Itinerant electrons in the Coulomb phase

1.4 MB·English

by  L. D. C. Jaubert

Itinerantelectrons inthe Coulombphase
L. D. C. Jaubert,1 Swann Piatecki,1,2 Masudul Haque,1 and R. Moessner1
1Max-Planck-Institut fu¨r Physik komplexer Systeme, 01187 Dresden, Germany.
2Laboratoire de Physique Statistique, E´cole Normale Supe´rieure, UPMC,
Universite´ Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France.
(Dated:January4,2012)
We study the interplay between magnetic frustration and itinerant electrons. For example, how does the
coupling to mobile charges modify the properties of a spin liquid, and does the underlying frustration favor
2
insulatingorconductingstates? SupportedbyMonteCarlosimulations,ourgoalisinparticulartoprovidean
1
analytical picture of the mechanisms involved. The models under considerations exhibit Coulomb phases in
0
2 twoandthreedimensions,wheretheitinerantelectronsarecoupledtothelocalizedspinsviadoubleexchange
interactions. Because of the Hund coupling, magnetic loops naturally emerge from the Coulomb phase and
n
serveasconductingchannelsforthemobileelectrons,leadingtodoping-dependentrearrangementsoftheloop
a
ensembleinorder tominimizetheelectronickineticenergy. Atlowelectrondensityρ, thedoubleexchange
J
coupling mainly tends to segment the very long loops winding around the system into smaller ones while it
3
graduallyliftstheextensivedegeneracyoftheCoulombphasewithincreasingρ.Forhigherdoping,theresults

arestronglylatticedependent,displayingloopcrystalswithagivenlooplengthforsomespecificvaluesofρ,
]
h whichcanmelt intoanother loop crystal byvarying ρ. Finally, wecontrast thistothequalitativelydifferent
c behaviorofanalogousmodelsonkagomeortriangularlattices.
e
m
I. INTRODUCTION Ourapproachtotheproblemconsidersaregimewherethe
-
t effectsoffrustrationareparticularlystrongbutwhereconsid-
a
erable progresstowards a detailed description is nonetheless
t The combination of magnetism and itinerant electrons is
s possible by analytical (or simple numerical) means. That a
. a multi-faceted field in the physics of correlated electrons,
t (non-trivial)regimewherethisispossibleexistsatallisapri-
a where our understanding is still remarkably patchy: even in
m thecaseofasquarelatticeHubbardmodel,welackconsensus orinotobvious,andwefindthatweneedtorestrictanumber
ofparameterstolimitingvalues:westudythelimitwherethe
- onadetailedphasediagraminthedoping-temperatureplane.
d Besides the cuprate superconductors, there are plenty of magneticenergyscalesaremuchlargerthanthehoppinginte-
n gralt,sothattheresultingproblemisoneofelectronshopping
other settings in which interesting questions arise, not least
o onaclassicalbackgroundspinconfiguration.Wedonotneed
popularizedoflatebyquestionsraisedbythepnictidesuper-
c
to restrict the electron density to be small, although for that
[ conductors,wheremagneticfrustrationandaccidentaldegen-
case,wehavethemostdetailedsetofresults.
eracieshavestartedtobeconsidered.
1 Wefindthatthelow-densitybehaviourcanbemappedonto
More broadly, there has been increased interest in the in-
v teractionoffrustratedmagnetismwithitinerantelectrons1–11. astudyofaclassicalloopmodelwithnon-trivialweightsaris-
7
ingfromtheadditionofelectronsHund’scoupledtothespins
7 Here, we take up the spirit of this thread of work and study
in the Coulomb phase. This results in phenomenasuch as a
6 itinerantelectronsonahighlyfrustratedlattice,thepyrochlore
0 lattice. We consider both three and two dimensions, the lat- transitionfroma(intwodimensions,critical)percolationsit-
1. tercasealsobeingknownasthesquarelatticewithcrossings, uationtooneinwhichtheloopsacquireanexponentiallength
distribution,therebyremovingallconductingpathsacrossthe
0 planarpyrochlore,orcheckerboardlattice.
sample.Asthedopingisincreasedfurther,wefindasequence
2 Westartwithanexoticfrustratedphaseofamagneticinsu-
1 lator,theCoulombphase,whichhasbeenextensivelystudied of density-dependentpreferredlooplengths, which lead to a
v: recently12–17. Thisphasehasanumberofunusualproperties, tendencytoformloopcrystalswhichmay,however,befrus-
tratedbythelatticegeometry.
i includingalgebraicspincorrelationsandtheemergenceofex-
X tendedone-dimensionaldegreesoffreedom18–20,thenatureof The organization of this article, and our main results, are
r whichisanindependentlyinterestingproblem21. summarizedbelow.
a
Especiallythe latter willplay animportantrole in thefol-
lowing analysis, given the ability of electronsto provideev-
idence for non-local structures through properties related to A. SummaryandOverview
transport phenomena. Indeed, it is the marriage of the lo-
cal constraints imposed by frustration with the ‘non-local’ We restrict ourselves to the limit of large exchange cou-
physicsdescribingmobileparticleswhichmakesupformuch pling interactions so that the ice-rules (see figure 1) them-
oftheinterestinthisfield. selvesarenevercompromised. Wemakeextensiveuseofthe
TheIsing-doubleexchangemodelwhichwestudyherehas loop picture encodingthe ice rules.20 The loopsserve as 1D
many parameters: electron density, ρ, temperature T, Ising channelsfor the electrons. The problemis thus transformed
anisotropy,Hund’scoupling,J ,magneticexchange,J,and to entropyand energyconsiderationsofpossible loopcover-
H
electronhoppingintegral,t. Afullstudyofgeneralparameter ings,withloopssupportingvaryingnumbersofelectron.This
choicesiswell-nighimpossibleinanydetailanalytically. “loop framework” for describing conduction electrons is in-
2
troducedinSectionII.
Ref.20hasdescribedinsomedetailtheloopdistributions
forboth2Dand3Dintheabsenceofelectrons.Onceelectrons
areadded,withineachloopelectronscanoccupystateswhose
energyisgivenbya1Ddispersion. Thedispersionminimum
is independentof loop length, so the first electron in a loop
has the same energy in all loops. As a result, there is a low
dopingregimewhereitispossibletofitatmostoneelectron
perloop. Allsuchconfigurationshavethesameenergy. This
istheentropicregime,becauseentropicargumentsdetermine
favorableconfigurationswithinanequal-energymanifold.
When the density of electrons ρ is larger, a sub-extensive FIG.1: IntheCoulombphase,eachfrustratedunit(crossedsquares
number of states, possibly even a unique one, tend to be fa- in2Dcheckerboard,left,andtetrahedrain3Dpyrochlore,right)pos-
vored, because they manage to minimize the kinetic energy; sessestwoupandtwodownspins,respectivelycoloredinblueand
thiswetermtheenergeticregime. red. Thosearetheso-calledice-rulesordivergencefreeconditions.
Connectingspinsofthesamecolorformsanetworkofloops,asillus-
In Section III, we presententropic considerationsrelevant
tratedforthecheckerboard. Thismodelisequivalenttothenearest
to the regime of low doping. The total entropy contribution
neighbourspinicemodel.
comesfromboththeloops(withparametersextractednumer-
ically),andtheelectrons(derivedanalytically). Inparticular,
thepresenceofelectronsactslikeacutoffonthetotalnumber
II. THESYSTEMANDTHELOOPFRAMEWORK
ofloopsinthesystem. In3D,forbiddingconfigurationswith
less loops than electrons suppresses the formation of exten-
siveloops(presentin thepeakofthe probabilitydistribution A. Themodel
function (PDF), see e.g. figure 5), but does not modify the
exponentsof thePDF. On theotherhandin 2D,thereisevi- We will focus primarily on the checkerboard and py-
denceforavariationintheexponentofthepowerlawofthe rochlore lattices, which are two lattices where a Coulomb
loopdistribution. phasecanappear(seefigure1). Thelocalizedmagneticmo-
Oncetheelectronicdensitygetstoolargeforloopstobere- ments are Ising spins Si all parallel to a global axis, while
strictedtoatmostoneelectron,weneedtoconsiderenerget- itinerantelectronscanhoponthe lattice sites. TheHamilto-
ics. Section IV presentsthese energyconsiderationsand the nianis
phase diagram obtained thereby. Ignoringlattice constraints
onloopcoverings,weuseenergycalculationsandaMaxwell H = J Si·Sj − t(c†i,αcj,α + c†j,αci,α)
constructiontoobtainthephasediagramofFigure9. Wever- hXi,ji hiX,ji,α
ify some of these results through Monte Carlo simulations. J c† (σ S )c (1)
Wealsopresentconstraintsimposedbythelatticesundercon- − h i,α α,β · i i,β
i,α,β
siderationsandidentifyloopcrystalsarisingasthedopingis X
varied.
wheretisthehoppingintegralbetweentwoneighboringsites,
Withperiodicboundaryconditions,loopsspanningthesys- c† (c ) are creation (annihilation) operators of itinerant
i,α i,α
temcanbedividedintosegmentsconnecting“opposite”faces,
electronsofspinαonsitei,andσ arethePaulimatrices.In
α,β
which we call filaments. Because the transmission of elec-
orderofappearance,thetermsinequation(1)aretheantifer-
trons through the system can only occur via these conduct-
romagneticnearest-neighbourexchangebetweenthelocalized
ing channels, section V is dedicated to their statistics, as a
spinsincorporatingmagneticfrustration,thehoppingtermal-
function of dopingρ and dimension. In absence of itinerant lowingmovementofitinerantelectrons,whosespinsinteract
electrons,thenumberoffilamentsgrowslinearlywith(cubic)
with the localized magnetic momentsthroughferromagnetic
system size in 3D but remains constant and of (1) in 2D. Hundcoupling(lastterm).
O
While these behaviors are qualitatively not modified at low
In this work, we focus on the limit t J J. In
h
doping,theconductingchannelsturnouttovanishatinterme- ≪ ≪
thislimit,thehighlydegenerategroundstateofthefrustrated
diatevaluesofρ.
systemservesasbackgroundforthemotionoftheelectrons.
Aseparatefinalsectionisdevotedtoanoutlookwhichalso Magneticexcitations(violationsoficerules)arenotpresentin
containssomewordsonthebehaviourofanalogousmodelson thislimit. Electronscanonlyhopbetweennearestneighbour
the triangular and kagomelattices, which turn out to exhibit spinshavingthesameorientation.
qualitatively different properties both from the pyrochlores Atzero temperature,a Ne´el orferromagneticorderwould
andfromeachother: wefindamagneticconductingsolidas give rise to an insulating or metallic state respectively, but a
wellasaninsulatingcooperativeparamagnet. spinliquidprovidesanetworkofconductingpathsforitiner-
The numerical component of our work involves Monte antelectrons. Thepresentworkunveilsthe geometryofthis
Carlosimulationsofseveraltypes;somedetailsareprovided network, as it is influencedby both the lattice and the mini-
intheAppendix. mizationofthehoppingenergy.
3
B. Emergenceofloops
The pyrochlore and checkerboard lattices are made of
corner-sharingunitswithfourspins;respectivelythetetrahe-
dron and the square with crossings. The antiferromagnetic
couplings impose the so-called “ice-rules” with zero mag-
netization per unit, obtained with two spins pointing up and
twospinspointingdown22. Inabsenceofitinerantelectrons,
2t 2t
this ground state is highly degenerate with 6 possible con-
figurations per unit. It correspondsto the 6-vertex model in
2 dimensions23 and can be mapped onto the nearest neigh- Ek
bor spin ice modelin 3 dimensions24,25. These spin systems es, k k
gi −π π −π π
serveasbackgroundfortheemergentphysicsoftheso-called er
Coulombphase16,agaugetheorywherethediscreteice-rules en
under coarse-grained lead to the emergence of a divergence
freeflux. −2t −2t
Joiningspinsofthesameorientationineveryunit,oneob-
tainsloopsofup spinsandloopsofdownspins(see Fig. 1). FIG.2: Single-particleenergylevelsofitinerantelectronsconfined
TheCoulombphasecanthusbedescribedasanensembleof toloopsoflengthℓ=8(left)andℓ=12(right),duetothehopping
possibleloopcoverings. Theresultingloopmodelpossesses terminHamiltonian(1).Therearenondegeneratelevelsatthehigh-
twoflavors(loopsofupanddownspins),whereeverysiteof estandlowestenergies,±2t,independent oflooplength; theother
thepremediallattice16 isoccupiedbytwoloops,oneofeach levelsaredoublydegenerate.
flavor,andeverybondisvisitedbyonelooponly.Inprevious
work by some of the authors20, a detailed account has been
givenofthestatisticsanddistributionoftheseloops,bothfor thetotalnumberofloopsinagivenconfigurationNℓ =
•
the2Dand3Dcases. ihi;
In the limit of large J , all up (down) electrons are only
h Ptheaveragelooplengthforagivenconfiguration
allowedtohopalonganup(down)loop,andareconstrained •
toremainwithinthisloop.Thisreducestheelectrondynamics ℓ = ihiℓi = N ;
tobeone-dimensionalwhateverthedimensionofthelattice. h N
P i i ℓ
The 1D hopping restriction allows us to describe electron
dynamics in terms of the dispersion of a 1D tight-binding thestatPisticalaveragelooplength ℓ overallloopcon-
• h i
problem, E = 2tcosk, with k the 1D momentum along figurations;
k
−
the loop. The 1D momentumalongthe loop, k = 2πq/ℓ, is
thestatisticalaveragenumberofloops N ;
discreteforaloopoffinitelengthℓ(q = ℓ/2,...,ℓ/2 1). • h ℓi
− −
Anup(down)loopoflengthℓcancontainbetween0toℓup thenumberoffilaments(seesectionV).
(down)electrons. Notethatdoubleoccupancydoesnotoccur •
inthelimitweareconsidering.
Thelowestandhighestsingle-particlelevelsinthedisper- III. LOWDOPINGREGIME
sion have energy 2t, independent of the loop length (fig-
±
ure2).
InthissectionweconsiderthelowdopingregimewhereN
e
Oursystemisthusdescribedby
issmallenoughtohaveloopconfigurationswith moreloops
thanelectrons,N >N . Sincethelowestsingle-electronen-
the numberof lattice sites N, equalto 4L2 and 16L3 ℓ e
• ergylevelinanyloopis 2t,theminimumaccessibleenergy
for the checkerboard and pyrochlore lattices respec- −
is the same ( 2tN ) for all such configurations. Therefore,
tively,whereListhelinearnumberofunitcells; − e
thegroundstatemanifoldconsistsofallsuchloopcoverings
the total electronnumberN , and the electrondensity with the same energy 2tN . The free energy within this
e e
• −
ρ=N /N; manifoldis thenonlydeterminedbyentropics. Atzerotem-
e
perature,entropyisunderstoodinthesensethatallconfigura-
the loop histogram of a given configuration, i.e. the
tionshaveanequalprobabilitytooccur.
•
number h of loops of length ℓ , ℓ being necessarily
i i i Inthefirstsubsectionbelow,wepresentsomeentropycal-
anevennumberonabipartitelattice;
culations,combiningloopandelectroniccontributionstothe
entropy,andshowhowthisdeterminestheaveragelooplength
the type of lattice, which will among other things de-
• termine the smallest possible loop in the system ℓ atnonzeroelectronicdensityρ. Inthesecondsubsectionwe
min
presentnumericalresultsontheeffectofelectronsontheen-
(4 for checkerboardand 6 for pyrochlore); the longest
tirelooplengthdistribution(loopPDF).Theeffectontheloop
possiblelooplengthisalways N/2.
∼ PDFisanaturalwaytocharacterizetheinfluenceofelectrons
Wedefineafewadditionalrelevantobservables: inthemagneticCoulomb-phasesystem.
4
3.5 7
3 6
2.5 5
N N
√ 2 √ 4
⋅ m ⋅ m
or 1.5 or 3
n n
H H
1 2
0.5 1
0 0
-1 -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4
(N - <N>) / √N (N - <N>) / √N
l l l l
FIG.3: HistogramofdistributionofthenumberofloopsN perconfigurationin2D(left)and3D(right),intheabsenceofelectrons. The
ℓ
distributionisscaledaccordingtotheGaussianexpressionofequation(2)collapsingallsystemsizesontothesamecurve(L=100to600in
2DandL=4to60in3D).In2D,thetwoGaussianshavethesamewidthandcorrespondstoevenandoddN (upperandlowercurve). In
ℓ
3D,finitesizecorrectionsarevisibleforL=4(redcrosses).
A. Entropy loopcontributiontotheentropy( lnP)is
∼
N 2
The total entropy consists of loop and electronic contri- S =S ℓ ℓ , (4)
loop 1− 2κ ℓ 4 −h i
butions. The loop distribution is of course itself affected by
h i
the itinerant electrons. However, in the limit of small ρ, the (cid:0) (cid:1)
whereS isaconstant.
changeinloopentropyissmall. Belowwecombinetheρ=0 1
Electronic entropy. Since we have at most one electron
loop entropy with the finite-ρ electronic entropy to approxi-
per loop, the number of possible combinations to put N
matethetotalentropyatsmallρ. e
electrons in N loops is the binomial (Pascal) coefficient
Loopentropy. Figure3showsthedistributionofthenum- ℓ
N !/[N !(N N )!]. Thelogarithmthengivestheelectronic
ber of loops N having length ℓ, in the absence of electrons l e l− e
ℓ contribution to the entropy. Using Stirling’s approximation
(ρ = 0), obtained from Monte Carlo simulations(Appendix
for the thermodynamiclimit (N 1, N 1), we get as
A).ThedistributionhasGaussianform: l ≫ e ≫
perusual
1 (N N )2
P(Nℓ) exp ℓ−h ℓi (2) N
∼ √N − 2κN S = [xlnx+(1 x)ln(1 x)] (5)
(cid:20) (cid:21) elec ℓ − −
whereκ =0.0384andκ =0.00423.In2D,therearetwo
2d 3d
Gaussians,correspondingtoevenandoddNℓ. wherex=Ne/Nl =ρℓ.
Forasystemwithshort-distancecorrelations,Gaussiandis- Total Entropy. For small electron densities, we can ap-
tributionsare naturalto expectfromtheCentralLimitTheo- proximatetheloopentropybytheρ=0expressioncalculated
rem,sincebydividingthesystemintosmallmesoscopicseg- above.Thetotalentropyofelectronsandloopsisthen
mentsthetotaldistributioncanberecastintoa sumofmany
random variables. In our case, however, we have a system N
S = ρℓlnρℓ+(1 ρℓ)ln(1 ρℓ)
with algebraic correlations and extended objects (loops), so tot
− ℓ − −
findingGaussiandistributionsisnotaprioritrivial.
(cid:2) N (cid:3) 2
Equation 2 can be expressed in terms of the average loop + S1− 2κ ℓ 4 ℓ−hℓi0 . (6)
lengthℓ=N/NℓinsteadofNℓ: h i0 (cid:0) (cid:1)
Thesubscript0representstheensembleaverageatρ=0.The
dN √N N 1 1 2
ℓ valueofℓthatmaximizesS isfound(inthelimitρℓ 1)
P(ℓ) P(N ) exp tot
≡ ℓ (cid:12) dℓ (cid:12) ∼ ℓ2 "−2κ(cid:18)ℓ − hℓi(cid:19) # tobe ≪
(cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)√N exp N ℓ−hℓi 2 (3) ℓopt ≈hℓi0 − ρκhℓi30. (7)
∼ ℓ2 "−2κ(cid:18) hℓi2 (cid:19) # Forρ ℓ 1,thedistributionstaysalmostGaussian,sothis
0
Herewehaveused1/ℓ ℓ 1/ ℓ 2,whichisvalidforlarge mosthpriob≪ablevalueofℓisapproximatelythemeanvalueof
h i ≈ h i
N intheregionwheretheGaussianisappreciable. Thusthe thedistribution ℓ . InFigure4,wecomparethisprediction
ρ
h i
5
225
24
22
200
20
175
<l>loop 18 <l>loop
16 150
14
125
12
10 100
0 0.01 0.02 0.03 0.04 0 0.001 0.002 0.003 0.004
ρ ρ
FIG.4: Meanvalueofthelooplengthhℓiasafunctionofdoping. Left: checkerboard, L = 10(reddots)and60(bluetriangles). Right:
pyrochlore,L=6(reddots)and20(bluetriangles).ThedashedstraightlinesarepredictionsfromEquation(7),withoutanyfittingparameters.
Forbothpanels,thebluetrianglesdatapointsareveryclosetothermodynamiclimit.
tonumericaldata.ForeachsystemsizeL,wecanextract ℓ 0 100
h i
fromthe numerical ℓ at ρ = 0. Eq. (7) then givesa linear
h i
prediction (dashed straight lines in Figure 4), which works
10
wellforsmallρ.
1
ρ)
B. Looplengthdistribution (l,D 0.1
P3
We nextexaminetheeffectofelectronsonthe entireloop 0.01
probability distribution function (PDF). We denote by τ the
power-law exponent, when the PDF has form ℓ−τ. Without 0.001
electrons(ρ = 0), the PDF follows P L2/ℓτ in 2D with
2D
∼
τ = 2+ 1/7. In 3D, the ρ = 0 PDF displays a crossover 0.0001
around ℓ L2 between two power laws, from L3/ℓ5/2 to 10 30 100 300 1000 3000 10000
∼ l
1/ℓ.(Fordetails,seeRef.20.)
Atfiniteelectrondensities,loopconfigurationswithN <
N are rejected due to energetics, as explained previouℓsly. FIG.5: LooplengthdistributionP3D(ℓ,ρ)in3Dfordifferentval-
e uesof doping (ρ = {0,0.00075,0.0015,0.002}]) and systemsize
Thuselectrondopingactsnotentirelyunlikea“chemicalpo-
L=10. ThedistributionisnormalizedsuchthatR ℓP3D(ℓ,ρ)dℓ=
tential” for loops, favoring configurations with more loops, 16L3 =N.Thedashedlineindicatesthepowerlawfitℓ−1atρ=0
andthusapriorishorterones.Inboth2Dand3D,thisimplies and the arrow shows the shifting of the distribution for increasing
a disappearanceoflongerloops,ascanbeseen inthe calcu- ρ. The exponents of the two power law regions (before and after
latedPDF’sofFigures5and6,wherethelarge-ℓpartsofthe ℓ ≈ L2 = 100) do not vary, but the peak for long winding loops
PDFareprogressivelydecimatedforincreasingρ. In3Dthe ℓ∼8L3 =N/2getssmallerwithincreasingρ.
formofthePDFisotherwiseunchanged(Figure5). In2Dthe
effectseemstobemoredrastic;theinsetoffigure6suggests
thattheentirepower-lawbehaviorofthePDFismodified.
To quantify how the 2D loop PDF changes qualitatively modelswithloopfugacityn(seee.g. Ref.23). Thepartition
at finite ρ, we define and compute a local exponent in ℓ: functionofthefullypackedloopmodelis = nNℓ,where
Z
τ (ℓ,ρ)=log(P (ℓ,ρ)/P (2ℓ,ρ))/log2. Thisisdis- the sum runsoverall possible configurations. The Coulomb
local 2D 2D
playedintheupperrightinsettoFigure6. Ourresultssuggest phasecorrespondstoafullypackedloopmodel2P0;atzerodop-
atrendtowardincreasingτ asafunctionofρ,consistent ing,thefreeenergyofourmodelistriviallyindependentofthe
local
withtheideathatitinerantelectronsfavorsmallloops. numberofloopsN andthuscorrespondstoafugacityn=1.
ℓ
This outcome deserves a few comments. In Ref. 20, the Highervaluesofthefugacitiesfavorconfigurationswithmore
loop statistics of the 2D Coulomb phase (zero doping) has loopsandtendtoincreasethevalueofτ27,inawayreminis-
beenshowntobeanalogoustotheStochasticLoewnerevolu- centoftheinfluenceofdopinghere. Theadditionofitinerant
tionprocessSLE withfractaldimensionD =1+κ/8= electronsremainsanon-trivialproblemandisnotexactlythe
κ=6 f
7/4. The SLE can be identified to various realizations of sameasafugacityforaloop,butatsmallandfinitedopingρ,
κ
the (n) model through the relation n = 2cos(4π/κ).26 somefeaturescouldbecapturedby (n(ρ) > 1) modelsor
O − O
The (n) modelis often used to describe fully packed loop SLE processes.
κ(ρ)<6
O
6
2 lengthℓ,theenergyis(seefigure2)
10
2.18 no 2πn sin(πρ)
101 τlocal 2.14 E(ρ,ℓ)=−2t n=X−no cos(cid:18) ℓ (cid:19)=−2t sin(π/ℓ) (8)
ρ) 100 2.10 Foranevennumberofelectrons2no,thisexpressionbecomes
P(l,2D 10-1 0 0.004ρ 0.008 E(ρ,ℓ) = 2t sinπ(ρ−1/ℓ) 2t cos(πρ) (9)
− sin(π/ℓ) −
-2 Moregenerally,if ρℓ is notaninteger, we definethe highest
10
oddintegerbelowρℓas
-3
10 ρℓ 1
10 30 100 300 1000 3000 η = 2E − +1 (10)
2
l (cid:18) (cid:19)
where E(.) is the floor function. Each loop is filled with at
FIG. 6: Main: Loop length distribution P2D(ℓ,ρ) in 2D for dif- leastηelectronsuptotheenergylevelsatk = ±π(η−1)/ℓ,
ferent values of doping (ρ = {0,0.0025,0.0056,0.0087}]) and while the remainingN(ρ η/ℓ) electronsin the system are
−
system size L = 40. The distribution is normalized such that distributed in the partially filled level at k = π(η +1)/ℓ.
R ℓP2D(ℓ,ρ)dℓ = 4L2 = N. Thedashedlineindicatespowerlaw Thetotalenergyis ±
L2/ℓ15/7andthearrowshowstheshiftingofthedistributionforin-
creasingρ. Onthisscale,itisnotobviouswhethertheexponentof sin(πη/ℓ) η η+1
E(ρ,ℓ)= 2tN + ρ cos π (11)
thepowerlawvariesornot. Inset: Thepower-lawexponentdefined − ℓsin(π/ℓ) − ℓ ℓ
locallyonℓ,τlocal forℓ = 8((cid:4)),16(N),32(•)forL = 40(similar (cid:20) (cid:16) (cid:17) (cid:18) (cid:19)(cid:21)
behaviorobtainedforL=20and60). Theseexpressionsareelectron-holesymmetric,i.e. invariant
underρ (1 ρ).
↔ −
Figure 7 displays the loop length ℓ(ρ) that minimizes the
IV. LARGEDENSITIES;PHASEDIAGRAMS energy (11) as a function of ρ, and thus corresponds to the
ground state if we impose a unique loop length in the sys-
tem. (Theelectron-holesymmetryshows up clearly through
In this section, we presentresults relevantfor higher den-
themirrorsymmetryoneithersideofρ=0.5. Therefore,we
sities, whereentropicconsiderationsarenolongersufficient,
shallfromnowononlyconsiderdensitiesbelow1/2.)
and non-trivial electronic hopping energies need to be con-
Sincethelowestpossibleenergyforanelectronis 2tand
sidered. Wecontinuetodescribethesystemintermsofloop −
is only accessible for one electron per loop, loops of length
coverings.
4 are favored up to ρ = 1/4. Of course for lower densities
Wefirstprovideananalysisbasedoncalculationsforloop
than1/4, otherconfigurationsmaybepossibleaslongasthe
coveringsof equal-lengthloops. Using a Maxwellconstruc-
numberofloopsislargerthanthenumberofelectrons,buta
tion,wecanusethisinformationtopredictrangesofelectron
systemwithonlyloopsoflength4willalwaysbepartofthe
densitywherethegroundstatemanifoldconsistsofcoverings
groundstatemanifoldforρ61/4.
byloopsoftwodifferentlengths.
Remarkably, for most densities, a finite loop length with
Theseconsiderations,describedinthefirsttwosubsections
discreteenergylevelsispreferredcomparedtoinfiniteloops,
below, do not take into account any lattice constraints other
exceptforρ=1/3and2/5.
than the fact that the minimum loop length is ℓ = 4(6)
min Weplotonfigure8theminimumenergycorrespondingto
forthe checkerboard(pyrochlore). Lattice constraints, disal-
thelooplengthℓ(ρ)offigure7: E (ρ)=E(ρ,ℓ(ρ)).
min
lowingsomeloopcoverings,aredifficulttoenumerateorlist
comprehensivelyonaccountoftheirnon-localnature. Inthe
finalsubsection,wepresentthelatticeconstraintsthatwehave
B. Maxwellconstruction;lattice-independentphasediagram
identified,andtheirimplications.
We now move beyond configurations with unique loop
length. Weneedtoconsidermixturesofelectrondensitiesin
A. Equal-lengthloopconfigurations thedifferentloops.ThisisdonethroughaMaxwellconstruc-
tion, similar to the physics of a liquid-gas first-order phase
Inthissubsectionandthenext, we imaginethatcoverings transition.
withanyuniquelooplengthℓ>ℓ arepossible. Later,we AsystemofN sitesanddensityρcanbedividedintotwo
min
will show that at certain fillings, the ground states exhibit a subsetsofsitesN1andN2,withdensityρ1andρ2ofelectrons
uniquelooplength,whileatothers,thebehaviourcanbeun- andlooplengthℓ1andℓ2respectively,with
derstoodthroughaMaxwellconstructionbasedonthesingle-
N = N + N
lengthresults. 1 2
N ρ = N ρ + N ρ . (12)
Foranoddnumberofelectronsρℓ = 2n +1inaloopof 1 1 2 2
o
7
1000 1000
100 100
) )
ρ ρ
( (
l l
10 10
6
4
1 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ρ ρ
FIG.7: Looplength ℓ(ρ)minimizing theenergy (11)for asystemfilledwithloopsof length ℓonly, assuming nolatticeconstraints other
than l = 4 (left panel, checkerboard) or 6 (right panel, pyrochlore). ρ takes all rational values p/q, withq ∈ {0,1,2,...,1000} and
min
p∈{0,1,2,...,q},andweconsideralllooplengthsfromℓ toℓ =100(redcrosses)and1000(bluesquares): plottingtwodifferent
min max
valuesofℓ providesagraphicalwaytovisualizethetwovaluesofρwherethemostfavorablelooplengthisinfinite,namelyρ=1/3and
max
2/5(andtheirsymmetricimageswithrespecttoρ=1/2).Ifseverallooplengthsgivethesameenergy,weplotthesmallestone.
Now if a straight line between E (ρ ) and E (ρ ) re- thatlatticeconstraints(nextsubsection)willmodifysomeof
min 1 min 2
mainsbelowthecurveE (ρ)onfigure8,i.e. if thephasediagram.
min
N E (ρ) > N E (ρ ) + N E (ρ ), (13)
min 1 min 1 2 min 2
thenthemixtureoftwodensitiesismorestablethanaunique C. Latticeconstraintsonloopcoverings
density and “phase separation” occurs. We thus construct
a phase diagram, separating regions of different loop length
Wenowconsidereffectsofthelatticeindisallowingsome
combinations.
oftheconfigurationspredictedbyouranalysisabove.Forthe
Onfigure9,thedotscorrespondtodensitieswhereaunique
specificdensitieswhereitispossibletocoverthelatticewith
loop length is favored, whereas the zones between them are
loopsofauniquelength,theresultisanorderedloopcrystal.
“phase mixed”, defined by the loop lengths and electron
Insomecases,however,acoveringbyauniquelengthorbya
densities of the surrounding dots. The ratio of one phase
combinationofloopsoftwolengths,isnotpossible.Wepoint
comparedtotheotherisgivenbyequations(12).Forexample
outsomesuchcasesbelow.
forρ=0.4,80%ofthesitesbelongtoloopsoflengthℓ =8
1
with electron density ρ = 3/8, while the remaining 20%
1
belong to loops of length ℓ = 6 with density ρ = 1/2.
2 2
1. Checkerboard
As shown on figure 7, loops of length 4 are particularly
robust over a wide range of ρ; preventing their formation,
e.g. in the pyrochlore lattice in d = 3 which permits loops Wefirstfocusontheregimeρ [1/4;3/8]favoringloops
∈
ofminimallength6,thusstronglymodifiesthephasediagram. of length 4 or 8 according to the Maxwell construction. As
illustratedinfigure10,notonlycanthelatticebecoveredby
These resultsindicatethatitinerantelectronstendto favor theseloops,butthetransformationfrom4loopsoflength4to
relativelysmallloopsandtopreventtheformationofinfinite 2oflength8isalsopurelylocalandallowsallpossibleratios
onesatzerotemperature. betweenthesetwophasesinthethermodynamiclimit. Apair
Atlowelectrondensities(upto1/ℓ ),theminimumen- ofloopsoflength8(onemadeofupspins,theotherofdown
min
ergy for a given ρ is degenerate: e.g. for ρ = 1/12, having ones) cannot be separated if there are no other lengths than
onlyloopsoflength4,6,8,10or12givesthesameenergy,as 4 and 8 in the system. We shallcall such pair a defect. The
thenumberofelectronsissmallerthanthenumberofloopson defectconcentrationisdeterminedbyρ.Defectsarenottopo-
thesystem,andeveryelectroncanfillthelowestenergylevel. logical in the sense that they can be created and annihilated
This explains why E (ρ) is a straight line in this region locally,theycanbeplacedanywhereinthebackgroundmade
min
(Figure8): as ρ decreasesfrom1/ℓ to 0, the degeneracy ofloopsoflength4.
min
of the groundstate increases untilonerecoversnaturallythe At quarter filling, there is one electron per loop of length
fulldegeneracyforρ = 0. Thisofcoursecorrespondstothe 4, its energy is 2t and their total number is N/4. Each
−
entropicregimementionedabove. additional pair of electrons fills the first excited level of the
We stress again that these results have been obtained in- newly created defect (3 electrons per ℓ = 8 loop) and gives
dependentlyofthelattice (exceptforthevalueofℓ ),and an energy 4t √2 1 . At a given ρ, the number of elec-
min
− −
(cid:0) (cid:1)
8
0 0
-0.1 -0.1
) )
ρ ρ
( (
E -0.2 E -0.2
-0.3 -0.3
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ρ ρ
FIG. 8: Minimum energy E(ρ) corresponding to the value of ℓ(ρ) plotted on figure 7, for a system filled with loops of length ℓ only,
assuming no lattice constraints other than l = 4 (left, checkerboard) or 6 (right, pyrochlore). ρ takes all rational values p/q, with
min
q ∈ {0,1,2,...,1000}andp ∈ {0,1,2,...,q}. Weconsideralllooplengthfromℓ toℓ = 1000. Wechosearbitrarilyt = 1/2. A
min max
Maxwellconstructioncanbevisualizedfromthisfigure.Thereddotsareasetofpoints,suchthatthedashedlinesconnectedthemarealways
belowtheredcurveE(ρ).Henceamixtureoftwophasescorrespondingtotwoconsecutivereddots(ℓ1,ρ1)and(ℓ2,ρ2)hasalowerenergy
thanasinglephasewithauniquedensityofelectronsρonauniquetypeoflooplengthℓ. Forincreasingvaluesofρ,thereddotscorrespond
tothemostfavoredlooplengthℓ={4,8,6,8,4}(leftpanel)andℓ={6,12,10,8,6,8,10,12,6}(rightpanel),ascanbereadfromfigure7.
Checkerboard
Pyrochlore
FIG.9: Zerotemperature“phasediagram”oflooplengthconfigurationasafunctionofelectrondensityρ,obtainedwithoutaccountingfor
latticeconstraintsotherthanℓ =4(6)forcheckerboard(pyrochlore).Aphaseisdefinedbythelengthofitsconstitutingloopsandthedensity
min
ofelectronsonthem.Thevaluesofρandℓ(ρ)foreachphasearegivenbelowandabovethelines.
betweenρ=1/4and3/8isthen
E(ρ)= 2tL2 1+(4ρ 1)(√2 1) (14)
− − −
(cid:16) (cid:17)
FIG.10: Schematicrepresentationof thecheckerboard latticewith
16sitesanddifferentloopcoverings. Left: 4loopsoflength4,cor-
respondingtothegroundstateatρ = 1/4with1electronperloop. Once normalizedpernumberof sites 4L2, thisexpressionis
Center: 2 loops of length 8, corresponding to the ground state at thedashedlineplottedonfigure8. Wehaveverifiedthisana-
ρ = 3/8with3electronsperloop. Between1/4and3/8,amixture
lyticalresultwithfinite-temperatureMonteCarlosimulations;
ofthesetwoconfigurationswilloccur,theirrespectiveratiobeingset
somedetailsofthemethodareinAppendixA.Extensivede-
by the total number of electrons N = ρN. This arrangement of
e generacyisrecoveredinthisregionbetween1/4and3/8.
twoloopsoflength8appearsasa“defect”inacrystalofloopsof
length4.Right:aredloopoflength6spannedbyanotherblueloop;
theshortest waytoclosethisbluelooprequires10sites. Itisthus At higher doping, aboveρ = 3/8, the Maxwell argument
impossibletohaveamixtureofloopsoflength6and8only. predicts a mixture of loops of length 8 and 6. However, as
we can see in figure 10, a single loop of length 6 imposes
the presenceof loopsoflength10at least. Thisimpliesthat
thisregionofthephasediagram(obtainedwithoutaccounting
tronsaddedwithrespecttotheloopcrystalatquarterfillingis for such lattice constraints) is further modified, in an as yet
(ρN N/4) = (4ρ 1)L2. Thetotalenergyofthesystem unknownmanner.
− −
9
2. Pyrochlore
100
Analogousmodificationofthephasediagramoffigure9is
moresevereforthepyrochlore,andshowsupalreadyatρ =
30
1/ℓmin = 1/6. We havefoundthatitisimpossibletocover ents
thepyrochlorelatticewithloopsoflength6only.However,it am
ispossibletodoitforhalfofthesystem,asexplainednext. of fil 10
Thepyrochlorelatticecanbeseenasanalternativestackof ber
m
kagomeandtriangularlayersorthogonaltooneoftheglobal Nu 3
[111]axes. Asdepictedonfigure13(d),2/3ofeachkagome
layer can be filled with loops of length 6 (blue hexagons), 1
3 6 10 30 60
while the other 1/3 of the kagome sites forms extensive
L
windingloopsalongthe[111]-axis,crossingalternativelythe
kagomeandtriangularlayers. Sinceallbluesitesformloops FIG.11: Averagenumberoffilaments,i.e.thosesegmentsofloops
of length 6, putting one electron per loop providesa ground spanningtheentiresystemfromoneborder totheoppositeone, as
stateconfigurationuptoρ=1/12atleast. Thisisapriorinot a function of system size L. We use two definitions of filaments;
theonlyone,butthisprovesitsexistence. they can either cross the orthogonal borders thanks to the periodic
boundaryconditions(•)ornot((cid:4)). Thecartoonontherightshows
Indeed, despite an intensive search by complete enumer-
suchaboundary-crossingfilament,whichwouldbeexcludedinthe
ation of configurations respecting the ice-rules on the py-
(cid:4) data. Dashed lines are guides tothe eye for the linear behavior
rochlorelattice,wehavenotdetectedanyrelevantloopcrystal
withL.Bothx-andy-axesareonalogarithmicscale.
or mixtureof themfor systemsup to 128sites. A system of
size 192 has been partially investigated, with the same out-
come. Thesmallestoccurrenceofasingle-lengthloopcover-
In 2D, there is a small but constant number ( 1.86)
ingisfor8loopsoflength16inasystemof128sites;however of winding loops on average in the thermodynamic≈limit20.
thisisnotarelevantlengthaccordingtoourphasediagramof
There is thus some probability to have filaments in the
Figure 9. Thus, the loop coverings on an actual pyrochlore
checkerboardcase,butthenumberoffilamentsdoesnotgrow
aremodifiedfromFigure9inmostorallofthedensityrange
withsystemsize.
ρ [1/6,1/2]. The details of this modification remains an
∈ In3D,thebackgroundofwindingloopsensuresthatthere
importantopenproblem.
arefilamentswhosenumbergrowswithsystemsize. Thedata
inFigure11showsthatthenumberoffilamentsincreaseslin-
earlywiththelinearsizeofthecubicsample.
V. CONDUCTINGCHANNELSOR“FILAMENTS”
Sinceoursimulationsandloopcountingareperformedwith
periodic boundary conditions, some of the filaments, while
Inthissection,wediscusstheeffectoftheCoulombphase spanning the sample in one direction, also cross one of the
and loop structure on the conductivity of mobile electrons. orthogonalboundaries. (Anexampleisshowninthecartoon
Since the electrons are confined to loops, they can conduct to the right of Figure 11.) One can argue that this type of
onlyifaloopconnectsoneedgeofthesampletotheopposite filamentwouldnotcontributetoconductioninareal-lifecubic
end.Therefore,westudythenumberofsuchsample-spanning pyrochloresample. Therefore, we show data both excluding
loops,which,followingRef.28,werefertoas“filaments”. andincludingthistypeoffilament,andtheyareseentohave
Whentherearenofilaments(e.g.inloopcrystalswithonly thesamepower-lawbehavior.
finite-length loops), the system is unambiguously an insula-
Since the data includingthese filamentshavebetter statis-
tor. Whentherearefilamentsspanningthesystem,thesystem
tics,forfiniteρwedisplaytheinclusivedata,withtheexpec-
cannotbeimmediatelycalledaconductor,becausetheactual
tationthatthereisnoqualitativedifference.
conductivitywilldependonscatteringmechanismsexteriorto
AccordingtoFig.5,itinerantelectronstendtomakedisap-
ourmodel.
pearextensiveloopsoflengthL3 in3D.Whetherornotthis
In the first subsection below, we consider small dopings,
preventstheformationoffilamentsisnotasstraightforwardas
where we present Monte Carlo results for the average num-
it seems. Indeed, one could naively assume that the number
ber of filaments as a function of system size. In the second
ofconductingchannelswilldecreaseandmaybeevenvanish
subsection, we comment on the consequences of our phase
inthethermodynamiclimit.Howeverthenumberoffilaments
diagram.
remainsapproximatelyconstantasplottedonfigure12. This
means that the average numberof loops increases with dop-
ingviadividingtheverylongonesintosmallerbutnonethe-
A. Filamentsatlowdensities less extensiveloopsspanningthe system. In the low doping
regimein3D,thenumberofconductingchannelsisapproxi-
We firstconsiderverylowdensitiesthatthe loopdistribu- matelyindependentofρ.
tioncanbeassumedtobelargelyunchangedfromtheρ = 0 Bycontrast,in2D,withrelativelylargeerror-bars,thenum-
case. beroffilamentsdecreasesbutremainsof (1)aselectronsare
O
10
added. A. Otherlattices:triangularandkagome
The Ising ground states observed for the pyrochlore lat-
B. Conductionchannelsatlargerfillings ticehavevanishingtotalspinoneachtetrahedron. These‘ice
rules’ ensure the existence of the Coulomb phase and states
InSectionIV,weidentifieddensities(forboth2Dand3D) obeyingthemamounttoamoderatezero-pointentropyofless
wherethesystemisaloopcrystaloriscoveredbyloopsoftwo thanathirdofthatofafreespin. Bycontrast,thezero-point
finitelengthsonly. Insuchcases, therearenofilaments,and entropyofthe triangularIsingmagnetis notfarfromhalfof
the system is truly insulating: lifting the frustration-induced that of a free spin, while that of the kagome magnet is over
degeneracy removes the non-locality of the loops. How the 70%oflog2.
energetics (and resulting degeneracies) imposed by lattice Most fundamentally, the single triangle is relatively much
constraints (see section IVC) manifest themselves in trans- more degenerate than a tetrahedron, with 6 out of 8 (rather
portpropertiesisanintriguingseparatequestion. thanoutof16)statesbeinggroundstates. Thetrianglestates
For some ranges of ρ, especially in 3D, the loop cover- have varying magnetizations of 1 (whereas states obeying
±
ingsindicatedbytheMaxwellconstructionaredisallowedby the ice rule have a uniquemagnetization, unless one tunes a
the lattice geometry (section IVC). In such cases it remains fieldtoatransitionbetweenmagnetizationplateaux30).
an openquestionwhetherornotthe lattice constraintsresult Tomovetowardsthefulltriangle-basedlattices,itisworth
inloopcoveringsincludinginfinite(sample-spanning)loops. noting that for any Ising antiferromagnet, the ‘hopping net-
Unfortunately, the issue of conducting channels depends on work’ formed by neighbouringaligned spins can have a co-
theanswertothisgenerallyunresolvedquestion. ordination of at most half that of z, the coordination of the
underlyinglattice–otherwiseitwouldbeenergeticallyfavor-
able to flip the highly coordinated spin. The concept of the
VI. SUMMARYANDOUTLOOK hoppingnetworkgeneralizestheloopsonwhichtheelectrons
hopontheplanarandthreedimensionalpyrochlores.
We have analyzed the double exchange model on the py- It now turns out that triangle and kagome lattices behave
rochlorelatticeintwoandthreedimensions. Wehavechosen entirelydifferentlyfrompyrochloreinbothtwoandthreedi-
toconsideraparameterrangeforwhich(i)magneticfrustra- mensions,onaccountofthenatureoftheirfrustratedground
tionisknowntogiverisetounconventionalgroundstateen- stateensembles.
semblesand(ii)whereitispossibletomakeconsiderablean-
alyticalprogressbymappingthesystemontoanensembleof
loops,thestatisticalpropertiesofwhichareinfluencedbythe 1. Triangularlattice
additionoftheelectrons.
We haveidentifieda numberofphenomenawhichdepend Theground-statedegeneracyofthetriangularlatticeisim-
onfeaturessuchasdimensionality,whichdetermineswhether mediately lifted by the addition of even a single hole. This
or notthere existloopsegmentswindingaroundthe system; resultisentirelyanalogoustothefrustratedNagaokatheorem
orlatticestructure,whichmayfrustratethegeometricpacking presented in Ref. 31, in the context of the magnetic super-
ofpreferredlooplengths. soliddiscussedthere,anditisalsoconnectedtothetriangular
The model studied here leaves unanswered a number of Bosonicsupersolids32–35.
questionsandimmediatelysuggestsmanygeneralizationsand Asz = 6forthetriangularlattice,thehoppingnetworkof
extensions. We have worked in a limit of parameters such alignedspinsnolongerhascoordinationtwoasintheloopsof
thatthemagneticexchangeandanisotropydominateoverthe thepyrochlorelattice. Infact,thecoordinationofasiteofthe
Hundcouplingwhichinturndominatesthehoppingintegral. hoppingnetworknolongerevenneedstobeuniform,sothat
Ouranalysisappliestozerotemperature. theremaybedanglingorevenisolatedsites,asonFig.13.(a).
It would be interesting to relax any of these choices, al- Thecoordinationcanrangeallthewayfromzero(foraspin
thoughtechnicallythis may notbe easy. In particular, given surroundedbyahexagonofoppositelyalignedspins)tomax-
thepresenceofgaplessexcitationsonlongloops,interesting imallyz/2=3.
low-temperature physics may appear. Canting can give rise The latter happenswhen the hole sits on a site experienc-
tonon-trivialBerryphasephysics,andfiniteHund’scoupling ingzeronetexchangefieldfromitssixneighbors. Thereex-
willenableelectronstohopbetweenloops. Forexample,re- ists a unique state (pictured in Fig. 13.(b)) in which there is
cent work at quarter filling on the pyrochlore Kondo lattice a network of three-fold coordinated sites. This state breaks
has shown the emergence of a chiral magnetic order in the translationalsymmetrybytriplingtheunitcellaswellastime-
weak-couplingregime29. reversalsymmetryasitcorrespondsto astate with amagne-
Finally,evenintheparameterrangediscussedhere,itwill tization of a third of the saturated value. We have not stud-
be interesting to ask how different frustrated lattices shape iedwhathappenstoafinitedopingbutaFermiliquidregime
up compared to the pyrochlores. We devote the final para- on the hexagonalbackboneat low dopinglooks likely. This
graphs of this paper to discussing this question for the case would imply a conducting state, with the possibility of ad-
oftriangle-basedlattices,thetriangularBravaislatticeandthe ditional low-energyexcitations in the form of defects of the
Archimedeankagomelattice. hexagonalbackbone.