Cellular Dynamical Mean Field Theory of the Periodic Anderson Model

0.22 MB·English

by  Lorenzo De Leo

CellularDynamicalMeanFieldTheory ofthe PeriodicAnderson Model
Lorenzo De Leo,1 Marcello Civelli,2 and Gabriel Kotliar1
1DepartmentofPhysicsandCenterforMaterialTheory, RutgersUniversity,Piscataway, NJ08854, USA
2Institut Laue Langevin, 6 rue Jules Horowitz - 38042 Grenoble Cedex, France
WedevelopaclusterdynamicalmeanfieldtheoryoftheperiodicAndersonmodelinthreedimensions,taking
aclusteroftwositesasabasicreferenceframe.ThemeanfieldtheorydisplaysthebasicfeaturesoftheDoniach
9 phasediagram:aparamagneticFermiliquidstate,anantiferromagneticstateandatransitionbetweenthem.
0
Incontrastwithspindensitywavetheories,thetransitionisaccompaniedbyalargeincreaseoftheeffective
0
masseverywhereontheFermisurfaceandasubstantialchangeoftheFermisurfaceshapeacrossthetransition.
2
Tounderstandthenatureandtheoriginofthephasesnearthetransition,weinvestigatetheparamagneticsolution

n underlyingtheantiferromagneticstate,andidentifythetransitionasapointwherethefelectronsdecouplefrom
a theconductionelectronsundergoinganorbitallyselectiveMotttransition. Thispointturnsouttobeintimately
J relatedtothetwoimpurityKondomodelquantumcriticalpoint. Inthisregime,nonlocalcorrelationsbecome

8 importantandresultinsignificantchangesinthephotoemissionspectraandthedeHaas-vanAlphenfrequencies.
1 Thetransitioninvolvesconsiderablef spectralweighttransferfromtheFermileveltoitsimmediatevicinity,
ratherthantotheHubbardbandsasinsinglesiteDMFT.

]
l
e PACSnumbers:71.27.+a,71.10.Hf,75.20.Hr,75.30.Mb
-
r
t
s I. INTRODUCTION tronsandconductionc electronswitha Fermisurfacewhich
.
t obeysLuttingertheorem,countingboththef andcelectrons.
a
Forsmallhybridizationthef electronsdecouplefromthecon-
m The heavy fermion materials have been one of the most
ductionelectronquasiparticles,andtheFermisurfacecontains
challengingproblemsforthecondensedmattercommunityin
-
d the lastthreedecades. The physicsof these systemsisdom- onlythelatter. Thedecouplingbetweenthef andcelectrons
n inated by the competition of many interactions, manifesting can be viewed as an orbitally selective Mott transition, and
o in a variety of phases1. Close to the zero-temperature tran- has been studied with single site DMFT12 and slave boson
c techniques13. As stressed in14, a precise formulation of KB
sition between two such phases the materials typically show
[
non-Fermiliquidbehavioronalargetemperaturerange. This requirestheeliminationofantiferromagneticlongrangeorder

3 behaviorhasbeenoften attributedto an underlyingquantum whichbreaksthetranslationalsymmetryoftheoriginallattice.
v Sofarthishasbeenachievedbystudyingmodelswithdisor-
criticalpoint(QCP).
9 derwithinDMFT15,orbymodifyingthesymmetrygroupof
In a subclass of these materials (including CeCu Au ,
5 6−x x thespinfromSU(2)toSU(N)14. ForearlierstudiesoftheKB
5 YbRh2Si2 and CeTIn5, with T=Ir, Co, Rh) it is possible to usingslavebosonssee16,17.
2 drive the system from an antiferromagnetic (AF) phase to a
0 paramagnetic (PM) phase by suppressing continuously the Single site Extended DMFT of various models have been
7 Ne´el temperature via application of pressure, magnetic field carried out. For the Anderson lattice model18,19, the KB
0 andtheonsetofAFshouldbeviewedasdistinctinstabilities
or doping. The conventional spin-density wave (SDW) the-
t/ oryofaQCP2,3,4 disagreeswiththeexperimentalevidencein which occur at two different values of the RKKY coupling
a manyofthese materials1,5,6,7,8. Thishastriggereda renewed (relativetotheKondocoupling).Thesamesituationoccursin
m
thefieldtheoryapproachofRef.14.
interestinmicroscopicdescriptionsoftheheavyfermionstate
d- and its possible instabilities to ordered states. More gener- On the other hand, extended DMFT studies of a Kondo
n ally, not only the critical point but also the formation of the lattice model, in which the conductionelectron band is con-
o heavy fermionstate in the regime where the RKKY interac- strainedsoastopreventitsmagneticpolarization,revealthat
c tionsaresizeablerequiresnontrivialtechniquesforitsmicro- the KB and the onset of magnetism take place at the same
v: scopicdescription9,10. Heavyfermionmaterialscanbemod- pointinthephasediagram20,21,22,23. SeehoweverRef.24. Jus-
i eledbytheperiodicAndersonmodelwhichdescribesawide tificationforthestudyofthismodelisgiveninRef.25.
X
conductionbandofspdelectronsandanarrowbandoff elec- InDMFTbasedapproachesthebreakdownofFermiliquid
r tronswhichisstronglycorrelatedandhybridizedwiththespd theoryatthetransitionisunderstoodintermsoflocalphysics.
a
electrons. In other works the same breakdownis explained in terms of
Doniach11 established that for large hybridization the f the coupling of the electronsto the long wave length fluctu-
electrons combine with the conduction electrons to form a ations of an emergent gauge field14 and to the Kondo order
heavy Fermi liquid, while for small hybridizationthe f mo- parameterfluctuations13,26.
mentsordermagneticallyandhencedonotundergotheKondo Inthispaperwehandlethecompetitionbetweenmagnetism
effect. Thus,changesinthehybridizationnotonlyleadtothe andformationofaheavyPMstateinanunbiasedwayusing
appearanceofmagneticlongrangeorderbutintroducealsoa CellularDynamicalMeanFieldTheory(CDMFT)27.Thisap-
changeintheelectronicstructurereferredtoasf −cdecou- proachimprovesthe treatmentof non-localcorrelationsover
plingorKondobreakdown(KB).Namely,forlargevaluesof single-site DMFT considering a cluster of sites and provid-
thehybridizationthequasiparticlesarecompositesoff elec- ingaself-consistencyconditionontheclusteraction. Allthe
2
correlations whose range is contained within the cluster are duction electron bandwidth W = 12|t| = 2 sets the energy
then treated beyonda simple mean-field level. In this paper unit.
we consider a cluster of two sites embedded in a three di- In the large U/V limit the model maps onto the Kondo
mensional lattice, which is the minimal unit able to capture lattice model, where each f orbital is replaced by a spin 1
2
the competitionbetweenPM andAF correlations. A similar whichisKondo-coupledtothelocalconductionelectronswith
approachwasintroducedinanearlierpublication10 basedon a coupling J = ( 1 + 1 )V2, where ρ is the
a quantum Monte Carlo impurity solver, but that study was conduction eKlectron|dEefn−sµit|y of|Ust+aEtefs−aµt|the chemical poten-
limitedtolargetemperaturesandsmallvaluesofU. Weover- tial. It is also possible to define two energy scales in the
comethislimitation byusingExactDiagonalization(ED)as problem. The first is the single-impurityKondo temperature
an impurity solver28. This solver gives access to the ground T0 = exp(−1/2ρJK). Itisnotknownapriorihowthe“lat-
state properties of the system with a finite energyresolution tice” Kondo energy is related to T0. The second scale is set
(typically0.005ofthebandwidthinourcase). bythemagneticinteractions. Thesearenotcontainedexplic-
InthetheoryoftheMotttransition,theinvestigationofthe itly in this Hamiltonian, but an effective RKKY coupling is
PMinsulatingphasehasbeenveryfruitfulandallowedanun- generatedinfourthorderperturbationtheoryinV (secondor-
derstandingofthefinitetemperaturestate,abovethemagnetic derintheKondocoupling,J ∼ J2/W). TheRKKY
RKKY K
orderin manycompounds. We pursuea similar approachto couplingoscillates in real space with a period that is related
the periodicAnderson modelby investigatingboth the mag- tothebareconductionbandFermimomentum,rangingfrom
neticanda“parentcompound”PMstate,tounderstandbetter antiferromagneticat half filling to ferromagneticin the limit
thecharacteroftheelectronicstructure. Thisallowsustode- of an emptyband. A comparisonof these two energyscales
scribethemagneticstateasaresultofanadditionalinstability allows to understandthe qualitative featuresof the phase di-
ofthePMstate. agramofthemodel11. TheratiobetweenKondotemperature
Weshowthataphasetransitionseparatestheheavyfermion andRKKY interactionis an increasingfunctionofV, hence
paramagnetic phase from an antiferromagnetic phase. Ap- forsmallV we willfindaphaseinwhichthemagneticfluc-
proachingthetransitionfromtheparamagneticsidetheFermi tuations dominate giving rise to an ordered state, while for
liquidisstronglyrenormalized.Ananalysisofthemeanfield large V the Kondo effect dominates and the f moments are
PM solution into the AF phase reveals an orbitally selective screenedbytheconductionelectrons.
Mott decouplingand localization of the f-electrons. Within
AthalffillingtheKondoscreeningofthemomentsbythe
theresolutionofourmethod,theantiferromagnetictransition
conductionelectronsresultsinaninsulatorwithagapinduced
andthedecouplingofthef andcelectronstakeplaceforthe
bythehybridization.Inthispaperweareinterestedinmetallic
samevalueofthehybridization. Throughacomparisonwith
heavyfermions,andwerestrictourselvestostudythesystem
aWilson’sNumericalRenormalizationGroup(NRG)calcula-
awayfromhalffilling.
tion,wefurtherrelatethenatureofthetransitiontotheprox-
Inthe pathintegralformalismthe conductionbandcan be
imity of the system in the parameter space to the quantum
integratedoutintheactionandreplacedbyaneffectivelong-
criticalpointofthetwo-impurityKondomodel(2IKM)29,30.
rangeretardedhoppingforf electrons.Afterthesemanipula-
Thetransitionhasdramaticconsequencesonthequasiparti-
tionstheLagrangianofthesystemis
clespectraandonthetopologyoftheFermisurfacewhichcan
bedetectedbyphotoemissionexperimentsandbyHalleffect
L = f† (τ)(∂ −µ+E )f (τ) (2)
anddeHaas-vanAlphenfrequenciesmeasures. iσ τ f iσ
Thepaperisorganizedasfollows. InSec. IIweintroduce Xi
the model Hamiltonian and the CDMFT method employed. + f† (τ)∆c (τ −τ′)f (τ′)+U nf nf
iσ ij jσ i↑ i↓
InSec. IIIweanalyzetheresultsoftheCDMFTcalculations ij i
X X
andcomparethemwithsinglesiteDMFTresultsandNumeri-
calRenormalizationGroupresultsontheunderlyingimpurity where the effect of the conduction band is absorbed in the
problem.ConclusionarepresentedinSec. IV. Weissfield
V2
∆c(iω,q)= (3)
II. METHODS iω+µ−ε
q
The three-dimensionalperiodicAndersonmodelHamilto- andεq =−2t α=x,y,zcos(qα).
nianis In CDMFT the original three-dimensional lattice is now
P
tiled with a superlattice of clusters and an effective Ander-
H = −t c† c −µ c† c +V f† c +h.c. son impurity action is derived for a single cluster. Finally a
iσ jσ iσ iσ iσ iσ
Xhiji Xi Xi (cid:16) (cid:17) self-consistencyconditionrelatestheclusterGreen’sfunction
to the local Green’s function of the superlattice. The self-
+(E −µ) f† f +U nf nf (1)
f iσ iσ i↑ i↓ consistencyismoreeasilywrittenasamatrixform
i i
X X
wσhaenrdecf†ii†σσcdroeeastetshaecsoanmdeucftoironthbeanlodcealleizcetrdonorabtistaitles.iTwhitehcsopnin- Gloc(iω)=X~k (cid:20)(cid:18)iω+µ0−Ef iω+µ0−Ef (cid:19) (4)
3
−11
− V2 iω+µ eikxε~k − Σ11(iω) Σ12(iω) TK/T0 (CDMFT)
(iω+µ)2−ε~2k (cid:18)e−ikxε~k iω+µ(cid:19) (cid:18)Σ21(iω) Σ22(iω)(cid:19)# 1 TK/T0 (DMFT)
0.8
0.5
wherespinindexhasbeensuppressedtosimplifythenotation.
It is very importantto notice that in the large U/V limit the 0.6
0
impurityactionbecomesequivalenttotheactionofatwoim- 0.6 0.8
purityKondomodelawayfromparticle-hole(p-h)symmetry.
Magnetization
Atp-hsymmetrythismodelisknowntohaveaQCPseparat- 0.4
Z (AF maj. spin)
ingaKondoscreenedphasefromaphaseinwhichtheimpuri-
tiesaredecoupledfromtheconductionband29,30. Awayfrom Z (AF min. spin)
p-hsymmetrytheQCPisnotaccessiblebutstillinfluencesthe 0.2 Z (PM)
physicsinanintermediateenergyrange31,32.
In order to solve the cluster-impurity problem we employ 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ED.Inthismethodtheclusteractionisrecastintheformofan V
AndersonimpurityHamiltonianwherethecontinuousdegrees
of freedom of the free electron bath are parametrized by a FIG.1:(Coloronline)MagnetizationMandaverageofthequasipar-
finitenumberN ofsites: ticleresidueZeigenvaluesfortheAF(bothformajorityandminor-
b
ityspin)andPMsolutionsasafunctionofV. Inset: Ratiobetween
latticeandsingle-impurityKondoenergiesinDMFTandCDMFT.
H = f† (E −µ)f +U nf nf (5)
ED iσ f iσ i↑ i↓
i=1,2 i=1,2
X X
+ ε a† a + V a† f +h.c. .
lσ lσ lσ ilσ lσ iσ III. RESULTS
l=X1,Nb Xi,l (cid:16) (cid:17)
We compute the staggered magnetization M = hnf i −
1↑
In this representation the hybridization-function ∆µν = hnf i = −(hnf i−hnf i) as a function of V (Fig.1). For
G0 −(iω+µ−E )δ , related to the noninteractingim- 1↓ 2↑ 2↓
µν f µν smallV theRKKYinteractionisdominatingandthesolution
purity Green’s function Gµ0ν = −hfµfν†iU=0 of the cluster describesanAFstatewithamagnetizationclosetoone.Inthe
impurityproblem,becomes oppositelimitoflargeV,wheretheKondoscreeningismore
effective, the system is a PM metal. IncreasingV in the AF
V∗ V phase,themagnetizationdecreasessmoothlyandapproaches
∆ (iω)= jlσ ilσ (6) zeroatavalueV∗ ∼ 0.585wherea phasetransitionoccurs.
ijσ
iω−ε
l=X1,Nb lσ Above this critical value the system becomes paramagnetic.
ThevalueofM atthetransitionjumpsfrom∼ 0.22forV =
0.58to∼0.006forV =0.59andinbetweentheconvergence
wheretheindicesi,j =1,2representtheclustersites. Inthe
becomes extremely slow. The numerical accuracy does not
figurespresentedinthisworkweused8sitesinthebath,but
allow us to exclude a weakly first order transition scenario,
theresultshavebeentestedinfewpointsusing10sites.
although in this case the coexistence region would be very
The ground state and Green’s functions of this Hamilto- small.
nian are determinedvia the Lanczos procedureand the self-
To clarify the nature of the transition we investigate the
consistency equation in turns allows to derive a new set of
bath parameters. The self-consistency condition is used to cluster quasiparticle residue Zˆ = (ˆ1 − ∂Σˆ/∂ω)−1|ω=0 (in
thiscase,amatrixinspinandclusterindices). TheAFthree-
determine the ∆ function on a finite mesh of points ω =
n dimensionalSDWtheorypredictsthatantiferromagnetismde-
(2n+1)π/β. This implies a finite resolution in frequency.
velopsfromawelldefinedFermiliquidstateandhenceafinite
We usedβ = 100. Theprocessisiterateduntilconvergence
quasiparticleresidueZ atthetransition. OnthecontraryKB
isreached. Tostudytheinterplayofantiferromagnetismand
scenariospredictthatZ goestozeroatthetransition. InFig.
Kondoscreeningwe restrictthe studyto the metallic regime
1weplottheZˆ eigenvalues. IntheAFphasethetwocurves
closeto half-fillingwheretheRKKY interactionispredomi-
correspondtominorityandmajorityspecies. InthePMphase
nantlyantiferromagnetic.Thisisachievedbyfixingthevalues
we plot only the average of the eigenvaluessince the differ-
U =10,E =−5.5andµ=0.2throughoutthepaper.Being
f
enceisnotappreciableonthescaleoftheplot. Inbothphases
ameanfieldtheory,DMFTgivesthepossibilitytoselectively
allowsomeinstabilitiesandforbidothers.Thuswewillallow theeigenvaluesoftheZˆ matrixdecreaseaswe approachthe
transition. Precisely at the transition one eigenvalue goes to
the developmentofAF orderandstudythe realistic material
zero,suggestingthattheKBisindeedthecorrectscenarioin
phasediagram. Howeverwewillalsoforcethesystemtore-
thismodel.
maininaPMstatetostudythe“normalstate”underlyingthe
AF phase. This will prove important to study the KB point ApproachingthetransitionfromthePMphasebyreducing
andtounderstandtheoriginofthetransitionsinthesystem. V,weobserveastrongreductionoftheZˆeigenvaluesleading
4
0
V=0.62 (PM)
0.4 V=0.58 (AF) -5
-10
0.2
0
-15
0 0.50
-10
0.55
V=0.62 (PM) -20
0.4 V=0.58 (PM) 00..5682 -20 0.26
0.36
-25 0.65 -30 0.45
0.50
0.2 0.55
-30 -40 0.58
0 1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -35
0 1 2
FIG. 2: (Color online) Spectral function of the f electrons on the
FIG.3: (Coloronline)Imaginarypartoftheevenself-energyeigen-
twosidesofthetransitionwithAForderallowed(upperpanel)and
valueontheimaginaryaxisfordifferentvaluesofthehybridization.
forbidden (lower panel). The two Hubbard bands are outside the
Inset:oddeigenvalue(inadifferentrangeofhybridizationvalues).
energyrange.
trastwiththeparamagneticsinglesiteDMFTsolutionwhere
to a large effective mass enhancement. This result suggests thereisonlyaFermiliquidKondoscreenedphaseandnosign
thatincreasingthecorrelationsbydecreasingV fromthePM ofsuchatransitionorofapseudogappedphase. Noticealso
side,theFermiliquidbecomesheavierandheavier,andeven- thatintheCDMFTtreatmentthef transferofspectralweight
tuallyitismorefavorabletogetridoftheentropybydevelop- takes place from the Fermi levelto its immediate vicinity as
ingAFlongrangeorder.Thefundamentaldifferencewiththe the hybridizationis reduced. This shouldbe contrasted with
SDW picture is that here the mechanism responsible for the thesinglesiteDMFTdescriptionoftheMotttransitioninthe
instability isshort-rangedin space. The cluster self-energies Hubbardmodelwherethespectralweightistransferredfrom
are in fact approximatelylocal on the PM side of the transi- theFermileveldirectlytotheHubbardbandsatmuchhigher
tion. This may lead to think that the results are essentially energies.
similartoasinglesiteDMFTsolutionoftheperiodicAnder- To clarify the Mott character of the transition we plot
son model. Howeverthere are essential differences. Indeed, in Fig.3 the imaginary frequency behavior of the two Σ-
evensettingtheoff-diagonalself-energyΣ12 =0theCDMFT eigenvalues within the PM solution, named respectively
self-consistencyequationsdonotreducetothecorresponding Σeven =Σ11+Σ12andΣodd =Σ11−Σ12.Approachingthe
single-site DMFT equations. To demonstrate this we com- transition from the PM phase the slope at zero frequencyof
parein theinsetof Fig.1theratio betweenthelattice Kondo botheigenvaluesincreases,signalingthegrowthoftheeffec-
temperatureobtainedthroughtherelationTK = ∆(0)Z and tivemassintheFermiliquid.Atthetransitiontheeveneigen-
thecorrespondingsingle-impurityenergyT0inthetwocases. value develops a divergence signaling the end of the Fermi
While the DMFT resultgivesa smoothand finite TK across liquid description and the beginning of the Mott insulating
V∗, in CDMFT TK appears to approach a zero value at the phase.Theoddeigenvaluehasasimilardivergenceatalower
transition. valueofthehybridizationasshownintheinset. Aqualitative
We take advantage now of the possibility of suppressing understanding of this behavior can be proposed considering
the AF long range order to study the normalPM solution in thatΣ andΣ representinaloosesensethelatticeself-
even odd
the AFphase. ForV < V∗ thissolutionisobviouslyunsta- energycalculatedaroundmomenta(0,0,0)and(π,π,π) re-
ble to antiferromagnetismand it correspondsphysically to a spectively. HencetheMotttransitioncorrespondstothefirst
highlyfrustratedsystem. Theanalysisofthespectralfunction appearance of a surface poles in the self-energy around the
alongthetwosolutions(allowingandforbiddingmagneticor- originoftheBrillouinzone.Decreasingfurtherthehybridiza-
dering, Fig.2) reveals that the transition takes place also in tionthesurfaceofpolesmovestowardsthecornerofthezone
absence of antiferromagnetism. Namely, at small V, a phase andfinallydisappearsforV ∼0.43.
with nospectralweightat the Fermilevelis realizedin both WhileforV >V∗theself-energywasapproximatelylocal,
cases. The Kondo peak present in the PM phase disappears inthePMsolutionthedevelopmentofmagnetismforV <V∗
in going across the transition, leaving behind a pseudogap, isreplacedbyadramaticincreaseofnon-localcorrelationsas
andthef electronseffectivelydecouplefromtheconduction showninFig.4,whereweplottherealpartofΣ12atthelowest
band,atleastwithintheenergyresolutionofourcalculations. availablefrequencyintheimaginaryaxis.
ThetransitionisanorbitallyselectiveMotttransitionforthe The behavior of T , of the spectral function and of the
K
f band. This implies that the f electrons do not contribute off-diagonal self-energy near the transition is very similar
to the Fermi surface for V < V∗. This is in marked con- to the corresponding behavior found in the 2IKM close to
5
30
c density (AF)
c density (PM)
20 f density
1.3
10
1.2
0
ReΣ (AF) 1.1
-10 12
ReΣ (PM)
12
ReΣ (PM-NRG)
12
-20
1
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1
V V
FIG.4: (Coloronline)Nonlocalcorrelations(ReΣ12,calculatedat FIG. 5: (Color online) f and c electron occupation per site. The
thelowestavailableimaginaryfrequency)alongtheAFandPMsolu- transitionismarkedbytheverticaldashedline.
tions. Theverticaldashedlinemarksthetransition. Forcomparison
withthePMcasetheNRGresults(withoutself-consistency).
criticalpoint. Genericproblems,astheoneconsideredinthis
its QCP31,32,33. To establish this connection we used Wil- paper, are p-h asymmetric, and therefore in these cases the
twoimpurityQCPisnotreachable. Howevertheanalysisof
son’sNumericalRenormalizationGrouptosolvetheimpurity
the NRG low-energy spectrum confirms that in going from
modelunderlyingtheself-consistencyinthePMcaseforthe
onesideto theotherofthecrossover(correspondingly,from
same setofparameters. Thedifferencewiththe ED calcula-
one side to the other of the transition in the CDMFT case)
tionis thatwe used a constantdensityofstates forthe NRG
the system evolves from a state in which the impurities are
bath,sotheresultsarenonself-consistent.Theresultsarenev-
screenedby the conductionelectrons(for largeV) to a state
ertheless qualitatively similar. The main difference between
inwhichtheimpuritiestakeadvantageoftheRKKYinterac-
theresultsofthetwocalculationsisthatthetransitionfound
tiontoformasingletanddecouplefromtheconductionband
inCDMFTisreplacedinNRGbyacrossover.Weascribethis
(forsmall V). Hence the low-energypicture for V < V∗ is
effecttothelackofaself-consistentbathintheNRGcalcula-
that of a conduction band asymptotically decoupled from a
tion.InFig.4wecomparethenonlocalcorrelations,findinga
collection of spins arranged in singlets. In the lattice model
verygoodagreement. Otherfeatureslikethepseudogapfor-
theself-consistencyconditiontransformsthecrossoverofthe
mationaroundV∗arealsoreproduced.Thesimilarityofthese
impuritymodelintoatruephasetransition.
tworesultssuggeststhatmostofthephysicsrelevanttothelat-
ticeisalreadycapturedatthelevelofthetwoimpurityprob- Followingtheparamagneticsolutionwecanalsoobservea
lem. Inthiscasetheonsetofmagnetisminthelatticemodel violation of the Luttinger theorem for V < V∗ in the form
canbeunderstoodasarisingfromthevicinitytotheimpurity of a mismatch between the number of particles and the vol-
QCP. Indeed at the 2IKM critical point there are three rele- ume of the Fermi surface. This is clear from the plot of the
vantoperatorsthatcorrespondtopossiblesymmetrybreaking numberofcandf electronspersite asafunctionofV (Fig.
instabilities. Susceptibilitiesin the antiferromagnetic,super- 5). In the PM phase both the conduction and the f bands
conductingandp-hsymmetrybreakingchannelsarediverging are doped (nf,nc 6= 1). In this case the Fermi surface has
attheimpurityQCP.Sincep-hsymmetryisalreadybrokenin a volume equal to the doping δ = nc +nf − 2. The total
the lattice system, only the other two instabilities are left to occupationisnotconstantbecausewefixedthechemicalpo-
provideawaytoquenchtheresidualentropyassociatedtothe tential rather than the density. For V < V∗ the occupation
QCP.Inthisworkwedidnotconsiderthesuperconductingin- in bothbandsbecomesessentially constant, implyingthatV
stabilitybutweexpectittoplayarelevantrole(andpossibly is no more effective. In this regime the only contribution to
competewithantiferromagnetism)dependingonthe specific the Fermi surface comes from the c electrons which, being
detailsofthebandstructureoftherealisticmaterial. non-interactinganddecoupledfromthef band,haveaFermi
Noticehoweverseveralimportantdifferenceswiththepic- surface with volume nc = 1+δ. These results once again
tureoflocalquantumcriticalityofRef.20. Thereferencesys- support the orbitally selective Mott transition picture of the
temdescribingthephysicsofthelatticemodelrequiresaclus- KB12,13. Namely,anf-likefluidthatbecomesincompressible
teroftwosites,ratherthanone,embeddedinamedium. The beyondtheKBpoint.
criticalpointofthistwoimpuritymodel,hastworelevantdi- In Fig.6 we plot the k-resolved total spectral density
rectionscorrespondingtovaryingthehybridizationandbreak- −π1Im(Gc(k,ω)+Gf(k,ω))atlowenergiesalongtheparam-
ing p-h symmetry. So the impurity model without the self- agneticsolutioninthetwophases. Weintentionallyconsider
consistency condition has a multicritical point rather than a pointsfarfromthetransitiontoshowthattheresultsarerobust
6
FIG.6: (Color online) Totalspectralfunction asafunctionof mo- FIG.7: (Coloronline) Totalspectral function asafunctionof mo-
mentumandenergyonthepathconnectingtheoriginΓ = (0,0,0) mentumandenergyalongthesamepathofFig.6fortheAFsolution.
andthecornerΠ = (π,π,π)oftheBrillouinzone. (a)V = 0.65 (V =0.50)
PMphase.(b)V =0.30PMsolutionintheAFphase.
peaksinFig.2,whilethebandscrossingthechemicalpotential
havemostlyccharacter.
anddonotdependonthevicinitytospecialpointsinthephase
diagram. Toobtainthisquantitywereconstructtherotational
andtranslationallatticesymmetry,brokenbythecluster-tiling
procedure,byperiodizationofthecumulants34. Thisscheme IV. CONCLUSIONS
ismoreappropriatewhencorrelationsarestronger.Indeedthe
simpleperiodizationoftheself-energydoesnotreproducethe Tosummarize,wehaveconstructedandinvestigatedaclus-
localquantitiescorrectlyuponintegrationovermomenta,in- ter dynamical mean field theory for the periodic Anderson
troducingspuriousstates in gapped regionsof the spectrum. modelusingalinkasareferenceframetocapturethecompeti-
In panel (a) the system is in the PM phase (V = 0.65). It tionoftheKondoeffectandtheRKKYinteraction.Themean
is possible to track the origin of the small value of Z in the fieldphasediagramawayfromhalf-fillingfeaturesatransition
narrowbandcrossingthechemicalpotential. TheFermisur- separatingaparamagneticheavyelectronphasefromananti-
faceenclosesa“small”regionoftheBrillouinzonecentered ferromagnetic phase. In the PM phase, the phase transition
around the origin. In panel (b) we show the PM solution in isaccompaniedbyastrongrenormalizationofquasiparticles,
the AF phase (V = 0.30). It is clear that the volume of the marked by a large enhancementof the effective mass on all
Fermisurfaceinthisphasehasincreased. Thedifferencebe- theFermisurface.
tween the Fermi surface volume and the density is equal to
By constraininga mean-fieldmetastable paramagneticso-
one within the numericalprecision, confirming the violation
lutionintotheAFregionofthephasediagram,wehavebeen
of the Luttinger theorem as expected in the orbital selective
able to show that the the transition can be described as an
Motttransition13.Itisalsopossibletotrackthenon-dispersive
orbitally selective Mott transition. The decouplingof the f-
f-bandataroundω ∼−0.2.
band from the conduction band marks a localization of the
Since the PM solution for V < V∗ is unstable to antifer- f-electrons, as shown by the opening of a pseudogap in the
romagnetism, to make contacts with experiments we plot in single particle spectrum and the appearance of poles in the
Fig.7thespectraldensityfortheAFsolution(V =0.50).The self-energy at the Fermi level, a clear fingerprint of “Mot-
bandsaredoubledbecauseoftheAFordering. Theseresults tness”. This is further supportedby the incompressibilityof
arerepresentativeforthewholeAFphasedowntosmallvalue thesystem, whichpresentsa constantdensitybyvaryingthe
ofV. ThestrikingdifferencewithrespecttothePMphaseis hybridization, and by the violation of the Luttinger theorem
the big rearrangementof the spectral weight: now the occu- forV <V∗alongthePMsolution.Itisimportanttocompare
piedregioniscenteredaroundthe(π,π,π)point.Thisresult thistransition,whichpresentsaorbital-selectiveMottcharac-
2 2 2
impliesasuddenjumpacrosstheantiferromagnetictransition ter, with the traditional DMFT Mott transition of the single
inallthepropertiesconnectedwiththetopologyoftheFermi bandHubbardmodel.Inthepresentcasetheweightisshifted
surfaceanditisconsistentwiththeexperimentalobservation from the chemical potential to the immediate vicinity form-
ofadiscontinuousbehavioroftheHallcoefficient35andofthe ingapseudogapratherthanbeingtransferredtotheHubbard
deHaas-vanAlphenfrequencies36. Thecontributionofthef bands at much higher energy. This feature is the productof
electrons in this phase is manifest in the two non-dispersing our link-DMFT analysis in three dimensions, strikingly dif-
bandsatω ∼ −0.1andω ∼ 0.45,correspondingto the two ferent from the infinite dimensional DMFT result where the
7
physicsispurelylocal.ItwouldbeimportantinfutureDMFT result of the response of the system to the instability of the
investigationstobetterunderstandthenatureofthisMott-like nearby decoupling QCP, rather than being an intrinsic fea-
transitionbystudyingitscluster-sizedependence. ture. Noticethatothercorrelationsaredivergingatthe2IKM
In the SDW scenario, the fermionswhich are notcoupled criticalpoint. We expectthatdependingonthedetailsofthe
bytheorderingvectorareunaffectedbythetransition.Onthe modelotherinstabilities(mainlysuperconductivity)compete
contraryinourCDMFT study,thetransitionisaccompanied with antiferromagnetismto orderthe system and quench the
by a dramatic rearrangementof the Fermi surface, in agree- residualentropyoftheunstableQCP.
mentwithrecentexperiments35,36.
Webelievethatourdescriptioncapturesthecorrectphysics
Ourfindingscan beunderstoodin termsof proximitytoa
oftheperiodicAndersonmodelinanintermediateasymptotic
decouplingQCP with the same propertiesof the QCP found
regimeofenergiesandproximitytothetransition. Athigher
inthe2IKM,namelytheimpuritymodelunderlyingtheself-
energies, the single site DMFT description is regained. At
consistency equations. Although the lattice system cannot
verylowenergiesandveryclosetothetransition,furthernon
reach that critical point due to the particle-hole symmetry
local effectscould turn out to be veryimportant. We do not
breaking, the region in which the effect of the critical point
knowifourtreatmentfitswithinthelocalquantumcriticality
isrelevantislargeenoughtotriggerthelatticeantiferromag-
paradigmwherenospatialanomalousdimensionsare gener-
neticinstability31. Ifwefollowtheparamagneticsolutionthe
ated.Toelucidatethesequestionswouldrequiretheinvestiga-
vicinity to the critical point results instead in an increase of
tionofthedynamicalsusceptibilitybeyondtheclusterDMFT
thenon-localcorrelationsΣ12inagreementwiththebehavior approximation,whichisbeyondthescopeofthiswork.
of the 2IKM. In real systems this effect is preemptedby the
lattice antiferromagnetic instability. It would be interesting We would like to thank M. Fabrizio, A. Georges, F.
to realize heavyfermionsystems in highlyfrustratedlattices Steglich, P. Coleman, C. Pepin, I. Paul, O. Parcollet, P. Sun,
tosuppresstheantiferromagneticinstability. Thiswouldpro- K.Haule,E.Lebanon,M.Capone,T.Stanescu,J.C.Domenge
videaphysicalrealizationoftheparamagneticsolutionofthe andL.de’Mediciforhelpfuldiscussionsandcomments.This
CDMFTequations. work was supported by the NSF under Grant No. DMR
In this framework the antiferromagnetic transition is the 0528969.
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