Polaron Crossover and Bipolaronic Metal-Insulator Transition in the half- filled Holstein model

0.17 MB·English

by  M. Capone

Polaron Crossover and Bipolaronic Metal-Insulator Transition in the half-filled
Holstein model.
M. Capone
Enrico Fermi Center, Roma, Italy
S. Ciuchi
Istituto Nazionale di Fisica della Materia and Dipartimento di Fisica
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Universit`a dell’Aquila, via Vetoio, I-67010 Coppito-L’Aquila, Italy
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(Dated: February 2, 2008)
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Theformation of afinitedensitypolaronic state isanalyzed inthecontextoftheHolstein model

n using the Dynamical Mean-Field Theory. The spinless and spinful fermion cases are compared to
a disentangle the polaron crossover from the bipolaron formation. The exact solution of Dynamical
J Mean-Field Theory is compared with weak-coupling perturbation theory, non-crossing (Migdal),

6 and vertex correction approximations. We show that polaron formation is not associated to a
metal-insulator transition, which is instead dueto bipolaron formation.

]
l PACSnumbers: 71.38.-k,71.30.+h,71.38.Ht,71.10.Fd
e
-
r
t
s Recentexperimentsstronglysuggestthattheelectron- In order to disentangle the polaron effects from bipo-
.
t phonon (e-ph) interaction is relevant in many materi- laronformation,wecomparethespinlessfermioncase,in
a
als, ranging from high T superconducting cuprates [1], which bipolarons are forbidden by Pauli principle, with
m c
to colossal magnetoresistance manganites [2], from the thespinfulcase,whichhasbeenextensivelystudiedinthe
-
d fullerenes [3] to magnesium diboride [4]. While the spe- recent past [14, 15, 16, 17, 18]. To be more explicit, we
n cific role of the e-ph interaction is certainly very differ- showthatthepolaroncrossoveratfinitedensityisnotby
o entfromcompoundtocompound,thepropertiesofthese itself a metal insulator transition (MIT), and insulating
c
materials are hardly explained in terms of standard ap- behavior can only be associated to localized bipolarons.
[
proachestothee-phproblem,liketheMigdal-Eliashberg This effect shows up in the non-vanishing of both the

2 theory, and pose a serious challenge to theories. quasiparticle renormalization factor, and of the renor-
v
malized phonon frequency in the spinless case. In the
7 More specifically, polaronic features have been ob-
spinfulcasethequasiparticleweightvanishesatsomedef-
7 served in lightly doped cuprates [5], in the manganites
6 inite coupling while the renormalized phonon frequency
[2], up to some indication in the fullerenes [6]. A small
4 remains finite.
polaronis a carrierso tightly interactingwith the lattice
0
We consider the Holstein molecular crystal model, in
3 that its effective mass is strongly enhanced therefore re-
which tight-binding electrons interact with local modes
0 ducing its mobility[7]. The single polaron problem has
/ beenextensivelystudiedallowingtounderstandindetail of constant frequency. The Hamiltonian is
t
a the polaron physics [8, 9]. Nevertheless, the experimen-
m talfindings ofpolaronic effects obviously dealwith finite H =−t X c†i,σcj,σ+H.c.−gXni(ai+a†i)+ω0Xa†iai,
- densitiesofcarriers,andpromptforananalogousunder- hi,ji,σ i i
d
standingofthefinitedensitypolaronproblem,wherethe (1)
n
o polarons are able to interact and to change drastically where ci,σ (c†i,σ) and ai (a†i) are destruction (creation)
c the phonon properties. In this regard,it is important to operators for fermions and for phonons of frequency ω0,
:
v stress the distinction between polaronic and bipolaronic ni the electron density, t is the hopping amplitude, g is
i states. If repulsion between the electrons is neglected, ane-phcoupling. Thedensity isfixedton=0.5(n=1)
X
two polarons (with opposite spin) tend in fact to bind, inthespinless(spinful)system,whichcorrespondstothe
r
a giving rise to a (local) pair, which is called bipolaron. particle-hole symmetric half-filled case. In analogy with
The bipolarons in turn may undergo a superconducting thestudiesoftheMotttransitionintheHubbardmodel,
transitionoftheBose-Einsteintype[10]. Ifsuperconduc- inwhichantiferromagnetismisneglected[19],werestrict
tivityisnotallowed,bipolaronsgiverisetoaninsulating ourselves to the state with no charge order [12]. Despite
stateoflocalizedpairs[11,12]whichmayeventuallycon- in the particle-hole symmetric case the ground state is
dense in a charge-ordered state [13]. In some important always ordered, the homogeneous solution may become
compoundsbipolaronicstatesareindeedunlikelyformed. representative of the groundstate whenever charge or-
Incupratesandfullerenes,thestrongelectroncorrelation dering is spoiled by some frustration effect like a next-
forbidsbipolaronformation,whileinthemanganites,the nearest-neighbor hopping. Moreover, this study allows
double-exchange mechanism favors ferromagnetic states to characterize how strong e-ph interaction may lead to
atlowtemperature,inwhichnobipolaroncanbeformed. the destruction of the metallic state (just like electron
2
correlation leads to the Mott state). The adiabatic ra-
tio γ = ω0/t has been shown to be an important pa-
rameter for the single polaron formation. In the adia-
batic regime γ < 1, the polaron crossover occurs when
λ=g2/ω0t 1, while in the antiadiabatic regimeγ >1,
the conditio≃n is instead (g/ω0)2 1 [8, 9]. (a) (b)
≃
We solve the model using the Dynamical Mean-Field
Theory(DMFT),anon-perturbativeapproachwhichbe-
FIG. 1: The NCA [(a)] and the VCA [(a) + (b)] diagrams
comes exact in the limit of infinite dimensions [19]. We
for electron (upper graphs) and phonon (lower graphs) self-
notice that in sucha limit the spinless fermion case does
energies. Wavy lines are phonon propagators while straight
not coincide with the infinite correlationlimit. Nonethe-
linesareelectronpropagators. Secondorderperturbationthe-
less, this system represents an instructive playground oryisobtainedreplacinginternallineswithfreepropagators.
where polaronic effects can be observed without too
many competing phases. In DMFT, the lattice model
is mapped onto an impurity problem subject to a self- self-energy, but the difference between this quantity and
consistency condition, which contains the information the result from second order perturbation theory. Zero
about the lattice. In our case the impurity model is temperature results are obtained by lowering the tem-
peratureuntilthephysicallyrelevantquantitiesconverge
H = −XVkc†k,σfσ+H.c.+XEkc†k,σck,σ (consistentlyincreasingthenumberofMatsubarafreque-
k,σ k,σ cies Nmax = 6/(2Tπ)). We checked the convergence to
− g(a+a†)Xσ fσ†fσ+ω0a†a, (2) TZ =defi0nebdy amsoZni−to1ri=ng(t1h−e q(Σua(siωipna=r1ti)cl−e Σsp(eiωctnr=a0l)w/2eπigTh)t
where ω = (2n 1)πT. The vanishing of Z is used
n
−
where the phonons live only onthe impurity (f)site, E within DMFT to characterize the Mott MIT in the re-
k
and V are the energy levels and the hybridization pa- pulsive [19], and the pairing transition in the attractive
k
rameters of the conduction bath. For the z-coordination Hubbardmodels[22]. AvanishingZ hasbeenfoundalso
Bethe lattice of half-bandwidth D = 2t√z = 1 the self- in the spinful Holstein model [12]. The convergence to
consistency equation in the z limit is T =0 turns out to depend onboth γ (as detailed below,
→∞
westudyγ =0.1andγ =1asrepresentativeofadiabatic
D42G(iωn)=XiωnVk2Ek. (3) aVnCdAnoTn/atd=iab3atic1r0e−g3imiesss,urffiescpieencttivteolyg)etanrdesλul.tsWreitphrien-
k − ×
sentative of the groundstate for weak/intermediatecou-
In this workwe use ExactDiagonalization(ED) to solve pling. Inthe spinful caseforγ =0.1T =10−3 isinstead
the impurity model (2) [20]. This method requiresto re- necessary since the polaron crossover is approached for
strict the sum in Eqs. (2,3) to a finite small number of smaller coupling (see below). Within NCA it is possible
levels N 1. The discretized model can then be solved to span the strong coupling regime, making an extrap-
s
−
at T = 0 using the Lanczos method. The convergence olation to T = 0 necessary in the adiabatic regime for
is exponential in N , and a few levels are enough to give λ>1(λ>0.5)inthespinless(spinful)case. Inthenon-
s
converged results. Results are obtained taking typically adiabaticregimetheextrapolationisrequiredforλ>1.6
N = 10, having checked that no significant change oc- (λ>1.2) in the spinless (spinful) case.
s
curs for larger N [21]. Error bars can be evaluated by Wefirstdiscussthespinlesscase. ExactDMFTresults
s
linearly extrapolating (in 1/N ) to N , and are of forZ areshowninFig. 2. Thelogarithmicscaleonthey-
s s
→ ∞
the size of symbols in the figures. We compare the ED axisevidencesthatinbothcasesZ,evenifexponentially
solutionofDMFT,withtwoapproximateschemes: aself- reduced, does never vanish by increasing λ, indicating
consistent Non Crossing Approximation (NCA) [11, 15] that no MIT is taking place. It is important to observe
(diagrams(a)inFig. 1)andaself-consistentapproxima- thatZ increaseswhenthetruncationerrorisreducedin-
tion including the first Vertex Correction beyond NCA creasingN . ThecomparisonwithNCAandVCAshows
s
(VCA) [15] (diagrams (a) and (b) in Fig. 1). Notice non trivial tendencies. Both approximations are accu-
that in the standard Migdal-Eliashberg approximation rate at weak coupling, but VCA remains closer to ED
the phonon spectrum is not self-consistently evaluated, forrelativelylargecouplingevenintheadiabaticregime,
but it is taken “from experiments”, while in our NCA where the Migdal approximation would be expected to
andVCAthephononself-energyisself-consistentlyeval- hold. Quitesurprisinglyeveninthenon-adiabaticregime
uated(onthe imaginaryfrequencyaxis)throughaniter- VCA improves NCA only at weak coupling. At strong
ative scheme, just like the electron self-energy. Actually, coupling NCA predicts a polaronic crossover,in qualita-
to allow for a safer truncationon the frequency axis, the tive agreement with exact results, even if, for γ =1, the
quantity which is iteratively determined is not the full crossover coupling is strongly underestimated by NCA.
3
1 1
0.1
0
Z Ωω/
0.01
2PT 2PT
NCA NCA
VCA VCA
ED ED
0.001 0.1
1
1
0
Z 1 Ωω/
2PT
0.1 0.1 NCA
VCA
0 1 2 3 4 5 6 ED
0.1

0 0.5 1 1.5 2 0 0.5 1 1.5 2
λ λ
FIG.2: Spinlessfermions. ThequasiparticleweightZ within FIG. 3: Spinless fermions. The renormalized phonon fre-
secondorderperturbationtheory(2PT),NCA,VCA,andED quencyΩ/ω0. Notations are the same of Fig. 2
for γ =0.1 (upperpanel) and γ =1 (lower panel) as a func-
tion of λ. The arrow in the upper panel marks the MIT for
γ = 0 [16]. The inset shows ED and NCA in a larger range
NCAandVCAdeviatefromEDislowerinnonadiabatic
of λ for γ =1 to makethepolaron crossover visible.
than in adiabatic regime. This “non uniform” behavior
is not present for a single polaron [9], and represents a
first peculiarity of the finite-density case.
On the other hand, VCA drastically diverges from ED,
Nowwecompareourfindingsforspinlessfermionswith
and gives unphysical negative mass renormalization at
the spinful fermion case. In the adiabatic regime a MIT
strong coupling both for γ = 0.1 and 1, in agreement
has been found around λ λ 0.68 [17], close to
MIT
withthedivergenceofvertexcorrectionspredictedinRef. ≃ ≃
the value (λ = 0.664) at which the density of state at
[11]. Thecouplingatwhichtheapproximatemethodsde-
Fermienergyvanishesintheadiabaticlimit(γ =0)[16].
viate fromexact results decreases with increasing γ, and
This transition has a precursor in the phonon softening
itis unrelatedwiththe polaroncrossovercouplingwhich
[12] but, contrary to the spinless case, the spin degrees
is instead larger for γ = 1, as shown in the inset of Fig.
of freedom prevent electron coherent hopping at strong
1, and in agreement with the single polaron case [8, 9].
coupling through a Kondo-like mechanism [23]. A MIT
Animportantdifference betweenthe finite-density sit- has been claimed to occur also within NCA [11].
uationandthesinglepolaronproblemisthatmanyelec- InFig. 4 we show results forγ =0.1. The logarithmic
tronsareabletorenormalizethephononproperties. The plot clearly shows that Z vanishes faster than exponen-
renormalized phonon frequency can be obtained from tial, signaling a MIT at λ 0.76, in agreement with
the phonon propagator D(iωn) = R0βdτeiωnτhT(a0(τ)+ Ref. [12]. Contrarytothespi≃nlesscase,hereZ decreases
a†0(τ))(a0+a†0) , as (Ω/ω0)2 = 2/ω0D(iωn =0)−1. As by increasing Ns. On the other hand, Ω/ω0 decreases
i −
it is shown in Fig. 3, Ω/ω0 never vanishes as a function with λ as in the spinless case, suggesting that, even if
of the coupling, but rather exponentially decreases. The stronglysoftened,the phononmode neverbecomes com-
way NCA and VCA results for Ω compare with ED is pletely soft. This behavior is more evident for larger
analogous to the results for Z. In the adiabatic regime phonon frequency γ = 1, as reported in Fig. 5. Also
NCA strongly overestimates Ω/ω0 around the polaron in this case Z vanishes much faster than exponential at
crossover. In the nonadiabatic regime NCA again pre- λ = 1.44, where the MIT takes place, while the phonon
dicts a phonon softening which does not actually occur renormalization is much less effective and leads to quite
at such small values of the coupling. In the same regime a large value of Ω/ω0 even at the MIT point (see inset).
VCA improves NCA result only at weak coupling and The comparison with approximate schemes qualita-
givesa phononhardeningat strongcoupling. It is worth tively resembles the spinless case. NCA underestimates
to note that for both Z and Ω the coupling at which Z for λ below the MIT, but it gives a finite weight also
4
above the exact MIT point. Our extrapolation to T =0
fromfinitetemperaturesdoesnotallowustodefinitively 1
rule out the existence of a MIT within NCA even if, in
contrastwithaclaiminRef. [11],Z seemstodecreaseex-
ponentiallywithλwithinthisapproximation[24]. Again
0.1
VCAgivesbetterresultsthanNCAevenintheadiabatic 1
regime up to intermediate coupling λ 0.55, but it be- Z
≃
comes completely unreliable at strong coupling.
We have studied the formation of a polaronic state in 0.01 0.1
2PT
theHolsteinmodelathalf-fillingwithinDMFT.Forspin-
NCA
lessfermions,a continuouscrossoverleads toa polaronic VCA
0.01
state by increasing the coupling constant. Despite the 0 0.4 0.8 1.2 1.6 ED
0.001
electroneffectivemassbecomesexponentiallylargeinthe 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
λ
strong-couplingregime,thegroundstateisalwaysmetal-
lic. The crossover is more abrupt in the adiabatic case.
In the spinful case, the polarons can bind to form bipo- FIG. 5: Spinful fermions. Z and Ω/ω0 (inset) for γ =1.0 as
larons, leading to a real MIT. The phonon renormaliza- a function of thecoupling.
tionismuchstrongerinthe adiabaticregimethaninthe
nonadiabatic case, but the phonons do not become com-
pletely soft at the MIT. Approximate treatments (NCA [5] P. Calvani, et al.,Phys. Rev.B 53, 2756 (1996)
and VCA) strongly deviate from exact DMFT above a [6] M. Ricc`o, et al., Phys.Rev.B 67, 024519 (2003).
couplingwhichdiminisheswithincreasingω0/t. Atweak [7] Herewewill onlybeconcernedwith smallpolaronsaris-
couplingVCAcorrectlyreproducesthequalitativetrends ing from Holstein-like interactions. For the definition of
of exact results, and improves on NCA. other kindof polarons see, e.g., J. T.Devreese, in Ency-
clopedia of Applied Physics (VCH,Weinheim (1996)).
M. C. thanks the hospitality and financial support of
[8] M.Capone,W.Stephan,andM.Grilli,Phys.Rev.B56,
the Physics Department of the University of Rome “La
4484(1997);S.Ciuchi,F.dePasquale,S.Fratini,andD.
Sapienza”,andofINFM,Unit´aRoma1. Weacknowledge Feinberg, ibidem 56, 4494 (1997).
financial support of MIUR Cofin 2001 and enlightening [9] M. Capone, C. Grimaldi, and S.Ciuchi,Europhys.Lett.
discussions with C. Castellani. 42, 523 (1998).
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Z Rev.B48,6302(1993); J.K.Freericks,ibidem48, 3881
0.0001
(1993).
0.1
[15] J. K.Freericks, Phys.Rev.B 50, 403 (1994)
1e-05 2PT
[16] A. J. Millis, R. Mueller, and B. I. Shraiman, Phys. Rev.
1e-06 0.01 NVCCAA [17] PB.5B4e,n5e3d8e9tt(i19a9n6d).R. Zeyher, Phys. Rev. B 58, 14320
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FIG. 4: Spinful fermions. Z and Ω/ω0 (inset) for γ =0.1 as Rev. Mod. Phys. 68 13 (1996).
a function of λ. The arrow marks theMIT for γ =0 [16]. [20] M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545
(1994).
[21] WealsotruncatethephononHilbertspaceintroducinga
cut-offinthenumberofphononsNph,andalwayscheck-
ing for convergency. The truncation is more delicate in
the adiabatic regime, where a large number of phonons
can be excited. When necessary, we used Nph =50.
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5
is hardly identified using a linear scale as in Ref. [11].