Table Of ContentFerromagnetism ofWeakly-Interacting Electrons inDisordered Systems
Xiao Yang1 and Chetan Nayak2
1 Physics Department, University of Virginia, Charlottesville, VA 22904-4714
2DepartmentofPhysicsandAstronomy,UniversityofCaliforniaatLosAngeles,LosAngeles,CA90095-1547
(Dated:February2,2008)
5 Itwasrealizedtwodecadesagothatthetwo-dimensionaldiffusiveFermiliquidphaseisunstableagainstarbi-
0 trarilyweakelectron-electroninteractions.Recently,usingthenonlinearsigmamodeldevelopedbyFinkelstein,
0 severalauthorshaveshownthattheinstablityleadstoaferromagneticstate.Inthispaper,weconsiderdiffusing
2 electronsinteractingthroughaferromagneticexchangeinteraction. UsingtheHartree-Fockapproximationto
directlycalculatetheelectronselfenergy,wefindthatthetotalenergyisminimizedbyafiniteferromagnetic
n
moment for arbitrarilyweakinteractions intwodimensions and for interactionstrengthsexceeding acritical
a
value proportional to the conductivity in three dimensions. We discuss the relation between our results and
J
previousones.
8
1
I. INTRODUCTION as we shall see in the next section, in a two-dimensional
]
n disorderedsystem, the interaction energygainvaries instead
n Since the discovery of Anderson localization, it has been as Jν2δµ2 ln(τδµ); consequently, it is always favorable
s- wellknownthatdisordercanprofoundlychangetheelectronic to ∼develop a ferromagnetic moment. (This is similar to
di properties of solids. For non-interacting electronic systems, what occurs in a quantum dot8.) We show this by directly
. the celebrated scaling theory of Abrahams et.al1 shows that calculating the disorder-averagedelectron self-energy, using
at electrons are always localized in one and two dimensions, the trick of ref. 9, in which itis relatedto the diffusivepole
m whileinthreedimensions,thereexistsametal-insulatortran- in the density-density correlation function. This method is
- sitionatacriticalvalueofthedisorderstrength. Ontheother physically transparent since one can directly see how the
d hand, wheninteractionsbetweenelectronsare takeninto ac- seemingly innocent diffusive pole leads to a logarithmic
n count, the results are much less clear. In a disordered sys- correctionto the self energy. We also avoid the need for the
o tem, at length scales beyond the mean free path, electrons replica trick used in the sigma model treatment. For a three
c
[ move diffusively. Since diffusion is slow, electron-electron dimensionalsystem,ourcalculationshowsthatferromagnetic
interactionsareconsequentlyenhanced.First,electrons’abil- order develops when J exceeds a critical value which is
1 ity to screen the long-ranged Coulomb interaction is ham- proportionalto the conductivityof the system. We compare
v
pered. Second, electrons stay in each other’s region longer theseresultswithpreviousones.
1
3 andhenceinteractwitheachothermorestrongly2. Asshown
4 byFinkelstein3andCastellanietal.4,thetwo-dimensionaldif-
1 fusiveFermiliquidphaseis,asaresult,unstable. However,it II. THEMODEL
0 isnotclearwhatisthestablefixedpointcontrollingthelow-
5 energy behavior of weakly-interacting electrons in the pres- We consider the following Hamiltonian for a weakly-
0
enceofdisorder. interactingelectronicsysteminthepresenceofarandomdis-
/
t Using the nonlinear sigma model developed by tributionofimpurities:
a
m Finkelstein3, several authors5,6,7 have shown that the en-
hancement of electron-electron interactions due to disorder H =H0+Hint (1)
-
d leads to the formation of ferromagnetic moment. The
where
n non-linearsigmamodeltakesasitsstartingpointarepulsive
o density-density interaction which is customarily split into
c small-angleandlargeanglescatteringwithcouplingconstants H0 = Emcˆ†mαcˆmα (2)
:
v Γ1 and Γ23. The latter coupling drives ferromagnetism. In mX,α
i thispaper,weconsideritinerantelectronsinteractingthrough
X isthenoninteractingHamiltonian,whichincludesthekinetic
aferromagneticexchangeH = J(r r′)S(r) S(r′).
r int − − · energy of electrons and the random potential. Em, ϕm(r)
a As in the case of a density-densty interaction, this is an arerespectivelythemth eigenvalueandeigenfunctionofH ,
R 0
interaction which, in the clean, disorder-free case, does not
whichweassumetobespin-independent.WewillcallE the
m
lead to ferromagnetismfor small J; if it occurs at all, it can
‘kinetic’energyoftheelectrontodistinguishitfromtheinter-
only be at large J. This is because a kinetic energy penalty
actionenergyassociatedwithH ,eventhoughE includes
int m
must be paid when the up- and down-spin Fermi surfaces
theeffectsoftherandompotential.c istheannihilationop-
mα
are split in order generate a net ferromagnetic moment.
eratorforanelectronofspinα= , intheeigenstateϕ (r).
m
The kinetic energy penalty for splitting the Fermi surfaces ↑ ↓
Theannihilationoperatorψ (r)foranelectronofspinαatr
by δµ is νδµ2, where ν is the density of states at the α
is:
∼
Fermi surface. On the other hand, the energy gain for non-
interacting electrons is Jν2δµ2. Hence, ferromagnetism ψ (r)= ϕ (r)cˆ (3)
can only occur for J >∼1, the Stoner criterion. However, α m mα
ν m
X
2
Asmentionedintheintroduction,theinteractionHamiltonian down-spinelectronsatthe Fermilevel: M = νδµ, where ν
H consideredhereis: isthesingle-electrondensityofstates. Now,wefindthatthe
int
disorder-averagedvariationalinteractionenergyis:
H = ddrddr′J(r r′)S(r) S(r′)
int
− − ·
Z Ψ H Ψ = ddrddr′J(r r′) Sz(r) Sz(r′)
0 int 0
h | | i − − h ih i
= ddrddr′J(r r′) Sz(r)Sz(r′) Z
− −
Z+2 S+(r)S−(r′)(cid:0)+S−(r)S+(r′) (4) + dEdE′ ddrddr′J(r−r′)F(E,E′,r,r′) NENE′
Z Z
(10)
Here J(r r′) > 0 is(cid:0)the exchangeinteractionwhic(cid:1)h(cid:1)is as-
sumedtob−eweak, ferromagnetic,andshort-ranged. S(r) = where
ψˆ†(r)~σ ψˆ (r) is the spin density operator, and ~σ are the
α αβ β Sz(r) = ϕ∗ (r)ϕ (r) N N (11)
Paulimatrices. h i m m h m↑− m↓i
m
X
III. HARTREEANDFOCKENERGIES
N = c† c =θ(n M m) (12)
h m↑,↓i m↑,↓ m↑,↓ 0± −
Asawarmup,letusfirstconsideracleansystem,forwhich D E
N isthenumberofelectronsinthestateatenergyE,namely
the ϕ (r) are plane waves. We look for ferromagnetism in E
m N = 2 for E < δµ, N = 1 for δµ < E < δµ, and
twhiitshmaofdererlobmyaagsnseutmicinmgotmakeinntg: atrialwavefunction|Ψ0iclean NEE = 0forE > δµ−. ThefEunctionF(E−,E′;r,r′)isdefined
as
Ψ = c† c† 0 (5)
| 0iclean k↑ k′↓| i F(E,E′;r,r′)= δ(E E )δ(E′ E )
k<YkF↑k′Y<kF↓ − m − n ×
m,n
X
and minimizing hΨ0|H|Ψ0i with respect to the magnetiza- ϕ∗m(r)ϕm(r′)ϕ∗n(r′)ϕn(r) (13)
tionM =νδµ,whereνisthesingle-electrondensityofstates
and2δµ = k2 k2 /2mis the differencein thekinetic Theupperbarindicatesaverageoverdisorder. Thefirstterm
F↑ − F↓
energiesofup-anddown-spinelectronsattheFermienergy. in (10) is similar in behavior to the clean case, and will be
(cid:0) (cid:1)
In evaluating the expectation value of the interaction en- neglectedinthefollowing. Thesecondtermismoreinterest-
ergy,wefindtwoterms: ing because the function F is closely related to the density-
density correlationfunction. The crucial differencewith the
Ψ H Ψ = J(0)M2+ clean case is that the latter has a diffusive form, as we now
0 int 0
h | | i −
describe.
3 1
J(k−k′) 8NkNk′ − 2SkzSkz′ (6) Weassumethatthedensitypropagatesdiffusively,sothat
k,k′ (cid:18) (cid:19)
X ∞
whereN , Sz =θ(k k) θ(k k).Theyareessentially Π(q,ω) = i dtddx [ρ(x,t),ρ(0,0)] e−iq·x+iωt
theHartrkeeakndFockFc↑o−ntrib±utionFs↓to−theenergy. Z0 Dq2 h i
IfwetakeforsimplicityJ(q)=J,then = ν (14)
iω+Dq2
−
3 3
H = JM2+ JN2 (7) where, again, ν is the density of states at the Fermi surface,
int
h i −2 8
andDisthediffusionconstant.Then
Ferromagnetism will only occur if this interaction energy is
∞
largerthanthekineticenergycost Π∗(q,ω)= i dtddx [ρ(0,0),ρ(x,t)] eiq·x−iωt
− h i
1 Z0
H H =νδµ2 = M2 (8) 0
h kiniM −h kini0 ν = i dtddx [ρ(x,t),ρ(0,0)] e−iq·x+iωt (15)
− h i
Z−∞
i.e. onlyfor3J/2 > ν. (Themagnetizationisstabilizedata
finitevaluebytheenergydependenceofthedensity-ofstates, On the second line, we have used the fact that after averag-
whichwehaveneglectedhere.) ing over disorder, the density-density correlation function is
Consider,instead,thedirtycase. Wetake translationallyinvariant. Thisassumptionisvalidinthehigh
conductivity regime of interest here. From this, we can ob-
Ψ = c† c† 0 (9) tain the Fourier transform of the density-density correlation
| 0idirty n↑ n′↓| i function,
n<Yn0+M n′<Yn0−M
∞
where E = E . Again, the magnetization M is propor-
n0 F A(q,ω)= dtddxe−iq·x+iωt [ρ(x,t),ρ(0,0)] (16)
tional to the difference in the ‘kinetic’ energies of up- and h i
Z−∞
3
byobservingthat Thus,theinteractionenergyis:
1
A(q,ω) = (Π(q,ω) Π∗(q,ω)) (17) −δµ E
i − hHintiM = dE dE′V(E−E′)·2·2
1 Z−Λ Z−Λ
= Im(Π(q,ω)) δµ −δµ
2 + dE dE′V(E E′) 2 1
Dq2ω − · ·
= ν Z−δµ Z−Λ
2(ω2+(Dq2)2) δµ E
+ dE dE′V(E E′) 1 1 (24)
WecanrelatethistoourfunctionF bynotingthat Z−δµ Z−δµ − · ·
Subtracting the interaction energyin the paramagnetic state,
[ρ(rt),ρ(r′,0)] = ϕ∗ (r)ϕ (r′)ϕ∗(r′)ϕ (r)
h i m m n n M =0,
m,n
X
Nm Nn (18) 0 E
×h − i H = dE dE′V(E E′) 2 2 (25)
Soweget h inti0 Z−Λ Z−Λ − · ·
∞ µ wehave
A(q,ω) = dE dE′F(E,E′;r,r′)
Zµ Z−∞ δµ E
= ωF(q,ω) × eiq(r−r′)δ(E−E′−ω()19) hHintiM −hHinti0 =−Z−δµdEZ−δµdE′V(E−E′)
νJ δµ E (E E′)2+Λ2
Where, on the last line, F(q,ω) is the Fourier transform of = dE dE′ ln −
−8πD (E E′)2
F(E,E′;r,r′), and ω is the energydifferenceE E′. µ is Z−δµ Z−δµ (cid:20) − (cid:21)
thechemicalpotential.Soweget − νJ δµ (E+δµ)2+Λ2
= dE(E+δµ) 1+ln
−8πD (E+δµ)2
Dq2 Z−δµ (cid:18) (cid:20) (cid:21)(cid:19)
F(q,ω)=ν (20) νJ Λ
2(ω2+(Dq2)2) = 6δµ2+4δµ2ln (26)
−8πD 2δµ
(cid:18) (cid:18) (cid:19)(cid:19)
andthesecondtermin(10)becomes
Includingthe‘kineticenergy’,wehave
Ψ H Ψ =...+
h 0| int| 0i 3νJ νJ Λ
ν dE µ dE′ ddq Dq2 J(q) (21) hHiM−hHi0 = ν− 4πD δµ2−2πD δµ2ln 2δµ
Z Z−∞ Z (2π)d(E−E′)2+(Dq2)2 (cid:18) (cid:19) (cid:18) (cid:19)(27)
The...referstothefirsttermin(10),whichissmallerbecause Even at small J, ferromagnetic order occurs because the
itdoesnothaveanyinfraredsingularitiesinitsintegral.Inthe totalenergyis minimalat a finite value of δµ and, therefore,
precedingderivation,weperformedthedisorderaverageofF, ofM:
by comparing it to the density-density correlation function,
νΛ
thereby avoiding the replica trick normally used in the field M = e−2πD/J (28)
theoreticaltreatment. 2
So we have shown that the diffusive fermi liquid is always
unstableagainsttheferromagneticorderintwodimensions.
IV. FERROMAGNETICGROUNDSTATE
Inthreedimensions,onecandotheanalogouscalculations,
leadingto
We now minimize H with respect to the magnetization
M. We define the effhectiive interaction V(E E′) between JνΛ1/2 4Jν
electronsinstatesatenergiesE,E′by − hHi−E0 = ν−ν2J − 2π2D3/2 δµ2+ 15πD3/2δµ5/2
(cid:18) (cid:19)
(29)
Ψ0 Hint Ψ0 = dEdE′V(E E′)NENE′ (22) In the first term, we have included the Hartree contribution
h | | i ZE>E′ − becausethe Fockpartisnon-divergentandcanbe viewedas
acorrection.Hence,thecriticalcouplingJ abovewhichfer-
Evaluatingthemomentumintegralin(21),weobtain c
romagnetismoccursismodifiedfromtheStonervalue:
νJ (E E′)2+Λ2
V(E E′)= ln − (23) 1 3√3
− 8πD (E E′)2 J = 1 (30)
(cid:20) − (cid:21) c ν − (kFℓ)2!
Here, we have ignored the momentum dependenceof J(q),
andasusual,Λistheultravioletcutoffwhichcanbetakento whereℓ=v τ istheelasticmean-freepathandwehaveused
F
be1/τ,whereτ istheelasticscatteringtime. D =v2τ/3inthreedimensions.
F
4
V. DISCUSSION trons approach each other. This strongly favors ferromag-
netism since electrons of the same spin cannot occupy the
Inthispaper,weuseastraightforwardFockapproximation sameenergylevel.Thisstrongdependenceontheenergysep-
toshowthatintwodimensionsanarbitrarilyweakferromag- arationoftheelectronsiscompletelyabsentinacleansystem.
netic interaction causes an instability of the diffusive Fermi Wenotethatonceferromagneticorderdevelops,thesystem
liquid. Inthreedimensions,thisinstabilityoccursonlywhen goesintotheuniversalityclassofelectronsinaZeemanfield4,
the interaction strength exceeds a critical value proportional whichflowstowardsaninsulatingstateintwodimensions.
to the conductivity. We worked directly with the electronic
Hamiltonian, without recourse to Finkelstein’s non-linear σ
model, which was used at an intermediate stage of previous
calculations5,6,7. Our results are consistentwith these earlier Acknowledgments
ones. Themodelandmethodofthispaperhastheadvantage
of physical transparency: it shows the direct connection be- We would like to thank C. Chamon and A. Kamenev for
tweendiffusivemotionandtheferromagneticinstability. theircommentsonanearlierversionofthismanuscript.C.N.
Byusingthemethodinref. 9,weavoidedusingthereplica was supported by the National Science Foundation under
trick, and saw the crucial feature of diffusive motion: it im- GrantNo.DMR-0411800.X.Y.wassupportedbytheChemi-
pliesaspatialcorrelationbetweensingle-particlestateswhich calSciences,GeosciencesandBiosciencesDivision,Officeof
arenearbyinenergy.Asaresultofthiscorrelation,theeffec- Basic EnergySciences, Office ofSciences, U.S. Department
tive interaction strength diverges as the energy of two elec- ofEnergy,andbyNSFunderGrantNo. DMR-0412936.
1 E. Abrahams, P.W. Anderson, D. C. Licciardello, and T. V. Ra- 5 D.BelitzandT.R.Kirkpatrick,Phys.Rev.B53,14364(1996).
makrishnan,Phys.Rev.Lett.42,673(1979). 6 C.ChamonandE.Mucciolo,Phys.Rev.Lett.85,5607(2000).
2 B. L. Altshuler and A. G. Aronov. Solid State Comm. 39, 115 7 C.NayakandX.Yang,Phys.Rev.B68,104423(2003).
(1979);Sov.Phys.JETP50,968(1979). 8 A.V.AndreevandA.Kamenev,Phys.Rev.Lett.81,3199(1998).
3 A.M.Finkelstein,Zh.Eksp.Teor.Fiz.84,168(1983),Sov.Phys. 9 E.Abrahams,P.W.Anderson,P.A.Lee,andT.V.Ramakrishnan,
JETP57,97(1983). Phys.Rev.B24,6783(1981).
4 C.Castellani,C.DiCastro,P.A.LeeandM.Ma,Phys.Rev.B30
527(1984).
