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A generalized thouless formula as a criterion for Anderson localization in two- and three-dimensional systems
Physica 131A (1985) 131-156 North-Holland, Amsterdam
A GENERALIZED THOULESS FORMULA AS A CRITERION FOR ANDERSON LOCALIZATION IN TWO- AND THREE-DIMENSIONAL SYSTEMS J. CANISIUS Sondetforschungsbereich
123 der Universitiit
Heidelberg,
6900 Heidelberg,
Fed. Rep. Germany
J.L. VAN HEMMEN* Imtitut fiir Physik der Universitiit
Maim,
6500 Maim,
Fed. Rep.
Germany
-I%. M. NIEUWENHUIZEN Instituut
voor Theoretische
Natuurkunde, Rijks-Universiteit Utrecht, The Netherlands
Utrecht,
Princetonplein
5, 3508 TA
Received 25 October 1984
A generalized Tbouless formula is proposed as an approximate criterion for Anderson localization in two- and three-dimensional systems. The criterion is exact for ordered systems and in the mean field limit of infinite dimensionality. For random systems it is expected that possible corrections are small. Ample numerical evidence is provided through an exact diagonalization of several large-size Hamiltonians. The predictions of the lltouless formula, whose evaluation only needs the eigenvalues, are critically checked against the inverse participation ratio (IPR) of the corresponding eigenvectors. Included are the Lloyd model (for sulliciently low disorder mobility edges cannot be excluded) and surprising, new results on the binary alloy problem. The criterion predicts four as the upper and two as the lower critical dimension.
1. Introduction
Ever since Anderson’s pioneering work’) on localization in disordered lattice systems a lot of effort has been devoted to obtaining precise localization criteria. In the one-electron approximation the question as to whether the eigenstates corresponding to a certain energy range are localized is of prime importance to many of the system’s properties; e.g. the conductivity. In the one-dimensional case a clear-cut answer is known [see, for example, Carmona’) *Now at the Sonderforschungsbereich Fed. Rep. Germany.
123 of the University of Heidelberg,
0378-4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
6900
Heidelberg,
J. CANISIUS
132
and references soon
quoted
as disorder
therein]:
is present,
et al.
all eigenstates
no matter
are exponentially
how small
localization length of the eigenstates can be given states only, through the Herbert-Jones-Thouless Jones3);
Thouless4)).
This
formula,
is rather
attractive
easier
to obtain
for harmonic
formula,
which
is commonly
in numerical
work
than the eigenstates
themselves.
Exact
(Dyson’);
Nieuwenhuizen6)),
as
The inverse
in terms of the density of formula (Herbert and
since
chains
localized
its magnitude.
called the
the Thouless
spectra
expressions
Kronig-Penney
binding electron models (Lloyd’); Nieuwenhuizer?). Also in quasi one-dimensional systems all eigenstates are localized (Gol’dSeid’); Johnson and Kunz”)) and a generalization
are much are known and
tight-
exponentially of the Thou-
less formula is easily obtained. In two and three dimensions the situation is less clear. It is widely believed that in two dimensions, if the degree of randomness W does not exceed a certain critical value, the states in the center of the band localized “logarithmically” and cross over into exponential localization at large distances, whereas the states
near the band
edge decay
exponentially
at small and large distances
alike (Nagaoka and Fukuyama”)). In three dimensions one expects a similar situation but now the states in the centre of the band are delocalized at all distances and the regions of localized and delocalized states are separated by mobility edges. In two dimensions a similar “mobility edge” is present between the regions where states are exponentially localized and where they are logarithmically localized at small distances. As the randomness is increased the mobility edges move towards the center of the band and at a critical WC they merge together, eliminating the region of non-exponentially ded states has been called
localized states. This disappearance of the extenthe Anderson transition. It may be well to point out
that in numerical calculations involving small two-dimensional systems it is hard to distinguish between exponential localization at large distances and logarithmic localization at small distances. In that case a careful (but rather expensive)
analysis
of the results
Anderson’) originally simple cubic lattice, E(n)+)+
considered
as a function a tight-binding
v c a(n +p> = J%(n) I!PlI= 1
)
of the system model
size is required.
of an electron
on a
(1.1)
where the I are independent, identically distributed random variables with a uniform distribution between -W/2 and + W/2, and estimated the critical value WC above which no extended states exist. For a review of subsequent developments, see Thouless”). In what follows we choose units such that v= 1.
CRITERION FOR ANDERSON LOCALIZATION
133
Using two assumptions Economou and Cohen13) derived a localization function which is defined on the real energy axis and exceeds one if the eigenstates are not exponentially localized whereas it is between zero and one if exponential localization holds. However, this function has a very complicated structure and even in the ordered case it has no explicit, simple form which allows direct verification of its predictions. Yet the idea of a localization function is rather appealing. In this paper we consider a generalization of the Thouless formula to dimensions higher than one. It is an approximate localization criterion which becomes exact in the ordered case. Its main practical merit is that it only depends on the eigenvalues, not the eigenvectors. The criterion seems to work quite well (certainly for not too large disorder) and we hope our work will stimulate further research in this direction. We first concentrate on the localization criterion itself in section 2 and present extensive numerical evidence in section 3. There one can also find some interesting new results on the existence of a mobility edge in tightbinding models with a binary distribution (the binary alloy problem). In addition we take the opportunity to scrutinize some recent arguments of Johnson and Kunz14) concerning the absence of a mobility edge in the two- and three-dimensional Lloyd model. Our numerical data (see subsection 3.1) raise serious doubt about their conclusions but do not rule them our completely. A summary may be found in section 4.
2. A generalized Thouless formula For every Y-dimensional, hypercubic,
lattice equation (1.1) may be written
(2.1)
Eu=(&+@)u,
where a is a vector with entries a(n), E a diagonal matrix with elements &(n) to be specified later, and @ the hopping matrix which connects each point n with its nearest neighbours n + p, i.e. @(n, n’) = XP 8n,,n+p. For a harmonic crystal with positive random masses (Lieb and Mattis”)) the eigenvalue equation takes a similar form,
(2.2)
&Mu=(@-2u)a=Au, where 5 = -o*, A is the discretized matrix with elements m(n).
Laplacian,
and M the diagonal
mass
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J. CANISIUS
For complex 0,(E)
for tight-binding 0,(t)
the characteristic function
E and 5 we define
= N-‘Tr
log[(E - F - @)/(-A)]
electron
= N-‘Tr
et al.
models,
log[(M[
(2.3)
and
- A )/(-A )]
(2.4)
for the random mass problem. They are primitives of the traces of the Green functions [E - c - @I-’ and [Mt - A]-‘, respectively. Along the real axis one has the following decomposition into real and imaginary parts: Ol(E _’ 10) = y”‘(E) k irr(l - H,(E))
(2.5)
and
L&(6 = -cd’* i0) = y(*)(w*) & ir&(w*)
,
(2.6)
where H,(E) and H2(w2) represent the integrated density work H is nothing but the empirical distribution function
of states. In practical of the eigenvalues (E
or w*). Using a basis with respect to which the arguments of the logarithms in (2.3) and (2.4) are diagonal one easily verifies that the quantities y(j) are given by
y”‘(E) = N-r 5 ln(E - Ejl - C, = 1 dH,(x) i=l
1nlE - XI - C,
(2.7)
and N
nmijw2-wfl-CY=N-‘~ln 1=l
i=l
2
%-l I Wf
I
(2.8) As N + 00, the yci) are self-averaging depends
on the dimension
C, = N-‘Tr
ln(-A)
In the infinite-volume
(van Hemmen*6)).
The constant
C,, which
V, equals
. limit C, may be written
(2.9)
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135
(2.10)
and we have Z,(2) = 0,
Z,(4) = 1.166243,
Z,(6) = 1.673338.
If v = 1, eq. (2.7) just gives the Thouless formula. Before proceeding we check that, without disorder,
(2.11)
y(E) s 0 for E inside the
spectrum, and y(E) > 0 outside. In the ordered case c(n) = 0 (m(n) = m) and while #*‘(o’) = the infinite system y’(E) = Re I,(E) - Z,(2v), for Re Z,(2v - mo2) - Z,(2v). We first turn to the tight-binding case. The spectrum is the closed interval and (2.10) we find
[-2v,
+2v].
Carrying
out one of the integrations
in (2.7)
y”‘=l~...I~Re(arccosh(~~-~cosq,)
- arccosh
u-l v- 2
COS
qi
(2.12)
.
i=l
Note that the sums in the integrand of (2.12) contain (v - 1) terms. Because arccosh(-x) = arccosh(x) 2 in, the function -y”‘(E) is even in E. It, therefore, suffices to take E positive. If E > 2v, the integrand is strictly positive and #“(E)>O. On the other hand, if 06 E <2v, the integrand is negative and y”‘(E) ~0, while it vanishes if and only if E = 2v, i.e. at the band edge. The vibration
spectrum
[0,4v/m].
Using the previous
with
respect
of a regular
to 2vlm.
Here
harmonic
crystal
is the
closed
w*-interval
argument too,
one directly verifies that yc2)(w2) is even only at the border of the y(‘) vanishes
spectrum, is negative inside and positive outside. Since the points where y = 0 mark the energies or frequencies where the solutions of (2.1) and (2.2) change from delocalized plane waves into exponentially decreasing functions, we pose the following conjecture for random systems: “The mobility edges for a v-dimensional system with (weak) disorder are (approximately) given by the points where y vanishes. Moreover, inside the spectrum a negative y signals eigenstates which are not exponentially localized (at small distances), whereas a positive y signals exponential localization at all distances.” This we call the y-criterion. It is expected to hold for general v. In the next section we present numerical evidence in favour of it. The criterion is exact for
136
J. CANISIUS
ordered
systems
sionality.
For
certainly As
and,
as will be shown
random
systems
et al
shortly,
correction
in a limit
terms
are
of infinite
expected
dimen-
to be small,
for not too large disorder.
Thouless
v-dimensional
(1979)
already
equation
observed
for
the
one-dimensional
(2.7) is close to the expression
derived
case,
our
by Economou
and Cohenj3), -log The
angle
L(E) = (1nlE - I brackets
- Sl) - In(K).
in (2.13)
denote
(2.13)
an average
over
a(n),
S is the
CPA
self-energy, and K is the connectivity of the lattice. An eigenstate with energy E is (not) exponentially localized if the right-hand side of (2.13) is greater (less) than zero. This criterion is not that easy to handle. A much simpler version was proposed Z(E)
by Ziman”), (2.14)
= (In/E - c(n)l> - In(z),
where z is the number of nearest neighbours. If Z(E) is positive, states with energy E are localized. However, for binary distributions Z will also go to minus infinity in the region of localized states where y(‘) may remain positive; see subsection 3.2. The constants C, is (2.7) and (2.8) seem to play a role similar to In(K) and In(z). In fact, since In(K) equals 0.97 if v = 2 and 1.54 if v = 3, we have that In(K) < C, (in(z) However,
In(K)
vanishes
In(z) do not. Probably subtracting C,. Indeed, as defined
.
(2.15) for quasi one-dimensional
systems
whereas
C, and
the corresponding y would have to be defined without the numerical calculations in section 3 will show that y
by (2.7) may become
negative
whereas
all eigenstates
are known
to
be localized exponentially (Gol’dSeidg); Johnston and Kunz”)). Introducing Lyapunov exponents yi (1
(2.16)
y = d-’ c yj - C,. j=l Without C, this is the appropriate generalization strip of d sites across. Note that here the Ziman
of the Thouless formula criterion fails completely.
to a
CRITERION FOR ANDERSON LOCALIZATION
137
Not only is the problem of a precise localization criterion still open, the values of the critical dimensions have not been determined unambiguously either. Though there is agreement about the value two for the lower critical dimension of the Anderson model, the upper critical dimension is believed to four (Nagaoka and Fukuyama”)) or even six. In order to investigate this question we use the y-criterion and expand (2.3) in powers of the randomness E, R,(E) = N-‘Tr{log(E
- @) - (EG + I*
+ am
+ am
+ . . .)} , (2.17)
where G = [E - @I-’ is the Green function of the model without disorder. If necessary, we may absorb the average value of 8 into E and assume (E) = 0. Carrying out the averages in (2.17) we obtain y”‘(E) = Re N-’ c [ [log(E - @)I;; - ~(.Y’)(G,)*- am I - a((~~)- 3(c*)‘)(G,)” - :(E*>*[~(G~~)*(G’),+ 2 (G6Gji)*]) i +
a(d, k2)(E3)) *
(2.18)
The diagonal elements of G and G* do not depend on the site i and are in the limit N + 00 given by (2.19)
For it = 1 and E close to the band edges -+2v, the integral diverges if Y 5 2; the divergence is logarithmic for Y = 2. Thus v = 2 indeed shows up as a (lower) critical dimension for the first terms in (2.18). On the other hand, if IZ= 2, the integral diverges at E = r2v for Y S 4; the divergence is logarithmic for v = 4. Also the higher order terms in (2.18) have v = 4 as critical dimension. We therefore may conclude that according to the y-criterion Y = 4 is the upper critical dimension. In the limit of infinite dimensionality (V + m) only the first term in (2.17) and (2.18) survives. This is just the well-known result that mean-field theory is exact in infinite dimensions. The proper scaling of the eigenvalues is given by X(q) =
$
E(q) Y
=$ $ zJI-1
COS qi *
(2.20)
J. CANISIUS
138
the distribution For V+W, (2/&)exp{ - fx’}. Calculating
of y”‘(E)
x
et al.
is
Gaussian,
for scaled energies
with
density
p(x) =
J? = E/L& we obtain
for large v
y”‘(i?)= _/dx
p(x) InIB - XI -
f ln
v + B( V-‘) ,
(2.21)
which approaches minus infinity as v + 00. Accordingly the y-criterion predicts delocalized states, which are indeed present because of translational invariance. If we do not scale E, we find for large v
y”‘(E) = In E I
which is positive is, the y-criterion
I
+ 6’(Y’) ,
(2.22)
outside the spectrum -2v
the band edges *2v. Finally we note that for harmonic systems with random masses and springs the characteristic function 0, is still defined by (2.4) but -A is now replaced by a random force matrix. The expression for 0, in tight-binding electron models with random hopping terms @(n, n’) can be obtained from (2.3) by making the replacement
A (n, n’)+ @(n,n’) - 6, ,,,2 @(n, n
+ ,I) .
(2.23)
P
3. Numerical
results
Our procedure to exhibit localization and to find a mobility edge is straightforward in principle, although computationally somewhat expensive. Through an exacr diagonalization we find all the eigenvalues and eigenvectors, and examine
the inverse
participation
ratio (IPR), (3.1)
of each eigenvector a. Delocalized states are expected to have small IPR, of order N-’ for N sites, while localized states show larger IPR values. In fact, the IPR equals one (the maximum) if a is localized at one site. In general, the IPR is between zero and one. Unless we state the contrary, one may assume that a
CRITERION FOR ANDERSON LOCALIZATION
139
scaling analysis has shown that the IPR’s in an, apparently, delocalized domain behave as N-l. We have performed the diagonalization in one, two, and three dimensions with lattice sizes up to 900 sites, usually employing periodic boundary conditions. We used the iterative OR algorithm to guarantee numerical stability, knowing that the eigenvalues are not degenerate because of the randomness. For a given probability distribution we need not average over many samples since the localization/delocalization structure should occur with probability one. To display the results we arrange the eigenvalues in an ascending sequence and plot both the IPR and the corresponding y against the label k. 3.1. Lloyd model For tight-binding tribution,
P(E) = ;
models whose site energies
(2+ P-l
)
I
have a Cauchy
dis-
(3.2)
Lloyd’) showed that the average one-particle Green function can be evaluated explicitly by making the substitution E-FE + iT in the analogous expression for the model without disorder. Also y(l) can be evaluated for the infinite system. It is given by (2.12) after we have made the same substitution E+E+iT. In figs. 1 and 2 both the IPR and the empirical y(l) are shown for lattice sizes 30 x 30, with r = 2.0, 2.65 (r,) and 7.6, and 9 x 10 x 10, with r = 3.5, 4.81 (r,) and 10.8. The difference between the empirical values of y(l) and the theoretical predictions was found to be small, as should be the case since y(l) is self-averaging. The existence of exponentially localized eigenfunctions for large positive and negative energies is seen clearly -in agreement with a result of Friihlich and Spencer”). For not too large r, the states in the center of the band have relatively small IPR values and are delocalized. For systems of the size 30 x 30, logarithmic localization at small distances (expected to be present in two dimensions) can hardly be distinguished from delocalization. A careful but expensive scaling analysis shows, however, that the average IPR in the plateau increases with the two-dimensional system size. This confirms the crossover to exponential localization at larger distances. It is seen in figs. la and 2a that the points where y(l) goes through zero coincide remarkably well with the regions where the IPR becomes appreciably different from zero-thus supporting the validity of the y-criterion. We note that the high IPR peak at number 387 in fig. 2a does not correspond to a
140
J. CANISIUS et al.
b 0.9
W h
O’S
ti
0.E
/ 9'0
r
LIdI
‘3 O’i
I
h'0
i’
/
2’0
!
I
I
CRITERION FOR ANDERSON LOCALIZATION
tl 0’1
I
0‘9
0.s
---_
ww O’b
tl3
O’F
0’2
0’1
I
c:
141
142
J. CANISIUS et al.
--
“..\
CRITERION FOR ANDERSON LOCALIZATION
i-
/
143
J. CANISIUS
144
et al.
Fig. 3. Mobility edge trajectories predicted by the y-criterion for the square (V = 2) and simple cubic (v = 3) lattices with a Cauchy distribution of the site energies (Lloyd model).
localized
eigenvector, but to a “resonance”: a state which is strongly peaked at one site and delocalized elsewhere. Due to the finite size of the sample it gives rise to a rather high IPR value, which is reduced to zero as the size of the system goes to infinity. In fig. 3 we have indicated the trajectories of the mobility edges EC, defined by y”)(E,) = 0, as a function of IY The critical r-values are r, = 2.657 for the square and r, = 4.813 for the simple cubic lattices. We think that the respective values 1.64 and 3.48 as estimated by Liciardello and Economou”) are somewhat too low. For r>r, extended eigenstates do not exist, as is clearly brought out by figs. lc and 2c. Note that, compared has changed rather dramatically. Recently exponentially and
Johnston localized,
independent
+2(v - 1). These
to figs. la and 2a, the IPR
and Kunz14) claimed that: (a) all eigenstates are whatever r, and (b) the localization length is constant
of the conclusions,
dimension
v for
energies
which were severely
between
criticized
-2(~
- 1) and
by Thouless”),
are
not directly supported by the IPRs of figs. la and 2a. To check them more carefully we made the Ansatz a(n) = exp{- y(E, T)lnl} in (3.1) with y(E, r) given by eqs. (3.16) and (3.17) of Johnston and Kunzi4), and calculated the corresponding IPR exactly for the eigenvalues E that were found by our diagonalization procedure. Though the resulting curve, labelled by JK in figs. 1 and 2, is consistent with both the existence and the width of our IPRits actual location does not agree that well. Moreover, we remind “plateau”, the reader that the IPR-plateau of fig. 2a, which is determined numerically, decreases as the inverse system size. We are inclined to interpret this as being Fig. 4. IPR and y(‘) for a 2~ 4.50 strip two sites across, with free boundary strip and Cauchy distribution (r = 2) for the site energies. Fig. 5. As fig. 4, but now for a 3 X 300 strip with periodic boundary Note that the eigenstates are not localized as well as those in fig. 4.
conditions
conditions
along the
in all directions.
CRITERION FOR ANDERSON LOCALIZATION
ti w w O‘h
O’S
”
n.F
‘-u
tr
3
0.2
0‘1
”
0’8
‘f
f m
5
145
146
J. CANISIUS
et al.
caused by the absence of exponential localization inside the plateau. As a further check, in particular of Thouless’ arguments, we have plotted in fig. 4 and IPR and y(‘) for a quasi one-dimensional system of size 2 x 450 with r = 2.0 and free boundary conditions along the strip. As expected, we find localized states only. In fig. 5 analogous results are shown for a 3 x 300 system with T = 2 and periodic boundary conditions in all directions. The states in the center of the band are seen to be localized not as well as those in fig. 4. The fact that y(l) becomes negative was anticipated in section 2. It shows that the -y-criterion cannot be valid for quasi one-dimensional systems. Since it holds in a case of infinite dimensions, a quasi one-dimensional system has a dimensionality which, in a sense, is too low for the criterion to be valid. More precisely, it is below the lower critical dimensionality, which is two. 3.2. Tight-binding
model with binary distribution
In fig. 6 we present y(l) and IPR for a 30 x 30 tight-binding system with a = &l. Contrary to previous expectations (Kirkpatrick and Eggarte?‘)) the IPR does not signal any localized state. The y-criterion predicts this type of state only near the band edge. Of course, it is possible that localized states occur only for energies so close to the band edges that they do not show up in our finite sample since the density of states falls off exponentially fast as one approaches the band edge (Lifshitz**)). Although such an explanation in terms of finite-size effects cannot be excluded, it seems that the y-criterion overestimates this region. A more detailed analysis by means of a perturbation expansion in E shows that in first approximation y is positive at the borders of the new spectrum, whereas in second approximation it is negative at ?2v, the edges of the original spectrum. Increasing the accuracy is a delicate matter, however. For a three-dimensional 9 X 10 X 10 sample with the same probability distribution as in fig. 6 even more striking results have been obtained: there is no localization whatsoever in fig, 7. Percolation effects (Canisius and van Hemmen23), Canisius et al.“)) may become important as the site energies increase in absolute value. A numerical simulation has been performed for a 30 x 30 system where &(n) = +4 with probability p = 0.3~~~ and I = -4 with probability q = 0.7>p,. As usual, p, is the percolation threshold, which is about 0.59 for site percolation in two dimensions. Since 2~ = 4 is the maximal value+ of the kinetic energy associated t
In
N x N
mathematicalterms, one matrix
(Wilkinsonz5)).
(aij)
must
lie
may apply the Gershgorin inside
the
circles
of
center
circle theorem: a;; and
The eigenvalues
radius
of an
IT?,+I /a;,(, 1 5 i 5 N
CRITERION
FOR ANDERSON
LOCALIZATION
147
with the hopping terms, the eigenstates with positive eigenvalues are expected to be localized on finite clusters with E = +4 whereas the eigenstates corresponding to negative eigenvalues may be spread out on the “infinite” percolating cluster of sites with E = -4. Fig. 8 confirms this intuitive picture: the states with positive energies are well-localized whereas the negative energy states have a rather small IPR and are (mainly) delocalized. The y(l) is seen to pass through zero very close to E = 0. The remaining error (y”‘(E) = 0 for E = -0.9) is explained by the extremely low density of states in the neighbourhood of E = 0; the eigenvalues closest to zero are -0.42 and 1.80 in this system. Nevertheless, the overall agreement is quite remarkable since, unlike in the random mass model (subsection 3.4) there is no a priori reason that forces As in Y (‘) to vanish at ES 0. This gives further support to the y-criterion. previous cases, y(l) is positive at the left band edge where the corresponding states are delocalized. 3.3. The Anderson model In the original Anderson (1958) model the a are independent, identically distributed random variables with a uniform distribution between -W/2 and + W/2. In fig. 9 we present results for a 30 x 30 system with W = 9. The same provisos as in the Lloyd case also apply here. Unfortunately, the results are somewhat hard to interpret. Although the IPR is quite peaked, we think that the majority of the eigenstates is not localized except for those close to the band edge. We, therefore, expect that in two dimensions the critical value WC slightly exceeds 9. The fact that a clear mobility edge is absent is somewhat puzzling. We suggest two explanations. First, the mobility edges may have a distance less than 2 from the band edges at a( W/2+ 2~). In these regions the density of states is so small (Lifshitz”)) that the eigenstates did not show up in our finite samples. Second, the mobility edges may move very fast to the center as W approaches WC so that no clear distinction between exponentially and nonexponentially localized states is possible. From the definition of y(l), eq. (2.7), one obtains an expansion in W-‘,
y”‘(E)=lnW-(ln2+1+C,)+2
(
$
>
*+$+fT(w_“).
Accordingly, the y-criterion predicts W = 2[ 1 - Y exp{-2(1+ C,)}] exp(1 + C,), so that for v = 2 we have WC= 17.0, which seems to be quite large, as recent estimates vary between 7 and 11 (Thouless’*); Liciardello and Economoulg)). In the same approximation as (3.3) the Ziman criterion (2.14) gives WC=
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J. CANISIUS
et al.
small values of the IPR and the complete
absence
of localized
e(n) = +l with probability p = 0.30 and E(R) = -1 with do not. Note the IPR scale and the absence of localized
Fig. 8. As fig. 6, for a 30 x 30 system with c(n) = k4 andp = 0.30. The lower horizontal axis is used to indicate the energy values. Note that only the sites n with e(n) = -4 percolate. The mobility edge has been indicated by an arrow. It is near to the location where y(‘) becomes positive.
Fig. 7. As fig. 6, for a 9 x 10 x 10 system with p = 0.25. Note the extremely states, in spite of the disorder.
Fig. 6. IPR and -y(‘)for a 30 x 30 tight-binding system with a binary distribution: probability q = 0.70. The sites n with c(n) = - 1 percolate but their counterparts states.
Fig. 8.
I5
OSb
flO@
OSE
OOE
OS2
002
OS1
01
OS
c
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FOR ANDERSON
LOCALIZATION
151
2 exp(1 + In z) = 22, which is even larger; cf. (2.15). Anderson himself originally estimated WCby 28. We remind the reader that in two dimensions all states are localized exponentially. WC denotes the transition from logarithmic to exponential localization at small distances for states in the center of the band. In fig. 10 another 9 x 10 x 10 case is shown with W = 17.25. Here the situations is similar to the one we just discussed. The critical value obtained from the y-criterion and (3.3) is WC= 28.6. Again, this is quite large compared to other recent estimates which give WC= 16.5 &OS (Mackinnon and Krame?)) and 14.5 < WC< 22 (Thouless”); Liciardello and Economou’g)), but smaller than Ziman’s”) estimate WC-32.4 and Anderson’s’) value WC-62. It is quite well possible that the y-criterion overestimates WC. As an explanation we suggest that we are already in the domain of large disorder. 3.4. Harmonic crystals with masses _tl Though negative masses may seem somewhat academic, they show up naturally in the study of spin waves in the quantum Heisenberg-Mattis model (Canisius and van Hemmen”“)). The problem of finding the low-lying elementary excitations of this model may be reduced to the eigenvalue problem (-A)x
=
wux,
(3.4)
where U = diag(m(n)) is a diagonal mass matrix with m(n) = +l. Let p be the probability of m(n) being +l, and 4 = 1 - p. If 5 = -w, the characteristic function for the problem (3.4) is given by n,(t)
= N-‘Trlog[(Ut
- A)/(-A)]
= -Y(~)(W) + i7rH3(o).
(3.5)
Expression (3.5) may be compared with (2.4). Note that ~‘~‘(0)= 0. In one dimension one expects, on the basis of the exponential growth phenomenon (Ishi?‘), YoshiokaB)), a pure point spectrum with well-localized eigenvectors. This is indeed what one finds [for details, see van Hemmenzg), Canisius and van Hemmenz3)] and also the y-criterion predicts localization for all eigenvectors. In two and three dimensions the results depend on whether p and q are above or below the site percolation threshold p, = 0.591 and 0.309,
Fig. 9. IPR and y(l) for the Anderson model with system size 30 X 30, W = 9.0, and periodic boundary conditions. The lower axis is used to indicate the energy values. The system seems near the Anderson transition. Fig. 10. As fig. 9, for a 9 X 10 x 10 system with W =
152
J. CANISIUS
et al.
CRITERION
FOR ANDERSON
LOCALIZATION
1.53
respectively. Localized states appear for w > 0 and p p, (or conversely), so that there is only one infinite cluster, one finds a mobility edge of o = 0. This is clearly brought out by figs. 11 and 12, and confirmed by the y-criterion.
4. Summary A mobility edge separates regions of delocalized (or logarithmically localized at small distances) and exponentially localized eigenstates of an infinite system. Usually the inverse participation ratio (IPR) is a suitable means of identifying a mobility edge. In this paper we have presented extensive numerical data concerning the IPR’s of the eigenstates in random tight-binding models and harmonic systems with “masses” 21. These results were obtained by an exact diagonalization of the appropriate matrices which, however, belong to finite samples. This restriction has to be constantly borne in mind. It was conjectured that mobility edges are (approximately) given by the zeroes of a slightly modified Herbert-Jones-Thouless expression which is commonly called the Thouless formula. The conjecture (the “y-criterion”) is exact for ordered systems or as the dimension Y+ 00, and reduces to the well-known Thouless formula if the dimension equals one. It predicts an Anderson transition in random tight-binding models with a continuous distribution and the absence of such a transition in harmonic systems with random masses and in tight-binding electron models with a discrete (binary) probability distribution. If the critical dimensions are not influenced by possible corrections to the y-criterion, we find two as the lower and four as the upper critical dimension. We now indicate our main conclusions. a) In the two- and three-dimensional Lloyd model the y-criterion seems to
Fig. 11. IPR and y”) versus eigenfrequency label and eigenfrequency o(lower horizontal axis) for a 25x 25 random mass system where m(n)= +l with probability 0.30 and m(n)= -1 with probability 0.70. The negative masses percolate and we have a mobility edge at o = 0, where y@) vanishes. Fig. 12. As fig. 11, for a 8 X 8 X 8 system with p = 0.25. Here too the negative masses percolate the positive masses do not.
and
154
identify
J. CANISIUS
et al.
the mobility edges rather well. Results of Johnston and Kunz14) their complete absence are only marginally supported. However, the -y-criterion is not valid for quasi one-dimensional systems, where all eigenstates are localized exponentially. b) In tight-binding systems with binary distributions a mobility edge at E = 0 is found if the probability distribution is chosen suitably and the strength of the random-site energies exceeds a critical value. Both above and below this critical value, the y-criterion predicts the presence or absence of a mobility edge correctly. Only for eigenstates with either very high or very low energies, at the border of the spectrum, the criterion always predicts that the states be localized whereas the IPR’s may indicate the opposite behaviour. This might be a finite-size effect, however. We will return to it shortly. c) The numerical data for the Anderson model do not show clear-cut mobility edges, but the y-criterion does on the basis of the very same data, i.e. the empirical distribution function of the eigenvalues. The criterion predicts WC/V= 17.0 in two and WC/V = 28.6 in three dimensions. Presumably these values are slightly too high but for getting better results much larger systems would have to investigated. At the moment these are out of question for an exact diagonahzation. d) In harmonic crystals with masses +l there exists a mobility edge at frequency w = 0 if one species (+l or -1) percolates and the other (-1 or +l) does not. This behaviour is reproduced by the -y-criterion as is the absence of a mobility edge in three dimensional systems where the masses + 1 and - 1 occur with equal probability and, thus, both percolate. For eigenfrequencies close to the band edges the y-criterion again predicts localized states whereas the IPR does not. We would like to analyze the behaviour of y near the band edges a bit further. As we already observed in subsection 3.2, expanding y in powers of the randomness a la (2.18) one finds in first order that y is positive at the new band edges-at least for 1/2 3, when the integrals converge. In second order one finds, again for v z 3, that y is negative at the band edge of the ordered system. Thus the y-criterion would predict a mobility edge inside the spectrum. However, because of the low density of states near the band edges (Lifshitz22)), its actual location in a numerical simulation may strongly depend on the system size. Indeed, this effect is much less pronounced in the three-dimensional tight-binding model with c(n) = 51 where the statistics is better than in the other systems. Apart from this the y-criterion seems to predict the localization/delocalization behaviour rather well and we hope that the present work will stimulate further research to provide a simple but accurate criterion for large disorder also. concerning
CRITERION
FOR
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LOCALIZATION
155
Acknowledgments Th.M. Nieuwenhulzen wishes to thank the members of the Subproject A2 of the Sonderforschungsbereich 123 at the University of Heidelberg for their hospitality during his stays in Heidelberg where most of this work was done. J.L. van Hemmen gratefully acknowledges two short visits to the Institute for Theoretical Physics at the University of Utrecht. Part of this work (J.C.) was made possible by the Deutsche Forschungsgemeinschaft (SFB 123) and (Th.M.N.) by the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is financially supported by the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Z.W.O.)“.
Note added in proof
After the completion of this manuscript a paper of MacKinnor?) came to our attention. Though using a completely different method, he fully confirms our results on the Lloyd model.
References 1) 2) 3) 4) 5) 6)
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
P.W. Anderson, Phys. Rev. 109 (1958) 1492. R. Camona, Duke Math. J. 49 (1982) 191. D.C. Herbert and R. Jones, J. Phys. C4 (1971) 1145. D.J. Thouless, J. Phys. C5 (1972) 77. F.J. Dyson, Phys. Rev. 92 (1953) 1331. Th.M. Nieuwenhuizen, Physica 125A (1984) 197; Phys. Lett. 103A (1984) 333; in: Localization, Interaction and Transport Phenomena in Impure Metals, B. Kramer, ed. (Springer, Heidelberg, 1984) and in the supplement volume, B. Kramer, ed. (P.T.B., Braunschweig, 1984). P. Lloyd, J. Phys. C2 (1%9) 1717. Th.M. Nieuwenhuizen, Physica 12OA (1983) 468. Ja. Gol’dSeid, Soviet Math. Dokl. 22 (1980) 670. R. Johnston and H. Kunz, J. Phys. Cl6 (1983) 3895. Anderson Localization, Y. Nagaoka and H. Fukuyama, eds. (Springer, Heidelberg, 1982). D.J. Thouless, in: Ill-Condensed Matter (Les Houches, 1978) R. Balian, R. Maynard and G. Toulouse, eds. (North-Holland, Amsterdam, 1979) pp. l-62. E.N. Economou and M.H. Cohen, Phys. Rev. B 5 (1972) 2931. R. Johnston and H. Kunz, J. Phys. Cl6 (1983) 4565. E.H. Lieb and D.C. Mattis, Mathematical Physics in One Dimension (Academic Press, New York, 1966), chap. 2. J.L. van Hemmen, J. Phys. A: Math. Gen. 15 (1982) 3891. J.M. Ziman, J. Phys. C2 (1%9) 1230. J. Frijhlich and T. Spencer, Commun. Math. Phys. 88 (1983) 151.
156
J. CANISIUS
et al.
19) 20) 21) 22) 23)
D.C. Licciardello and E.N. Economou, Phys. Rev. B 11 (1975) 3697. D.J. Thouless, J. Phys. Cl6 (1983) L 929. S. Kirkpatrick and T.P. Eggarter, Phys. Rev. B 6 (1972) 3598. I.M. Lifshitz, Adv. Phys. 13 (1964) 483. Phys. Rev. Lett. 46 (1981) 1487, 47 (1981) 212; J. Canisius and J.L. van Hemmen, manuscript in preparation. 24) J. Canisius, J.L. van Hemmen and Th.M. Nieuwenhuizen, in: Localization, Interaction and Transport Phenomena in Impure Metals, B. Kramer, ed. (Springer, Heidelberg, 1984) and in the supplement volume, B. Kramer, ed. (P.T.B., Braunschweig, 1984). 25) 26) 27) 28) 29) 30)
J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford A. Mackinnon and B. Kramer, Z. Phys. BJ3 (1983) 1. K. Ishii, Progr. Theor. Phys. Suppl. 53 (1973) 77. Y. Yoshida, Proc. Jpn. Acad. (1973) 665. J.L. van Hemmen, Z. Phys. B40 (1980) 55. A. MacKinnon, J. Phys. C7 (1984) L 289.
Univ. Press, London,
1965).
A GENERALIZED THOULESS FORMULA AS A CRITERION FOR ANDERSON LOCALIZATION IN TWO- AND THREE-DIMENSIONAL SYSTEMS J. CANISIUS Sondetforschungsbereich
123 der Universitiit
Heidelberg,
6900 Heidelberg,
Fed. Rep. Germany
J.L. VAN HEMMEN* Imtitut fiir Physik der Universitiit
Maim,
6500 Maim,
Fed. Rep.
Germany
-I%. M. NIEUWENHUIZEN Instituut
voor Theoretische
Natuurkunde, Rijks-Universiteit Utrecht, The Netherlands
Utrecht,
Princetonplein
5, 3508 TA
Received 25 October 1984
A generalized Tbouless formula is proposed as an approximate criterion for Anderson localization in two- and three-dimensional systems. The criterion is exact for ordered systems and in the mean field limit of infinite dimensionality. For random systems it is expected that possible corrections are small. Ample numerical evidence is provided through an exact diagonalization of several large-size Hamiltonians. The predictions of the lltouless formula, whose evaluation only needs the eigenvalues, are critically checked against the inverse participation ratio (IPR) of the corresponding eigenvectors. Included are the Lloyd model (for sulliciently low disorder mobility edges cannot be excluded) and surprising, new results on the binary alloy problem. The criterion predicts four as the upper and two as the lower critical dimension.
1. Introduction
Ever since Anderson’s pioneering work’) on localization in disordered lattice systems a lot of effort has been devoted to obtaining precise localization criteria. In the one-electron approximation the question as to whether the eigenstates corresponding to a certain energy range are localized is of prime importance to many of the system’s properties; e.g. the conductivity. In the one-dimensional case a clear-cut answer is known [see, for example, Carmona’) *Now at the Sonderforschungsbereich Fed. Rep. Germany.
123 of the University of Heidelberg,
0378-4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
6900
Heidelberg,
J. CANISIUS
132
and references soon
quoted
as disorder
therein]:
is present,
et al.
all eigenstates
no matter
are exponentially
how small
localization length of the eigenstates can be given states only, through the Herbert-Jones-Thouless Jones3);
Thouless4)).
This
formula,
is rather
attractive
easier
to obtain
for harmonic
formula,
which
is commonly
in numerical
work
than the eigenstates
themselves.
Exact
(Dyson’);
Nieuwenhuizen6)),
as
The inverse
in terms of the density of formula (Herbert and
since
chains
localized
its magnitude.
called the
the Thouless
spectra
expressions
Kronig-Penney
binding electron models (Lloyd’); Nieuwenhuizer?). Also in quasi one-dimensional systems all eigenstates are localized (Gol’dSeid’); Johnson and Kunz”)) and a generalization
are much are known and
tight-
exponentially of the Thou-
less formula is easily obtained. In two and three dimensions the situation is less clear. It is widely believed that in two dimensions, if the degree of randomness W does not exceed a certain critical value, the states in the center of the band localized “logarithmically” and cross over into exponential localization at large distances, whereas the states
near the band
edge decay
exponentially
at small and large distances
alike (Nagaoka and Fukuyama”)). In three dimensions one expects a similar situation but now the states in the centre of the band are delocalized at all distances and the regions of localized and delocalized states are separated by mobility edges. In two dimensions a similar “mobility edge” is present between the regions where states are exponentially localized and where they are logarithmically localized at small distances. As the randomness is increased the mobility edges move towards the center of the band and at a critical WC they merge together, eliminating the region of non-exponentially ded states has been called
localized states. This disappearance of the extenthe Anderson transition. It may be well to point out
that in numerical calculations involving small two-dimensional systems it is hard to distinguish between exponential localization at large distances and logarithmic localization at small distances. In that case a careful (but rather expensive)
analysis
of the results
Anderson’) originally simple cubic lattice, E(n)+)+
considered
as a function a tight-binding
v c a(n +p> = J%(n) I!PlI= 1
)
of the system model
size is required.
of an electron
on a
(1.1)
where the I are independent, identically distributed random variables with a uniform distribution between -W/2 and + W/2, and estimated the critical value WC above which no extended states exist. For a review of subsequent developments, see Thouless”). In what follows we choose units such that v= 1.
CRITERION FOR ANDERSON LOCALIZATION
133
Using two assumptions Economou and Cohen13) derived a localization function which is defined on the real energy axis and exceeds one if the eigenstates are not exponentially localized whereas it is between zero and one if exponential localization holds. However, this function has a very complicated structure and even in the ordered case it has no explicit, simple form which allows direct verification of its predictions. Yet the idea of a localization function is rather appealing. In this paper we consider a generalization of the Thouless formula to dimensions higher than one. It is an approximate localization criterion which becomes exact in the ordered case. Its main practical merit is that it only depends on the eigenvalues, not the eigenvectors. The criterion seems to work quite well (certainly for not too large disorder) and we hope our work will stimulate further research in this direction. We first concentrate on the localization criterion itself in section 2 and present extensive numerical evidence in section 3. There one can also find some interesting new results on the existence of a mobility edge in tightbinding models with a binary distribution (the binary alloy problem). In addition we take the opportunity to scrutinize some recent arguments of Johnson and Kunz14) concerning the absence of a mobility edge in the two- and three-dimensional Lloyd model. Our numerical data (see subsection 3.1) raise serious doubt about their conclusions but do not rule them our completely. A summary may be found in section 4.
2. A generalized Thouless formula For every Y-dimensional, hypercubic,
lattice equation (1.1) may be written
(2.1)
Eu=(&+@)u,
where a is a vector with entries a(n), E a diagonal matrix with elements &(n) to be specified later, and @ the hopping matrix which connects each point n with its nearest neighbours n + p, i.e. @(n, n’) = XP 8n,,n+p. For a harmonic crystal with positive random masses (Lieb and Mattis”)) the eigenvalue equation takes a similar form,
(2.2)
&Mu=(@-2u)a=Au, where 5 = -o*, A is the discretized matrix with elements m(n).
Laplacian,
and M the diagonal
mass
134
J. CANISIUS
For complex 0,(E)
for tight-binding 0,(t)
the characteristic function
E and 5 we define
= N-‘Tr
log[(E - F - @)/(-A)]
electron
= N-‘Tr
et al.
models,
log[(M[
(2.3)
and
- A )/(-A )]
(2.4)
for the random mass problem. They are primitives of the traces of the Green functions [E - c - @I-’ and [Mt - A]-‘, respectively. Along the real axis one has the following decomposition into real and imaginary parts: Ol(E _’ 10) = y”‘(E) k irr(l - H,(E))
(2.5)
and
L&(6 = -cd’* i0) = y(*)(w*) & ir&(w*)
,
(2.6)
where H,(E) and H2(w2) represent the integrated density work H is nothing but the empirical distribution function
of states. In practical of the eigenvalues (E
or w*). Using a basis with respect to which the arguments of the logarithms in (2.3) and (2.4) are diagonal one easily verifies that the quantities y(j) are given by
y”‘(E) = N-r 5 ln(E - Ejl - C, = 1 dH,(x) i=l
1nlE - XI - C,
(2.7)
and N
nmijw2-wfl-CY=N-‘~ln 1=l
i=l
2
%-l I Wf
I
(2.8) As N + 00, the yci) are self-averaging depends
on the dimension
C, = N-‘Tr
ln(-A)
In the infinite-volume
(van Hemmen*6)).
The constant
C,, which
V, equals
. limit C, may be written
(2.9)
CRITERION FOR ANDERSON LOCALIZATION
135
(2.10)
and we have Z,(2) = 0,
Z,(4) = 1.166243,
Z,(6) = 1.673338.
If v = 1, eq. (2.7) just gives the Thouless formula. Before proceeding we check that, without disorder,
(2.11)
y(E) s 0 for E inside the
spectrum, and y(E) > 0 outside. In the ordered case c(n) = 0 (m(n) = m) and while #*‘(o’) = the infinite system y’(E) = Re I,(E) - Z,(2v), for Re Z,(2v - mo2) - Z,(2v). We first turn to the tight-binding case. The spectrum is the closed interval and (2.10) we find
[-2v,
+2v].
Carrying
out one of the integrations
in (2.7)
y”‘=l~...I~Re(arccosh(~~-~cosq,)
- arccosh
u-l v- 2
COS
qi
(2.12)
.
i=l
Note that the sums in the integrand of (2.12) contain (v - 1) terms. Because arccosh(-x) = arccosh(x) 2 in, the function -y”‘(E) is even in E. It, therefore, suffices to take E positive. If E > 2v, the integrand is strictly positive and #“(E)>O. On the other hand, if 06 E <2v, the integrand is negative and y”‘(E) ~0, while it vanishes if and only if E = 2v, i.e. at the band edge. The vibration
spectrum
[0,4v/m].
Using the previous
with
respect
of a regular
to 2vlm.
Here
harmonic
crystal
is the
closed
w*-interval
argument too,
one directly verifies that yc2)(w2) is even only at the border of the y(‘) vanishes
spectrum, is negative inside and positive outside. Since the points where y = 0 mark the energies or frequencies where the solutions of (2.1) and (2.2) change from delocalized plane waves into exponentially decreasing functions, we pose the following conjecture for random systems: “The mobility edges for a v-dimensional system with (weak) disorder are (approximately) given by the points where y vanishes. Moreover, inside the spectrum a negative y signals eigenstates which are not exponentially localized (at small distances), whereas a positive y signals exponential localization at all distances.” This we call the y-criterion. It is expected to hold for general v. In the next section we present numerical evidence in favour of it. The criterion is exact for
136
J. CANISIUS
ordered
systems
sionality.
For
certainly As
and,
as will be shown
random
systems
et al
shortly,
correction
in a limit
terms
are
of infinite
expected
dimen-
to be small,
for not too large disorder.
Thouless
v-dimensional
(1979)
already
equation
observed
for
the
one-dimensional
(2.7) is close to the expression
derived
case,
our
by Economou
and Cohenj3), -log The
angle
L(E) = (1nlE - I brackets
- Sl) - In(K).
in (2.13)
denote
(2.13)
an average
over
a(n),
S is the
CPA
self-energy, and K is the connectivity of the lattice. An eigenstate with energy E is (not) exponentially localized if the right-hand side of (2.13) is greater (less) than zero. This criterion is not that easy to handle. A much simpler version was proposed Z(E)
by Ziman”), (2.14)
= (In/E - c(n)l> - In(z),
where z is the number of nearest neighbours. If Z(E) is positive, states with energy E are localized. However, for binary distributions Z will also go to minus infinity in the region of localized states where y(‘) may remain positive; see subsection 3.2. The constants C, is (2.7) and (2.8) seem to play a role similar to In(K) and In(z). In fact, since In(K) equals 0.97 if v = 2 and 1.54 if v = 3, we have that In(K) < C, (in(z) However,
In(K)
vanishes
In(z) do not. Probably subtracting C,. Indeed, as defined
.
(2.15) for quasi one-dimensional
systems
whereas
C, and
the corresponding y would have to be defined without the numerical calculations in section 3 will show that y
by (2.7) may become
negative
whereas
all eigenstates
are known
to
be localized exponentially (Gol’dSeidg); Johnston and Kunz”)). Introducing Lyapunov exponents yi (1
(2.16)
y = d-’ c yj - C,. j=l Without C, this is the appropriate generalization strip of d sites across. Note that here the Ziman
of the Thouless formula criterion fails completely.
to a
CRITERION FOR ANDERSON LOCALIZATION
137
Not only is the problem of a precise localization criterion still open, the values of the critical dimensions have not been determined unambiguously either. Though there is agreement about the value two for the lower critical dimension of the Anderson model, the upper critical dimension is believed to four (Nagaoka and Fukuyama”)) or even six. In order to investigate this question we use the y-criterion and expand (2.3) in powers of the randomness E, R,(E) = N-‘Tr{log(E
- @) - (EG + I*
+ am
+ am
+ . . .)} , (2.17)
where G = [E - @I-’ is the Green function of the model without disorder. If necessary, we may absorb the average value of 8 into E and assume (E) = 0. Carrying out the averages in (2.17) we obtain y”‘(E) = Re N-’ c [ [log(E - @)I;; - ~(.Y’)(G,)*- am I - a((~~)- 3(c*)‘)(G,)” - :(E*>*[~(G~~)*(G’),+ 2 (G6Gji)*]) i +
a(d, k2)(E3)) *
(2.18)
The diagonal elements of G and G* do not depend on the site i and are in the limit N + 00 given by (2.19)
For it = 1 and E close to the band edges -+2v, the integral diverges if Y 5 2; the divergence is logarithmic for Y = 2. Thus v = 2 indeed shows up as a (lower) critical dimension for the first terms in (2.18). On the other hand, if IZ= 2, the integral diverges at E = r2v for Y S 4; the divergence is logarithmic for v = 4. Also the higher order terms in (2.18) have v = 4 as critical dimension. We therefore may conclude that according to the y-criterion Y = 4 is the upper critical dimension. In the limit of infinite dimensionality (V + m) only the first term in (2.17) and (2.18) survives. This is just the well-known result that mean-field theory is exact in infinite dimensions. The proper scaling of the eigenvalues is given by X(q) =
$
E(q) Y
=$ $ zJI-1
COS qi *
(2.20)
J. CANISIUS
138
the distribution For V+W, (2/&)exp{ - fx’}. Calculating
of y”‘(E)
x
et al.
is
Gaussian,
for scaled energies
with
density
p(x) =
J? = E/L& we obtain
for large v
y”‘(i?)= _/dx
p(x) InIB - XI -
f ln
v + B( V-‘) ,
(2.21)
which approaches minus infinity as v + 00. Accordingly the y-criterion predicts delocalized states, which are indeed present because of translational invariance. If we do not scale E, we find for large v
y”‘(E) = In E I
which is positive is, the y-criterion
I
+ 6’(Y’) ,
(2.22)
outside the spectrum -2v
the band edges *2v. Finally we note that for harmonic systems with random masses and springs the characteristic function 0, is still defined by (2.4) but -A is now replaced by a random force matrix. The expression for 0, in tight-binding electron models with random hopping terms @(n, n’) can be obtained from (2.3) by making the replacement
A (n, n’)+ @(n,n’) - 6, ,,,2 @(n, n
+ ,I) .
(2.23)
P
3. Numerical
results
Our procedure to exhibit localization and to find a mobility edge is straightforward in principle, although computationally somewhat expensive. Through an exacr diagonalization we find all the eigenvalues and eigenvectors, and examine
the inverse
participation
ratio (IPR), (3.1)
of each eigenvector a. Delocalized states are expected to have small IPR, of order N-’ for N sites, while localized states show larger IPR values. In fact, the IPR equals one (the maximum) if a is localized at one site. In general, the IPR is between zero and one. Unless we state the contrary, one may assume that a
CRITERION FOR ANDERSON LOCALIZATION
139
scaling analysis has shown that the IPR’s in an, apparently, delocalized domain behave as N-l. We have performed the diagonalization in one, two, and three dimensions with lattice sizes up to 900 sites, usually employing periodic boundary conditions. We used the iterative OR algorithm to guarantee numerical stability, knowing that the eigenvalues are not degenerate because of the randomness. For a given probability distribution we need not average over many samples since the localization/delocalization structure should occur with probability one. To display the results we arrange the eigenvalues in an ascending sequence and plot both the IPR and the corresponding y against the label k. 3.1. Lloyd model For tight-binding tribution,
P(E) = ;
models whose site energies
(2+ P-l
)
I
have a Cauchy
dis-
(3.2)
Lloyd’) showed that the average one-particle Green function can be evaluated explicitly by making the substitution E-FE + iT in the analogous expression for the model without disorder. Also y(l) can be evaluated for the infinite system. It is given by (2.12) after we have made the same substitution E+E+iT. In figs. 1 and 2 both the IPR and the empirical y(l) are shown for lattice sizes 30 x 30, with r = 2.0, 2.65 (r,) and 7.6, and 9 x 10 x 10, with r = 3.5, 4.81 (r,) and 10.8. The difference between the empirical values of y(l) and the theoretical predictions was found to be small, as should be the case since y(l) is self-averaging. The existence of exponentially localized eigenfunctions for large positive and negative energies is seen clearly -in agreement with a result of Friihlich and Spencer”). For not too large r, the states in the center of the band have relatively small IPR values and are delocalized. For systems of the size 30 x 30, logarithmic localization at small distances (expected to be present in two dimensions) can hardly be distinguished from delocalization. A careful but expensive scaling analysis shows, however, that the average IPR in the plateau increases with the two-dimensional system size. This confirms the crossover to exponential localization at larger distances. It is seen in figs. la and 2a that the points where y(l) goes through zero coincide remarkably well with the regions where the IPR becomes appreciably different from zero-thus supporting the validity of the y-criterion. We note that the high IPR peak at number 387 in fig. 2a does not correspond to a
140
J. CANISIUS et al.
b 0.9
W h
O’S
ti
0.E
/ 9'0
r
LIdI
‘3 O’i
I
h'0
i’
/
2’0
!
I
I
CRITERION FOR ANDERSON LOCALIZATION
tl 0’1
I
0‘9
0.s
---_
ww O’b
tl3
O’F
0’2
0’1
I
c:
141
142
J. CANISIUS et al.
--
“..\
CRITERION FOR ANDERSON LOCALIZATION
i-
/
143
J. CANISIUS
144
et al.
Fig. 3. Mobility edge trajectories predicted by the y-criterion for the square (V = 2) and simple cubic (v = 3) lattices with a Cauchy distribution of the site energies (Lloyd model).
localized
eigenvector, but to a “resonance”: a state which is strongly peaked at one site and delocalized elsewhere. Due to the finite size of the sample it gives rise to a rather high IPR value, which is reduced to zero as the size of the system goes to infinity. In fig. 3 we have indicated the trajectories of the mobility edges EC, defined by y”)(E,) = 0, as a function of IY The critical r-values are r, = 2.657 for the square and r, = 4.813 for the simple cubic lattices. We think that the respective values 1.64 and 3.48 as estimated by Liciardello and Economou”) are somewhat too low. For r>r, extended eigenstates do not exist, as is clearly brought out by figs. lc and 2c. Note that, compared has changed rather dramatically. Recently exponentially and
Johnston localized,
independent
+2(v - 1). These
to figs. la and 2a, the IPR
and Kunz14) claimed that: (a) all eigenstates are whatever r, and (b) the localization length is constant
of the conclusions,
dimension
v for
energies
which were severely
between
criticized
-2(~
- 1) and
by Thouless”),
are
not directly supported by the IPRs of figs. la and 2a. To check them more carefully we made the Ansatz a(n) = exp{- y(E, T)lnl} in (3.1) with y(E, r) given by eqs. (3.16) and (3.17) of Johnston and Kunzi4), and calculated the corresponding IPR exactly for the eigenvalues E that were found by our diagonalization procedure. Though the resulting curve, labelled by JK in figs. 1 and 2, is consistent with both the existence and the width of our IPRits actual location does not agree that well. Moreover, we remind “plateau”, the reader that the IPR-plateau of fig. 2a, which is determined numerically, decreases as the inverse system size. We are inclined to interpret this as being Fig. 4. IPR and y(‘) for a 2~ 4.50 strip two sites across, with free boundary strip and Cauchy distribution (r = 2) for the site energies. Fig. 5. As fig. 4, but now for a 3 X 300 strip with periodic boundary Note that the eigenstates are not localized as well as those in fig. 4.
conditions
conditions
along the
in all directions.
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O’S
”
n.F
‘-u
tr
3
0.2
0‘1
”
0’8
‘f
f m
5
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caused by the absence of exponential localization inside the plateau. As a further check, in particular of Thouless’ arguments, we have plotted in fig. 4 and IPR and y(‘) for a quasi one-dimensional system of size 2 x 450 with r = 2.0 and free boundary conditions along the strip. As expected, we find localized states only. In fig. 5 analogous results are shown for a 3 x 300 system with T = 2 and periodic boundary conditions in all directions. The states in the center of the band are seen to be localized not as well as those in fig. 4. The fact that y(l) becomes negative was anticipated in section 2. It shows that the -y-criterion cannot be valid for quasi one-dimensional systems. Since it holds in a case of infinite dimensions, a quasi one-dimensional system has a dimensionality which, in a sense, is too low for the criterion to be valid. More precisely, it is below the lower critical dimensionality, which is two. 3.2. Tight-binding
model with binary distribution
In fig. 6 we present y(l) and IPR for a 30 x 30 tight-binding system with a = &l. Contrary to previous expectations (Kirkpatrick and Eggarte?‘)) the IPR does not signal any localized state. The y-criterion predicts this type of state only near the band edge. Of course, it is possible that localized states occur only for energies so close to the band edges that they do not show up in our finite sample since the density of states falls off exponentially fast as one approaches the band edge (Lifshitz**)). Although such an explanation in terms of finite-size effects cannot be excluded, it seems that the y-criterion overestimates this region. A more detailed analysis by means of a perturbation expansion in E shows that in first approximation y is positive at the borders of the new spectrum, whereas in second approximation it is negative at ?2v, the edges of the original spectrum. Increasing the accuracy is a delicate matter, however. For a three-dimensional 9 X 10 X 10 sample with the same probability distribution as in fig. 6 even more striking results have been obtained: there is no localization whatsoever in fig, 7. Percolation effects (Canisius and van Hemmen23), Canisius et al.“)) may become important as the site energies increase in absolute value. A numerical simulation has been performed for a 30 x 30 system where &(n) = +4 with probability p = 0.3~~~ and I = -4 with probability q = 0.7>p,. As usual, p, is the percolation threshold, which is about 0.59 for site percolation in two dimensions. Since 2~ = 4 is the maximal value+ of the kinetic energy associated t
In
N x N
mathematicalterms, one matrix
(Wilkinsonz5)).
(aij)
must
lie
may apply the Gershgorin inside
the
circles
of
center
circle theorem: a;; and
The eigenvalues
radius
of an
IT?,+I /a;,(, 1 5 i 5 N
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with the hopping terms, the eigenstates with positive eigenvalues are expected to be localized on finite clusters with E = +4 whereas the eigenstates corresponding to negative eigenvalues may be spread out on the “infinite” percolating cluster of sites with E = -4. Fig. 8 confirms this intuitive picture: the states with positive energies are well-localized whereas the negative energy states have a rather small IPR and are (mainly) delocalized. The y(l) is seen to pass through zero very close to E = 0. The remaining error (y”‘(E) = 0 for E = -0.9) is explained by the extremely low density of states in the neighbourhood of E = 0; the eigenvalues closest to zero are -0.42 and 1.80 in this system. Nevertheless, the overall agreement is quite remarkable since, unlike in the random mass model (subsection 3.4) there is no a priori reason that forces As in Y (‘) to vanish at ES 0. This gives further support to the y-criterion. previous cases, y(l) is positive at the left band edge where the corresponding states are delocalized. 3.3. The Anderson model In the original Anderson (1958) model the a are independent, identically distributed random variables with a uniform distribution between -W/2 and + W/2. In fig. 9 we present results for a 30 x 30 system with W = 9. The same provisos as in the Lloyd case also apply here. Unfortunately, the results are somewhat hard to interpret. Although the IPR is quite peaked, we think that the majority of the eigenstates is not localized except for those close to the band edge. We, therefore, expect that in two dimensions the critical value WC slightly exceeds 9. The fact that a clear mobility edge is absent is somewhat puzzling. We suggest two explanations. First, the mobility edges may have a distance less than 2 from the band edges at a( W/2+ 2~). In these regions the density of states is so small (Lifshitz”)) that the eigenstates did not show up in our finite samples. Second, the mobility edges may move very fast to the center as W approaches WC so that no clear distinction between exponentially and nonexponentially localized states is possible. From the definition of y(l), eq. (2.7), one obtains an expansion in W-‘,
y”‘(E)=lnW-(ln2+1+C,)+2
(
$
>
*+$+fT(w_“).
Accordingly, the y-criterion predicts W = 2[ 1 - Y exp{-2(1+ C,)}] exp(1 + C,), so that for v = 2 we have WC= 17.0, which seems to be quite large, as recent estimates vary between 7 and 11 (Thouless’*); Liciardello and Economoulg)). In the same approximation as (3.3) the Ziman criterion (2.14) gives WC=
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small values of the IPR and the complete
absence
of localized
e(n) = +l with probability p = 0.30 and E(R) = -1 with do not. Note the IPR scale and the absence of localized
Fig. 8. As fig. 6, for a 30 x 30 system with c(n) = k4 andp = 0.30. The lower horizontal axis is used to indicate the energy values. Note that only the sites n with e(n) = -4 percolate. The mobility edge has been indicated by an arrow. It is near to the location where y(‘) becomes positive.
Fig. 7. As fig. 6, for a 9 x 10 x 10 system with p = 0.25. Note the extremely states, in spite of the disorder.
Fig. 6. IPR and -y(‘)for a 30 x 30 tight-binding system with a binary distribution: probability q = 0.70. The sites n with c(n) = - 1 percolate but their counterparts states.
Fig. 8.
I5
OSb
flO@
OSE
OOE
OS2
002
OS1
01
OS
c
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2 exp(1 + In z) = 22, which is even larger; cf. (2.15). Anderson himself originally estimated WCby 28. We remind the reader that in two dimensions all states are localized exponentially. WC denotes the transition from logarithmic to exponential localization at small distances for states in the center of the band. In fig. 10 another 9 x 10 x 10 case is shown with W = 17.25. Here the situations is similar to the one we just discussed. The critical value obtained from the y-criterion and (3.3) is WC= 28.6. Again, this is quite large compared to other recent estimates which give WC= 16.5 &OS (Mackinnon and Krame?)) and 14.5 < WC< 22 (Thouless”); Liciardello and Economou’g)), but smaller than Ziman’s”) estimate WC-32.4 and Anderson’s’) value WC-62. It is quite well possible that the y-criterion overestimates WC. As an explanation we suggest that we are already in the domain of large disorder. 3.4. Harmonic crystals with masses _tl Though negative masses may seem somewhat academic, they show up naturally in the study of spin waves in the quantum Heisenberg-Mattis model (Canisius and van Hemmen”“)). The problem of finding the low-lying elementary excitations of this model may be reduced to the eigenvalue problem (-A)x
=
wux,
(3.4)
where U = diag(m(n)) is a diagonal mass matrix with m(n) = +l. Let p be the probability of m(n) being +l, and 4 = 1 - p. If 5 = -w, the characteristic function for the problem (3.4) is given by n,(t)
= N-‘Trlog[(Ut
- A)/(-A)]
= -Y(~)(W) + i7rH3(o).
(3.5)
Expression (3.5) may be compared with (2.4). Note that ~‘~‘(0)= 0. In one dimension one expects, on the basis of the exponential growth phenomenon (Ishi?‘), YoshiokaB)), a pure point spectrum with well-localized eigenvectors. This is indeed what one finds [for details, see van Hemmenzg), Canisius and van Hemmenz3)] and also the y-criterion predicts localization for all eigenvectors. In two and three dimensions the results depend on whether p and q are above or below the site percolation threshold p, = 0.591 and 0.309,
Fig. 9. IPR and y(l) for the Anderson model with system size 30 X 30, W = 9.0, and periodic boundary conditions. The lower axis is used to indicate the energy values. The system seems near the Anderson transition. Fig. 10. As fig. 9, for a 9 X 10 x 10 system with W =
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respectively. Localized states appear for w > 0 and p
4. Summary A mobility edge separates regions of delocalized (or logarithmically localized at small distances) and exponentially localized eigenstates of an infinite system. Usually the inverse participation ratio (IPR) is a suitable means of identifying a mobility edge. In this paper we have presented extensive numerical data concerning the IPR’s of the eigenstates in random tight-binding models and harmonic systems with “masses” 21. These results were obtained by an exact diagonalization of the appropriate matrices which, however, belong to finite samples. This restriction has to be constantly borne in mind. It was conjectured that mobility edges are (approximately) given by the zeroes of a slightly modified Herbert-Jones-Thouless expression which is commonly called the Thouless formula. The conjecture (the “y-criterion”) is exact for ordered systems or as the dimension Y+ 00, and reduces to the well-known Thouless formula if the dimension equals one. It predicts an Anderson transition in random tight-binding models with a continuous distribution and the absence of such a transition in harmonic systems with random masses and in tight-binding electron models with a discrete (binary) probability distribution. If the critical dimensions are not influenced by possible corrections to the y-criterion, we find two as the lower and four as the upper critical dimension. We now indicate our main conclusions. a) In the two- and three-dimensional Lloyd model the y-criterion seems to
Fig. 11. IPR and y”) versus eigenfrequency label and eigenfrequency o(lower horizontal axis) for a 25x 25 random mass system where m(n)= +l with probability 0.30 and m(n)= -1 with probability 0.70. The negative masses percolate and we have a mobility edge at o = 0, where y@) vanishes. Fig. 12. As fig. 11, for a 8 X 8 X 8 system with p = 0.25. Here too the negative masses percolate the positive masses do not.
and
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the mobility edges rather well. Results of Johnston and Kunz14) their complete absence are only marginally supported. However, the -y-criterion is not valid for quasi one-dimensional systems, where all eigenstates are localized exponentially. b) In tight-binding systems with binary distributions a mobility edge at E = 0 is found if the probability distribution is chosen suitably and the strength of the random-site energies exceeds a critical value. Both above and below this critical value, the y-criterion predicts the presence or absence of a mobility edge correctly. Only for eigenstates with either very high or very low energies, at the border of the spectrum, the criterion always predicts that the states be localized whereas the IPR’s may indicate the opposite behaviour. This might be a finite-size effect, however. We will return to it shortly. c) The numerical data for the Anderson model do not show clear-cut mobility edges, but the y-criterion does on the basis of the very same data, i.e. the empirical distribution function of the eigenvalues. The criterion predicts WC/V= 17.0 in two and WC/V = 28.6 in three dimensions. Presumably these values are slightly too high but for getting better results much larger systems would have to investigated. At the moment these are out of question for an exact diagonahzation. d) In harmonic crystals with masses +l there exists a mobility edge at frequency w = 0 if one species (+l or -1) percolates and the other (-1 or +l) does not. This behaviour is reproduced by the -y-criterion as is the absence of a mobility edge in three dimensional systems where the masses + 1 and - 1 occur with equal probability and, thus, both percolate. For eigenfrequencies close to the band edges the y-criterion again predicts localized states whereas the IPR does not. We would like to analyze the behaviour of y near the band edges a bit further. As we already observed in subsection 3.2, expanding y in powers of the randomness a la (2.18) one finds in first order that y is positive at the new band edges-at least for 1/2 3, when the integrals converge. In second order one finds, again for v z 3, that y is negative at the band edge of the ordered system. Thus the y-criterion would predict a mobility edge inside the spectrum. However, because of the low density of states near the band edges (Lifshitz22)), its actual location in a numerical simulation may strongly depend on the system size. Indeed, this effect is much less pronounced in the three-dimensional tight-binding model with c(n) = 51 where the statistics is better than in the other systems. Apart from this the y-criterion seems to predict the localization/delocalization behaviour rather well and we hope that the present work will stimulate further research to provide a simple but accurate criterion for large disorder also. concerning
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Acknowledgments Th.M. Nieuwenhulzen wishes to thank the members of the Subproject A2 of the Sonderforschungsbereich 123 at the University of Heidelberg for their hospitality during his stays in Heidelberg where most of this work was done. J.L. van Hemmen gratefully acknowledges two short visits to the Institute for Theoretical Physics at the University of Utrecht. Part of this work (J.C.) was made possible by the Deutsche Forschungsgemeinschaft (SFB 123) and (Th.M.N.) by the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is financially supported by the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Z.W.O.)“.
Note added in proof
After the completion of this manuscript a paper of MacKinnor?) came to our attention. Though using a completely different method, he fully confirms our results on the Lloyd model.
References 1) 2) 3) 4) 5) 6)
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
P.W. Anderson, Phys. Rev. 109 (1958) 1492. R. Camona, Duke Math. J. 49 (1982) 191. D.C. Herbert and R. Jones, J. Phys. C4 (1971) 1145. D.J. Thouless, J. Phys. C5 (1972) 77. F.J. Dyson, Phys. Rev. 92 (1953) 1331. Th.M. Nieuwenhuizen, Physica 125A (1984) 197; Phys. Lett. 103A (1984) 333; in: Localization, Interaction and Transport Phenomena in Impure Metals, B. Kramer, ed. (Springer, Heidelberg, 1984) and in the supplement volume, B. Kramer, ed. (P.T.B., Braunschweig, 1984). P. Lloyd, J. Phys. C2 (1%9) 1717. Th.M. Nieuwenhuizen, Physica 12OA (1983) 468. Ja. Gol’dSeid, Soviet Math. Dokl. 22 (1980) 670. R. Johnston and H. Kunz, J. Phys. Cl6 (1983) 3895. Anderson Localization, Y. Nagaoka and H. Fukuyama, eds. (Springer, Heidelberg, 1982). D.J. Thouless, in: Ill-Condensed Matter (Les Houches, 1978) R. Balian, R. Maynard and G. Toulouse, eds. (North-Holland, Amsterdam, 1979) pp. l-62. E.N. Economou and M.H. Cohen, Phys. Rev. B 5 (1972) 2931. R. Johnston and H. Kunz, J. Phys. Cl6 (1983) 4565. E.H. Lieb and D.C. Mattis, Mathematical Physics in One Dimension (Academic Press, New York, 1966), chap. 2. J.L. van Hemmen, J. Phys. A: Math. Gen. 15 (1982) 3891. J.M. Ziman, J. Phys. C2 (1%9) 1230. J. Frijhlich and T. Spencer, Commun. Math. Phys. 88 (1983) 151.
156
J. CANISIUS
et al.
19) 20) 21) 22) 23)
D.C. Licciardello and E.N. Economou, Phys. Rev. B 11 (1975) 3697. D.J. Thouless, J. Phys. Cl6 (1983) L 929. S. Kirkpatrick and T.P. Eggarter, Phys. Rev. B 6 (1972) 3598. I.M. Lifshitz, Adv. Phys. 13 (1964) 483. Phys. Rev. Lett. 46 (1981) 1487, 47 (1981) 212; J. Canisius and J.L. van Hemmen, manuscript in preparation. 24) J. Canisius, J.L. van Hemmen and Th.M. Nieuwenhuizen, in: Localization, Interaction and Transport Phenomena in Impure Metals, B. Kramer, ed. (Springer, Heidelberg, 1984) and in the supplement volume, B. Kramer, ed. (P.T.B., Braunschweig, 1984). 25) 26) 27) 28) 29) 30)
J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford A. Mackinnon and B. Kramer, Z. Phys. BJ3 (1983) 1. K. Ishii, Progr. Theor. Phys. Suppl. 53 (1973) 77. Y. Yoshida, Proc. Jpn. Acad. (1973) 665. J.L. van Hemmen, Z. Phys. B40 (1980) 55. A. MacKinnon, J. Phys. C7 (1984) L 289.
Univ. Press, London,
1965).