Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions

0.88 MB·English

by  Laszlo Ujfalusi

Finite-size scaling and multifractality at the Anderson transition for the three
Wigner-Dyson symmetry classes in three dimensions
L´aszl´o Ujfalusi and Imre Varga
∗
Elm´eleti Fizika Tansz´ek, Fizikai Int´ezet, Budapesti Mu˝szaki ´es Gazdasa´gtudoma´nyi Egyetem, H-1521 Budapest, Hungary
(Dated: May 27, 2015)
Thedisorderinducedmetal–insulatortransitionisinvestigatedinathree-dimensionalsimplecubic
latticeandcomparedforthepresenceandabsenceoftime-reversalandspin-rotationalsymmetry,i.e.
inthethreeconventionalsymmetryclasses. Largescalenumericalsimulationshavebeenperformed
on systems with linear sizes up to L = 100 in order to obtain eigenstates at the band center,
5 E = 0. The multifractal dimensions, exponents Dq and αq, have been determined in the range of
1 −1 ≤ q ≤ 2. The finite-size scaling of the generalized multifractal exponents provide the critical
0 exponents for the different symmetry classes in accordance with values known from the literature
2 based on high precision transfer matrix techniques. The multifractal exponents of the different
symmetry classes provide further characterization of the Anderson transition, which was missing
y
from the literature so far.
a
M
PACSnumbers: 71.23.An,71.30.+h,72.15.Rn

6
2
I. INTRODUCTION ventions are the same in the field of disordered systems.

Theclassificationconsiderstwoglobalsymmetries: time-
]
n reversal and spin-rotational symmetry. As it turns out,
The metal-insulator transition (MIT), and disordered
n beside these symmetries there are three further symme-
systems have been at the forefront of condensed matter
- try classes according to the presence of chiral symme-
s research since the middle of the last century1, and yet
i try,andinadditiontherearefourBogoliubov-deGennes
d this topic still has several open questions and is still ac-
classes also, corresponding to particle-hole symmetry12
. tively investigated. In the last few years experimental
t prominent in hybrid (superconductor-normal) systems.
a evidence has been obtained about this topic; in particu-
m The effect of symmetry classes at the Anderson transi-
lar, reporting Anderson localization of ultrasound in dis-
tion has already been investigated earlier13 using spec-
- ordered elastic networks2,3, light in disordered photonic
d tral statistics, but there is much less work based on the
lattices in the transverse direction4, or in an ultracold
n multifractal analysis of the eigenstates, and multifractal
atomic system in a disordered laser trap5. Richardella
o exponentsareknownnumericallyonlyfortheorthogonal
c et al.6 examined the MIT in a dilute magnetic semi- class14.
[ conductor Ga Mn As, which is a strongly interacting
and disordered1−sxystexm. They found a clear phase tran- Our goal in this article is to fill in this gap and apply
2
sition together with multifractal fluctuations of the local multifractal finite-size scaling (MFSS), developed orig-
v
7 density of states (LDOS) at the Fermi energy, showing, inally by Rodriguez, Vasquez, R¨omer and Slevin14, to
4 that multifractality is a robust and important property the Anderson models in the three conventional Wigner-
1 of disordered systems. Multifractal properties consistent Dyson (WD) classes. The organization of the article is
2 with the theory of Anderson localization are also found the following. In Sec. II we define the model and de-
0 in the ultrasound system3. On the theoretical side, we scribe its numerical representation. In Sec. III we briefly
.
1 knowthatdisorderplaysacrucialroleinintegerquantum describe the finite-size scaling analysis of the general-
0 Hall effect7, and recently it was shown that an enhanced ized multifractal exponents of the critical eigenstates, in
5 correlation of multifractal wave-function densities in dis- Sec. IV we give the results obtained for the three uni-
1 ordered systems can increase the superconducting criti- versality classes and finally in Sec. V we summarize our
:
v cal temperature8 or the multifractal fluctuations of the results.
i LDOS close to criticality may lead to a new phase due
X
to the presence of local Kondo effects induced by local
r
a pseudogaps at the Fermi energy9. Moreover, Anderson
localizationhasalsobeenreportedinthespectrumofthe
II. MODELS AND NUMERICAL
Dirac operator within the lattice model of QCD at high REPRESENTATION
temperaturesusingspectralstatistics10,andmultifractal
analysis seems to corroborate it, as well11.
A. The model
These models show an increased interest in under-
standing the nature of the Anderson transition in the
presence of various global symmetries. A comprehensive In this article we investigate Anderson models belong-
review of the current understanding is given in Ref. 12. ing to the three WD classes, without chiral and particle-
Thesesymmetryclasseshavebeenintroducedfirsttode- holesymmetry. Weinvestigatethecaseofdiagonaldisor-
scribe random matrix ensembles, but the naming con- der and nearest-neighbor hopping, therefore the Hamil-
2
tonian reads as of the Hamiltonian reads as
ε U(cid:2) W,W(cid:3), if i=j
H=(cid:88)iσ εic†iσciσ−i(cid:88)jσσ(cid:48)tijσσ(cid:48)c†iσcjσ(cid:48) +h.c., (1) HU =−i1∈, iifnit−hane2dxjo2raryednieriegchtbioonring sites (3)
wherei,j andσ,σ(cid:48) standforsite-andspinindex,εi-sare i≤j −ei2sπiΦtexs, iifnithanedzjdiarreectnioeinghboring
0, otherwise
randomon-siteenergies,whichareuniformlydistributed
over the interval (cid:2) W,W(cid:3), W acts as disorder strength.
− 2 2 Complexhermiticitysetstheoff-diagonalelementsinthe
Using a uniform distribution is just a convention, other
lower triangular part, j < i. Periodic boundary con-
distributions of disorder, e.g. Gaussian, binary, etc. can
ditions and flux quantization force a restriction for the
be used as well.
magnetic flux namely, that Φ L must be an integer.
·
In the orthogonal class time-reversal and spin- In the thermodynamic limit arbitrarily small magnetic
rotational symmetry are preserved. In this case the field drives the system from the orthogonal to the uni-
Hamiltonian is invariant under orthogonal transforma- tary class. However, in a finite system the relationship
tions–hencethename–,thereforeitisarealsymmetric between the system size, L, and the magnetic length,
matrix. Since spin does not play a role, we consider a LH = √21πΦ matters. In the case of weak magnetic field,
spinless Anderson model. In the numerical simulations L L , the system belongs to the orthogonal class, in
H
(cid:28)
the Hamiltonian is represented by an N N real sym- the case of strong magnetic field, L L , it belongs to
H
metricmatrix,whereN =L3,andListh×elinearsystem theunitaryclass. Sinceweusesystem(cid:29)sizesthataremul-
size in lattice spacing. The diagonal elements, are are tiples of 10 lattice spacings, see Tab. I, we chose Φ= 1.
5
uniformly distributed random numbers, the off-diagonal This leads to L 0.892 therefore this choice clearly
H
≈
elementsarezero,exceptifiandj arenearestneighbors: fulfills the two conditions above.
In the symplectic class time-reversal symmetry is
present, and spin-rotational symmetry is broken, which
HO =εi1∈, iUf i(cid:2)−anW2d,jW2ar(cid:3)e,nifeiig=hbjoring sites (2) dcaessecrtihbeesHaamsyilsttoenmianwiisthinvsparinia-notrbuintdienrtesryamctpiolenc.ticIntratnhsis-
ij −0, otherwise formations therefore it is a quaternion hermitian matrix.
For the numerical simulations we followed Asada, Slevin
and Ohtsuki16. Since in this case we have to deal with
thespinindexalso,theHamiltonianisan2N 2N com-
The energy unit is fixed by setting the hopping elements ×
plex hermitian matrix. Diagonal elements corresponding
to 1. To avoid surface effects, we use periodic bound-
to the ith site and hopping elements between sites i and
ary conditions. However, this case was investigated very
j are 2 2 matrices because of the spin indexes, having
carefully by Rodriguez et al.14, we consider this symme- ×
a form
try class to verify our numerical method, and to obtain
(cid:18) (cid:19) (cid:18) (cid:19)
a complete description of all the WD classes. (cid:15) = εi 0 t = eiαijcosβij eiγijsinβij ,
i 0 εi ij e−iγijsinβij e−iαijcosβij
In the unitary class time-reversal symmetry is broken, −
(4)
which can be realized physically by applying a magnetic
where ε is an uniformly distributed random on-site en-
field. It can be shown, that either spin rotational sym- ergy froim the interval (cid:2) W,W(cid:3), α , β and γ were
metry is broken or not, the model will belong to the uni- − 2 2 ij ij ij
chosentoformanSU(2)-invariantparametrization,lead-
tary class12. The Hamiltonian is invariant under unitary
ingtotheso-calledSU(2)model: α andγ areuniform
transformations therefore it is a complex hermitian ma- ij ij
random variables from the interval [0,2π], and β has a
trix. Wediscussthecasewhenspin-rotationalsymmetry
probability density function p(β)dβ = sin(2β)dβ in the
is present, because this way we can use spinless fermions range (cid:2)0,π(cid:3). The upper triangular of the Hamiltonian
again, which keeps the matrix size N N. However, one 2
× has the following form:
has to store about twice as much data compared with
the orthogonal case, because here every off-diagonal ma- 
trix element is a complex number. Obviously finding an (cid:15)i, if i=j
HS = t , if i and j are neighboring sites (5)
eigenvalue and an eigenvector takes more time, too. i j ij
≤ 0, otherwise
For the numerical simulations we followed Slevin and
Ohtsuki15. Let us consider a magnetic field pointing in The off-diagonal elements are defined following complex
they directionwithfluxΦ, measuredinunitsoftheflux hermiticity. To store the Hamiltonian requires about
quantum, h/e. Its effect can be represented by a unity eight times more memory compared to the orthogonal
phase factor, the Peierls substitution for the hopping el- case, because here every off-diagonal element contains
ementsoftheHamiltonianmatrix. Theuppertriangular four complex numbers. Finding an eigenvalue is much
3
system size (L) number of samples Considering a d-dimensional cubic lattice with linear
20 15000 sizeL,onecandividethislatticeintosmallerboxeswith
30 15000
linearsize(cid:96). IfΨisaneigenfunctionoftheHamiltonian,
40 15000
the probability corresponding to the kth box reads as
50 15000
60 10000 (cid:88)
µ = Ψ 2. (6)
70 7500 k | i|
80 5000 i∈boxk
90 4000
Onecanintroducetheqthmomentoftheboxprobability
100 3500
(frequently called generalized inverse participation ratio,
GIPR), and its derivative:
TABLE I: System sizes and number of samples for the simu-
lation for each WD symmetry class.
λ−d λ−d
R =(cid:88)µq S = dRq =(cid:88)µqlnµ . (7)
q k q dq k k
k=1 k=1
slower than for the unitary case, mainly because of the
The average of R and S follows a power-law behavior
q q
linear size of the matrix is twice as large.
as a function of λ= (cid:96), with exponent τ and α :
L q q
ln R dτ S
q q q
B. Numerical method τq = lim (cid:104) (cid:105) αq = = lim (cid:104) (cid:105) . (8)
λ 0 lnλ dq λ 0 Rq lnλ
→ → (cid:104) (cid:105)
MFSS deals with the eigenvectors of the Hamiltonian, τ can be rewritten in the following form:
q
which is a large sparse matrix. Recent high precision
calculations14 useJacobi-Davidsoniterationwithincom- τq =Dq(q 1)=d(q 1)+∆q, (9)
− −
pleteLUpreconditioning,thereforewedecidedtousethis
combination. For preconditioning the ILUPACK20 was where Dq is the generalized fractal dimension, and ∆q is
used, for the JD iteration the PRIMME21 package was theanomalousscalingexponent. EmployingaLegendre-
transform on τ , we obtain the singularity spectrum,
used. Since the metal-insulator transition occurs at the q
band center12 (E = 0) at disorder WO 16.5 for the f(α):
c ≈
orthogonal, at WU 18.3 for the unitary (depending
on the strength ocf m≈agnetic field), at WcS ≈ 20 for the f(αq)=qαq−τq. (10)
symplectic class (for our parameters), most works study
τ , α , D and ∆ are often referred to as multifractal
q q q q
the vicinity of these points. To have the best compari-
exponents.
son,weanalyzedthisregime,therefore20disordervalues According to recent results18 a symmetry relation ex-
were taken from the range 15 W 18 for the orthogo-
≤ ≤ ists for αq and ∆q given in the form:
nal class, 23 disorder values were taken from the interval
17 W 20 for the unitary class, and 20 disorder val- ∆ =∆ α +α =2d (11)
≤ ≤ q 1 q q 1 q
ues were taken from the interval 19.4 W 20.5 for − −
≤ ≤
the symplectic class. System sizes were taken from the Fornumericalapproachesonehastodefinethefinite-size
range L=20..100, and the number of samples are listed version of these MFEs at a particular value of disorder:
in Tab. I. We considered only one wave-function per re-
alization, the one with energy closest to zero in order α˜ens(W,L,(cid:96)) = (cid:104)Sq(cid:105) (12)
toavoidcorrelationsbetweenwave-functionsofthesame q R lnλ
q
system14. (cid:104) 1(cid:105) ln R
D˜ens(W,L,(cid:96)) = (cid:104) q(cid:105), (13)
q q 1 lnλ
−
III. FINITE SIZE SCALING LAWS FOR whereensstandsforensembleaveragingoverthedifferent
GENERALIZED MULTIFRACTAL EXPONENTS disorder realizations. One may define typical averaged
versions also:
Inrecenthigh–precisioncalculations14 themultifractal (cid:28) (cid:29)
S 1
exponents (MFEs) of the eigenfunctions of the Hamilto- α˜typ(W,L,(cid:96)) = q (14)
q R lnλ
nian have been used to describe the Anderson metal– q
insulator transition. We use almost the same notation D˜typ(W,L,(cid:96)) = 1 (cid:104)lnRq(cid:105). (15)
and methods as Ref. 14, but for better understanding q q 1 lnλ
−
here we introduce shortly the most important quantities
and notations. The method has recently been success- Similarly to α˜ and D˜ , ∆˜ or τ˜ can be defined, which
q q q q
fully extended for the investigation of the quantum per- are called generalized multifractal exponents (GMFEs).
colation transition in three dimensions17. Every GMFE approaches the value of the corresponding
4
MFEatthecriticalpoint,W =W ,onlyinthelimitλ Our central goal is to fit the above formulas to the nu-
c
→
0. We would like to emphasize, that MFEs are defined merically obtained data, where W , ν, y and G appear
c q
through ensemble averaging in principle (see Eq. (8)), amongthefitparameters. Thisfitprocedurewillprovide
and ensemble and typical averaged MFEs are equal only us the physically interesting quantities and their confi-
in a range of q, q < q < q 12, defined by the two dence intervals. In the next sections we present different
+
zeros of the singula−rity spectrum, f(α )= f(α )= 0. methods for the finite-size scaling.
q− q+
Therefore when in Sec. IVB we compute MFEs, we will
use ensemble averaged quantities only.
The choice of the investigated range of q is influenced A. finite-size scaling at fixed λ
bythefollowingthreeeffects. Ifq islarge, the qthpower
in Eq. (7) enhances the numerical and statistical errors,
At fixed λ, G in Eq. (18) can be considered as the
q
leading to a noisy dataset. If q is negative with large
constant term of , therefore
q
absolute value, the relatively less precise small wave- G
functionvaluesdominatethesumsinEq.(7), whichalso (cid:18)L(cid:19)
results in a noisy dataset. These two effects together G˜q(W,L)= q , (19)
G ξ
lead to a regime q q q , where GMFEs be-
min max
≤ ≤
have numerically the best. The third effect is coarse where the constant λ has been dropped. can be ex-
q
graining which suppresses the noise. For (cid:96) > 1 in an panded with one relevant, (cid:37)(w), and one irGrelevant oper-
(cid:96) (cid:96) (cid:96) sized box positive and negative errors on the ator, η(w), the following way by using w =W W :
wa×ve-f×unctions can cancel each other. Moreover, in a − c
bgeotxhlearrgweitahndhisgmhapllrwobaavbe-ifluitny,ctaionndatmhipsliwtuadyesthaeppreealartitvoe- Gq(cid:16)(cid:37)Lν1,ηL−y(cid:17)=Gqr(cid:16)(cid:37)Lν1(cid:17)+ηL−yGqir(cid:16)(cid:37)Lν1(cid:17) (20)
error of a µ box probability is reduced. In other words
k
All the disorder-dependent quantities in the above for-
coarse graining has a nice smoothing effect, which can
mula can be expanded in Taylor-series:
help to widen the range of q that can be investigated.
The renormalization flow of the AMIT has three fixed
pmoeitnatlsl:icafimxeedtapllioci,natnevinesruylasttiantgeaisndexatecnridteicdalwointhe.pIrnotbhae- Gqr(cid:16)(cid:37)Lν1(cid:17) = (cid:88)nr ai(cid:16)(cid:37)Lν1(cid:17)i (21)
i=0
bility one therefore the effective size of the states grows
pthreopinorstuiloantainlgtofixtehdepvooinlutmeve,erlyeasdtaintegitsoexDpqmoneten≡tiadll.y lIon- Gqir(cid:16)(cid:37)Lν1(cid:17) = (cid:88)nir bi(cid:16)(cid:37)Lν1(cid:17)i (22)
i=0
calized, the effective size of a state does not change with n(cid:37) nη
changingsystemsize,resultinginDqins ≡0forq >0,and (cid:37)(w)=w+(cid:88)ciwi η(w)=1+(cid:88)diwi (23)
Dins for q < 0. Renormalization does not change
q ≡ ∞ i=2 i=1
thesystematcriticality,thereforeitisscaleindependent,
whichmeansself-similarity. Thereforewave-functionsare Theadvantageofthismethodis,thatintheTaylor-series
supposed to be multifractals, in other words generalized only one variable appears, (cid:37)Lν1, therefore the number of
fractals19. parameters(includingWc,ν and y)isnr+nir+nρ+nη+
Close to the critical point due to standard finite-size 4, which grows linearly with the expansion orders. This
scaling arguments one can derive the following scaling methodisveryeffectiveforcomputingWc, ν,andy,but
laws for the exponents α˜ and D˜ defined above as: sinceλisfixed,onecannotobtaintheMFEs. Inallcases
q q
we used λ = 0.1, because it leads to excellent results in
(cid:18) (cid:19)
1 L (cid:96) Ref. 14. It seems, that it is small enough to capture the
α˜ (W,L,(cid:96))=α + , (16a)
q q lnλAq ξ ξ details of a wave-function, and it allows many different
(cid:18) (cid:19) system sizes in the range of 20 L 100, which we
D˜ (W,L,(cid:96))=D + q L, (cid:96) (16b) investigated. This way we can al≤so com≤pare our results
q q q
lnλT ξ ξ
to those of Ref. 14 very well.
Equations (16a)–(16b) can be summarized in one equa-
tion:
B. Finite size scaling for varying λ
(cid:18) (cid:19)
1 L (cid:96)
G˜ (W,L,(cid:96))=G + , (17)
q q q
lnλG ξ ξ
In order to take into account different values of λ the
(cid:16) (cid:17) scaling law given in Eq. (17) has to be considered. The
(L,(cid:96))ontheleft-handsideand L, (cid:96) ontheright-hand
ξ ξ expansion of in (17) is
(cid:16) (cid:17) G
side can be changed to (L,λ) and L,λ :
ξ (cid:18) (cid:19) Gq(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1,η(cid:48)L−y(cid:48),η(cid:96)−y(cid:17)=Gqr(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17)+
G˜q(W,L,λ)=Gq+ ln1λGq Lξ,λ (18) +η(cid:48)L−y(cid:48)G(cid:48)iqr(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17)+η(cid:96)−yGqir(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17).
5
According to Rodriguez et al.14 the most important ir- and the correlation matrix of the numerically obtained
relevant term is the one containing the finite box size, (cid:96), data points by C, which can be computed numerically
therefore we took into account that one only. This leads with a similar expression to the variance. With these
to notations χ2 reads as
G˜q(W,L,(cid:96))=Gq + ln1λ(cid:16)Gqr(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17)+ χ2 =(cid:88)i,j (yi−fi)(cid:0)C−1(cid:1)ij(yj −fj), (28)
(cid:16) (cid:17)(cid:17)
+ η(cid:96)−yGqir (cid:37)Lν1,(cid:37)(cid:96)ν1 . (24) for more details see Ref. 14. If the data points are not
correlated, C is a diagonal matrix, and the expression
The Taylor expansions of the above functions are leads to the usual form:
Gqr(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17) = (cid:88)nr (cid:88)i aij(cid:37)iLνj(cid:96)i−νj (25) χ2 =(cid:88)i (yi−σi2fi)2. (29)
i=0j=0 The number of degrees of freedom, N is the number of
df
Gqir(cid:16)(cid:37)Lν1,(cid:37)(cid:96)ν1(cid:17) = (cid:88)nir (cid:88)i bij(cid:37)iLνj(cid:96)i−νj (26) dχa2t/a(Npdofint1s)min1usmtehaensnuthmabtetrheofdfievtipaatiroanmseftreorms.tAherbaetsiot
− ≈
i=0j=0 fit are ofthe order of the standarddeviation (correlation
n(cid:37) nη matrix). The second criterion was that the fit has to be
(cid:88) (cid:88)
(cid:37)(w)=w+ ciwi η(w)=1+ diwi (27) stableagainstchangingtheexpansionorders,i.e. adding
i=2 i=1 a few new expansion terms. From the fits that fulfilled
the first two criteria we chose the simplest model, with
The advantage of this method is, that it provides the
thelowestexpansionorders. Sometimeswealsotookinto
MFE, G , since it is one of the parameters to fit. There
q account the error bars, and we chose the model with the
are many more data to fit compared to the fixed λ case.
lowesterrorbarforthemostimportantquantities(W ,ν,
Fixed λ means that at a given system size one can use c
etc...), if similar models fulfilled the first two criteria.
GMFEs obtained at a certain value of (cid:96) – the one that
Theerrorbarsofthebestfitparameterswereobtained
leads to the desired λ – , while in this case one can fit
byaMonte-Carlosimulation. Thedatapointsareresults
to GMFEs obtained at different values of (cid:96). However,
of averaging so due to central limit theorem, they have a
these GMFEs are correlated, because they are the re-
Gaussian distribution. Therefore we generated Gaussian
sults of the coarse graining of the same wave-functions
random numbers with parameters corresponding to the
with different sizes of boxes. During the fitting proce-
mean of the raw data points and standard deviation (or
dure one has to take into account these correlations, see
correlationmatrix)ofthemean,andthenfoundthebest
Sec. IIIC. Since the relevant and irrelevant scaling func-
fit. RepeatingthisprocedureN =100timesprovided
tions have two variables, (cid:37)Lν1 and (cid:37)(cid:96)ν1, one has to fit the distribution of the fit paraMmCeters. We chose 95%
a two-variable function with the number of parameters
confidence level to obtain the error bars.
(n +1)(n +2)/2+(n +1)(n +2)/2+n +n +3. We
r r ir ir ρ η
cansee,thatthenumberofparametersgrowsas n2 ,
∼ r/ir
insteadofas nr/ir asforfixedλ. Thismakesthefitting IV. RESULTS OF THE MFSS FOR THE
∼
procedure incorporating the correlations definitely much ANDERSON MODELS IN THE WD SYMMETRY
more difficult. CLASSES
With the numerical method described in Sec. II we
C. General principles for the FSS fit procedures computed an eigenvector for every disorder realization
of the Hamiltonian. From the eigenvectors every GMFE
In this section we discuss the details of the methods is computable, for the orthogonal and unitary class the
and criteria we used during the MFSS. In order to fit Ψ 2 expression in Eq.(6) is trivial, and it means sum-
i
| |
the scaling law Eq. (19) and (24) we used the MINUIT mation for the spin-index for the symplectic class, since
library22. To find the best fit to the data obtained nu- spatial behavior is in our interest. At fixed q exponents
merically the order of expansion of qr/ir, (cid:37) and η must τq and ∆q are linear transforms of Dq, so we used only
be decided by choosing the values oGf nr,nir,n(cid:37) and nη. theα˜q andD˜q GMFEsfortheMFSS.Weinvestigatethe
Since the relevant operator is more important than the range 1 q 2, because GMFEs behave the best in
− ≤ ≤
irrelevantonewealwaysusedn n andn n . To this regime for the reasons described in Sec. III.
rel ir (cid:37) η
≥ ≥
choosetheorderoftheexpansionweusedbasicallythree
criteria. The first criterion we took into account was to
checkhowclosetheratioχ2/(N 1)approachedunity, A. Results of the MFSS at fixed λ=0.1
df
−
where N stands for the number of degrees of freedom.
df
Letusdenotethenumericallyobtaineddatapointsbyy , The typical behavior of the GMFEs is presented in
i
the fit function value at the ith parameter value by f , Fig.1. Inallcasesthereisaclearsignofphasetransition:
i
6
5.4 2.6 3.9
α˜ens0 44445.....4524682 αln˜ens011111111111.....23456......23456755555-4 -3 -2 -1ln̺L0ν1 1 2 3 ˜Dtyp1.511122.....246824 L ˜Dlntyp1.5-0000000000..........011123456789-3 -2 -1 ln0̺Lν11 2 3 ˜Dens0.5−3333....5678 ˜Dlnens0.5−11111L11111.....12345.....1234555555-5-4-3-2l-n1̺L0ν1 1 2 3 4
3.8 L
1.2
3.6 orthogonal unitary 3.4 symplectic
3.4 1
3.2 0.8 3.3
15 15.5 16 16.5 17 17.5 18 17 17.5 18 18.5 19 19.5 20 19.4 19.6 19.8 20 20.2 20.4 20.6
W W W
FIG.1: DotsaretherawdatafordifferentGMFEsintheconventionalWDsymmetryclasses. Redlineisthebestfitobtained
by MFSS. Insets are scaling functions on a log-log scale, after the irrelevant term was subtracted. Error bars are shown only
on the large figures, in order not to overcomplicate the insets.
With increasing system size the GMFEs tend to oppo- yOλ =1.67(1.53..1.80),obtainedfromα˜ withthesame
Rod 0
site direction on both sides of their crossing point. Note method (fixed λ). This agreement verifies our numerics
that there is no well-defined crossing point due to the and fit method, and makes it reliable for the other two
irrelevant term in Eq. (20). Applying the MFSS method universality classes.
described in Sec. IIIA with the principles of Sec. IIIC
In the unitary class the critical parameters match
to the raw data leads to a well- fitting function, see red with the results of Slevin and Ohtsuki15, WU =
lines in Fig. 1. After the subtraction of the irrelevant 18.375 (18.358..18.392) and νU = 1.43 (1.37..1.c49S)l,eob-
part from the raw data, plotting it as a function of (cid:37)Lν1 tainedbytransfermatrixmethSoled(theydidnotpublished
results a scaling-function also, see insets of Fig. 1.
the value of the irrelevant exponent). They used mag-
TheMFSSprovidedusthecriticalpoint, Wc, thecrit- netic flux Φ = 1, while we used Φ = 1, and according
ical exponent, ν, and the irrelevant exponent, y at every 4 5
to Dr¨ose et al.23, WU depends on the applied magnetic
investigated values of q, the results are given in Fig. 2. c
flux. However, in Fig.2. of Ref. 23 it can be seen that
The parameters of the critical point correspond to the the critical points at Φ = 1 and Φ = 1 are very close
systemitself,thereforeitshouldnotdependonthequan- 4 5
to each other, hence the agreement between our critical
tity we used to find it. In other words, it should be in-
point and the result of Slevin and Ohtsuki.
dependent of q, the averaging method and the GMFE
In the symplectic class the critical parameters agree
we used. From Fig. 2 it is clear that this requirement
more or less with the results of Asada et al.16, WS =
is fulfilled very nicely. There is a small deviation for the c Asa
irrelevantexponent,y,obtainedfromαtyp atq = 1and 20.001 (19.984..20.018), νASsa = 1.375 (1.359..1.391)
− and yS = 2.5 (1.7..3.3), obtained by transfer matrix
q = 0.75 in the unitary and symplectic class, but since Asa
− method. However,thedifferencedoesnotseemtobevery
y describes the subleading part, it is very hard to deter-
large, our critical point is considerably different, even
mine,andwecannotexcludesomesortofunderestimati-
though we used exactly the same model. Due to big-
ionoftheerrorbarofthisexponent. Anotherinteresting
ger computational resources we could investigate much
feature of the results is that the error bars get larger as
bigger system sizes than they did, therefore it is possi-
q goesabove1. AswritteninSec.IIIC,largeq enhances
ble that they underestimated the role of the irrelevant
the errors through the qth power in Eq. (7), leading to
scaling, resulting in a somewhat higher critical point.
bigger error bars. A similar effect can be seen around
q 1, where the relatively less precise small wave- Thecriticalpointsarehigherintheunitaryandinthe
≈ −
function values dominate the sums in Eq. (7), which can symplectic class, than in the orthogonal class, showing
also contribute to the deviation of y obtained from α˜typ thatbrokentime-reversalorspin-rotationalsymmetryre-
in this regime. These two effects together lead to our quiresmoredisordertolocalizewave-functions. Sincethe
investigated interval 1 q 2, where GMFEs behave value of the critical point in the unitary and symplectic
− ≤ ≤
the best. The results are strongly correlated, since they class can be influenced by the strength of the applied
were obtained from the same wave-functions, therefore magnetic flux and spin-orbit coupling, the relationship
they cannot be averaged. We chose a typical q-point for between WUλ and WSλ probably depends on these two
c c
everysymmetryclasstodescribethevaluesofthecritical parameters. However, because of their close value of the
parameters, see Tab. II. critical exponents, νUλ and νSλ are the same within our
In the orthogonal class the critical parameters are confidence interval, and the following relation appears:
in excellent agreement with the most recent high νOλ > νUλ νSλ. The situation for the irrelevant ex-
≥
precision results of Rodriguez et al.14, WOλ = ponent is similar namely, that they are the same within
c Rod
16.517 (16.498..16.533), νOλ = 1.612 (1.593..1.631) and error bar, but yOλ seems to be slightly higher than yUλ,
Rod
7
16.65 1.9 3.5
1.85 αeqns αeqns
16.6 αtqyp 3 αtqyp
1.8 Dqens Dqens
16.55 1.75 Dqtyp 2.5 Dqtyp
Wc 16.5 ν 1.7 y 2
1.65
16.45 αeqns 1.5
16.4 Dαqetqnyps 11.5.65 1
16.35 Dqtyp 1.5 0.5
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
18.44 1.6 2.2
18.42 αeqns 2
18.4 1.55 Dαqetqnyps 1.8
18.38 1.5 Dqtyp 1.6
18.36
Wc 18.34 ν y 1.4
1.45 1.2
18.32 αeqns αeqns
18.3 αtqyp 1.4 1 αtqyp
18.28 Dqens 0.8 Dqens
18.26 Dqtyp 1.35 0.6 Dqtyp
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
19.95 2.2 3
19.9 2 αeqns αeqns
19.85 1.8 Dαqetqnyps 2.5 Dαqetqnyps
19.8 1.6 Dqtyp 2 Dqtyp
1.4
Wc 19.75 ν 1.2 y 1.5
19.7
αeqns 1 1
1199.6.65 Dαqetqnyps 00..68 0.5
19.55 Dqtyp 0.4 0
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
FIG. 2: Critical parameters of the Anderson models in WD classes obtained by MFS at fixed λ=0.1. First row corresponds
to the orthogonal class, second row corresponds to the unitary class, and third row corresponds to the symplectic class.
class exp Wλ νλ yλ N χ2 n n n n
c df r ir (cid:37) η
ort α˜ens 16.524 (16.511..16.538) 1.598 (1.576..1.616) 1.763 (1.679..1.842) 172 176 3 2 1 0
0.6
uni α˜ens/typ 18.373 (18.358..18.386) 1.424 (1.407..1.436) 1.633 (1.516..1.751) 198 179 4 2 1 0
0
sym D˜typ 19.838 (19.812..19.869) 1.369 (1.305..1.430) 1.508 (1.309..1.743) 171 151 4 2 1 0
−0.25
TABLE II: Result of the MFSS at fixed λ=0.1 for the selected values of q.
which is a bit higher than ySλ. that, we choose a range of box size (cid:96), which is used for
the MFSS. We always use the widest range of (cid:96), that re-
sults in convergence, χ2/(N 1) 1. We find that for
df
− ≈
B. Results of the MFSS at varying λ ourdatasetfordifferentvaluesofq forαq orDq different
ranges of (cid:96) were the best. We used minimal box sizes
(cid:96) =2 or (cid:96) =3 and maximal box sizes correspond-
AsmentionedinSec.IIIC,GMFEsobtainedbytypical min min
ing to λ = 0.1 or λ = 0.066. At α and α
averagingareequaltoensemble-averagedGMFEsonlyin max max 0.4 0.6
the fitting method sometimes suffered from convergence
a range of q, q < q < q . Since we intend to compute
theMFEsalso,−werestric+touranalysistoensembleaver- troubles and resulted in large error bars, because these
points are close to the special case of q = 0.5 where, by
aged GMFEs, and drop the label ens from the notation.
definition, α =d. Artifacts from this regime were also
We fit the formula Eq. (24)to the raw data. To do 0.5
8
reported in Ref. 14, so we decided not to take into ac- C. Analysis of the multifractal exponents
countthesepointsforα. Wetriedseveralcombinationsof
(cid:96) , λ and expansion orders in the symplectic class
min max
for α and α , but none of them resulted in stable
1.75 2
fit parameters. Therefore values computed from these
MFSS for varying λ provided us the MFEs in all WD
points are also missing from our final results, which are
classes,whicharelistedinTab.V,anddepictedinFig.4.
visible in Fig. 3. The results are independent of q and
For the orthogonal class one can find matching results
the GMFE we used, similar to the fixed λ method. In
withthelistedMFE-sinRef.14. Sincetheprecisevalues
Sec.IVAwealreadysawthataccordingtothearguments
oftheMFEsinthreedimensionsweredeterminedfirstin
of Sec. IIIC error bars get bigger, if q grows beyond 1.
Ref. 14 for the orthogonal class only, the lack of reliable
This phenomenon is more amplified here, especially for
analytical and numerical results for the other symmetry
values coming from fits for α , but larger error bars on
q classesmakesourresultsmoreimportant. Themostcon-
values corresponding to D are present on a moderate
q spicuous thing in Fig. 4 is that curves for different sym-
level also. Since Fig.8 of Ref. 14 shows results for this
metryclassesareveryclosetoeachother,theyarealmost
regime only for values corresponding to ∆ , which is a
q indistinguishable at the first sight. This shows that the
lineartransformofD ,wecancomparetheirresultsonly
q broken time-reversal or spin rotational symmetry has a
to ours corresponding to D . One can see that our error
q very small effect on the MFEs in three dimensions. Tak-
bars are similar, even though there are differences prob-
ing a closer look (or from Tab. V) one can see that the
ably due to the fact that they used system sizes up to
curve of D and α are the steepest in the symplectic,
L = 120, which was not possible for us, mainly because q q
the second steepest in the unitary, and the less steep in
ofthelongruntimeandlargememoryusageforthesym-
the orthogonal class. From Tab. V it is also clear that
plectic model. They also use (cid:96) = 1 and (cid:96) = 2,
min min at most of the q values there is a significant difference
while(cid:96) =1wasneversuitableforourdataset. Wedo
min between the MFEs of different symmetry classes.
notknowthepreciseoriginofthisbehavior, butwehave
a few possible explanations. We experience that larger
system sizes allow a wider range of (cid:96) to be used. We
have smaller system sizes than Ref. 14, and fewer sam- There are no critical states in the two dimensional or-
ples for the largest systems sizes. Noise also gets bigger thogonal class12, but one can find values of α0 for the
as (cid:96) decreases, because of the smoothing effect of box- two dimensional unitary class (Integer Quantum Hall),
ing described in Sec. IIIC, which can also explain partly αU =2.2596 0.000424, and symplectic class, αS =
02D ± 02D
our experience. Another important difference is that in 2.172 0.00225. Comparing the difference between these
±
Eq. (37) of Ref. 14 the authors use an expression in the exponents in two dimensions we get αU αS =
02D − 02D
expansion of the scaling function, which is proportional 0.0876 0.0024, while our result for three dimensions is
±
to the square of the irrelevant term, (η(cid:96)−y)2. According α0U3D−α0S3D =−0.03±0.015. Thereisaboutafactorof
to our experience the inclusion of this term produced no 3 between the magnitude of these values, and even their
improvementinthescalinganalysis,soweusethescaling signisopposite,whichshowsverydifferenteffectofpres-
function described in Eq. 24. Such a difference might be ence or absence of spin rotational symmetry in different
explained again by our different dataset. dimensions.
As written in Sec. IVA, the results for different values
We tested the symmetry relation Eq. (11) for α and
q
of q are strongly correlated, therefore we chose one of
∆ , the results are listed in Tab. V and depicted in
q
them with the lowest error bars that represents well the
Fig. 5. The symmetry relation is fulfilled in the range
results for that universality class.
0.25 q 1.25 (in the symplectic class only for
− ≤ ≤
0.25 q 1), and small deviations are visible out-
− ≤ ≤
side this interval. In this regime error bars are growing
The critical parameters listed in Tab. III are in a very
verylarge, comingmainlyfromthelargeerrorsofα
nice agreement with our previous results for the fixed q 1.5
andD . Similareffectswerealreadyseenforthec≥rit-
method of λ = 0.1, see Sec. IVA, and also with the q 1.5
icalpar≥ametersinFig.3. Itisreallyhardtoestimatethe
results of Refs. 14–16. Comparing the critical param-
correct error bars in this large q case, and the deviations
eters for the orthogonal case with the results of Ro-
driguez et al.14 obtained by the same method, WO = from symmetry are small, therefore we believe that dif-
16.530 (16.524..16.536), νO = 1.590 (1.579..1.6c02R)o,dwe ferences appear only because of slightly underestimated
Rod error bars of α and D . All in all we find nu-
see a nice agreement again. Moreover these results are q 1.5 q 1.5
merical results b≥asically mat≥ching with Eq. (11).
more accurate with this method compared to the fixed λ
method, leading to (for yO and yU only almost) signif-
icantly different critical exponents and irrelevant expo-
nents for the different WD classes, νO > νU > νS and Assuming, that ∆ is an analytic function of q, and
q
yO yU >yS. using the symmetry relation, Eq. (11), one can expand
≥
9
16.68 1.75 2.8
16.66 αeqns 1.7 2.6 αeqns
16.64 Dqens 1.65 2.4 Dqens
16.62
1.6 2.2
16.6
1.55 2
Wc 16.58 ν 1.5 y 1.8
16.56
1.45 1.6
16.54
16.52 1.4 1.4
16.5 1.35 αeqns 1.2
16.48 1.3 Dqens 1
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
18.55 1.8 4.5
18.5 Dαqeqennss 1.7 4 Dαqeqennss
3.5
1.6
18.45 3
Wc ν 1.5 y
18.4 2.5
1.4
2
18.35 1.3 αeqns 1.5
18.3 1.2 Dqens 1
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
19.9 1.55 3
1199..8889 Dαqeqennss 1.5 22..68 Dαqeqennss
19.87 1.45
19.86 2.4
19.85 1.4 2.2
Wc 19.84 ν 1.35 y 2
19.83 1.8
19.82 1.3
1.6
19.81
19.8 1.25 αeqns 1.4
19.79 1.2 Dqens 1.2
-1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2
q q q
FIG. 3: Critical parameters of the Anderson models in WD classes obtained by two-variable MFFS with varying λ. First row
corresponds to the orthogonal class, second row corresponds to the unitary class, and third row corresponds to the symplectic
class.
class exp W ν y N χ2 n n n n
c df r ir (cid:37) η
ort α˜ 16.524 (16.513..16.534) 1.595 (1.582..1.609) 1.749 (1.697..1.786) 241 267 3 2 1 0
0
uni D˜ 18.371 (18.363..18.380) 1.437 (1.426..1.448) 1.651 (1.601..1.707) 275 232 4 2 1 0
0.1
sym α˜ 19.836 (19.831..19.841) 1.383 (1.359..1.412) 1.577 (1.559..1.595) 361 352 3 2 1 0
0
TABLE III: Critical parameters of the Anderson models in the WD symmetry classes obtained by two-variable MFSS with
varying λ.
∆ in Taylor series around q = 1: wherethecondition∆ =∆ =0enforcedbythedefini-
q 2 0 1
tionof∆ (seeEq.(9))wasusedinthelaststep,leading
q
(cid:88)∞ (cid:18) 1(cid:19)2k (cid:88)∞ (cid:18) 1(cid:19)k to k =1 as the lower bound for the summation. Similar
∆ = c q = c q(q 1)+ =
q k k expression can be derived for α by using the connection
− 2 − 4 q
k=0 k=0 α =d+ d ∆ derived from Eqs. (8)–(9):
q dq q
k (cid:18) (cid:19) (cid:18) (cid:19)k i
= (cid:88)∞ c (cid:88) k (q(q 1))i 1 − =
k=0 k i=0 i − 4 αq =d+(1−2q)(cid:88)∞ ak(q(1−q))k−1, (31)
k=1
= (cid:88)∞ d (q(1 q))k, (30)
k where a = kd , and a = d = α d. One can
− k k 1 1 0
−
k=1
10
4 orthogonal 6 orthogonal 1.25 orthogonal 1.2 orthogonal
3.5 symupnlietcatriyc 5 symupnlietcatriyc 11.1.25 symupnlietcatriyc 1.15 symupnlietcatriyc
3 4 1.1 1.1
Dq 2.5 αq 3 αdq−12q−1.015 ∆qq(1q)−1.05
2 2 0.95 1
0.9
1.5 1 0.85 0.95
0.8 0.9
1 -1 -0.5 0 0.5 1 1.5 2 0 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 -2 -1.5 -1 -0.5 0 0.5
q q q(1 q) q(1 q)
− −
0.5 3
FIG.6: Dotsanderrorbarsarenumericalvaluesforthecor-
0 2.5 respondingquantities, α(q)−d and ∆(q) ,fortheWDsymme-
2 1−2q q(1−q)
-0.5 try classes. Lines are the best fits. Several points are shifted
∆q -1 fα() 1.51 horizontally a bit for better viewing.
0.5
-1.5 orthogonal orthogonal
unitary 0 unitary
-2 symplectic -0.5 symplectic ort uni sym
-1 -0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6
q α d1 1.044 (1.041..1.047) 1.097 (1.095..1.098) 1.123 (1.122..1.125)
d 0.095 (0.085..0.105) 0.096 (0.091..0.100) 0.088 (0.084..0.093)
2
FIG.4: MFEsoftheAndersonmodelsintheWDuniversality d 0.018 (0.011..0.025) 0.017 (0.014..0.020) 0.014 (0.010..0.017)
3
classes. Corresponding data are listed in Tab. V. a 1.045 (1.042..1.048) 1.099 (1.096..1.102) 1.124 (1.123..1.126)
1
a 0.182 (0.168..0.195) 0.185 (0.174..0.197) 0.185 (0.179..0.191)
2
a 0.044 (0.035..0.053) 0.043 (0.035..0.050) 0.044 (0.038..0.049)
3
6.4 0.2
6.35 orthuongiotanrayl 0.18 orthuongiotanrayl TABLEIV:ExpansioncoefficientsofEqs.(30)–(31)obtained
6.3 symplectic 00..1146 symplectic by a fit depicted in Fig. 6.
α1q− 66.2.25 ∆1q− 00.1.12
+ 6.15 − 0.08
αq 6.1 ∆q 0.06
0.04
6.05 0.02 a are significantly different for the different symmetry
6 0 1
5.95 0.8 1 1.2 1.4 1.6 1.8 2 -0.02 0.6 0.8 1 1.2 1.4 1.6 1.8 2 classes, while d2, d3, a2 and a3 are the same within er-
q q ror bars. Their real value is probably different, but the
relative error of the expansion coefficients naturally in-
FIG.5: TestforsymmetryrelationEq.(11)intheWDsym- creasesask grows,leadingtoindistinguishablevaluesfor
metry classes. Points are shifted horizontally a little bit for the different symmetry classes for k 2.
bettervisualization. Onlytherangeq≥0.5isvisiblebecause Wegner computed analytically26 t≥he value of ∆ with
q
expressionα +α (∆ −∆ )issymmetric(antisymmet-
q 1−q q 1−q ε expansion using nonlinear σ-model up to fourth-loop
ric) for q=0.5.
orderfortheorthogonalandtheunitarysymmetryclass,
resulting an expansion in dimensions d = 2+ε for ε
(cid:28)
112:
obtain the d and a coefficients by fitting the expres-
k k
sions Eq. (30)–(31). We used only the range q 1.25, ζ(3)
≤ ∆O =q(1 q)ε+ q(q 1)(q2 q+1)ε4+ (ε5)
because beyond this regime error bars are growing ex- q − 4 − − O
tremely large, and there are small deviations from the (cid:18) ζ(3) (cid:19) ζ(3)
symmetry relation Eq.(11) also. We plotted ∆(q) and = ε ε4 q(1 q)+ ε4(q(1 q))2+ (ε5)
q(1 q) − 4 − 4 − O
−
α(q)−d in Fig. 6 to make the presence of higher-order (32)
1 2q
ter−ms of the expansion visible. (cid:114)ε 3
We fit expressions Eq. (30)–(31) up to third order in ∆Uq = 2q(1−q)− 8ζ(3)ε2(q(1−q))2+O(ε25) (33)
allcases, theresultingexpansioncoefficientsarelistedin
Tab. IV. From the data listed one can see that the ex- Even though ε 1 should hold, one can try to extrapo-
(cid:28)
pansion coefficients fulfill the relation a = kd . How- latetothree-dimensionsbyinsertingε=1. Thisleadsto
k k
ever α and ∆ were obtained from the same wave- dO 0.699, dO 0.301, dU 0.707 and dU 0.451.
q q 1 ≈ 2 ≈ 1 ≈ 2 ≈ −
functions, they are results of completely independent fit- As one can see, these values are rather far from our
procedures. Therefore the fact, that they satisfy the numerical results, but this is not surprising for an ε-
equation a = kd further confirms our result for their expansion at ε = 1. These results capture well the ten-
k k
value listed in Tab. V for q 1.25 and shows the consis- dencyatleastthatdOisslightlysmaller,thandU. Onthe
≤ 1 1
tency of the MFSS. other hand it leads to dO and dU having opposite sign,
2 2
Asonewouldexpectforexpansioncoefficients, d and which is highly inconsistent with our numerical results.
k
a show decreasing behavior as k grows. Only d and Itisinterestingthatthefirst-loopterm, whichispropor-
k 1