Study of the triangular lattice tV model near x=1/3

0.23 MB·English

by  O. I. Motrunich

Study of the triangular lattice tV model near x = 1/3
O. I. Motrunich and Patrick A. Lee
Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139
(Dated: January 11, 2004)
We study extended Hubbard model on a triangular lattice near doping x = 1/3, which may be
relevantfortherecentlydiscoveredsuperconductorNaxCoO2 yH2O.Bygeneralizing thismodelto
·
4 N fermionic species, we formulate a meanfield description in thelimit of large N. In meanfield, we
0 find two possible phases: a renormalized Fermi liquid and a √3 √3 charge density wave state.
×
0 Thetransitionbetweenthetwophasesisdrivenbyincreasingthenearest neighborrepulsionandis
2 found to be first order for doping x = 1/3, but occurs close to the point of the local instability of
the uniform liquid. We also study fluctuations about the uniform meanfield state in a systematic
n
a 1/N expansion, focusing on the residual interaction of quasiparticles and possible superconducting
J instabilities due to this interaction. Upon moving towards the CDW instability, the increasing
charge fluctuations favor a particular f-wave triplet state. (This state was recently discussed by
1
Tanaka et al., cond-mat/0311266). We also report a direct Gutzwiller wavefunction study of the
1
spin-1/2 model.

]
l
e
- I. INTRODUCTION triangular lattice. RPA treatment in the spirit of
r Scalapino et al.4 has been done by Tanaka et al.8 very
t
s recently. They find that in the regime of interest for
Motivated by the recently discovered
.
at superconductivity1 in NaxCoO2 yH2O (x 1/3) NaxCoO2·yH2O, a particular f-wave triplet channel is
m and the unusual electronic propert·ies2 of the Na≈CoO favored close to the CDW instability.
x 2
Here,we reporta Kotliar-Liutype analysisforthe tri-
series, we began a study of he tV model on a triangular
-
d lattice,3 focusing primarily on the renormalized Fermi angular lattice, which has not been done so far. The
n liquid regime. We pointed out that nearest neighbor advantageofourapproachovertheRPAisthatthetreat-
o ment of the strong on-site repulsion is better controlled.
repulsion can lead to significant renormalization of the
c In ourstudy, we also find that the preferredsupercon-
effective hopping amplitude, which seems to be the case
[
in these materials. We also argued that strong repulsion ductingchannelnearthetransitiontothe√3 √3CDW
1 drives this model into a √3 √3 charge ordered state is the f-wave triplet state discussed by Tan×aka et al.8.
v × This state has the lobes of large ∆ oriented towards M
near commensurate dopings x=1/3 and 2/3.
0 points of the Brillouin zone (BZ) boundary as depicted
We continue this study here and perform a system-
6
in Fig. 1. The origin of this result can be understood
1 atic slave boson meanfield analysis of the Fermi liquid
from the following Fermi surface nesting argument.
1 closetothechargedensitywave(CDW)statefordopings
In an RPA-type treatment, the effective interactionat
0 near x = 1/3. We also consider the residual interaction
4 between the quasiparticles and study which supercon- wavevectorq is
0
ductivity channels are favored in the tV model on the
/ V(q)
at triangular lattice. Of particular interest here is possible Veff(q)= 1+2χ (q)V(q) (1)
enhancementofsomechannelsduetochargefluctuations 0
m
upon approaching the CDW phase.
where χ (q) is positive. This can be also written as
- 0
d Such studies of superconductivity due to residual in-
n teraction in the models with strong local repulsion are V (q)=V(q) 2χ(q)V(q)2 (2)
eff
o familiarinthehigh-T field. Scalapinoet al.4 studiedthe −
c
c Hubbard model on a 3D cubic lattice in a randomphase where χ(q)=χ (q)/[1+2χ (q)V(q)] is the full suscepti-
0 0
:
v approximation(RPA)andfoundthatd-wavepairingbe- bilityinRPA.Now,V(q)isnegativeoversomeportionof
i comes attractive close to the spin density wave transi- the BZ, in particular at the √3 √3 ordering wavevec-
X
tion. Kotliar and Liu5 studied the infinite-U Hubbard tors such as Q = 4π/(3a) xˆ, so×for sufficiently large V
r
a model on a 2D square lattice in a systematic large-N there is an instability at which χ(Q) diverges. Eq. (2)
treatment and found that the residual interaction from shows that close to the instability the effective interac-
theno-double-occupancyconstraintfavorsd-wavesuper- tionbecomesstronglyattractiveattheorderingwavevec-
conductivity close to half filling. More recently, McKen- tors. If there are sections of the Fermi surface that are
zie et al.6 applied the analysis of Kotliar and Liu to ex- nearly connected by such wavevectors, then we expect
tended Hubbard model on the square lattice at quarter enhanced superconductivity in channels that utilize this
fillingandfoundatransitiontothe√2 √2CDW,which nesting. Our calculation using slave boson theory also
×
is driven by the nearest neighbor repulsion. Merino and produces attractive effective interactions due to charge
McKenzie7 studied superconducting instabilities of the fluctuations. The difference is that now the effective po-
Fermi liquid near this transition. tentialdepends onk andk of the electrons,ratherthan
′
It is of interest to perform similar studies on the q = k k. Near the Fermi surface we write this as
′
−
2
model with N fermion species. Near x = 1/3, two can-
didate states are studied—a uniform Fermi liquid and a
x=1/3
- state with √3 √3 charge order. The phase diagram is
×
established. Wethenstudyfluctuationsovertheuniform
+ M saddlepointin1/N expansion,andfocus ontheresidual
+ θ
M quasiparticle interactions from such fluctuations. This
approach can be viewed as a version of RPA that treats
θ
K K theon-siteconstraintsystematically,andprovidesamore
quantitative justification of the above nesting argument.
- -
We conclude this Introduction with one remark. Ear-
lier works10,11,12,13,14 considered tJ model on the trian-
+ gular lattice and found that the dominant superconduc-
tivity from the J interaction is d+id singlet state. This
conclusionwasalsoreachedinourearlierreport3inwhich
westudiedtJV modelandtreatedJ asthemainresidual
FIG. 1: Schematics of the nesting argument for doping x = interaction. The tV model with no J terms predicts f-
1/3. ExternalhexagonshowstriangularlatticeBrillouinzone, wave triplet state near the CDW order,and offers a way
whileinternalhexagonshowsreducedBZforthe√3 √3or- todistinguishbetweenthesetwodifferentpairingmecha-
×
der. Flatsurfaces ofthereducedBZboundaryareconnected
nisms. The experimental situation regarding the pairing
by the √3 √3 ordering wavevectors perpendicular to the
surfaces. N×ear these momenta transfer, Veff(θ,θ′) becomes symmetryofNaxCoO2·yH2Oremainsinconclusive,with
controversy still surrounding whether there is a jump in
attractive close to the CDW instability. We also show the
the Knight shift below T .15,16
f-wave triplet gap function that we find to be the dominant c
superconducting channel from the residual interaction; this
channel utilizes the near nesting of θM points indicated with
black diamonds. II. EXTENDED HUBBARD MODEL ON
TRIANGULAR LATTICE
Veff(θ,θ′) and we find that Veff has attractive compo- We consider extended Hubbard model in the limit of
nents when θ and θ′ areconnectedby anorderingvector large onsite repulsion, i.e., the following tV Hamiltonian
Q.
Figure1showsthesituationfordopingx=1/3(drawn
1
tlioesscfaalier)ly. IctlotuserntsootuhtethreadtuactetdhiBsZfillcinorgrtehspeoFnedrminigstuorftahcee HˆtV =−PG tijc†iσcjσPG+ 2 Vijninj (3)
ij ij
√3 √3order. Wealsoshowthegapfunction∆ ofthe X X
k
×
dominantf-wavestate. Thishaslargeabsolutevaluesin withnearest-neighborrepulsionV =V. P projectsout
ij G
the directions of the M points of the full BZ boundary double occupation of sites. The band is less than half-
and signs as shown— positive signnear black diamonds filled, with the averagefermion density of 1 x per site,
−
andnegativesignforwhitediamonds. Withanattractive and we specifically consider the case t>0. See Refs. 3,9
Veff(Q), pairing between the black diamonds and also for a more detailed discussionofthe possible application
pairing between the white diamonds are very favorable, of this Hamiltonian to the Na CoO system. Here, we
x 2
since the black diamonds are nearly connected among are primarily interested in the doping near x=1/3.
themselvesbyorderingwavevectors,andsoarethewhite
diamonds.
We shouldof course considerthe near nesting of other A. Meanfield formalism
Fermi surface segments as well. For example, take the
pointθ andthepointoppositetoitontheFermisurface
K Our meanfield treatment follows closely McKen-
shownbythefilledcirclesinFig.1. Thesepointsarealso zie et al.6. The general formalism is the same except
nearly connected by an ordering vector and this pairing
for an arithmetic difference in Eq. 14 and some minor
would favor a spin singlet state. As an example, an s-
differences in the analysis. We apply this formalism to
wave state gains from all points θ,θ that are connected
′ the triangularlatticetV model. Slavebosonformulation
by V (θ,θ ) < 0. The reason why the s-wave channel
eff ′ is used to treat the no-double-occupancy constraint. We
is still disfavoredis that there remains significant overall
repulsionovernon-nestedV (θ,θ )thatitcannotavoid. write c†iσ = fi†σbi, and the slave boson Hamiltonian acts
eff ′
Theabovef-wavestateappearstobebestfortheoverall in the Hilbert space with fi†σfiσ +b†ibi = 1; the boson
field b keeps track of the empty sites.
V .
eff
To formulate meanfield description and also in order
The paper is organized as follows. We first perform to go beyond the meanfield in a systematic manner, we
a slave boson meanfield treatment of a generalized tV consider a generalized model with N fermionic species.
3
The slave boson Hamiltonian is written as (our large N limit is thus like a thermodynamic limit in
the fermion flavors). At the end of the calculation, we
1 1
Hˆ =−NS tijfi†σfjσb†jbi+2NS Vijfi†σfiσ(NS−b†jbj),will put N = 2, S = 1/2, and our specific choices when
defining H are such that this will reproduce the slave
ij ij fb
X X
(4) boson Hamiltonian for the spin-1/2 model.
which now acts in the space Proceeding as in McKenzie et al.6 and Kotliar and
Liu5, we write the path integral in the radial gauge17
fi†σfiσ+b†ibi =NS (5)
(thespinindexσ runsfrom1toN). Forconvenience,we β
introduced parameter S, which is kept fixed as we take Z = DfDfDrDλ exp dτL(τ) . (6)
N . Wealsousedaparticularformfortherepulsion Z "−Z0 #
→∞
term. To study the system behavior with doping, we fix
the totalfermionnumber to NS(1 x) fermions per site The imaginary time Lagrangian
−
1 1
L= f (∂ µ+iλ )f t r r f f + f f V (NS r2)+ iλ (r2 NS)+µ NS(1 x).
iσ τ− i iσ−NS ij i j iσ jσ 2NS iσ iσ ij − j i i− −
i ij ij i i
X X X X X
(7)
To obtain this, auxiliary field λi(τ) was introduced to Here, t˜k = r′trr′e−ik(r−r′) and similarly for V˜k (we
enforce the constraint; the phase of the original boson willoftendroptildeswhenthemeaningisunambiguous);
fieldb wasgaugedaway,while the absolutevalue is now Ld is the nuPmber of lattice sites; f(ξ) = 1/(eβξ +1) is
i
a real field r = b [the corresponding measure is Dr = theFermidistribution;andξ isthequasiparticleenergy
i i k
| |
r (τ)dr (τ)]. Chemical potential µ sets the correct measured relative to the Fermi level
i,τ i i
fermion density.
Q
We now formally integrate out the fermions and seek
1
saddlepointsoftheresultingactionintermsofthefields ξk = xt˜k+ϕ+ V˜0(1 x) µ . (11)
− 2 − −
r andλ . Thenaturalscalefortherfieldsisr2 N,and
i i
∼
uponsuchrescaling(whichwedonotperformexplicitly)
the saddle point analysis can be cast into a formal large The chemical potential is tuned so that
N procedure.
The saddle point conditions read 1
f(ξ )=S(1 x) . (12)
Ld k −
hfiσfiσi+ri2 =NS , (8) Xk
i.e., the constraints are satisfied on average,and
The uniform saddle point represents a renormalized
ϕ iλ = 1 rj t f f +t f f Fermi liquid with effective hopping xtij. We recognize
i ≡ i 2NS r ijh iσ jσi jih jσ iσi suchhoppingenergyrenormalizationascomingfromthe
i
j
X (cid:2) (cid:3) configurational constraints imposed by the no-double-
1
+ V f f . (9) occupancycondition. Atthis level,the repulsiveinterac-
2NS jih jσ jσi tion leads only to the shift in the bottom of the band.
j
X
We seek time-independent saddle points. However, the
fieldsmaybespatiallyvaryingtoallowforpossiblecharge
inhomogeneity. Inthiscase,ϕ + 1 V (NS r2)is
i 2NS j ij − j
aneffectivepotentialonsitei(cf.Eq.7),while 1 r r t B. Fluctuations over the uniform state
P NS i j ij
is an effective hopping amplitude in such meanfield.
Among all saddle points, we are to take the one that
Away from half-filling and for small V, the uniform
minimizes the free energy specified by Eq. (7).
saddle point has the lowest free energy. We can estab-
We first consider the uniform saddle point, which has
lish the region of its local stability by considering small
r2 b2 =NSx and
i ≡ fluctuations above the uniform state. Proceeding as in
Refs. 5,6, we obtain the following quadratic action for
1 N 1
iλ ϕ= (t˜ + V˜ )f(ξ ) . (10) the fluctuations iλ =ϕ+iδλ , r =b(1+δr )
i ≡ NS Ld k 2 0 k i i i i
k
X
4
1 Γ Γ δr(q,ω )
S(2) = δr( q, ω ) δλ( q, ω ) rr rλ n , (13)
2 − − n − − n Γλr Γλλ δλ(q,ωn)
qX,ωn(cid:0) (cid:1)(cid:18) (cid:19)(cid:18) (cid:19)
where ω is bosonic Matsubara frequency, and the “inverse RPA propagator”is given by
n
2b2 N 1 b4 N f(ξ ) f(ξ )
Γ (q,ω ) = 2b2ϕ (t + V˜ )f(ξ ) k+q − k (t +t +V )2 , (14)
rr n − NSLd k+q 2 0 k − (NS)2Ld iω +ξ ξ k k+q q
n k k+q
k k −
X X
b2 N f(ξ ) f(ξ )
Γ (q,ω ) = Γ (q,ω )=i 2b2+ k+q − k (t +t +V ) , (15)
rλ n λr n NSLd iω +ξ ξ k k+q q
" n k k+q #
k −
X
N f(ξ ) f(ξ )
k+q k
Γ (q,ω ) = − . (16)
λλ n Ld iω +ξ ξ
n k k+q
k −
X
The above expressions coincide with Eq. (17) in Ref. 6 instability is towardsphaseseparationinto hole-richand
upon replacements t/(NS) t/N, V/(NS) 2V/N, hole-poor regions. This can be seen by examining the
→ →
except that the V˜ term in Γ replaces V˜ there. energy of the uniform state (measured per site)
0 rr k
We can write more compactly
1
H(x)=E (x)+ V˜ (1 x)2 , (24)
2b2 b4 t 2 0 −
Γ = Y (V2X +2V X +X ) ,(17)
rr NS − (NS)2 q 0 q 1 2
where E (x) is the hopping energy per site in the free
b2 t
Γ = Γ =i 2b2+ (V X +X ) , (18) fermionproblemwithhoppingamplitudetxandfermion
rλ λr q 0 1
NS density (1 x). After some analysis, the condition (23)
(cid:20) (cid:21) −
Γλλ = X0 , (19) is seen to be equivalent to H′′(x) < 0, which indeed
leads to phase separation. Note that moderate nearest-
where we used Eq. (10) and introduced neighbor repulsion V stabilizes the uniform saddle point
against phase separation, and the corresponding bound-
X (q,ω ) = N (t +t )pf(ξk+q)−f(ξk) (,20) ary is shown in the lower left hand corner of Fig. 2.
p n Ld k k+q iω +ξ ξ TheinstabilityoftheuniformstateforlargeV isfound
n k k+q
k −
X to always occur at the √3 √3 ordering wavevector.
N ×
Y(q) = (t t )f(ξ ) . (21) The corresponding critical (V/t) is shown with the dot-
Ld k− k+q k c
ted line in Fig 2. In the region designated renormalized
k
X
Fermi liquid the uniform state is stable towards static
As discussed in Ref. 5, the localstability of the saddle fluctuations at any wavevector.
point is determined by the condition detΓ = Γ Γ
rr λλ
−
Γ Γ >0, which translates to
rλ λr
4b2 2Y b2 X X 2 C. √3×√3 CDW saddle points
X V + 2 +b2 2+ 1 >0 .
0 q
NS NS − NSNS NS
(cid:18) (cid:19) (cid:18) (cid:19) However,localstability does not guaranteeglobalsta-
(22)
bility ofthe saddle point. As a specific example, we con-
We now focus on the stability to static perturbations,
sider x = 1/3. The above local stability analysis gives
setting ω = 0. Since X > 0 and V is negative in
n 0 q
some portion of the Brillouin zone, there is clearly an
instability for sufficiently large V.
V (x=1/3)=2.11 . (25)
The local stability analysis specialized to N = 2,S = c
1/2(andinthe zero-temperaturelimit) foralldopingsis
The uniform saddle point has the energy per site
summarized in Fig. 2.
For x < 0.14 there is an instability at q = 0 for small
V, which persists as long as Euniform(x=1/3)=−6txhfi†σfjσi+3V(1−x)2 (26)
detΓ(q =0)=const×[1+2tFν(ǫF)+V˜0ν(ǫF)]<0 , with hfi†σfjσi = 0.337 for the uniform hopping at this
(23) doping.
where we used X (q=0)=ν(ǫ ), etc., ν(ǫ ) is the spin- On the other hand, consider a competing state with
0 F F
ful density of states per site at the Fermi level. The complete √3 √3 order which has all charges on the B
×
5
isfy b2 + b2 + b2 = 3x. Similarly, we allow differ-
5 A B C
local stability of the uniform state ent ϕ and ϕ = ϕ . For convenience, we can take
1st-order phase boundary A B C
ϕ +ϕ +ϕ = 0, since there is also the chemical po-
2nd-order phase boundary A B C
4 tential degree of freedom which is tuned to obtain the
√3 × √3 CDW correct fermion density for each set bA,ϕA. Now, for
c3 each bA we tune ϕA until b2A+hfA†σfAσi=1 is satisfied,
) and finally adjust b so that the other self-consistency
V/t q=(4π/3, 0) condition is satisfiedA. We find all such saddle points and
(2
compare their free energies obtained from Eq. (7). The
result is shown in Fig. 2 with circles. Filled circles indi-
1 Renormalized Fermi liquid catethesituationsimilartox=1/3,whenthetransition
is first order. For open circles, we tentatively conclude
q=0 that the transition point coincides with that determined
0 PS
bythelocalstabilityanalysisandthetransitionissecond
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
doping x order.
We considered in detail the uniform and the √3 √3
FIG. 2: Meanfield phase diagram of the tV Hamiltonian. ×
CDWsaddlepoints. Sinceweareinterestedinthedoping
For x < 0.14, the uniform state is unstable at q = 0 (to-
wards phase separation) for small repulsion V, but is stabi- regimenearx=1/3,wherewedonotexpectsomeother
lized for moderate V. For large V, the instability occurs at state to enter the competition, the presented meanfield
the √3 √3 ordering wavevector and the critical (V/t)c is analysis is complete.
×
shown with the dotted line. In the region between the dot-
tedlines,theuniformstateislocallystableatallwavevectors.
Thecirclesshowtheactualmeanfieldtransitionswithgeneral
√3 √3 ansatz; filled circles indicate first-order phase tran- III. RESIDUAL SUPERCONDUCTING
×
sitions, while open circles indicate second-order transitions. INSTABILITIES
We now aim to go beyond the meanfield description.
and C sublattices, b = b = 0, while the sublattice A
B C Thefreeenergyoftheuniformstateatnextorderin1/N
is not occupied, bA =1. The energy per site is canbeobtainedfromthequadraticactionS(2),Eq.(13).
In terms of the fermionic quasiparticles, this contains
E (x=1/3)=V. (27)
√3 √3 further effective mass renormalization as well as resid-
×
ualinteractionof quasiparticlesmediated by the bosons.
It is simple to see that the latter becomes energetically
We focus on the residual interaction, in order to decide
preferred over the uniform state for V = 2.00, i.e., be-
which pairing channel is favoredat low temperatures for
lowthelocalinstabilitypointoftheuniformstate,which
our triangular lattice with hard repulsion.
meansthatthetransitionisfirstorder. Thefollowingde-
tailedanalysisfinds thatthe the actualtransitionoccurs The verticescouplingthe fermions andthe bosonscan
at V1storder(x = 1/3) = 1.98 to a state which is close to be obtained by examining the LagrangianEq. (7), while
c
the above state with complete CDW order. thebosonpropagatorsaregivenbytheinverseofthema-
To study such possibilities in detail, we consider sad- trix Γˆ(q,ωn) in the quadratic action, e.g., Drr(q,ωn)
≡
dle points that have the √3 √3 ordering pattern. The δr( q, ωn)δr(q,ωn) = Γλλ/detΓˆ, etc. From the ex-
procedure is as follows: We×select the A-sublattice of phres−sion−s for Γ and uising b2 N, all propagators are
∼
the three sublattices, and seek saddle points that have O(1/N).
b = b on the A sublattice and b = b = b on Forour study, we singleout interactionterms relevant
i A i B C
the B and C sublattices. Of course, these must sat- for the BCS instability, obtaining
b4 b2
V (k ,k )=D D t +t +V 2+i [D (V +t +t )+D (V +t +t )] , (28)
eff 1 2 λλ− (NS)2 rr| k1 k2 q| NS λr −q −k1 −k2 rλ q k1 k2
where all boson propagators are at wavevector q = tonian
k k . The interactionisoforderO(1/N). Ifwe ignore
1 2
−
the frequency dependence of the boson propagators, the 1
relevanttermscorrespondtothestandardpairingHamil- Hˆpairing = 2Ld Veff(k1,k2)fk†1,σf−†k1,σ′f−k2,σ′fk2,σ .
kX1,k2
(29)
6
to add more repulsion at zero momentum transfer. In-
2.0
x=0.33 V=0.0 creasing V further and approaching the √3 √3 CDW
0.5 ×
instability, there is a dramatic development in the fea-
1.5 1.0
1.5 turesassociatedwithscatteringatthe√3 √3ordering
2.0 wavevectors. Asdiscussedintheintroduct×ion,thequasi-
1.0
)K particle interactionat suchmomentum transfer becomes
θθ, 0.5 attractive close to the critical point, and this is clearly
(eff seenintheplots. InFig.3,thecriticalpointVc =2.11is
V defined from the quadratic fluctuation analysis. As dis-
0.0
θ M cussedearlier,inthe meanfield,wefindinstead1st-order
-0.5 θK K phase transition at a somewhat lower Vc1storder = 1.98.
However the features in V (θ,θ ) are already enhanced
eff ′
atthe actualtransitionsince it occursclose to the insta-
-1.0
bilitypointandhassignificantcriticalfluctuations,sowe
0 60 120 180 240 300 360
θ - θ (degrees) will mostly ignore the distinction in what follows.
K
We now study BCS instabilities due to this residual
2.0
x=0.33 V=0.0 interaction; the analysis below is valid for either singlet
0.5
or S = 0 triplet superconducting channels. We follow
1.5 1.0 z
1.5 Refs. 4,5 and define coupling constant associated with
2.0 each channel ∆ g (k)
1.0 k ∼ α
)M
θθV(, eff 00..05 θ θ M cα = − Z d|vσkk| Z (2πd)σdk|v′dkσ′|kgαgα(k(k)∗)V2eff(k,k′)gα(k′) ,
M K Z |vk|| |
-0.5 (30)
where the integration is over the Fermi surface elements
-1.0 dσ, and v(k) = kξ(k) is the Fermi velocity. For an
∇
0 60 120 180 240 300 360 attractive channel, we must have c > 0, and the tran-
θ - θ (degrees) α
M sitions temperature is given by the weak coupling BCS
FIG. 3: Residual quasiparticle interaction relevant for the expression (which is appropriate in the present case)
pairing instability (cf. Eq. 29) plotted for the momenta k1
and k2 on the Fermi surface. One angle is kept fixed while Tc[α]=Ωexp[ 1/cα] ; (31)
theotherisvariedovertheFSasindicatedintheinsets,which −
alsoshowthetriangularlatticeBZandthereducedBZforthe the frequency cutoff is roughly Ω xt, since the energy
√3 √3 CDW order (see Fig. 1). The data is for x = 1/3, ∼
× integration is over the entire band.
and the uniform state becomes unstable at Vc = 2.11. The Representative triangular lattice tight binding har-
vertical scale is in units of t. Top panel: The fixed angle is
monics that cover main symmetry classes are listed in
in the direction of the K point of the full BZ boundary. As
we approach theCDW instability, 180◦ scattering amplitude Table I. Four of the ansatze have their real-space ∆rr′
becomesattractive,whichisassociated withthefactthatthe nonzeroonnearest-neighborbondsonly,theNNNf-wave
corresponding Fermi surface points (black circles) are nearly ansatz has next-nearest-neighbor bonds, and the i-wave
connectedbyoneofthe√3 √3orderingwavevectors. Bot- ansatz requires even further neighbor bonds. Since V
× eff
tompanel: ThefixedangleisinthedirectionoftheM point; is real, only real ansatze need to be considered. In-
◦
thethreenearlynestedpointsoccurat120 from eachother. deed, in this case the problem of finding a harmonic
with the largest coupling constant is equivalent to a real
symmetric eigenvalue problem; the easiest way to see
Fromnowon,wefixN =2,S =1/2. Wefirstconsider this is to discretize the dσ integrals and define ψα(k) =
in detail doping x=1/3. We are primarily interested in gα(k)(dσk/vk )1/2.
| |
thescatteringprocesseswhenmomentak andk lienear Figure 4 shows the coupling constant c evaluated for
1 2 α
the Fermi surface. Therefore, we visualize the effective the harmonics in Table I. We find that for all V (except
interaction by fixing the momenta on the Fermi surface very close to V ) the NNN f-wave triplet is the dom-
c
and plotting V (θ ,θ ) as a function of polar angles. inant channel, while the other ansatze are repulsive or
eff 1 2
ThisisshowninFig.3,whereweseehowV evolvesfor become repulsive upon increasing V. The performance
eff
increasing V approaching the CDW transition. Observe of the NNN f-wave is understood by looking at the bot-
thatthereisresidualrepulsionevenwhenV =0,whichis tom panel of Fig. 2. The three marked points 120 from
◦
entirely due to the no-double-occupancy constraint, and each other are nearly connected by √3 √3 ordering
×
that this repulsion already has some momentum space wavevectors and the scattering among these points be-
features. As we increase V from zero,the initial effect is comesattractiveuponapproachingtheCDWregime. As
7
Ansatz gk Label 0.10
cosk e1+cosk e2+cosk e3 s x=1/3
· · ·
sink e2 sink e3 px
· − · 0.05 NNN f-wave
cosk e2 cosk e3 dxy = 1st eigval
· − ·
sink e1 sink e2+sink e3 f
· − · · 2nd
sink·(e1+e2)−sink·(e2+e3)+sink·(e3−e1) NNNf cα 0.00 i
cosk (2e1+e2) cosk (e1+2e2)+cosk (2e2+e3) i dxy f
· − · ·
cosk (e2+2e3)+cosk (2e3 e1) cosk (e3 2e1)
− · · − − · −
-0.05 py
TABLE I: Triangular lattice harmonics evaluated in Fig. 4.
e1, e2, e3 = e2 e1 refer to unit triangular lattice vectors.
−
The labels are to betaken as descriptiveonly.
s
-0.10
0.0 0.5 1.0 1.5 2.0 Vc
V
displayedin Fig. 1, the NNN f-waveansatz has its posi-
FIG.4: Couplingconstantsforthetriangularlatticeharmon-
tivelobes orientedpreciselyinthesethree directionsand
ics in Table I (the s-wave ansatz is below the bottom of the
is able to utilize this attraction.
plot). The doping is x=1/3, and the critical Vc =2.11 sets
Here, we note that generic next-nearest-neighbor theright-handplotboundary. Wealsoshowthemaximaland
ansatze utilize this nesting better than the nearest- thesecondeigenvalue;theNNNf-wavecouplingconstantco-
neighborones,andwe triedthe correspondingNNN ver- incides with themaximal eigenvalue for all V shown.
sions of the s, p, and d-wave; the coupling constants are
improved (not shown), but are still far from approach-
ing the NNN f-waveansatz. This is becauseeachansatz 0.20
also has to pay the cost of repulsive scattering on large x=0.20
non-nested portions of the Fermi surface, and the NNN e 0.30
v
a
f-waveansatzappearstobebesthereaswell. Forexam- w0.15 0.40
ple, the coupling constant for the s-wave remains nega- N f-
N
tive and appears on the scale of Fig. 4 only very close to N
the critical point, despite the fact that it gains from all =
e 0.10
attractive V (θ,θ )<0. u
eff ′ al
In fact, we also solve the full eigenvalue problem spec- nv
e
ified by Eq. (30), and plot the first (maximal) and the g
ei0.05
second eigenvalue in the same figure. We find that the x
a
NNN f-wave coupling constant coincides with the max- m
imal eigenvalue in the entire range of V < V (except
c 0.00
maybe very close to V ). The second eigenvalue is well
c 0.0 0.5 1.0 1.5 2.0 2.5
separated from the first; the corresponding eigenvector V
over a large range of V does not have one of the sim-
FIG. 5: Maximal eigenvalue for dopings x = 0.20, 0.30, and
ple s, p, d or i character, but instead has eight lobes in 0.40. The critical Vc from the quadratic fluctuation analysis
momentum space, four of each sign. is indicated by vertical line in each case, while the 1st-order
Finally,Fig.5showsthemaximaleigenvalueasafunc- transition isindicated withthecorrespondingarrownearthe
tion of V for several dopings. In each case, the maximal bottom of thegraph (cf. Fig. 2).
eigenvalue corresponds to the NNN f-wave ansatz. The
nestingdisplayedinthebottompanelofFig.3thatfavors
the NNN f-wave improves as we lower the doping, since
we use Eq. (31) literally, the obtained scale of T is still
theFermisurfacepassesthroughtheendpointsofthere- c
very small: for example, if we take c = 0.1, which is
duced BZ when x=0.194 (see also Fig. 1). From Fig. 5 α
fairly large, then exp[ 1/c ] 5 10 5, which is tiny.
we see that the enhancement in the coupling constant − α ∼ · −
Here we remark that one specific ingredientthat canen-
is strongest and over the broadest range for x = 0.20.
hanceT ismissinginthepresenttreatment,namely,the
Observe also that the residual interaction from the no- c
enhancement of the effective mass by the nearest neigh-
double-occupancyconstraintonly(V =0)alsofavorsthe
bor repulsion while remaining in the Fermi liquid. More
discussed NNN f-wavechannel onthe triangularlattice.
precisely, at the presented level of analysis, the effective
Weconcludebydiscussingwhattheseresultsmeanfor hopping t = xt is not renormalized by V, whereas it
eff
the scale of superconductivity. We see that because of can become severaltimes smaller when additionalshort-
the nesting the maximal c can be relatively large com- rangecorrelationsareincludedasdiscussedinRef.3and
α
paredtosimilarpredictionsinthenon-nestedcases(such in the following section. Since 1/c t /V , this can
α eff eff
∼
as square lattice at finite doping, Ref. 5). However, if have dramatic effect on the calculated T .
c
8
IV. JASTROW-GUTZWILLER Hamiltonian is suggested by the meanfield treatment of
WAVEFUNCTION STUDY. DISCUSSION the Fermi liquid - CDW competition in Sec. II, and in
fact by any such meanfield. However, our wavefunctions
We now consider in some detail how the above results have further local correlations built in by the Jastrow
apply to the original spin-1/2 model. To this end, we factors which allow the liquid to better accommodate
have performed a systematic trial wavefunction study thenearest-neighborrepulsion(alsoleadingtosignificant
of the tV model. Specifically, we consider a family of renormalizationofthebandwidthasdiscussedinRef.3),
Jastrow-Gutzwiller wavefunctions3,18 while this is missing in the meanfield.
Wesummarizeourwavefunctionstudiesforthreedop-
ΨJG(r1σ1,...)=e− i<ju(ri−rj)det[ψα(Rj)]det ψα(Rj′) ,ings x = 0.24, 0.33, and 0.40. We report only the re-
P (32) sultofthethree-parameterW,κ,ϕ optimization. When
A
(cid:2) (cid:3)
where R and R denote the positions of spin-up and κ=0,ϕ =0, we obtain the Fermi liquid state. As dis-
′ A
{ } { }
spin-downfermionsrespectively,andthe wavefunctionis cussed earlier, this restricted study should suffice for an
nonzero only when the two sets do not overlap, which is accurate determination of the uniform liquid to CDW
the result of Gutzwiller projection. ψ refer to appro- transition.
α
{ }
priatesingle-particlestatesthatareoccupiedinthe“pre- For dopings x = 0.33 and 0.40, we find rather abrupt
projected”wavefunction. Eachconfigurationoffermions transitions to the √3 √3 state at V (x = 0.33) 3.5
c
× ≈
is also weighted by a Jastrow factor defined via two- and V (x = 0.40) 4.0. On the other hand, for the
c
≈
particlepseudopotentialsu(ri rj),whichputsadditional doping x = 0.24, the system shows no ordering till very
−
correlations into the wavefunction. large V = 15 to 20 and maybe even higher. Compar-
The Fermi liquid state is obtained by occupying ap- ing with Fig. 2, the behavior at this lowest doping is
propriate plane wave states ψk, k kF. From our very different from the meanfield prediction. This is an
| | ≤
studies of the tV model, we conclude that already the extreme manifestation of the fact that the charges are
nearest-neighbor Jastrow factor gives good control over abletoeffectivelyavoideachotherandremainintheliq-
localcorrelationsandissufficientforanaccurateenerget- uid state, which is captured by our Jastrow-Gutzwiller
ics distinction between the Fermi liquid and competing wavefunction, while the meanfield uniform state cannot
charge ordered state. accommodate this and pays large repulsion energy cost.
As discussed at length in Ref. 3, lattice gas system Finally,havingestablishedthedominantlocalenerget-
described by the nearest-neighbor u(ri,rj) W under- ics,we alsotriedaddingsuperconducting correlationson
≡
goes a transition to a √3 √3 state for large W. Our topofthe renormalizedFermiliquidstate. We triedsev-
×
directoptimizationwithsuchsingle-parameterwavefunc- eral superconducting ansatz including extended s-wave,
tion showed that this state is driven into the charge- p+ip,d+id,f-wave,andalsotheirNNNversions. How-
ordered regime for sufficiently strong repulsion V and ever, in our studies we were not able to detect any im-
dopingsintherange0.27<x<0.50. Wealsomentioned provement in the energetics upon adding superconduc-
that once this happens, we are no longer justified in us- tivity. From this we conclude that if some such state
ing plane wavesfor the preprojectedorbitals. We should appears at low energies, the condensation energy is still
instead consider more general single-particle states; for verysmallto be detectedby directnumericalstudies. In
example, we consider orbitals obtained by diagonalizing this situation, we are left to rely on approximateanalyt-
a “trial” Hamiltonian ical calculations such as discussed in the main body of
this paper.
HˆCDW =− χijc†iσcjσ+ ϕic†iσciσ , (33)
ij i
X X
where we allow generic hopping amplitudes χ and site Acknowledgments
ij
potentials ϕ that follow the √3 √3 ordering pattern.
i
×
Specifically, we select the A sublattice and take χ = We thank Stuart Brown for bringing Ref. 7 to our at-
AB
χ =1+κ,χ =1 κ;κ>0makesthehoppingmore tention. This work was supported by the National Sci-
AC BC
−
dice-lattice-like, while κ < 0 makes it more honeycomb- ence Foundation under grant DMR–0201069. OIM also
like. We also use the convention ϕ = ϕ = ϕ /2 wants to thank his family for support during his stay in
B C A
−
and vary ϕ . Observe that this structure of the trial Ukraine.
A
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