Ground-state phase diagram of the one-dimensional half-filled extended Hubbard model

0.66 MB·English

by  M. Tsuchiizu

Ground-state phasediagram ofthe one-dimensional half-filled extended Hubbard model
M. Tsuchiizu1,2 and A. Furusaki1,3
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan
3Condensed-MatterTheoryLaboratory,TheInstituteofPhysicalandChemicalResearch(RIKEN),Saitama351-0198,Japan
(Dated:January26,2004)
4 Werevisittheground-statephasediagramoftheone-dimensional half-filledextendedHubbardmodelwith
0
on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-
0
coupling renormalization-group approach (g-ology) including second-order corrections to the coupling con-
2
stants,weshowthatbond-charge-density-wave(BCDW)phaseexistsforU ≈2V inbetweencharge-density-

n wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin
a electronsdisfavorstheBCDWstateandleadstoabicriticalpoint wheretheCDW-BCDWandSDW-BCDW
J continuous-transitionlinesmergeintotheCDW-SDWfirst-ordertransitionline.Inthesecondhalfofthepaper,

6 weinvestigatethephasediagramoftheextendedHubbardmodelwitheitheradditionalstaggeredsitepotential
2 ∆orbondalternationδ.Althoughthealternatingsitepotential∆stronglyfavorstheCDWstate(thatis,aband
insulator),theBCDWstateisnotdestroyedcompletelyandoccupiesafiniteregioninthephasediagram. Our

] resultisanaturalgeneralizationoftheworkbyFabrizio, Gogolin, andNersesyan[Phys.Rev.Lett.83, 2014
l
e (1999)],whopredictedtheexistenceofaspontaneouslydimerizedinsulatingstatebetweenabandinsulatorand
- aMottinsulatorinthephasediagramoftheionicHubbardmodel. Thebondalternationδ destroystheSDW
r
t state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the
s modelwithδcontainsonlyasinglecriticallineseparatingthePeierlsinsulatorphaseandtheCDWphase.The
.
t additionof∆orδ changestheuniversalityclassoftheCDW-BCDWtransitionfromtheGaussiantransition
a
intotheIsingtransition.
m
- PACSnumbers:71.10.Fd,71.10.Hf,71.10.Pm,71.30.+h
d
n
o
I. INTRODUCTION repulsion U, the ground state is in the Mott insulating state
c
[ wherethespinsectorexhibitsquasi-long-rangeorderofspin-
densitywave(SDW);wecallittheSDWstate.Intheopposite

2 It is well known that a one-dimensional (1D) spin sys-
limitofstrongV,thegroundstateofthehalf-filledEHMhasa
v tem has instability to dimerization that changes the system
long-rangeorderofthecharge-densitywave(CDW);wecall
7 intoanonmagneticinsulatingstate,theso-calledspin-Peierls
this state the CDW state. Furthermore, in the atomic limit
5 state.1 Indeed the spin-Peierls state is realized in many sys-
1 where the electron hopping t is ignored, the CDW state ap-
tems including quasi-one-dimensionalorganic compounds2,3
8 pearsforU <2V whereastheuniformstatecorrespondingto
and the inorganicmaterial4 CuGeO , andits propertieshave
0 3 theSDWstateisstableforU >2V inonedimension.Strong-
3 been studied extensively both experimentally and theoreti- couplingperturbationtheory in t has established that a first-
0 cally.Ofparticularinterestisasituationinwhichadimerized
order phase transition between the SDW state and the CDW
t/ state appears spontaneously due to strong correlations and state occurs at U 2V.18,19,20,21 As for the weak-coupling
a frustration.5 A well-known example is the frustrated spin-1 ≃
m 2 regime,perturbativerenormalization-group(RG)approachor
Heisenbergchainwithnearest-neighbor,J ,andnext-nearest-
1 g-ologyledtoasimilarconclusionthatthegroundstateathalf
- neighbor,J ,antiferromagneticexchangeinteractions,where
d 2 filling is either in the SDW state or in the CDW state with a
n a0.s2p4oJnt.a6neOotuhselrysdyimsteemrizsedofphcausrereinstreianltiezreedstfoarreJ2qu≥asJi-2ocn≃e- continuous phase-transition line at U = 2V.18 Thus, it had
o 1 been considered for a long time that the ground-state phase
c dimensionalelectronsystemsinorganicmaterials,wherethe diagram of the EHM at half filling has only two phases, the
: spin-Peierlsstateappearsduetostrongelectroncorrelationat
v SDW and CDW states, and that the order of the phase tran-
halffilling7,8,9,10,11,12,13,14andatquarterfilling.15,16
i sitionatU 2V changesfromcontinuoustofirstorderata
X ≃
RecentlyitwaspointedoutbyNakamura17andco-workers tricriticalpointwhichwasspeculatedtoexistintheintermedi-
ar thataspontaneouslydimerizedstateoccupiesafiniteparam- atecouplingregime.20,22,23,24 Thiscommonviewwasrevised
eter space in the ground-statephase diagram of the 1D half- by the Nakamura’s discovery that the BCDW state exists at
filledHubbardmodelwiththenearest-neighborrepulsionV, U 2V inbetweentheSDWandCDWphasesintheweak-
i.e., the extended Hubbard model (EHM). This spin-Peierls cou≃pling region,17 which is supported by recent large-scale
stateisoftencalledbond-charge-density-wave(BCDW)state Monte Carlo calculations.25,26 Related studies of the dimer-
or bond-ordered-wave state. The appearance of the BCDW izedstate in theEHM with additionalcorrelationeffectscan
state in the purely electronic model is nontrivial and has at- befoundinRefs.[27,28,29,30,31].
tractedmuchattentionfromtheoreticalpointofview. Toap- A related and still controversial issue of current interest
preciate this surprising result, let us consider some limiting is whether or not a spontaneously dimerized phase exists in
cases. Inthelimitofweaknearest-neighborrepulsionV,orin the 1D Hubbard model with alternating site potential, the
thehalf-filledHubbardmodelwithonlytheon-siteCoulomb so-calledionicHubbardmodel.32,33,34,35,36,37,38,39,40,41,42,43,44,45
2
Thissystemwasintroducedasasimpleminimalmodelforthe the 1D EHM with additionalbonddimerization, butwithout
neutral-ionic transitions observed in quasi-one-dimensional thestaggeredpotential. Thismodelexhibitsaquantumphase
organicmaterials46,47,48 and forferroelectricperovskites.49,50 transition between a dimerized Peierls insulator and a CDW
Obviouslythe modelhastwo insulatingphases. The ground state. SectionVIisdevotedtoconclusions,anddetailsofthe
stateis(i)abandinsulatorwiththeCDWorderwhenthestag- technicalcalculationsaregiveninAppendixes.
geredsitepotentialismuchlargerthantheon-siterepulsionor
(ii) a Mottinsulatorwith quasi-long-rangeSDW orderwhen
thestaggeredsitepotentialisnegligible.Earlyexactdiagonal- II. EXTENDEDHUBBARDMODEL
izationstudies49,50,51ofsmallsystemshavefoundatransition
betweenthe twophasesandalso reporteddramaticenhance- In the first half of this paper (Secs. II and III), we con-
mentoftheelectron-latticeinteractionbystrongelectroncor- siderthestandard1DEHMwhichhason-site,U,andnearest-
relation near a boundary between the band insulating phase neighbor,V,interactions.TheHamiltonianisgivenby
(theBIstate)andtheMottinsulatingphase(theSDWstate).
Mostly through bosonization analysis of the ionic Hubbard H = −t (c†j,σcj+1,σ+H.c.)
model, Fabrizio, Gogolin, and Nersesyan recently argued32 j,σ
X
thataphaseofaspontaneouslydimerizedinsulator(SDI)in-
+U n n +V n n , (2.1)
j,↑ j,↓ j j+1
tervenes between the ionic insulating phase (band insulator)
j j
and the Mott insulating phase. The SDI state is closely re- X X
latedtotheBCDWstatementionedabove. Earliernumerical where n c† c 1, n n +n , and c† de-
studies34,35,36,38,39,51 havedrawncontradictoryconclusionsas notes thej,σcre≡atiojn,σojp,eσra−tor2ofjan≡elecj,t↑ron wji,t↓h spin σj,σ(= ,
↑
to whether the SDI phase exists or not, but more recent nu- )onthejthsite. We assumerepulsiveinteractions,i.e., the
↓
mericalstudiesfind two phase transitionsand the SDI phase couplingconstantsU andV arepositive.NotethattheHamil-
in between.37,40,45 Nevertheless there still remain unresolved tonianhasglobalSU(2)spinsymmetry. Followingtheprevi-
issuesonthecriticalpropertiesnearthequantumphasetran- ousstudiesonmodelswithcorrelated-hoppinginteractions,28
sitions. weconsidertheCDW,SDW,BCDW,andbond-spin-density-
wave (BSDW) phases as potential ordered ground states at
In thispaperwe givesupportingtheoreticalargumentsfor
halffilling. Theyarecharacterizedbytheorderparameters
theexistenceofthespontaneouslydimerizedinsulatingstates
inthe1Dhalf-filledextendedHubbardmodelwithandwith- ( 1)j(n +n ), (2.2a)
CDW j,↑ j,↓
O ≡ −
outstaggeredpotentials. We adoptthestandardbosonization ( 1)j(n n ), (2.2b)
SDW j,↑ j,↓
approachandperformbothperturbativeRG analysisvalidin O ≡ − −
the weak-coupling regime and semiclassical analysis which OBCDW ≡(−1)j(c†j,↑cj+1,↑+c†j,↓cj+1,↓+H.c.), (2.2c)
is expectedto givea qualitativelycorrectpicture evenin the ( 1)j(c† c c† c +H.c.). (2.2d)
strong-couplingregime. This paper is organizedas follows. OBSDW ≡ − j,↑ j+1,↑− j,↓ j+1,↓
SectionsIIandIIIaredevotedtotheanalysisofthestandard The order parameter of the BCDW state corresponds to the
EHM,i.e.,thesystemwithoutthestaggeredpotential. Some Peierls dimerization operator. We note that the BCDW
oftheresultsofthispartarealreadypresentedinRef.52. In state can be also regarded as the p-density-wave state,53 as
Sec. II, we introduce the model and reformulate the weak- the order parameter of the BCDW state can be written as
couplingtheory, the g-ology, to include higher-ordercorrec- sin(ka)c† c , where c =
jOBCDW ∝ k,σ k,σ k+(π/a),σ k,σ
ttiivoensHtaomcioltuopnliianngacnodnsdtearnivtse. thWeerebnoosromniazleizlaotiwo-ne-ngerorguypeefqfueac-- PN−1/2 je−ikRPjcj,σ with Rj = ja (a: the lattice spacing,
N:thenumberofsites). TheBSDWstatedescribesasite-off-
tions. In Sec. III, we determine the ground-state phase dia- diagonaPlSDWstate.28
gram. First, fromtheperturbativeRG analysiswe showthat
the BCDW phase occupies a finite region near the U = 2V
line in the weak-coupling limit. Next, from the semiclassi- A. g-ologyapproach
calanalysiswearguethattheumklappscatteringofparallel-
spinelectronsdestabilizestheBCDWphaseandgivesriseto
The hopping t generates the energy band with dispersion
a bicritical point where the CDW-BCDW and SDW-BCDW
ε = 2tcoska,wheretheFermipointsareatk = k =
continuous-transition lines merge into the CDW-SDW first- k − ± F
π/2a at half filling. In order to analyze the low-energy
ordertransitionline. Finally,combiningtheperturbativeRG ±
physics near the Fermi points, we introduce a momentum
equationswiththesemiclassicalanalysis,weobtaintheglobal
cutoff Λ (0 < Λ < k ) and divide the momentum space
phase diagram of the 1D EHM. In Sec. IV we study the 1D F
into the three sectors (Fig. 1) (i) k R, (ii) k L, and
EHMwiththestaggeredsitepotential.Wetakethesamestrat- ∈ ∈
(iii) k / (R L), where R = [k Λ,k + Λ] and
egy as in the previous sections and perform a semiclassical ∈ ∪ F − F
L = [ k Λ, k +Λ]. We then introduce the follow-
analysis of the bosonizedHamiltonian. With the help of the − F − − F
ingfermionoperators:
perturbativeRGanalysisweobtaintheglobalphasediagram
thatindeedhastheSDIphase. WefindthattheBCDWphase a fork R
k,+,σ
∈
oftheEHMiscontinuouslydeformedtotheSDIphaseupon c = a fork L (2.3)
k,σ  k,−,σ ∈
introducingthealternatingsitepotential. InSec.V,westudy b otherwise.
 k,σ

3
ε
Theindex ( )ofthecouplingconstantsdenotesthescatter-
k
k ⊥
ingofelectronswithsame(opposite)spins.
−k k
F F k
−π/a 0 Λ π/a
B. Vertexcorrections
In the conventional weak-coupling approach to the 1D
FIG.1: Single-particleenergy band. The annihilation operator of EHM,17,18 one estimates the coupling constants in Eq. (2.4)
an electron near the Fermi points with momentum k ∈ [−kF − onlyuptothelowestorderinU andV:
Λ,−kF+Λ](k∈[kF−Λ,kF+Λ])isdenotedak,−,σ(ak,+,σ),and
thatofanelectronfarawayfromtheFermipointsisdenotedb .
k,σ
g = g =(U 2V)a, (2.5a)
1⊥ 3⊥
−
Electrons near the Fermi points are shuffled by the two- g = g =(U +2V)a, (2.5b)
2⊥ 4⊥
particlescattering: H = U n n +V n n .
int j j,↑ j,↓ j j j+1 g = g = 2Va, (2.5c)
Following the standard g-ology approach,18,54 we will focus 1k 3k −
P P g = g =+2Va. (2.5d)
onthescatteringprocessesbetweenelectronsneartheFermi 2k 4k
points,i.e.,thescatteringprocesseswhichinvolvea only.
k,±,σ
TheHamiltonianforsuchinteractionprocessesis
g In analyzing the low-energy physics of Eq. (2.4), one then
Hint = + 21Lk :a†k1,p,σ ak2,−p,σ a†k3,−p,σ ak4,p,σ: employs the standard g-ology approach,54 i.e., the perturba-
kXi,p,σ tiveRG method,andobtainsflowequationsforthemarginal
+ g1⊥ :a† a a† a : termsin Eq. (2.4). From this RG analysis18,54 one finds that
2L k1,p,σ k2,−p,σ k3,−p,σ k4,p,σ the g term generates a gap in the charge excitation spec-
kXi,p,σ trum3if⊥g > (g +g g ) andg = 0, whereas
g 3⊥ 2k 2⊥ 1k 3⊥
+ 2k :a† a a† a : the g |term| yie−lds a gap in t−he spin excitatio6n spectrum if
2L k1,p,σ k2,p,σ k3,−p,σ k4,−p,σ 1⊥
kXi,p,σ |g1⊥| > −(g2k − g2⊥ − g1k) and g1⊥ 6= 0. Hence, with
g thelowest-ordercouplingconstantsEq.(2.5),onewouldcon-
+ 2⊥ :a† a a† a :
2L k1,p,σ k2,p,σ k3,−p,σ k4,−p,σ clude that the charge (spin) excitations become massless at
kXi,p,σ U 2V = 0 (U 2V 0). This would mean that, as
+ g3k :a† a a† a : U −increases, both t−he char≥ge and spin sectors become criti-
2L k1,p,σ k2,−p,σ k3,p,σ k4,−p,σ calsimultaneouslyatU =2V,whereadirectandcontinuous
kXi,p,σ
g CDW-SDWtransitiontakesplace.Thisanalysisisfoundtobe
+ 3⊥ :a† a a† a :
2L k1,p,σ k2,−p,σ k3,p,σ k4,−p,σ insufficientfromthefollowingargument.The(accidental)si-
kXi,p,σ multaneousvanishingofg3⊥andg1⊥resultsfromthelowest-
g
+ 4k :a† a a† a : order estimate in U and V and there is no symmetry princi-
2L k1,p,σ k2,p,σ k3,p,σ k4,p,σ plethatenforcesg andg tovanishsimultaneously. Itis
kXi,p,σ possible thatthe hi1g⊥her-ord3e⊥rcorrectionsto g liftthe degen-
g
+ 4⊥ :a† a a† a :, eracyofzerosandchangethetopologyofthephasediagram.
2L k1,p,σ k2,p,σ k3,p,σ k4,p,σ
kXi,p,σ Therefore,inordertoanalyzethephasediagramatU ≈ 2V,
(2.4) weneedtogobeyondthelowest-ordercalculationofthecou-
pling constants in the g-ology. In this section, we compute
whereσ = ( )forσ = ( ),Listhelengthofthesystem, thevertexcorrectionsduetovirtualprocessesinvolvinghigh-
and::denot↓es↑normalorde↑rin↓g.Thesummationoverthemo- energystates55 byintegratingoutb . Thisprocedureallows
k,σ
mentumk istakenundertheconditionofthetotalmomentum us to obtain the effective coupling constants g’s that include
i
beingconserved(equalto 2π/afortheumklappscattering). higher-ordercorrections.
±
Theindexp = +/ denotesthe right-/left-movingelectron.
−
Thecouplingconstantsg andg (g andg )denotethe The second-order vertex diagrams for the coupling con-
1k 1⊥ 3k 3⊥
matrixelementsofthebackward(umklapp)scattering,while stants are shown in Fig. 2. The solid lines denote the low-
g and g (g and g ) denote the matrix element of the energy states a , while the dashed lines denote high-
2k 2⊥ 4k 4⊥ k,±,σ
forward scattering with the different (same) branch p = . energy states b . The nonzero contributions from the
k,σ
±
4
Λonlyweakly,andwecansetΛ=π/4inthefollowinganal-
ysisasweareinterestedinthequalitativefeatureofthephase
= + +
diagram(differentchoiceswillonlyleadtosmallquantitative
(cid:1) (cid:2) (cid:3) (cid:4) changes in phase boundaries). Incidentally, the logarithmic
a) b)
divergenceofC (Λ)inthelimitΛ 0leadstothefamiliar
1
→
one-loopRGequations.
+ + +
(cid:5) (cid:6) (cid:7) C. Bosonization
c) d) e)
FIG. 2: Vertex diagrams with second-order corrections [(a)-(e)]. Having integrated out the high-energy virtual scattering
Solid lines denote electron states in the momentum space k ∈ R processes, we now focus on the low-energy states and lin-
ork ∈ L,whilethedashedlinesdenoteelectronstatesintheother earizethedispersionofa aroundtheFermipoints. The
k,±,σ
momentumspace. kinetic-energytermwiththelinearizeddispersionisgivenby
H = v (k k )a† a
second-ordervirtualprocesses(a)-(e)are 0 F − F k,+,σ k,+,σ
k∈R,σ
X
δg1(a⊥) = −δg3(b⊥) =−4Uπ2tD1a+ Vπt2D2a, (2.6a) + vF(−k−kF)a†k,−,σak,−,σ, (2.9)
k∈L,σ
X
V(U 2V)
δg(c) = +δg(c) =+ − D a, (2.6b)
1⊥ 3⊥ πt 1 wherevF = 2taistheFermivelocity. Thefieldoperatorsof
U2 V2 theright-andleft-movingelectronsaregivenby
δg(a) = δg(b) = D a D a, (2.6c)
2⊥ − 2⊥ −4πt 1 − πt 2
1
δg(a) = +V2D a, (2.6d) ψ+,σ(x) ≡ √L eikxak,+,σ, (2.10a)
1k πt 2 kX∈R
δg(c) = (U −2V)2+4V2D a V2D a, (2.6e) ψ−,σ(x) 1 eikxak,−,σ. (2.10b)
1k − 4πt 1 − πt 2 ≡ √L
k∈L
X
V2
δg(a) = D a, (2.6f)
2k −πt 2 We apply the Abelian bosonization method and rewrite the
δg(c) = (U −2V)2+4V2D a+ V2D a, (2.6g) kfiienldetsica-se(nseeregyAtpeprmendHix0A=) dxH0 intermsofbosonicphase
3k − 4πt 1 πt 2 R
v
where = F (2πΠ )2+(∂ θ)2
0 θ x
H 4π
D1(Λ) ≡ π/2−aΛ codskk, (2.7a) + v4hFπ (2πΠφ)2+(∂xφi)2 , (2.11)
Z−π/2+aΛ h i
π/2−aΛ sin2k whereθ(φ)isthebosonicfieldwhosespatialderivativeispro-
D (Λ) dk . (2.7b)
2 ≡ cosk portionalto the charge(spin) density,[θ(x),φ(y)] = 0. The
Z−π/2+aΛ operatorsΠ andΠ arecanonicallyconjugatevariablestoθ
θ φ
By introducing C (Λ) 2ln[cot(aΛ/2)] and C (Λ) andφ,respectively,andsatisfytheconventionalcommutation
1 2
2cosaΛ,D1(Λ)andD2(≡Λ)arerewrittenasD1(Λ)=C1(Λ≡) relations,[θ(x),Πθ(x′)] = [φ(x),Πφ(x′)] = iδ(x−x′). We
and D (Λ) = C (Λ) C (Λ). In terms of C and C , the alsointroducechiralbosonicfields
2 1 2 1 2
−
couplingconstantswithsecond-ordercorrectionsaregivenby
1 x
θ (x) θ(x) 2π dx′Π (x′) , (2.12)
± θ
≡ 2 ∓
C C (cid:20) Z−∞ (cid:21)
g =(U 2V)a 1 1 (U 2V) 2V2a, (2.8a) 1 x
1⊥ − − 4πt − − πt φ±(x) φ(x) 2π dx′Πφ(x′) . (2.13)
(cid:20) (cid:21) ≡ 2 ∓
C C (cid:20) Z−∞ (cid:21)
g = 2Va 1 (U 2V)2a 2V2a, (2.8b)
1k − − 4πt − − πt One can easily verify that these chiral fields sat-
C C isfy the commutation relations [θ (x),θ (x′)] =
g =(U 2V)a 1+ 1 (U +6V) + 2V2a, (2.8c) ± ±
3⊥ − 4πt πt [φ±(x),φ±(x′)] = i(π/2)sgn(x x′) and
(cid:20) (cid:21) [θ (x),θ (x′)] = [φ (x),±φ (x′)] = iπ/−2. In terms
C C + − + −
g = 2Va 1 (U 2V)2a+ 2V2a, (2.8d) ofthesefields,thekinetic-energydensityreads
3k − − 4πt − πt
v
and g2k = +2Va, g2⊥ = (U +2V)a, g4k = +2Va, and 0 = F (∂xθp)2+(∂xφp)2 . (2.14)
g =(U +2V)a. ExceptwhenaΛ 1,theC sdependon H 2π
4⊥ ≪ i p=X+,−h i
5
To express the electron field operators ψ with the low-energypropertiesofthe charge(spin)modes,18,54 where
p,σ
bosonicphasefields,weintroduceanewsetofchiralbosonic g =g +g g ,g =g ,g =g g +g ,and
ρ 2⊥ 2k 1k c 3⊥ σ 2⊥ 2k 1k
− −
fields g = g . The g , g , g , andg termswith scaling di-
s 1⊥ cs ρs cσ ρσ
mension4couplethespinandchargedegreesoffreedom.The
ϕp,↑ =θp+φp, ϕp,↓ =θp−φp, (2.15) gcscouplingcomesfromtheumklappscatteringg3k. Thegρs
(g ) coupling is generated from the backward scattering of
ρσ
whichobeythecommutationrelations
antiparallel-(parallel-)spin electronswhile the g coupling
cσ
isgeneratedfromtheumklappscatteringofelectronswithan-
[ϕ (x),ϕ (x′)] = iπsgn(x x′)δ , (2.16a)
±,σ ±,σ′ ± − σ,σ′ tiparallel spins (see Appendix A). These coupling constants
[ϕ+,σ(x),ϕ−,σ′(x′)] = iπδσ,σ′. (2.16b) aregivenbyg = g = g = g = 2Vatolowestor-
cs ρs cσ ρσ
−
derinV. CannonandFradkinexaminedtheeffectoftheg
Intermsofthephasefieldsϕ ,theelectronfieldoperators cs
p,σ term and argued that it plays a crucial role in the first-order
canbewrittenas
CDW-SDWtransition.22Voitincludedtheg andg terms,
ρs cσ
ψ (x)= ησ exp[ipk x+ipϕ (x)], (2.17) aswellasthegcsterm,intheperturbativeRGanalysisofthe
p,σ √2πa F p,σ couplingconstants,butdidnotconsiderthegρσ term.24 Here
wenotethatitisimportanttokeeptheg termaswell,since
ρσ
where the Klein factor η satisfies the anticommutation re-
σ the global SU(2) symmetry in the spin sector is guaranteed
lation η ,η = 2δ . One can verify that the operator
σ σ′ σ,σ′ onlywheng =g ,g =g ,andg =g .
{ } σ s cs cσ ρs ρσ
definedinEq.(2.17)satisfiesthesameanticommutationrela-
tionas the fermionfield operator. ItfollowsfromEq.(2.17)
thattheorderparametersinEq.(2.2)arerewrittenas
(x) cosθ(x) sinφ(x), (2.18a)
SDW
O ∝
D. Renormalization-groupequations
(x) sinθ(x) cosφ(x), (2.18b)
CDW
O ∝
(x) cosθ(x) cosφ(x), (2.18c)
BCDW
O ∝
(x) sinθ(x) sinφ(x). (2.18d) We perform a perturbative RG calculation to examine the
BSDW
O ∝ low-energypropertiesof the 1D EHM in the weak-coupling
regime,takingintoaccountquantumfluctuationsofthephase
TheinteractionpartoftheHamiltonianH ,Eq.(2.4),can fields. The operator product expansion (OPE) technique al-
int
bealso expressedin termsofthebosonfieldsθ andφ . It lows us to systematically handle the higher-order terms in
± ±
has been suggested that, besides the marginaloperators, op- the bosonized Hamiltonian (2.19). The one-loop RG equa-
erators with higher scaling dimensions can play an impor- tions that describe changes in the coupling constants during
tant role in the first-order CDW-SDW transition22,24 which the scalingof theshort-distancecutoff(a aedl) are given
→
is known to occur in the strong-coupling region of the 1D by(seeAppendixBfortheirderivation)
EHM.18,19,20,21Wethusincludeallthetermsofscalingdimen-
sion 4 [= 2(chargesector)+2(spinsector)]. We also note
d
thattherearesomecomplicationsandsubtletiesinbosonizing G = +2G2+G2 +G G , (2.20)
the off-site interaction term, i.e., the nearest-neighbor inter- dl ρ c cs s ρs
action term V (see Appendix A for detail). We obtain the d
G = +2G G G G G G , (2.21)
c ρ c s cs cs ρs
bosonizedHamiltoniandensity, dl − −
d
= 1 v (∂ θ )2+v (∂ φ )2 dlGs = −2G2s−GcGcs−G2cs, (2.22)
ρ x p σ x p
H 2π d
pg=Xρ+,−(cid:2) gσ (cid:3) dlGcs = −2Gcs+2GρGcs−4GsGcs
+ (∂ θ )(∂ θ ) (∂ φ )(∂ φ )
2π2 x + x − − 2π2 x + x − 2GcGs 2GcGρs 4GcsGρs, (2.23)
− − −
g g
c s d
cos2θ+ cos2φ
− 2π2a2 2π2a2 Gρs = 2Gρs+2GρGs
dl −
g
cs cos2θ cos2φ 4G G 4G2 4G G , (2.24)
− 2π2a2 − c cs− cs− s ρs
g
ρs
(∂ θ )(∂ θ ) cos2φ
− 2π2 x + x −
whereG aredimensionlesscouplingconstantswith theini-
g ν
cσ
+ 2π2 (∂xφ+)(∂xφ−) cos2θ tial values Gν(0) = gν/(4πta). The number of the inde-
pendentcouplingconstantsisfive,sincetheSU(2)spinsym-
g
+ 2πρσ2 a2(∂xθ+)(∂xθ−)(∂xφ+)(∂xφ−). (2.19) metry guarantees the relations Gσ = Gs, Gcσ = Gcs, and
G =G toholdinthescalingprocedure.Fromthesescal-
ρσ ρs
The renormalized velocities are v = 2ta + (g + g ing equations, one finds that the G , G , and G terms are
ρ 4k 4⊥ ρ c s
g )/2πandv =2ta+(g g g )/2π. Themargin−al marginal (the scaling dimension=2),56,57 while the G and
1k σ 4k 4⊥ 1k cs
− −
terms with the couplings g and g (g and g ) determine G termsareirrelevantoperatorsofthescalingdimension4.
ρ c σ s ρs
6
III. PHASEDIAGRAMOFTHEHALF-FILLED
TABLE I: Possible ground-state phases and positions of (quasi)
EXTENDEDHUBBARDMODEL
lockedphasefieldsdeterminedonlyfromthemarginaltermsinEq.
(2.19).
A. Bond-charge-density-wavestate
Phase (θ,φ) (gc,gs)
Inthissection,weshowthattheBCDWphaseexistsinbe- SDW (0,±π/2),(π,±π/2) (+,+)
tweentheCDWandSDWphasesintheweak-couplingregion CDW (±π/2,0),(±π/2,π) (−,−)
ofthe1DEHM. BCDW (0,0),(π,π),(0,π),(π,0) (+,−)
Firstwefocusontheweak-couplinglimitU,V t,where BSDW (±π/2,±π/2) (−,+)
≪
wecanneglecttheirrelevanttermsofscalingdimension4and
restrictourselvestothemarginalterms g , g , g , andg .
ρ σ c s
∝
Effectsofthe irrelevanttermsarediscussed later in this sec-
(ii) g < 0 and g > 0: The phase fields are locked at
tion. Within this approximation,the Hamiltonianreduces to s c
(θ,φ) = (πI ,πI ). The nonvanishing order parameter is
two decoupledsine-Gordonmodels, and we can analyze the 1 2
then ,andthegroundstateistheBCDWstate. Both
propertiesofthespinandchargemodes,separately. Theone- BCDW
O
chargeandspinexcitationsaregapped.
loopRG equationsfor these couplingconstantsare givenby
Eqs.(2.20)–(2.22)withGcs =Gρs =0: (iii) gs > 0 and gc < 0: The field θ is locked at θ =
(π/2)+πI ,andthefieldφtendstobearoundφ=(π/2)+
1
dG (l) = 2G2(l), (3.1) πI2 althoughitis notlockedin the low-energylimit. Inthis
dl ρ c casethedominantcorrelationisthatoftheBSDWstate. The
d chargeexcitationsaregappedwhereasthespinexcitationsare
G (l) = 2G (l)G (l), (3.2)
dl c ρ c gapless.
d (iv)g > 0andg > 0: Thefieldθ islockedatθ = πI ,
G (l) = 2G2(l). (3.3) s c 1
dl s − s whereasthe field φ tends to be near φ = (π/2)+πI2. The
dominant correlation is the SDW order. The charge excita-
The spin excitations are controlled by the G coupling,
s tionsaregappedwhilethespinexcitationsaregapless.
which is marginally relevant (marginally irrelevant) when
CombiningtheresultsofTableIandthecouplingconstants
G < 0 (G > 0). If g < 0, then G (l) increases with
s s s s
| | Eqs.(2.8a) and(2.8c), we obtaintheground-statephasedia-
increasingl. Inthiscasethephasefieldφislockedatφ = 0
gramofthe1DEHMintheweak-couplinglimit. ForU larger
modπ to gain the energy[see Eq. (2.19)], andconsequently
than2V suchthatg >0andg >0,wehavetheSDWphase,
thespinexcitationshaveagap. Ontheotherhand,ifg > 0, c s
s
whileforU sufficientlysmallerthan2V (g <0andg <0)
then G (l) decreasesto zero as l increases, and the φ field c s
s
| | we have the CDW phase. At U = 2V, we see from Eqs.
becomesafreefield;thespinsectorhasmasslessexcitations.
(2.8a)and(2.8c)thatg (=g )<0andg (=g )>0due
The approachof G to zero is very slow ( 1/l), and the φ s 1⊥ c 3⊥
s
∼ totheC term. Thisimpliesthatanewphasedifferentfrom
field has a strong tendency to be near φ = π/2 mod π. Al- 2
theCDWandSDWstatesappearsforU 2V. FromTableI,
thoughiteventuallyfailsto lockthe phaseφ, the marginally
≈
weidentifythenewphasewiththeBCDWphase. Withinthe
irrelevant coupling still has an impact on low-energy prop-
approximationweemployhere,thephaseboundarybetween
erties by givingrise to logarithmiccorrectionsto correlation
functions.58 the BCDW phase and the CDW (SDW) phase is located at
The charge sector is governed by the two couplings G
c
andGρ,whoseRGflowdiagramisoftheKosterlitz-Thouless φ
type. Sinceg = (U +6V)a > 0,G isarelevantcoupling
ρ c
SDW
and always flows to strong-couplingregime, unless g = 0.
c π CDW
This means that G (l) has two strong-couplingfixed points,
c
G (l) andG (l) ,dependingonitsinitialvalue BCDW
c c
g > 0→an∞dg < 0. As→see−n∞fromEq.(2.19),therelevantg BSDW
c c c
withpositive(negative)signimpliesthephaselockingofθat
thepositionθ =0(π/2)modπ.
Fromtheabovestandardarguments,thegroundstatecanbe 0 π θ
identifiedbysimplylookingattheinitialvalueofthecoupling
constants g and g . The ground state is classified into four
c s
cases as summarized in Table I, and the positions of locked
phases(θ,φ)forrespectivecasesareshowninFig.3.
(i) g < 0 and g < 0: The phase fields are locked at
s c
(θ,φ)= (π/2)+πI ,πI ,whereI andI areintegers.In
1 2 1 2
thiscase,amongtheorderparametersinEqs.(2.18),onlythe
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
CDWorderparameterhasa finite expectationvalue,andthe
groundstate is foundto be the CDW state. Both chargeand FIG.3: PositionsoflockedphasefieldsθandφintheSDW,CDW,
spinexcitationsaregapped. BCDW,andBSDWstates.
7
g =0(g =0).Inthisphasediagram,thechargeexcitations g
c s s
are gapful except on the CDW-BCDW transition line, while
thespinexcitationsaregaplessintheSDWphaseandonthe
BSDW
SDW-BCDW transition line. From Eqs. (3.1)–(3.3), we can
SDW
estimatethechargegap∆ andthespingap∆ as
c s
|g |
cs
g 2πta/gρ 2πta
∆c ≈t |tac| , ∆s ≈texp g (3.4) |gcs|
(cid:18) (cid:19) (cid:18) s (cid:19) g
0 c
for g g taand0< g ta,respectively.
c ρ s
| |≪ ≪ − ≪
Nextweexamineeffectsoftheparallel-spinumklappscat-
teringg ontheBCDWstate. Weconsiderthesituationvery
cs
CDW
closetotheCDW-BCDWtransitionbyassumingg 0and
c
gs <0,i.e.,U 2V = C2V2/πt+O(V3/t2). Int≈hiscase BCDW
− −
thespingapisformedfirstastheenergyscaleislowered.For
energies below the spin gap, we can replace cos2φ with its
average cos2φ (∆ /t)2. This means that the coupling
s
h i ≈
constantg ismodifiedas
c FIG.4: PhasediagramobtainedbyminimizingthepotentialV(θ,φ)
g∗ =g +g cos2φ . (3.5) forgcs <0. Thedoublelinedenotesthefirst-ordertransition,while
c c csh i thesinglelinedenotesthesecond-ordertransition. Bicriticalpoints
Thus we find that the BCDW state, which is realized for areat(gc,gs)=(±|gcs|,∓|gcs|).
g∗ > 0, becomes less favorable due to the g (< 0) term.
c cs
We note, however,that the CDW-BCDW boundarydoesnot
case,weperformasemiclassicalanalysis: weneglectspatial
move across the U = 2V line because g cos2φ
2Vaexp[ c(t/V)2]ismuchsmallerthanth|eCcshtermiin|E≈q. variationsofthefieldsinEq.(2.19)andfocusonthepotential,
2
−
(2.8c) for V t, where c is a positive constant. A similar
≪
argumentapplies to the regionnear the SDW-BCDW transi-
V(θ,φ)= g cos2θ+g cos2φ g cos2θ cos2φ, (3.7)
tion.SupposethatU 2V =+C V2/πt+O(V3/t2)where − c s − cs
2
−
gs ≈ 0 andgc > 0. Inthis case, asthe energyscale is low- where gcs = g3k < 0. The order parameters of the
ered, the charge gap opens first and the θ field is pinned at SDW, CDW, BCDW, and BSDW states take maximum am-
θ = 0modπ. Belowthecharge-gapenergyscale,theφfield plitudes when the fields θ and φ are pinned at (θ,φ) =
issubjecttothepinningpotentialgs∗cos2φwith πI1,(π/2) + πI2 , (π/2) + πI1,πI2 , (πI1,πI2), and
(π/2)+πI ,(π/2)+ πI , respectively, where I and I
g∗ =g g cos2θ , (3.6) (cid:0)(cid:0)(cid:0) 1 (cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0) 2 (cid:1)(cid:1)(cid:1) 1 2
s s− csh i are integers. The potentialenergyin these states is obtained
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
where cos2θ (∆c/t)2(1−Gρ). Thus the BCDW phase, by inserting these pinned fields into Eq. (3.7), e.g, VSDW =
whichihsnowriea≈lizedforg∗ <0,alsobecomeslessfavorable V πI1,(π/2)+πI2 ,yielding
s
bnyotthmeo−vegdcsbhceoyson2dθit(h>e0U)t=erm2.VAglianiensthinecpehagscesbcoousn2dθaryis (cid:0)(cid:0)(cid:0) VSD(cid:1)(cid:1)(cid:1)W = −gc−gs−|gcs|, (3.8a)
2Va(c′V/t)πt/V is much smaller than the C| 2 therm ini|E≈q. VCDW = +gc+gs−|gcs|, (3.8b)
(2.8a), where c′ is a constantof order 1. Thereforewe con- V = g +g + g , (3.8c)
BCDW c s cs
− | |
cludethat the BCDW phase is robustagainstthe gcs term in V = +g g + g . (3.8d)
BSDW c s cs
the weak-coupling limit. The analysis in this section estab- − | |
lishes the existence of the BCDW phase near U 2V for Wefindthattheg termstabilizestheSDWandCDWstates
≈ cs
0<U,V t. whileitworksagainsttheBCDWandBSDWstates. Compar-
≪
ing these energies, we obtain the phase diagramin the g -g
c s
planeata fixedg (Fig.4). Inthe presenceoftheg term,
cs cs
B. First-orderSDW-CDWtransition thedirectCDW-SDWtransitionlineappearsinthisphasedi-
agram.
Inthissection,wediscusshowtheBCDWphasebecomes We now discuss the nature of the phase transitions. The
unstableatstrongcouplingandhowthetwocontinuoustran- potentialV(θ,φ) onvarioustransition linesis shownin Fig.
sitionschangeintothefirst-orderSDW-CDWtransition. 5. On the boundary between the SDW and BCDW phases,
To our knowledge, Cannon and Fradkin were the first to whichislocatedatg = g andg > g ,thepotential
s cs c cs
−| | | |
arguethattheg term(describingtheumklappscatteringof takestheformV(θ,φ)= g cos2θ+g cos2φ(1 cos2θ)
3k c s
− −
parallel-spinelectrons), which is conventionallyignoreddue [Fig. 5(a)], which pins the θ field at θ = πI and leaves the
1
to its large scaling dimension, can become relevant at large φ field completely free. We thus find that the SDW-BCDW
U andV andcausethefirst-orderCDW-SDWtransition.22To transitioniscontinuous,i.e.,theSDWandBCDWphasesco-
get an insight into the effect of the g term in the relevant existwithoutpotentialbarrieronthephaseboundary. Onthe
cs
8
(a)
V π V π
−π 0 φ −π 0 φ
0
0
(cid:13) θ
θ
π−π π−π
(b)
FIG. 6: The potential V(θ,φ) on the bicritical point (gc,gs) =
(|gcs|,−|gcs|). The potential minima are the lines θ = πI1 and
φ=πI2.
V π where the potential minima are given by the isolated points
(θ,φ) = πI ,(π/2)+πI and (π/2)+πI ,πI . These
1 2 1 2
minimacorrespondtotheSDWstateandtheCDWstate,see
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
−π 0 φ Fig.3.Thepointtonoteisthatthereisafinitepotentialbarrier
ofheightmin(g ,2g 2g )betweenthecorresponding
cs cs c
| | | |− | |
0 minimafor the SDW andCDW phases. Hence we conclude
θ thatthe CDW-SDWtransitionis first orderwhen g is rele-
cs
π−π vant.
From the above arguments, we find that strong umk-
(c)
lapp scattering of the parallel-spin electrons destabilizes the
BCDW and BSDW states and gives rise to bicritical points
(g ,g ) = (g , g )wherethetwocontinuous-transition
c s cs cs
± −
linesmergeintotheCDW-SDWfirst-ordertransitionline.Let
us take a closer look at these bicritical points. Taking into
V π account the fact that g > 0 and g < 0 for U 2V in
c s
≈
the original EHM, we will focus on the bicritical point at
(g ,g ) = (g , g ). The effective potential at the bi-
c s cs cs
−π 0 φ criticalpoint|take|s−th|efo|rm
0 V(θ,φ)= g(cos2θ+cos2φ cos2θcos2φ), (3.9)
θ − −
π−π which is shown in Fig. 6. This potential has an interesting
feature that its potential minima are not isolated points but
FIG. 5: The potential V(θ,φ) on the SDW-BCDW (a), BCDW- the crossing lines θ = πm or φ = πn (m, n: integer). On
CDW(b),andCDW-SDW(c)transitionlines. these lines either θ or φ becomesa free field; the theoryhas
more freedom than a single free bosonic field, but less than
twofreebosonicfields. Wethusexpectthatthetheoryofthe
bicriticalpointshouldhaveacentralchargelargerthan1but
boundary between the BCDW and CDW phases, located at smallerthan2. Detailed analysisof thecriticaltheoryisleft
g = g and g < g , the potential now takes the forafuturestudy. We notethatwheng = 0thefirst-order
c cs s cs cs
| | −| |
form V(θ,φ) = g cos2θ(1 cos2φ) + g cos2φ [Fig. CDW-SDWtransitionlinecollapsesintoatetracriticalpoint,
c s
− −
5(b)]. The potential locks the φ field at φ = πI , where it (g ,g ) = (0,0), and the phase boundariesin Fig. 4 reduce
2 c s
has no effect on the θ field. Thus, we find that the CDW- to the lines g = 0 and g = 0 where all the transitions are
c s
BCDW transitionisalso continuous. Fromsimilar consider- continuous.
ations, we find thatthe SDW-BSDWandBSDW-CDWtran- Fabrizio et al.32 and Bajnok et al.59 discussed effects of
sitions are continuous as well. In Fig. 4, the phase bound- higher-frequencyterms,suchassin3θ andcos4θ, whichare
aries of continuous transitions are shown by the solid lines. generatedthroughthe renormalization-grouptransformation.
On the contrary, the phase boundary shown by the double From the semiclassical arguments, it can be seen that these
line in Fig. 4 is of different nature from the others. The termscanalsochangeasecond-ordertransitiontoafirst-order
potential V(θ,φ) on the double line is shown in Fig. 5(c), transition.59Infact,itwasarguedthatthesehigher-frequency
9
4 φfieldismarginallyirrelevantandthusthespinsectorshould
becomegapless.
The phase diagram obtained in this manner is shown in
t
/ CDW Fig. 7. The single lines denote continuous transitions, and
V
thedoublelinedenotesthefirst-ordertransition. Intheweak-
couplinglimit,theBCDWphaseappearsatU 2V andthe
≈
successivecontinuoustransitionsbetweenthe SDW, BCDW,
2 and CDW states occur as V/U increases. When U and V
increase along the line U 2V, the BCDW phase first ex-
BCDW
≈
pands and then shrinks up to the bicritical point (U ,V )
c c
≈
(5.0t,2.3t) where the two continuous-transition lines meet.
SDW Beyondthis pointthe BCDW phase disappearsand we have
the direct first-order transition between the CDW and SDW
phases. Thephasediagram(Fig.7)issimilartotheonesob-
tained by using more sophisticated numerical methods.17,25
0
0 4 8 We note that the position of the first-order transition line in
U/t Fig. 7 is not reliable quantitativelyas we have used the per-
turbativeRGequations.TherecentMonteCarlocalculation25
givesthemostreliableestimateforthepositionofthebicrit-
FIG. 7: Phase diagram of the half-filled 1D extended Hubbard
icalpoint, (U ,V ) (4.7 0.1)t,(2.51 0.04)t , which
model. Thedoublelinedenotesthefirst-ordertransition, whilethe c c ≈ ± ±
agrees with our estimate in Fig. 7 within 10%. The semi-
singlelinesdenotethesecond-ordertransitions. Thebicriticalpoint
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
quantitativeagreementgivesusconfidencethatourapproach,
isat(Uc,Vc)≈(5.0t,2.3t).
semiclassical analysis of the low-energy effective Hamilto-
nianderivedwithuseoftheperturbativeRG,isreliableeven
termsmaketheSDW-CDWtransitionfirstorderinthestrong- inthestrong-couplingregimenearthemulticriticalpoint.
couplingregimeofthe1DEHM.32However,wehaveshown
thattheSDW-CDWfirst-ordertransitioncanoccursimplydue
totheg termwhichistheleadingirrelevantterminthissys- IV. EFFECTOFSTAGGEREDSITEPOTENTIAL
cs
tem. Since the higher-frequencytermsare evenless relevant
than the gcs term, we expectthat the gcs term should play a In this section, we examine effects of alternating on-site
dominantroleinthefirst-ordertransitioninthe1DEHM. modulation of the chemical potential, i.e., the staggered site
potential, in the half-filled 1D EHM. The Hamiltonian to be
consideredisgivenbyH′ = H +H withH definedinEq.
∆
C. Globalground-statephasediagram (2.1)and
To obtain the global phase diagram of the 1D EHM, we H =∆ ( 1)jn . (4.1)
∆ j,σ
have numericallysolved the scaling equations(2.20)–(2.24). −
j,σ
X
We findoutwhichphaseisrealizedbylookingatwhichone
of the couplings G , G , and G becomes relevant first, as The model is called the ionic Hubbard model if V = 0.
c s cs
we have discussed in Secs. IIIA and IIIB. First, if G When U = V = 0, the system is a trivial band in-
c
| |
grows with increasing l and reaches, say, 1 first among the sulator, since the ∆ term induces a gap 2∆ at k =
| |
three couplings, then we stop the integration and compute π/2 in the single-particle spectrum and the lower band is
±
G∗ = G G sgn(G ). Since the charge fluctuationsare fully filled. For many years effects of the on-site repul-
s s − cs c
suppressedbelowthisenergyscale,weareleftwithEq.(3.3), sive interaction U on the band insulator have been investi-
whereG isreplacedbyG∗. WeimmediatelyseefromTableI gated intensively32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51
s s
thatapositive(negative)G∗ leadstotheSDW(BCDW)state from both numerical and analytical approaches. Using the
s
forG >0andtheBSDW(CDW)stateforG <0. Second, standard bosonization method, Fabrizio, Gogolin, and Ners-
c c
if G becomes1 first, or more precisely, if G reaches 1 esyanrecentlyarguedthatthegroundstate oftheionicHub-
s s
| | −
first,thenweareleftwithEqs.(3.1)and(3.2),whereG and bardmodelexhibitsthreephasesasU increases: thebandin-
ρ
G are replaced by G∗ = G G and G∗ = G +G , sulator,theSDI,andtheMottinsulator.32Theorderparameter
c ρ ρ − ρs c c cs
respectively. We see that a positive (negative) G∗ leads to oftheSDIstateisnothingbutthatoftheBCDWstate,andwe
c
theBCDW(CDW)state. Finally,when G reaches1first, canregardthe two states asessentially identical. Itwasalso
cs
| |
we stopthe calculationandcompareG andG . Since both arguedthatthequantumphasetransitionfromthebandinsu-
c s
charge and spin fluctuations are already suppressed by the lator to the SDI state belongs to the Ising universality class
G cos2θcos2φpotential,wecandeducethephasefromthe whereastheothertransitionfromtheSDIstatetotheMottin-
cs
semiclassicalargument. FromFig.4weseethatwehavethe sulator is of the Kosterlitz-Thouless type. Recent numerical
SDWstateforG > G andtheCDWstateforG < G . studies,34,35,36,37,38,39,40,41,45however,havereportedcontrover-
s c s c
− −
HerewenotethatintheSDWstatethepinningpotentialtothe sialresultsontheexistenceoftheSDIphase. Someclaimed
10
tofindtwoquantumphasetransitionswhileothersfoundev-
TABLEII:Possibleorderedgroundstatesandthepositionof(quasi-
idences of only one phase transition. With this issue of the
)lockedphasefieldsdeterminedfromEq.(4.4).
SDIphaseinmind,inthissectionweinvestigatethephasedi-
agramofthe1DextendedHubbardmodelwiththestaggered Phase (θ,φ)
site potentialand examinecritical propertiesof the quantum
SDW (0,±π/2),(π,±π/2)
phasetransitions.
BI(forg∆ >0) (+π/2,0),(−π/2,π)
We take into account the staggered site potential and the
correlationeffectsonequalfootingbytreatingthemasweak BI(forg∆ <0) (+π/2,π),(−π/2,0)
BCDW +(π/2)±α0,0 , −(π/2)±α0,π
perturbations. WeuseEq.(2.17)torewriteH inthecontin- θ θ
uumlimitasH∆ = dx ∆,where32,33 ∆ BSDW (cid:0)((cid:0)(cid:0)+π/2,0±α0φ),(cid:1)(cid:1)(cid:1)−(cid:0)(cid:0)(cid:0)π/2,±(π−α0φ)(cid:1)(cid:1)(cid:1)
H
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
R g
= ∆ sinθ cosφ (4.2)
H∆ −2(πa)2
phases θ and φ are determined from the saddle-point equa-
withg =4π∆a.NotethattheCDWorderparameter
nisalprfoopr∆coertcioonuaplletodHto∆, and g.∆ThciasnhbaesrtehgeacrdoendseaqsueannOceeCxtDtheWar-t tgi∆onssi:ncθo)s=θ(40g.cIsninorθd−ergt∆osciomspφl)if=yt0heanndotsaitnioφn(s−,l4egtsucsoisnφtro+-
OCDW duce
acquiresa nonvanishingexpectationvalueforany fi-
CDW
O
niteU andV,aslongasg = 0. Inthissectionwewillde-
∆
6
notetheinsulatingphaseconnectedtothefree-electronband g g
itnhsaunlathtoerC(DUW=phVas=e. 0 and ∆ 6= 0) by the BI phase, rather α0θ ≡(cid:12)cos−1(cid:18)4g∆c(cid:19)(cid:12), α0φ ≡(cid:12)cos−1(cid:18)4g∆s(cid:19)(cid:12), (4.5)
(cid:12) (cid:12) (cid:12) (cid:12)
ThebosonizedformoftheHamiltonianH′ canbethought (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
of as a generalization of the so-called double sine-Gordon
where g /g 4, g /g 4, and 0 α0,α0 π are
(DSG) model as H′ contains sine/cosine terms with differ- | ∆ c| ≤ | ∆ s| ≤ ≤ θ φ ≤
assumed.Thesolutionsofthesaddle-pointequationsyieldthe
ent frequencies (sinθ and cos2θ, cosφ and cos2φ). The
followingfourstateswithdistinctconfigurationsofthelocked
DSGtheoryitselfhasbeeninvestigatedintensively32,59,60and
phase fields θ and φ (modulo2π): (i) the SDW state with θ
shown to have a critical point belongingto the Ising univer-
andφlockedat(θ,φ) = (0, π/2)or(π, π/2);(ii)theBI
salityclass[c= 1 conformalfieldtheory(CFT)].Toobtaina ± ±
2 statewith(θ,φ) = (+π/2,0),( π/2,π)ifg∆ > 0andwith
qualitativeunderstandingofthecriticalpropertiesinoursys- −
(θ,φ) = (+π/2,π),( π/2,0)ifg < 0;(iii)the“BCDW”
∆
tem,wefirstperformasemiclassicalanalysisinasimilarway −
statewheretheBCDWorderandtheCDWordercoexistand
to Sec. III B, before examining the global phase diagram of
which is realized when (θ,φ) = (π/2 α0,0) or ( π/2
H′withuseoftheRGmethod. α0,π);(iv)the“BSDW”statewherethe±BSDθWandth−eCDW±
θ
ordercoexistandwhichisrealizedwhen(θ,φ) = (π/2,0
α0)or π/2, (π α0) .TableIIandFig.8summarizeth±e
A. Semiclassicalanalysis poφssible−ordered±gro−undφstatesandcorrespondingpositionsof
(cid:0)(cid:0)(cid:0) (cid:1)(cid:1)(cid:1)
lockedphasefields. Thepotentialenergiesinthesestatesare
In this section, we performa semiclassical analysis to the
Hamiltonian ′ = + , where and aregivenby
∆ ∆
H H H H H
Eqs. (2.19) and (4.2), respectively. We neglectspatial varia-
φ
tionsofthefieldandfocusonthelockingpotential:
SDW
V∆(θ,φ) = gccos2θ+gscos2φ gcscos2θcos2φ π BI
− −
g sinθ cosφ. (4.3) BCDW
∆
−
BSDW
First, we examinethe case g = 0, whichcorrespondsto
cs
thesituationwheretheg termbecomesirrelevantintheRG
cs
scheme.Thepotentialtobeconsideredis 0 π θ
V0(θ,φ) V (θ,φ)
∆ ≡ ∆ |gcs=0
= g cos2θ+g cos2φ g sinθ cosφ.
c s ∆
− −
(4.4)
Due to its double-frequency structure, possible locations of
the phase locking are different from the ones we found in
Sec. III B. For example, when g > 0 (g > 0), the two
c s
kinds of potentials proportional to sinθ and cos2θ (cosφ FIG.8: Positionsoflocked phasefieldsθ andφinthefourstates
andcos2φ) competewitheach other.59,60 Thelockingof the wheng∆ >0.