Flat bands with non-trivial topology in three dimensions

0.61 MB·English

by  C. Weeks

Flat bands with non-trivial topology in three dimensions
C. Weeks and M. Franz
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
We construct a simple model for electrons in a three-dimensional crystal where a combination
of short-range hopping and spin-orbit coupling results in nearly flat bands characterized by a non-
trivial Z topological index. The flat band is separated from other bands by a bandgap ∆ much
2
largerthanthebandwidthW. Whentheflatbandispartiallyfilledweshowthatthesystemremains
non-magneticforasignificantrangeofrepulsiveinteractions. Inthisregimeweconjecturethatthe
true many-body ground state may become a three-dimensional fractional topological insulator.
2
1 Introduction. – When single-particle states of bosons rochlore and the perovskite lattice.18,20,21 In their sim-
0
or fermions comprise a flat band, the effect of interac- plestformhowevertheabovetight-bindingmodelsdonot
2
tionsisgenericallynon-perturbativeandcanleadtonew constituteaplatformsuitableforthestudyofexoticcor-

n states of quantum matter. For 2D electrons in a strong related phases for at least two reasons. First, in all cases
a
perpendicular magnetic field the flatness of Landau lev- the flat bands touch other dispersing bands, usually in
J
els combined with non-zero Chern numbers leads to the a quadratic fashion. The interactions are thus bound to
2
spectacularfractionalquantumHall(FQH)effects1,2 and mix states from adjacent bands and this is clearly detri-
2
otherexoticstatessuchastheWignercrystal.3 Recently, mental to the emergence of correlated phases. Second,

] models for spinless fermions moving in two-dimensional evenwhentheHamiltonianismodifiedtoopenagap,e.g.
ll lattices have been constructed that exhibit nearly-flat byaddingSOCasinRef.20,theresultingflat(ornearly-
a bands with non-zero Chern numbers, but remarkably in flat)bandsareinallknowncasestopologicallytrivial. In
h
zero external magnetic field.4–7 It has been proposed, such topologically trivial flat bands the eigenstates can
-
s on the basis of heuristic arguments4,5 and numerical be chosen as exponentially localized around lattice sites
e simulations,6,8,9 that at partial filling repulsive interac- and, at partial filling, interactions will typically select a
m
tions may drive these systems into the zero-field FQH crystalline ground state with broken translational sym-
t. states. Althoughitisnotclearapriorihowtoexperimen- metry and not an exotic featureless liquid that underlies
a
tallyrealizetheabovesituationinaphysicalsystem,the theFQHeffect. AsalreadyarguedinRefs.4–6,20anat-
m
prospect of engineering systems that can support frac- uralplacetoseeksuchexoticphasesisinaflatbandthat
- tional excitations without involving a magnetic field has is topologically non-trivial.
d
led to significant interest recently in flat or nearly flat
n It is well known that in 2D a non-zero Chern number
o bands in various lattice models. represents a topological obstruction to the formation of
c In this Communication we ask whether it is possi- exponentially localized Wannier states.22 It is precisely
[
ble to achieve a flat band with non-trivial topology in this obstruction that tilts the balance in favor of FQH

2 a system of electrons in three spatial dimensions. It is states in partially filled Landau levels (although Wigner
v well known that the concept of the first Chern number crystal phases are still known to arise at very low fill-
7 (and the FQH physics) does not generalize to three di- ing fractions). For Z -odd phases of -invariant band
4 2 T
mensions. However, recent advances in the theory of insulators in 2D and 3D a similar topological obstruc-
4
time-reversal ( ) invariant band insulators have estab- tionexists23 ifoneinsistsonWannierstatesthatrespect
1
. lished a uniqueTly three-dimensional Z2-valued topologi- . Thisconsiderationsuggeststhatinteractingelectrons
1 calinvariant10–12 calledν which,aswearguebelow,can Tpartiallyfillingaflatbandwithanon-trivialZ indexin
1 0 2
playasimilarrole. Takingaspecificexampleofelectrons a 3D crystal could either (i) spontaneously break and
1
T
1 moving in the 3D edge-centered cubic (perovskite) lat- form a Wigner crystal, or (ii) remain -invariant and
T
: tice we show that a suitable combination of short-range form one of the proposed 3D fractional topological in-
v
hoppings and spin-orbit coupling (SOC) terms can give sulators characterized either by spin-charge separation13
i
X rise to a nearly flat band with a non-trivial Z index orbyfractionalmagneto-electriceffectcontrolledbyfrac-
2
r ν0 = 1. With the chemical potential inside the bandgap tional values of the axion parameter θ accompanied by
a suchcrystalwouldbeastrongtopologicalinsulatorwith ground state degeneracy.14–16
an odd number of topologically protected gapless states
Inthefollowingwelaygroundworkforfutureinvestiga-
associated with all of its surfaces. At partial filling and
tions of these 3D exotic phases by constructing a simple
in the presence of repulsive interactions we argue that
tight-bindingmodelwhosespectrumhasaflatbandchar-
the system may become a three-dimensional ‘fractional’
acterized by ν = 1 and is separated from other bands
topological insulator.13–16 0
by a large gap. We remark that constructing a flat band
Many lattice models are known to support completely with non-trivial topology in a 3D crystal is mathemati-
flat bands. In 2D these include p-orbitals in the hon- callymoreconstrainedproblemthanin2Danditisbyno
eycomb lattice,17 as well as s-orbitals in the kagome, meansobviousthatthiscanbeachievedwithshort-range
dice and Lieb lattices.18–20 3D examples include the py- hoppings only.
2
The model. – As a first step in the program outlined (a) (c) 4
a
abovewestudyasimpletight-bindingmodelwithintrin- 2
sicSOCtermsonthe2DLieblattice,seeninFig.1a,and 2
E
the 3D peroviskite lattice seen in Fig. 2a. The relevant 1 2 t
a
Hamiltonian reads 1
0
H = (cid:88) t c† c δ (cid:88) c† c 3
− ij iα jα− iα iα
(cid:104)ij(cid:105)α i∈corner −20 π/2 π
(cid:88) (1) kx
+iλ (d1ij ×d2ij)·σαβc†iαcjβ, (b) kx (d)
(cid:104)(cid:104)ij(cid:105)(cid:105)αβ
101
where c† creates an electron of spin α on site i of the
iα ∆
lattice, tij are the hopping amplitudes which we take W
equaltot , t ,andt forthefirst, secondandthirdnear- 100
1 2 3
est neighbor sites respectively, and δ is an onsite energy
for the corner sites (indicated as filled circles in Figs. 1a 10−1
and2a). Lastly, λrepresentstheamplitudeforthenext- ky 0.0 0.2 0.4 x 0.6 0.8 1.0
nearest neighbour SOC where d1 and d2 are the two
ij ij
unit vectors along the nearest neighbour bonds connect-
FIG. 1: (Color online)(a) Lieb lattice showing 3-site ba-
ing site i to its next-nearest neighbour j and σ is the
sisinunitcellandbasisvectorsa anda . Secondandthird
1 2
vector of Pauli spin matrices. neighborhoppingsareindicatedbyredandgreendottedlines,
In Ref. 20, we showed that for nearest neighbor hop- respectively. (b) Tight-binding dispersion for H after tun-
k
ping (t only) and non-zero λ both Lieb and perovskite ingparameterstoachievemaximalfigureofmerit∆/W. For
1
lattices became topological insulators possessing a non- claritythebulkenergybandsareplottedovertheshiftedBril-
trivial Z2 invariant. In both cases the bands carrying louin zone 0 ≤ kx ≤ π/a, 0 ≤ ky ≤ π/a where a represents
non-trivial invariants were strongly dispersing. From the length of the nearest neighbor bond. (c) Bandstructure
forstripofN =10unitcellswithopenboundaryconditions
here then, our goal is to flatten out these bands by tun- y
along the y direction and infinite along x with the same pa-
ing the hopping strengths t , t , λ and the onsite energy
2 3 rametersasin(b). (d)Semi-Logplotshowingtheevolutionof
δ without leaving the topological phase. Also, we want
thegapasonemovesbetweenaknowntopologicalinsulating
the flat band to be separated from all other bands by a
phase and the final set of finely tuned parameters.
large gap ∆ so that interactions do not mix states from
different bands. The relevant figure of merit,4–6 then, is
the ratio of the flat-band width W to the bandgap ∆.
Lieb lattice. – As a warm-up exercise we first consider the topological phase.20 The result for ∆/W, starting at
the valence band in the 2D Lieb lattice. Going over to
λ = 0.1 (t = 1) then adiabatically tuning x from 0 to
1
momentum space and diagonalizing the Hamiltonian
k 1, for the same final set of parameters in H, can be seen
H
that follows from Eq. (1), the result of the tuning proce-
in Fig. 1(d). The ratio increases from its initial value
dure can be seen in Fig. 1b.
∆/W 0.17 to its largest value 40.45 at x=1, without
The procedure used for the tuning was the simplest ≈
closing the gap. The system is seen to remain in the
available, namelyintroducingtheparametersonebyone
topological phase.
and modifying the values by hand until the band was as
Whatcanonesayaboutthefateofthegroundstatein
flat as we could achieve. For the following set of val-
the presence of repulsive interactions when the flat band
ues: t = 1, t = 0.17, t = 0.114, λ = 0.303 and
1 2 − 3 − is partially filled? Close to half filling, since the spin
δ = 0.725, the ratio of the bandwidth to the bandgap,
− up and down electrons are decoupled, one expects the
∆/W 40.45. A systematic numerical minimization
≈ Stoner instability towards the ferromagnetic state. If we
technique could probably improve upon our result, but
parametrizetheinteractionbytheusualon-siteHubbard
comparing to previous results on the kagome, honey-
U thenagap U shouldopenbetweenthemajorityand
comb, square, checkerboard and ruby lattices,4–7 this ∼
minority bands when W < U < ∆. For large enough U
value is already quite respectable.
one can focus on the spin-polarized electrons in the ma-
To show that the system has not migrated away from
jorityband. Inthepresenceofsufficientlystrongresidual
thetopologicalphaseduringtheparametertuningstage,
interactions (e.g. nearest-neighbor repulsion V) and at
we first solve the system in a strip geometry numerically
fractional filling, this problem becomes similar to that
using the same parameters as above. This verifies that
considered in Refs. [4–6] with spinless fermions. Anal-
the spin-filtered gapless edge states, seen in Fig. 1c, in-
ogous arguments then suggest the emergence of FQH
deed persist. We also consider the Hamiltonian
states under favorable conditions. Another possibility
H(cid:48)(x)=(1 x)H +xH, (2) is the formation of a -invariant fractional topological
− 0 insulator24 whichcanbTepicturedastwodecoupledFQH
whereH includesonlyt andλandisknowntosupport states for the two spin species. Such a state might be
0 1
3
remainsfiniteacrosstheentirerangefromitsinitialvalue
(a) (c)
∆/W 0.15 to its largest value 17.56 at x = 1. We
4 conclud≈e that the flat-band carries Z index (1;111).
2
a3 E 2
t Now imagine that the (doubly degenerate) flat band
4 0 is partially filled and consider the effect of repulsive in-
a teractions. Unlike the 2D case discussed above where
3 2 (cid:239)2
the residual U(1) spin symmetry is preserved despite the
1 2 a1 Γ A L H Γ presence of SOC, in the 3D strong topological insula-
tor the SU(2) spin symmetry is completely broken. In
(b) kz (d)
this situation it is not clear what the leading instabil-
A 101
ity might be. In a similar flat-band setting describing
Ir-based pyrochlore Y Ir O Pesin and Balents13 argued
L H ∆ 2 2 7
W foranexotic -invariantspin-chargeseparatedtopologi-
100 T
cal Mott insulator while others found more conventional
Γ ky magneticphases.25,26 Below,weinvestigatethemagnetic
instabilitiesofourmodelinthesimplestcasewithon-site
X M 10(cid:239)1 repulsion and at exact half filling of the flat bands. We
kx 0.0 0.2 0.4 x 0.6 0.8 1.0 show that the system remains non-magnetic when inter-
action strength is below the critical value which is large
FIG.2: (Coloronline)(a)Perovskitelatticeshowing4sitesin compared to the bandwidth W. In this regime, there-
unitcellalongwithbasisvectors. Secondandthirdneighbor fore, the true many-body ground state could become a
hoppingsareindicatedbyredandgreendottedlines,respec- -invariant fractional topological insulator.
tively. (b) High symmetry points in the Brillouin zone. (c) T
Bandstructure inside bulk along path of high symmetry for Magnetic Instabilities. – We extend the Hamiltonian
H withfinelytunedparameterset. (d)Semi-Logplotshow-
k
ing the evolution of the gap as one moves between a known
topological insulating phase and the final set of finely tuned
parameters.
(a) 12 (c) 4
favored if U V. 8 2
jecPtievreovosfktithei(cid:28)sLwatotrikce:.co–nWstreucntoiwngaflpaptrobaacnhdsthweitmhaainnoobn-- U/W Et 0
4
trivial Z index in a 3D lattice. Our starting point is
2
again the Hamiltonian given in Eq. (1) but now on the (cid:239)2
perovskite lattice, shown in Fig. 2(a). In Ref. 20 we 00 4 8 12 Γ X M Γ
U/W
showed that for nearest-neighbor hopping and λ>0 the
(b) 20 (d)
system becomes a (1;111) strong topological insulator
when the lowest doubly degenerate band is filled. In an 16 4
analogous fashion to the two dimensional case then, we
aim to flatten out the lowest energy band by means of W12 E 2
including additional parameters t2, t3 and δ. The band- U/8 t 0
structure for the set of parameters that maximize the
4 (cid:239)2
bandgap to bandwidth ratio is shown in Fig. 2c, over a
pathofhighsymmetryintheBrillouinzoneillustratedin 00 1 2 Γ A L H Γ
Fig. 2b. Thefollowingsetofvalues: t =1,t = 0.416, U/W
1 2
−
t = 0.047,λ= 0.331andδ =1.318,yieldedtheratio
3
− − FIG. 3: (Color online)(a) Phase diagram for Lieb lattice
∆/W 17.56. We remark that the ratio was calculated
≈ (W ≈0.034),and(b)perovskitelattice(W ≈0.072). Critical
over the entire Brillouin zone and in Fig. 2c we have
valuesU ,U(cid:48)fortheonsetofmagneticorderreflectthecharge
simply chosen one possible path here to illustrate the c c
distribution on the basis sites. The latter is approximately
result. Including additional parameters in H, such as uniformfortheLieblatticeimplyingU ≈U(cid:48). Onperovskite
c c
longer range hoppings, could no doubt further improve latticemostofthechargedensityisonsite1causingU (cid:28)U(cid:48).
c c
the above figure of merit but we do not pursue this here. (c) Band structure for Lieb lattice with U/W =U(cid:48)/W =15,
To ensure that the system remains in the topological m(1) ≈ 0.35 and m(2) = m(3) ≈ 0.32, and (d) perovskite
phase,weadiabaticallytunetheparameterxfrom0to1 lattice with U/W = 5, U(cid:48)/W = 25, m(1) ≈ 0.62, m(2) =
in the Hamiltonian H(cid:48)(x) defined in Eq. (2). The result m(3) ≈0.11 and m(4) ≈0.05.
ofthisprocedurecanbeseeninFig. 2d. Theratio,again,
4
(1) above by including a Hubbard term Unlike the 2D case where the Laughlin liquid2 fur-
nishes a well established paradigm for the topologically
(cid:88) (cid:88)
H =U n(1)n(1)+U(cid:48) n(l)n(l) (3) ordered correlated ground state, in 3D such a template
int i↑ i↓ i↑ i↓
is presently lacking. Therefore, understanding the fate
i i,l>1
of the ground state of electrons in the topologically non-
with the goal of mapping out the magnetic phases as trivial3Dflatbandinthepresenceofstronginteractions
a function of U and U(cid:48) by means of a standard mean- beyond the simple mean-field analysis is clearly a prob-
fieldcalculation. Thesuperscriptldenotesthebasissites lemoutsidethescopeofthisstudy. Inthisworkwemade
in the lattice, Figs. 1a and 2a. We use the following afirststepinthisdirectionbydemonstrating,withinthe
decoupling mean-fieldapproximation,thatthegroundstateremains
non-magneticoverasignificantportionoftheU-U(cid:48)phase
ni↑ni↓ ni↑ ni↓ +ni↓ ni↑ ni↑ ni↓ (4) diagram (Fig 3b). Since mean-field calculations tend to
→ (cid:104) (cid:105) (cid:104) (cid:105)−(cid:104) (cid:105)(cid:104) (cid:105)
overestimate the stability of ordered phases the actual
and definine the magnetization m(l) = n(l) n(l) and non-magnetic region will likely be somewhat larger and
i (cid:104) i↑(cid:105)−(cid:104) i↓(cid:105)
occupationnumbern¯(l) = n(l) + n(l) . Furthermore,we can in principle host a -invariant fractional topologi-
i (cid:104) i↑(cid:105) (cid:104) i↓(cid:105) cal insulator. Our resultTin 3D thus provides a concrete
focus on uniform magnetic phases, such that m(l) =m(l)
i foundation for addressing these interesting questions us-
independent of the site index i.
ingthesuiteofanalyticalapproachesornumericalmany-
To map out the phase diagram, we minimize the
bodytechniquesandwehopethatitmightinspirefurther
ground state energy with respect to m(l). The result
studies.
for half filling of the flat bands can be seen in Figures 3a
Finallyweremarkthatthereexistmanyperovskitesin
and 3b, for the Lieb and perovskite lattices respectively.
nature. Our theoretical results demonstrate that a near-
Here m = (cid:80) m(l) is the total magnetization per unit
l ideal situation (in terms of strong interaction physics)
cell. The key observation to be made here is that both
can arise in a very simple model in this family of lat-
systems remain non-magnetic over a significant range of tices. Also alluded to previously20 was the possibility of
interaction strengths.
engineering a relevant 2D system by modulating a two-
Conclusions. – We have established that it is possible
dimensionalelectrongaswithaperiodicpotentialhaving
toengineernearlyflatbandscharacterizedbynon-trivial
Lieblatticesymmetry,inananalogousfashiontothe‘ar-
Z2 topological invariants in both the 2D Lieb lattice and tificial graphene’ created recently.27 Another possibility
its three dimensional counterpart, the edge centered cu-
lies with cold fermionic or bosonic atoms in optical lat-
bic (perovskite) lattice. With our 2D example we have
tices as discussed in Refs. [28,29], with a detailed study
simply added to the growing number of lattices capa-
on the Lieb lattice provided in Ref. [30].
ble of producing this behavior, whereas the 3D result is
to the best of our knowledge completely novel. What
makes these findings potentially broadly relevant is that
they provide a concrete framework for future studies ad- Acknowledgment.—The authors have benefited from
dressing the nature of the many-body ground state that discussions with D. Bercioux, C. Chamon and J.E.
occurs in the presence of repulsive interactions and at Moore. This work was supported by NSERC and CI-
partial filling. fAR.
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