Electronic Properties of the Hubbard Model on a Frustrated Triangular Lattice

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by  Bumsoo Kyung

Electronic Properties of the Hubbard Model on a Frustrated Triangular Lattice
Bumsoo Kyung
D´epartement de physique and Regroupement qu´eb´ecois sur les mat´eriaux de pointe,
Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1, Canada
(Dated: February 6, 2008)
We study electronic properties of the Hubbard model on a triangular lattice using the cellular
7 dynamicalmean-fieldtheory. Theinterplayofstronggeometricfrustrationandelectroncorrelations
0 causes a Mott transition at theHubbard interaction U/t=10.5 and an unusual suppression of low
0 energy spin excitations. Doping of a triangular Mott insulator leads to a quasiparticle peak (no
2 pseudogap) at the Fermi surface and to an unexpected increase of low energy spin excitations, in
stark contrast to the unfrustrated square lattice case. The present results give much insight into
n
a strongly frustrated electronic systems. A few predictions are made.
J
PACSnumbers: 71.10.Fd,71.27.+a,71.30.+h,71.10.-w
6
2

] Geometric frustration with strong electronic correla- (a) (b)
l tionsisoneofthemainissuesinmoderncondensedmat- K
e
- terphysics. Thesimplestexampleisthetwodimensional
r Heisenberg model (large U limit of the half-filled Hub- M
t
s bard model) on a triangular lattice in which all three BATH BATH
. Γ
t spins cannot be antiparallel at the same time. The frus-
a
m trated triangularlattice geometry was arguedby Ander-
son[1]toprovideanidealbackgroundforthelongsought
- BATH
d resonating valence bond (RVB) state. The interplay of
n stronggeometricfrustrationandelectroniccorrelationsis
o expectedtoleadtosomeexoticphases. Recentdiscovery FIG.1: (a)Three-sitequantumclusterembeddedinaneffec-
c of superconductivity in the triangular lattice compound tive medium (called bath) and (b) first Brillouin zone of the
[
NaxCoO2 · yH20 [2] and a possible quantum spin liquid triangular lattice.

3 state in the anisotropic triangular lattice organic mate-
v rialκ-(ET)2Cu2(CN)3 [3]hasfurther stimulatedinterest
2 in strongly frustrated electronic systems. electrons of spin σ, niσ = c†iσciσ is the density of σ spin
0
The electronic properties of the Hubbard and t − J electrons, U is the on-site repulsive interaction, and µ is
2
models on a triangular lattice have been studied using the chemical potential controlling the electron density.
8
various analytical and numerical techniques. These in- To describe strongelectroncorrelationsandgeometric
0
6 clude a high-temperature expansion [4], a slave-boson frustration accurately, we use cellular dynamical mean-
0 mean-field [5, 6, 7], an RG method [8], an exact diag- field theory (CDMFT) [16], a quantum cluster approach
/ onalization(ED)technique[9,10,11],aquantumMonte that allows one to extend DMFT [17] to incorporate
t
a Carlo (QMC) simulation [12], a dynamical mean-field short-range spatial correlations explicitly. CDMFT has
m
theory (DMFT) [13, 14], and a variational Monte Carlo been benchmarked and is accurate even in one dimen-
- approach [15]. However, some of the central issues on sion [18, 19]. The infinite lattice is tiled with identical
d
n a triangular lattice have not been clearly addressed and clusters of size Nc, and the degrees of freedom in the
still remain an open question: what is the nature of the cluster are treated exactly while the remaining ones are
o
c ground state of a doped triangular Mott insulator? is replacedbyabathofnon-interactingelectronsthatisde-
: it a non-Fermi liquid (manifested as the presence of a terminedself-consistently(Fig.1(a)). Tosolvethe quan-
v
pseudogapatsmalldopinglikeonanunfrustratedsquare tum cluster embedded in an effective medium, we con-
i
X lattice) or a correlated Fermi liquid? what are qualita- sider a cluster-bath Hamiltonian of the form [16, 18, 20]
r tive differences between the triangular and square lat-
a tice systems? In this paper, utilizing recent theoretical H = X tµνc†µσcνσ+UXnµ↑nµ↓+ X εαmσa†mασaαmσ
progress in quantum cluster methods, we address these hµνi,σ µ m,σ,α
issuesandprovidemuchinsightintoelectronicproperties
in strongly frustrated electronic systems. + X Vmαµσ(a†mασcµσ +H.c.). (2)
The two-dimensional Hubbard model on a triangular m,µ,σ,α
lattice is described by
Here the indices µ,ν = 1,··· ,N label sites within the
c
H = X tijc†iσcjσ +UXni↑ni↓−µXc†iσciσ, (1) cluster,andcµσ andaαmσ annihilateelectronsontheclus-
terandthebath,respectively. t arethehoppingmatrix
hiji,σ i iσ µν
elements within the cluster (the chemical potential µ is
where c† (c ) are creation (annihilation) operators for absorbed here), εα are the bath energies and Vα are
iσ iσ mσ mµσ
2
0.3 0.2
U/t=9, n=1 U/t=9
(a)
0.2
U/t=10
)
ω
0.1 χ(ch0.1 U/t=10.5
m n=1
0 I
U/t=10, n=1
) 0.2 0
ω 0 5 10 15
N( ω/t
0.1
4
0
(b)
U/t=10.5, n=1
0.2 )
ω
(
0.1 χsp 2
m
I
0
−12 −8 −4 0 4 8 12
ω/t
0
0 0.5 1 1.5
ω/t
FIG.2: DensityofstatesN(ω)onatriangularlatticeathalf-
filling (n=1) with increasing U/t.
FIG. 3: (color online) Imaginary part of the local dynamical
(a) charge and (b) spin correlation functions at half-filling
with increasing U/t. The dashed, dotted and solid curves
the bath-cluster hybridization matrices. The exact diag- correspond to U/t=9, 10 and 10.5, respectively.
onalization method [21] is used to solve the cluster-bath
Hamiltonian Eq. 2 at zero temperature, which has the
advantage of computing dynamicalquantities directly in angular lattice is dramatically different from the unfrus-
realfrequencyandoftreatingthelargeU regimewithout trated square lattice case. In the latter [20], with in-
difficulty. In the present study we used N = 3 sites for creasing U/t the spectral weight always becomes a local
c
theclusterandN =9sitesforthebathwithm=1,2,3, minimum at the Fermi energy until it completely van-
b
α=1,2,3. Althoughthepresentstudyfocusesonasmall ishes near U/t=6.
clusterwithadditional9bathsites,weexpectourresults Next we examine the local dynamical charge and spin
to be robust with respect to an increase in the cluster correlation functions near the Mott transition shown in
size. This was verified by our recent low (but finite) Fig. 3. With increasing U/t the low energy charge ex-
temperature CDMFT+QMC calculations [19] where at citations are gradually depleted and transfered to high
intermediate to strong coupling a 4-site cluster accounts energynearω =U/t. Onthe otherhand,the lowenergy
formorethan95%ofthe correlationeffectofthe infinite spectrum in the spin correlation function undergoes a
size cluster in the single-particle spectrum. Quantita- dramatic change near the Mott transition. As the Mott
tively similar results were found for different bath sizes transition is approached, the primary low energy exci-
(Nb = 6). In this paper we focus on the normal state tation near w/t = 0.2 moves rapidly to the secondary
properties at and near half-filling. peak near w/t = 0.5 − 0.6 and eventually disappears.
We first study the Mott transition of the half-filled Namely, geometric frustration suppresses the low energy
Hubbard model on a triangular lattice. Among several spin excitations, which is most strongly manifested in
physicalquantitiesonecangaugetoinvestigatetheMott the insulating state (U/t = 10.5). We found similar re-
transition, we present the density of states N(ω) as a sultsforthenearestneighborchargeandspincorrelation
function of U/t shown in Fig. 2. For U/t <10.5 a small functions. This feature is in stark contrast to the square
spectral weight remains near the Fermi energy reminis- lattice case. In the latter, for the same range of U/t
cent of the Kondo resonance observed in DMFT [17]. (9−10.5) the low energy spin spectra are almost identi-
When U/t approaches 10.5, the spectral weight of the calwith asharpprimarypeak near w/t=0.25−0.3 [22]
centralpeakiscompletelypushedawayfromω =0lead- andasecondarypeak nearw/t=0.6−0.7similarto the
ing to aninteraction-drivenMott insulator. This critical caseofU/t=9onatriangularlattice. Neartherangeof
value of U/t is somewhat smaller than 12 obtained in U/t(4−6)whereacontinuouscrossoverofametaltoan
the ED study of an isolated12-site cluster [9] and in the insulator occurs on a square lattice, with increasing U/t
DMFT analysis [14]. Beyond U/t = 10.5 it becomes in- the low energy spin spectrum near w/t = 0.25 becomes
creasingly more difficult to find a convergent solution, stronger, which is opposite to the behavior found on a
since the groundstate starts to become degenerate. The triangular lattice.
way how the Mott transition occurs on a frustrated tri- Figure 4 shows the spectral function A(~k,ω) for both
3
0.2
(a) U/t=10.5 n=0.9
n=1
)
ω
n=1.1
(a) U/t=10.5, n=0.9 χ(ch0.1
Γ m
I
K 0
0 5 10 15
M ω/t
4
Γ (b)
-10 -5 0 5 10
)
ω
(b) U/t=10.5, n=1 χ(sp 2
Γ m
I
K 0
0 0.5 1 1.5
M ω/t
Γ FIG. 5: (color online) Imaginary part of the local dynamical
-10 -5 0 5 10
(a) charge and (b) spin correlation functions for U/t = 10.5
with doping. The inset in (b) is the imaginary part of the
(c) U/t=10.5, n=1.1
localdynamicalspincorrelationfunctionfortheunfrustrated
Γ square lattice case. The dashed, solid, and dotted curves
correspond to n=0.9, 1 and 1.1, respectively.
K
M lattice [20]. The absence of the pseudogap is ascribed to
toomuchfrustrationofshort-rangespincorrelationsona
Γ triangularlatticeasshowninFig.6. Herewepredictthat
-10 -5 0 5 10
ω/t near half-filling A(~k,ω) has a quasiparticle peak at the
Fermi surface, which should be tested by angle resolved
FIG.4: SpectralfunctionA(~k,ω)for(a)n=0.9,(b)n=1,and photoemission spectroscopy (ARPES) experiments on a
(c)n=1.1alongthepathΓ-M-K-ΓforU/t=10.5. Γ,M,and triangular lattice compound such as NaxCoO2 at small
K are definedin Fig. 1(b). doping.
Next let us examine the local dynamical charge and
spin correlation functions upon doping shown in Fig. 5.
10% hole- and electron-dopings. Electron-doping corre-
As expected, with doping the high energy charge exci-
spondstothenegativesignofthehoppingintegral(t<0)
tations at half-filling move to the low energy due to the
in the t − J model. At half-filling (Fig. 4(b)) A(~k,ω) metallic nature of electrons. The most surprising result
shows several features, which are similar to those on a comes from the spin correlation function. The primary
square lattice [20]. Although they are weaker than on a low energy peak near w/t = 0.2, which has vanished in
squarelattice,thelowenergyinnerbandsareclearlyseen the insulating state due to strong geometric frustration
near the Fermi energy, which are caused by short-range shown in Fig. 3, reappears upon doping. This is com-
spin correlations [20]. Upon hole- or electron-doping the pletely opposite to the square lattice case (the inset in
Fermi level jumps to the nearest low energy band just Fig. 5(b)) where the primary low energy peak weakens
likeonasquarelattice. Thisindicates the importanceof rapidly with doping. On a triangular lattice geometric
short-rangecorrelationsevenonahighlyfrustratedtrian- frustration is released by doping. Here we also predict
gular lattice. The most dramatic difference, however, is that inelastic neutron scattering experiments should ob-
thatnearhalf-fillingthereisnoevidenceofa(strongcou- serve the increase of low energy spin excitations with
pling) pseudogap unlike on a square lattice. It is more doping if the undoped system is a Mott insulator (see
evident from the fact that A(~k,ω) becomes sharper as caveat [23]).
~k approaches the Fermi surface. This is true for other Finally we study the static local and nearest neigh-
fillings (5−30% hole- and electron-doping). Thus the borspincorrelationswithdopingforbothtriangularand
presentresultclearlyshowsthatadopedtriangularMott square lattices shown in Fig. 6. For the local quantity
insulator is a correlated Fermi liquid unlike on a square hSz · Szi they are similar in both cases, although the
1 1
4
6 1 such as doping level, our present result offers a natural
(a) triangle (b) routeto spin triplet superconductivity in this compound
5
square 0
within the one-band Hubbard model (without assuming
> 4 > −1 the existence of unobserved hole pockets near~k =K).
zS1 3 zS2
zS1 zS1−2 SodiumcobaltateNaxCoO2 hasshownarichphasedi-
< 2 < agram[24]. These include a superconducting phase near
−3 x = 0.3−0.35 upon hydration, a charge-ordered insu-
1 U/t=10.5
lating phase at x=0.5 and a spin-density-wave metallic
0 −4 phase above x = 0.75. Since the charge-ordered insulat-
0.5 1 1.5 0.5 1 1.5
n n ing phase in cobaltates is observed at x = 1/2 doping
which is not a natural commensurate value (x = 1/3
FIG.6: (coloronline)Static(a)localand(b)nearestneighbor or 2/3) of the triangular lattice, the one-band Hubbard
spin correlations for U/t = 10.5 with doping. The dashed modelalone wouldnot be enoughto describe this phase.
curvesare the corresponding results on a square lattice. It requires longer-range Coulomb interaction V , which
ij
we did not consider in this work.
In conclusion, we have studied electronic properties of
particle-hole symmetry is broken on a triangular lattice.
theHubbardmodelonatriangularlatticeusingthecellu-
Themostdramaticdifferenceisseeninthenearestneigh-
lar dynamical mean-fieldtheory. The interplay of strong
bor spin correlation hSz ·Szi. First of all the nearest
1 2
geometricfrustrationandelectroncorrelationsisrespon-
neighborspincorrelationismuchreducedcomparedwith
sibleformanyunconventionalfeatures,whichareinstark
a square lattice because of strong geometric frustration.
contrast to the unfrustrated square lattice case. In par-
While it remains essentially antiferromagnetic (negative
ticular the appearance of a Fermi liquid upon doping is
sign) on a square lattice, it becomes rapidly ferromag-
ascribedto too much frustrationof short-rangespin cor-
netic onthe electron-dopedside ofthe triangularlattice.
relations on a triangular lattice. We predict that upon
The strong asymmetric behavior on a triangular lattice
doping a triangular Mott insulator ARPES should ob-
that the spin correlation becomes ferromagnetic only on
serve a quasiparticle peak (no pseudogap) at the Fermi
the electron-doping side is in agreement with the high
surface and that inelastic neutron scattering should find
temperature expansion study of the t−J model by Ko-
an increase of low energy spin excitations.
retsuneet al.[4]andtheanalysisofanisolatedthree-site
cluster by Merino et al. [13]. It is also noteworthy that We thank P. W. Anderson, S. R. Hassan, and A. -M.
the ferromagnetic correlationis maximum near n=1.35 S. Tremblay for useful discussions, and A. -M. S. Trem-
(x = 0.35), where superconductivity has been observed blay for critical reading of the manuscript. Computa-
in NaxCoO2 · yH20 [2]. Although the small cluster used tions were performed on the Elix2 Beowulf cluster and
in this work (N = 3) does not allow us to study which onthe DellclusteroftheRQCHP.Thepresentworkwas
c
gap symmetry (p or f) is more stable and how robust supportedbyNSERC(Canada),FQRNT(Qu´ebec),CFI
this result would be with respect to several parameters (Canada), and CIAR.
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2
[22] The position of thispeak scales as J =4t /U.
[23] If the undoped system is a metal, the opposite result