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Excitonic Phase Transition in Electronic Systems

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Excitonic Phase Transition in Electronic Systems Mucio A.Continentino Instituto de F′?sica,Universidade Federal Fluminense Campus da Praia Vermelha,Niter′o i,24.210-340,RJ,Brazil and National High Magnetic Field Laboratory Florida State University,1800E.Paul Dirac Dr.Tallahasse,Fl 32306,USA Gloria M.Japiass′u Instituto de F′?sica,Universidade Federal do Rio de Janeiro Caixa Postal 68.528,Rio de Janeiro,21.945-970,RJ,Brazil Am′o s Troper Centro Brasileiro de Pesquisas F′?sicas Rua Dr.Xavier Sigaud 150,Rio de Janeiro,22.290-180,RJ,Brazil

Abstract

Although predicted theoretically excitonic phase transitions have never been observed.Recently it has been claimed that they can occur in some strongly correlated materials.We study the possibility of an excitonic transition in a two-band model and show that a true phase transition never occurs in the presence of hybridization.We suggest an

alternative interpretation for these experiments based on the opening or closing of a hybridization gap at a critical pressure.

PACS:71.30+h;71.28+d;71.35+z;75.30Mb;68.35Rh

Recently Wachter and colaborators[1,2]have claimed to observe a tran-sition to an excitonic insulator in a doped narrow-band-gap semiconductor at moderate pressures and in a strongly correlated metal[2].The evidence for this transition is obtained by monitoring the resistivity close to the metal-insulator transition as a band-gap is continuously closed or opened by ex-ternal pressure.The resistivity at low temperatures showed a huge peak as a function of applied pressure which they attibuted to the long predicted excitonic phase transition[3].They also presented data of the Hall constant which reveals that this resistivity anomaly is caused by a reduction in the number of carriers.

With renewed interest in this subject motivated by these challenging ex-periments[1,2]and the close connection of the materials which have been investigated with Kondo insulators[4],we study the possibility of this exci-tonic transition in presence of hybridization in a two-band model.We point out that no sharp excitonic phase transition occurs when external pressure is applied.The reason is that the hybridization which depends on pressure acts as a conjugate?eld to the order parameter of the excitonic phase and destroys this transition[5].The e?ect is similar to that of a ferromagnetic system in the presence of an uniform external magnetic?eld.The mixing term however is strongly renormalized by excitonic correlations.We suggest an interpretation for the experiments of Wachter et al.on Sm0.75La0.25S [2]on the basis of a metal-insulator(MI)transition associated with the ap-

pearance of a hybridization gap at a critical pressure.This metal-insulator transition is in the universality class of the density-driven transitions,for which the relevant critical exponents have been obtained previously[5].This seems to hold even in the presence of correlations,as we?nd here,as long as the MI transition is not accompanied by the appearance of long range magnetic order[5].A characteristic feature of this transition is that the gap opens or closes linearly with pressure near the critical pressure.This is a direct consequence of the fact that the gap exponent[5]assumes the value νz=1.

The Hamiltonian which describes our system is:

H= k?a k a?k a k+ k?b k b?k b k+ k V k(a?k b k+b?k a k)? k,k′,q G(q)a?k+q a k b?k′?q b k′(1) where?a k and?b k represent the energies for electrons in the narrow a?band and in the large conduction b?band,respectively.The operators a?k,a k create and destroy electrons in the narrow band and b?k,b k are creation and annihilation operators of electrons in the wide conduction band.V is the mixing term, which arises from the crystalline potential and G is the e?ective attractive interaction between spinless electrons and holes[6].For the situation we are interested this is the most important interaction.Note that we should have taken into account in Eq.1,the Coulomb repulsion between the electrons in the narrow a-band.Since,however,as we argued before[5]these interactions are irrelevant,in the renormalization group sense,for the metal-insulator transition we are going to study we do not consider it for simplicity.In fact

within the approximation which is generally used to treat the Hamiltonian above[3]and that we also use in this Letter,these interactions merely shift the energy of the a-electrons.

We point out that we adopt here a di?erent approach to the excitonic problem[3].Instead of starting with two hybridized bands and the electron-hole attraction[3],we include explicitly the hybridization in the starting Hamiltonian to imply that it should be treated together with the many-body correlations and not separetely.The main di?erence is that within our approach the gap which eventually opens or closes as will be shown below and is experimentally acessible turns out to be a truly many-body quantity. In the standard treatment[3]where the bands are diagonalized separately from the correlations one arrives at the arti?cial situation where the gap is a simple one-body e?ect.

If it were not for the attractive term,the Hamiltonian given by Eq.1 could be exactly diagonalized giving rise to two hybrid bands[7].However the many-body term due to the e?ective attractive interaction G makes this a di?cult problem for which an approximation must be introduced.We shall employ the Green’s function method[8]to obtain the order parameter associ-ated with the excitonic phase,namely,?= k b?k a k .When calculating the

equation of motion for the Green’s function a k;b?k ωwe?nd it gives rise to new propagators.Introducing a a convenient mean-?eld approximation[3]:

a k?q b?k′?q

b k′;b?k ω≈ a k?q b?k′?q b k′;b?k ω

a?k′+q a k′b k′+q;b?k ω≈? a?k′+q b k+q a k′;b k ω.(2) we obtain a closed set of equations which can be solved to yield

a k;b?k ω=?V k

2{?a k+?b k±

ω1(k)?ω2(k){1

ω?ω2(k)}(6)

from which we obtain

?k=?V k

The quantityα(α<1)may be interpreted as taking into account the di?er-ent e?ective masses of the electrons in the narrow a?band and the large b?band,i.e.(m b/m a)=α.The quantityβgives the shift of the narrow band with respect to the large band.

Now we introduce two new functions g1(ω)and g2(ω)[7]through the following equation

[ω?ω1(k)][ω?ω2(k)]=α[g1(ω)??k][g2(ω)??k](8)

from which we get

g2,1(ω)=1

[(α?1)ω+β]2+4α?V2}(9)

The energies of the bottom of the hybrid bands correspond to g i(E i B)=0 with i=1,2

E2,1B=1

2}(10)

while the energies of the tops are obtained when g i(E i T)=D,where D is the bandwidth of the large conduction b?band,

E2,1T=D

D±[((α?1)+

)2]1

α dω

where N(ω)= kδ(ω??k)and

|g1(ω)?g2(ω)|=12.

We also obtain an expression for the gap?G between the two bands as a function of the hybridization V and the electron-hole interaction G.It corresponds to the di?erence in energy between the top of the?rst hybrid band E1T and the bottom of the second E2B:

2{[(α?1+β

)2]1

)2+4(

2?(1+α)}.(12)

Consequently for a two-band system the opening of a hybridization gap, contrary to what occurs for the Anderson lattice model[10],requires a critical value of renormalized hybridization ?V

D c=1D(α?1)]2D)2}1

D c we?nd?G

D ?

D ? ?V

variable here is ?V

D,such thatα=1andβ=D

.(14)

G?G c

Then there is a critical value for the electron-hole attraction G c,G c=D,for which the excitonic order parameter is di?erent from zero even in the absence of hybridization.This is formally similar to a Stoner-like criterion for the appearance of ferromagnetic order in a metallic system.In the present case, the hybridization plays the role of the uniform magnetic?eld conjugate to the order parameter?.The fact that a k-independent,local hybridization V acts as the conjugate?eld of the excitonic order parameter,independently of any particular approximation,can be directly seen from the Hamiltonian, Eq.1,with the one-body mixing term written as V k(a?k b k+b?k a k)=V?. This shows that V couples directly to?.For the case of d?f hybridiza-

tion there are di?erent mechanisms,for example phonons,which can provide

local mixing between these orbitals with di?erent symmetries[13].Conse-

quently we do not expect a sharp phase transition to an excitonic phase to

occur whenever an external parameter which changes the hybridization,as

external pressure for example is varied.There is no singularity in any phys-

ical quantity when G approaches G c,whenever V=0,contrary to what the

simple mean-?eld equation above suggests.However we expect strong renor-

malization of the physical quantities,as the hybridization,due to excitonic

correlations for G≈G c.The only zero temperature phase transition which

can occur in our model as pressure is varied is a metal-insulator transition

associated with the opening of a hybridization gap at a critical value of the

ratio(V/D)c.For the divalent semi-metal discussed above this occurs at √3/4)?(G/D)?],where due to the excitonic correlations this (V/D)c=[(

value is renormalized with respect to that of the non-interacting system[5].

For completeness we give the self-consistency equation for the renormal-

ized hybridization for generalαandβand square bands

V=?V 1?G B| (15) with

A(α,β)=E2B(1?α)?β+ E2B(1?α)?β 2+4α?V2 1

with E1B and E2B given before.We notice that this is independent ofμand forβ=D/2and in the limitα→1it gives correctly the previous equation for?in the symmetric,divalent semi-metal case.In the limitα→0,it reduces to the expression for the renormalized hybridization in the Anderson lattice,i.e.,for a collection of local levels[13][14].

We give now the scaling results for the properties of the Fermi liquid in the metallic phase close to the metal-insulator transition.We?nd that the thermal mass m T,de?ned as the coe?cient of the linear term of the speci?c heat scales as m T∝|P?P c|d

2.The behavior of the thermal mass obtained above is opposite to that found in heavy fermions[15],where it is enhanced as the system approaches the critical point.The characteristic or coherence temperature in the metallic phase scales as T c∝(P?P c) sinceνz=1.The existence of a small coherence temperature close to the density-driven phase transition gives rise in the presence of electron-electron interactions to a signi?cant T2term in the resistivity even in wide band materials as Y b[16].If we writeρ≈ρ0+AT2for T<

At the critical pressure,P=P c,we?nd non-Fermi liquid behavior with the speci?c heat vanishing with temperature as C∝T3/2andχ0∝T1/2.In

actual systems,states in the tails of the density of states due to impurities or disorder and which give rise to a saturation of the resistivity at the lowest temperatures may spoil this simple behavior.

In conclusion the possibility of an excitonic phase transition in a two-band model has been investigated.The dominant interaction was taken to be the electron-hole attraction.We argued that a phase transition to an excitonic phase never occurs in the presence of hybridization since the one-body mix-ing term acts as a conjugate?eld to the order parameter of this phase.We suggested an alternative interpretation of the experiments of Wachter et al. on Sm0.75La0.25S.Our approach gives rise to a gap that varies linearly close to the critical pressure,i.e.,?G∝(P?P c)as observed[2,11].We have derived the properties of the Fermi liquid in the metallic phase close to the transition and predicted non-Fermi liquid behavior at the critical pressure.

Acknowledgments:We thank Conselho Nacional de Desenvolvimento Cien-t′??co e Tecnol′o gico-CNPq-Brazil,for partial?nancial support.

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