*Corresponding author.E-mail address: fxd@cyber100.com (X. Feng).

Physica B 291 (2000) 280}284

Slave boson mean "eld study of the periodic Anderson modelwith correlation in the conduction band

Minghui Qiu, Junxia Mu, Xiaobing Feng*

Department of Physics, Dalian Railway Institute, 116028 Dalian, People's Republic of China

Received 9 June 1999; received in revised form 15 November 1999

Abstract

The periodic Anderson model with correlation among conduction electrons is studied in slave boson mean "eldapproximation. The results show that there is a bandwidth narrowing e!ect accompanied by an enhancement of thedensities of states of the two kinds of electrons. The e!ect may play a very important role in some heavy-fermion systemswhich are nearly half "lling of conduction electrons. ( 2000 Elsevier Science B.V. All rights reserved.

PACS: 71.27.#a; 71.10.Fd; 65.40.#g

Keywords: Slave boson mean "eld; HFS; Conduction band correlation

1. Introduction

As a typical strongly correlated electronic sys-tem, heavy-fermion system (HFS) has manyanomalous properties, such as the small antifer-romagnetically ordered magnetic moment [1],non-Fermi liquid behavior [2,3], metal}insulatortransition [4,5] and the coexistence of antifer-romagnetism and superconductivity [6,7], etc. Itis important to study these phenomena in orderto give a thorough understanding of HFSs, andit would help understand the other stronglycorrelated electronic systems. The periodicAnderson model (PAM) and its variants (withgeneralization to multi-conduction bands or SU(N)

symmetry, etc.) have been used in most of thetheoretical works [8,9]. The conduction bandsin HFS are formed by 3d electrons. Apart fromthe degeneracy there are also strong on-siteCoulomb interactions between 3d electrons.The Coulomb correlation between 3d electronsplay an important role in the Kondo insulators[10] and the many anomalous properties ofhigh-temperature superconductors are also causedby the strong Coulomb interaction between 3delectrons of Cu. Using multi-channel Kondo latticemodel, Granath et al. [11] have studied the e!ectof Coulomb correlation in the conduction bandon the electronic speci"c heat. The result showsthat the correlation in conduction band could re-sult in non-Fermi liquid behavior. In this paperusing slave boson mean "eld (SBMF) method, westudy the e!ect of the conduction band correlationon the electronic spectrum and electronic speci"cheat.

0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 3 0 0 6 - 9

o#(u)"G

1

2Db2#

, u3[u2(e8 (!D)), u

2(e8 (D))] or u3[u

1(e8 (!D)), u

1(e8 (D))],

0 others,

(9)

2. The model

We use the following Hamiltonian,

H"+kp

ekpcskpckp#+ip

edf sip f

ip

#

<

JN+kp

(cskp fkp#H.c.)

#;+i

nfit

nfis#;

#+j

ncjt

ncjs

, (1)

in which ckp and fkp represent the annihilation oper-ators of the conduction band electrons and thef-electrons, respectively. ek is the dispersion relationof the itinerant electrons. < is the hybridizationpotential.; and;

#are the on-site Coulomb repul-

sion energy of f-electrons and itinerant electrons,respectively. In this paper we study only the case of;

#";"#R. According to the SBMF method

[12,13] we introduce two slave bosons, bs#, bs

&. In the

Hamiltonian, the annihilation operator, cip and f

ip ,should be replaced by c

ipbsi# and fipbsi& , respectively,

and the following constraint conditions must besatis"ed.

bsi#bi##+

pcsipcip"1, (2)

bsi&bi&#+

pf sipfip"1. (3)

In the SBMF approximation [13], one can obtainthe following equations satis"ed by b

#, b

&and

chemical potential k.

j##

1

N+kp

ekScskpckpT#<b&

b#

1

N+kp

S f skpckpT"0, (4)

j&#<

b#

b&

1

N+kp

Scskp fkpT"0, (5)

in which j#

and j&

are the Lagrangian factors. Inthe SBMF approximation b

i#and b

i&are replaced

by their average values, which is the same for all thelattice sites in the paramagnetic state.

Now, the Hamiltonian is of a single-particle form

HMF

"+kp

e8 kpc skpckp#+kp

e8&f skp fkp

#<I +kp

(cskp fkp#H.c.) (6)

in which, e8 k"b2#ek!k#j

##e

#, e8

&"e

&!k#j

&,

<I "b#b&<. Eq. (6) can be strictly diagonalized. The

Green's functions of the two kinds of electrons canbe written as

G##(k, iun)"

A1

iun!u

1

#

A2

iun!u

2

, (7)

G&&(k, iun)"

B1

iun!u

1

#

B2

iun!u

2

(8)

in which u1,2

"12[e8 k#e8

&$J(e8 k!e8

&)2#4<I 2],

which are the dispersion relations of the two hy-bridized bands,

A1"

e8&!u

1u

2!u

1

, A2"

u2!e8

&u

2!u

1

,

B1"

e8 k!u1

u2!u

1

, B2"

u2!e8 k

u2!u

1

.

Taking the density of states (DOS) of ek as a rec-tangle centered at e

#, with the bandwidth 2D and

the DOS 1/2D. The DOSs of the correspondingtwo kinds of electrons have simple forms in theSBMF approximation:

o&(u)"

<I 2(u!e8

&)2

o#(u) (10)

The general magnetic susceptibility can be ob-tained,

s&&lk(q,u)"k2B<

Tr(plpk)

] +i,j,k

Ai(k)A

j(k!q) [n

F(u

i(k))!n

F(u

j(k!q))]

u#uj(k!q)!u

i(k)#id

(11)

M. Qiu et al. / Physica B 291 (2000) 280}284 281

Fig. 1. The DOSs of the c- and f-electrons with b"10 000,k"0, e

&"!0.4 eV, <"0.25 eV, e

#"0.4 eV. The unit of

DOS is eV~1 per site per spin.

in which nF

is the Fermi distribution function, i andj represent subband indexes. Because the trace ofpl is zero, slk is diagonal and isotropic. Since thereare two subbands due to the hybridization interac-tion the slk has a contribution from the electronhopping between the two subbands. The thermo-dynamic potential X of the system can be obtained,

X"!

1

b+ikp

ln[1#e~b(uikp~k)]. (12)

3. Results and discussions

The DOSs of the two kinds of electrons areshown in Fig. 1. In the SBMF approximation thelifetime e!ect of quasiparticles has been neglected,the DOS has sharp edges at the two sides of thegap. From Eqs. (9) and (10) we can see that the totalwidth of the two subbands is 2b2

#D. It has been

narrowed by a factor of b2#. At the same time the

DOSs have been enhanced by a factor of 1/b2#. The

band gap D between the two subbands formed bythe hybridization interaction may be easily ob-tained,

D"12[J(e8 (!D)!e8

&)2#4<I 2

#J(e8 (D)!e8&)2#4<I 2]!b2

#D. (13)

When < is small enough, the gap is approxim-ately equal to 2Db2

#<2/[(e8 (D)!e8

&) (e8

&!e8 (!D))].

The results show that the band gap is also nar-rowed by the Coulomb correlation in the conduc-tion band. In addition, DJ<2, there is no criticalvalue of hybridization potential <

#for the opening

of the gap. This is di!erent from the models withdispersion relation of the f-electrons [14]. The re-normalized f-electron energy level always lie in thegap. The condition for the Fermi energy k to lie inthe gap (i.e. the system is insulating) is

u2(e8 (D))(k(u

1(e8 (!D)).

In Fig. 1, the k is very close to the lower edge of thegap, and the DOS is large. This would lead to theheavy quasiparticle states. The Coulomb repulsioninteraction among the conduction band electrons

has an enhancement e!ect on the DOSs. Under thechoices of parameters in Fig. 1 the e!ect is evident,the enhancement factor, i.e. 1/b2

#, is about three.

When the conduction band is near half "lling (i.e.,+p Sn#

ipT+1) b2#, which represents the average hole

number at a site, is very small, and the enhance-ment e!ect will be very large. We think that in somecases this mechanism should not be omitted inexplaining the heavy electron behavior of HFS. Ifthe on-site energy e

&or e

#was further lowered, the

k would lie in the gap, and the system becomesinsulating. We "nd that even the parameters havevery small changes we could not solve the b

#, b

&,

j#, j

&numerically. We consider that the metal}in-

sulator transition is accompanied by the paramag-netism}antiferromagnetism transition. In anti-ferromagnetic phase one should introduce di!erentslave bosons on di!erent sublattice sites. In thismodel the enhancement of e!ective mass mH arisesfrom two factors. On the one hand, a DOS peak isformed due to the hybridization of the two kinds ofcarriers and the peak is most enhanced when thesystem is near metal}insulator transition which isaccompanied by a transition into antiferromagneticphase, and this is why HFSs usually have strongantiferromagnetic #uctuations. On the other hand,the bandwidth is reduced due to the conductionband correlation which results in the enhance-ment of DOS and the reduction of the gap widthbetween the two subbands. The conduction band

282 M. Qiu et al. / Physica B 291 (2000) 280}284

Fig. 2. The temperature dependence of the speci"c heatsc("C/¹) of the f-electrons. The parameters are the same as inFig. 1 and the unit of c is mJ/mol K2.

correlation could result in a antiferromagneticphase with very small energy gap, we think that theanomalously small ordered moment in some HFSsmay have relationship with this e!ect.

In SBMF approximation b#, b

&, j

#and j

&are

temperature dependent, so the electronic spectrumis temperature dependent. The thermodynamicproperties, such as electronic speci"c heat C andmagnetic susceptibility s will vary with temper-ature. The C reads as

C"!2kBbPo(E)(E!k)2

RfRE dE, (14)

in which o(E) is the DOS per site and per spin.When the o(E) is taken as a constant at aboutFermi energy, one has the usual result,s/c"3g2k2

B/k2

Bp2, which is temperature indepen-

dent.It is shown in Fig. 2 the temperature dependence

of c("C/¹) of the f-electrons. One can see that thec has a strong temperature dependence due to thevariation of the DOS of the f-electrons with temper-ature. The c can be expressed as c"0.00025¹6 at¹(6 K. This is di!erent from the results due toantiferromagnetic #uctuations because of the weak-ness of SBMF approximation in dealing with mag-netic correlations.

Eq. (6) is of single-particle form, when the #uctu-ations of the slave bosons and the Langrangianfactors are taken into account there would be inter-actions between the quasiparticles. The #uctuatione!ect is very important for studying the magneticproperties of the system. According to Ref. [15] wetake b#(f)

i"db#(f)

i#b#(f), j#(f)

i"dj#(f)

i#j#(f), in

which db, dj represents the #uctuations aroundtheir average value in the SBMF approximation.The Hamiltonian can be decomposed into twoterms, i.e., H

MFand H

FL. H

FLrepresents the contri-

butions from #uctuations, it can be written as

HFL

" +WijX

tij(db#

i#db#s

j)c

ipscjp

#+ip

b&ed(dbs

i#db&s

i)f sip f

ip

#

<

JN+ip

(b&db#i#b#db&s

i)cs

ipfip#H.c.

#+i

j#b#(db#si#db

i#)

#+i

j#db#si

db#i#+

i

j&db&sidb&

i

#+i

j&b&(dbsi&#db

i&)

#+i

dj#iAb#2#+

pcsipcip!1B

#+i

dj&iAb&2#+

pf sip f

ip!1B (15)

In Eq. (15) we discarded the second-order termsof db. Through canonical transformation or usingfunctional integral method when the integrals overdb is performed, one can obtain the following in-teractions between quasiparticles, cs

ipcjpcslpcip ,csipfipf sipcip . The #uctuations can renormalize the

on-site energy and the transfer integral. In addition,the #uctuations can also introduce the interactionamong c and f electrons, and the interaction be-tween c and f on the same sites. The properties ofthe interactions depend on the properties of the

M. Qiu et al. / Physica B 291 (2000) 280}284 283

bosonic #uctuations. One can see from HFL

that c-and f-electrons are directly coupled to the disper-sionless bosons. The #uctuations have similar ef-fects as the optical phonons had on electrons. So farwe have neglected the #uctuations of dj

i, which is

important near the half "lling, meanwhile the e!ec-tive interaction between f-electrons caused by the#uctuations may be neglected because of the b

#b&

factor.

4. Conclusion

Using the SBMF method the PAM with correla-tion between conduction electrons is studied in thecase of ;

#"#R. There is a narrowing e!ect on

the bandwidth of the two kinds of electrons becauseof the correlation among conduction electrons, andthe factor is b2

#. Meanwhile the DOSs are enhanced

by a factor of 1/b2#. This e!ect may become very

important when the system is near half "lling ofconduction band electrons.

References

[1] W.J.L. Buyers, Physica B 223}224 (1996) 9.[2] M.B. Maple et al., J. Low Temp. Phys. 95 (1994) 225.[3] F. Steglich et al., Physica B 237}238 (1997) 192.[4] G. Aeppli, Z. Fisk, Comments Cond. Mat. Phys. 16 (1992)

155.[5] Z. Fisk et al., Physica B 223}224 (1996) 409.[6] A. Amato, Rev. Mod. Phys. 69 (1997) 1119.[7] F. Steglich et al., Physica B 223}224 (1996) 1.[8] M. Lavagna, A.J. Millis, P.A. Lee, Phys. Rev. Lett. 58

(1987) 266.[9] F.B. Anders, M. Jarrell, D.L. Cox, Phys. Rev. Lett. 78

(1997) 2000.[10] M.V. Tovar Costa, A. Troper, N.A. de Oliveira, G.M.

Japiassu, M.A. Continentino, Phys. Rev. B 57 (1998) 6943.[11] M. Granath, H. Johannesson, Phys. Rev. B 57 (1998) 987.[12] S.E. Barnes, J. Phys. F 61 (1976) 1375.[13] P. Coleman, Phys. Rev. B 29 (1984) 3035.[14] M.A. Continentino, G.M. Japiassu, A. Troper, Phys. Rev.

B 49 (1994) 4432.[15] R. Citro, J. Spalek, Physica B 230}232 (1997) 469.

284 M. Qiu et al. / Physica B 291 (2000) 280}284