Anderson transition in systems with chiral symmetry

0.18 MB·English

by  Antonio M. Garcia-Garcia

Anderson transitioninsystemswithchiral symmetry
Antonio M. Garc´ıa-Garc´ıa
Physics Department, Princeton University, Princeton, New Jersey 08544, USA and
TheAbdusSalam International CentreforTheoretical Physics, P.O.Box586, 34100Trieste, Italy
Emilio Cuevas
7 Departamento de F´ısica, Universidad de Murcia, E30071 Murcia, Spain
0 (Dated:February5,2008)
0
Andersonlocalizationisauniversalquantumfeaturecausedbydestructiveinterference. Ontheotherhand
2
chiralsymmetryisakeyingredientindifferentproblemsoftheoreticalphysics: fromnonperturbativeQCDto

n highlydopedsemiconductors. Weinvestigatetheinterplayofthesetwophenomenainthecontextofathree-
a dimensional disordered system. We show that chiral symmetry induces an Anderson transition (AT) in the
J
regionclosetotheband center. Typical propertiesat theATsuch asmultifractalityand criticalstatisticsare

1 quantitativelyaffectedbythisadditionalsymmetry.TheoriginoftheAThasbeentracedbacktothepower-law
decayoftheeigenstates;thisfeaturemayalsoberelevantinsystemswithoutchiralsymmetry.
]
n
n PACSnumbers:72.15.Rn,05.40.-a,05.45.Df,71.30.+h
-
s
i Thecombinationofglobalsymmetriesanddimensionality sionalityandsymmetriesofthesystem.
d
. providesuswithausefulclassificationschemeforthestudyof In contrast, much less is known about the region close to
at Andersonlocalization.1Accordingtotheone-parameterscal- theoriginwherethechiralsymmetryplaysacrucialrole. An
m ingtheory(ST),2alleigenstatesofadisorderedsystemareex- exceptionisthemetalliclimit,wheretheeffectofthissymme-
- ponentiallylocalizedinfewerthantwodimensionsinthether- tryonlevelstatistics13 hasbeeninvestigatedindetailedwith
d modynamic limit and undergo a localization-delocalization the help of the powerful analytical techniques of RMT. Be-
n
transition[orAndersontransition(AT)]inhigherdimensions, yond this region we are still far from even a qualitative un-
o
ifthe strengthofdisorderis sufficientlyweak. However,de- derstandingoftheinterplaybetweendisorderandchiralsym-
c
[ viationsfromSTpredictionshavebeenfoundinsystemswith metry. As a general rule, chiral symmetry tends to delocal-
additional discrete symmetries. A typical example is a dis- izeeigenstatesclosetotheoriginsinceweak-localizationcor-
3
ordered lattice without on-site disorder but random hopping rections vanish.14 Indeed in the one-dimensional case it can
v
amplitudes.3 Formally,thesystemmaybeconsideredastwo berigorouslyproved3 that, indisagreementwith the ST pre-
1
3 independent sublattices with nonzero matrix elements con- diction, eigenstates at the band center are not localized ex-
3 nectingonlysitesofdifferentsublattices. Thecorresponding ponentially. In two dimensions, localization propertiesseem
2 Hamiltonianisinvariantunderasimultaneoussignflipofan to be very sensitive to the microscopic form of the random
0 arbitraryeigenvalueandtheeigenstatecomponentsofoneof potential.15ForthespecificcaseofaweakGaussiandisorder,
6 the associated sublattices. As a consequenceof this discrete it is well established that eigenvectorsat the band center re-
0
symmetry,referredto aschiralsymmetry,eigenvaluesof the maindelocalized.16Inthree-dimensions(3D),forweakdisor-
/
at Hamiltoniancomeinpairsof±ǫi,thatis,thespectrumisin- derandtime-reversalinvariance,theresultsofRefs.17,18,19
m variantaroundthepointǫ=0.Thissymmetryisanimportant suggestthatchiralsymmetrymayinducepower-lawlocaliza-
ingredientin differentproblemsof theoreticalphysics: from tionofthezero-energyeigenstates.
d- theQCDDiracoperator4tobosonsinrandommedia5andthe Surprisingly,withtheexceptionofcertainone-dimensional
n spectrumofahighlydopedsemiconductor.6 systemswith-longrangedisorder,20,21 the transitionto local-
o
From the definitionof chiralsymmetryit is clear thatone izationinsystemswithchiralsymmetryhashardlybeenstud-
c
: must distinguish between eigenvalues near zero (the origin) ied despite itspotentialapplicationsto thedescriptionof the
v anddistantfromit(thebulk).Inthebulk,spectralcorrelations chiralphasetransitioninQCD,22 orthemetal-insulatortran-
i
X in the metallic limit are described by the universalresults of sitioninhighlydopedsemiconductors. ThepresentBriefRe-
randommatrixtheory(RMT),usuallyreferredtoasWigner- portisastepinthisdirection.Weinvestigatetheinterplaybe-
r
a Dyson (WD) statistics.7 Weak-localization corrections to tweenchiralsymmetryandlocalizationinathree-dimensional
these universalresults are obtainedby mappingthe localiza- systemwithshort-rangedisorder.Ourmainfindingisthatchi-
tionproblemontoasupersymmetricnonlinearσmodel.8The ralsymmetryinducesanATintheregionclosetotheorigin.
transitiontolocalizationoccurringforstrongerdisorderisbe- Eigenstatesarepower-lawlocalizedwithanexponentthatin-
yond the reach of current analytical techniques. Numerical creasesaswemovefromtheorigin.TheAToccurswhenthis
results suggest that the AT is characterized by multifractal9 exponentmatchesthedimensionalityofthespace. We recall
eigenfunctions,ascale-invariantspectrum,10andspectralcor- that, since the AT is to a great extent universal, the validity
relations (usually referred to as ”critical statistics”11) differ- of our results does not depend on the microscopic details of
entfromPoissonandWDstatistics.10,12 Thesepropertiesare themodelbutonlyonthedimensionalityofthespaceandthe
supposedto beuniversal,namely,theydo notdependonthe (chiral)symmetryoftheHamiltonian.
microscopicdetailsoftheHamiltonianbutonlyonthedimen- We consider the following 3D tight-binding Hamiltonian
2
withonlyoff-diagonaldisorder:

H=X[tijeiθija†iaj +H.c.], (1) 1.0
hiji
0.8
where the sum is restricted to nearest neighborsand the op-
eratora (a†) destroys(creates)an electronat the ithsite of
i i
the 3D cubic lattice. We break time-reversal invariance by 0.6
)
W
introducing a random magnetic flux described by the above L,
Peierlsphaseeiθij withθij uniformlydistributedintheinter- (
0.4
val [−π,π]. Although we mainly focus on the broken time-
L=10 (square)
reversalcase,wehavealsocheckedthatourmainconclusions 12 (star)
arevalidiftime-reversalinvarianceispreserved.Thehopping 0.2 16 (circle)
integralst arerealrandomvariablessatisfyingtheprobabil- 20 (diamond)
ij
itydistribution
0.0
12 16 20 24 28 32
P(lnt )=1/W for −W/2≤lnt ≤W/2, (2)
ij ij
W
and zero otherwise. The above exponentialdistribution pro-
videsan effectiveway of setting thestrengthof off-diagonal
FIG.1: ScalingvariableηasafunctionofdisorderW fordifferent
disorder.23 volumes; onlyeigenvalues inthewindow 10−4 ≤ ǫ ≤ 10−3 have
Inordertoproceed,wecomputeeigenvaluesandeigenvec-
beenconsidered.ThesystemundergoesanATatW =Wc ∼18±1.
torsoftheHamiltonian(1)fordifferentvolumesL3 byusing Inallcasesthestatisticalerror(notshown)is∼0.003.
standard numerical diagonalization techniques. The number
of disorderrealizationsforeach volumeL3 rangesfrom104
(L=10)to500(L=20).Hard-wallboundaryconditionsare
otherintermediatevalueofηinthethermodynamiclimitisan
imposed in order to better examine the asymptotic decay of
indicationofamobilityedge.
eigenstatesinrealspace(seebelow). Sinceweareinterested
In Fig. 1 we plot the W dependence of η for different
in the effect of chiral symmetry we mainly focuson a small
system sizes. It is clear that, in the window of energy stud-
energywindowclosetotheoriginǫ=0.Theeigenvaluesthus
ied ǫ ∈ (10−4,10−3), the mobility edge signaling an AT is
obtained are appropriately unfolded, i.e., they were rescaled
located around W ∼ 18. We have also found that W in-
so thatthe spectraldensityona spectralwindowcomprising c c
creases as the energy window of interest gets closer to the
severallevelspacingsisunity.
origin.Foraweaker(stronger)disorder,ηtendsslowlytothe
Our first task is to look for a mobility edge in a small
WD (Poisson) results. This slow convergenceto WD statis-
spectral window close to the origin. The chosen interval,
tics(seeFig. 1)suggeststhateigenstatesmaystillhavesome
ǫ ∈ (10−4,10−3) in ourcase, isnotimportantprovidedthat
kindofstructureevenonthedelocalizedsideofthetransition.
it is close to the origin. For other intervals one gets similar
Theanalysisofeigenfunctionswillshowthatthisisthecase.
results but for a different critical disorder. We recall that in
We have checkedthat the use of scaling variablesotherthan
generalthe value W = W for which the system undergoes
c η10,25 do not alter the results. For the sake of completeness
anATisquitesensitivetothedetailsoftheHamiltonianand
werepeatthecalculationforthecaseoftime-reversalinvari-
consequentlyisnotuniversal. Inordertolocatethemobility
ance, θ = 0 in Eq. (1). We have also observeda mobility
edgewe use the finite size scaling method.10 Firstwe evalu- ij
edgewithsimilarpropertiesbutataslightlyweakerdisorder
ateacertainspectralcorrelatorfordifferentsizesanddisorder
W ∼15,inagreementwithpreviousresultsforthestandard
strengths W. Then we locate the mobility edge by finding c
3DAT.26Finally,weremarkthatthestudyofη(W,L)fordif-
thedisorderW suchthatthespectralcorrelatoranalyzedbe-
c
ferentdisordersandsystemsizesisequivalenttothestandard
comessizeindependent.10
renormalizationgroupanalysisofthedimensionlessconduc-
In our case we investigate the level spacing distribution
tance g. For instance, according to the one-parameter scal-
P(s)[theprobabilityoffindingtwoneighboringeigenvalues
ingtheory,gshouldbescaleinvariantattheAT.Anyspectral
at a distance s = (ǫ − ǫ )/∆, with ∆ being the local
i i+1 i correlator, η in particular, is a function of g only and, con-
mean levelspacing]. The scaling behaviorof P(s) is exam-
sequently, it shouldbe scale invariantat the AT as well. We
inedthroughthefollowingfunctionofitsvariance:24
havechosenηinsteadofgfornumericalreasons.Theformer
iseasiertocalculateandprovidesmuchmorestableresults.
η(L,W)=[var(s)−var ]/[var −var ], (3)
WD P WD
We now studywhetherlevelstatistics aroundthe mobility
whichdescribestherelativedeviationofvar(s)fromtheWD edgearecompatiblewiththoseofanAT.Spectralcorrelations
limit. In Eq. (3) var(s) = hs2i−hsi2, where h···i denotes at the AT (critical statistics) combine typical properties of a
spectral and ensemble averaging, and var = 0.286 and metal(WD) with those of an insulator(Poisson). Thuslevel
WD
var = 1arethevariancesofWDandPoissonstatistics, re- repulsionP(s)→0fors→0,typicalofWDstatisticsisstill
P
spectively. Hence η = 1(0) for an insulator (metal). Any presentattheAT.However,along-rangecorrelatorsuchasthe
3
spectral window ǫ ∈ (10−4,10−3) we have found that the

eigenfunctions are indeed multifractal (D = 0.69 ± 0.02,
2
0.4 1.0 D =0.45±0.01,andD =0.340±0.007).Thesevaluesof
30 (a) (b) 3 4
D are,roughlyspeaking,afactorof2smallerthantheones
q
25 2 l() 0.2 P(s)0.5 atthestandard3DAT.Thisisconsistentwithourresultsfrom
levelstatistics,whereasimilarfactorwasfoundfortheslope
χofthenumbervarianceΣ2(ℓ) ∼ χℓ. Qualitatively,thisnu-
) 20 0.00 2 4 6 0.00 1 2 3 4 mericaldifferencecanbetracedbacktothechiralsymmetry
2 l( l s of our model: according to the ST, parameters such as χ or
15 D are functionsof the dimensionlessconductanceg at the
q c
AT. Although an explicit expression is not known, it is ex-
10 L=10 (square) pectedthat,atleastforsufficientlyhighdimensions,bothD
2
12 (star) and 1−χ will be proportionalto g . On the other hand, it
16 (circle) c
5 20 (diamond) iswellestablished14,16thatweak-localizationcorrectionsvan-
ish in systems with chiral symmetry. Hence, a stronger dis-
10 15 20 25 30 order is needed to reach the AT region. As a consequence,
l thechiralgcwillbesmaller,inqualitativeagreementwithour
results. Having discussed the details of the AT close to the
origin, we now clarify the physical mechanism causing this
FIG.2:Thel≫1behaviorofthenumbervarianceΣ2(l)andP(s)
transition. Chiralsystemsdonothaveweak-localizationcor-
[inset (b)] inthe spectral window ǫ ∈ (10−4,10−3) at the critical
rections,sothetransitiontolocalizationmustbeinpartdueto
point Wc ∼ 18fordifferent volumes. Σ2(l)islinearwithaslope
somenonperturbativeeffect,suchastunnelingcombinedwith
(solidline)χ ∼ 0.49±0.01, P(s) ∼ sfor s ≪ 1, and bothare
theprogressiveweakeningoftheeffectofthechiralsymmetry
size independent. These features are typical signatures of critical
statistics. For W = 0.2 ≪ Wc [inset (a)] the number variance aswemoveawayfromtheorigin.Ontheotherhand,theslow
(dottedline)iswelldescribedbythepredictionofchiralRMT(solid rate of convergencetoward WD statistics observedin Fig. 1
line). suggeststhat, althougheigenstatesseem to be delocalizedin
the thermodynamiclimit, they still have some kind of local-
ization center. This is in agreement with the results of Ref.
thenumbervarianceΣ2(ℓ)=h(N −hN i)2i∼χℓforℓ≫1 17, where it was found that the zero-energy wave function
ℓ ℓ
(N isthenumberofeigenvaluesinanintervaloflengthℓ)is of a time-reversed, weakly disordered 3D chiral system was
ℓ
asymptoticallylinear,12asforaninsulatorΣ2(ℓ)∼ℓ,butwith power-lawlocalizedψ(r)∼1/rα(W). Werecallthat,accord-
aslopeχ<1 (∼0.27for3DAT). ingtoRefs. 28and30,power-lawlocalizationinducesanAT
AsshowninFig. 2allthesefeaturesarealsopresentinour inanydimensionprovidedthatthedecayexponentαmatches
model. However,wehavefoundthattheslopeofthenumber thedimensionalitydofthespace. Ifα < (>) d, eigenstates
varianceχ=0.49±0.01isroughlytwicethatofthestandard tendtobedelocalized(localized)inthethermodynamiclimit
3D AT. We refer to the analysis of eigenvectors(see below) butthedegreeofconvergenceisslow.31
foraqualitativeexplanationofthisfeature. Basedontheabovefactsweclaimthatthetransitiontolo-
calizationclosetotheoriginobservedintheHamiltonian(1)
We have verified that the number variance in the limit of
weakdisorderagreeswiththeprediction13ofchiralRMT[see hasitsorigininthepower-lawlocalizationoftheeigenstates.
We remark that this mechanism should be at work for any
inset (a) Fig. 2]. For weak enough disorder results depend
systemwithchiralsymmetryanddisordersuchthatamobil-
onwhetherthetime-reversalinvarianceisbroken[inset(a)in
ityedgeappearsclosetotheorigin. TheHamiltonian(1)was
Fig. 2] or not (not shown). As a general rule, as disorder
chosenforthesakeofsimplicity. Itisjustoneofthesimplest
is increased the magnetic flux gets weaker, for W ≥ 10 we
representativesoftheuniversalityclassassociatedwiththeAT
couldnotobserveanydifference.Additionally,wehavefound
insystemswithchiralsymmetry.
that, if chiral symmetry is broken, by for instance adding a
Moreover, for a given spectral window close to the ori-
weak diagonal disorder, level statistics are very close to the
gin, we expect the exponent α describing the decay to in-
Poissonlimittypicalofaninsulator.
crease with the disorder strength W. The AT will occur for
To conclude, the analysis of level statistics shows that, as
a W = W so that α ∼ d = 3. For stronger disorder
a consequenceof the chiral symmetryof the model, there is c
W > W eigenstates must be eventually exponentially lo-
anATclosetothebandcenterwithpropertiessimilarbutnot c
calized,thoughtheymaystillpossessapower-lawtailfordis-
equaltothoseofthestandard3DAT.
tancessmallerthanthelocalizationlength.
One of the signaturesof an AT transition is the multifrac-
Wehavetestedthisconjecturebyanalyzingthetypicalde-
talityoftheeigenstates. Aneigenfunctionissaidtobemulti-
cay of eigenstates in the window 10−4 ≤ ǫ ≤ 10−3 for
fractal(foranalternativedefinitionseeRef. 27)iftheeigen-
L = 20anddifferentW’s(similar resultsare obtainedifW
functionmomentsPq =R ddr|ψ(r)|2q ∝ L−Dq(q−1) present iskeptfixedandtheenergywindowismodified)throughthe
anomalous scaling with respect to the sample size L, where
followingcorrelationfunction:32
D isasetofdifferentexponentsdescribingthetransition.28,29
q
After performing ensemble and spectral averaging in the g(r)=hln(|ψ |/|ψ |)i, r=|r −r |, (4)
jmax j j jmax
4
thedecayincreaseswithW andtendstotheconjecturedvalue

α = 3 as we get close to the critical disorder W ∼ 18.
c
5 Thuschiralsymmetrycombinedwithstrongdisorderinduces
20 4 power-lawlocalizationandeventuallyanATclosetotheori-
W)
( 3 gin.Thisisthemostrelevantresultofthepaper.
2 We remark that the relation between chiral symmetry and
16 1 power-law localization has already been established in the
121416182022
) contextofQCD(seeRef. 18andreferencestherein). Specif-
r W
(
g ically,itwasfoundthatthelow-lyingeigenstatesoftheQCD
operator (with gauge configurations given by the instanton
12
liquid model) are power-law localized due to the long-range
behavior of certain nonperturbative solutions (instantons) of
theclassicalequationsofmotion. Wespeculatethatasimilar
8
mechanismisatworkinourcase.
A finalcommentis in order: the claim thatthe power-law
0 4 8 12 16 20
localizationisthe precursoroftheATis notincontradiction
r with the multifractalstructure of the eigenstates. Since g(r)
is averaged over many realization of disorder, all multifrac-
tal fluctuationsare washed out. What remainsis the smooth
FIG.3:Thecorrelationfunctiong(r)asafunctionofr(dashedlines)
skeletonoftheeigenstatewhich,inourcase,hasapower-law
forasystemsizeL= 20andW =12,14,16,18,20,and22from
tail.
bottom to top. The numerical curves were fitted to a curve of the
formg(r) = ln(Arα+B)(solidlines). Thebest-fittingparameter To conclude, we have investigated the interplay between
α is represented as a function of W in the inset. In all cases the
chiralsymmetryandAndersonlocalizationina3DAnderson
statisticalerror(notshown)is∼α/50.
modelwithoff-diagonaldisorder.Closetotheoriginwehave
found a mobility edge induced by the chiral symmetry with
propertiesquantitativelydifferentfromthoseofastandard3D
wherej denotesthelatticesitewiththelargestamplitude
max AT.ThischiralATcanbetracedbacktothepower-lawnature
and the angular brackets stand for the ensemble average. In
of eigenstates close to the origin. Our results are relevant
order to examine the asymptotic decay of g(r) (we restrict
to chiral systems undergoing a metal-insulator transition as
ourselvestothe range8 ≤ r ≤ 22), wefitthenumericalre-
a function of disorder. Typical examples include the QCD
sultstoacurveoftheformg(r) = ln(Arα +B)withA,B,
Dirac operator around the chiral phase transition and highly
and α fitting parameters. The parameters A and B describe
doped semiconductors where off-diagonal bond disorder
finite-sizeeffectsandremnantsoftheeigenstatecore(nonde-
playsadominantrole.
caying)part. Theexponentα(depictedintheinsetofFig. 3)
providesuswithvaluableinformationaboutthedecayofthe
eigenstates. A.M.G. acknowledges support from the European Union,
Theexcellentagreement(seeFig. 3)betweenthenumeri- MarieCurieprogram,ContractNo. MOIF-CT-2005-007300.
cal results and the fitting curve indicates that eigenstates are E.C. thanks the FEDER and the Spanish DGI for financial
power-law localized. Moreover, the exponent α controlling supportthroughProjectNo. FIS2004-03117.
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