Mott transition and magnetism of the triangular-lattice Hubbard model with next-nearest-neighbor hopping

2.8 MB·

by  Kazuma Misumi

Mott transition and magnetism of the triangular-lattice Hubbard model
with next-nearest-neighbor hopping
Kazuma Misumi, Tatsuya Kaneko, and Yukinori Ohta
Department of Physics, Chiba University, Chiba 263-8522, Japan
(Dated: February 1, 2017)
The variational cluster approximation is used to study the isotropic triangular-lattice Hubbard
model at half filling, taking into account the nearest-neighbor (t ) and next-nearest-neighbor (t )
1 2
hopping parameters for magnetic frustrations. We determine the ground-state phase diagram of
the model. In the strong correlation regime, the 120◦ N´eel and stripe ordered phases appear, and
a nonmagnetic insulating phase emerges in between. In the intermediate correlation regime, the
nonmagneticinsulatingphaseexpandstoawiderparameterregion,whichgoesintoaparamagnetic
7 metallic phase in the weak correlation regime. The critical phase boundary of the Mott metal-
1 insulator transition is discussed in terms of the van Hove singularity evident in the calculated
0 density of states and single-particle spectral function.
2
n PACSnumbers: 71.10.Fd75.10.Jm71.30.+h75.10.-b
a
J
I. INTRODUCTION the strong correlation limit, has on the other hand been
1
studiedmuchindetail,mostlyfromthetheoreticalpoint
3
The physics of geometrical frustration in strongly cor- ofview[24]. IntheclassicalHeisenbergmodelwherethe

] related electron systems has long attracted much atten- spins are treated as classical vectors, it is known that a
l
e tion [1–3]. In particular, possible absence of magnetic single phase transition occurs at J2/J1 = 1/8 between
- the three-sublattice 120◦ N´eel ordered state and an in-
r long-range orders at zero temperature in the Heisenberg
t and Hubbard models defined on frustrated lattices, or finitely degenerate four-sublattice magnetically ordered
s
. the realization of a spin liquid phase as an exotic state states [25]. This degeneracy is lifted by quantum fluctu-
t
a of matter, has been one of the major issues in this field. ations, thereby selecting a two-sublattice stripe ordered
m state through the so-called “order-by-disorder” mecha-
The Mott metal-insulator transition is also a fundamen-
nism[25–28]. Onemaythenexpectinthecorresponding
- tal phenomenon in the field of strongly correlated elec-
d quantum Heisenberg model that an intermediate phase
tronsystems[4,5],whichhasattractedmuchexperimen-
n can appear near the classical critical point at J /J =
talandtheoreticalinterestaswell. Asoneofthesimplest 2 1
o
1/8,forwhichmanystudieshavebeencarriedouttopre-
c modelswithgeometricalfrustrationandMotttransition,
dict that the nonmagnetic disordered phase bordered by
[ we therefore study the Hubbard model at half filling de-
the 120◦ N´eel ordered phase at J /J (cid:39)0.05−0.12 and
fined on the triangular lattice in this paper, where not 2 1
1 the stripe ordered phase at J /J (cid:39)0.14−0.19 actually
only the nearest-neighbor hopping parameters but also 2 1
v
emerges. In particular, recent studies actually predict
4 the next-nearest-neighbor ones are included.
the emergence of either a gapless or gapped spin liquid
0 Much effort has so far been devoted in the study of
9 the triangular-lattice Hubbard model with anisotropic phase in this intermediate region [29–35]. These results
8 nearest-neighbor hopping parameters [6–14], which was of the Heisenberg model may be compared with those of
0 ourHubbardmodelinthestrongcorrelationlimit(aswe
motivated by experimental findings of possible spin liq-
. will see below).
1 uid states in some organic Mott insulators such as κ-
0 (ET)2Cu2(CN)3 [15–18] and EtMe3Sb[Pd(dmit)2]2 [19, In this paper, motivated by the above developments
7
20]. The triangular-lattice Heisenberg model with the in the field, we will study the triangular-lattice Hubbard
1
anisotropic exchange interactions has also been studied model at half filling with the isotropic nearest-neighbor
:
v tofindavarietyoforderedphasessuchasN´eelandspiral and next-nearest-neighbor hopping parameters in its en-
Xi orders,aswellasthequantumdisordered(orspinliquid) tire interaction strength. We use the variational cluster
phases in-between [21–23]. approximation(VCA),oneofthequantumclustermeth-
r
a However, to the best of our knowledge, the isotropic odsbasedontheself-energyfunctionaltheory(SFT)[36–
triangular-lattice Hubbard model with both the nearest- 40], which enables us to take into account the quantum
neighbor(t )andnext-nearest-neighbor(t )hoppingpa- fluctuations of the model with geometrically frustrated
1 2
rameters has not yet been addressed, the study of which spin degrees of freedom. We thereby calculate the grand
will therefore provide useful information on the physics potential of the system as a function of the Weiss fields
of magnetic frustrations and Mott metal-insulator tran- for spontaneous symmetry breakings; here, we take the
sition in strongly correlated electron systems. 120◦ N´eel and stripe magnetic orders and evaluate the
The isotropic Heisenberg model with the nearest- order parameters and critical interaction strengths. We
neighbor (J ) and next-nearest-neighbor (J ) exchange also calculate the charge gap as well as the density of
1 2
interactions, which may be derived by the second-order states (DOS) and single-particle spectral function using
perturbation of the above-mentioned Hubbard model in the cluster perturbation theory (CPT) [40] and deter-
2
mine the ground-state phase diagram of the model in its
entire parameter region.
We will thereby show that in the strong correlation
regime the 120◦ N´eel and stripe ordered phases appear,
andin-between,thenonmagneticinsulatingphasecaused
by the quantum fluctuations in the frustrated spin de-
grees of freedom emerges, in agreement with the Heisen-
berg model studies. We will also show that in the in-
FIG.2. (Coloronline)(a)Thereferencesystemofthe12-site
termediate correlation regime the nonmagnetic insulat-
clusterusedinouranalysis;thethree-sublatticesystemcorre-
ing phase expands to wider parameter regions located spondingtothe120◦ N´eelorder(left)andthetwo-sublattice
around 0 ≤ t2/t1 (cid:46) 0.3 and 0.4 (cid:46) t2/t1 ≤ 1, which go system corresponding to the stripe order (right). (b) The
into a paramagnetic metallic phase in the weak correla- first Brillouin zone of our triangular-lattice Hubbard model:
√
tion regime via the second-order Mott transition. The Γ(0,0), K(4π/3,0), and M(π,π/ 3).
characteristic behavior of the critical phase boundary of
theMotttransitionisdiscussedintermsofthevanHove
singularity appearing in the calculated DOS and single-
where c† (c ) creates (annihilates) an electron with
particle spectral function. iσ iσ
spin σ at site i, and n = c† c . (cid:104)i,j(cid:105) indicates the
Therestofthepaperisorganizedasfollows. InSec.II, iσ iσ iσ
nearest-neighbor bonds with the hopping parameter t
we introduce the model and discuss the method of cal- 1
and (cid:104)(cid:104)i,j(cid:105)(cid:105) indicates the next-nearest-neighbor bonds
culation briefly. In Sec. III A, we present our results
with the hopping parameter t . We consider the param-
obtained in the strong correlation regime and compare 2
eter region 0 ≤ t /t ≤ 1, including two limiting cases,
them with those of the Heisenberg model. In Sec. III B, 2 1
t =0 (isotropic triangular lattice) and t =t . U is the
we present our results obtained in the intermediate to 2 2 1
on-site Coulomb repulsion between two electrons and µ
weak correlation regime and discuss the phase diagram
is the chemical potential maintaining the system at half
of our model. The critical phase boundary of the Mott
filling.
transition is also discussed. A summary of the paper is
In the large-U limit, this model may be mapped onto
given in Sec. IV.
the triangular-lattice Heisenberg model of spin-1/2 de-
fined by the Hamiltonian
(cid:88) (cid:88)
H =J S ·S +J S ·S (2)
1 i j 2 i j
(cid:104)i,j(cid:105) (cid:104)(cid:104)i,j(cid:105)(cid:105)
with the exchange coupling constants of J =4t2/U and
1 1
J = 4t2/U for the nearest-neighbor and next-nearest-
2 2
neighbor bonds, respectively. The spin operator is given
by S = (cid:80) c† σ c /2 with the vector of Pauli ma-
i αβ iα αβ iβ
trices σ . The results obtained for the Hubbard model
αβ
[Eq. (1)] in the strong correlation regime are compared
FIG. 1. (Color online) Schematic representations of (a) the
with those of the Heisenberg model [Eq. (2)].
triangular-lattice Hubbard model with the nearest-neighbor
LetusdescribetheVCAbriefly,whichisamany-body
(t ) and next-nearest-neighbor (t ) hopping parameters, (b)
1 2 variational method based on the SFT, where the grand
the 120◦ N´eel order, and (c) the stripe order. The arrows
potential of the system is formulated as a functional of
represent the directions of electron spins on the A, B, and C
the self-energy [36–38]. The ground state of the original
sublattices defined by different colors.
systeminthethermodynamiclimitcanthusbeobtained
viathecalculationofthegrandpotentialΩofthesystem
withtheexactself-energy. Then,intheVCA,restricting
the trial self-energy to the self-energy of the reference
systemΣ(cid:48),weobtaintheapproximategrandpotentialas
II. MODEL AND METHOD
Ω[Σ(cid:48)]=Ω(cid:48)+Trln(G−1−Σ(cid:48))−1−Trln(G(cid:48)−1−Σ(cid:48))−1,
0 0
(3)
Weconsiderthetriangular-latticeHubbardmodel[see
Fig. 1(a)] defined by the Hamiltonian where Ω(cid:48) is the grand potential of the reference system,
and G and G(cid:48) are the noninteracting Green’s functions
0 0
(cid:88)(cid:88) (cid:88) (cid:88)
H =−t c† c −t c† c of the original and reference systems, respectively. The
1 iσ jσ 2 iσ jσ HamiltonianofthereferencesystemH(cid:48) isdefinedbelow.
(cid:104)i,j(cid:105) σ (cid:104)(cid:104)i,j(cid:105)(cid:105) σ
Notethattheshort-rangecorrelationswithintheclusters
(cid:88) (cid:88)
+U ni↑ni↓−µ niσ, (1) of the reference system are taken into account exactly.
i i,σ See Refs. 39 and 40 for recent reviews of the method.
3
The advantage of the VCA is that the spontaneous
symmetrybreakingcan be treatedwithintheframework
ofthetheory,whereweintroducetheWeissfieldsasvari-
ationalparameters. Inthepresentcase,theHamiltonian
of the reference system is taken as H(cid:48) = H +H with
M
the Weiss fields
H =H +H (4)
M 120◦ str
(cid:88)
H =h(cid:48) e ·S (5)
120◦ 120◦ ai i
i
(cid:88)
H =h(cid:48) eiQstr·riSz, (6)
str str i
i
whereh(cid:48) andh(cid:48) arethestrengthsoftheWeissfields
120◦ str
for the 120◦ N´eel and stripe ordered states, respectively.
For the N´eel order, the unit vectors e are rotated by
ai
120◦ to each other, where a (= 1,2,3) is the sublattice
i
index of site i. For the stripe order, the wave vectors
√
can be taken equivalently as either Q = (π,π/ 3),
√ √ str
(π,−π/ 3),or(0,−2π/ 3). Thevariationalparameters
are optimized on the basis of the variational principle,
i.e., ∂Ω/∂h(cid:48) = 0, for each magnetic order, where the
solution with h(cid:48) (cid:54)=0 corresponds to the ordered state.
FIG.3. (Coloronline)Calculatedground-statephasediagram
In our VCA calculations, we use the 12-site cluster
of our model in the strong correlation regime (U/t = 60).
1
shown in Fig. 2 as the reference system. This is the best Upperpanel: theorderparametersofthe120◦N´eelandstripe
appropriate and feasible choice of the reference cluster
ordered phases. Solid (open) symbols indicate that the state
because we can treat the two-sublattice order (stripe or- is stable (metastable). Lower panel: the ground-state ener-
der)withanequalnumberofup-anddown-spinelectrons gies (per site) of the ordered phases compared with that of
and the three-sublattice order (120◦ N´eel order) with an the disordered phase. Inset shows the enlargement of the en-
equalnumberofthethreesublatticesitesa =1,2and3. ergy difference ∆E between the 120◦ ordered and disordered
i
Thecluster-sizeandcluster-shapedependencesofourre- phases.
sultsarediscussedinAppendix. Notethatlongerperiod
phases such as the spiral phase mentioned in a different
system [11] cannot be treated in the present approach;
Att /t =0,the120◦N´eelorderedstatehasthelowest
in our analysis, we fix the pitch angle of the spiral order 2 1
to be 120◦ (or the three-sublattice of a = 1,2,3) even energyandwithincreasingt2/t1itapproachestheenergy
i
of the nonmagnetic disordered state gradually. Then, at
for t (cid:54)= 0. The charge orderings discussed in the ex-
2 t /t =0.20,the120◦ N´eelorderedstatedisappearscon-
tendedHubbardmodelwithintersiteCoulombrepulsions 2 1
tinuously. The calculated order parameter M also
[41] are also neglected. To our knowledge, no other or- 120◦
indicates the continuous (or second-order) phase tran-
dershavebeenpredictedinthepresenttriangular-lattice
sition. On the other hand, at t /t = 1.0, the stripe
Hubbard and Heisenberg models. 2 1
ordered state has the lowest energy and, with decreasing
t /t , the energy of stripe order crosses to that of the
2 1
nonmagneticstateatt /t =0.50,indicatingthediscon-
2 1
III. RESULTS OF CALCULATION
tinuous (or first-order) transition between the stripe and
disordered phases. The calculated order parameter M
str
A. Strong correlation regime also disappears discontinuously at t /t =0.50.
2 1
Theseresultsmaybecomparedwiththepreviousstud-
First, let us discuss the strong correlation regime, ies on the J -J triangular-lattice Heisenberg model [29–
1 2
U/t = 60. We calculate the ground-state energies 34]. The transition point between the 120◦ N´eel and
1
E =Ω+µ (per site) and magnetic order parameters M nonmagnetic phases has been estimated to be J /J =
2 1
defined as M =(2/L)(cid:80) e ·(cid:104)S (cid:105) for the 120◦ N´eel 0.05 − 0.12, which corresponds to t /t = 0.22 − 0.35
order and M1s2t0r◦= (2/L)(cid:80)iieiaQistr·rii(cid:104)Siz(cid:105) for the stripe of our Hubbard model parameters. A2 r1easonable agree-
order, where (cid:104)···(cid:105) stands for the ground-state expecta- ment is thus obtained. The transition point between
tion value. The results are shown in Fig. 3, where we the stripe and nonmagnetic phases has also been esti-
find three phases: the 120◦ N´eel ordered phase around mated to be J /J = 0.14−0.19, which corresponds to
2 1
t /t =0,thestripeorderedphasearoundt /t =1,and t /t =0.37−0.44ofourHubbardmodelparameters. We
2 1 2 1 2 1
the nonmagnetic disordered phase in-between. again find a reasonable agreement with our estimation.
4
FIG.4. (Coloronline)Calculatedground-statephasediagram
of our model in the intermediate to weak correlation regime,
which includes the 120◦ N´eel ordered, stripe ordered, non-
magnetic insulating, and paramagnetic metallic phases. Cir-
cle and triangle at U/t = 60 indicate the calculated phase
1
boundaries of the 120◦ N´eel order and stripe order, respec-
tively, shown in Fig. 3.
FIG. 6. (Color online) Calculated DOSs of our model in the
metallicstate(leftpanels)andinsulatingstatewithoutlong-
Theordersofthephasetransitions,i.e.,thesecond-order
range magnetic orders (right panels). η/t =0.1 is assumed.
for the 120◦ N´eel phase and the first-order for the stripe 1
The vertical line in each panel indicates the Fermi level.
phase, are also in agreement with the previous study of
the Heisenberg model [31]. We may point out that the
strong quantum fluctuations in the frustrated spin de-
grees of freedom causes this nonmagnetic phase because
B. Intermediate to weak correlation regime
the classical spin model predicts either the 120◦ N´eel or
four-sublattice ordered phase without any intermediate
Next, let us discuss the intermediate to weak correla-
nonmagnetic phases [25–28].
tion regime 0 ≤ U/t ≤ 10. We here calculate the total
1
energies,orderparameters,andchargegapsofthemodel,
as well as the grand potential as a function of the Weiss
fields,andsummarizethemastheground-statephasedi-
agram in the parameter space (t /t ,U/t ), as shown in
2 1 1
Fig. 4. We find four phases: the 120◦ N´eel and stripe or-
dered phases at large U/t , which are continuous to the
1
phases at U/t = 60 discussed above, and nonmagnetic
1
insulating phase in-between, as well as the paramagnetic
metallic phase in the weak correlation regime. In the
intermediate correlation regime, the nonmagnetic insu-
lating phase expands to wider parameter regions, which
are around 0 ≤ t /t (cid:46) 0.3 and around 0.4 (cid:46) t /t ≤ 1.
2 1 2 1
Wenotethatthepresenceofthenonmagneticinsulating
phase around 0 ≤ t /t (cid:46) 0.3 is in agreement with pre-
2 1
vious studies of the triangular-lattice Hubbard model at
t /t =0 [12, 13, 42–44].
2 1
The calculated order parameters of the 120◦ N´eel and
stripephasesareshowninFig.5asafunctionofU/t for
1
FIG. 5. (Color online) Calculated charge gap ∆ (upper pan-
severalvaluesoft /t . Wefindthatthetransitiontothe
els) and order parameters M120◦ and Mstr (lower panels) of stripe ordered pha2se1is continuous, irrespective of t /t ,
our model as a function of U/t . 2 1
1
up to a large value of U/t ∼ 30, but it changes to the
1
discontinuous transition as seen in Fig. 3 at U/t = 60.
1
Wealsofindthatthetransitiontothe120◦ N´eelordered
5
FIG. 7. (Color online) Calculated single-particle spectral function A(k,ω) in the paramagnetic state of our model. The wave
vector k is chosen along the line connecting Γ, K, and M points of the Brillouin zone [see Fig. 2 (b)]. η/t =0.1 is assumed.
1
Thenoninteractingbanddispersionisalsoshownbyathinsolidcurveineachoftheupperpanels. TheFermilevel(indicated
by the vertical line) is set at ω/t =0.
1
phase is discontinuous at 0 < t /t ≤ 0.35 for U/t (cid:39) 6 with the CPT Green’s function [40]
2 1 1
but it is continuous for larger values of U/t . The tran-
1
sitionatU/t1 =60isalsocontinuous(seeFig.3). These 1 (cid:88)Lc
behaviors are observed also in the calculated Weiss-field GCPT(k,ω)= L Gij(k,ω)e−ik·(ri−rj), (9)
c
dependence of the grand potentials of our model. i,j=1
where we define the L ×L matrices for the cluster of
The charge gap is evaluated from the total number of c c
size L as G(k,ω) = [G(cid:48)−1(ω)−V(k)]−1 with V(k) =
electrons as a function of µ (see Fig. 5) to examine the c
G(cid:48)−1−G−1. The exact Green’s function of the reference
Mott metal-insulator transition of the system. We find 0 0
system G(cid:48)(ω) is given by
that the transition is continuous (or second-order) and
the phase boundary is located around U/t (cid:39) 4−6, as
1 1
shown in Fig. 4. We note that the phase boundary de- G(cid:48)ij(ω)=(cid:104)ψ0|ciσω−H(cid:48)+E c†jσ|ψ0(cid:105)
creases(shiftstoalowerU/t side)withincreasingt /t 0
1 2 1 1
up to t2/t1 (cid:39) 0.5, but it increases for larger values of +(cid:104)ψ0|c†jσω+H(cid:48)−E ciσ|ψ0(cid:105), (10)
t /t . This behavior is in contrast to that of the square- 0
2 1
lattice Hubbard model with the next-nearest-neighbor
where|ψ (cid:105)andE arethegroundstateandgroundstate
hoppingparameters,whereamonotonousincreaseinthe 0 0
energy of H(cid:48).
critical interaction strength is observed [45–47], which is
The calculated results for the DOS and single-particle
duetothemonotonousincreaseinthebandwidthofthe
spectral function of our model are shown in Figs. 6 and
model.
7, respectively. We find that the sharp peak appeared
above the Fermi level at t /t = 0, which is caused by
Tofindouttheoriginofthisbehavior,wecalculatethe 2 1
the van Hove singularity in the triangular lattice, shifts
DOS ρ(ω) and single-particle spectral function A(k,ω)
to the lower energy side with increasing t /t , and at
in the paramagnetic state of the system using the CPT, 2 1
t /t = 0.5, the peak position coincides with the Fermi
which are defined as 2 1
level (see Fig. 6). This situation of the high DOS at
the Fermi level is energetically unstable [48], so that the
bandgapopenstogaininthebandenergyinthepresence
1 (cid:88) of the Hubbard interaction U. With further increasing
ρ(ω)= A(k,ω) (7)
L t2/t1, the peak shifts to the higher energy side again.
k The Hubbard band gap is then the largest at t /t =0.5
2 1
1
A(k,ω)=− lim(cid:61)G (k,ω+iη) (8) as seen in Figs. 5 and 6. This singularity is also seen in
π η→0 CPT thesingle-particlespectralfunctionasthepresenceofthe
6
flat-bandregionaroundtheKpointoftheBrillouinzone the SFT, which has not been used for the present pur-
(seeFig.7). Thisbehaviorthusexplainswhythecritical poses. Wehavetherebycalculatedthegrandpotentialof
interactionstrengthbecomessmallataroundt /t (cid:39)0.5. the system as a function of the Weiss fields for the 120◦
2 1
N´eel and stripe magnetic orders, and have determined
the order parameters. We have also calculated the DOS
andsingle-particlespectralfunctionaswellasthecharge
gap of the system. These results have been summarized
as the ground-state phase diagram of the system.
We have found four phases: In the strong correlation
regime, there appear (i) the 120◦ N´eel ordered phase in
a wide parameter region around t /t (cid:39) 0 and (ii) the
2 1
stripe ordered phase in a wide parameter region around
t /t (cid:39) 1, and in-between, (iii) the nonmagnetic insu-
2 1
lating phase caused by the quantum fluctuations in the
geometricallyfrustratedspindegreesoffreedomemerges.
The obtained phase boundaries in the strong correlation
limithavebeencomparedwiththoseofthecorresponding
Heisenberg model to find a reasonable agreement. The
orders of the phase transitions of the two magnetically
ordered phases have also been determined. In the in-
termediate correlation regime, the nonmagnetic insulat-
ing phase expands to a wider parameter region of t /t .
2 1
Then, decreasing the interaction strength further, the
system turns into (iv) the paramagnetic metallic phase
intheweakcorrelationregimeviathesecond-orderMott
metal-insulator transition. The characteristic behavior
FIG. 8. (Color online) Calculated generalized magnetic ofthe critical phaseboundaryof theMott transitionhas
susceptibility in the noninteracting limit χ0(q) defined in also been discussed in terms of the shift in the van Hove
Eq. (11). The corresponding Fermi surface is shown in each singularity due to the presence of t , as seen in the cal-
2
panel,wherethefirstBrillouinzoneisindicatedbyahexagon.
culated DOS and single-particle spectral function.
Wesuggestthatthephasediagramobtainedheremay
contain different types of nonmagnetic insulator (or spin
To confirm the absence of any magnetic instability in
liquid) states depending on the region in the parame-
our model in the weak correlation regime, we here calcu-
ter space. The characterization of the states is however
latethegeneralizedsusceptibility(orLindhardfunction)
beyondthescopeoftheVCAapproachbasedontheself-
in the noninteracting limit, which is defined as
energy(orsingle-particleGreen’sfunction),forwhichwe
χ (q)= 1 (cid:88)f(εk)−f(εk+q), (11) hope that our results will encourage future studies.
0 L ε −ε
k+q k
k
ACKNOWLEDGMENTS
where ε is the corresponding noninteracting band dis-
k
persion and f(ε) is the Fermi function. The calculated
results at temperature 0.01t are shown in Fig. 8, where WethankS.Miyakoshiforenlighteningdiscussionsand
1
we find that, in accordance with the absence of signif- K. Seki for careful reading of our manuscript. This work
icant Fermi-surface nesting features, no singular behav- was supported in part by a Grant-in-Aid for Scientific
iors actually appear in χ (q), indicating the absence of Research (No. 26400349) from JSPS of Japan. T. K. ac-
0
magneticlong-rangeordersintheweakcorrelationlimit. knowledges support from the JSPS Research Fellowship
This result supports the validity of our phase diagram for Young Scientists.
shown in Fig. 4 in the weak correlation regime.
Appendix: Cluster dependence of the phase
IV. SUMMARY boundaries
We have studied the Mott metal-insulator transition In the VCA, or in any other quantum cluster meth-
and magnetism of the triangular-lattice Hubbard model ods, we can in principle calculate the physical quantities
at half filling in the entire region of the interaction in the thermodynamic limit, but the calculated results
strength, taking into account the next-nearest-neighbor necessarily depend on the size and shape of the solver
hopping parameters for the effects of magnetic frustra- cluster. Thus, the choice of the solver cluster is impor-
tions. We have employed the method of VCA based on tant in the present approach. In the main text, we have
7
chosen the 12-site cluster shown in Fig. 2, which is the
best appropriate one because it is first of all computa-
tionally feasible and also because it fits with both the
three-sublattice 120◦ order and two-sublattice stripe or-
der without introducing unnecessary frustrations. How-
ever, it seems instructive to check the cluster-size and
cluster-shapedependencesofourresultspresentedinthe
main text.
Here, we choose several clusters [see Fig. 9(b)] that fit
eitherwiththe120◦orderorwiththestripeorder,andwe
calculatethephaseboundariestocheckthesolvercluster
dependence of the ground-state phase diagram shown in
Fig. 4. The calculated results for the phase boundaries
are shown in Fig. 9(a). We thus find that the phase
boundary between the 120◦ N´eel ordered and nonmag-
netic insulating phases is located around 0≤t /t (cid:46)0.4
2 1
inanintermediatetolargeU/t ((cid:38)6)regionandthatthe
1
phaseboundarybetweenthestripeorderedandnonmag-
netic insulating phases is located around 0.5(cid:46)t /t ≤1
2 1
in an intermediate to large U/t ((cid:38) 7) region, irrespec-
1
tive of the appropriate choices of the solver cluster. We
also find that the phase boundary of the Mott metal-
insulator transition is located around U/t (cid:39)4−6 with
1
a minimum at t /t (cid:39) 0.5, irrespective of the choices of
2 1
FIG. 9. (Color online) (a) Cluster-size and cluster-shape de- the solver cluster.
pendences of the phase boundaries between the 120◦ N´eel
ordered, stripe ordered, nonmagnetic insulating, and param-
agneticmetallicphases. Unlabeledlinesaretheresultsshown
inFig.4. (b)Schematicrepresentationsoftheclustersusedin
the calculations; in the 6-site cluster calculation, we combine
two of them to reproduce the 120◦ N´eel order.
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