Novel Phase Between Band and Mott Insulators in Two Dimensions

0.21 MB·English

by  S. S. Kancharla

Novel Phase Between Band and Mott Insulators in Two Dimensions
S. S. Kancharla1 and E. Dagotto1,2
1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 32831
2Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996
(Dated: February 6, 2008)
Weinvestigate theground state phasediagram of thehalf-filled repulsive Hubbardmodel in two
7 dimensions in the presence of a staggered potential ∆, the so-called ionic Hubbard model, using
0 clusterdynamicalmeanfieldtheory. Wefindthatforlarge Coulomb repulsion,U ≫∆,thesystem
0 isaMottinsulator(MI).Forweaktointermediatevaluesof∆,ondecreasingU,theMottgapcloses
2 atacriticalvalueUc1(∆)beyondwhichacorrelatedinsulatingphasewithpossiblebondorder(BO)
n is found. Further, this phase undergoes a first-order transition to a band insulator (BI) at Uc2(∆)
a with a finite charge gap at the transition. For large ∆, there is a direct first-order transition from
J a MI to a BI with a single metallic point at thephase boundary.

8
PACSnumbers: 71.10.-w,71.10.Fd,71.10.Hf,71.27.+a,71.30.+h,71.45.Lr
1

]
l The Mott metal-insulator transition is a paradigmatic where the combined density of A and B sublattices is 1
e
problemincondensedmatter physicsthathas been real- and so we set the chemical potential, µ, to U/2. In the
-
r izedinabroadsetofexperiments,rangingfromthe high atomic limit (t = 0) and for U/2 < ∆, the ground state
t
s temperature superconductors to recently observed phe- has two electronson the B sublattice andnone on the A
.
t nomena involving cold atoms trapped in optical lattices. sublattice resulting inchargedensity wave(CDW) order
a
Mott insulators (MI) are characterized by a density of withabandgapof∆−U/2. Inthe oppositelimit where
m
one electron per unit-cell and strong Coulomb interac- U/2>∆, eachsite is occupied by one electronand a MI
-
tions that often leads to symmetry breaking via antifer- isformedwithagapU. Therefore,intheatomiclimitat
d
n romagnetic (AF) order. In contrast, charge localization exactly Uc = 2∆ the system is gapless1,2,3,4. This tran-
o in band insulators (BI) occurs when the number of elec- sition from a BI to a MI is likely to persist when t 6= 0,
c tronsperunitcelliseven,sothatthebandsareeitherfull butnon-trivialchargefluctuationsintheintermediateto
[
or empty. These two types of insulators can be realized strongcouplingregime(U ∼2∆)renderstaticmean-field

2 within a single system, the Ionic Hubbard model (IHM) and perturbative treatments of the transition invalid.
v at half-filling, by tuning the strength of Coulomb inter-
Bosonization calculations for the IHM in one
8 action. TheBI-MIquantumphasetransitionintheIHM dimension4,validintheweakcouplingregime(∆≪U ≪
6 has been well studied in the limit of either one1,2,3,4 or
t), find a transition from a BI to a bond-ordered (BO)
5 infinite dimensions5,but significantuncertaintiesremain
7 insulatoratU =Uc,wherethechargegapvanishes. The
in the phase diagram. Originally suggested to describe
0 BOphaseischaracterizedbyfinitechargeandspingaps.
6 charge-transfer organic salts6, there has been renewed Increasing U further, the spin gap vanishes at U = U
s
0 interestinthemodelduetopotentialapplicationstofer- in a Kosterlitz-Thouless-type transition leading to a MI.
/ roelectric perovskites7. Confinement of Fermionic atoms
t Numerical studies for long chains using the density ma-
a in an optical lattice to realize the ionic Hubbard model trixrenormalizationgroup(DMRG)2,3 inthestrongcou-
m
is also conceivable in the future, because recent experi-
plingregimefindthatthechargegapremainsfiniteatthe
- mental progress has allowed for Fermionic atoms to be transitionpointU totheBOphase,althoughtheoptical
d cooled well below the degeneracy temperature8. c
n gap does vanish. DMRG is unable to identify a second
Our objective in this letter is to obtain the ground
o transition point towards a MI because extrapolation to
state phase diagram of the IHM at half-filling in two
c large system sizes in the critical region is a challenge to
v: dimensions using cluster dynamical mean field theory numerical methods.
(CDMFT)9. The IHM is realized on a bipartite lattice
i Our key findings for the IHM in two dimensions are
X by the assignment of an alternating chemical potential
the following. A nonzero staggered chemical potential
r (±∆) for each sublattice on top of the regular Hubbard
a model. The Hamiltonian can be written as induceschargedensitywave(CDW)orderovertheentire
rangeofU, with the CDW orderparameterapproaching
H = −t c† c +h.c +U n n zero asymptotically for large U. In the large U regime,
iσ jσ i↓ i↑
i∈XAj∈Bh i Xi the system behaves as a MI with AF order. For weak
to intermediate∆, decreasingthe value ofU,the system
+∆ n −∆ n −µ n . (1)
i i i undergoes a transition to a correlated insulating phase
Xi∈A iX∈B Xi suggesting BO with a closing of the charge gap at the
Here, t and U denote the hopping amplitude between transition point U (∆). The transition also manifests
c1
nearest-neighborsitesandthe onsiteCoulombrepulsion, itselfvia anabruptincreaseinthe slope ofthe staggered
respectively. +∆(−∆)denotesthelocalpotentialenergy density, double occupancy, and staggered magnetization
for the A(B) sublattice. We are interested in the case as a function of U. Further, the BO phase undergoes a
2
first-order transition to a BI at a lower critical coupling 1.5 0.5
fUocu2n(∆d)onwlhyeirnetthheeMchIaragnedgBapOrpemhaasienss.fiFnoirte∆. A∼F4o.5rtd,etrhies n-nAB
0.4
system undergoes a direct first-order transition from the
1
BI to the MI, with a finite charge gap everywhere.
B
CDMFT is a non-perturbative technique where the - nA 0.3 5 U 6
full many-body problem is mapped onto local degrees n
of freedom treated exactly within a finite cluster that 0.5 BI
BO
is embedded in a self-consistent bath. It is a natural
MI
generalization of single-site DMFT to incorporate spa-
tial correlations. The method has passed rigorous tests
0
for the 1DHubbardmodel where it compareswellto ex- 0 2 4 6 8 10
U
act solutions10. CDMFT has also been used to elucidate
various aspects of the phase diagram of the cuprates,
including the pseudogap11 and superconducting phases FIG. 1: Staggered charge density as a function of U in the
within the 2D Hubbard model12. BI, MI, and BO regions for ∆=2t. Inset shows an enlarged
Using CDMFT, the IHM on the infinite lattice in two view of the abrupt changeof slope at the MI-BO boundary.
dimensions reduces to the cluster-bath Hamiltonian be-
low, that is subject to a self-consistency condition: 0.8
H = t c† c +U n n +∆ (−1)µn 0.6
µν µσ νσ µ↑ µ↓ µ 0.6
hXµνiσ Xµ Xµ BI
M
BO
+ ǫmσa†mσamσ+ Vmµσ(a†mσcµσ+h.c). (2) M0.4 MI
Xmσ mXµσ
0.4
0.2 5 6
U
µ,ν =1,···,N denote indices labelling the cluster sites
c
and m = 1,· · ·,N represent those in the bath. The
b 0
self-consistentcalculationproceedsbyaninitialguessfor
the cluster-bath hybridization V and the bath site 0.15
mµσ
energies ǫ to obtain the cluster Green’s function Gµν. D
mσ c
Applying the Dyson’s equation, Σ =G−1−G−1, where
c 0c c 0.3
G denotes the non-interacting Green’s function for the D 0.1
0c BI 5 6
Hamiltonianin Eq.2, the cluster self-energyis obtained. BO U
Σ isthenusedintheself-consistencyconditionbelowto MI
c
determine a new G :
0c
0
N 1 −1 0 2 4 6 8 10
G−1(z)−Σ (z)= c dK . U
0c c (cid:20)4π2 Z z+µ−t(K)−Σc(z)(cid:21)
(3)
Here,K denotesamomentumvectorinthereducedBril- FIG. 2: (a) Staggered magnetization, and (b) double occu-
louin zone of the cluster superlattice and z = iω is the pancy as a function of U, for ∆=2t.
n
Fermionic Mastubara frequency. From G in Eq. 3 a
0c
new set of V and ǫ ’s is generated closing an it-
mµσ mσ
erative loop. In this work, the Lanczos method is used presents long-range charge density wave (CDW) order
to solve the cluster-bath Hamiltonian. The cluster size over the entire range of U. The CDW order parameter
is fixed to N = 4 and the bath size to N = 8. The asymptotically approaches zero for increasing U but re-
c b
Lanczos method can access both the strong and weak mains finite everywhere. Starting from the MI phase at
coupling regimes with equal ease and it is well suited to largeU, the staggeredchargedensity increasessmoothly
compute dynamical quantities directly in real frequency. as U is reduced, but at U=U =5.4t there is an abrupt
c1
Rotational symmetries of the cluster on a square lattice, increase in the slope. This anomaly is clearly identified
together with particle-hole symmetry at half-filling, re- in the inset and it is believed to be a signature of the
duce the number ofbath parameterssignificantly. These transition to a Bond-Ordered(BO) phase. This result is
bath parameters can depend on spin, allowing for sym- similar to observations in earlier DMRG calculations for
metry breaking solutions. For details of the method, we the one dimensional IHM2,3. The BO phase extends up
refer to earlier work9,10. to U=U =4.6t,where there is a first-ordertransitionto
c2
We analyze our results by starting with Fig. 1, which a BI signalledby a jump in the staggereddensity. Start-
shows the density difference between sublattices A and ing from U = 0, the BI shows a continous decrease in
B as a function of U, for ∆=2t. Clearly, the system the staggered density and persists up to U=6.55t. Con-
3
sequently, we have the coexistence of a BI and BO for in the gap as compared to U = 4.60 signalling a tran-
4.6t < U < 5.4t and a BI and MI for 5.4t < U < 6.55t. sition to a BI, which also shows an increasing gap with
We computed the total energy in the coexistence region decreasing U. For ∆>4.5t, the BI-MI transition has no
(notshown)andfindthattheBIsolutionfortheCDMFT interveningphaseanditisaccompaniedbytheclosingof
equations is always higher in energy than the BO or the thechargegap. Thistransitionscenarioisreminiscentof
MI phase for the same U. thephasediagramoftheextendedHubbardmodelinone
dimension,whereanintermediatephasewithbondorder
isstabilizedbetweentheCDWandSDWphasesonlyfor
BI U=0.30
1.2 intermediate strength of Coulomb repulsion U when the
nearest neighbor Coulomb repulsion V is varied13.
0
BI U=3.00
1
(a)
0 (π,π) U=0.3t
ω) 0.8 BI−BO UU==44..6505
A(
0
U=5.40
0.6 BO U=5.20 (0,0)
U=5.00
0 (π,π) (b) U=5.4t
0.8 MI U=6.00 ω)
k,
0 (
A
U=8.00
0.8 MI (0,0)
0 (c) U=10t
-4 -2 0 2 4 (π,π)
ω
FIG. 3: Local density of states as a function of U for ∆=2t (0,0)
shown with a broadening factor of 0.02t. -5 0 5
ω
To further characterizethe nature ofthe transitionlet
us consider the staggered magnetization, M=n − n ,
↑ ↓ FIG. 4: Momentum dependent density-of-states along the
andthedoubleoccupancy,D=hn n i,asafunctionofU
↑ ↓ (0,0)to(π,π)directionfor(a)theBI,(b)thephaseboundary
for ∆=2t shown in Fig. 2. With reducing U both these
of the MI-BO transition, and (c) theMI for ∆=2t.
quantities show an abrupt increase in slope at the same
transition point U=U =5.4t (see inset) where the BO
c1
To characterize the distinct phases observed, we have
insulatoremergesfromthe MI.BothM andD presenta
computed the momentum-dependent density-of-states
jump at the first-order transition from the BO phase to
using the following relation between the the lattice
a BI.The staggeredmagnetizationabruptly goesto zero
Green’s function and the cluster self-energy:
upon entering the the BI phase signifying an absence of
AF order. The double occupancy increases monotoni-
1 1
cally with reducing U in the BI state due to the increas- G(k,z)= eik·(µ−ν) .
N (cid:20)z+µ−t(k)−Σ(z)(cid:21)
ing tendency towards CDW order. c Xµν µν
InFig.3,resultsforthelocaldensityofstates(LDOS) (4)
as a function of U for ∆ = 2t are presented. Note that In Fig. 4, we show the evolution of the density of states,
the LDOS shows particle-hole symmetry consistent with A(k,ω), in the first quadrant of the Brillouin zone, from
half-filling because it is averaged over the A and B sub- (0,0) to (π,π). At large U (Fig. 4(c)) sharp coherent
lattices. For U equalto the bandwidth of8t, the spectra peaks that define the Mott gap are observed. In addi-
show lower and upper Hubbard bands consistent with a tion,the lowerandupper Hubbardbandsarepresent,as
MI. In addition, sharp coherent peaks are observed (ab- well as high energy features separated from them by ∆.
sent in single-site DMFT). They correspond to the mo- Theminimumofthegapisfoundatk=(π/2,π/2)where
tionofacarrierinanAFbackgroundanddefinetheMott the low energy features disperse towards zero frequency,
gap. AsU islowered,theMottgapreducesitsvalueuntil while the maximum of the gap is found at k=(π,π) or
it closes at the transition point U = 5.4t, which mani- (0,0). ForsmallU (Fig.4(a)),wherewehaveaBI,adis-
fested itself earlier as an abrupt change in slope for the persion characteristic of AF order is observed but with
staggereddensity,doubleoccupancy,andstaggeredmag- the notable absence of Hubbard bands. By contrast, for
netization. WithfurtherreductioninU,thetrendforthe intermediate U, on the phase boundary of the MI-BO
Mott gap is reversed, with an increase in the gap in the transition at U = 5.4t, we found a metallic density of
BO phase. Further, atU =4.55there is anabruptjump states with distinct higher energy bands at (π,0) and
4
(0,π) that emerge from spontaneous dimerization and at k=(0,π) or (π,0), consistent with the spontaneously
persist into the BO phase. Therefore, the BO phase, al- dimerized BO phase with incommensurate order.
thoughsimilartotheBIinthebehaviorofthechargegap
To summarize, we have studied the quantum phase
as a function of U, shows a distinct signature of strong
transitionbetweenaBI andaMIinthetwodimensional
correlations. Themetallicpeakitselfsplitstoopenagap
IHMusingCDMFT.CDWorderpersiststhroughoutthe
right below the phase boundary.
entire range of U. For ∆<∆ ∼4.5t, the large U Mott
c
insulator state undergoes a transition to a correlatedin-
(a) U=0.3t sulatingphase BOwith reducingU,accompaniedby the
(π,0)
closing of the charge gap at the transition point U (∆).
c1
The BO phase undergoes a first-order transition to a
BI at U (∆) upon further reduction of U. As shown
(0,π) c2
in the phase diagram depicted in Fig. 6, the width of
(b)-3 0 3
(π,0) U=5.2t the BO phase shrinks as both small and large ∆ are ap-
) proached. For ∆ > ∆ , a direct first-order MI-BI tran-
ω c
k, sition with no intervening BO phase is observed. AF
(
A order is found only in the MI and BO phases. Our re-
(0,π)
(c)-0.3 0 0.3 sults differ significantly from earlier ones that used the
(π,0) U=10t single-site DMFT technique5 where a paramagnetic so-
lution must be enforced even though the ground state is
AF. Single-siteDMFT misses spatialcorrelationscrucial
(0,π) to the physics of this model. Moreover, the single-site
-2 0 2 DMFT finds a finite metallic region between the BI and
ω
MI whereas we find a single metallic point at the MI-
BO (∆ < ∆ ) or MI-BI (∆ > ∆ ) transitions. Longer
c c
range interactions could give rise to a metallic phase be-
FIG. 5: Momentum dependent density-of-states along the
(0,π) to (π,0) direction for ∆=2t. tween the insulators14. Our results are fairly similar to
those obtained for this model using DMRG2,3 in one di-
mension. An important difference is that at the metallic
16 point we observe a closing of the charge gap whereas
DMRG finds only a closing of the optical gap, with the
12 MI
charge gap remaining finite over the entire range of U.
U 8 U (∆) The two-transition scenario moving from the BI to the
4 BI Uc2(∆) MI was observed earlier in the weak coupling regime us-
c1 ing Bosonization4 for this model in one dimension. We
BO
00 2 4 6 8 have shown that a similar transition persists in two di-
∆
mensions, even in the strong coupling regime when ∆ is
comparable to half the bandwidth. Modulation of the
local potential with an alternating value could be real-
FIG. 6: Proposed CDMFT phase diagram for the2D IHM.
ized in an two dimensional optical lattice loaded with
ultracoldFermionicatoms. Noisecorrelationsintime-of-
In Fig. 5, A(k,ω) is shown, from (0,π) to (π,0). The
flightimagescanbe usedto distinguishbetweenthe AF,
uncorrelatedBI at U =0.3t (Fig. 5(a)) shows no disper- BI and BO phases15. Energy gaps can be obtained, in
sion of the low lying features and the gap becomes in- principle,usingrfspectroscopy16. Weexpectourresults,
dependent of momentum. For the MI phase at U = 10t
predicting a novel correlatedinsulating phase with bond
(Fig. 5(c)) we see an inward dispersion in the occupied
order,tomotivatesuchexperimentstoverifyourfinding.
band atk=(π/2,π/2)where the gapis a minimum, con-
sistent with commensurate AF fluctuations. In contrast S.S.K. is supported by the LDRD program at Oak
tothisresult,Fig.5(b)forU =5.2tshowsamaximumof Ridge National Laboratory. E.D. acknowledges support
the charge gap at k=(π/2,π/2) while the minimum lies by the NSF Grant No. DMR-0454504.
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