Table Of ContentLocalization in random fractal lattices
Arkadiusz Kosior1,2 and Krzysztof Sacha1,3
1Instytut Fizyki imienia Mariana Smoluchowskiego,
Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
2ICFO-Institut de Ciencies Fotoniques, Av. Carl Friedrich Gauss, 3, 08860 Castelldefels (Barcelona), Spain
3Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
(Dated: March 28, 2017)
We investigate the issue of eigenfunction localization in random fractal lattices embedded in two
7 dimensional Euclidean space. In the system of our interest, there is no diagonal disorder – the
1 disorder arises from random connectivity of non-uniformly distributed lattice sites only. By adding
0 or removing links between lattice sites, we change the spectral dimension of a lattice but keep the
2 fractionalHausdorffdimensionfixed. Fromtheanalysisofenergylevelstatisticsobtainedviadirect
diagonalizationoffinitesystems,weobservethateigenfunctionlocalizationstronglydependsonthe
r
a spectral dimension. Conversely, we show that localization properties of the system do not change
M significantly while we alter the Hausdorff dimension. In addition, for low spectral dimensions, we
observe superlocalization resonances and a formation of an energy gap around the center of the
7 spectrum.
2
PACSnumbers: 03.65.Aa,72.15.Rn,05.45.Df
]
s
a
I. INTRODUCTION
g
-
t
n Anderson localization (AL) is a single-particle disor-
a derinducedeffectwhichleadstoexponentiallocalization
u
of particles’ eigenfunctions [1–5]. In his groundbreaking
q
work,Andersonconsiderednon-interactingelectronicgas
.
t in a tight-binding model in the presence of on-site dis-
a
m order. Since then, AL was investigated in many differ-
entmodels,includingoff-diagonaldisorder[6–8],disorder
-
d correlations [9–12], random fluxes [13, 14], localization
n in the momentum space of classically chaotic systems
o [15, 16] and, recently, localization in the time domain Figure1: (coloronline)Anexampleofaminimalrandomfrac-
c [17, 18]. tal lattice (RFL) mapped on the 2D square lattice for η=1,
[
TheinterestinALrenewedafterthefirstexperimental see text. The solid blue lines indicate that the neighboring
2 observationofthephenomenoninultracoldatomicgases sites are linked, i.e. the quantum tunneling between these
v sites is possible. Red dotted lines represent the lack of quan-
[19–22]. AlthoughtheAndersonmodelwascreatedtode-
4 tumtunnelingbetweennearestneighbors. Byaddinglinksto
scribe electronic gases, AL is difficult to observe in met-
7 the lattice (i.e. replacing a number of red dotted lines with
2 als due to electron-phonon and electron-electron interac- bluelines)oneincreasesthespectraldimensionbutleavesthe
4 tions. On the contrary, interatomic interactions can be Hausdorff dimension unchanged.
0 switchedoffinasystemofultracoldatomicgasestrapped
. inopticallatticepotentials[23]. Anopticallatticeserves
1
0 as an artificial, phononless crystalline structure, whose
7 geometry and properties can be easily changed [24, 25]. [32,33]. WhereastheHausdorffdimensiondescribeshow
1 Therefore,inrecentyears,ultracoldatomsinopticallat- thenumberofsitesscaleswiththesystemsize, thespec-
v: tices have become a very important toolbox used to test tral dimension is related to a random walk on the lat-
i diverse physical models and phenomena [26–28]. tice: the number of distinct sites Sn visited by a random
X
Thedimensionalityofasystemplaysanimportantrole walker in n steps scales as Sn ∝ nds/2, provided that
ar in the context of AL [2, 3, 5]. In particular, in three ds < 2. The studies proved that it is the spectral di-
dimensional (3D) space a phase transition occurs at a mension ds that is relevant in AL and that ds =2 is the
critical energy, called the mobility edge, separating lo- lower critical dimension, below which all the states are
calized and extended states [29, 30]. It is therefore natu- localized in the presence of a disorder potential [34].
ral to investigate AL phenomenon in systems with non- Anotherimportanttheoreticalmodelinstudyoflocal-
integer dimension, i.e. in fractals [31, 32]. Whereas in izationpropertiesisthequantumpercolationmodel(QP)
Euclideanspaceitissufficienttodefineonekindofspace [35–38]. Anderson model describes particles in the pres-
dimension,inthecaseoffractalsoneneedstodistinguish: ence of potential disorder (purely diagonal), whilst QP
the dimension of the embedding Euclidean space D, the involvesthebinarykineticdisorder(purelyoff-diagonal).
Hausdorff dimension d and the spectral dimension d In QP models the disorder comes from random geome-
H s
2
try: a QP lattice, which is a subset of the D-dimension 1.7
lattice, arises by a removal of a number of sites or links
1.6
with a probability q (being the only parameter of the
model). Despite its simplicity, QP models still arise con- 1.5
troversies. The main concern is the question of existence ds
1.4
ofthelocalization-delocalizationtransitionin2Dmodels
for q > 0 (see [39–41] and references therein). In par- 1.3
ticular, this issue might be important in the context of
1.2
application of QP models in the description of transport
properties in e.g. manganite films [42], granular metals 0.0 0.2 0.4 0.6 0.8 1.0
[43] and doped semiconductors [44]. p
Inthepresentpaperweinvestigateeigenfunctionlocal-
ization in random fractal lattices (RFL), i.e. in fractal Figure 2: The plot illustrates the change of the spectral di-
objects with random site connectivity in the absence of mensionofRFLsfortheHausdorffdimensiond =1.75±0.02
H
anydiagonaldisorder,Fig.1. Wewouldliketostressthat when increasing the number of lattice links. The dimension-
QP systems do not belong to fractal objects. That is, in lessparameterpisthefractionoflinksaddedtotheminimal,
single connected lattice. A parameter value p=0 represents
the QP case only an infinite cluster of a percolated lat-
RFLs with the minimal number of links, i.e. no link can be
tice at the percolation threshold is a fractal object [45].
removed without disconnecting a part of the lattice. Con-
Here, on the contrary, we consider a family of lattices
versely, p = 1 represents RFLs with the maximal number of
with well defined Hausdorff dimension d [46]. Starting
H links,i.e. nolinkcanbeaddedwithoutcreatinganextralat-
with the minimal, connected lattice (i.e. a lattice with-
tice site. The value of the spectral dimension d for a given
s
out loops) and adding links between nearest neighbors p was obtained from the exponent of the number of distinct
wecanincreasethespectraldimensionds ofalatticeand sites visited by a random walker Sn ∝ nds/2, and averaging
keep the Hausdorff dimension d fixed. Therefore, the over 2000 independent RFLs and 500 realizations of random
H
main focus of this paper is to investigate the presence or walks of length 2500.
absence of localization in RFLs while changing the spec-
tralandHausdorffdimensionsindependently. Itisworth
noting that the theoretical model investigated here can
in n steps of a random walk is proportional to nds/2, if
be realized in ultra-cold atoms laboratories where lattice
d < 2. From a general analysis, the spectral dimension
geometry can be nearly arbitrarily shaped [47–49]. s
is never larger than the Hausdorff dimension [32, 33].
This paper is organized as follows. In Section II we
RFLsunderconsiderationareembeddedinthe2DEu-
describe the growth algorithm of random fractal lattices
clidean space and have their Hausdorff and spectral di-
and how the spectral dimension changes when new lat-
mensions smaller than 2. A minimal RFL, i.e. a lattice
tice links are created. In Section III we focus on local-
with the smallest number of lattice links, is a single con-
ization properties of RFLs and on their dependence on
nectedlatticegeneratedbythegrowthalgorithmdefined
the spectral and Hausdorff dimensions. Particularly, we
inRef.[46]. Inanutshell,anewlatticesite(i(cid:48),j(cid:48))ischo-
analyzesuperlocalizationresonancesandformationofan
sen and linked to the existing lattice at site (i,j) with
energy gap which emerges in the system for small spec-
the probability
traldimension. InSectionIVweinvestigatetransmission
probabilities through the system and quantum evolution (cid:16) (cid:17) (φ )η
of initially localized particles. Finally, in Section V we P (i,j)→(i(cid:48),j(cid:48)) = (cid:80)(φi(cid:48),j(cid:48) )η, (1)
conclude. i(cid:48),j(cid:48)
where the summation goes over all possible choices, η is
a free parameter and φ is a function fulfilling discrete
i,j
II. RANDOM FRACTAL LATTICES Laplace equation:
1
We consider a family of lattices that we call random φi,j = 4(φi,j+1+φi,j−1+φi+1,j +φi−1,j). (2)
fractal lattices (RFLs),whichfirstaroseinamodelofdi-
electric breakdown [46]. The RFLs are lattices with ran- At start, we set φ = 0.5 everywhere. When a site
i,j
domsiteconnectivityandwithwell-definedHausdorffd (i(cid:48),j(cid:48))isbeingconnectedtothelattice, thenthevalueof
H
andspectrald dimensions. TheHausdorffdimension,or φ is changed to zero. Before linking another lattice
s i(cid:48),j(cid:48)
thecapacitydimension,describeshowthenumberofsites site,thevaluesofφintheneighborhoodof(i,j)needsto
scales with the system size. In other words, if a lattice be updated [typically in 5-20 iterations of Eq. (2)]. The
in Fig. 1 is a fractal object, then the number of sites in- algorithm is stopped after reaching N lattice sites. The
side a sphere o radius r is proportional to rdH, where in latticesgrowninthismannerhavenonuniformgeometry,
general d is a noninteger exponent [45]. On the other bothinthelatticesitesandlatticelinksoccurrence,asin
H
hand,thespectraldimensionisrelatedtoarandomwalk Fig. 1. In particular, the two neighboring lattice sites do
on the lattice. The number of distinct sites S covered notnecessarilyneedtobeconnectedandtheclosedloops
n
3
are forbidden, i.e. a minimal RFL is created. By adding
linksbetweennearestneighborstoagivenminimalRFL,
one opens up new possibilities for a random walker to
exploreandthereforeincreasesthespectraldimensiond
s
of the system while keeping the Hausdorff dimension d
H
intact.
TheHausdorffdimensionofRFLsdependsonthevalue
of the parameter η [46]. For example, setting η = 1
one can generate a minimal RFL with the Hausdorff di-
mension d = 1.75 ± 0.02 and the spectral dimension
H
d = 1.33±0.03. Adding links to a minimal RFL re-
s
sults in an increase of the spectral dimension, Fig. 2. At
the same time the Hausdorff dimension d remains un-
H
changed.
III. LOCALIZATION PROPERTIES
InthefollowingweanalyzesolutionsoftheSchrödinger
equation
(cid:88)
Eψ =− ψ , (3)
(i,j) (i(cid:48),j(cid:48))
i(cid:48),j(cid:48)
where (i,j) denotes a position on a RFL embedded in
the 2D Euclidean space and the sum runs over nearest
neighbor sites if there is a link between (i,j) and (i(cid:48),j(cid:48)).
We assume that the tunneling amplitudes of a particle
between neighboring sites and the Planck constant are
equal to unity.
A. Analysis of energy level statistics
In our analysis we investigate two distinct scenarios:
Figure 3: (color online) A plot of the averaged ratio of con-
• First, we fix the Hausdorff dimension by setting secutive energy level spacings r(E) of RFLs with the fixed
η = 1 in (1), which corresponds to d = 1.75± Hausdorff dimension d = 1.75±0.02 (top panel), and two
H H
0.02. Then, we change the spectral dimension in cuts for the extreme cases (middle panel). The vertical axis
therangebetween1.33and1.55byaddinglinksto in the top panel shows the impact of the spectral dimension
the minimal RFLs, see Fig. 2. ds,throughthedimensionlessparameterp,seeFig.2,onthe
level statistics. With increasing d the system gradually de-
s
• In the second scenario, we do the opposite, i.e. we localizes. For low values of p (corresponding to ds ≈ 1.35)
fix the spectral dimension (d ≈ 1.35 or d ≈ 1.5) and near E = 0 there is a narrow energy gap emerging, see
s s
thediscussioninthemaintext. Thelocalizationisnotmuch
andchangetheHausdorffdimensionbyvaryingthe
influenced by a change of the Hausdorff dimension (bottom
parameter η in (1).
panel). TheRFLsystemswithdifferentvaluesofd andvery
H
similard possesssimilarlocalizationproperties,showingthat
In order to explore the presence or absence of local- s
it is the spectral dimension d that is the relevant dimension
ization of eigenfunctions of a particle in RFLs, we use s
in this context. All RFLs analyzed here consist of N =5000
a convenient method engaging the energy level statistics
sites.
obtained via direct diagonalization of finite systems [50–
52]. Since the localized states usually have very small
overlaps (as they may be localized in different parts of
the system), the energy levels can be nearly degenerate. ordered spectrum of energy levels {Ei}, we can calculate
This is why the localization of eigenstates can be ob- a quantity
served directly from the spectrum. Therefore, we expect
min(δ ,δ )
that in the localized phase the energy level statistics fol- ri = max(δi,δi−1), (4)
lowthePoissondistribution,andinthedelocalizedphase i i−1
fulfilltheWignerDysondistribution[50–52]. Havingthe where δ =E −E . Next, we average the results over
i i i−1
4
2000 realizations of RFL and over neighboring energies 0.30
r(E)=(cid:104)r (cid:105). The distinction between the two regimes is
i 0.25
possible since for the Poisson distribution r(E)≈0.3863
localized phase) and for the Wigner Dyson distribution 0.20
r(E)≈0.5359 (delocalized phase) [50–52].
Δ0.15
To investigate the localization properties of wavefunc-
tionsunderthechangeofthespectraldimensionofRFLs, 0.10
we plot r(E) in Fig. 3 (top panel). The vertical axis
0.05
expresses the spectral dimension via the dimensionless
parameter p, see Fig. 2. The panel illustrates strong de- 0.00
1500 4500 7500 10500 13500 16500 19500
pendenceofthelocalizationpropertiesonthespectraldi-
N
mension d . While increasing the spectral dimension we
s
observeasmoothtransitionfromthelocalizedtodelocal- Figure 4: The change of the energy gap ∆ near E = 0 with
ized phase, what is evident in the middle panel of Fig. 3, the increasing size of the system N for the minimal RFLs.
where the averaged ratio r(E) is plotted for two extreme The values were averaged over 200 realizations. The error
bar indicates one standard deviation, and the black dashed
values of d . Furthermore, we observe nonmonotonous
s lines represent the minimal and the maximal value obtained
dependence of r(E) on energy (i.e. the localization is
in 200 realizations. The numerical values were obtained via
stronger near the edges and the center of the spectrum),
the direct diagonalization.
which has been also observed in QP models [37, 40].
In the bottom panel of Fig. 3 we present r(E) for a
few different Hausdorff dimensions d and very similar
H
spectraldimensiond . Thedatawithdifferentd donot 0.35
s H
Δ
differ significantly apart from the edges of the spectra. 0.30 E=1 E= 2 E= 3
o
Tishteheresspuelcttsrsahlodwimtehnasti,osnimdsiltahralyttisotthheerAelLevmanotddeilms[e3n4s]i,oint onRati00..2205 E= 52-1 E= 52+1
inthecontextofthelocalizationpropertiesofthesystem. ati
p
ci0.15
arti
P0.10
B. Energy gap and superlocalization resonances 0.05
0.00
0.0 0.5 1.0 1.5 2.0
In Fig. 3 we can observe a peculiar narrow energy gap
E
∆ around E =0 for d (cid:46)1.4, which eventually closes up
s
when the spectral dimension is being increased. We find Figure5: AplotoftheparticipationratioPR(E)ofthemin-
thatthevalueofenergygap∆fortheminimalRFLs(i.e. imal RFLs (averaged over 2000 realizations). The participa-
d =1.33±0.03) of length N =5000 is tion ratio PR confirms the existence of the energy gap ∆.
s
Also, an additional structure is revealed - we observe some
∆=0.114±0.017. (5) very narrow superlocalization resonanc√es for a dis√crete set of
degenerate energies, where E = 0,± 5±1,±1,± 2 are the
r 2
The energy gap (5) decreases slightly in larger systems, mostdominant. FortheresonanteigenenergiesEr thesystem
however, it seems that ∆ survives in the thermodynami- reducestoafewverysmallclusters,seeFig.6. Notethatthe
energygap∆isaveragedovermanyrealizationsandasingle
cal limit, see Fig. 4. In order to illustrate the energy gap
realizationvaluemightdiffer,seeFig.4. Duetothisfact,the
more clearly we show in Fig. 5 the participation ratio
averaged PR(E) is smeared around E =∆.
PR(E)
(cid:88)(cid:12) (cid:12)4
PR(E)= (cid:12)(cid:104)(i,j)|ψ(E)(cid:105)(cid:12) , (6)
(cid:12) (cid:12)
(i,j) resonances is not visible in Fig. 3 because in Eq. (4) the
degenerate levels are discarded to avoid divergence. The
where|ψ(E)(cid:105)isaneigenstatecorrespondingtoanenergy namesuperlocalizationstemsfromthefactthattheeigen-
E and |(i,j)(cid:105) is a state localized at a lattice site denoted states localize on very small (a few lattice site) disjoint
by (i,j) in the 2D Euclidean space. The participation clusters. For example, a zero energy state can be local-
ratio is yet another measure of the localization [2, 3]. ized on two sites only, as long as a certain building block
That is, the inverse of PR estimates a number of fractal appears on the lattice boundary. In Fig. 6 (a) we see a
points on which an eigenstate is localized on. block with one vertex and four lattice sites. Note, that
What is the most striking in Fig. 5 is the emergence itszeroenergyeigenstatehasnon-zerovaluesontwosites
of peaks that are related to superlocalization resonances only and the other two sites form an „empty leg”. Now
also observed in the QP model [37]. The resonances ap- notice, that a structure in Fig. 6 (c) must have similar
pear for a discrete s√et of energie√s Er and they are dom- zeroenergyeigenstatesbecauseconnectingemptylattice
inant for E = 0,± 5±1,±1,± 2. The presence of the to an empty leg of a block like in Fig. 6(a) does not
r 2
5
0
-10
]
)-20
E
(
t
n[-30
l
-40
-50
-4 -2 0 2 4
Figure6: (coloronline)Asamplebuildingblocksresponsible E
for the superlocalization resonances: an eigenstate of a 4-site
0.100
block corresponding to E = 0 (a) and an eigenstate of a 10-
√
siteblockrelatedtoE = 3(b). Adotrepresentsanon-zero 0.010
value of an eigenstate on a given sites ψ : a dot’s size scales
i
with|ψ |andred/bluecolorrepresentsplus/minussignofψ . 0.001
i i
Notice that a state of block (a) has the same energy as an t 10-4
eigenstate of a lattice in panel (c) (connecting empty sites to
an empty leg of block (a) does not change its energy). 10-5
10-6
0.0 0.2 0.4 0.6 0.8 1.0
change a zero energy eigenstate localized on two sites. p
Therefore, if a lattice geometry allows for small blocks
(like e.g. in panel (a) or (b) of Fig.6), then some eigen- Figure 7: (color online) The energy dependent transmission
states of the lattice coincide with those of small blocks probability of a quantum particle between the most distant
and superlocalization resonances emerge. sites of a lattice of length 500 for η = 1 (top panel) and the
same transmission probability averaged over the correspond-
If some blocks are frequently occurring in the lattice,
ing energy spectra (bottom panel). The curves on the top
the corresponding superlocalization resonances can be
panel were plotted for p = 0.95 (solid red) p = 0.55 (black
extremely degenerate. For example, for the minimal
dashed) and p = 0.05 (blue dotted). We observe that the
RFLs with d = 1.75±0.02 about 10% of eigenstates
H transition probability strongly increases while we add links
have zero eigenenergy. The zero energy manifold is thus
to the lattice, which is in the agreement with Fig. 3. Also,
extendedoverasubstantialnumberoflatticepointsthat one can distinguish the termination of transport for p=0.05
is related to the appearance of the energy gap in the around E = 0 which corresponds to the energy gap in the
spectrum. That is, the zero energy manifold is large and system and for the resonant energies E (most critically for
r
othereigenstateswithnon-vanishingoverlapontheman- E =±1). The data were averaged over 2000 realizations.
ifold must necessary possess different energies.
Thefour-sitestructureshowninFig.6(a)hasalsonon-
zero energy eigensolutions, for instance corresponding to
√
tems are closer to the experimental reality in ultra-cold
E = − 3. However, such eigenstates do not have an
atomic gases.
„empty leg” and therefore if the block is connected to an
empty big lattice, these eigenstates are disturbed. Nev- The transmission probability from site r to site r(cid:48) of
ertheless, it is possible to build a 3-vertex block, see a quantum particle with the energy E is defined [2] as:
Fig. 6(b), where there are eigenstates corresponding to
E = −√3 which possess „empty legs” and are not dis- t(r,r(cid:48),E)=(cid:68)(cid:12)(cid:12)(cid:104)r|G+(E)|r(cid:48)(cid:105)(cid:12)(cid:12)2(cid:69), (7)
turbedwhentheblockisattachedtoabigemptylattice.
Such a 3-vertex structure is far less common√in RFLs, where (cid:104)..(cid:105) denotes average over different realizations of
which explains low abundance of the E = − 3 super- RFL, G+(E)=lim (E+iη −H)−1 is the retarded
localization resonance (less then 1%(cid:24)). one-particleGreen’sη→fu0n+ction[53]. Weplotthetransmis-
sion probability throughout considered lattices, i.e. be-
tween the most distant sites, in Fig. 7. The top panel
IV. TRANSMISSION AND QUANTUM presents the dependence of the transmission probabil-
EVOLUTION ity on energy for different spectral dimension of RFLs
whereasinthebottompaneltherearetransmissionprob-
In this section we investigate transport properties of abilities averaged over entire energy spectra for different
a quantum particle on RFLs: transmission probability number of links in the system. In an agreement with the
through the lattice and evolution of a particle initially resultspresentedinFig.3,weobserveadrasticreduction
localized on a single lattice site. Here, we focus on small (5 orders of magnitude) of the transport while decreas-
systems with 500 lattice sites only because smaller sys- ing the number of links in the system (bottom panel).
6
of the evolution of a quantum particle in lattices for the
twoextremecases: theminimalandthemaximalnumber
of links for a given geometry. In panels (c) and (d) the
timeaveragedresultsareshown. Startingfromthesame
initial, fully localized (on a single lattice site) state, we
obtain the opposing results: the probability density of
finding a particles is either localized around the initial
state or explores the whole lattice.
V. CONCLUSIONS
We have investigated localization and transport of a
quantumparticleinlatticeswithafractalstructure. The
lattices consist of points that form a connected cluster.
Sites of the lattices are generated so that their fractal
(Hausdorff) dimension d is controlled. Independently
H
one can control the spectral dimension d of the systems
s
by choosing how many nearest neighbor sites are linked
to a given lattice point. It allows us to analyze how the
localization properties vary with independent changes of
the Hausdorff and spectral dimensions.
Analysisofenergylevelstatisticsandparticipationra-
tio of eigenstates shows that while the localization prop-
erties depend very weakly on d , they change strongly
H
with d . For the smallest spectral dimension of the sys-
s
tems we observe strong localization of eigenstates. With
anincreaseofd ,eigenstatesloosetheirlocalizationprop-
s
Figure8: (coloronline)Theevolutionofaquantumparticlein
erties and become extended over the entire finite lattices
asampleRFLlatticeforthetwoextremecases: theminimal
that we consider. Disorder in our systems stems from a
numberoflinks(a)andthemaximalnumberoflinks(b)ina
non-uniform distribution of lattice points and from their
givenfractalgeometry. Wechoseasystemof500latticesites
random connections. When d approaches d all near-
generatedforη=1,Eq.(1). Theinitialstatewaslocalizedon s H
asinglelatticesite. Panels(a)and(b)presenttheprobability est neighbor sites become connected and the random-
densities of finding a particle for different evolution times. ness is related to the non-uniform distribution of lattice
Panels (c) and (d) present the time averaged densities for points only. The latter introduces too weak dephasing
50<t<150. and eigenstates do not localize.
We observe also eigenstates that are strongly localized
onsmallpartsoftherandomfractallattices. Thesmaller
partofthefractal,thehigherchanceforsucheigenstates
Furthermore, we can see a number of strong dips in the
to occur. The zero energy eigenstates can occupy two
plot of t(E) (top panel), especially for p = 0.05. These
sites only and consequently they form the largest degen-
dipscorrespondtotheenergygaparoundE =0andthe
erate manifold. At low spectral dimension they are so
superlocalization resonances for discrete degenerate en-
many that an energy gap around E = 0 is created. The
ergies, see Fig. 5. The most pronounced dips are related
presence of strongly localized eigenstates is imprinted in
to: E = 0 (about 10% of all energy levels correspond to
√ the transport properties of the systems, i.e. the parti-
E = 0), E = ±1 (4%) and E = ± 2 (1%). Note, that
cle transmission probability drops at the corresponding
anincreaseofp(blackdashedandredsolidcurvesinthe
energies.
top panel) narrows the dip around E = 0 significantly
down because the gap is disappearing.
Furthermore, the transport properties of a quantum
Acknowledgments
particle can be investigated more directly by solving the
time dependent Schrödinger equation
We are grateful to Marcin Płodzień for encouraging
(cid:88)
i∂ ψ =− ψ , (8) discussions.
t (i,j) (i(cid:48),j(cid:48))
AK acknowledges a support of the National Science
i(cid:48),j(cid:48)
Centre,PolandviaprojectDEC-2015/17/N/ST2/04006.
cf. Eq. (3). KS acknowledges a support of the National Science Cen-
Inpanels(a)and(b)ofFig.8wepresentthesnapshots tre, Poland via project No.2015/19/B/ST2/01028.
7
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