Adiabatic approximation for localized electrons in periodic Anderson model

Physica C 296 Ž1998. 298–306

Adiabatic approximation for localized electrons in periodic Anderson model E. Kochetov a , V. Yarunin a

a,)

, M. Zhuravlev

b

BogoluboÕ Theoretical Laboratory, Joint Institute for Nuclear Research, Dubna 141980, Russia b Institute of General and Inorganic Chemistry, Moscow 117907, Russia Received 3 January 1997; revised 28 October 1997; accepted 30 October 1997

Abstract The partition function of the periodic Anderson model for an infinite U-term is represented by the path integral over the Grassmann variables for band s-electrons and supercoherent SUŽ2 < 1. variables for localized d-electrons. The effective electron action is obtained for d-electrons with a level E - 0 through the averaging of the thermal distribution over the s-electron trajectories. Due to the spinon–charge separation in the path integral over the d-variables, the low-temperature and spin mean-field approximations are introduced and are shown to lead to the adiabatic approximation suggested earlier. Non-linear dependence of the chemical potential m on the electron concentration n - 1 is established for the narrow s-electron band w < < E < and the Kondo-like temperature behavior is found for n ) 1 in wide w 4 < E < s-electron band. q 1998 Elsevier Science B.V. Keywords: Anderson lattice; Band structure; Hubbard variables; SUŽ2 < 1. coherent states

1. Introduction The properties of mixed valence systems, transition metal compounds, doped fullerites are described by the periodic Anderson model ŽPAM.. The generic feature of this model is the hybridization V between localized Ž d . and band Ž s . electrons q d s H s Ý En di s q V Ž sq i s d i s q d i s si s . y m Ž n i s q n i s . y t

i,s

Ý

d d sq i s s js q U Ý n i n i ,

² i , j:

Ž 1.

i

< : state, t is the where sq i s , s i s are creation and annihilation operators of the itinerant s-electrons in the s s hopping amplitude between the neighbor sites, dq , d are the creation and annihilation operators of the is s correlated electrons in the orthogonal to < ss : local d-orbital, m is the chemical potential. Important realistic parameters of the PAM Ž1. are the half-bandwidth w s Zt Ž Z is the number of nearest neighbor in a given crystal lattice., the single-electron energy E - 0 of the d-orbital and the temperature by1 for a thermal

)

Corresponding author. Fax: q7 09621 65084; E-mail: [email protected].

0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 8 2 7 - 3

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equilibrium state. We assume that the Hubbard energy U ) w and that other interatomic correlations are strong enough to quench out charge and orbital fluctuations from the state of d-occupation and to make d-electrons heavier than s-ones. In this case the adiabatic approximation may be introduced for a d-electron subsystem in Ž1. like for nuclei in the theory of molecules. This idea can be realized via the path-integral representation of the partition function Q H s Sp s, d4 exp Ž yb H . s H DmŽ s. DmŽ d . exp S,

Ž 2.

where Dm is the functional measure in the space of trajectories for s- and d-electrons with boundary conditions d antiperiodical on w0, b x. The motion of ‘slow’ d-electrons is described by the effective action Seff obtained by integrating over variables of the ‘fast’ s-electrons, so that the partition function Q H becomes d Q H s H DmŽ d . exp Seff .

Ž 3.

Evidently, the partition function Q H can always be represented in the form Ž3., whereas the representation d Q H s Sp exp Ž yb Heff .

Ž 4. d Seff ,

holds only for the time-local functional and Eq. Ž4. is referred to as the adiabatic approximation. This idea was suggested in w1x and developed further in w2,3x by the direct calculation of Sp in Ž2. over s-variables with the d-electrons trajectories taken as constants. The Hubbard-like and Kondo-like behavior of d-electrons for certain values of E, w and by1 was shown to occur, while the U-term was assumed to be responsible for the quenching of d-electron motion. The advantage of this treatment is a result of the non-perturbative origin of Ž4., however, the conditions for d-electrons to be ‘slow’ should be analyzed more carefully. In the present paper this approximation is examined in the case of the infinite Hubbard ŽU s `. repulsion by using of the Hubbard X-operators for localized electrons 0s s0 h s H < Us ` s Ý Ž E y m . X iss y m Ý n iss q V Ý Ž sq i s X i q X i si s . y t

is

is

is

Ý

sq i s s js ,

Ž 5.

² ij :s

where the conditions s s ", s / 2 in Ž5. ensure the single-occupancy of a lattice site, which corresponds to the case U ™ ` in Ž1.. We will use the supersymmetric SUŽ2 < 1. representation of Hubbard variables in the path integral for the partition function Q h . The charge–spinon separation along with the spin mean-field approximation in the low-temperature limit determine the properties of localized electrons, close to those predicted in w1–3x in the adiabatic approximation. 2. SU(2 < 1) path integral for localized electrons We use the ordinary Grassmann variables si s , si s for band electrons and follow the idea w4x of the supercoherent state representation for Hubbard degrees of freedom developed recently in terms of the SUŽ2 < 1. path integral w5x Žsee also Appendix A. Q h s Sp exp Ž yb h . s

HŁ is

d m i d si s d si s exp

1 2

b

ÝH i

0

z˙i z i y z i z˙i q j˙i j i y j i j˙i 1 q < zi < 2 q ji ji

dt

1 q

b b s˙i s si s y si s s˙i s / d t y Ý H h Ž z i , j i , z i , j i . d t Ý H ž 2 0

is

b

H0

q

ž

t

Ý ² ij :s

i

/

si s s js q m Ý si s si s d t , is

0

Ž 6.

E. KochetoÕ et al.r Physica C 296 (1998) 298–306

300

where hŽ zi ,ji , zi ,ji . s

² zi ,ji < h < zi ,ji : ² zi ,ji < zi ,ji :

and for the supercoherent state < z, j : the matrix element is hŽ zi , x i , zi ,ji . s

Ž E y m . Ž1 q < zi < 2 . 2

1 q < zi < q ji ji

qV

si ≠ j i q si x j i z i q j i si ≠ q z i j i si x 1 q < zi < 2 q ji ji

.

The SUŽ2 < 1.-invariant integration measure in Ž6. reads d mi s

d zi d zi

d ji d ji

2p i

1 q < zi < 2 q ji ji

.

Ž 7.

We can simultaneously bring the measure Ž7. and the kinetic SUŽ2 < 1. term to a form corresponding to independent SUŽ2. spin and spinless fermions. Performing two successive transformations in Eq. Ž6.

(

z ™ z 1 q jj ,

(

j™j 1q< z<2

one reduces the measure Ž7. and the kinetic term in the action on a site to d mi ™

S kin ™

d zi d zi d ji d ji 2p i Ž 1 q < z i < 2 . 1

Ý 2

,

2

z˙i z i y z i z˙i 1q< z<

i

2

1 q 2

Ý ž j˙i j i y j i j˙i / .

Ž 8.

i

In view of Eq. Ž8. the Grassmann trajectories j i , j i correspond to spinless variables and satisfy the antiperiodic boundary conditions j Ž0. s yj Ž b ., j Ž0. s yj Ž b ., whereas the complex trajectories z i , z i with periodic boundary conditions z Ž0. s z Ž b ., z Ž0. s z Ž b . correspond to spin variables of the d-subsystem. It means that the d-electron amplitude ds is decoupled into the j and z fields, the former carrying the electron charge while the latter correspond to the spin degree of freedom. The point is that the spin of d-electron is written as a polynomial function of z, z, the latter are referred to as spinon variables. The Hamiltonian function takes the form h Ž z i , j i , z i , j i . s Ž E y m . Ž1 y j i j i . y

V

(1 q < z < ž s 2

i≠

j i q si x j i z i q j i si ≠ q z i j i si x

.

/

i

The path integration over s-variables is of a Gaussian form and can be performed explicitly Q h s Sp z , z , j , j 4 Sp s, s4 exp Ž yb h . s Det 5 d 2 S Ž s . 5

H Łi d m exp S

eff ,

i

where Det 5 d 2 S 5 stands for the functional determinant of the second variation of the s-electron action and does not depend on the d-variables. As a result, we obtain the effective action for the subsystem of quasilocalized heavy electrons: d Seff s

1

b

ÝH 2 i

0

qV 2 Ý ² ij :

z˙i z i y z i z˙i 1q< z< b

b

H0 H0

2

dtq

1

b

ÝH 2 i

0

ž j˙ j y j j˙ / d t q N Ž E y m . y Ž E y m . Ý H i

i

d t d t X j i Ž t . j j Ž t X . Gi j Ž t y t X .

i

i

i

1 q z i Ž t . z j Ž tX .

(Ž1 q < z Ž t . < . ž1 q < z Ž t . < / . X

2

i

j

2

b

0

ji ji d t

Ž 9.

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Now let us note that the last term in the effective action Ž9. can be written as the SUŽ2. matrix element:

Ž z i Ž t . < z j Ž tX . . s

1 q z i Ž t . z j Ž tX .

(Ž1 q < z Ž t . < . ž1 q < z Ž t . < / . X

2

i

Ž 10 .

2

j

The Green function Gi j Žt y t X . appears as the propagator of s-electrons:

ž

d dt

q m Gi j Ž t y t X . y t Ý G k j Ž t y t X . s d i j d Ž t y t X . ,

/

G Ž 0,t . s yG Ž b ,t . .

Ž 11.

k

It is important to note that the effective action Ž9. is of a time-nonlocal form, so that no time-independent Hamiltonian can be associated with Ž9..

3. Adiabatic approximation In this section we address ourselves to the following problem: under which conditions does Eq. Ž11. result in a local form for the effective action Ž9., thereby leading to the adiabatic approximation d Q h , Sp exp Ž yb h eff .

Ž 12 .

for the system Ž5.. The solution to Eq. Ž11. reads 1

X

Gi j Ž t y t . s

vn s

ž

2nq1

b

Ý

Nb

/

exp yi v n Ž t y t X . q i q Ž i y j . i vn q t q y m

q, n

,

Ž 13 .

t q s yt Ý e i q D Z ,

p,

Z

here D is the lattice intersite distance. It is seen from Ž13. that the effective action Ž9. becomes local if the inequality < b Ž wym. <41

Ž 14 .

holds and in this case we arrive at the low-temperature approximation. It then follows that G™d Žtyt X .

1

exp i q Ž i y j .

Ý N

tq y m

q

and d Seff s

1

b

ÝH 2 i

z˙i z i y z i z˙i

0

1q< z<

Ý

H0

V2 q N

b ² ij :

2

dtq

1 2

b

ÝH i

d t ji Ž t . jj Ž t . Ý

0

ž j˙ j y j j˙ / d t q N Ž E y m . y Ž E y m . Ý H i

i

i

i

exp i q Ž i y j .

q

i

tq y m

Ž zi Ž t . < z j Ž t . . .

b

0

ji ji d t

Ž 15 .

It can be shown Žsee Appendix B. that the static version of the mean-field approach to the spin variables in Ž15. results in the approximation Ž z i Ž t .. < z j Ž t .. ™ 1. Therefore, we get for the effective Hamiltonian d h eff s Ž Eeff y m . Ý dq i diy

i

Ý t ieffj dqi d j , ² ij :

Ž 16 .

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Ž . Ž . where dq i d i are the creation destruction operators of spinless fermions and Eeff s E y

V2 N

1

Ýt q

t ieffj s

,

qym

1

exp i q Ž i y j .

Ý N

tq y m

q

.

Eq. Ž16. is the adiabatic approximation Ž12. for d-electron subsystem. Using a specific model of the s-electron energy band in the next section we will transform the sums over q to integrals and examine the energy shift d E s Eeff y E as a function of parameters w, E and b . 4. Temperature anomalies in a d-electron subsystem Let us consider the two-dimensional conduction s-electron band with a density of states r s Ž e . s Ž1r2w .u Ž w 2 y e 2 .. In this case the zero-temperature limit of the energy shift d E of the d-electron level looks like rs 1 d E s yV 2 d e , rs s . Ž 17 . eym 2w

H

Within the adiabatic approximation in the low-temperature spin mean-field treatment Ž16. of d-electrons the situations of their deep Ž w < < E <. or shallow Ž w 4 < E <. d-electron level can be considered. These correspond to the cases of the narrow and wide s-electron bands with different intervals of e in the integral Ž17.. So, the adiabatic approximation for the deep d-level leads to the formula

dEsy

V2 2w

½H

w

myT

q

yw

HmqT

V2

de

5

sy

eym

2w

log

wym wqm

Ž 18 .

for the conditions E ) m , w < < E <, T - < E <, d E - 0, and electron concentration number n - 1. The diagonal part of the hopping term in Ž16. t eff s

3 4

V 2m

ž

w2

2V 2

wym

log

q wqm

w

/

.

Ž 19 .

gives the non-linear equation for the chemical potential

ž

m s weff

3 2

n d y 1 q Eeff ,

/

weff s zteff

instead of the usual linear m Ž n d . dependence w6x, valid for the occupation number n d F 1 and the equality m ; E. One can see, that in this case the d-electron energy Eeff s E q V 2rE coincides with the heavy branch of excitations known earlier w7x. Then, the adiabatic approximation Ž14. for the shallow d-level leads to the formula

dEsy

V2

wy2T

H 2 w yw

V2

de ,y

eym

2w

log

2T wqm

.

Ž 20 .

for the conditions < E < < w, T - w, n ) 1. The reorganization of spectrum d E ) 0 occurs and the ‘Kondo temperature’ Tk s

wqm 2

exp

ž

Eym

r sV 2

/

Ž 21 .

follows from the condition E y m q d E s 0. The temperature dependence of the energy shift d E arising due to the hybridization effect in the model Ž5., was determined earlier in w8x by the perturbation theory in small Vr< E <.

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In contrast to the s–d exchange integral that appeared to explain this effect in w8x, our SUŽ2. spinon scalar product Žresulting from the spinon–charge separation. is reduced to unity in the mean-field spin approximation. So, the ‘Kondo-like’ effect here does not depend on the details of spin structure, but originates from the effective interaction between s- and d-electrons. We note that formulae Ž17. – Ž19. were deduced in w1x and discussed later in w2,3x for the hybridization part of the Hamiltonian Ž1. with the spinless d-electrons. These formulae follow from the direct application of the adiabatic hypothesis for the deep-level d-electrons in a low-temperature limit w1–3x, and they are justified here for the same conditions for the model Ž5.. Note that one may also employ the standard method perturbative in V to second order w9x to deduce the result Ž17.. As to the temperature reorganization of spectrum, represented for the model Ž5. by the Kondo-like effect Ž20–21., it takes place for the shallow-level d-electrons and occurs also at low temperatures. The latter result does not confirm the high-temperature condition for the same effect in w3x, the corresponding point of which must be revised.

5. Conclusion To conclude, we have shown that the supercoherent SUŽ2 < 1. representation of the Hubbard operators in the path integral for the partition function of the U ; ` limit PAM Ž5. leads to a separation of charge and spinon variables for local electrons. Namely, they are the trajectories z and j corresponding to the SUŽ2. spin Žspinon variable. and a spinless fermion. The low-temperature and spin mean-field approximations for this representation justify the adiabatic approximation w1–3x for d-electrons in the partition function of the system Ž1. and leads to the effective Hamiltonian Ž16.. The shift d E of the d-electron level is the result of the averaging over degrees of freedom of s-electrons. Two different situations for the cases of deep Ž w < < E <, n - 1. and shallow Ž w 4 < E <, n ) 1. levels of d-electrons are possible. Temperature independent behaviour of d E Ž18. is found in the first case and the d E Kondo-like log T behaviour Ž21. in the second. The present analysis shows that the low temperature is necessary for localized electrons to be treated adiabatically, that is, as the ‘slow’ particles in the periodic Anderson model Ž1. for the infinite U-term with the spin mean-field effective Hamiltonian Ž16.. We hope that this analysis may provide an adequate basis for non-perturbative study of the spin-fermion properties of correlated electrons.

Acknowledgements The authors thank Prof. V.A. Ivanov for the discussion on the properties of the Hamiltonian Ž5. of the present paper and thank ‘Cyclone’ LTD for financial support. E.K. and V.Y. thank the Russian Foundation for Fundamental Research for the support under grant No. 96-01-00223, M.Zh. thanks Russian Ministry of Science and Technology for the financial support of the Project No. 96149.

Appendix A To make the exposition self-contained and for the reader’s convenience we place in this appendix some information concerning a definition of the suŽ2 < 1. superalgebra and its representations as well as of related coherent states and path integrals.

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As is known, the SUŽ2 < 1. supergroup in the fundamental representation is the group of Ž2 q 1. = Ž2 q 1. unitary, unimodular supermatrices with the Hermitian conjugate operation. It is generated by even generators  B,Q3 ,Qq,Qy 4 and the odd ones Wq,Wy,Vq,Vy 4 which satisfy the following commutation rules w10x:

w Q3 ,Q " x s "Q " , w Qq ,Qy x s 2 Q3 , w B,Q " x s w B,Q3 x s 0, w B,V " x s 12 V " , w B,W " x s y 12 W " , w Q3 ,V " x s " 12 V " , w Q3 ,W " x s " 12 W " , w Q " ,V. x s V " , w Q " ,W. x s W " , w Q " ,V " x s w Q " ,W " x s 0,  V " ,V " 4 s  V " ,V. 4 s  W " ,W " 4 s  W " ,W. 4 s 0,  V " ,W " 4 s "Q " ,  V " ,W. 4 s yQ3 " B. Let < b,q,q3 : stand for a vector of any abstract representation of suŽ2 < 1., where b, q and q3 denote the eigenvalues of the operators B, Q 2 and Q3 , respectively. When considering the highest-weight state as the fiducial state <0:, the typical SUŽ2 < 1. coherent state reads < z , j , u : s N exp Ž yu Wyy j Vyq zQy . < b,q,q : , Ž A.1 . where Ž z, j , u . g SUŽ2 < 1.rUŽ1. = UŽ1.. We are interested in the so-called degenerate b s q s 1r2 representation which happens to be relevant for the Hubbard operators. It is specified by Wy < q,q,q : s 0 and is called the Ž q s 1r2, q s 1r2. representation. The even Žbosonic. states <1r2, 1r2, 1r2: and <1r2, 1r2, y1r2: are identified with the spin up and spin down states, < q : and < y :, respectively, whereas the odd Žfermionic. state <1, 0, 0: with the doped state <0:. The dimension of this representation is equal to 3 as it should be. The coherent state ŽA.1. is reduced in this representation to < z , j : s Ž 1 q < z < 2 q jj

y1 r2

.

ey j Vy qz Q y < q,q,q : ,

where we have evaluated the normalization factor explicitly. The variables z and j are parametrized in the N s 2 supersphere S 2 < 2 s SUŽ2 < 1.rUŽ1 < 1. w11x. Resolution of unity in the Ž1r2, 1r2. representation space holds H < z , j :² z , j < d mSUŽ2 < 1. Ž z , j . s I, provided the SUŽ2 < 1. invariant measure Ž7. is used. Evaluating a partition function in the < z, j :4 basis results eventually in the SUŽ2 < 1. path integral representation w5x tr exp w yb H x ' ZSUŽ2 <1. s

HzjŽ0.0szsyŽ bj. b Dm Ž .

Ž

.

SUŽ2 < 1.

Ž z , j . exp w A x ,

where DmSUŽ2 < 1.Ž z, j . stands for an infinite pointwise product of the SUŽ2 < 1. invariant measures and the classical action on S 2 < 2 with a Hamiltonian function H cl s ² z, j < H < z, j : reads 1 As

b

H 2 0

˙ y jj˙ ˙ y zz˙ q jj zz 2

1 q < z < q jj

dty

b

H0

H cl Ž z , j . d t.

To explicitly evaluate H cl one needs the SUŽ2 < 1. covariant symbols of the generators. These are found to be Ž Acl ' ² z, j < A < z, j :.: Q3cl s 12 Ž 1 y < z < 2 . w,

cl cl Ž Qq . s zw, Ž Qy . s zw,

B cl s 12 Ž 1 q < z < 2 q 2 jj . w,

cl cl Ž Vq . s yz j w, Ž Vy . s j w,

cl cl Ž Wq . s yj w, Ž Wy . s yz j w, w s Ž 1 q < z < 2 q jj .

y1

.

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The Hubbard operators in the basis <0:, < s :4 takes the form X

X ss s < s :² s X < ,

X s 0 s < s :²0 < ,

Ž A.2 .

where <0: stands for a doped site Žhole. and < s : for the state having an electron occupied with spin s . It is clear that there are eight linearly independent operators since X 00 q Ý X ss s I,

Ž A.3 .

s X

X s 0 appearing as a fermionic operator, whereas X ss correspond to bosonic degrees of freedom. In fact, representation ŽA.2. means that the X-operators are closed into the uŽ2 < 1. superalgebra, which in view of ŽA.3. is reduced to the eight-dimensional suŽ2 < 1. superalgebra. The algebra of the X-operators can explicitly be identified with the degenerate Ž1r2, 1r2. representation of suŽ2 < 1. in the following way, Q3 s 12 Ž Xqqy Xyy . ,

Qqs Xqy,

Qys Xyq,

B s 12 Ž Xqqq Xyy . q X 00

and Vqs X 0y ,

Vys yX 0q ,

Wqs Xq0 ,

Wys Xy0 .

Appendix B In this appendix we perform the averaging Žmean-field. treatment Žover variables z, z . of the effective action Ž15.. To this end, let us first introduce the so-called Darboux variables Õ,Õ related to z, z by zs

Õ

(1 y < Õ <

2

,

zs

Õ

(1 y < Õ <

2

< Õ < F 1.

,

As a result, one has Seff s V 2 Ý ij

1 q 2

b

H0

j i Ž t . j j Ž t . t ieffj

½ (Ž1 y < Õ Ž t . < . ž1 y < Õ Ž t . < / q Õ Ž t . Õ Ž t . 5 d t 2

i

2

j

i

j

b Ý H ž Õ˙i Õi y Õi Õ˙i / d t q PPP , i

0

where we have dropped terms independent of Õ,Õ. This can be rewritten in a trigonometric form Ž Õi s cos u i e i f i , u i ' u i Ž t ., f i ' f i Ž t .. Seff s V 2 Ý ij

b H0 j Ž t . j Ž t . t  sin u i

j

eff ij

i

sin u j q cos u i cos u j eyiŽ f iy f j . 4 d t y i Ý i

b

H0

f˙ i cos 2u i d t q PPP

Ž B.1 . Let us consider Eq. ŽB.1. in the so-called quasistatic mean-field approximation. This means that the following two assumptions are supposed to be fulfilled. First, though all dynamical variables in Eq. ŽB.1. depend on time, the dependence for f Ž t . is assumed to be weak: f˙ ; 0 and, second, a spatial distribution of f values at any given time moment is such that for any pair of nearest neighbors i and j one has f i y f j ' D f i j < f i, j .

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As the result, Eq. ŽB.1. takes on the form Seff s V 2 Ý ij

b

H0

j i Ž t . j j Ž t . t ieffj cos Ž u i y u j . d t q O Ž D f . .

The stationary-phase equation d Seff s 0 results in V 2 Ý t keffj sin Ž u k y u j . j k j j q j j j k s 0,

½

j

5

which has an obvious solution u i y u i s p n i j , n i j g N. Thus, we have finally arrived at qs Seff sÝ

ij

b

H0

j i Ž t . Ž y1 .

ni j

j j Ž t . t ieffj q PPP

Ž B.2 .

Let us further represent n i j s m i y m j , m i ,m j g N, in other words we assign to any lattice site i an integer number m i . Eq. ŽB.2. becomes qs Seff sÝ

ij

b

H0

j i Ž t . Ž y1 .

m iym j

j j Ž t . t ieffj q PPP s Ý ij

b

H0 j Ž t . e i

ip Ž m iym j .

j j Ž t . t ieffj q PPP

By making finally change of variables j i ™ j i eyi p m i , j i ™ j i e ip m i , one gets Eq. Ž16. of Section 3.

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