Table Of ContentPlaquette bond order wave in the quarter-filled extended Hubbard model on the
checkerboard lattice
Martin Indergand,1 Carsten Honerkamp,2 Andreas La¨uchli,3 Didier Poilblanc,4 and Manfred Sigrist1
1Institut fu¨r Theoretische Physik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland
2Theoretical Physics, Universit¨at Wu¨rzburg, D-97074 Wu¨rzburg, Germany
3Institut Romand de Recherche Num´erique en Physique des Mat´eriaux (IRRMA), CH-1015 Lausanne, Switzerland
7 4Laboratoire de Physique Th´eorique, CNRS & Universit´e de Toulouse, F-31062 Toulouse, France
0 (Dated: February 6, 2008)
0
2 An extended Hubbard model (including nearest-neighbor repulsion and antiferromagnetic spin
exchange) is investigated on the frustrated checkerboard lattice, a two-dimensional analog of the
n
pyrochlore lattice. Combining Gutzwiller renormalized mean-field (MF) calculations, exact diago-
a
nalization (ED) techniques, and a weak-coupling renormalization group (RG) analysis we provide
J
strong evidence for a crystalline valence bond plaquette phase at quarter-filling. The ground state
2
istwofold degenerateandbreakstranslation symmetry. The bondenergies showastaggering while
2
thecharge distribution remains uniform.
]
l PACSnumbers: 71.27.+a,73.22.Gk,71.30.+h
e
-
r
t I. INTRODUCTION estingly,althoughtheGSistwofolddegenerateduetothe
s
. breaking of inversion symmetry that leads to two types
t
a ofnon-equivalentunits,latticetranslationalsymmetryis
Many frustrated quantum magnets exhibit complex
m preservedandallsitesremainequivalent,i.e.,there isno
behaviordeeply rootedinthe macroscopicdegeneracyof
charge density wave (CDW) ordering. Each site of the
-
their classical analog. Spin liquids characterized by the
d lattice belongs to two non-equivalent units and the two
n absence of any kind of symmetry breaking and valence
types ofunits havedifferent bond strength,i.e., different
o bond crystals (VBC) characterized by translation sym-
expectation values for the kinetic and magnetic energy,
c metry breaking only (SU(2) symmetry is preserved) are
[ which is the typical characteristic of a bond order wave
exotic phases possibly realized in some two-dimensional
2 (2D)orthree-dimensional(3D)frustratedmagnets.1The (BOW).
Thecheckerboardlatticecanbeviewedasa2Dcorner-
v VBCinstability in2Dand3Darehigherdimensionalre-
sharing array of tetrahedra. As shown in Ref. 6 the t-J
9 alizationsoftheone-dimensional(1D)spin-Peierlsmech-
model with two electrons on a single tetrahedron has a
2 anism which is known to occur, e.g., in the Heisen-
1 very robust non-degenerate and non-frustrated ground
bergspin-1/2chainwithnearestandnext-nearestneigh-
9 statewhichisgivenbyanequalamplitude superposition
bor exchange interaction above some critical frustration
0 ofthesixstateshavingasingletononebond. Therefore,
(even in the absence of lattice coupling).2 In that case,
6 it was conjectured in Ref. 6 that a BOW might occur
0 the chain spontaneously dimerizes leading to a twofold
quitegenerallyonabipartiteandcorner-sharingarrange-
/ degenerate gapped ground state (GS). Similarly, the GS
t ment of tetrahedra at quarter-filling, i.e., that the BOW
a ofthe Heisenbergmodelonthecheckerboardlattice (the
m square lattice with diagonal bonds on every second pla- instability occurs not only in the 3D pyrochlore lattice
but also in the 2D checkerboardlattice.
- quette, see Fig. 1) was shown to be twofold degenerate
d with long range plaquette VBC order characterized by a In the present paper, we provide numerical and ana-
n lytical evidence for this new type of spontaneous BOW
Q = (π,π) ordering wavevector.3 In this state the spins
o instability for correlated electrons on the checkerboard
pair up in four-spin singlet units located on every sec-
c lattice at quarter-filling (n = 1/2). The spontaneous
: ondvoidplaquettegivingrisetostrongerbondsonthose
v symmetry breaking produces a doubling of the unit cell
plaquettes.
i like in the spin-Peierls and VBC scenarios and does not
X
So far the investigation of charge degrees of freedom lead to a breaking of inversion symmetry as on the py-
ar in geometrically frustrated systems has been poorly ex- rochlore lattice. The BOW instability occurs on the
plored. Most computations deal with a small fraction checkerboard lattice at quarter-filling in the presence of
of holes doped into a Mott insulator4 or with itinerant nearest-neighbor(n.n.) interactions (repulsion and mag-
systems at half-filling.5 netic exchange).
It has been suggested recently,6 that for strongly cor- TheBOWphasebearssimilaritieswiththeVBCphase
related electrons on frustrated lattices such as the 2D at half-filling (Heisenberg model), namely long range
kagom´e or the 3D pyrochlore lattice a related sponta- plaquette-plaquette correlations with the same wavevec-
neous symmetry breaking away from half-filling might tor, but the spin singlets are formed on the crossed pla-
occur at the fractional filling n = 2/3 and n = 1/2, re- quettes for the BOW at quarter-filling whereas they are
spectively. These so-called bisimplex lattices are arrays formed on the empty plaquettes for VBC at half-filling.
of corner sharing triangular or tetrahedral units. Inter- We report the results of strong-coupling (U = ) ap-
∞
2
proaches in Sec. II, where Sec. IIA is dedicated to a
Gutzwiller MF calculation and Sec. IIB contains ED re-
sults.
In Sec. III we derive an effective weak-coupling model
on the square lattice and analyze it with RG methods.
TheSec.IIIAcontainsaMFandtwo-patchRGanalysis
and in Sec. IIIB the results of the more sophisticated
N-patch RG are presented.
Thedifferentapproachesprovideaveryconsistentpic-
ture and show that the BOW instability on the checker-
board lattice at quarter-filling occurs both for weak and
strong coupling and for quite general repulsive interac-
tions.
A. The Hamiltonian FIG. 1: (Color on-line) Schematic pattern of the plaquette
phase. Different line styles (colors) correspond to different
electron hopping amplitudes and spin-spin correlations (see
We study the Hamiltonian H =H0+Hint at quarter- Fig. 2).
filling(n=1/2)onthecheckerboardlattice. Thekinetic
part is given by
andadecouplingintheparticle-holechannelleadstothe
H = t c† c +h.c. (1) following MF Hamiltonian,
0 − X X (cid:16) iσ jσ (cid:17)
hijiσ=↑↓ H = t gt c† c +h.c.
MF − XX ij(cid:0) iσ jσ (cid:1)
with the positive hopping matrix element t. The sum hiji σ
runs over all bonds on the checkerboard lattice. 3
PThheijiinteraction part of the Hamiltonian is given by − X (cid:16)4giJjJ +V(cid:17)(cid:0)χjic†iσcjσ +h.c.−|χij|2(cid:1)
hijiσ
Hint =UXni↑ni↓+JXSi·Sj +V Xninj (2) + V X(hniinj +hnjini−hniihnji), (5)
i hiji hiji hiji
where the Gutzwiller weights have been expressed in
with onsite repulsion U, n.n. repulsion V, and n.n. spin
terms of local fugacities z = (1 n )/(1 n /2),
eHxacmhailntgoeniJan. For U = ∞ we obtain the strong coupling gitj =√zizj andgiJj =(2−zii)(2−zj−),htoiiaccou−nthforiipos-
sible (small) non-uniform charge modulations (if any).8
Ht-J-V = PH0P +JXhijiSi·Sj +V Xhijininj, (3) Thch†jσecsieσlif.-cRonesciesnttelnyc,ythciosnadpiptiroonascahrweaimsspulecmceesnsftueldlyaussχejdit=o
investigatethepropertiesof4 4checkerboard-likeBOW
= Ht-J +V′Xninj, (4) in the lightly doped t-J mod×el on the (non-frustrated)
hiji square lattice.9
The MF equations are solvedself-consistently on 16
where is the projectionoperatorthat enforces the sin- ×
gle occPupancy constraint and V′ =V +J/4. For V′ =0 16and48×48lattices startingfromarandombondand
site configuration. We assume a fixed physical value of
thestrongcouplingHamiltonianreducestotheusualt-J
J/t = 1/3 and systematically vary the ratio V/t. We
model.
find that in the range 0.3 V/t 1.75 the MF equa-
≤ ≤
tions convergeto a unique solutiongivenby the checker-
board pattern shown in Fig. 1. Interestingly, finite size
II. STRONG COUPLING
effects are weak for this range of parameters. As far as
symmetrypropertiesareconcerned,themodulatedstruc-
A. Gutzwiller renormalized mean-field turefoundherecorrespondstoadoublingoftheunitcell
withtwonon-equivalent“weak”and“strong”tetrahedra
We study the t-J-V model Hamiltonian (3) on the (crossedplaquettes)henceleadingtofourtypesofbonds
checkerboard lattice. This Hamiltonian describes the schematicallyrepresentedbydifferentlinestyles(colors).
physical processes in the restricted Hilbert space of con- Each bond is characterized by the expectation values of
figurationswithno doubly occupiedsites. Physically,we the hopping and of the spin-spin correlations which are
expect V > J but we do not necessarily assume V large plotted in Fig. 2. It is important to emphasize here that
compared to t. The local constraints of no doubly occu- noparticularsuperstructurewasassumedinthe calcula-
pied sites are replaced by statistical Gutzwiller weights7 tion(the equationsaresolvedin realspacestarting from
3
(a) N =20
FIG.2: (Coloron-line)Expectationvaluesvs.V/tofthehop-
ping |˙(c†iσcjσ+h.c.)¸| (left) and exchange |˙Si·Sj¸| (right)
operators on the four non-equivalent bonds of the plaquette
phase (see Fig. 1). A different line style and color is used (b) N =24
for thedifferentplaquettesand thethicker(thiner)lines cor-
respond to the outer (crossed) bonds of the plaquette. The
results are obtained from an unrestricted MF calculation on
a 48×48 lattice for J/t=1/3.
a random state) and that our solution space a priori al-
lows for CDW’s and/or bond currents (complex values
for χ ). However, our solution exhibits real values of
ij
χ (it does not break time-reversal symmetry) and has
ij
a uniform charge density on all sites. It is therefore a
pure BOW. Note that two types of bonds (with slightly
different amplitudes) are found within eachtetrahedron.
Although the MF treatment, very likely, overestimates FIG.3: (Coloronline)Correlationfunctionofthekineticen-
the amplitude of the modulations,10 these results con- ergy (Eq. 6) of (a) a 20 sites and (b) a 24 sites checkerboard
sample at n = 1/2, J/t = 1/3, and V′/t = 7. The black,
vincingly suggest a quite strong BOW instability of the
empty bonds denote the same reference bond, the red, full
checkerboard lattice. Note also that for large V/t val-
bonds negative and the blue, dashed bonds positive correla-
ues our MF approach fails to converge as the system of
tions. The line strength is proportional to the magnitude of
non-linear equations becomes unstable and oscillatesbe- thecorrelations.
tween two attractors. In fact, in the strong V/t limit,
the “tetrahedron rule”11 (or ice rule) should be obeyed,
namelythereshouldbeexactlytwoparticleswithineach where
crossed plaquette. It is clear that the simple decoupling
(i,j)= c† c +h.c., (7)
scheme of the V term does not fulfill this “hard-core” K − X iσ jσ
σ
constraint.
and (i,j) is a n.n. bond while (k,l) denotes any other
bondwhichislinkedintheHamiltonian. Weevaluatethe
correlationfunctionforallpossiblecombinationswithout
B. Exact Diagonalizations
common sites in the ground state of several samples at
n = 1/2 and J/t = 1/3, for varying V′/t.20 Note that a
We study the strong coupling t-J-V model (4) within BOW of the type presented in Section IIA is character-
ED on finite-size checkerboard samples of N = 16,20, ized by long range correlations of C corresponding, in
K
and 24 sites at the filling n=1/2.19 the thermodynamiclimit, to amodulationof (i,j) to
hK i
In finite, periodic systems the BOW instability can be compared directly to the MF values in the left panel
be detected using a correlation function of the bond of Fig. 2 (the long distance correlation function is pro-
strengths. Here we chose to work with the kinetic term. portionaltothe magnitude-squaredofthe kineticenergy
The correlationfunction is defined as: modulation).
Firstweplotarealspacepictureofthecorrelationson
C [(i,j),(k,l)]= (i,j) (k,l) (i,j) (k,l) , the N =20 and N =24 samples at a value of V′/t= 7.
K
hK K i−hK ihK i(6) As discussed below this is the value of V′/t where the
4
ical considerations14 show further that this second type
of BOW phase (not shown) is also realized for fermions
when spin degrees of freedom are included. The investi-
gationofthetransitionoccuringbetweenthesetwotypes
of plaquette phases is left for future work.
III. WEAK COUPLING
A. Two-patch RG and MF analysis
We turn now to the weak-coupling analysis of the
Hamiltonian H = H + H with H and H given
0 int 0 int
by Eq. (1) and Eq. (2), respectively. The quadratic
Hamiltonian, H , in reciprocal space consists of a dis-
0
persing band and a flat band lying on top (for t > 0)
FIG. 4: Behavior of the absolute value of a particular bond of the dispersing band. If we introduce a chemical po-
correlationfunction(Eq.6)atlargestdistanceasafunctionof tential term with µ = 2t we can write H in the form
0
V′/t at fixed J/t=1/3 and n=1/2 for several system sizes. ξ γ† γ +4tN −where the operator N counts
Correlations are weak at small V′/t, peak around V′/t ≈ 7 Pkσ k kσ kσ flat flat
the number of electrons in the flat band at 4t. In the
and then decrease again towards the extremely large V′/t
weak-coupling limit these states do not affect the low
limit, where another phasemight appear.
energy physics of the system and therefore they will be
dropped in the following analysis. The dispersion of the
other band is given by ξ = 2t(cosk +cosk ) and is
k 1 2
−
correlations are strongest. The reference bond uniquely identicaltothetight-bindingdispersiononthesquarelat-
belongstoacertainclassofcrossedplaquettes. Basedon tice. It is particle-hole symmetric, perfectly nested with
the theoretical picture one expects the correlation func- the nesting vector Q=(π,π), and has a logarithmically
tion to be positive for all bonds on the same type of divergent density of states at the Fermi energy.
crossed plaquettes and negative on the others. This is ForsmallcouplingsU,V,andJ weobtainaneffective
indeed what is seen on both samples shown in Fig. 3. interaction Hamiltonian (Appendix A) given by
Note also that the correlations are rather regular and
1 ′
uniform throughout the system. Heff = g γ† γ† γ γ , (8)
Int 2N X k1...k4 X k1σ k2σ′ k3σ′ k4σ
We now track the evolution of the correlations as a k1...k4 σσ′
function of V′/t by choosing the value of the correlation
with
function at the largest possible distance on a given sam-
ple. We compare the evolution on the three finite size 2
samples in Fig.4. The correlationsallstartat smallval- gk1...k4 = X(Ueνq+2V˜eνq23 −Jeνq24)eνk1eνk2eνk3eνk4
ues for V′/t<2, and in that region an inspection of the ν=1
spatial struct∼ure similar to Fig. 3 reveals that the bond 2
correlationpatternhasdefects,i.e.,correlationswiththe + X(4V˜fkν1k4fkν¯2k3 −2Jfkν1k3fkν¯2k4) (9)
wrong sign. This makes it difficult to conclude firmly on ν=1
the presence of the BOW phase at small repulsionbased
with eν = cos(k /2), fν = eν eνeν , V˜ = V J/4,
on the ED simulations only. The abrupt jump seen on k ν kk′ k−k′ k k′ −
q=k +k k k ,andq =q 2(k k ). Thetwo
theN =16sampleisduetoalevelcrossing,whichisab- 1 2− 3− 4 ij − i− j
saddle points P = (0,π) and P = (π,0) of the disper-
sentonthelargersamples. Forintermediatevaluesofthe 1 2
sion lead to the logarithmic divergence of the density of
repulsion the pattern becomes correct on all bonds and
states. Therefore,wecancharacterizeveryweakinterac-
theamplitudespeakaroundarepulsionofV′ /t 7for
the largest two samples. Beyond this valuemtahxe c≈orrela- tions by the values of the function gk1...k4 where all four
momenta k k lie on one of the two saddle points.
tionsweakenagain,butthepatternremainsqualitatively 1··· 4
Adopting the notation of Ref. 15 we have the following
intact. ThephysicsatverylargeV/t,wherestatesfulfill-
four coupling constants:
ing the ”tetrahedronrule” are dominant, is of particular
interest but beyond the scope of this work. Calculations g = 2J k ,k P
1 1 3 1
for spinless fermions (i.e. polarized electrons)12 reveal a g =−+4V˜ k ,k ∈P
more complicated translation symmetry breaking state gk1...k4 = g2 =0 k1,k4 ∈P1 (10)
3 1 2 1
with a larger supercell. In the case of bosons however, a g =U +2V˜ J k ...k∈ P
similar plaquette BOW is found although involving the 4 − 1 4 ∈ 1
void plaquettes where cyclic exchange terms act in this From the RG equations of Ref. 15 we see that for posi-
limit13. Preliminary numerical calculations and analyt- tive values of U, V, and J the coupling g flows to
1
−∞
5
and the coupling g flows to + ,21 whereas the other
2
∞
couplings flow to 0. This shows that the charge density
wave susceptibility is diverging most rapidly under the
RGflow. As shownin Fig.5 the CDW instability onthe
squarelatticecorrespondstotheBOWinstabilityonthe
checkerboardlattice,i.e.,bothinstabilities breakexactly
the same symmetries. Note, that within the framework
ofthisRGscheme,wecannotdeterminewhetherausual
CDW or a so-called charge flux phase, which is a CDW
with a d-wave form factor, is stabilized. In the follow-
ing we will denote these two phases with s-CDW and FIG. 5: (Color on-line) Correspondence between the s-CDW
phase on the effective square lattice (left) and the plaque-
d-CDW. (Note, that for J = 0 only the coupling g di-
2
ttephaseor BOW phaseon theoriginal checkerboard lattice
verges. Inthiscasewecannotevendeterminewhethera
(right).
CDWofSDWphaseisstabilized.) Inordertoshow,that
the s-CDW phase is in fact favored over the d-CDW, at
least in a mean-field analysis, we restrict the interaction
know that the critical temperature of the s-CDW phase
Hamiltonian (8) to the CDW channel and obtain:
is higher than the critical temperature of the d-CDW
1 phase. Including the non-negative second term in (14)
HCDW = 4N XVkk′Xγk†σγk+Q,σγk†′+Q,σ′γk′σ′ (11) would only lead to a further increase of the critical tem-
kk′ σσ′ perature. Note,thatthed-andthes-CDWpotentialsare
with identical for the momenta on the saddle points. There-
foreitisnotsurprisingthattheycannotbedistinguished
Vkk′ = 2gk,k′+Q,k′,k+Q gk,k′+Q,k+Q,k′ (12) bythetwo-patchRGmethod. However,inthemean-field
−
= Vksk′ +Vkdk′ +Vkdk′′ +Vkpk′ (13) swteatcea,nasseiettohpaetnsthaefisn-CitDeWgapisaflaovnogretdheoveenrtitrheeFdS-.CIDtWis
possible to perform such a mean-field analysis also for
Vksk′ = −V0ˆ(1−cosk1cosk2)(1−cosk1′ cosk2′) (14) SDW and superconducting instabilities. For supercon-
+(sin2k1+sin2k2)(sin2k1′ +sin2k2′)/2˜ ductivity we have strong repulsion (∝ U) in the d- and
in the s-wave channel, and weaker ( V,J) and mainly
Vkdk′ = −V0(cosk1−cosk2)(cosk1′ −cosk2′) (15) repulsive interactions in the d′- an t∝he p-wave channel.
Vkdk′′ = −V0sink1sink2sink1′ sink2′ (16) It is therefore quite clear, that the superconducting in-
U stabilities can not compete with the s-CDW instability.
Vkpk′ = +4(sink1sink1′ +sink2sink2′) (17) ForthepairpotentialoftheSDWinstabilities,weshould
replace only V by V˜/2 and U by U in Eqs. (14-17).
whereV0 =(V˜+J)/2. In(15)wedroppedatermpropor- For J >0 we h0ave V =(V˜ +J)/2−>V˜/2 and therefore
0
tional to cosk1+cosk2, as it vanishes along the FS, and the s-, d- and d′-CDW instabilities are favored over the
in (17) we dropped terms proportional to V/U or J/U.
correspondingSDW instabilities. Thep-SDWinstability
With ∆k = N1 Pk′Vkk′Fk′ and Fk = Pσhγk†+Q,σγkσi is strongly attractive but has the handicap, that it does
we obtain the linearized gap equation not open a gap at the saddle points, therefore, for weak
interactions the s-CDW state will still be favored over
∆k =−N1 Xk′ Vk,k′tanh(2ξξkk′′/2Tc)∆k′. (18) tghiveepn-SbDyW(αsut,aαtev,,iα.ej.),fworeicnatenraficntdionfopraarlalmpeotseirtisv(eUv,aVl,uJe)s
of (u,v,j) an α such that for 0 < α < α the s-CDW
0 0
Note,thatthe d′-(16)andthe p-wave(17)instabilitydo state is stabilized.
not open a gap at the saddle points, furthermore, the p-
In conclusion, in this section we showed with RG and
wave instability is strongly repulsive. For the d-CDW
mean-field arguments, that for weak enough repulsive
state the pair potential (15) is separable and the lin-
and antiferromagnetic interactions the s-CDW phase,
earized gap equation can be written in the simpler form
which corresponds to a bond order wave state on the
checkerboardlattice, is stabilized.
V tanh(ξ /2Td)
1= 0 (cosk cosk )2 k c . (19)
N X 1− 2 2ξk
k
B. N-patch functional RG method
The pair potential for the s-CDW state (14) is not sep-
arable. But if we neglect for a moment the second line
in(14) weobtainananalogousexpressionto(19)forthe As the two-patch model in the last section could not
critical temperature of the s-CDW with cosk cosk determine whether the s- or the d-CDW ground state is
1 2
−
replaced by 1 cosk cosk . As on the FS we have favored,weadvancetoamorerefinedN-patchfunctional
1 2
cosk = cos−k and as 2cosk 1 + cos2k we renormalizationgroup(fRG)studyoftheeffectivemodel
2 1 1 1
− | | ≤
6
forthedispersiveband. Thisallowsustostudythewhole 0
Fermisurfaceand,inparticular,thecompetitionbetween −5 a.u.] 40
stshe-CsessDeaWcnhoaannn-ndterlidsv-iaCanlDdkWa-dlsesoptatehtneed.ebnTacrheeeeoeffuffetesccitdtivieveetihnientetsreaardacdtciltoeionpnposoinisnt- V / t−−1150 χ, [WdCDW2300
regions which can be taken into account in the N-patch −20 CD10
χs
fRG scheme. 0
−1 0 1 10−1 100
We set up a one-loop N-patch fRG calculation of Angle around FS / π Λ / t
the one-particle irreducible interaction vertex using the
methods described in Refs. 16. This fRG scheme inte-
FIG.6: fRGresultsfora32×3discretizationoftheBrillouin
grates out all intermediate one-loop processes (particle- zone, for the half-filled band, T =0.001t, V˜ =0.6t, J =0.4t
holediagramsandparticle-particlecontributions)witha and U =0. Left: Effective interactions VΛ(k,k′,k+Q) with
decreasing energy scale Λ. In the one-loop diagrams at Q=(π,π) and k near the (π,0)-point at angle 0 (solid line)
scale Λ, one intermediate particle is in the energy shell ornearthe(π/2,π/2)-point atangleπ/4(dashedline) vs.k′
with band energy Λ, while the other particle is further movingaroundtheFSatscaleΛ=0.33t wherethestrongest
away from the Fe±rmi surface. The RG flow is started attractive component of the coupling function exceeds −24t.
Right: Flow of the zero-frequency s-CDW (solid line) and
at Λ bandwidth. The integration of the one-loop
0 ∼ d-CDW susceptibility for the same parameters. The inset
processes in the flow down to lower Λ renormalizes the
showsthe32FSpoints(filledcircles)andtheadditionalpatch
wavevector-dependenceoftheeffectiveinteraction. From
centers at band energy ±0.4t (open circles).
the growth of certain components toward low scales, the
dominant correlations can be read off. In parallel with
the running interactions, the flow of various static sus-
patchingscheme)areapparentlynotsufficienttoproduce
ceptibilities can be obtained. These quantities are again
an instability at a higher scale.
renormalizedbythescale-dependentinteractionsviaone- Fornonzeronearest-neighborinteractionsV˜ andJ the
loopprocesses. Theirgrowthcanbecomparedasafunc-
fRG produces run-away flows at measurable scales. For
tionoftheRGscale. Inthecaseofadivergenceofthein- instance, for U = 0, V˜ = 0.6t and J = 0.4t we find a
teractionsata nonzerocriticalscaleΛ , the fastedgrow-
c flow to strong coupling, i.e., a divergence of some com-
ing susceptibility determines the dominant instability or ponents ofV (k,k′,k+q) inthe one-loopflow,atscales
Λ
ordering tendency of the system.
Λ 0.033t. Thisscalesetsanuppertemperaturebound
c
For the spin-rotationally invariant system, the scale- for≈possible long range order. The divergence is dom-
dependent interaction vertex can be expressed using a inated by scattering processes V (k,k′,k + Q) with k
Λ
running coupling function VΛ(k,k′,k+q). The initial andk′attheFSandwithwavevectortransferQ=(π,π)
coupling function, VΛ0(k,k′,k + q), is obtained from whichflowtostronglynegativevalues. Forincomingand
Eq.(9)asgk+q,k′−q,k′,k. Herethefrequencydependence outgoingparticlesnearthesaddlepoints,theseprocesses
hasbeenneglectedasinmostpreviousapplicationsofthe belongtotheg -typescatteringknowntodivergeto
1
method for correlated fermions.17 In order to describe from the previous section. In the two-patch model,−al∞so
the wavevector dependence, we use 32 angular patches theg -typeprocessesseemedtodiverge. Thistendencyis
2
aroundtheFermisurface.18 Thesepatchesareagainsplit notseeninthefRGforthefullFermisurface,nowweob-
up in 3 patches, one above the Fermi surface down to serveonlyasmallgrowthoftheg -typeprocesses. InFig.
2
band energies of 0.2t, one including the Fermi surface in 6 we display the diverging component V (k,k′,k+Q)
Λ
thebandenergywindowbetween 0.2tand0.2t,andone of the effective interactions for two different choices of
−
below this energy window. We have checked our results k vs. k′ moving around the FS at a low scale Λ where
using other partitions. themaximalcomponentofthecouplingfunctionexceeds
We have run the fRG for half band filling and various 24t. Oneclearlyobservesthatthedataforkfixednear
choices of the interaction parameters U, V˜ and J in (8). −(π,0) is basically a superposition of an attractive offset
From a perturbative point of view, the model with pure andacosk cosk modulation. Theattractiveoffsetbe-
1 2
onsiterepulsionandV˜ =J =0isremarkablystable. For comesmore−negativewithdecreasingscaleanddrivesthe
instanceatU =2tandhalffillingofthedispersiveband, s-CDW susceptibility. The d-wave part also grows and
the fRG flow of the interactions remains finite down to feedsthed-CDWchannel. ForkfixedneartheBZdiago-
low scales and temperatures 10−4 t without any indi- nal,i.e.,nearthenodesofthe cosk cosk formfactor,
1 2
∼ −
cationforaninstability. Thisstabilityisquitesurprising the d-wave part is missing or strongly deformed, while
for a perfectly nested Fermi surface with van Hove sin- the attractive offset is more or less unchanged. Hence
gularities at the Fermi level. However, as already found we can expect a stronger growth in the s-CDW chan-
in the two-patch model, for many scattering processes nel. This is also clearly reflected in the data for the
within the dispersive band between states near the van flow of the susceptibilities shown in the right panel of
Hove points, the bare interactions vanish, and the inter- Fig. 6. The growth of the s-CDW susceptibility by far
actionsawayfromthese points andawayfromthe Fermi exceeds the d-CDW channel and all other channels such
level (which are kept in approximate form in our refined as spin density waves. Hence the fRG treatment agrees
7
withthemeanfieldpicturedevelopedintheprevioussec- supportby the SwissNationalFund andNCCRMaNEP
tion. Bothsuggestas-CDWorderedgroundstateinthe (Switzerland). Computations were performed on the
effective one-band model, translating into a bond-order IBM Regatta machines of CSCS Manno andIDRIS (Or-
wave as primary order in the full checkerboardsystem. say). D.P. thanks the Agence Nationale de la Recherche
The s-CDW divergence described here occurs in a (France)forsupportandtheITPatETH-Zu¨richforhos-
rather large parameter window. We increased the value pitality.
ofU from0upto2tandfoundanincreaseofthe critical
scalefrom0.033tto0.037ttogetherwithanevenstronger
dominance of the s-CDW over the d-CDW channel.
APPENDIX A: CHECKERBOARD LATTICE
IV. CONCLUSIONS The elementary unit cell of the checkerboard lattice
contains two lattice sites situated at x /2 (ν = 1,2)
ν
We studied the Hubbard model including n.n. repul- where the vectors xν are two primitive lattice vectors.
sion and n.n. antiferromagnetic exchange for strongly With this convention we choose the origin of the lattice
correlated electrons at quarter-filling (n = 1/2) on the atthecenterofacrossedplaquette,wherethereisnolat-
checkerboard lattice in the strong-coupling (U = ) tice site. The tight-binding Hamiltonian for this system
and in the weak-coupling regime. The MF calculatio∞ns, is given by
the ED simulations and the RG analysis all predict a
spontaneousbreakingofthetranslationsymmetryinthe H = t c† (c +c (A1)
ground state, that is characterized by the nesting vec- 0 − Xh rνσ r+xν,νσ r−xν¯,ν¯σ
rνσ
tor Q = (π,π). The ground state is twofold degenerate
+c )+h.c. µNˆ,
as there are two types of crossed plaquettes (“strong” r−x1+xν,ν¯σ i−
and “weak” plaquettes). These ground states can be
approximatively described as product states of localized where ν¯ = 2,1 if ν = 1,2, Nˆ is the number operator
two-particlestates onthe “strong”plaquettes,whichare
and µ = 2t in the following. We introduce Fourier
equalamplitudesuperpositionsofasingletwavefunction −
transformed operators as
on each of the six bonds. In this way the two electrons
forming a singlet gain the magnetic energy J and the
kinetic energy −4t. Furthermore,the presenc−e of a third crνσ = √1N Xeik·(r+x2ν)ckνσ, (A2)
electrononthecrossedplaquettewhichwouldcostan.n. k
repulsion energy 2V is avoided. Therefore, the BOW in-
stability isa cooperativeeffectthat resultsfroma simul- whereN isthenumberofunitcells. Withtheseoperators
taneous optimization of kinetic and interaction energy. the tight-binding Hamiltonian reads
Note, that for negligible kinetic energy (V t) other
phasescanbe realizedonthe quarter-filledch≫eckerboard Hˆ =4t δ cos kµ cos kν c† c , (A3)
lattice.22Aninterestingissueforfurthertheoreticalstud- 0 X h µν − (cid:0) 2 (cid:1) (cid:0) 2 (cid:1)i kµσ kνσ
kµνσ
ies are the properties and the excitations of the weakly
doped BOW phase, e.g., the question whether the ad-
where k =k x . Diagonalizing this Hamiltonian leads
ditional holes or particles pair and lead to a supercon- ν · ν
to a flat band at 4t and to a band with the dispersion
ducting phase where the U(1) gauge symmetry and the
ξ = 2t cosk which is nothing but the nearest-
translation symmetry are simultaneously broken. k − Pν ν
neighbor tight-binding dispersion of the square lattice.
Inconclusionwiththediscoveryandthedescriptionof
The operators of this dispersive band are expressed in
the BOW phase on the quarter-filled checkerboard lat-
terms of the original operators by
tice we show how a strongly frustrated system can avoid
thelocalfrustrationatfractionalfillingbyexploitingthe
1 k
bipartitearrangementoflargerunits. Wehopethatwith γ = cos ν c (A4)
our work we can stimulate further progress in the chal- kσ √rk X (cid:0) 2 (cid:1) kνσ
ν
lengingandexciting fieldofstronglycorrelatedelectrons
on frustrated lattices. with r = cos2(k /2). In weak coupling, we can re-
k ν ν
P
strict our attention to the states close to the Fermi sur-
face,wherer =1andforeveryoperatoronthechecker-
k
Acknowledgments board lattice we can obtain an effective operator on the
square lattice by the substitution c cos(k /2)γ .
kνσ ν kσ
→
We thank T.M. Rice, K. Wakabayashi, K. Penc, and InthiswaytheeffectiveHamiltonian(8)isobtainedfrom
S. Capponi for stimulating discussions. We acknowledge the interaction Hamiltonian (2).
8
1 For a review see, e.g., G. Misguich and C. Lhuillier 15 B.Binz,D.Baeriswyl, andB.Dou¸cot,Eur.Phys.J.B25,
in “Frustrated spin systems”, H.T. Diep editor, World- 69 (2002).
Scientific (2005). 16 C. Honerkamp, M. Salmhofer, N. Furukawa, and
2 F.D.M. Haldane, Phys. Rev.B 25, 4925 (1982). T.M. Rice, Phys. Rev. B 63, 035109 (2001); C. Hon-
3 J.B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, erkamp,Euro. Phys.J. B 21, 81 (2001).
Phys.Rev.B67, 054411 (2003); seealso S.E.Palmer and 17 For short reviews, see W. Metzner, Prog. Theor. Phys.
J.T. Chalker, Phys.Rev.B 64, 94412 (2001). Suppl.160,58(2005);C.Honerkamp,cond-mat/0411267.
4 A. L¨auchli and D. Poilblanc, Phys. Rev. Lett. 92, 236404 18 D.ZanchiandH.J.Schulz,Europhys.Lett.44,235(1997);
(2004); D. Poilblanc, Phys. Rev. Lett. 93, 197204 (2004); Phys. Rev.B 61, 13609 (2000).
D.Poilblanc,A.L¨auchli,M.MambriniandF.Mila,Phys. 19 TheHilbertspaceofthelargestsamplehasadimensionof
Rev.B 73, 100403 (2006). approximately 100 million after symmetry reduction.
5 Y. Imai, N. Kawakami and H. Tsunetsugu, Phys. Rev. B 20 Note, that in this section we work with the Hamiltonian
68, 195103 (2003); E. Runge and P. Fulde, Phys. Rev. B 4 and use the parameter V′ = V +J/4 instead of V. For
70245113(2004);N.Bulut,W.KoshibaeandS.Maekawa, V′/t = 7 and J/t =1/3 the difference between V and V′
Phys. Rev.Lett. 95, 037001 (2005). is very small ((V′−V)/V =0.012).
6 M.Indergand,A.L¨auchli,S.CapponiandM.Sigrist,Phys. 21 For V˜ < 0 g2 flows to 0, but also in this case the CDW
Rev.B 74, 064429 (2006). phase is stabilized.
7 M.C.Gutzwiller,Phys.Rev.Lett.10,159(1963);D.Voll- 22 As an example consider the case t = 0 and U = V = ∞,
hardt, Rev. Mod. Phys. 56, 99 (1984); F.C. Zhang, where the constraint of two particles per crossed plaque-
C.Gros,T.M.Rice,andH.Shiba,Supercond.Sci.Technol. tte allows only states where occupied sites form closed
1, 36 (1988). (or infinite) loops on the checkerboard lattice. For J = 0
8 P.W. Anderson, cond-mat/0406038; B.A. Bernevig, all these states are degenerate but for finite J the states,
G. Chapline, R.B. Laughlin, Z. Nazario, and D.I. Santi- where all loops have the minimal length four, are ground
ago, cond-mat/0312573. states,asthemagneticenergyperelectronfortheloopsof
9 D. Poilblanc, Phys. Rev. B 72, 060508(R) (2005); C. Li, length four is minimal. The closed loops of four electrons
S. Zhou,and Z. Wang, ibid.73, 060501(R) (2006). around open plaquettes can be placed in a periodic pat-
10 C. Weber, D. Poilblanc, S. Capponi, F. Mila, C. Jaudet, tern where the centers of the loops are connected by the
Phys. Rev.B 74, 104506 (2006). vectors 2x1,2x2 (cf. App. A). Note, that in this periodic
11 P.W. Anderson, Phys.Rev. 102, 1008 (1956). arrangementallfourelectronplaquettesonadiagonalcan
12 F.PollmannandP.Fulde,Europhys.Lett.75,133(2006). simultaneouslyslidealongadiagonallinebyx1orx2atno
13 N. Shannon, G. Misguich, and K. Penc, Phys. Rev. B 69, energycost.Therefore,thegroundstateisinfactinfinitely
220403 (2004). degenerate.
14 D. Poilblanc, K. Pencand N.Shannon,in preparation.
