Flat band analogues and flux driven extended electronic states in a class of geometrically frustrated fractal networks

0.68 MB·English

by  Atanu Nandy

Flat band analogues and flux driven extended electronic states in a class of
geometrically frustrated fractal networks
Atanu Nandy,∗ Biplab Pal,† and Arunava Chakrabarti‡
Department of Physics, University of Kalyani, Kalyani, West Bengal-741235, India
5
We demonstrate, by explicit construction, that a single band tight binding Hamiltonian defined
1
on a class of deterministic fractals of the b = 3N Sierpinski type can give rise to an infinity of
0
dispersionless, flat-bandlike states which can beworked out analytically using thescale invariance
2
of the underlying lattice. The states are localized over clusters of increasing sizes, displaying the
n existenceofamultitudeoflocalizationareas. Theonsetoflocalizationcan,inprinciple,be‘delayed’
a
inspacebyanappropriatechoiceoftheenergyoftheelectron. Auniform magneticfieldthreading
J
the elementary plaquettes of the network is shown to destroy this staggered localization and gener-

7 ate absolutely continuous sub-bands in the energy spectrum of these non-translationally invariant
networks.
]
n
PACSnumbers: 71.23.An,72.15.Rn,73.20.Hb,73.23.Ad
n
-
s
i I. INTRODUCTION tron infinite, making the electron having such an energy
d
practically immobile in the lattice. The corresponding
.
at The interplay of geometry and quantum interference single particle state is sharply localized at a point, or
m in a finite cluster of points in the lattice. Such clusters
leading to exotic band structure in a class of quasi
are separated from the neighboring clusters of the lat-
- one-dimensional geometrically frustrated lattices (GFL)
d ticebyverticeswheretheamplitudeofthewavefunction
has drawn considerable interest in recent times. The
n is zero. The energy eigenvalue corresponding to such a
GFL havebeenquite exhaustivelystudiedoverthe years
o
construction belongs to a flat band. Fig. 1 displays one
c mainly in the context of localized spin systems [1], using
such case, where the central hexagonal cluster has non-
[ the Heisenberg model for studying frustrated antiferro-
zeroamplitudeatthevertices,whilethesubsetofatomic
magnetism[2]-[7],oritinerantferromagnetismwithinthe
2 sitesresidingattheverticesofthetrianglearezero. This
Hubbardmodel[8,9]. Thedegeneracyinducedbythege-
v
figure corresponds to energy E = 2 in a tight binding
1 ometricfrustrationandnon-trivialgroundstatessuchas −
description where the ‘on-site’ potential at every lattice
3 thespinliquidstatesasitsconsequence[1]haveunveiled
pointhasbeensetequaltozero,andthenearestneighbor
1 rich physics of the spin systems.
1 hoppingintegralhasbeenassignedthe valueunity. Such
Parallely, electronic spectra of several simple lattices,
0 situations canalso be realizedin the kagom´elattice [21],
studied mainly within a single band tight binding for-
. or other FB systems [22].
1 malism and only with nearest neighbor hopping, have
0 displayed exotic features such as the appearance of dis-
5
persionless, flat bands (FB) [10]-[14]. Such frustrated
1
hopping models are important as they offer prospects of
:
v stronginteractionphysicssuchasthefractionalquantum
Xi Hall effect. Graphene anti-dot lattices [15] and post-
graphene materials such as the phosphorene [16] have
r
a been proposed as systems exhibiting the qualities for
quantumcomputation[17]andanelectricaltunabilityof
quasi-flat bands respectively. Recent study on ultracold
fermionicatomspatternedintofinitetwodimensionalop-
tical lattices of a kagom´e geometry has added to such
excitements [18]. Advancement in controlled growth of
artificial, tailor-made lattices having atoms trapped in
optical lattices with the complications of even a kagom´e
FIG. 1: (Color online) Schematic view of a part of an el-
structure [19, 20] has contributed towards increasing in-
ementary star-network illustrating the localized distribution
terest in these deceptively simple looking systems.
of amplitudes within a finitecluster.
The non-dispersive character of the energy (E)-wave
vector (k) curve makes the effective mass of the elec-
The FB systems have been exemplified so far in terms
of periodic lattice models. The number of flat bands in
these systems is always finite. Comparatively speaking,
practically nothing is known about the existence of dis-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] persionless, flat band modes in hierarchically generated
‡Electronicaddress: arunava˙[email protected] artificiallattices,where,becauseofthe inherentscalein-
2
variance, it might be possible to identify an infinity of of n. Using an RSRG scheme it is possible to unravel
such modes. an infinity of such localized states and the correspond-
HereweaddressthisissueintermsofaclassofSierpin- ing spanning clusters for every member of the b = 3N
ski gasket (SPG) lattices, which have been widely stud- family. We present explicit results for the b = 3 gasket.
ied over the years [23]-[26] in respect of their electronic The remaining members can be easily dealt with using
properties. As one should appreciate, the FB states are the same formalism, but with increasingly cumbersome
localized states with finite support in a lattice. Though, algebra.
Anderson localization with weak disorder has been ad- In the secondphase of our workwe examine the effect
dressedfortightbindingmodelswithflatbands[27],the of a perturbation on the FB states in such hierarchical
effect of self similar lattice topology in this context is networks. The perturbation is chosen in the form of a
something which we find practically unaddressed. uniform magnetic field piercing each elementary plaque-
Thisleadstoanother,andaveryimportantadditional tte in the b = 3 gasket. The flux threading each pla-
motivation for the present work, namely, an exact eval- quette produces remarkable changes in the energy spec-
uation of at least a subsection of localized eigenstates in trum, generatingapparentabsolute continuity inthe en-
a hierarchical lattice. As such lattices are not periodic, ergy spectrum. Such subbands are populated only by
one generally has to resort to an exact diagonalization extended wavefunctions, as revealed by exhaustive nu-
of the Hamiltonian to know the energy eigenvalues, and merical study based on the RSRG scheme. The fractal
because of the fragmented, Cantor set-like energy spec- characterof the energy spectrum is removed, at least lo-
trumusuallyofferedbytheselattices[23]theeigenvalues cally, and the lattice becomes completely transparent to
obtained from one finite generation of the lattice in gen- an incoming electron when the Fermi level is chosen to
eral,slipoutofthespectrumonincreasingthesizeofthe lie in such continuous subbands. The FB states appar-
system. A precise knowledge of the energy spectrum in ently disappear (at least, they could not be explicitly be
thethermodynamiclimitthusturnsouttobeanontriv- constructed any more).
ial problem. Though in recent times the existence of an In what follows we describe our findings. In section II
infinity of extended, fully transmitting eigenstates have wedescribetheHamiltonianandtheRSRGscheme. Sec-
been proven within the framework of real space renor- tion III contains a discussion of the construction of the
malization group (RSRG) schemes [26, 28], the exact localizedeigenstatesonfinitesupports. Acriticaldiscus-
evaluation of localized state-eigenvalues has still largely sion on the flat band status of such states are provided
remainedanunresolvedissue,untilveryrecently[30,31] inSection IV.Section Vhighlights the effect ofinserting
and,foraspecialkindofdeterministicfractalgeometries. a uniform magnetic flux in each elementary plaquette of
theb=3gasket,andthetwo-terminaltransportcalcula-
tions are discussed in Section VI. Finally, in section VII
we draw our conclusions.
II. THE HAMILTONIAN AND THE RSRG
SCHEME
We refer to Fig. 2 where a portion of a b = 3 SPG
network (left) and its renormalized version are shown
(right). The electron’s hopping is restricted only along
FIG. 2: (Color online) Portion of an infinite b=3 Sierpinski each edge, and between nearest neighbors only. We dis-
gasket (left) and its renormalized version. The circles col- tinguishbetweentwotypesofsites,viz.,onewithcoordi-
ored red and black have coordination numbers four and six nationnumberfourandcoloredinred(henceforthcalled
respectively. The encircled vertices survive after decimation, C sites) and the other, black circles,having a coordina-
4
leading tothe scaled lattice (right).
tion number six (C sites). The system is described by
6
the tight binding Hamiltonian,
Inthiscommunicationwefocusonthesocalledb=3N
Sierpinskigasketfamily[32]. Weshowthat,itispossible H = ǫ c†c + t c†c +h.c. (1)
i i i ij i j
to explicitly ‘construct’localizedeigenstatesspanning fi- Xi Xhiji h i
nitesizedclustersonsuchlattices. Theclusterssupport-
ing such localized distribution of amplitudes of the wave ǫ is the on-site potential at each vertex, and will be as-
i
function are separated from neighboring clusters by ver- signed symbols ǫ and ǫ for the C and C sites respec-
4 6 4 6
tices where the amplitude of the wavefunction is zero at tively. t is the nearest neighbor hopping integral. We
thespecifiedenergies. Asaresult,theelectronwithsuch shall be interested only in the localization induced by
energies are forced to hop around and remain confined the topology of the lattice, and hence, in all numerical
inthese finite clusters. The minimumsize ofthe clusters results we shall set ǫ =ǫ = 0, and t = 1. However, on
4 6
supportingtheselocalizedstatesincreaseswiththevalue renormalization,the on-sitepotentials deviate fromeach
3
other at every stage, and that is why it is necessary to the amplitude of the wave function ψ is zero at the C
j 6
distinguish them at the very beginning. vertices with coordination number six and lying at the
Using the set of difference equations satisfied by the center of the shaded hexagon, as well as on a subset of
electrons, viz., C vertices. The non-zero amplitudes are designated by
4
x (which can be set equal to unity, for example). This is
(E ǫ )ψ = t ψ (2) illustratedintheleftpanelofFig.3. Thislattersubsetof
i i ij j
−
Xj C4 sites sitatthe ‘connecting’points betweenhexagonal
plaquettes atdifferentportions ofthe infinite lattice. By
weobtainascaledversionoftheoriginallattice,asshown
making the amplitude vanish at these strategic sites it
in Fig. 2 . An appropriate subset of the C and C sites
4 6 is possible to confine the hopping of the electron around
are decimated to yield an identical geometry with the
the periphery of the shaded hexagons. The eigenstate
renormalizedon-sitepotentialsandthehoppingintegral.
resembles the spirit of a molecular state [29] localized
The ‘surviving’ sites on the renormalized lattice are en-
partially by quantum interference andpartially by phys-
circledinthe left portionofFig. 2. The RSRG recursion
ical boundary formed by the sites with zero amplitude.
relations for the on-site potentials and the hopping inte-
The wave function corresponding to this energy eigen-
grals are given by,
value does not have any overlap with the wave function
described on the next neighboring hexagons. The local-
ǫ = ǫ +3
6,n+1 6,n Fn ization area is the shaded hexagonal plaquette confining
ǫ4,n+1 = ǫ4,n+2Fn the electron’s motion for E = ǫ4−2t along the ‘periph-
β E2+γ E+δ ery’,andisdistributedthroughouttheinfinite geometry.
2,n 2,n 2,n
t = (3)
n+1
∆
n
where, =(α E3+β E2+γ E+δ )/∆ , and, B. Exploiting the self similarity
n 1,n 1,n 1,n 1,n n
F
α1,n = 2t2n Oncesuchaconstructionoflocalizedstate-amplitudes
β = 2t2(t +2ǫ +ǫ ) is made possible in the bare length scale, the self sim-
1,n − n n 4,n 6,n ilar structure of the SPG lattices can be exploited to
γ = 2t2[5t2 t (ǫ +ǫ )
1,n − n n− n 4,n 6,n − construct wave functions, and to ‘work out’ the corre-
ǫ4,n(ǫ4,n+2ǫ6,n)] sponding areasoflocalizationatvariousscalesoflength.
δ = 2t2[2t3 +t ǫ ǫ +ǫ2 ǫ The eigenvalues corresponding to such cases are easily
1,n − n n n 4,n 6,n 4,n 6,n− extracted from the solutions of the equation
t2(4ǫ +ǫ )] (4)
n 4,n 6,n
E =ǫ 2t (6)
4,n n
The other symbols used are, −
at any desired n-th stage of RSRG. For every real root
β2,n =t3n of this equation, a construction similar to that in the
γ =t3(4t ǫ ǫ ) originallatticecanbemadeinrespectofthevaluesofthe
2,n n n− 4,n− 6,n amplitudes of the wave function at different C and C
δ =t3(ǫ ǫ +2t2 4t ǫ ) 4 6
2,n n 4,n 6,n n− n 4,n verticesonthescaledversionofthelattice. Onethenhas
∆ =[(E ǫ )2 t2][(E ǫ )(E ǫ 2t ) 6t2] toworkinthereversedirectiontolayouttheamplitude-
n − 4,n − n − 6,n − 4,n− n − n
(5) distribution on the original lattice at the bare scale of
lengthbysimultaneouslysolvingthedifferenceequations
The ‘stage’ of renormalization is denoted by n 0, and Eq.(2)for the respectiveenergyeigenvalue. The scheme
≥
we have chosen to stick to ǫ4,0 = ǫ6,0 = ǫ and t0 = t at is illustrated in Fig. 3(b). With ǫ4 = ǫ6 = 0 and t = 1,
the bare scale of length. we get E = 2 at the original length scale, while the
TherecursionrelationsEq.(3)willnowbeexploitedto same paramet−ers yield localized states at E = 2 and
construct and analyze a class of localized and extended E = 1 √2 when we set n = 1, that is, when w−e solve
states in such systems. the equ±ation E = ǫ 2t . Let us now explain the
4,1 1
−
construction of localized states for the new roots, viz.,
E =1 √2.
III. CONSTRUCTING THE LOCALIZED The±hexagon in Fig. 3(b) whose sides are highlighted
STATES
bythickgreenlineisthe‘smallest’hexagononaonestep
renormalized gasket. We choose the distribution of am-
A. The basic scheme plitudes atE =ǫ 2t aroundthishexagonexactlyin
4,1 1
−
the same way as we did for E = 2 in the bare length
−
Let us begin with the b=3 gasketand refer to Fig. 3. scale (Fig. 3(a)). Such a construction, as we now appre-
It is simple to work out that, if we set E = ǫ 2t, a ciate, can be made consistently throughout the infinite
4
−
consistent solution of the difference equations (or equiv- system. Thenextjobistoworkout,inthereversedirec-
alently, the Schr¨odinger equation) is obtained in which tion, the amplitudes of the wave function at all the ver-
4
0 0
x −x x −x
0 0
−x x −x x
e d
0
f c
0 x −x 0 0 a b x −x 0
(a) (b)
FIG. 3: (Color online) (a) Distribution of a flat-band state amplitudes on a b = 3 SPG network at its bare scale of length,
and (b) the distribution on a one step renormalized gasket. The localized state eigenvalues are extracted from the equations
E =ǫ4,0 2t0 and E =ǫ4,1 2t1 respectively.
− −
tices of the original gasket with the prefixed amplitudes of length.
(with values x at the edges of the green highlighted
±
hexagon and zero at its center) for energy eigenvalues
E =1 √2. C. The ‘staggered’ character of localization
±
The amplitudes are denoted by the letters a-f in
Fig.3(b). Wehaveshownexplicitlythesmallerhexagons
An interesting qualitative distinction between the en-
(blue highlighted) that are embedded in the larger one
ergy eigenvalues corresponding to these localized states
(green highlighted). It is possible to obtain a consis-
can be unravelled if we look at the evolution of the hop-
tent setof values for the amplitudes atallvertices ofthe
ping integral under RSRG steps. For any energy eigen-
smaller (blue) hexagons. For example, with x = 1, set
value obtained from the n-th stage, and by solving the
arbitrarily,thedistributionofamplitudesforE =1+√2 equation E = ǫ 2t , the hopping integral remains
4,n n
(one solution of E = ǫ4,1 2t1 with ǫ = 0 and t = 1) is finite upto n-th RS−RG steps. This means that one has
−
given by, a = f = (√2 1)/2, b = e = 1/√2 and non-zero overlap of the wave function between nearest
− − −
c = d = 1/2. The amplitude at the center of a small neighboring sites at that length scale. When mapped
−
hexagonis zerofor this particulardistribution. An iden- back onto the original lattice it amounts to a non-zero
tical pattern of distribution, but with a different set of overlapof the wave function between sites much beyond
values for a, b, ... f, can be easily obtained if one sets the nearest neighbors. However, the hopping integral
E =1 √2. starts decaying after the n-th RSRG step implying that
−
Needless to say, the scheme can be extrapolated to thewavefunctionlooks‘extended’onlylocally,andinre-
unravel other localized states by solving the equation ality such wave functions are localized on a finite cluster
E = ǫ 2t at any desired level. The roots increase oftheinfinitelattice. Theoverlapoftheenvelopesofthe
4,n n
−
in number, and the set of solutions obtained at the n-th wave function for a localized state-energy extracted at
stage of renormalization contains all the roots obtained the n-th stage weakens beyond a certain scale of length.
from the (n 1)-th stage. In every case one has to iden- Such finite sized clusters span larger and larger areas in
−
tify the smallest hexagon in that scale of length, set the real space as one extracts the energy eigenvalue by solv-
amplitudeofthewavefunctionequaltozeroatitscenter. ing the above equation at deeper and deeper scales of
The sites decorating the periphery of the basic hexagon length.Thus simply by ‘choosing’ the value of stage of
then should be assigned amplitudes equal to x as we renormalization ’n’ one can delay, in space, the localiz-
±
did in Fig. 3. Such a basic hexagon at the renormalized ingcharacterofthewavefunction. Inarecentstudy[30]
scalemustbe separated fromits neighboringhexagons,a we encountered such staggered localization in a special
conditionthatiseasilysatisfiedbysettingtheamplitude class of fractal lattices.
at the ‘connecting’ sites equal to zero. The final task is Before we end this section, it is pertinent to point out
tomapthedistributionbackontotheoriginallatticeand that the size of the spanning clusters even at the bare
to solve the set of equations Eq.(2) to work out the am- length scale depends on the value of n in the class of
plitudes at the inner sites of the lattice at its bare scale b = 3N SPG fractals. Fig. 4 depicts three other cases,
5
(a) (b) (c)
FIG. 4: (Color online) Distribution of amplitudes for a non-dispersive eigenstate at E = ǫ4 2t on b = 6, b = 9 and b = 12
−
Sierpinski gaskets. With increasing scale factor, the amplitudes assume non-trivial distribution over larger and larger clusters
which ultimately get decoupled from therest of the lattice, courtesy,the zero amplitudes at theconnecting vertices.
viz., b=6, b=9 andb=12. We stick to the same value As is evident, such a construction requires the presence
of energy, viz., E = ǫ 2t. The localization areas even of a top vertex with coordination number two (colored
4
−
at the bare length scale are already larger than that in cyan in Fig. 5). It is important to appreciate that for
the b = 3 case. In each case, the electron’s hopping is such finite systems the distribution of amplitudes does
confined around the peripheries of clusters of finite size, not follow the method prescribed for Fig. 3, which was
shown by the shaded areas. Such finite sized clusters typical of an infinite system where the ‘end points’ were
growbiggerandbiggeras one goes fromb=6 to b=12, not ‘visible’.
and beyond. The clusters, however big, are ultimately As we gradually increase the value of n, the unit cell
separated from each other by a subset of C sites with itself turns out to be a reasonably good approximant of
4
ψ = 0, and are distributed throughout the infinite ge- the infinite system, and becomes the true representative
j
ometry. in the limit n . For a periodic array of n-th gener-
→ ∞
ation SPG networks one can renormalize each ‘unit cell’
to n-times to convert the array into an array of trian-
IV. THE FLAT BAND ANALOGUES gles with the potentials at the vertices equalto C2,n and
C respectively. The nearestneighbor hoppingintegral
4,n
A pertinent questionis, whether these infinite number alongtheedgesofsucheffectivetrianglesaretn. Thisar-
of localized eigenstates are non-dispersive in character, ray of triangles is then converted into an array of single
in the spirit of the flat band states observedin a class of ‘effective’atomicsites(encircledbydottedlinesinFig.5)
periodic lattices as mentioned in the introduction. The withenergydependentpotential,andinter-connectedby
absence of translational invariance in the SPG class of anenergydependenthopping integral. The effective,en-
lattices prevents a derivation of the usual energy-wave ergy dependent on-site potential and the nearest neigh-
vector dispersion relation. At the same time, the fact bor hopping integral in the linear chain of renormalized
that the localized states discussed so far have been ‘con- atoms are given by,
structed’soastodisplayakindofself localization,where 2t2
theamplitudesaredistributedoverafiniteclusteroflat- ǫ˜ = ǫ4,n+ n
E ǫ
tice points, effectively ‘decoupled’ from the rest of the − 2,n
system. This inspires us to undertake a couple of indi- t˜ = t2n (7)
rect,buteffectivewaysofdiscerningtheactualcharacter E ǫ2,n
−
of these states.
Inthiscase,weneedanadditionalrenormalizationofthe
top vertex with coordination number two, viz.,
A. Dispersion relation for a periodic array of ǫ =ǫ +
2,n+1 2,n n
F
fractal approximants
The definition of is already given in earlier section
n
F
(Eq.(4)andEq.(5)). TheE-kdispersionrelationforsuch
First, we construct a periodic arrayof b=3 SPG net-
aperiodicchainistheneasilyobtainedthroughtheequa-
works. Each‘unitcell’consistsofafiniteSPGatitsn-th
tion,
generation. One such array comprising the first genera-
tionb=3SPGnetworkisshowninFig.5forillustration. E =˜ǫ+2t˜coska (8)
6
FIG.5: (Coloronline)Aperiodicarrayoffinitegeneration b=3SPGnetworks. Withincreasinggeneration eachunitcellwill
be representativeof an infinite SPGnetwork.
In Fig. 6 we depict the dispersion relation when the shown in Fig. 7(a) and (b) respectively. This is done
following the standard procedure, and using the RSRG
recursion relations Eq. (3), which finally yield the lo-
cal Green’s function (LGF) at the C and C sites as,
4 6
G(4) = (E ǫ∗)−1 and G(6) = (E ǫ∗)−1 respectively.
Th00e symbo−l ǫ∗4indicates th00e fixed p−oin6t value of the re-
j
spectiveparameter. Finally,theLDOSatthedesiredsite
isobtainedas,ρ =( 1/π)lim ImG(j)(E+iη),with
j − η→0 00
j =4 or 6 depending on the coordination number.
The LDOS (in the absence of an external magnetic
field) exhibits a highly fragmented distribution as seen
inFig.7(a)and(b). Thisisexpected. Thefinestructure
is related to the distribution of spectral weight [33], and
is controlledby the lattice topology. One should observe
that the j-th site (C or C ) having ψ =0 can not con-
4 6 j
tributetotheLDOSatthatparticularsite. Nevertheless,
non-trivialcontributioncomesfromtheneighboringsites
yieldingafinitevalueofthedensityofstatesataC site.
4
It is important to appreciate the role of the numeri-
FIG. 6: (Color online) The dispersion relation for the array
cal value of the imaginary part η added to the energy.
depicted in Fig. 5. The flat, dispersionless modes are ap-
η determines the width of the LDOS around an energy.
parent. The flat band at E = 2, the root of the equation
− We have carefully examined the growthof the LDOS for
E =ǫ4 2twithǫ4 =ǫ6 =0andt=1hasalreadybeenshown
− E = 2, 1 √2 and at several other localized flat band
to exist, by explicit construction, for an infinitesystem.
− ±
statesasobtainedbyexplicitconstructionatbarelength
scaleandfromaone-steprenormalizedlattice. Theresult
unitcellconsistsof(a)thefirstgeneration,(b)thesecond
is tabulated in Table I for the above three energy eigen-
generation, (c) the third generation and (d) the fourth
values. It is explicitly seen that, for each of these energy
generationb=3SPGnetworks. Wehavesetǫ =ǫ =0
4 6
eigenvalues, with a sequential decrease in the value of η,
and t = 1. It is clearly seen form Fig. 6(c) and (d)
theLDOSattheC site,viz.,ρ increasesexponentially,
that flat, non-dispersive bands appear at E = 2 and at 4 4
E =1 √2–thethreerootsobtainedfromthe−equation indicating that in the limit η 0, the LDOS diverges
→
± at these special energy values. As we know, the average
E =ǫ 2t at the bare length scale and from the one
4,n n
− density ofstates(ADOS) is relatedto the groupvelocity
steprenormalizedlattice,asdiscussedinthelastsection.
v of the electron by the relation
Evenwith amodest unitcell structurefor the triangular g
array, the self (staggered) localized states corresponding
to the truly infinite system are showing up. Thus, we
feel tempted to identify these self-localized states as flat ρav ∝Z vg−1d3k (9)
bandanaloguesofsimilarstatesinquasionedimensional
systems [10]-[13]. However, we are now in a position to
appreciate that such states will be infinite in number as where, v dE/dk. Therefore, a diverging LDOS, lead-
g
∝
the size of the system approaches the thermodynamic ingto adivergingADOS,is indicativeofthe zero(orex-
limit. tremely low) group velocity and hence, a non-dispersive
character of the state. The energy eigenvalues, obtained
by construction, and corresponding to the self-localized
B. A study of the density of states eigenfunctions in the present example can thus be con-
sidered as the so called flat band states. We can thus
As a second check, we evaluate the local density of unravel infinite number of such non-dispersive modes in
states (LDOS) at the C and C sites. The results are an infinite b=3N SPG network.
4 6
7
E η ρ4(E) ρ6(E) lapbetweentheenvelopesofthewavefunctionatvarious
10−2 3.28 0.03 scalesoflength,andthat, thesubbandsaredenselypop-
10−3 15.66 0.07 ulated by extended wave functions. In addition, the en-
10−4 74.38 1.5 10−3 ergy E = 0 lying at the center of the spectrum exhibits
× a one cycle fixed point in the parameter space, that is,
2 10−5 355 3.2 10−4
− 10−6 1703.2 6.8×10−5 (ǫ4,n,ǫ6,n,|tn|)=(ǫ4,n+1,ǫ6,n+1,|tn+1|), for n≥0.
×
10−7 8701 1.34 10−5
×
10−8 40419 2 10−6
×
10−2 1.68 1.06
10−3 5.2 0.5
10−4 22.5 0.11
1 √2 10−5 106.4 2 10−2
− ×
10−6 506.24 5.4 10−3
×
10−7 2402.8 1.13 10−3
×
10−8 11454 2.4 10−4
×
10−2 1 7.5 10−2
×
10−3 4.41 1.75 10−2
×
10−4 20.91 3.8 10−3
×
1+√2 10−5 100 8 10−4
×
10−6 473 1.7 10−4
×
10−7 2249 3.6 10−5
×
10−8 10747 7.4×10−6 FIG. 7: (Color online) Local density of states at the (a) C4
and (b) C6 sites of an infinite b = 3 SPG network in the
TABLE I: The values of the local densities of states ρ4(E) absenceofamagneticfield. (c)LocaldensityofstatesataC4
and ρ6(E) at special energy eigenvalues E = 2, and 1 √2 sitewheneachelementaryplaquetteispiercedbyamagnetic
− ±
as obtained from the original lattice and its one-step renor- fluxΦ=Φ0/4, and(d)thesameat aC6 sitewithΦ=Φ0/4.
malized version corresponding to localized flat-band states. Wehavesetǫ4 =ǫ6 =0andt=1. Theimaginarypartadded
We have set the on-site potential ǫ=0 at each site, and the to theenergy is η=10−4.
energy is measured in unit of thehopping integral t.
The existence of the subbands of extended states does
not rule out the other localized states, though the non-
V. EFFECT OF A UNIFORM MAGNETIC dispersive,flatbandsapparentlydonotexist(or,atleast,
FIELD become undetectable). As, for an electron circulating
aroundaclosedloop,themagneticfluxplaysaroleequiv-
Asiswellknown,theFBSareunstableagainstpertur- alent to the wave vector [34], we illustrate the E-Φ dia-
bation. We examine the general character of the energy gram (Fig. 8) as an analogue of the dispersion relation
spectrum of a b = 3 SPG network when each elemen- in Fig. 6. The energy eigenvalues of the fractal network
taryplaquette isthreadedbyauniformmagneticfluxΦ. are seen to form mini bands as a function of the flux Φ
The presence of a magnetic field breaks the time rever- with period Φ . There are multiple, inter-twined band
0
sal symmetry of the electron-hopping along a bond, and crossings, and quite a dense distribution of eigenvalues,
this is taken care of by introducing a Peierls’ phase in forming almost continuous quasi-flat E-Φ ‘bands’. The
the hopping integral, viz., t texp(i2πΦ/3Φ ), where pattern is flux periodic with a period equal to one flux
0
→
Φ =hc/e is the fundamental flux quantum. quantum, and close observation of the E-Φ ‘band’ sub-
0
The RSRG method is used again to evaluate the sections reveals formation of interesting variants of the
LDOS, which is shown in Fig. 7(c) and (d) for the C Hofstadter butterflies [35]. The spectral landscape is a
4
and C sites respectively for Φ=Φ /4. We draw the at- quantumfractal,andencodingthegapswithappropriate
6 0
tentionofthe readertothe absolutely continuousshaded topological quantum numbers remains an open problem
portions in the LDOS. This is a generic character for all for such deterministic fractals.
values of the piercing flux. We have carefully scanned
the continuous subbands in arbitrarily small energy in-
tervals, and observed the flow of the hopping integral VI. TWO-TERMINAL TRANSMISSION
under the RSRG iterations. Extensive numerical search CHARACTERISTICS
reveals that for any energy belonging to the continua,
the hopping integral keeps on oscillating for an indefi- For the sake of completeness of the above discussion,
nite number of iteration loops. This implies that, for wehavecomputedthetwo-terminaltransmissioncharac-
any energy forming the continua there is non-zero over- teristicsofafinitesystem. Theprocedureisstandardand
8
FIG. 8: (Color online) Distribution of energy eigenvalues
against the magnetic flux Φ (in unit of Φ0 = hc/e) in an
infiniteb=3 SPGnetwork.
is often used to evaluate the transmission coefficient of
suchhierarchicalfractalstructures[30]. Thebasicideais FIG.9: (Color online) Transmission coefficient ofa4-thgen-
toclampthesysteminbetweentwosemi-infiniteperiodic erationb=3SPGnetworkwith(a)Φ=0,and(b)Φ=Φ0/4.
leads, the so called ‘source’ and the ‘drain’. The on-site Wehavechosen ǫ4 =ǫ6 =0, t=1, UL =0 and TL =3.
potentialatthe atomicsitesinthe leadsis . Thehop-
L
U
ping integral is in between the nearest neighboring
L
T
atomic sites in the leads. The finite sized network sand-
value. However, to save space, we refrain from showing
wiched in between the two ordered leads is then succes-
these figures.
sively renormalized to reduce it to an effective diatomic
molecule(dimer)[30]. Thetransmissioncoefficientofthe
lead-network-lead system then is given by a well-known
formula [36],
4sin2ka VII. CONCLUSION
T = (10)
2+ 2
|A| |B|
with, =[(M M )+(M M )coska] In conclusion, we have unravelled an infinity of flat
12 21 11 22
A − −
band-like, dispersionless eigenstates in the b = 3N class
and =[(M +M )sinka]
11 22
B of Sierpinski gasket fractals. The states are localized in
where, M refer to the dimer-matrix elements, written clustersoffinitesize,whichareeffectivelyseparatedfrom
ij
appropriately in terms of the on-site potentials of the similar clusters in the lattice by atomic sites where the
finalrenormalizedleft (L)andright (R)atomsatthe ex- amplitude of the wave function is zero. The topology
tremitiesofthefiniteSPGnetworkandtherenormalized of the lattice confines the dynamics of the electron in
hopping between them. coska = (E )/2 , and a such local clusters for these special sets of energy eigen-
L L
−U T
is the lattice constant in the leads which is set equal to values. The states have been obtained by explicit con-
unity throughout the calculation. struction and the inherent self similarity of the lattice is
It is apparent from Fig. 9 that, the low transmitting exploited through a real space rescaling of the system.
characterofthegasketintheabsenceofanexternalmag- The dispersionless character of the flat band states are
netic field (Fig. 9(a)) is enriched by high transmitting confirmedbyobtainingtheE-kdispersioncurvesofape-
windows in the presence of a magnetic flux (Fig. 9(b)). riodic array of the approximants of the true gasket, and
This corroborates the argument about the existence of also from the diverging character of the local density of
absolutely continuous subbands populated by extended statesatthosespecialenergyeigenvalues. Thetwotermi-
eigenfunctions. nal transport and the magnetic field induced absolutely
The transmission coefficient, as expected, is flux peri- continuoussubbandshavebeendiscussedindetails. The
odic, with period equal to one flux quantum. The fine intricate mixture of complexity, frustration and a deter-
structures of the Aharonov-Bohm oscillations in trans- ministic geometry may throw up new challenges in the
port are of course, sensitive to the chosen energy eigen- quantum Hall physics in such fractal geometries.
9
Acknowledgments fellowship. AC thanks Bibhas Bhattacharyya and San-
tanu K. Maiti for enlightening discussions.
AN acknowledgesa researchfellowship fromUGC, In-
dia and BP is thankful to DST, India for an INSPIRE
[1] R. Moessner and A. P. Ramirez, Physics Today, Febru- [18] G.-W.Chern,C.-C.Chien,andM.DiVentra,Phys.Rev.
ary,p.24 (2006). A 90, 013609 (2014).
[2] H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, [19] G.-B. Jo, J. Guzman,C. K.Thomas, P.Hosur, A.Vish-
T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. wanath,andD.M.Stamper-Kurn,Phys.Rev.Lett.108,
Ohta,Phys. Rev.Lett. 94, 227201 (2005). 045305 (2012).
[3] A. M. S. Macˆedo, M. C. dos Santos, M. D. Coutinho- [20] K.Shiraishi,H.Tamura,andH.Takayanagi,Appl.Phys.
Filho, and C. A. Macˆedo, Phys. Rev. Lett. 74, 1851 Lett. 78, 3702 (2001).
(1995). [21] I. Syˆozi, Prog. Theo. Phys.6, 306 (1951).
[4] O. Derzhko and J. Richter, Phys. Rev. B 70, 104415 [22] S. Miyahara, K.Kubo, H.Ono, Y.Shimoumura, and N.
(2004). Furukawa, J. Phys. Soc. Jpn. 74, 1918 (2005).
[5] O. Derzhko and J. Richter, Phys. Rev. B 72, 094437 [23] E. Domany, S. Alexander, D. Bensimon, and L. P.
(2005). Kadanoff, Phys. Rev.B 28, 3110 (1983).
[6] O. Derzko, J. Richter, and A. Honecker, J. Phys.: Conf. [24] R. Rammal, J. Phys. (Paris) 45, 191 (1984); Phys. Rev.
Series 145, 012059 (2009). B 28, 4871 (1983).
[7] O. Derzhko, J. Richter, A. Honecker, M. Maksymenko, [25] J. R. Banavar, L. P. Kadanoff, and A. M. M. Pruisken,
and R.Moessner, Phys.Rev. B 81, 014421 (2010). Phys. Rev.B 31, 1388 (1985).
[8] A. Mielke, J. Phys. A: Math. Gen. 24, L73 (1991); 24, [26] X. R.Wang, Phys. Rev.B 51, 9310 (1995).
3311 (1991). [27] J. T. Chalker, T. S. Pickles, and P. Shukla, Phys. Rev.
[9] H.Tasaki, Phys. Rev.Lett. 69, 1608 (1992). B 82, 104209 (2010).
[10] A. A. Lopes and R. G. Dias, Phys. Rev. B 84, 085124 [28] A. Chakrabarti, J. Phys.: Condens. Matter 8 10951
(2011). (1996).
[11] M. Hyrk¨as, V. Apaja, and M. Manninen, Phys. Rev. A [29] S.Kirkpatrickand T.P.Eggarter, Phys.Rev.B6, 3598
87, 023614 (2013). (1972).
[12] A. A. Lopes, B. A. Z. Ant´onio, and R. G. Dias, Phys. [30] B. Pal and A. Chakrabarti, Phys. Rev. B 85, 214203
Rev.B 89, 235418 (2014). (2012).
[13] S.Flach,D.Leykam,J.D.Bodyfelt,P.Matthies,andA. [31] A. Nandy, B. Pal, and A. Chakrabarti, Phys. Lett. A
S.Desyatnikov,Europhys.Lett. 105, 30001 (2014). 378, 3144 (2014).
[14] D. Leykam, S. Flach, O. B.-Treidel, and A. S. Desyat- [32] T.Sto˘si´c,B.Sto˘si´c,S.Milo˘sevi´c,andH.E.Stanley,Phys.
nikov,Phys.Rev.B 88, 224203 (2013). Rev. E49, 1009 (1994).
[15] M. Vanevi´c, V. M. Stojanovi´c, and M. Kindermann, [33] K. Nikoli´c and A. MacKinnon, Phys. Rev. B 47, 6555
Phys.Rev.B 80, 045410 (2009). (1993).
[16] M. Ezawa, arXiv:1404.5788v1 [cond-mat.mes-hall] [34] H.-F. Cheung, Y. Gefen, E. K. Riedel, and W.-H. Shih,
(2014). Phys. Rev.B 37, 6050 (1988).
[17] T.G.Pedersen,C.Flindt,J.Pedersen,N.A.Mortensen, [35] D. R.Hofstadter, Phys. Rev.B 14, 2239 (1976).
A.-P. Jauho, and K. Pedersen, Phys. Rev. Lett. 100, [36] Y. Liu and K. A.Chao, Phys.Rev. B 34, 5247 (1986).
136804 (2008).