Table Of ContentRole of domain wall fluctuations in non-Fermi liquid behavior of metamagnets
Vladimir A. Zyuzin1 and A.Yu. Zyuzin2
1Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA
2A.F. Ioffe Physical-Technical Institute, 194021, St.Petersburg, Russia
Westudyresistivity temperaturedependenceofa threedimensional metamagnet nearthemeta-
magnet phase transition point in the case when magnetic structure tends to split into regions with
high and low magnetization. We show that in the case of weak pinning the spin relaxation time of
0 domain wall is much larger than that of the volume spin fluctuations. This opens a temperature
1
rangewhereresistivitytemperaturedependenceisdeterminedbyscatteringofconductingelectrons
0
bythedomainwallfluctuations. Weshowthatitleadstoquasi-linearlowtemperaturedependence
2
of resistivity.
c
e PACSnumbers:
D
3 I. INTRODUCTION phase boundary can strongly influence temperature de-
1 pendence of electronic kinetic properties of material.
Inthis paper weconsiderthe temperature dependence
] UnderstandingthedeviationfromtheFermiliquidbe-
l of resistivity due to scattering of conducting electrons
l havior at phase transition critical points is a current
a by domain walls fluctuations in broadened metamag-
research question. The most important are deviations
h netic first order phase transition. Domain walls corre-
- from quadratic temperature dependence of resistivity
spondtocrossoverfromstateswithlowmagnetizationto
s near quantum critical points. Theory proposes that ori-
e highmagnetization. Theoreticallydomainwallsbetween
ginofnonFermiliquidbehaviorisscatteringofconduct-
m phases appear in regions of space where two minima of
ing electrons by bosonic critical soft modes. In case of
. free energy have equal value. The contribution to resis-
t ferromagnetic transition these modes are collective spin
a tivity will only be significant when the wall fluctuations
excitationswhoserelaxationtime divergesnearthetran-
m relaxation time is comparable or larger than that of the
sition point. In nearly magnetic metals at temperatures
- volume magnetic fluctuations.
larger than inverse spin relaxation time resistivity due
d
The paper is organized as following. In section II we
n toelectronscatteringbyspinfluctuations,stronglydevi-
give a description of the system of coupled conducting
o atesfromFermiliquidquadraticdependence. Itbecomes
electrons and itinerant electrons responsible for metam-
c linear or even saturates.1–4 In strongly Stoner enhanced
[ agnetic state. We propose a model where the magnetic
paramagnetic metals deviation might starts at very low
state splits in to domain walls. In section III we obtain
1 temperature.
solutionforthedomainwallprofileandconsiderthefluc-
v It was experimentally shown that in the region near
tuations around it. We then obtain dynamical magnetic
7
quantum critical point, magnetic state of many systems
1 susceptibility of the domain wall fluctuations. In section
experiencefirstorderphasetransitionwithseparationof
6 IV the temperature dependence of the resistivity caused
2 different phases.5–14 In this case the temperature depen- by the scattering of conducting electrons by the domain
2. dence of resistivity is characterized by non-Fermi liquid wall fluctuations is considered. We find transition from
resistivity exponents, which depend on proximity of the
1 quadratictolineartemperaturedependencewithincreas-
system to the quantum critical point. And it was shown
0 ingtemperature. Whilethistemperaturedependenceco-
1 that dynamics of magnetic state in this case is not criti- incides with that due the volume spin-fluctuations.3, it
v: cal. starts at much lower temperature. In section V we dis-
i Nonuniformmagneticstatecanbe consideredasadis- cuss the interaction correction18 at small temperatures.
X tinctive phase itself.15,16 For example in metamagnetic Itisknownthatconductingelectronsscatteringbyimpu-
r transitionofthinmagnetic filmsmagneto-dipoleinterac- ritiesandspinfluctuationsresultsinimportantcontribu-
a
tioncangiverisetoformationofmagneticdomains.16 In tionstothetemperaturedependenceofconductivity.19–22
anotherexampledomainscanbeformedinspatiallyran- We obtain that for domain wall fluctuations the interac-
domvariationofeffectivemagneticfield. Thismodelwas tion correction to conductivity has temperature depen-
proposed to explain unusual magnetotransporteffects of dence predicted for the 2D volume spin fluctuations.21
ferromagnetic alloys.17
The phase separation in first order magnetic phase
transitionmightexists in considerablerangeoftempera- II. DESCRIPTION OF THE MODEL
ture,pressureandotherparameters. Inbroadenedphase
transitionthe ratio ofphases volume,structure and area We consider a three dimensional metallic metamagnet
of phase boundary depend on temperature. In broad- where s electronsare consideredto be conducting,and
−
ened phase transition such factors as large difference in d electronsresponsibleforthemagneticstate. Coupling
−
phases resistivity, or strong electron scattering by the of conducting electrons with bosonic modes is obtained
2
by integrating out d electrons.23 Action describing con- T
−
ducting electrons in random impurity potential V(r) is
∇2
Ss =T drψα†(r,Ωn) iΩn+ +µ−V(r) ψα(r,Ωn),
Z (cid:20) 2m (cid:21)
XΩn
(1)
where ψ describes electrons with spin α and mass m,
Matsubara frequency Ω = πT(2n+1) (T is tempera-
n
ture), with Fermi level µ, and we have set ¯h = 1. We
assume impurity potential to satisfy < V(r) >= 0, and h
<V(r)V(r′)>=1/(2πντ)δ(r r′), where ν - density of ∆h
−
electronsper onespin, τ -isthe electronmeanfreetime.
The free energy of the magnetic state is described by
FIG.1: Schematicphasediagramofthebroadenedfirstorder
two energetically unequal minima. Application of mag-
phase transition. The solid curve shows putative transition,
netic field brings the system in to a state at which these while area enclosed by dashed curves represents the phase
two minima have the same energy, this point is called separated state.
metamagnetic phase transition point and the value of
the magnetic field at which it occurs is called metamag-
neticvalueofmagneticfield. Atthemetamagneticphase ing the damping is Γ(Q,ωn) = γD|ωQn2| which is valid for
transition point we will be describing the free energy of DQ2 > ω where D is the diffusion constant.25
n
| |
the metamagnetic state by two parabolas with minima Finally, action describing coupling of conducting elec-
at m(r,τ) = m , corresponding to high and low mag- trons with magnetization is
0
±
netizationstates. Inthis casethe domainwallsoriginate
1/T
from the spatial deviation of the magnetic field from its S =G dτ dr s(r,τ)m(r,τ), (4)
int
metamagnetic value, and we introduce such deviation as Z0 Z
h(r) (see figure 1). Assuming strong magnetic field we
heres(r,τ)isoperatorofspindensityofconductingelec-
only consider longitudinal component of magnetization
tronsalongthelongitudinalcomponentofmagnetization,
density m(r) (in units ofgµ 1) inthe action. Having
B ≡ G is a phenomenological coupling constant.
aboveassumptionsinhandwewritetheactiondescribing
the magnetization density m(r,τ) in the form
III. DOMAIN WALL FLUCTUATIONS
S [m]= 1/T dτ dr K ( m(r,τ))2
mag 0 2χ0 ∇
(cid:16) In this section we give a solution to the mean field
+ α (m(r,τR) m )R2 h(r)m(r,τ) +S , (2)
2χ0 | |− 0 − D equation for the domain wall and discuss fluctuations
(cid:17) around this solution. Let x be a coordinate normal to
here coefficient K−1/2 is of the order of electrons Fermi the domain wall and h(x0(ρ)) = 0, so that at x < x0
wavelength,α−1 isaStonerenhancementfactor,andχ0 it is state with high magnetization, and at x > x0 it is
is conducting electron spin susceptibility which we have a low magnetization state. Here ρ is a 2D coordinate
assumed to be equal for both high and low magnetiza- along the two dimensional domain wall. Equation for
tion phases. This assumption greatly simplifies further magnetization is obtained by varying the action (2) and
calculations, and does not affect main conclusions about we get
resistivity temperature dependence. We assume the spa-
d2
tial distribution of the h(r) to be of the Gaussian form K m+α(m m sign(m))=χ h(x). (5)
− dx2 − 0 0
and give the averaging procedure in the appendix.
The last term in (2) is the Landau damping described Whenh(r)isslowlyvaryingfunctiononascaleof K/α
by the domain wall can be approximated as flat. With this
p
assumptionthesolutionof(5)describingthedomainwall
is
S [m]= drdr T m(r,ω )Γ(r r,ω )m(r,ω ),
D ′ n ′ n ′ n
−
Z Xωn m= m (1 e √α/Kx x0(ρ))sign(x x (ρ))+χ0h(x).
(3) − 0 − − | − | − 0 α
here ω is the bosonic Matsubara frequency. At this (6)
n
point it is necessary to distinguish two cases of the form Letus considerfluctuations near this solution. Taking
ofthe damping. When the scatteringofconducting elec- second derivative of free energy we obtain equation for
trons by impurities is ballistic the Fourier image of Γ is eigenfunctions, which describe fluctuations
given by Γ(Q,ωn) = γv|FωQn|, which is valid for large mo- ( K 2+α)δm(r)
menta v Q > ω where v is Fermi velocity, and γ − ∇
is a damFping co|nnst|ant.3,24 IFn case of diffusive scatter- −|ddx2(αmm(0x))|x=x0δ(x−x0(ρ))δm(r)=ǫδm(r). (7)
3
Deriving this equation we have used the equality much smaller than domain wall curvature. Typical scat-
tering of conducting electrons by fluctuations of the do-
δ(x x (ρ))
0 mainwallischaracterizedbythemomentumoftheorder
δ(m(x))= − . (8)
d (m(x)) of β. Which is much larger than the momentum of the
dx x=x0
dynamical susceptibility of fluctuations. This allows to
(cid:12) (cid:12)
δ type potential in equa(cid:12)tion (7) is(cid:12)related to non ana- separately average over the domain wall direction and
lytical dependence of free energy on magnetization. The position, introducing concentration of wall n . The av-
W
equation(7)hasonlyoneboundedsolution,thusstrongly erageoverrandomǫ isapproximatedbysubstitutingthe
0
simplifyingconsiderationoffluctuations. Atslowlyvary- average value of ǫ in to the susceptibility. The details
0
ingx0(ρ)and ddxm(x0(ρ))onecansearchforthesolution of the described above averaging steps are given in the
of equation (7) in the form of a plane wave in ρ appendix to the paper.
From (17) we see that relaxation time of domain wall
δm(r)=Ψ0(x−x0(ρ))eiQρ, (9) fluctuations is proportional to ǫ−01 which is much larger
than that of the volume fluctuations since ǫ << α. At
where 0
small ǫ contribution of the domain wall fluctuations to
0
Ψ (x)= βe βx x0 , (10) the total susceptibility of the system is approximately
0 − | − |
and assuming ǫ=KQ2+pǫ0 we find δχ0 nW dxdx′Ψ0(x)Ψ0(x′)χ(0,0) χ0nW, (18)
∼ ∼ βǫ
Z 0
χ0 d −2
ǫ =α 1 1+ h(x) , (11) which can be of the same order as χ depending on the
0 " −(cid:18) m0√Kα(cid:12)dx (cid:12)(cid:19) # parameters. 0
(cid:12) (cid:12)
(cid:12) (cid:12)
whereβ = (α ǫ )/K. Atslo(cid:12)wlyvary(cid:12)ingh(r)wefind
0
−
that ǫ0 <<pα and we approximate β = α/K. In this IV. RESISTIVITY TEMPERATURE
case DEPENDENCE
p
2χ d
0
ǫ = h(x) . (12) Let us consider contribution to the resistivity due to
0
m β dx
0 (cid:12) (cid:12) domain wall scattering of conducting electrons. There
(cid:12) (cid:12)
Dynamics of spin fluctuati(cid:12)ons is go(cid:12)vernedby the Lan- are three temperature dependent contributions to the
(cid:12) (cid:12)
daudamping(3)whichforexcitationdescribedbyΨ (x) conductivity. They are due to scattering of conducting
0
translates to electronsby fluctuations ofdomainwall, due to domains
wallsshapechangewhentemperatureincreases,anddue
γ(Q)= γ ∞ dxdxΨ (x)Ψ (x) ∞ dq eiq(x−x′) , to variationof concentrationofwallsnW. Firsttwo con-
vF Z−∞ ′ 0 0 ′ Z−∞ Q2+q2 tmriobnutniaotnusrae.reTchoinrsdidceornetdriibnutthioisnsdeecptieonndasnodnhpaovseitaiocnomon-
(13)
where as an example we used ballistic casep. At small the phase diagram. We discuss its contribution in the
momenta β >Q we have for the ballistic case conclusions of the paper.
Resistivity due to electron scattering by spin fluctua-
4γ tion is obtained in second order perturbation theory in
γ(Q)= ln(β/Q). (14)
πβv interaction (4) and is expressed through imaginary part
F
of averagedsusceptibility as2,3
Same procedure for the diffusive case gives
2pF
2γ 1 dqq3 ∞ dωω
γ(Q)= . (15) ρ(T)=R Imχ(q,ω),
DQβ 0T p4 sinh2(ω/2T)
Z F Z
0
The dynamical susceptibility of one domain wall fluctu- −∞ (19)
ations is represented in the form here
χ(r,r′,ω)= (d22πQ)2e−iQ(ρ−ρ′)Ψ0(x)Ψ0(x′)χ(Q,ω), R0 = m32Ge22nν, (20)
Z
(16)
where ν is density of states of conducting electrons per
where
spin, p , n, m are Fermi momentum, density, and mass
F
χ
χ(Q,ω)= 0 . (17) of conduction electrons respectively.
ǫ0+KQ2+iωγ(Q) In addition the fluctuations give temperature depen-
dent contribution to the average magnetization of wall.
We consider a locally flat domain boundary. This can
Fluctuation part of magnetization is determined by
be justified if scale along wall L K 1 is derivative δm(r)= δ∆Ω of fluctuation part of the free
k ∼ ǫ0 ∼ pF√ǫ0 −δh(r)
q
4
energy ∆Ω = 1T ln(ǫ + γ(Q)ω ). This leads here we also averaged over the positions of the do-
2 ωn,Q Q n
to a change of the domain wall profile and gives addi- main walls after which the equation above became
P
tionaltemperaturedependenceofresistivityproportional isotropic. In the ballistic regime Tτ > 1 one can
to G2m(Q)δm(Q,T). Sum of both contributions to the use the following approximation B(q,ω) 2/(v q)2
F
≈
resistivity is given by which is valid for v q > ω , and the expression
F
| |
for the domain wall profile fluctuations propa-
R= 4πβR0nW ∞dω ω 2coth( ω )+2 gator at small momenta q < β is χ(q,ω) =
p4F 0 (cid:16)Tsinh2(ω/2T) − 2T (cid:17) 4nWχ0/β ǫ0+KQ2+i(4γω/(πvFβ))ln α/KQ2 −1.
R d2Q Imχ(Q,ω). (21) We only consider temperature dependent terms of the
(2π)2 (cid:0) (cid:0) (cid:1)(cid:1)
correction.
Term, proportionaRl to ( coth( ω ) + 1), is due to We find that at temperatures T > πv βǫ /(2γ) T
− 2T F 0 ≃ 0
δm(Q,T). and T <v ǫ /K the correction to the conductivity is
F 0
Our calculations show that despite of the difference logarithmic in temperature
in damping in ballistic (14) and diffusive (15) regimes p
the temperature dependence of resistivity has the same δσ = χ0 e2νG2n ǫ /Kv τln ǫ0/KvF .
form for both of the cases. It is quadratic at tempera- −(cid:18)2π2γ W(cid:19) 0 F p T !
tureslowerthantheinverserelaxationtimeandlinearat p (25)
largertemperature. Wegetthatintheballisticscattering At higher temperatures T > v ǫ /K and T < 2vFγ
regime at temperatures T <T resistivity has quadratic F 0 πKβ
0 the correction to the conductivity is linear
temperature dependence p
5χ
2πβR n χ T2 δσ = 0 e2νG2n τT. (26)
R= 0 W 0 , (22) 12π2γ W
p4 KT (cid:18) (cid:19)
F 0
At higher temperatures the correctiondecays as 1/T.
and at temperatures T >T the dependence is linear
0 Now let us discuss the diffusive regime at Tτ <
2πβR n χ T 1. In this case B(q,ω) = 4 Dq2 and
R= 0 W 0 , (23) 3(Dq2+iω)3
p4F K χ(q,ω) = 4nWχ0/β ǫ0+KQ2+i2γω/(DβQ) −1 ap-
proximated at small momenta q < β. Most singular
here T0 = βvFǫ0/(5ln(β2K/ǫ0)). In case of diffusive contribution arises fro(cid:0)m ω /D < q < min((cid:1)β,1/ℓ),
scattering regime we get that the resistivity dependence | |
and q > ǫ /K. Where ℓ is the electron‘s mean
of the temperature has the same form with T0 modified 0 p
as T Dβǫ K/ǫ . The transition from quadratic T2 free path. Calculations show that at temperatures
to li0ne≡ar T d0pepende0nce corresponds to transition from p = min(Ωp1,Ω2,1/τ) > T, where Ω1 = 2Kγ22Dβ2, Ω2 =
quantum to thermal fluctuations of domain wall.26 √2β4DK/γ the contribution to the conductivity is
We would like to notice that the same temperature
dependence holds for scattering by volume spin fluctu- δσ = 27/2χ0e2νG2n ln p ln pT . (27)
ations, except for the difference of effective T in bal- − 9π3γ W T Ω2
listic and diffusive cases.24,25 Let us compare r0esistivity (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) 1(cid:19)
We would like to point out that we have not considered
temperature dependence (22) and that ̺ (T) due to
vol allofthe possible regimesofparameters,focusingonthe
volume spin fluctuations. In the considered temperature
most interesting cases. Expressions (25), (26), and (27)
rangethevolumecontributionisquadraticand̺ (T)
R χ T2/(p v √α).2,3 Ratio R(T)/̺ (T) vnoWl ca∼n have a 2D temperature behavior while obtained for the
0 0 F F vol ∼ pFǫ0 3D electron system.21 The resistivity correction is ob-
be of order of one.
tainedbyδρ=1/σ (1 δσ/σ ),whereσ isthe Drude
D D D
−
conductivity.
V. INTERACTION CORRECTION
VI. CONCLUSION
Atlowtemperaturesimportantresistivitytemperature
dependence is related to weak localization and electron
We considered the temperature dependence of resis-
interaction corrections.18 Here we are going to discuss
tivity due to electron scattering by spin fluctuations of
contribution to the conductivity originating from the in-
domain walls. We showed that in case of weak pin-
terplay between electron inelastic scattering by domain
ning α>>ǫ the relaxation of domain wall fluctuations
0
wall fluctuations and elastic scattering by impurities.
is much slower than relaxation of volume spin density.
Thetripletchannelcontributiontotheconductivityafter
Therefore, classical regime of fluctuations characterized
disorder averaging is given by21,27
by linear temperature dependence starts at much lower
δσ =2πe2v2τνG2 dω ∂ ωcoth ω temperaturethanforthevolumefluctuations. Inconsid-
F 4π2 ∂ω 2T ered temperature range contribution of volume fluctua-
×Im (2dπ3q)3RB(q,ω(cid:2))χ((cid:0)q,ω), (cid:1)(cid:3) (24) tions to resistivity is always ∼T2.
R
5
It is reasonable to assume, that additional resistivity Appendix A: Averaging over random surfaces
in considered system is proportional to n . Depending
w
on relation between average magnetic field and metam-
Here we show the averaging procedure over the direc-
agnetich (T)theconcentrationn canchangewiththe
m w
tion of domain walls and random ǫ . We assume that
temperature. In considered temperature range the con- 0
magneticfieldisdescribedbyGaussiandistributionsuch
centration changes with the temperature as n (T) =
W
that
n (0)(1 + a(Tτ )2), where a 1 and the sign of
W S
| | ∼
a depends on the average magnetic field. The τ
S
∼
1/(αpFvF) is relaxation time of volume fluctuations. ∆h(r)∆h(r′) =Aexp( ζ(r r′)2) (A1)
Parameter Tτ determines renormalization of free en- h i − −
S
ergy due to volume spin fluctuations.3 For example in
schematic phase diagram shown in fig. 1 the nW de- and h∆h(r)i = 0, where ∆h(r) = h(r)−h0. When pa-
creases in the region h < 0 when h (T) increases with rameter ζ is small the magnetic field becomes a slowly
m
temperature. varying random function h(r). Typical phase boundary
In case when a < 0 there can be a cancelation of insucharandomfieldissmooth. Positionofdomainwall
quadratic temperature dependence of volume contribu- is defined by the following equation
tion to resistivity by the contribution of domain walls
n (T). Incase ofthe cancelation,obtainedresults (23),
W h(r)=h +∆h(r)=0 (A2)
0
(25), and (26) will be dominant. And the total resistiv-
ity will have quasi-linear non-Fermi liquid temperature
dependence. Note, that under this definition we neglect cases when
in small closed regions there is no solution of mean field
equation(5)forfavoritephase,orenergyassociatedwith
Acknowledgments it is too high.
Fourier transform of any quantity V(r) which is
This work was financially supported by ARO under nonzero near the domain wall surface and slowly vary-
grant No. W911NF-09-1-0527. ing along it, is calculated as
d
V(q)= dr exp(iqr)V(r)= dS exp(iqr )V(qn(r ),r )= dr V(qn(r),r)exp(iqr) h(r)δ(h(r)), (A3)
S S S |dr |
Z Z Z
here r is a point on the surface, and n(r ) is a vector normal to the surface at the point r defined as n(r) =
S S S
dh(r)/ d h(r). In (A3) V(qn(r),r) is a one dimensional Fourier transform in the direction normal to the surface.
dr |dr |
We now can average the dynamical susceptibility of the number of domain wall. For one domain wall the sus-
ceptibility is given by expression (16). Scale along the surface of the domain wall that we are interested in is
L K 1 , here δh √A αm0 and the ǫ is estimated as
k ∼ ǫ0 ∼ pF√ǫ0 ∼ ∼ χ0 0
q
χ dh
ǫ 0 √αK ζ. (A4)
0
∼ m α/K|dx|∼
0
p
p
Therefore L 1 β/α√ζ is smaller than the radius of surface curvature L √ζ √ζ/β << 1, and under
k ∼ pF k ∼
averaging procedure we can use the expression (16). Then the average susceptibility is
p p
d2Q
χ(q,ω)= d3R dbΠ(R,b)exp(iqR)b2Ψ (qn)Ψ ( qn) exp(iQR)χ(Q,ω), (A5)
0 0 − (2π)2
Z Z Z
where Ψ (q)=2β3/2/(q2+β2) is a Fourier transform of Ψ (x) defined by (10), and
0 0
1 d d
Π(r r ,b)= δ(h(r ))δ(h(r ))δ(b ( h(r )+ h(r ))) . (A6)
1− 2 h 1 2 − 2 dr 1 dr 2 i
1 2
At R<<√ζ the b is a normal to the surface and there- fore is also normal to R . Therefore in (A5) we can sep-
6
aratelyaverageoverthedirectionandvalueofb,consid- ing of the domain wall concentration and is defined as
ering Π(R,b) only in the limit of R√ζ <<1. In case of
Gaussian distribution Π(R,b) is calculated analytically drδ(h(r)) d h(r) √8ζ h2
as nW h |dr |i = exp( 0) (A10)
≡ V π −A
R
n
Π(R,b)= W P(b )P(b ), (A7)
16RAζ ⊥ k where V is the volume of the system. The factor b2
undertheintegralofexpression(A5)isestimatedasb2
where ǫ2. So averagingof quantities like susceptibility does no∼t
0
divergeatsmallǫ andcanbeapproximatedbyinserting
1 b2 0
P(b )= exp( ⊥ ) (A8) average ǫ0. Finally, the averaged susceptibility over the
⊥ 2πAζ −2Aζ domain wall positions is given by the next expression:
and
1 n χ
χ(q,ω)= d2nΨ (qn)Ψ ( qn) W 0 ,
3 3b2 4π 0 0 − ǫ +KQ2+iωγ(Q)
P(bk)= πAζ3R4 exp(−Aζ3Rk4), (A9) Z 0 (A11)
r
where ǫ is approximated by its average value. At small
0
here b and b are perpendicular and parallel compo- momenta q < β Fourier transform Ψ (qn) is approxi-
0
nents t⊥o R conskequently. The quantity n has a mean- mated as Ψ =2/√β.
W 0
1 A.I. Shindlerand M.J. Rice, Phys. Rev.164, 759 (1967). 15 E.L. Nagaev, Phys.-Usp.,38, 497 (1995).
2 K.UedaandT.Morya,J.Phys.Soc.Jpn.39,605(1975). 16 B. Binz, H.B. Braun, T.M. Rice, and M. Sigrist, Phys.
3 T. Moriya, Spin fluctuations in itinerant electron mag- Rev.Lett. 96, 196406 (2006).
netism, Spinger-Verlag (1987). 17 A.T.Burkov,A.Yu.Zyuzin,T.Nakama,andK.Yagasaki,
4 G.R. Stewart, Rev.Mod. Phys., 73, 797 (2001). Phys. Rev.B, 69, 144409 (2004).
5 W.Yu,F.Zamborszky,J.D.Thompson,J.L.Sarrao,M.E. 18 B.L. AltshulerandA.G.Aronovin ”Electron-Electron In-
Torelli, Z. Fisk, and S.E. Brown, Phys. Rev. Lett. 92, teractions in Disordered Systems”, edited by A.L. Efros
086403 (2004). and M. Pollak (North-Holland, Amsterdam, 1985).
6 C.Pfleiderer,D.Reznik,L.Pintschovius,H.v.Lohneysen, 19 A. Rosch,Phys. Rev.Lett. 82, 4280 (1999).
M. Garst, and A.Rosch, Nature,427, 227 (2004). 20 YongBaek Kim and A.J. Millis, Phys. Rev.B 67, 085102
7 S. A.Grigera et al, Science, 306 , 1154 (2004). (2003).
8 R.S. Perry, K. Kitagawa, S.A. Grigera, R.A. Borzi, A.P. 21 I.Paul,C.Pepin,B.N.Narozhny,andD.L.Maslov,Phys.
Mackenzie,K.Ishida,andY.Maeno,Phys.Rev.Lett.92, Rev. Lett. 95, 017206 (2005); I. Paul, Phys. Rev. B 77,
166602 (2004). 224418 (2008).
9 Z.Q. Mao, M. Zhou, J. Hooper, V. Golub, and C.J. 22 T.R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. 104,
O’Connor, Phys.Rev. Lett.96, 077205 (2006). 256404 (2010).
10 Y.J.Uemura et al., NaturePhysics, 3, 29 (2007). 23 Ar. Abanov, A.V. Chubukov, and J. Schmalian, Adv.
11 M. Uhlarz, C. Pfleiderer, and S. M. Hayden , Phys. Rev. Phys. 52, 119 (2003).
Lett. 93, 256404 (2004). 24 R.Jullien, Journal of Low Temp.Physics, 42, 207 (1981).
12 S.V. Maleyev, S. V. Grigoriev , arXiv:0704.0683 (unpub- 25 P. Fuldeand A. Luther,Phys. Rev.170, 570 (1968).
lished). 26 A.Onuki,PhaseTransitionDynamics,CambridgeUniver-
13 T. Goto, K. Fukamichi, T. Sakakibara, and H. Komatsu, sity Press, (2004).
Solid State Commun. 72, 945 (1989). 27 Gabor Zala, B.N. Narozhny, and I.L. Aleiner, Phys. Rev.
14 S. Takashima, M. Nohara, H. Ueda,N. Takeshita, C. Ter- B 64, 214204 (2001)
akura, F. Sakai, and H. Takagi, J. Phys. Soc. Jpn. 76,
043704 (2007).
