Spectral and transport properties of the two-dimensional Lieb lattice

2.4 MB·English

by  M. Niţă

Spectral and transport properties of the
two-dimensional Lieb lattice
M. Ni¸t˘a1, B. Ostahie1,2 and A. Aldea1,3
1National Institute of Materials Physics,
POB MG-7, 77125 Bucharest-Magurele, Romania.
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1 2Department of Physics, University of Bucharest
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3Institute of Theoretical Physics, Cologne University, 50937 Cologne, Germany.

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(Dated: January 10, 2013)
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Abstract

]
l
l The specific topology of the line centered square lattice (known also as the Lieb lattice) induces
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remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac
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e cone in the low energy spectrum, and the peculiar Hofstadter-type spectrum in magnetic field. We
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. studyherethepropertiesofthefiniteLieblatticewithperiodicandvanishingboundaryconditions.
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m We find out the behavior of the flat band induced by disorder and external magnetic and electric
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d fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux
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there are edge states with nontrivial behavior. The specific class of twisted edge states, which
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have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to

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v the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where
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0 these states are located is revealed. The diamagnetic moments of the bulk and edge states in the
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Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.
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3 PACS numbers: 73.22.-f, 73.23.-b, 71.70.Di, 71.10.Fd
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1
I. INTRODUCTION
The interest in the line centered square lattice, known as the 2D Lieb lattice, comes from
the specific properties induced by its topology. The lattice is characterized by a unit cell
containing three atoms, and a one-particle energy spectrum showing a three band structure
with electron-hole symmetry, one of the branches being flat and macroscopically degenerate.
Fortheinfinitelattice, thethreeenergybandstoucheachotheratthemiddleofthespectrum
(taken as the zero energy ), and the low energy spectrum exhibits a Dirac cone located at
the point Γ = (π,π) in the Brillouin zone. Except for the presence of the flat band, the
Lieb lattice shows similarities with the honeycomb lattice in what concerns both spectral
and transport properties. For instance, besides the presence of the Dirac cone, the energy
spectrum in the presence of the magnetic field shows also a double Hofstadter picture, with
√
the typical B dependence of the relativistic bands on the magnetic field B [1]. The Hall
resistance of the two systems in the quantum regime behaves alike, but the step between
consecutive plateaus equals h/e2 in the Lieb case (instead of h/2e2 for graphene) because of
the presence of a single Dirac cone per BZ. An all-angle Klein transmission is proved by the
relativistic electrons in the Lieb lattice [2–4].
There are more lattices that support flat bands, however it is specific to the Lieb lattice
that the band is robust against the magnetic field, while other lattices develop dispersion
at any B (cid:54)= 0. The intrinsic spin-orbit coupling does not affect the flat band either, but
opens a gap at the touching point Γ, the Lieb system becoming in this way a quantum spin
Hall phase [3, 5]. Topological phase transitions driven by different parameters are studied
in [6, 7]. The zero-energy flat bands became a topic of intense study also for other reasons:
they may allow for the non-Abelian FQHE [8–10] or for ferromagnetic order and surface
superconductivity [11–13].
In this paper we address the properties of the finite (mesoscopic) Lieb lattice with em-
phasis on some features of the flat band and of the edge states which are specific to this
lattice. We adopt the spinless tight-binding approach, and the spectral properties are exam-
ined under both periodic and vanishing boundary conditions applied to the system described
in Fig.1. In section II we find that the zero energy flat band exists independently of the
boundary conditions. It turns out, however, that in the periodic case the band is built up
only from B- and C- orbitals, while in the other case the A-type orbitals are also involved
2
(seeFig.1). Weprovethisanalyticallybycalculatingtheeigenfunctioninbothsituations. In
thiswaywe alsofind out thatfor confinedsystems (i.e., withvanishingboundary conditions)
the degeneracy of the flat band equals N + 1 (N is the number of cells of the meso-
cell cell
scopic plaquette). We find in section IIIA that for a confined plaquette two levels separate
from the bunch when a perpendicular magnetic field is applied, such that the degeneracy is
reduced by 2. This is proved in a perturbative manner for the general case, however it can
be observed more easily by the use of the toy model consisting of two cells only.
Next, we study how the flat band degeneracy is lifted by disorder and by an external
electric field applied in-plane. An exotic result is that the extended states of the disordered
flat band in the presence of a magnetic field behave according to the orthogonal Wigner-
Dyson distribution although the unitary distribution is expected. When an electric field
is applied, the flat band splits in a Stark-Wannier ladder whose structure is analyzed by
calculating the diamagnetic moments of the states in crossed electric and magnetic fields.
In section IIIB we study the edge states which fill the gaps of the double Hofstadter
butterfly when the magnetic field is applied on the confined Lieb plaquette. We identify
three types of such states. The conventional edge states located between the Bloch-Landau
bands and also between Dirac-Landau bands (i.e., the relativistic range of the spectrum)
differ, as expected, in their chirality. Additionally, we detect twisted edge states situated in
the magnetic gap which protect the zero-energy band, coming in bunches and characterized
by an oscillating chirality as function of the magnetic field. The twisted edge states show
remarkable properties: surprisingly, they are not robust to disorder, as the other types of
edge states are, and does not carry transverse current (i.e., the QHE vanishes in the energy
range covered by these states). The last property comes from a specific symmetry of the
transmittance matrix which is discussed in section IV.
Finally, one has to note that the line centered square lattices are found in nature as
Cu−O [14] planes in cuprate superconductors, and can be engineered as an optical lattice
2
[3, 15].
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II. THETIGHT-BINDINGMODELFORTHELIEBLATTICE:PERIODICVER-
SUS VANISHING BOUNDARY CONDITIONS
Our aim is to point out specific aspects of the confined Lieb plaquette from the point
of view of spectral and transport properties. In order to allow for a comparison we shortly
describe also the case of the infinite system, with and without magnetic field, although the
eigenvalue problem is already known from the literature. We remind that the continuous
√
model for the infinite Lieb system in perpendicular magnetic field [3], shows the B depen-
dence on the magnetic field of the eigenenergies in the relativistic range. The information
obtained in the long-wave approximation of the Schro¨dinger equation, concerning the de-
pendence on B of the Bloch-Landau or Dirac-Landau bands are recaptured in the spectrum
of the discrete tight-binding model (Fig.5a) together with effects coming from the periodic
lattice and finite edges.
Inthissection, startingfromthetight-bindingHamiltonian, webuiltuptheeigenfunction
of the periodic and finite Lieb plaquette, and prove the degeneracy and structure of the zero-
energy flat band. The crossover from the simple Hofstadter spectrum of the simple square
lattice to the Lieb spectrum characterized by a double butterfly, magnetic gap and a flat
band is shown in Fig.3.
Y
C (n,m+1) (n+1,m+1) (n+2,m+1)
(m+1)a A
C (n,m) (n+1,m) (n+2,m)
t
y t
x
ma
A B A B
O na (n+1)a (n+2)a X
FIG. 1: (Color online) The Lieb lattice: the unit cell contains three atoms A,B and C; indices
(n,m)identifythecell; t ,t arethehoppingintegralsalongthedirectionsOxandOy,respectively;
x y
a is the lattice constant.
The Lieb lattice is a 2D square lattice with centered lines as shown in Fig.1. It is
characterized by three atoms (A,B,C) per unit cell, the connectivity of the atom A being
4
equal to four, while the connectivity of atoms B and C equals two.
Introducing creation {a† ,b† ,c† } and annihilation {a ,b ,c } operators of the
nm nm nm nm nm nm
localized states |A >,|B >,|C > (where (nm) stands for the cell index and the
nm nm nm
letters A,B,C identify the type of atom), the spinless tight-binding Hamiltonian of the
Lieb lattice in perpendicular magnetic field reads:
H = (cid:80) E a† a +E b† b +E c† c
nm a nm nm b nm nm c nm nm
+t e−iπmφa† b +t eiπmφa† b +t a† c +t a† c
x nm nm x nm n−1,m y nm nm y nm n,m−1
+t e−iπmφb† a +t eiπmφb† a +t c† a +t c† a , (1)
x nm n+1,m x nm nm y nm nm y nm nm+1
where φ is the flux through the unit cell of the Lieb lattice measured in quantum flux units;
(cid:126)
we mention that the vector potential has been chosen as A = (−By,0,0).
The presence of a spectral flat band can be noticed already in the simplest case of the
periodic boundary conditions and vanishing magnetic flux. Assuming that the lattice is
composed of Nx = N cells along Ox and Ny = M cells along Oy, the Fourier transform
cell cell
c = c = √1 (cid:80) c ei(kxn+kym) (and similarly for all the other operators) yields the
(cid:126)k kx,ky NM n,m nm
k-representation of the Hamiltonian described by a 3×3 matrix:
  
E ∆∗(k ) Λ∗(k ) a
a x y (cid:126)k
(cid:88)(cid:16) (cid:17)  
H = a† b† c† ∆(k ) E 0  b , (2)
(cid:126)k (cid:126)k (cid:126)k  x b  (cid:126)k 
  
(cid:126)k Λ(k ) 0 E c
y c (cid:126)k
where k = 2πp/N (p = 1,..,N), k = 2πq/M (q = 1,..,M), and the notations ∆(k ) =
x y x
t (1+eikx) , Λ(k ) = t (1+eiky) has been used. With the choice E = E = E = 0, one
x y y a b c
obtains the following eigenvalues:
Ω ((cid:126)k) = ±(cid:112)|∆|2 +|Λ|2 = ±2(cid:113)t2cos2(k /2)+t2cos2(k /2),
± x x y y
(cid:126)
Ω (k) = 0, (3)
0
where Ω are the energies of the upper and lower band, respectively, and Ω is the nondis-
± 0
persive (flat) band of the Lieb lattice. The most interesting point in the BZ is the point
Γ = (π,π), where in the case of the infinite lattice the three branches are touching each
other. The expansion of the functions ∆(k ) and Λ(k ) about this point gives rise to a Dirac
x y
cone (massless) spectrum:
(cid:113)
Ω = ± t2k2 +t2k2. (4)
± x x y y
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On the other hand, the expansion of the same functions about R = (0,0) shows a parabolic
dependence:
k2 k2
Ω = ±(cid:0) x + y (cid:1), (5)
±
2m 2m
x y
where m ,m are effective masses along the two directions.
x y
Other relevant points in the BZ are M = (π,0) and (0,π), which prove to be saddle
points in the spectrum as it can be noticed also in Fig.2. Above and below the corresponding
energy E = ±2t (where we considered t = t = t) the effective mass exhibits opposite signs
x y
inducing the change of sign of the Hall effect which is visible in Fig.15.
For comparison’s sake, we remind that the energy spectrum of the honeycomb lattice
contains two cones per BZ, and that the saddle point occurs at the energy E = ±t. The
tight-binding spectrum of the graphene extends over the interval [-3t,3t], while for the Lieb
√ √
lattice the interval is [−2 2t,2 2t].
FIG. 2: (Color online) The energy spectrum of the infinite Lieb lattice. (left) The case E =
a
E = E = 0 when the three bands (two dispersive and one flat) get in contact at (cid:126)k = (π,π).
b c
At low energy the dispersion is linear giving rise to Dirac cones. (right) The staggered case
E = 0,E = E = 1 when the spectrum is gapped and the rounding of cones is obvious.
a b c
Inordertogetsupplementaryinformationabouttheoriginoftheflatbandletusconsider
the staggered case E = 0,E = E = E . Then, a gap is expected in the energy spectrum,
a b c 0
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and, indeed, the eigenvalues are now [4]:
1 (cid:113)
Ω ((cid:126)k) = (cid:2)E ± E2 +4(|∆|2 +|Λ|2) (cid:3), Ω ((cid:126)k) = E . (6)
± 2 0 0 0 0
The new spectrum is shown in Fig.2b, where one notices the persistence of the flat band,
which is however shifted to E = E . Since the energy E = E corresponds to the atomic
0 0
level of the orbitals B and C, the result argues that the flat band states are created only
by this type of orbitals. The gap induced by the staggered arrangement is accompanied by
the rounding of the cones, that indicates a non-zero effective mass in the low-energy range
of the two spectral branches, as it can be observed in Fig.2b.
In what follows we shall calculate the eigenfunctions of the finite Lieb lattice, imposing
first periodic conditions, and then the vanishing boundary conditions proper to the confined
plaquette. LetΨ betheeigenfunctionsoftheLieblatticewithperiodicboundaryconditions
(cid:126)k
built up as the linear combination:
Ψ = α a†|0 > +β b†|0 > +γ c†|0 >, (7)
(cid:126)k (cid:126)k (cid:126)k (cid:126)k (cid:126)k (cid:126)k (cid:126)k
where the coefficients α ,β ,γ satisfy the equations:
(cid:126)k (cid:126)k (cid:126)k
E α + ∆∗(k )β +Λ∗(k )γ = Eα
a (cid:126)k x (cid:126)k y (cid:126)k (cid:126)k
∆(k )α +E β = Eβ
x (cid:126)k b (cid:126)k (cid:126)k
Λ(k )α +E γ = Eγ . (8)
x (cid:126)k c (cid:126)k (cid:126)k
Then, the functions corresponding to the eigenvalues Ω and Ω in Eq.(3) read:
0 ±
1
Ψ0((cid:126)k) = (cid:0)Λ∗(k )b† −∆∗(k )c†(cid:1)|0 > (9)
(cid:112)|∆|2 +|Λ|2 y (cid:126)k x (cid:126)k
1 ∆(k ) Λ(k )
Ψ±((cid:126)k) = (cid:0)±a† + x b† + y c†(cid:1)|0 > . (10)
2 (cid:126)k (cid:112)|∆|2|+|Λ|2 (cid:126)k (cid:112)|∆|2 +|Λ|2 (cid:126)k
In the case of periodic conditions applied to the finite plaquette there are some subtleties
concerning the band degeneracy which become unimportant in the limit of infinite system.
(cid:126)
It is obvious from Eqs.(9-10) that the three bands come into contact at k = (π,π), however
(cid:126)
this value of k is allowed only if both N and M are even. In this case the flat band at E = 0
is (N +2)- fold degenerate, otherwise all the three bands are N -fold degenerate (where
cell cell
the number of cells N = NM).
cell
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The expression of Ψ0((cid:126)k) in Eq.(9) indicates again that the flat band of the periodic lattice
is composed only from orbitals of the type B and C. On the other hand, we shall see below
that in the case of vanishing boundary conditions the zero-energy eigenfunction may sit also
on the A−type sites, and that the degeneracy of the flat band becomes N +1.
cell
The periodic boundary conditions can be used in the presence of a uniform perpendicular
magnetic field for rational values of the magnetic flux φ = p/q resulting in a spectrum
composedoftwoHofstadterbutterfliessimilartothecaseofthehoneycomblattice. However,
in contradistinction to the honeycomb lattice, one notices the presence of a dispersionless
band at E = 0, which is flat with respect to the variation of the magnetic flux, and is
protected by a gap opened at B (cid:54)= 0 [2, 3] . The spectrum exhibits Bloch-Landau bands at
the extremities and also relativistic Dirac-Landau bands towards the middle. The two types
of bands are distinguished by opposite chirality dE/dφ and by different dependence on the
magnetic field.
The periodic boundary conditions discussed above can be properly used for describing
infinite lattices, however when interested in mesoscopic plaquettes they have to be replaced
with vanishing boundary conditions. We intend to identify the differences introduced by the
finite size, which will turn out to be non-trivial in the case of the Lieb lattice.
For the confined Lieb lattice, the eigenfunctions can be obtained as combinations of
functions Eq.(9) or Eq.(10) with coefficients that ensure the vanishing of the eigenfunction
along the edges. As a technical detail we mention that (along the Ox direction, for instance)
the finite plaquette begins with the atom A in the first cell, and also ends with an atom A
(cid:126)
whichbelongtothe(N+1)-thcell. Thismeansthatthewavefunction|Φ(k) >shouldvanish
at the site B in the 0-th and (N +1)-th cell, i.e.: < Φ((cid:126)k)|b† |0 >=< Φ((cid:126)k)|b† |0 >= 0.
N+1,m 0,m
Similarly, the vanishing condition along Oy occurs at the site C in the 0-th and (M +1)-th
cell along this direction, i.e.: < Φ((cid:126)k)|c† |0 >=< Φ((cid:126)k)|c† |0 >= 0.
n,M+1 n,0
In the localized representation, which is the proper one in the case of confined systems,
the eigenfunctions |Φ0((cid:126)k) > corresponding to E = Ω = 0 look as follows:
0
|Φ0((cid:126)k) >= (cid:114) 2 (cid:114) 2 N(cid:88)+1M(cid:88)+1(cid:0) 2tycosk2y sink n sink (m− 1) b† |0 >
N +1 M +1 (cid:112)|∆|2 +|Λ|2 x y 2 nm
n=1 m=1
− 2txcosk2x sink (n− 1) sink m c† |0 > (cid:1), (11)
(cid:112)|∆|2 +|Λ|2 x 2 y nm
wherek ,k areobtainedfromtheconditionthatthewavefunctionvanishesattheboundary,
x y
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and equal k = pπ/(N+1) (p=1,..,N+1) and k = qπ (q=1,..,M+1). Since the situations
x y M+1
p = N +1 (at any q) and q = M +1 (at any p), generate |Φ0 >= 0, we are left in Eq.(11)
with only NM non-vanishing degenerate orthogonal eigenfunction.
The eigenfunctions |Φ±((cid:126)k) > corresponding to the other two energy branches can be
written similarly as:
(cid:115)
N+1M+1
|Φ±((cid:126)k) >= 2 (cid:88) (cid:88) (cid:0)±sink (n− 1)sink (m− 1) a† |0 >
(N +1)(M +1) x 2 y 2 nm
n=1 m=1
2t coskx 1
+ x 2 sink n sink (m− ) b† |0 >
(cid:112)|∆|2 +|Λ|2 x y 2 nm
2t cosky 1
+ y 2 sink (n− )sink m c† |0 > (cid:1), (12)
(cid:112)|∆|2 +|Λ|2 x 2 y nm
where states of the type A are this time also present. One can readily see that the number of
non-vanishing states in each spectral branch is (N+1)(M+1)−1, since the point Γ = (π,π)
has to be treated separately. This is because its corresponding energy vanishes and the state
should be counted in the flat band. In this case the wave function becomes:
(cid:114) (cid:114) N+1M+1
1 1 (cid:88) (cid:88)
|Φ0 >=: |Φ±(π,π) >= (−1)n+m a† |0 > . (13)
a N +1 M +1 nm
n=1 m=1
For the finite Lieb plaquette with vanishing boundary conditions, one may conclude that the
flat band degeneracy equals NM+1, while each other branch contains NM+N+M states,
so that the total number of states equals indeed the number of sites 3NM +2(N +M)+1.
In the presence of the magnetic field, the vanishing boundary conditions give rise to edge
states which fill the gaps of the Hofstadter spectrum corresponding to the periodic system.
Besides the edge states existing in the energy range of the Bloch-Landau levels (which are
the only met for the finite plaquette with simple square structure), there are edge states
in the relativistic range which show opposite chirality [16], but also non-conventional edge
states lying in the central gap which protects the zero energy dispersionless band. This
last new class of edge states exhibits oscillating chirality when changing either the magnetic
flux or the Fermi energy. These states will be studied in the next chapter. The fate of the
zero-energy states in the presence of confinement will be discussed in the next section.
The Lieb lattice can be generated from the simple square lattice by extracting each the
second atom when moving along both Ox and Oy direction. Formally, this means either
to push to infinity the energy E of these atoms or to cut down the hopping integrals t(cid:48)
d
9
connecting them to the neighboring atoms, and it is instructive to follow the change of the
spectrum when E /t(cid:48) → ∞. By driving the system in this way from 1 to 3 atoms/unit cell,
d
the lattice periodicity is doubled along both directions, and the flat band is generated. The
middle panel of Fig.3 shows how the butterfly wings break off during the process giving rise
to the relativistic range.

4 3
2.4
3 2 1.6
2
Eigenvalues-101 Eigenvalues-011 Eigenvalues-000...088
-2 -1.6
-2
-3
-2.4
-40.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -30.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
FIG. 3: The energy spectrum as function of the magnetic flux for three values of the hopping
integral t(cid:48) (see text) : (left) t(cid:48) = 1, corresponding to the simple square lattice, (middle) t(cid:48) = 0.5,
(right) t(cid:48) = 0, corresponding to the square lattice with centered lines (Lieb lattice). φ is the
magnetic flux through the unit cell of the simple lattice measured in quantum flux units. The
Hofstadter butterfly is obvious for t(cid:48) = 1, while a doubled butterfly results for t(cid:48) = 0 in each of the
intervals φ ∈ [0,0.25], φ ∈ [0.25,0.5], etc. (one has to keep in mind that the flux through the Lieb
unit cell is four times larger than φ). The energy is measured in units of hopping integral t.
III. SPECIFIC ASPECTS OF THE FINITE LIEB PLAQUETTE IN MAGNETIC
FIELD: ZERO ENERGY FLAT BAND AND TWISTED EDGE STATES
A. The properties of the flat band
There are several pertinent questions which can be asked concerning the flat band in the
energy spectrum of the Lieb finite system: what is the degeneracy, what is the response to
the magnetic and electric field and to the disorder?
Let us find first the conditions which should be satisfied by the zero energy eigenfunction
Ψ . Let H be the tight-binding Hamiltonian of a finite system and Ψ its eigenfunctions:
E=0 E
(cid:88) (cid:88) (cid:88)
H = E |n >< n|+ t |n >< m|, Ψ = α |n >, (14)
n nm E n
n n,m n
10