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Level curvature and metal-insulator transition in 3d Anderson model
K. Zyczkowski, L. Molinari, F. Izrailev
To cite this version:
K. Zyczkowski, L. Molinari, F. Izrailev. Level curvature and metal-insulator transition in 3d Anderson model. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1469-1477. �10.1051/jp1:1994201�. �jpa- 00247006�
J. Phys. I France 4 (1994) 1469-1477 OCTOBER 1994, PAGE 1469
Classification Pllysics Abstracts
71.31~ 71.55
Level curvature and metal-insulator transition in 3d Anderson
mortel
K. Zyczkowski (1), L. Molinari (~) and F. M. Izrailev (~)
(~ Uniwersytet Jagiellonski, Instytut Fizyki, uI.Reymonta 4, 31~-059 Krakôw, Poland (~) Dipartimento di Fisica and INFN, Via Celoria 16, 21~133 Milano, Italy
(~) Budker Institute of Nuclear Physics, 631~090 Novosibirsk, Russia
(Received 4 January 1994, revised 19 May1994, accepted 5 July 1994)
Abstract. The level curvature in the Anderson model on a cubic lattice is numerically
investigated as an indicator of the metallic-insulator transition. It is shown that trie
mean
curvature obeys a scaling law in the whole range of the disorder parameter. In the metallic
regime, the distribution of rescaled curvatures is found to be well described by a formula proposed
by Zakrzewski and Delande [1] for random matrices, implying a relation similar to that by
Thouless. In the localized regime the distribution of curvatures is approximated by a log-normal
distribution.
Introduction.
The numerical investigation of transport properties of disordered 3D systems is severely limited
by the problem of diagonalizing large matrices. This occurs for exarnple in the identification and the study of the metal-insulator transition, which shows in a structural change of the eigen-
functions from extended to localized. It is therefore desirable to devise appropriate indicators that capture the structure of eigenfunctions with the minimal computational effort.
One such quantity is level curvature, which measures the response of energy levels to variations of boundary conditions, and obviously relates to trie taris of eigenfunctions, responsible for con-
duction. This quantity was first proposed and investigated by Edwards and Thouless [2, 3].
The parametric dependence of boundary conditions can be introduced by replicating the cubic
sample in a periodic sequence, so that each eigenvalue becoInes a function of a Bloch phase.
An equivalent procedure is to connect two opposite faces of the cube and allow aInagnetic flux through the ring, involving a phase change of the wave function across trie cut that affects the
eigenvalues.
TO probe the bulk properties of eigenfunctions, it is possible to resort on first order perturba-
tion theory, by applying externat weak fields [4]. The response of levels to a uniform electric field provides the average position (ni in the lattice. This, and the response to a linear field,
1470 JOURNAL DE PHYSIQUE I N°10
provide the variance in space of the probability distribution of a single eigenvector: a measure of its localization [Si.
An interesting feature of the metallic-insulator transition is the change of the level spacing
statistics from the Wigner distribution, characteristic of the Gaussian orthogonal ensemble
(GOE), to the Poisson distribution. Sl~klovskii et al.
[GI investigated the Anderson transition by looking at a parameter j which weights the tait of the spacing distribution P(s), being
Gaussian or exponential, respectively for GOE (delocalized states, small disorder) and Poisson statistics (localized states, large disorder). In the large saInple size limit the quantity j, as a
function of the disorder parameter W, is expected to approach a step function centered on the critical value W~ of the Anderson transition. Even for cubes with side lengths L
= 10-16 the transition is evidenced by a modal structure in the superposition of the curves j(W, L). In
computing the tait of P(s), the central half of the energy spectrum was used.
In this paper we investigate numerically the properties of trie distribution of level curvatures m the 3D cubic Anderson model. Curvature of energy levels appears to be a good indicator for the metal-insulator transition, even with relatively small sizes of the sarnples. The mean curvature satisfies a scaling law in the saine parameter that describes the scaling of localiza- tion lengths, numerically investigated by MacKinnon and Kramer [7]. In theInetallic regime
a simple rescaling of curvatures allowed us to reproduce the universal curvature distribution
proposed by Zakrzewski and Delande iii for GOE. The form of the rescaling and the existence of a universal curve imply a relation which has the same content as the famous Thouless for-
mula. In the localized regime the distribution of curvatures is well described by a log-normal
distribution.
Level curvature.
The 3D Anderson model on a cubic lattice is described by an ensemble of tight binding Hamilto- nians, which are the sum of a kinetic term accounting for the hopping among nearest neighbour sites, and a random potential. Labelling a site with a triple of integers: n = (ni, n2>n3) the
Hamiltonian is
Ù
=
£ 1ni(ml + W £ VI n)luiInl (i)
(n,m) n
where in, mi are nearest neighbours, W > 0 is the disorder parameter and (V(n))n is a set of values of independent random variables with uniform distribution m i-1/2,1/2]. In the free situation (W = o), the eigenfunctions are plane waves with energies in a baud -6 < E < 6.
The metallic-insulator transition takes place at a critical value W~ m 16.5, where for (E( below
a mobility edge E~ the eigenfunctions are exponentially localized [7].
When deahng with finite samples of size L~, one has to specify boundary conditions for the
eigenfunctions ~fi(n). We have chose~l periodic conditions along the ~ and y directions, and introduced a flux parameter in the z direction, so that
il(L
+ Î,n2,'l3)" iÎ(Î,n2,'l3)
~fi(ni, L + 1,n3) # lk(ni,1,n3) (2)
iÎl'il,'l2, L + Î)
= e~~~$(nli'l2,1)
Introducing an ordering of sites through the integer a(n) = ni + (n2 1)L + (n3 -1)L~ and
defining ~fi~ = ~fi(n), the eigenvalue equation É~fil~) = Er~fi(~), r = i L~, cari be put in matrix
form, with the matrix Hab mcorporating the boundary conditions. The Hamiltonian matrix takes the form
H(çi)
=
H1°) + e~~V + e~~~V°~ (3)
N°10 LEVEL CURVATURE AND 3D ANDERSON TRANSITION 1471
10 < P~>
~ ~ ll.114
6
4 0.03
2
0 0.02
2
0.01 4
o-o
'3 '2 1 0 2 3 o-o 0.2 0.4 0.6 0.8 1,o 1.2 1.4x/2
§~
Fig. 1 Fig. 2
Fig. 1. Level motion in a sequence of eigenvalues, changing the phase from -~ to ~. Trie eigenvalues
are rescaled to have a unitary average spacing at çi = 0. Trie value of the disorder pararneter is W
= 10,
with extended states sensitive to the boundary.
Fig. 2. Mean kinetic energy (P~) versus trie phase parameter çi, for dilferent values of disorder.
From above: W
= 12, 16, 20, 24. Inspection of trie curves indicates that (P~) at çi
= ~/2 is, up to a
constant, a good measure of the çi-averaged value of the mean kinetic energy.
where H1°J
is a real symmetric L3 x L3 matrix, V is a real Inatrix with nonzero entries (b = i for a
= i...L~, b
= a + L3 L~, and V°~ is its transpose. The V Inatrices describe the couplings between the n3
" 1 and trie n3
= L faces of trie cube.
The eigenvalues of the Inatrix are functions of the externat parameter çi. In figure i we display the motion of a few levels in the metallic phase, where states are extended and sensitive to
boundary changes. Note the symmetry E(çi) = E(-çi). The dynarnics involves the definition for each eigenvalue of a Inomentum and a curvature
vr(1)
=
~(j'~
,
kr(1)
= ~~)jÎ'~. (4)
Perturbation theory around çi = 0 yields pr
= 0 and
~kr
"
~ lk~~~ (ni, n2,1)~fi("(ni, n2,L)+
2
ni,n2
~ ~lnl,n2Î~~~~~~~'~~'~~~~~~~~~'~~'~~ ~~~~~~~'~~'~~~~~~~~~'~~'~~~~~
j~)
~ç ~ç
s#r ~ ~
At çi = 0 the response of the energy level is therefore proportional to curvature.
It is worth to recall some recent results which will be of interest in the following discussion.
The introduction of a complex phase factor changes the symmetry of the random Hamiltonians from real symmetric to complex Herlnitian: Dupuis and Montambaux [8] have shown that the spectral properties neon the transition are functions of the scaling parameter (k~)~Hçi. Szafer and Altshuler [9], and Beenakker [10] have shown that the correlation of1nomenta
c(jj =
)
/~"jpjj')p(j
+ j')jdj' j6)
exhibits trie universal behaviour C(çi) m çi~~ for # larger than a critical value. A simple analytical expression for the whole range of çi has been recently proposed by Delande and
1472 JOURNAL DE PHYSIQUE I NO ID
Zakrzewski il Ii. In the above formula and in the following, (. .) denotes an average on disorder, and A is the local spacmg between eigenvalues. The quantity C(0) measures the average kinetic energy of the gas of levels, rescaled in order to have local spacing equal to one; for the Anderson Inodel it is plotted in figure 2. Simon and Altshuler [12, 13] bave trier shown that defining trie
rescaled flux pararneter 4l = çi@@, the rescaled eigenvalues er(4l)
= A~lEr(4l/@@) are
random functions with universal statistical properties. This strong stateInent is consistent with trie hypothesis of a universal distribution of curvatures that is discussed in the next section.
The perturbative formula (5) and certain assumptions led Thouless to the fundamental relation that in the metallic regime the mean curvature divided by trie level spac1~lg is proportional
to conductance, as give~l by Kubo's forInula. This relation bas been recently reco~lsidered by
Akkerma~ls and Mo~ltambaux [14], who restated Thouless' forInula with greater generality and showed its equivale~lce with the follow1~lg relation
@
= ~~aA(k~)~/~ I?i
in which a is a model depe~lde~lt constant and the overbar de~lotes an average over the flux çi of squared mome~lta.
Numerical results.
We constructed matrices represe~lting trie 3D Anderson mortel for the cube size varying from 4 to 12, with various values of the disorder paraIneter W and the phase çi and diagonalized
them numerically. The energy levels were then rescaled so that the average spacing at # = 0 is set to unity. We accordingly define tl~e normalized moInenta and curvatures P
= p/A and
K = kIA.
Curvatures in # = 0 are computed for each level by fitting a parabola to three energy values obtained for # = -à,0,à. The accuracy of computation was verified by comparing with the
curvature obtained for #
= -26,o,26. Tl~e optimal value of à varies from lo~l to 10~3 and depends on the pararneters W and L.
Tl~e density of states p(E) is displayed in figure 3 for W > W~, for whicl~ localization occurs
(Fig. 3b), and for W < W~ (Fig. 3a), for which states are extended. It is worth to note that
the mean spacing A
= 1/p is practically constant over an extended energy range: this will
< k >
~Î~Î~ ~
(hi
~
~~~ 02
006 0.2
11.06
o.15 o.15
0.04 0.04
Ù-1 o-1
~~~ 0.02 0.02
Ù-Ù Ù-Ù Ù-o Ù-Ù
10 5 0 5 10 10 5 0 5 10
E E
Fig. 3. Trie energy density p (fuit fine) aud trie mean absolute value of curvature (dotted fine) as
functions of rescaled energy. Lattice size 5 x 5 x 5, disorder average on 100 samples. a) W
= 12, b)
W = 20.
N°10 LEVEL CURVATURE AND 3D ANDERSON TRANSITION 1473
allow to take it out from average symbols. The same graph also shows the depeudeuce of the absolute value of curvature ((K(E) ii. The observation that both distributions are almost flat in
a wide range su ggests to consider the whole central third fraction of eigenvalues for computiug
average quantities. The probability distribution P(K) of curvature is shown in figures 4a, b, c;
it broadens with decreasing disorder and is always syInmetric around K
= 0. Au approximate number of 4000 level curvatures are computed in each hystograrn.
The metallic-1~lsulator transition is well reflected in the bel~aviour of average curvatures. In a
truly insulator phase, occurriug at L - cc, the curvature should be zero due to the exponential
localizatio~l of states. For fi~lite but 1~lcreas1~lg values of L, the shape of the curve log (K( versus
W shows a more pronounced inflectio~l at the critical value W~. This elfect is euha~lced by superposing tl~e curves, as in figure 5, where a ~lodal structure appears ~lear the transition value. This patter~l is very similar to that produced by Shklovskii et al. [GI, at the cost of collect1~lg ma~ly more data to get a reliable tait for the level spac1~lg distribution.
The same data are show~l in figure 6, as a fuuction of the scaling ratio (ccIN. Tl~e localizatiou
lengtl~ for the infinite sample (cc(W) has bee~l take~l from the paper by MacKi~1~1o~1 aud Kramer [7]. As in the previous picture, a transition is evident at the value (logK) m -3.5. This figure
demonstrates the scaling property of the average curvature, s1~lce ail points are distributed
along a fine. A similar behaviour was observed for the quantity j by Shklovskii et al. [GI.
Next we exhibit other 1~lterest1~lg properties, which concern the distribution of curvatures in the two regimes of the Anderson model.
PjK)
40 ~~~
30
20
io
0
Ù-1 0.05 Ù-Ù 0.05 Ù-1
K
P(Kj 8
~~~
P(K)
j~~
2_0 6
5
~
i o
2 0.5
0 Ù-Ù
0.2 Ù-1 Ù-1 0.2 0.6 0.4 0.2 Ù-Ù 0.2 0.4 0.6
K K
Fig. 4. Distribution of curvature for decreasing disorder. The curves are symmetric about K = 0.
Only the middle third of the energy spectrum is considered hereafter. Trie lattice size is 5 x 5 x 5, disorder average on 100 samples. Notethe change ofscales. a) W = 26 (localized regime), b) W
= 16.5
(near metallic-msulator transition), c) W
= 10 (metallic regime).
1474 JOURNAL DE PHYSIQUE I N°10
~~Î~Î> 0 <in[K[>~
O , é a
,2 ~
o
aoà~°.° ° ~
,4 4 ~$~~~~
,6 ÎÎÎ
,,
~"..-.( 6 ~?
o -~~ ~
,8 . 1=9 O
, 8
. =>o .. ., .
, i=,i .,,.
ÎÙ ~"'~ ~ ifl ,a
5 10 15 20 25 30
00 0.5 1.Ù 1.5 2.0 2.5 3.0 3.5
W (L/pcc)~~~
Fig. 5 Fig. 6
Fig. 5. The behaviour of trie mean curvature (log [K[) with disorder W, for various sizes of the cube. The fines cross at the critical value Wc m 16.5 and change their order approaching the step
curve, expected in the limit of large size L.
Fig. 6. One parameter scaling of curvatures. Data of figure 5 are plotted against the scahng parameter (L/jac(W))~/~, with the localization length fac(W) taken from [7].
In the recent work by Gaspard et ai. ils] the following dimensio~lless curvature is defiued:
k =
~
K (8)
~v(P
where (P2)
= C(0) is the average kiuetic e~lergy. The factor v is equal to 1 for the level
dy~lamics inside the orthogonal ensemble, and v
= 2 for the level dynamics inside the unitary ensemble. Note that this curvature definition coincides with the second derivative of rescaled energy with respect to rescaled flux, as defined in [12].
For the rescaled curvature k of the level dynamics inside the canonical enseInbles, Zakrzewski and Dela~lde iii proposed the following distribution
~~~~ ~"
(i + jÎ)1+u/2' ~~~
which well reproduces numerical results obtained for random matrices. N~ is a uorInalization constant equal to 1/2 (v
= 1) and 2/~ (v
= 2). The sonne distribution was later derived by
Takami a~ld Hasegawa [11].
The distribution implies a Ri ~("+~) tait, already predicted by Gaspard et ai. ils]. that excludes
the existence of moments of degree higher thon v.
The universal formula for the curvature distribution in the metallic regime has been found to hold also for the level dynamics of baud random matrices [16].
It is interesting to verify whether it holds for the Anderson Inodel, which is described by baud and sparse matrices. In both cases the level dynaInics is driver by a coInplex phase, which produces a motion from real symmetric matrices, at # = 0, where curvatures are sampled,
to the Hermitian ores (0 < # < ~). We have considered a rescaling with v
= i, a choice that revealed to be correct by measuring the tait behaviour of the experimental curvature
distribution.
The rescah~lg in equation (8) requires the evaluation of the average of the function (P~(#)),
shown in figure 2, over the whole range of #. This evaluation is a tiIne-cousum1~lg procedure
N°10 LEVEL CURVATURE AND 3D ANDERSON TRANSITION 1475
11.5 In<[K[>
P(Ln[K[) 0
11.4
i
11.3 ,2
11.2 '~
4
°.1 %
5
o o 7 ,6 5 -4 .3 -2 -1
5 4 ,3 ,2 11 2 In< P~>
Ln([K[)
Fig. 8
Fig. 7
Fig. 7. Distribution of the rescaled curvature K for W
= 10, in the metallic regime (see aise Fig.
4c). The histogram is well fitted by the universal distribution (9).
Fig. 8 Curvature and squared momentun1for sizes L
= 4 (circle), L
= 5 (square) and L
= 6
(diamond). The disorder parameter ranges from 30 (left) to 5 (right). The solid fine reproduces the relation (11).
in numerical computations. It turned out to be a good criterion to consider (P~(~/2)) as a good represe~ltative of (P2), since the two values are almost in constant ratio. Therefore we
introduce the rescaled curvature
'~
IP~Î~/211
~ ~~~~
which dilfers from the defi~litio~l equatio~l (8) o~lly by a constant factor.
It tur~led out that the rescal1~lg (10) actually produced a good correspo~ldence of ~lumerical data, in the metallic regime, with the u~liversal distribution (9), with tt replac1~lg k.
In order to demo~lstrate this correspo~ldence a~ld to exhibit properties of curvature distribution,
it is co~lve~lient to use a logarithmical scale. Figure 7 shows the distribution P(log [tt[) in the metallic regime (W = 10). Note the fine agreeme~lt with the Zakrzewski-Delande distribution.
A diIfere~1t rescal1~lg would manifest in a shift in figure 7. In the range i < logtt < 2.5 our
~lumerical results follow the [tt[~3 law. However, the statistics obta1~led is Dot suliicie~lt to exte~ld the validity of this statement to larger values of tt a~ld to co~lclude that the second
moment of the curvature distribution in the Anderso~l model does Dot exist.
The rescah~lg equatio~l (10) a~ld the assumption of trie existence of a universal curve for tt
in the metallic regime imply a relation similar to that by Akkerma~ls a~ld Montambaux (7).
Taki~lg the mean value in both sides of equation (10), and evaluatiug ([tt[) = i by means of the distribution (9) with v
= 1, we obtai~l
iP~i))) = )ilKl). (III
The l1~lear relation is show~l in the logarithmic plot in figure 8, the straight l1~le represe~lt1~lg equatio~l (II). The agreement with the ~lumerical data, good in the metallic phase represe~lted
in the right side of the figure, is lost for the disorder parameter close to the critical value W~
for the Anderson transition.
The relation (II) should be compared with equation (7) with A = 1, where the effective value of curvature is measured by the variance of the curvature distribution, and trot by the
