Wave function statistics at the symplectic 2D Anderson transition: bulk properties

0.17 MB·English

by  A. Mildenberger

2D-sp-AM/INT
Wave function statistics at the symplectic 2D Anderson transition: bulk properties
A. Mildenberger1 and F. Evers2,3
1Fakult¨at fu¨r Physik, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
2Institut fu¨r Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany
3Institut fu¨r Theorie der Kondensierten Materie, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
7 (Dated: February 6, 2008)
0
The wave function statistics at the Anderson transition in a two-dimensional disordered elec-
0
tron gas with spin-orbit coupling is studied numerically. In addition to highly accurate exponents
2
(α0=2.172±0.002,τ2=1.642±0.004), wereportthreequalitativeresults. (i)Theanomalousdimen-
n sions areinvariant under q→(1−q) which is in agreement with a recent analytical prediction and
a supports the universality hypothesis. (ii) The multifractal spectrum is not parabolic and therefore
J
differs from behavior suspected, e.g., for (integer) quantum Hall transitions in a fundamental way.

8 (iii) The critical fixed point satisfies conformal invariance.

] PACSnumbers: 72.15.Rn,05.45.Df
l
l
a
h
Disordered electron systems that are confined to two scrutinize the nature of τ more closely. For one thing,
- q
s spatial dimensions (2D) cannot support a true metallic thewavefunctionstatisticscanbemeasured,inprinciple,
e
state because of Anderson localization.1 The underlying andpromisingsteps inthis directionweremade notlong
m
physics relates to an interference-enhanced return prob- ago.6
.
t ability of quantum mechanical particles due to repeated But also important questions concerning our con-
a
backscattering of the same (quenched) disorder configu- ceptual understanding of the localization-delocalization
m
ration. There are exceptions to the rule, however. For transitionarecloselyrelatedtomultifractality. First,the
- instance, if spin-orbit scattering exists, the return prob- analytic structure of τ is a specific characteristic of the
d q
n ability is not enhancedbut evendepleted andthe metal- criticalfieldtheoryofthe transitiondescribingscalingof
o lic state survives.2 Universal properties of such metals the localdensity of states. For example,it has been pro-
c are described by the symplectic symmetry class of Gaus- posedthatthe(integer)quantumHalltransitionexhibits
[ sian random matrix theories. By increasing the disorder reduced anomalous dimensions δ ,
q
2 strength W, a metal-insulator (i.e. Anderson transition)
v canbe driveninthese materials. Its universalproperties τq =d(q−1)+δqq(1−q), (2)
0 have been studied intensively in the last two decades.
6 One of the controversial questions in the late 1990s with a special property: δq does not depend on q, such
5 concerningthesymplectictransitionin2Dwasaboutthe that τq is parabolic and also invariant under q → 1−q.
8 numerical value of the critical exponent ν that describes Very recently, it has been predicted7 – based on exact
0
the divergence of the localization length when the disor- results for the nonlinear σ model and invoking the uni-
6
0 der approaches its critical value: ξ ∼ |W −Wc|−ν. In versalityhypothesis–thatthislastsymmetryisageneral
/ recent work, Asada et al. have made a very convincing property of all transitions belonging to the conventional
at case in favor of ν=2.75 (overview in Table I) employing Wigner-Dyson classes. That is
m the SU(2) model.3 A work by Markos and Schweitzer4
δ =δ (3)
- comes to a similar conclusion, ν ≈2.8±0.04,within the q 1−q
d Ando model and the debate is now settled.
n should hold. A numerical verificationbeyond the frame-
However,this latter worknotonly has helped to fixν,
o workofthe powerlawrandombandedmatrix modelhas
it also has reemphasizedthat another important topic is
c not been reported yet. This would be an interesting test
: still unresolved. Recall that the critical wave functions,
v Ψ(x) atthe boundary between insulator and metal obey of universality, since it does not only rely on compar-
i ing quantitative values for some few exponents – which
X a multifractal statistics.5 This implies that the moments
has been the usual procedure – but rather refers to the
r
a hh|Ψ(x)|2qii∼L−d−τq, q ∈R (1) analytic structure of an exponent spectrum. Note that
Eq. (3) does not generally hold outside the conventional
scale with system size L, introducing the exponent spec- symmetry classes. The spin quantum Hall effect is an
trum τ . (The angular brackets denote a combined spa- example for a transition in a nonstandard universality
q
tialandensembleaverage.) Aprecisenumericaldetermi- class, where Eq. (3) is manifestly violated.8,9
nationofτ hasnotbeenundertakenyet. Thenumerical Second, lately it has become clear that near bound-
q
work presented in this letter is an attempt to close this aries multifractality differs from the bulk: flat interfaces
gap. supporttheirown“surface”spectrumτs;inthepresence
q
There are several good reasons why one would like to of corners yet another spectrum is superimposed, etc.10
2
model method Wc Λc α0 =2+δ0 δq ν reference
SU(2) TM 5.953±0.001 1.843±0.0013 2.746±0.009 3
AM TM 5.838±0.007 1.87±0.02 2.8±0.04 4
MAt 5.838±0.007 2.107±0.005 δ1=0.111 4
AM MAt 5.86±0.04 δ2 =0.19±0.005 2.41±0.24 23
AM wave-packet propagation 5.74 δ2=0.15±0.02 24
AM MAt 5.74 2.19±0.03 δ2=0.17±0.025 15
EZM MAa δ1=0.16±0.02 25
δ2=0.185±0.01 26
network model 1.83±0.03 2.51±0.18 27
TABLE I: Overview of results for the symplectic transition in two dimensions. AM: Ando model16; EZM: Evangelou-Ziman-
model28; MAt (MAa): multifractal analysis based on scaling of typical (average) amplitudes; TM: transfer matrix; SU(2):
SU(2) model18; δ : reduced anomalous dimension, see Eq. (2). Entriesfor thesame model are in chronological order, starting
q
with thelatest work.
Also, in principle, an edge could break a bulk symmetry and the earlier result3 Λ =1.843 we arrive at πΛ δ =
c c 0
andthuswouldnotevensharethebulkuniversalityclass. 0.996±0.012. Thus numerical evidence is provided that
In fact, the unraveling of surface multifractality could the symplectic fixed point obeys CI, in agreement with
leadtoaparadigmaticshiftofourpresentunderstanding general expectations.
ofcriticalwavefunctionstatistics. Clearly,aprerequisite Models: We consider a tight-binding Hamiltonian on
for all this is a detailed knowledge of bulk properties. a two dimensional square lattice with nearest neighbor
Third, finally, a relation between δ and the ratio Λ coupling
0 c
ofwidthandlocalizationlengthofquasi-1Dstripsexists:
H = ǫ c† c + V c† c , (5)
X i i,σ i,σ X i,σ;j,σ′ i,σ j,σ′
Λc =1/πδ0, (4) i,σ hi,ji,σ,σ′
whichisexactifthecritical2Dfixedpointisconformally where c† (c ) denotes a creation (annihilation) opera-
invariant.11 It is believed thatconformalinvariance(CI) i,σ i,σ
tor of an electron with spin σ on site i.
is a generic property of localization-delocalization tran- In the Ando model16, the on-site energies ǫ are taken
i
sitions in 2D. For instance, it has been demonstrated to
independently fromthe interval[−W/2,W/2] with a ho-
hold at the integer quantum Hall transition.12,13 Excep-
mogenous distribution. The hopping matrix Vi,σ;j,σ′ re-
tions are not known so far, but Eq. (4) can be used
flecting the spin-orbit coupling is chosen as
as a test of CI. In this respect, recent numerical results
are alarming. It is reported4 that δ0=0.107±0.005 and Vi,σ;i+k,σ′ =(V0exp(iθkσk))σ,σ′, k =x,y, (6)
Λ =1.87±0.02; thus the product πΛ δ =0.629±0.036
c c 0
would signal a strong violation of Eq. (4) and therefore with σ ,σ denoting Pauli matrices and the parameters
x y
absence of CI.14 V = 1 and θ = π/6. We have determined the critical
0 k
In this Rapid Communication,we present a numerical disorder strength independently via analysis of the crit-
high-precision study of δ at the 2D-symplectic transi- ical level statistis.17 Our finding W =5.85±0.025 agrees
q c
tion. Our particular aim is to answer three qualitative well with earlier work.4
questions. (i) Is δ a constant, so τ is parabolic? (ii) If The second model, the SU(2) model, has been intro-
q q
not,doesitobeythesymmetryrelationEq. (3)confirm- duced by Asada, Slevin, and Ohtsuki18. In addition
ing the universality hypothesis? (iii) Is the fixed point to the on-site energies ǫ , now also the hopping matrix
i
conformally invariant? Vi,σ;j,σ′ israndom. Itistakentobeuniformlydistributed
Most earlier works analyzed typical moments in small over the entire group SU(2) using the group invariant
ensembles, where finite-size effects make it difficult to (Haar) measure.18
obtainreliableerrorbars. Bycontrast,weemployscaling H is implemented on square L×L-size lattices with
of typical and average moments in very large ensembles periodic boundary conditions. For our numerical diag-
with big system sizes. Errors can thus be reduced by onalization of the resulting 2L2 ×2L2 matrices we use
almost an order of magnitude. In order to cross-check, an inverse iteration routine coupled with direct sparse
we analyze the two most important microscopic models. solversin orderto obtainthe eigenvaluesandwavefunc-
Results thus obtained agree very well. Specifically, we tions with energies closest to zero.19 (Cf. Ref. 20.)
find that δ is not a constant and the symmetry relation Multifractal analysis: Our multifractal analysis pro-
q
(3) is satisfied. ceeds by analyzing the scaling behavior of the average
On a quantitative level, we obtain δ =0.180±0.002 moments of wave function amplitudes, Eq. (1).
2
(both models), δ =0.173±0.003 (Ando model), and In order to analyze the critical behavior we take the
0
δ =0.172±0.002 (SU(2) model). Together with Eq. (4) disorder value W =5.84 (for states at energy zero being
0 c
3
0.2
0.18 SU(2)-model Ando-model
0.176
q 1-q
0.174
0.16
)
0.19 (L W=5.84, q=1.5, 0.0042
0.172 Ωq W=5.84, q=2.0, 0.0087
n 0.14 q=1.5, 0.0040 W=5.86, q=1.5, 0.0042
) l q=2.0, 0.0083 W=5.86, q=2.0, 0.0087
q 0.17
δ( -0.5 0 0.5 1 1.5
0.18 0.12
3 4 5 3 4 5
ln L ln L
FIG.2: TestfunctionΩ (L)highlightingvariabilityofδ with
q q
0.17
q for two q values. Solid lines represent power law fits, with
a slope representing δ −δ ; for values, see legend. Slight
0 1 2 3 q 1/2
q deviations between models are due to larger errors in finite-
sizeextrapolationofAndomodel. Forthatmodel,resultsfor
two values of W are given to illustrate that the uncertainty
FIG. 1: Reduced anomalous dimension δ as defined in Eq.
q in W is not a precision-limiting factor. Note that δ −δ
c 3/2 1/2
(3) for the Ando model (dashed, Wc=5.84) and the SU(2) agrees well with curvature δ′′ seen in Fig. 1.
model (solid, W =5.953). Additionally, anomalous dimen- 1/2
c
sionsδ˜ obtainedfromtypicalinverseparticipation ratiosare
q
shown (⋄) for the latter model. Dashed lines indicate the
require a coarse graining in order to overcomethe diver-
estimated error (2σ) in δ0. Inset: blowup of the solid line
gence of the moments (1) related to zeros of the wave
behavior near q=0.5 now represented by ◦. Data near q=0
and q=1 suffer from noise amplification (dividing by q(1−q) functions.) A symmetric shape of the curve is clearly
in Eq. (2)) and have therefore been omitted. Filled symbols displayedinthe regimeofbestaccuracy,−0.5>q >1.5.
(•) show original trace after reflection at q=0.5. Dot-dashed
line indicates parabolic fit (offset: 10−3) with δ1/2=0.1705 (iii) The set of exponents δq does not reduce to a con-
and curvatureδ1′′/2=0.0043. stant, e.g., δq has a small but nonzero curvature δ1′′/2.
Detecting δ′′ requires high-precision data, because the
1/2
numericalwindowislimited toq >2.0. Atlargervalues,
critical) in the Ando model. For the SU(2) model we
(a)finite-Leffectsproliferate(inAndomodelfasterthan
employW =5.953inordertohaveamobilityedgeaten-
c in SU(2)), so deviations between solid and dashed lines
ergy ǫ=1.3 The average (1) has been performed over an
increase. And(b)momentshh|Ψ|2qiiforlargeq probethe
ensemble of wave functions that have been calculated in
tails of the distribution function, so that typical values
systemsofsizesL=16,24,32,48,64,96,128,192,256(the
and averages differ from each other. Then, error bars
last two values were not used in all cases). For each dis-
tend to become large due to undersampling.20 The part-
order realization64 wave functions closestto the critical
ing of the three curves at q ? 2 visible in Fig. 1 is a
energy have been taken into account; all together the
consequence of these effects.
number of wave functions in the ensemble is typically
As a sensitive test for variability of δ we investigate
4×107 (L=16) to 3×105 (L=256). q
in Fig. 2 the ratio
The exponents τ are readily extracted from a power-
q
law fit as suggested by Eq. (1).21 In Fig. 1 we plot the 1/q(1−q) 4
reduceddimensionsδq definedin(2)asobtainedforboth Ωq(L)=hhh|Ψ|2qiiLdqi /hhh|Ψ|iiLd/2i (7)
models. It incorporates our three main results.
(i)We determine δ =0.172±0.002. The valuesatisfies encompassing only unprocessed data. It scales as
0
Eq. (4) and thus the consistencycheck on CI is positive. Ωq(L)∼L−δq+δ1/2 and therefore any slope in lnΩ signal-
Thegoodaccuracystemsmainlyfromlargestatisticsand izes that δ deviates from δ =0.1705±0.001. Data for
q 1/2
the fact, that finite-size corrections in the SU(2) model Ω at q=1.5,2.0 are shown in Fig. 2. It clearly exhibits
q
turnouttobeextremelysmallatq >1.5. Ascanalsobe a linear trace with the nonzero slope indicative of curva-
seen from Fig. 1, the Ando model gives a similar result. ture in δ . Note that finite-size effects are very small, so
q
(ii)Thefunctionδ satisfiesthesymmetryrelationEq. that δ −δ can be extracted with good accuracy.
q q 1/2
(3). Thus the universality postulate is confirmed. The A more conventional object than δ to characterize
q
inset of Fig. 1 shows that part of the full curve δ , for the wave function statistics is the Legendre transformed
q
which numericaldata areavailable at both points, q and f(α) = qα−τ , α =∂τ/∂q, displayed in Fig. 3. Even
q q
its image 1−q. (The numerical procedure that we work thoughwe haveobtainedτ onlyfor q?−1/2andthere-
q
with is limited to q?−1; more negative values would fore are restricted to α>α , the spectrum can be re-
1/2
4
constructed also at values α?α by making use of Eq.
2 1/2
(3).7 Then deviations from parabolicity obtrude.
Summary: The multifractal spectrum of wave func-
1
tions at the 2D symplectic Andersontransitionhas been
α) calculatedin the Ando andSU(2) models with high pre-
(
f cision. On a qualitative level, our results demonstrate
0
thatthe criticalfixedpointis conformallyinvariantwith
a nonparabolic spectrum τ . Furthermore, δ =δ , as
q q 1−q
predictedfromcalculationswithinthenonlinearσ model
-1 parabolic approx., δ =0.1715
0 and thus supports the universality hypothesis.
1 α2 3 We thank L.Schweitzer andK. Yakubo for useful cor-
respondence and A. D. Mirlin for valuable discussions
FIG. 3: f(α) spectrum from data of Fig. 1, SU(2) model. and suggestions on the manuscript. While finalizing the
f(α) is slightly asymmetric and not a parabolic function, manuscript, we learned about a closely related project,
which would havemeant f(α)=2−(α−2−δ0)2/4δ0. with partly overlapping results.22
1 P. Anderson,Phys. Rev. 109, 1492 (1958). fields is J(L,W) = J˜ (φL1/ν) + χL−yJ˜(φL1/ν) + ...
R I
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c c
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c
035115 (2004). error of theexponentsresult from the uncertaintyin W .
c
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0.19±0.03 (Table I), which would appear to give a much and in contrast to earlier claims made in the literature31.
betterfit,1/πδ0=1.68±0.3 (Ref.15).However,errorbars 22 H.Obuse,A.R.Subramaniam,A.Furusaki,I.A.Gruzberg,
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c
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c
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c
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5
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