Table Of ContentDiluted banded random matrices: Scaling behavior of eigenfunction and spectral
properties
J. A. M´endez-Bermu´dez,1 Guilherme Ferraz de Arruda,2,3 Francisco A. Rodrigues,2 and Yamir Moreno3,4,5
1Instituto de F´ısica, Benem´erita Universidad Auto´noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
2Departamento de Matema´tica Aplicada e Estat´ıstica,
Instituto de Ciˆencias Matema´ticas e de Computa¸ca˜o,
Universidade de Sa˜o Paulo - Campus de Sa˜o Carlos,
Caixa Postal 668, 13560-970 S˜ao Carlos, SP, Brazil
3Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain
4Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain
5Complex Networks and Systems Lagrange Lab, Institute for Scientific Interchange, Turin, Italy
7 (Dated: January 9, 2017)
1
Wedemonstratethatthenormalisedlocalizationlengthβoftheeigenfunctionsofdiluted(sparse)
0
bandedrandommatricesfollowsthescalinglawβ =x∗/(1+x∗). Thescalingparameterofthemodel
2
n iNs disefitnheedmaastrxi∗x∝siz(eb,2efaf/nNd)δδ,∼w1h.eAredbdeiftfioisnathlley,awveerashgoewnuthmabtexr∗ofalnsoons-czaelreosethleemsepnetcstrpaelrpmroapterrixtiersowof,
a
the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.
J
5 PACSnumbers: 89.75.Hc
]
n
Introductionandoutlook.–Randommatrix(RM)mod- Thus, motivated by the ample interest on banded RM
n
elsservetodescribestatisticalpropertiesofcomplexsys- models and the recent attention on diluted versions of
-
s tems and related processes: From the original Gaussian them [48–50] in this paper we study scaling properties
i
d ensembles of Wigner and Dyson [1, 2] (which reproduce of a diluted version of the BRM model. In particu-
t. the statistics of energy levels of complex nuclei, quan- lar we demonstrate that both eigenfunction and spectral
a
tized chaotic systems, disordered systems, random net- properties scale with a parameter that relates the model
m
works, etc.) to very recent and more elaborated ensem- attributes (matrix size, bandwidth, and sparsity) in a
-
bles e.g. relevant to the problem of many-body localiza- highly non-trivial way.
d
n tion [3]. Model definition and statement of the problem.– The
o Even at the early years of RM modeling, Wigner BRM ensemble is defined as the set of N ×N real sym-
c himself realized the need to refine the generic Gaus- metric matrices whose entries are independent Gaussian
[
sian ensembles in order to incorporate properties of re- random variables with zero mean and variance 1+δi,j
1 alistic physical systems. In this respect he introduced if |i−j| < b and zero otherwise. Hence, the bandwidth
v the so-called Wigner-banded RM model [4, 5] (see also b is the number of nonzero elements in the first matrix
4
Refs. [2, 6–13]), a model including a bandwidth and an row which equals 1 for diagonal, 2 for tridiagonal, and
8
4 increasing diagonal. In particular the bandwidth, which N for matrices of the GOE. There are several numerical
1 quantifies the range of interactions, has been the main andtheoreticalstudiesavailableonthismodel,seeforex-
0 ingredient of other RM models proposed to deal with ample Refs. [16–32]. In particular, outstandingly, it has
1. explicit applications: As examples we can mention the been found [16–18, 21] that the eigenfunction properties
0 power-law banded RM model [14, 15] (used to simu- of the BRM model, characterized by the scaled localiza-
7 latetheAndersonmetal-insulatortransition),thebanded tion length β (see Eq. (6) below), are universal for the
1
randommatrix(BRM)model[16–32](introducedtoem- fixed ratio
:
v ulate quasi-one-dimensional disordered wires), the em-
i bedded ensembles [33–35] (which take into account the X =b2/N . (1)
X
many-body interactions in complex nuclei and many-
r More specifically, it was numerically and theoretically
a body systems), system-specific banded Hamiltonian RM
shown that the scaling function
models [36, 37] (where the bandwidth of the Hamilto-
nian matrix can be obtained by means of semiclassical
ΓX
arguments [7, 38]), among many others [2, 39–50]. β = , (2)
1+ΓX
On the other hand, there exist several works dealing
with diluted RM models, see for example Refs. [51–62]. with Γ ∼ 1, holds for the eigenfunctions of the BRM
However, we know just a few RM models including, in model, see also Refs. [22–25]. It is relevant to men-
addition to sparsity, an effective bandwidth: i.e. the tion that scaling (2) was also shown to be valid, when
Wigner-banded RM model with sparsity [47], diluted the scaling parameter X is properly defined, for the
power-law RM models [48, 49], and a diluted block- kicked-rotatormodel[21,63,64](aquantum-chaoticsys-
banded RM model [50]. tem characterized by a random-like banded Hamiltonian
2
matrix), the one-dimensional Anderson model, and the ensemblesofdBRMscharacterizedbytheparametersN,
Lloyd model [65]. b, and α. We perform the average (cid:104)S(cid:105) taking half of the
We define the diluted BRM (dBRM) model by includ- eigenfunctions, around the band center, of each random
ing sparsity, characterized by the parameter α, in the matrix. Each average is computed with 5 × 105 data
BRM model as follows: Starting with the BRM model values.
we randomly set off-diagonal matrix elements to zero InFig.1(a)wepresentβ asafunctionofx,seeEq.(3),
such that the sparsity is defined as the fraction of the forensemblesofmatricescharacterizedbythesparsityα.
N(b−1)/2 independent non-vanishing off-diagonal ma- We observe that the curves of β vs. x have a functional
trix elements. According to this definition, a diagonal form similar to that for the BRM model (corresponding
random matrix is obtained for α=0, whereas the BRM to α = 1). In addition, in Fig. 1(b) the logarithm of
model is recovered when α=1. β/(1−β)asafunctionofln(x)ispresented. Thequantity
Therefore, inspired by scaling studies of the BRM β/(1−β)wasusefulinthestudyofthescalingproperties
model [16–18, 21–23, 26, 28, 30], here we propose the of the BRM model [16, 22] because β/(1 − β) = γx,
study of eigenfunction and spectral properties of the which is equivalent to scaling (2), implies that a plot of
dBRM model as a function of the parameter ln[β/(1−β)] vs. ln(x) is a straight line with unit slope.
Even though, this statement is valid for the BRM model
x=b2eff/N , beff ≡αb , (3) inawiderangeofparameters(i.e.,forln[β/(1−β)]<2)
it does not apply to the dBRM model; see Fig. 1(b). In
where the effective bandwidth of the dBRM model b is,
eff fact,fromthisfigureweobservethatplotsofln[β/(1−β)]
in analogy to the bandwidth b of the BRM model, the
vs. ln(x) are straight lines (in a wide range of x) with a
average number of nonzero elements per matrix row.
slope that depends on the sparsity α. Consequently, we
Eigenfunction properties.– A commonly accepted tool
propose the scaling law
to characterize quantitatively the complexity of the
eigenfunctions of random matrices (and of Hamiltonians β
=γxδ , (7)
corresponding to disordered and quantized chaotic sys-
1−β
tems) is the information or Shannon entropy S. This
measures provides the number of principal components where both γ and δ depend on α. Indeed, Eq. (7) de-
of an eigenfunction in a given basis. The Shannon en- scribeswellourdata,mainlyintherangeln[β/(1−β)]=
tropy, which for the eigenfunction Ψl is given as [−2,2], as can be seen in the inset of Fig. 1(b) where
we show the numerical data for α = 0.6, 0.8 and 1 and
N
(cid:88) include fittings with Eq. (7). We stress that the range
S =− (Ψm)2ln(Ψm)2 , (4)
n n ln[β/(1−β)]=[−2,2] corresponds to a reasonable large
n=1
rangeofβvalues,β ≈[0.12,0.88],whoseboundsareindi-
allows to compute the so called entropic eigenfunction catedwithhorizontaldot-dashedlinesinFig.1(a). Also,
localization length, see e.g. [64], we notice that the power δ, obtained from the fittings of
the data using Eq. (7), is very close to unity for all the
(cid:96)N =Nexp[−(SGOE−(cid:104)S(cid:105))] , (5) sparsity values we consider here (see the upper inset of
Fig. 1(b)).
where S ≈ ln(N/2.07), which is used here as a ref-
GOE Therefore, from the analysis of the data in Fig. 1, we
erence, is the entropy of a random eigenfunction with
are able to write down a universal scaling function for
Gaussian distributed amplitudes (i.e., an eigenfunction
the scaled localization length β of the dBRM model as
of the GOE). With this definition for S when α = 0 or
b=1, since the eigenfunctions of the dBRM model have β
=x∗ , x∗ ≡γxδ . (8)
onlyonenon-vanishingcomponentwithmagnitudeequal 1−β
toone,(cid:104)S(cid:105)=0and(cid:96) ≈2.07. Ontheotherhand,when
N
α = 0 and b = N we recover the GOE and (cid:104)S(cid:105) = S ; TovalidateEq.(8)inFig.2(a)wepresentagainthedata
GOE
so, the fully chaotic eigenfunctions extend over the N forln[β/(1−β)]showninFig.1(b)butnowasafunction
available basis states and (cid:96) ≈N. of ln(x∗). We do observe that curves for different values
N
Here, as well as in BRM model studies, we look for ofαfallontopofEq.(8)forawiderangeofthevariable
thescalingpropertiesoftheeigenfunctionsofthedBRM x∗. Moreover, the collapse of the numerical data on top
model through the scaled localization length ofEq.(8)isexcellentintherangeln[β/(1−β)]=[−2,2]
for α≥0.5, as shown in the inset of Fig. 2(a).
β =(cid:96)N/N , (6) Finally, we rewrite Eq. (8) into the equivalent, but ex-
plicit, scaling function for β:
which can take values in the range (0,1].
Inthefollowingweuseexactnumericaldiagonalization x∗
to obtain the eigenfunctions Ψm (m = 1...N) of large β = 1+x∗ . (9)
3
α = 0.2
1
4 α = 0.4
α = 0.6
0.8 α = 0.2 β)] αα == 01.8 2
0.6 −
α = 0.4 1 0
β α = 0.6 β/( A) 0
0.4
A) α = 0.8 n[
l
0.2 α = 1 -4 -2 0 2 -2
0 -4 0 4
0 1 2 3 4
x ln(x*)
1
B)
4 1.2δ
0.8
1.1 1
β)] 1 2 0.6
β/(1− 0 0.α5 1 0 β0.4
n[ B)
l -4 -2 0.2 0
-2 0 0 3 6
0
-8 -4 0 4 0 2 4 6 8
ln(x) x*
FIG. 1: (Color online) (a) Scaled localization length β as FIG. 2: (Color online) (a) Logarithm of β/(1−β) as a func-
a function of x = b2 /N [see Eq. (3)] for ensembles of di- tion of ln(x∗) [see Eq. (8)]. Inset: Enlargement in the range
eff
luted banded random matrices characterized by the sparsity ln[β/(1−β)] = [−2,2] including curves for α ∈ [0.5,1] in
α. Horizontalblackdot-dashedlinesatβ ≈0.12and0.88are stepsof0.05. Greendashedlinesinmainpanelandinsetare
shownasareference,seethetext. (b)Logarithmofβ/(1−β) Eq.(8). (b)β asafunctionofx∗. Inset: Dataforα∈[0.5,1]
asafunctionofln(x). Upperinset: Powerδ,fromthefittings in steps of 0.05. Green dashed lines in main panel and inset
ofthedatawithEq.(7),asafunctionofα. Lowerinset: En- are Eq. (9).
largement in the range ln[β/(1−β)]=[−2,2] including data
for α = 0.6, 0.8, and 1. Lines are fittings of the data with
Eq. (7). Izrailev’s distribution [26, 66]:
(cid:34) (cid:32) (cid:33) (cid:35)
1 β(cid:101)
P(s)=B1zβ(cid:101)(1+B2β(cid:101)z)f(β(cid:101))exp −4β(cid:101)z2− 1− 2 z ,
In Fig. 2(b) we confirm the validity of Eq. (9). We
(10)
would like to emphasize that the universal scaling given
in Eq. (9) extends outsize the range β ≈ [0.12,0.88], for where z = πs/2, f(β(cid:101)) = β(cid:101)−12β(cid:101)(1−β(cid:101)/2)−0.16874, and
which Eq. (7) was shown to be valid, see the main panel theparametersB1,2 aredeterminedbythenormalization
(cid:82)∞ (cid:82)∞
of Fig. 2(b). Furthermore, the collapse of the numerical conditions P(s)ds= sP(s)ds=1. We call β(cid:101)the
0 0
dataontopofEq.(9)isremarkablygoodforα≥0.5, as spectral parameter. In fact, Eq. (10) has been shown to
shown in the inset of Fig. 2(b). beusefultocharacterizetheP(s)oftheBRMmodel[26],
soweexpectEq.(10)withβ(cid:101)∈[0,1]toproperlydescribe
Spectralproperties.–Forcompleteness,nowweanalyze
the P(s) of the dBRM model.
the spectral properties of the dBRM model. To this end
Thus,weconstructhistogramsofP(s)foralargenum-
wechooseP(s),thenearest neighbor energy-level spacing
ber of combinations of the parameters of the dBRM
distribution. For α = 0 or b = 1, i.e. when the dBRM
model (α,b,N) and by fitting them with Eq. (10) we
model produces diagonal matrices, P(s) follows the ex-
extractsystematicallythecorrespondingvaluesofβ(cid:101). We
ponential distribution P(s) = exp(−s); better known in
always construct P(s) from half of the total unfolded [1]
RM theory as Poisson distribution or the spacing rule
spacings s =(Em+1−Em)/∆ around the band center,
for random levels [1]. In the opposite limit, for α = 1 m
where the density of states is approximately constant.
and b = N, i.e. when the dBRM reproduces the GOE,
Here, Em is the m-th eigenvalue and ∆ the mean level
P(s) closely follows the Wigner-Dyson distribution [1]:
spacing. Eachhistogramisconstructedwith5×105spac-
P(s) = (π/2)sexp(−πs2/4). Then, by moving α and b
ings.
in the intervals (0,1) and (1,N), respectively, the P(s)
In Fig. 3(a) we present the spectral parameter β(cid:101) as a
of the dBRM model should have a shape in-between the
functionofthescaledlocalizationlengthβ forthedBRM
Poisson and Wigner-Dyson distributions.
model. As in Figs. 1 and 2, here we label different spar-
Here, in order to characterize the P(s) for our RM sities α with different colors (symbols). It is interesting
modelweusethephenomenologicalexpressionknownas to note that even though the relation between β(cid:101) and β
4
scales the spectral properties of the dBRM model.
1 α = 0.2
While general diluted RM models have direct applica-
α = 0.4
0.8 α = 0.6 0 tions to random networks (i.e. the adjacency matrices of
α = 0.8 ~β
0.6 α = 1 ln complex networks are, in general, diluted random matri-
~β A) -3 ces), the dBRM model may be used to model multilayer
0.4 randomnetworkssincethebandwidthbandthesparsity
α can be associated, respectively, to the size and con-
-6
0.2 -6 -3
nectivity of the subnetworks composing a multilayer, see
ln β
0 e.g. [50].
0 0.2 0.4 0.6 0.8 1
Finally,wewanttorecallthatthescaling(2),validfor
β
the BRM model (and other disordered systems [63, 65]),
1
wasrewritteninamoreelegantwayasarelationbetween
0.8 properly-defined inverse lengths [22, 65]
1
0.6 [d(N,W)]−1 =[d(∞,W)]−1+[d(N,0)]−1 , (11)
~β
0.4 B)
were d(N,W) ≡ exp[(cid:104)S(N,W)(cid:105)] and W represents b for
0.2 0 the BRM model or the localization length for the one-
0 2 4 6
dimensional Anderson model and Lloyd’s model. Here,
0
0 2 4 6 8 in the case of the dBRM model, scaling (9) can also be
x*
written in the “model independent” form (11) as
FIG.3: (Coloronline)(a)Spectralparameterβ(cid:101)[seeEq.(10)] [d(N,b,α)]−1 =[d(∞,b,α)]−1+[d(N,N,1)]−1 , (12)
as a function of the scaled localization length β for ensem-
bles of diluted banded random matrices characterized by the with d(N,b,α) ≡ exp[(cid:104)S(N,b,α)(cid:105)] and d(N,N,1) =
sparsityα. Inset: lnβ(cid:101)vs.lnβ forα∈[0.5,1]instepsof0.05. exp[S (N)] (the reference entropy).
GOE
Green dashed lines in main panel and inset are the identity. We hope our results may motivate a theoretical ap-
(b) Repulsion parameter β(cid:101)as a function of x∗. Inset: Data proach to the dBRM model.
for α ∈ [0.5,1] in steps of 0.05. Green dashed lines in main
Acknowledgements.– This work was partially sup-
panel and inset are Eq. (9).
ported by VIEP-BUAP (Grant No. MEBJ-EXC17-I),
Fondo Institucional PIFCA (Grant No. BUAP-CA-169),
and CONACyT (Grant No. CB-2013/220624). FAR ac-
is not simple, e.g. linear as reported for other disordered
knowledges CNPq (Grant No. 305940/2010-4), FAPESP
systems[9,64,67–69],thecurvesβ(cid:101)vs.β areindependent
(Grant No. 2011/50761-2 and 2013/26416-9), and NAP
of α once α > 0.4, see inset of Fig. 3(a). This allows us
eScience - PRP - USP for financial support. GFA would
to guess that x∗ can also serve to scale the spectral pa-
like to acknowledge FAPESP (grants 2012/25219-2 and
rameter β(cid:101), at least for α>0.4. Accordingly, in Fig. 3(b)
2015/07463-1) for the scholarship provided. Y. M. ac-
weshowthatthecurvesofβ(cid:101)vs.x∗ falloneontopofthe knowledges support from the Government of Arag´on,
other mainly for α>0.4, see the inset of the figure.
Spain through a grant to the group FENOL, and by
FromFig.3(b)wealsoobservethatthecurvesβ(cid:101)vs.x∗ MINECO and FEDER funds (grant FIS2014-55867-P).
areaboveEq.(9),thatweincludeasdashedlines,except
for very small values of x∗ where they coincide. This
fact has already been reported for the BRM model in
Ref. [21]. This also means that the spectral properties
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