σ β Z (2π)22[iω+iΩ+µ−Σ(iω+iΩ)]2[iω+µ−Σ(iω)]
Xiω
G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]+G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]
f f f f
n
+G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)]+G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)]
f f f f
o
[iω+iΩ+µ−Σ(iω+iΩ)][iω+µ−Σ(iω)]−[v |k+q|+Φ(iω+iΩ)][v |k|+Φ(iω)](kˆ+qˆ)·kˆ δ
f f ij
n(cid:16) (cid:17)
+[vf|k+q|+Φ(iω+iΩ)][vf|k|+Φ(iω)][ǫinǫjn′(kˆn+qˆn)kˆn′ +ǫjnǫin′(kˆn+qˆn)kˆn′] , (A4)
o
where ǫ is an antisymmetric tensor with i,j =x,y, and
ij
1
G[iω,X]= .
iω+µ−Σ(iω)−X
We rearrange the above expression as
g2 1 d2k
Π (q,iΩ)=N δ
ij σ 4 β Z (2π)2 ij
Xiω
G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]+G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]
f f f f
n
+G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)]+G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)]
f f f f
o
+Nσg42β1 Xiω Z (d22πk)2n−(kˆ+qˆ)·kˆδij +[ǫinǫjn′(kˆn+qˆn)kˆn′ +ǫjnǫin′(kˆn+qˆn)kˆn′]o
G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]−G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,v |k|+Φ(iω)]
f f f f
n
−G[iω+iΩ,v |k+q|+Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)]+G[iω+iΩ,−v |k+q|−Φ(iω+iΩ)]G[iω,−v |k|−Φ(iω)] ,
f f f f
o
(A5)
where the frequency term in the numerator is reorganized.
It is convenient for the momentum integration to linearize the band dispersion near the chemical potential
ˆ ˆ
µ−v |k+q|≈−v k ·(k−k )−v k ·q, (A6)
f f f f f f
ˆ
where the Fermi momentum k =|k |k is defined from µ−v |k |=0.
f f f f f
10
Inserting the linearized band into Eq. (A5) and performing the decomposition for the denominator, we obtain
g2 1 d2k 1
Π (q,iΩ)≈N δ
ij σ 4 β Z (2π)2 ijiΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
−
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
+Nσg42β1 Xiω Z (d22πk)2n−(kˆ+qˆ)·kˆδij +[ǫinǫjn′(kˆn+qˆn)kˆn′ +ǫjnǫin′(kˆn+qˆn)kˆn′]o
1
ˆ
iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v k ·q
f f
1 1
− (A, 7)
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
where only dominant contributions are selected, which means that all terms with the chemical potential in the
denominator are safely neglected.
Before evaluating integrals, we summarize each component for the spin-fluctuation self-energy as
g2 1 d2k 1
Π (q,iΩ)=N
xx σ 4 β Z (2π)2iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
−
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
g2 1 d2k −(kˆ+qˆ)·kˆ+2(kˆ +qˆ )kˆ
y y y
+N
σ 4 β Z (2π)2iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
− ,
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
g2 1 d2k 1
Π (q,iΩ)=N
yy σ 4 β Z (2π)2iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
−
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
g2 1 d2k −(kˆ+qˆ)·kˆ+2(kˆ +qˆ )kˆ
x x x
+N
σ 4 β Z (2π)2iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
− ,
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
g2 1 d2k 2kˆ kˆ +(qˆ kˆ +qˆ kˆ )
Π (q,iΩ)=Π (q,iΩ)=−N x y x y y x
xy yx σ 4 β Z (2π)2iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v kˆ ·q
Xiω f f
1 1
− (.A8)
ˆ ˆ ˆ
niω−Σ(iω)−Φ(iω)−vfkf ·(k−kf) iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−vfkf ·(k−kf)−vfkf ·qo
First, we consider the diagonalcomponent of the boson self-energy. It is convenient for the momentum integration
to take the angular coordinate
Π (q,iΩ)
xx
g2 1 2π ∞ 2sin2θ
=N dθ dε
σ4v (2π)2β Z Z iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v q cosθ−v q sinθ
f Xiω 0 −∞ f x f y
1 1
−
iω−Σ(iω)−Φ(iω)−ε iω+iΩ−Σ(iω+iΩ)−Φ(iω+iΩ)−ε−v q cosθ−v q sinθ
n f x f y o
iπg2 ∞ dω 2π 2sin2θ[sgn(ω)−sgn(ω+Ω)]
=N dθ , (A9)
σ4v (2π)2 Z 2π Z iΩ−[Σ(iω+iΩ)−Σ(iω)]−[Φ(iω+iΩ)−Φ(iω)]−v q cosθ−v q sinθ
f −∞ 0 f x f y
where we considered the zero temperature limit in the last equality.