Hubbard-I approach to the Mott transition

0.17 MB·English

by  Przemyslaw R. Grzybowski

Hubbard-I approach to the Mott transition
Przemysl aw R. Grzybowski1 and Ravindra W. Chhajlany1
1Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland
(Dated: October13, 2011)
A Hubbard-I like approach is reconsidered in the context of the metal-insulator transition in
strongly correlated electron systems. We show that this approach yields a simple treatment of the
transition (at half-filling) and the metallic phase (for arbitrary fillings) in a constrained Hubbard
model, that is equivalent to a particular case of a Hubbard bond charge Hamiltonian on bipartite
lattices. These calculations, which are to our knowledge the first Hubbard-I calculations for this
1 model, yield results which compare well with known results for this model. For half filling, we also
1
comparetheseresultswiththoseobtained usingtheGutzwiller approximation. Finally,wepropose
0
a candidate variational wave function connecting the Gutzwiller wave function approach with the
2
schemepresented here.

t
c
PACSnumbers: 71.30+h,71.10.Fd,71.10.Ay
O

2
Introduction The metal to Mott insulator (MI) tran- Gutzwiller approximation [5], that increase in U is asso-
1
sition, envisaged by Mott [1], is one of the striking ef- ciated with the diminishing of the quasiparticle residue

] fects induced by strong electronic correlations in many of the Fermi liquid, or equivalently with the increase
l
e electronsystems. TheCoulombinteractionbetweenelec- in quasiparticle effective mass. In this framework, a
- trons U leads to highly correlatedgroundstates in these metal-insulatortransitionoccurswhentheeffectivemass
r
t systems, rendering their description quite challenging. diverges and is thus driven by quasiparticle localisa-
s
. As a result there have been various theoretical attempts tion. The critical value U obtained in the (Gutzwiller)-
t c
a at providing a satisfying description of the Mott transi- Brinkman-Rice (GBR) approach is much larger than in
m
tion. the Hubbard approach. This essentially weak coupling
d- The first of these was due to Hubbard [2], who pro- methodgivesagoodlowenergydescriptionofthemetal,
n videdaseminalnonperturbativeapproach–theso-called but does not describe the precursors of the Hubbard
o Hubbard-I(H-I)approximation–toasimplifiedinteract- bandswhichshouldplausiblyappearonthemetallicside.
c ing electron problem described by the Hubbard model. Moreinvolvedcontemporarymethods,suchasdynam-
[
H-I describes both (i) the atomic limit (i.e. limit of van- ical mean field theory (DMFT) [7] or Green’s function

1 ishing bandwidth W 0) exactly, in particular yielding ladder diagram techniques[8], provide a connection be-
v two atomic levels corr→esponding to single and double lo- tween a strongly correlated fermi liquid picture on the
9
cal occupancy, and (ii) the non-interacting case (U =0) onehand,andformationofHubbardbandsontheother.
7
6 exactly, and so held some promise of describing the in- While providing greater insight into the physics at play,
2 termediate physics in a consistent interpolating fashion. thesemethodsdonotprovideasimplepicturecharacter-
. For finite bandwidth, the atomic levels broadeninto two istic of more analytical techniques. In this paper, we re-
0
1 “dynamic”(sub)bandswhichareoccupation-numberand consider the Hubbard-I approach as a systematic means
1 interaction dependent. These are always split by a gap of obtaining a picture, complementary to the one ob-
1 for all U > 0 indicating an insulator phase. Unfortu- tained using the Gutzwiller approximation, of the Mott
v: nately, a metal-insulator transition is predicted exactly transition from the point of view of strong coupling the-
i atU =0,associatedwithgapclosure. Thesituationwas ory. We shall show that there are strongly correlated
X
somewhat improved in the Hubbard-III approximation Hamiltonians for which the Hubbard-I approach yields
ar [3], where scattering and resonance broadening correc- good qualitative and quantitative results.
tions shift the transition point to U W. However, Inspection of the Hubbard-I approach Contrary to
c
∼
this approach does not predict the expected Fermi Liq- early expectations, the Hubbard approach, based on a
uid propertiesonthe metallic side, [4]. These drawbacks Green’s function decoupling scheme, is now seen as a
not withstanding, the Hubbard approaches are very im- large–U approximation, despite its non perturbative na-
portant for the conceptual introduction of the Hubbard ture. One may therefore suspect that some problems
subbands and also because they give a basic description encountered in the Hubbard approach may originate in
of the insulating phase. limitations of strong-coupling perturbation theory.
A complementary approach, starting from the weak Consider therefore the Hubbard model: H = Tˆ +
coupling limit, is based on the Gutzwiller variational UDˆ, where Dˆ = ini↑ni↓ is the number of dou-
wave function [5], which describes an increasingly cor- bly occupied sites anPd Tˆ = t c† c is the ki-
i,j,σ i,j iσ jσ
related Fermi liquid with increasing interaction U. netic energy. The large-U Pperturbational expansion
Brinkman and Rice [6] showed, using the so-called starts from splitting the kinetic energy into three parts
Tˆ = Tˆ + Tˆ + Tˆ . Tˆ = Tˆ + Tˆ is the The electron Green’s function thus has the character-
0 +1 −1 0 UHB LHB
double occupancy conserving hopping (commuting per- istic Hubbard two pole form describing two subbands,
turbation) in the Upper and Lower Hubbard Bands howeverthe poles here have an elementary form in com-
Tˆ = t n c† c n ,Tˆ = t (1 parisonwith the solutionfor the full Hubbardmodel [2].
UHB i,j,σ ij iσ¯ iσ jσ jσ¯ LHB i,j,σ ij −
n )c† c (P1 n ), corresponding to hopPping of pro- The main result stemming from these Green’s functions
iσ¯ iσ jσ − jσ¯ is that a Mott transition is elevated to finite interaction
jected electrons on doubly and singly occupied sites re-
spectively, and interband hopping terms Tˆ and Tˆ = U. Indeed, for a paramagnetic configuration at half fill-
+1 −1
Tˆ+†1 (off-diagonal perturbation). The perturbational ex- Uing hnW↑/i4=whhni↑lei t=he1/h2igthheestloewneesrtgyenienrgtyheinLtHhBe UisHWB/i4s
pansionfortheHubbardmodeliswellknownandcanbe −
(where W is the free electron bandwidth), hence a gap
performed e.g. using the method of canonical transfor-
opens at a critical value U = W/2. A fundamental
mations [9, 10] which to second orderyields the effective c
drawback of the Hubbard-I approach to the full Hub-
Hamiltonian:
bardmodelisthattheresultantGreen’sfunctionsdonot
1
H(2) =Tˆ +UDˆ + [Tˆ ,Tˆ ]. (1) conserve the electron-hole symmetry at half-filling. Im-
eff 0 U +1 −1
portantly,thepresentedsolutionsEqs.(3,4)areexplicitly
Recall that the perturbation expansion to any or- electronholesymmetricanddonotsufferfromthisdraw-
der leads to an effective Hamiltonian eliminating mix- back. Accordingly, we show below that the gap opening
ingbetweenthedegeneratesubspacesoftheunperturbed at U = W/2 is associated with a Mott transition only
Hamiltonian Uˆ. Due to this property, the averageof op- exactly at half-filling. These results provide explicit evi-
erators jointly changing the total number of doubly oc- dence that the source of problems in the Hubbard-I ap-
cupied sites, such as e.g. hcjσnjσ¯c†lσhlσ¯i,(hjσ¯ = 1−njσ¯ proach stem from the effects of the T±1 terms.
is the on-site hole occupation number), are identically TheMotttransition TheconstrainedHamiltonianH
c
zero leading to the vanishing of the associated Green’s Eq.(2) is interesting in itself because it can be viewed
functions. Thus in this framework, the single particle as a particular model of extremely correlated electrons.
Green’s function G(j,l) = c c† decomposes into Recall that the term extremely correlated electron sys-
hh jσ| lσii
a sum of two Green’s functions Γσ = c n c† n tems has recently been introduced by Shastry [11], em-
lj hh jσ jσ¯| lσ lσ¯ii
andΓσ = c h c† h relatedtothe propagationof phasizing the appearance of noncanonical fermions re-
jl hh jσ jσ¯| lσ lσ¯ii sultingfromtheprohibitionofdoubleoccupanciesinthe
fermionicquasiparticles,intheupperandlowerHubbard
e U = limit, which is a special case of symmetry of
bands.
∞
conserveddoubleoccupancies. The modelH considered
Considerthe Hubbard-Iapproachtothesimplestnon- c
here, that allows for double occupancies which are con-
triviallevel of strong-couplingperturbationalexpansion,
servedand thus describes hopping of noncanonical (con-
described by the first two terms in the expansion Eq.(1)
strained) fermions c n and c h can therefore be
jσ jσ¯ jσ jσ¯
Hc =Tˆ0+UDˆ. (2) considered a generalization of the problem of extremely
correlated fermions for general interaction strength U.
In the frequency domain, the equations of
NotethatthemodelH containstheU limitofthe
motion for the basic Green’s function are c →∞
Hubbard model. Even for small U and low fillings, we
ωΓσ(ω) = 1 n δ + [c n ,H ]c† n and
ωpeΓerσllfjjo(rωm) =the212ππfhohhllloσ¯lσw¯iδiilnjlgj+thyhhp[hcelσlσholfσ¯l,σ¯Hdecc]co|cu|†jpσjlhσinjσ¯gjiσ¯iiωoi.ωn tWhee tdhiseWcugesrsonuobnwedlocwsotnatsthiedasetrotfhthtehegedroUeutan=idls∞sotfaHttehusebobHfauHrdbcbmcaororddr-eeIls.dpeosncdritpo-
higher order Green’s function in the equations for tionofthezerotemperatureMotttransitioninthemodel
Γσlj(ω): hhcl+δσnl+δσ¯nlσ¯|c†jσiiω ≈ hnlσ¯iΓσl+δ,j(ω), Hc ofextremelycorrelatedelectrons. Theanalysisiscar-
c c† c n c† c c† c n c† , riedoutonlyforthe paramagneticphases,withn =n .
hh lσ lσ¯ l+δσ¯ l+δσ| jσiiω ≈ h lσ lσ¯ihh l+δσ¯ l+δσ| jσiiω ↑ ↓
h c† c c c† c c c† h c† , From the band structure of the one particle Green’s
hwhitlh+δσanla+loδσ¯goluσ¯slσc|omjσpiileωm≈entharlyσ¯ ltσriehahtml+eδnσ¯t l+ofδσt|hjeσiisωet function,itisevidentthattheinsulatorphaseexistsonly
Γσ(ω). The last two decouplings lead to magnetic and if U > Uc and the lower band is completely filled while
lj
superconducting order parameters are set here to zero, theupperbandisempty. TheGreen’sfunctionsEqs.(3,4)
e
as we are interested in the description of a pure Mott leadtothefollowingequationforthenumberofelectrons
transition. Solving the equations of motion, one obtains of a given spin species:
the following momentum (or Fourier transformed)
W
Green’s functions for the system: n =(1 n ) f((1 n )ǫ µ)ρ(ǫ)dǫ
σ − σ¯ Z − σ¯ −
−W
1 n
Γσk(ω)= σ¯ (3) W
2πω−(nσ¯ǫk+U −µ) +nσ¯Z f(nσ¯ǫ+U −µ)ρ(ǫ)dǫ (5)
1 1 n −W
Γσk(ω)= − σ¯ (4)
2πω ((1 nσ¯)ǫk µ) where f(...) is the Fermi-Dirac distribution and ρ(ǫ) is
− − −
e
2
the density of states. The boundaries of the insulator 0,75
1
phase are obtained, independently of the lattice, when MI 1d DOS
eitherµ=(1 n )W/2isattheendofthelowerbandor 0,8 Sq. DOS
σ¯ S. C. DOS
µ=−nσ¯W/2−+U is at the beginning ofthe upper band. W0,6 Metallic n<1 R1de cetx. aDcOtS
Forboththesecases,atzerotemperature,Eq.(5)reduces 0,5 U/
0,4
tonσ =1 nσ¯,showingthatthetransitiononlyoccursat Metallic n>1
− 0,2
half-filling. The jump in µ at half-filling n↑ = n↓ = 1/2 U/W
reduces to zero at U = W/2 indicating the transition 0
-0,25 0 0,25 0,5 0,75
point. The Mott”lobe” is showninFig.1. Note that the 0,25 µ/W
kineticenergyandinteractionenergyarebothzerointhe
Metallic U=inf.
Mott phase, which is an exact feature of the model.
Outside the Mott phase, the system is in a metallic
phasedescribedbyfour(twoperspinspecies)dispersion-
0
0 0,2 0,4 0,6 0,8 1
less extremely correlated Fermi liquids (ECFLs) associ-
n
atedwith the lowerandupper Hubbardbands. Depend-
ing on the filling factor and value of U, for each spin Figure 1: Ground state phase diagram in the U −n plane.
species one can obtain a situation where only one kind Inallcalculationsweassumeonlynearestneighbourhopping
of Fermi liquid appears to which there pertains a single tij =−t. AmetallicECFLexistsforalln≤1. Atlowfilling,
the metal resembles the U =∞ with no double occupancies.
well defined Fermi surface (see Fig. 1). For n < 1, this
Closer to n = 1, the metal has double occupancies and is
case physically resembles the ground state properties of
described by two Fermi liquids per spin species. In the H-
the infinite U limit of the Hubbard model. The single I calculation, the boundary between these two behaviours is
Fermi liquid comes about because for small n↑,n↓ the a Lifshitz-type transition, shown for hypercubic lattices in
lower Hubbard band is wide while the upper Hubbard 1,2,3 dimensions and for the rectangular DOS.Shown is also
band is very narrow. Even for U < W/2, although the the exact result for 1d, from [17] with rescaled U → U/2,
for comparison. All transition lines cross the n = 1 line at
two bands overlap, the upper band centered around U
finite U = W/2 marking the point of Mott transition. Solid
is too narrow to be occupied in the ground state. On
line depicts the Mott phase. The n > 1 can be obtained by
the other hand, closer to half filling n 1 for U < W/2
≤ invoking electron-hole symmetry. Inset: Ground state phase
when the upper band can also be occupied, two coexis- diagramintheU−µplane. Theelectronholesymmetryline
tant Fermi liquids emerge, per spin species, associated (dotted) separates n < 1 and n > 1 cases. The n = 1 case
with two Fermi surfaces. The boundary between these coincides with this line for U < W/2 whereas for U > W/2
twocasesisaFermisurfacetopologychangingtransition opens intoa Mott insulator ”lobe” of finitewidth.
of the Lifshitz type.
WenowfocusontheinteractiondrivenMotttransition
the band, as ǫ W/2 ∆ǫk k2, we have
fromthemetallicphase. UnlikeintheGutwzillerorslave → − ⇒ ∝
for 1 dimension ρ( W/2 + ∆ǫ) 1/√∆ǫ, 2 dimen-
boson methods, the hopping in the approach discussed − ∝
sions that ρ( W/2 + ∆ǫ) const., and 3 dimensions
here cannot be used as an indicator of the transition, as − ∝
ρ( W/2 + ∆ǫ) √∆ǫ. Then we obtain that in 1d:
itremainsconstant. Thedensityofdoublyoccupiedsites − ∝1
aDg=oohdDˆpia/Nram(oectceurpaatthioanlfofiflltihnegucpappteurrHinugbtbhaerdtrbaannsidt)ioins DD ∝[[((UUcc−UU))//UUcc]]223., 2d: D ∝(Uc−U)/Uc while in 3d:
∝ −
Relation to the Bond-charge Hubbard model While
for this model. It takes a particularly simple form
the results presented in the previous paragraph provide
D = Dˆ /N = 1 −U ρ(ǫ)dǫ (6) aclearandsimplequantitativepictureofaparamagnetic
h i 2Z transition as well as the metallic phase, it is not a pri-
−W/2
ori clear how good an approximation the Hubbard-I ap-
and depends only on U and the density of states.
proachyieldsfortheconstrainedmodelH . Thismodelis
We show results for a few different density of states c
related to the class of generalized Hubbard models, dif-
(DOSes)in Fig.2.
fering from the Hubbard Hamiltonian by an additional
Note that an important feature of our Hubbard-I cal-
bond charge interaction term (see e.g. [13–15]):
culation is that the critical point does not depend on
the lattice type. Furthermore, lattice dependent proper- H =Tˆ+UDˆ + X (n +n )c† c
ties emerge for the number of doubly occupied sites. A bond−charge X ij iσ¯ jσ¯ iσ jσ
i,j,σ
linear dependence is associated here only with a rectan-
(7)
gular density of states. In fact, the critical behaviour
of D has universal properties depending only on spa- At a symmetry point, when the bond charge interaction
tial dimensions of the lattice. It is governed by the isequaltothehoppingX = t ,thisgeneralizedHub-
ij ij
−
behaviour of the DOS at the bottom of the band. In- bardmodelconservesthe numberofdouble occupancies.
deed for parabolic energy profiles near the bottom of In fact, the resultant symmetric model can be mapped
3
0,3 as in 1 dimension. We thus see, that the phase diagram
1d DOS derivedbytheHubbard-Iapproachisingoodqualitative
0,25 Sq. DOS agreement with exact results.
S. C. DOS
Rect. DOS Comparison withtheGutzwillerapproach Itisworth-
0,2 while to compare our H-I results, for half filling, with
thoseobtainedusingtheGutzwillerapproximationtothe
D0,15 model H . We perform a standard calculation (see e.g.
c
[12]), i.e assuming the reference state, that is Gutzwiller
0,1 projected, to be the product of Fermi seas of the two
spin species. Within the GA, the average energy of the
0,05 Hamiltonian Hc can be written as:
H = q ǫ +UD, (8)
00 0,1 0,2 0,3 0,4 0,5 h ci X σ 0σ
σ
U/W
where ǫ is the energy of the Fermi sea of a given spin
0σ
Figure 2: The double occupancy per site D at half-filling
species. The band narrowing factor q is easily found to
σ
n = 1 as a function of U. Shown are results for hypercubic
be given by:
lattices in 1, 2, 3 dimensions and for rectangular DOS. For
all DOSes D = 1/4, at U = 0, as in the uncorrelated metal,
(n D)(1+D n n )+D(n D)
although it is a ECFL here. The Mott transition takes place q = σ − − σ− σ¯ σ¯ − . (9)
σ
n (1 n )
at U = W/2. Note different critical behavior in 1,2 and 3 d. σ σ
−
Thecurvecalculatedfor1dcoincideswithknownexactresult
[17] for rescaled U →U/2. Minimizing the average energy with respect to D, at
half filling, we obtain a Gutzwiller-Brinkman-Rice tran-
sition(GBR) at a finite U = 4ǫ , where ǫ is the
c 0 0
− | |
ontothemodelH onbipartitelattices,viathecanonical summedgroundstateenergyofthenoninteractingFermi
c
transform U = exp[ iπ R n n ], where R = 1 liquids. The GA density of doubles on the metallic side
on different sublatti−ces,Pasj shjowj↑n ji↓n [16]. Anjalyti±cal is linearly dependent on the interaction U, reducing to
ground states of the symmetric bond-charge model were zero at the transition point:
obtained in [15] in the regime U > 2W at half filling,
1
for arbitrary dimensions d. These results were improved D = 1 U/U , U U . (10)
c c
4(cid:16) − (cid:17) ≤
upon in [17] revealing that a metal-insulator transition
occurs atU =W, probably with ferromagneticpolariza- The GBR transition is of a quantitatively different
tion on the metallic side (for d>1), which is supported character than the obtained H-I transition. Indeed, the
by numerical evidence [18]. distinct points are that the transition point is lattice de-
The special case of d = 1, was exactly solved in [17] pendent, while the metallic side has lattice independent
leading to the following main results: the metal to in- double occupancies (which seem to be doubtful in the
sulator transition occurs at U = W, and also the num- light of some exact results summarized above). As an
c
ber of double occupancies is D = 1/2πArcCos(U/U ). aside,notehoweverthat,thechoiceofthereferencestate
c
In d = 1, the H-I approach used here underestimates as noninteracting uncorrelated Fermi seas is of doubtful
the transition point, but remarkably the average num- applicability,in the Gutzwiller analysis,ofthe model H
c
ber of double occupancies calculated for the 1-d DOS given that it certainly does not even correspond to the
ρ(ǫ) = 1/(π√1 ǫ2), using Eq.(6) leads to exactly the ground state of a correlated hopping Hamiltonian any-
−
same function and prefactor as the exact solution (with where, including in the limit U = 0. This is analogous
U = W/2 now being the H-I critical value). This sug- to the Gutzwiller study of the bond-charge Hamiltonian
c
gests,thattheH-Iworksparticularlywellin1dimension. Eq.(7)performedin[20],wheretheresultsforlargebond
Furthermore, exact results away from half-filling n = 1 charge interaction X cannot be considered reliable. In-
reveal,thatthereisacriticalboundaryseparatingstates deed, in particular at the symmetry point (which is of
withdoublyoccupiedsitesfromstateswithnosuchsites, direct relevance to the model H ), X = t , Kollar
c ij ij
−
which in the H-I framework considered here corresponds and Vollhardt in [20] obtain a critical point U = 0 us-
c
to the Lishitz lines U depicted in Fig.1. The exact ing the GA. Thus, rather interestingly, we observe that
L
boundary is given by the relation U =W cos(π(1 n)) the Gutzwiller approximation yields a qualitatively bet-
L
−
[17]. Notice that, apartfrom the value of the Mott tran- ter picture of the phase transition itself in the model
sition point, the Lifshitz line obtained using the H-I ap- H than in the bond-charge Hamiltonian. Finally, the
c
proachhereisingoodquantitativeagreementwiththese GAshowsaferromagneticinstability inthe bond-charge
results (see Fig.1). In general dimensions, the expected model, for certain lattices [20] before the Mott insulator
phasediagramisexpectedtobequalitativelysimilar[19] transition. However, this is not the case for the GA in
4
theH Hamiltonian. Indeed,asusual[6,20]onecancal- macroscopicseparationofthe system into two ferromag-
c
culate the bulk magnetic susceptibility χ, which is here netic domains,Eqs. (13, 14)showthatthereis norenor-
given by: malizationofthebandwidthintheHubbardbands,and
thus U = W. As all Mott states have the same energy,
1 (U/U )2 c
χ−1 = − c (1 ρ(ε )U ) (11) this analysis indicates that the Mott phase is unstable
F c
2ρ(εF) − already at Uc = W (which is in good agreement with
known results summarized earlier), and changes proba-
where ρ(ε ) is the density of states at the Fermi sur-
F
bly into a ferromagnetic metallic state.
face. OnlyonefactordependsonU andcandivergehere,
Conclusions and Outlook In this work, we have an-
accompanying the metal insulator transition U U .
c
→ alyzed the Hubbard I approximation, explicitly showing
Thus,the GA,likethe H-I calculation,doesnotdescribe
that its known drawbacks originate from the interband
a ferromagnetic metal before the Mott transition. How-
hopping T terms in the Hubbard model. We proposed
ever, apart from this the phases and critical points are ±1
to use the H-I approach in conjunction with perturba-
better described by the H-I approach.
tionalexpansion. TheH-Iapproach,incombinationwith
TheRoth–correctedapproach Wehaveshownthatthe
lowest order perturbational expansion, leads to a physi-
Hubbard-I approach to the model H gives the correct
c
cally appealing, picture of the Mott transition including
scale of energy for the Mott transition. For general di-
the appearance of a extremely correlatedFermi liquid in
mensions,however,thecriticalpointU doesnotdepend
c
its vicinity, which is complementary to the Gutzwiller-
on magnetic polarization, since the gap is always given
Brinkman-Rice picture.
by U W/2. Recall, that in obtaining the descriptionof
− It is natural to wonder how well this approach fits as
the Mott transitionwe neglectedterms, in the equations
a description of Mott transitions and the surrounding
of motion, related to magnetic order and other terms.
ECFLinrealisticstronglycorrelatedelectronmodels. In
All these terms can be treated in a unified manner by
thisregard,weemphasizethatthedoubleoccupancycon-
the procedure invoked by Roth [21], which removes am-
serving Hamiltonian H analyzed here is equivalent to
biguitiesinthedecouplingofGreen’sfunctionsequations c
the Hubbardmodelwith bondchargeinteractionsatthe
of motion method. Applying the Roth procedure at the
symmetry point X = t – which has been argued to be
Hubbard-I level of equations of motion for the consid- −
aquiterealisticvalue(seee.g. [15,17]). OurH-IorRoth
ered Hamiltonian H , we obtain the following Green’s
c
improvedH-Icalculationscomparequitefavourablywith
functions:
knownandexpectedresultsforthelattermodel,andare
Γσ(ω)= 1 nσ¯ , Γσ(ω)= 1 hσ¯ (12) remarkablyconsistentwiththemin1dimension. Indeed,
k k
2πω−Eσ e 2πω−Eσ othneenweoiguhldboeuxrpheocotdtohfetphiectsuyrmemperetrsyenpteodinth.erOentothheooldtheinr
where, the quasi-particle energy factors are e
hand,forpureHubbardlike–systems,i.e. X closeto0,of
coursethisstrongcouplingpictureoftheMotttransition
Eσ =(U µ)+ǫk n0σ¯nδσ¯ /nσ¯
− h i canonlybequalitativelycorrect(whenantiferromagnetic
−ǫk(hTDi+hTXi)/nσ¯ +(hTUHBi−hTLHBi)/nσ¯, (13) orderissuppressed)andthattooonlynearthetransition
Eσ = µ+ǫk h0σ¯hδσ¯ /hσ¯ point. Thisgivesrisetoanopenquestionconcerningthe
− h i
possibility of obtaining an interpolating scheme between
ǫk( TD + TeX )/hσ¯ +( TUHB TLHB )/hσ¯. (14)
− h i h i h i−h i these two extreme cases.
TofhientteerrcmhanhTgXeiof≡sphinc†0sσ¯bceδtσ¯wc†δeσenc0nσeiigdhebscoruibriensgtshiteesp,rwocheisles tioInnalthgirsourengdarsdta,tweefoprrsoypsotesemtsoexuhseibtithiengfomlloewtailntgovMaroiatt-
hTDi ≡ hc†0σ¯cδσ¯c†0σcδσi describes the process of doublon insulator transitions:
transfer. The averagesin Eqs. (13, 14) are yet to be de- Ψ =exp[ αTˆ ]exp[ ηDˆ]Ψ =exp[ αTˆ ]GWF ,
0 0 0
terminedquantities,whichisastandardfeatureofRoth’s | i − − | i − | i
(15)
method. Notice, that disregarding the second lines in
Eqs. (13, 14), andconsideringsite occupationsasuncor- where α,η are variational parameters and Ψ is an
0
| i
related, one recovers the Hubbard I results. appropriate reference state. This state is an extension
AfullanalysisoftheRothsolutionsshallbeconsidered of the standard Gutzwiller Wave Function (GWF) and
elsewhere [22], while here we only consider implications should provide an improved description of the metal-
forthe Mottphaseinwhich T , T , T , T lic side of the transition. Note that the two exponen-
X D UHB LHB
h i h i h i h i
vanish identically for the considered model H . The clo- tial terms commute and together form the exponential
c
sureofthespectralgaptoexcitationsindicatestheMott of αH (U = η/α), which may be viewed as a partial
c
−
phase instability. Consider the case half-filled case with projection on to the ground state of H (U =η/α). This
c
n = n = 1/2. If there are no intersite correlations in makes a connection with the ECFL properties described
↑ ↓
the Mott states, U = W/2 as indicated above. On the in this paper and allows for dressing of the GWF with
c
other hand for saturated ferromagnetic correlation, i.e. precursorsoftheHubbardbands. Itisworthmentioning
5
here that a similar function (containing hopping only of
the lower Hubbard band in T ) has already been used
0
in [23] and has provided excellent results in comparison
withexactresultsfor2electronsontheHubbardsquare.
Acknowledgements We thank Roman Micnas for
many interesting discussions.
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