Functional expansion of the Anderson and SU(N) lattice models

Physica B 171 (1991) 126-130 North-Holland

Functional M.E.

expansion of the Anderson

and SU(iV) lattice models

Foglio

Institute de Fisica Gleb Wataghin, Universidade Estadual de Campinas,

13081 Campinas, Snb Pa&o, Brazil

We employ the functional expansion of Kadanoff and Baym to study the Anderson and SU(N) lattice models in the infinite correlation limit (Li+ 30). We use Hubbard operators that describe real localized electrons and forbid their double occupation at each site. In the lowest nontrivial approximation, our expressions are similar to (but different from) those derived by several ‘effective Hamiltonian’ techniques, like the Mean Field Slave Boson (MFSB). In the usual large N limit, our results coincide with those of the equations of motion and Brillouin-Wigner expansions, which are exact in that limit. Our results at T= 0 K are compared to those of. the MFSB quasiparticle description, and we discuss the two approximations in the region in which they are different. We conclude that the two treatments are complementary: the quasiparticle description should give better results for the thermodynamic properties but our treatment describes in a more physical way the overall behavior of the spectral density of the localized electrons. The Kondo resonance is not obtained in our treatment, and we conjecture that it should appear in a higher-order approximation.

1. Introduction

tice as H = Ho + H’ where

The study of the Periodic Anderson Model (PAM) is very relevant to the heavy-fermion problem, and a variety of treatments have been employed to calculate its properties (cf. the many references mentioned in the paper by Brandow [l]). In the present work we shall apply the functional technique of Kadanoff and Baym [2] to the PAM in the U -+ UJlimit, with the same type of approximations employed by Ruckenstein and Schmitt-Rink [3] to treat the Hubbard model. We shall compare our results to those obtained by other methods, in particular to the slave boson treatment [4] in the mean field approximation [5] (MFSB), and we shall use the variational treatment of Brandow [l] as a reference frame to understand the differences between these methods. We have also studied the SU(N) model, which gives an exact solution in the N+ 00 limit when NV2 is kept constant (V is the hybridization parameter).

HO = c E(f&,,

2. The functional expansion for the SU(iV) model We write the Hamiltonian

for the SU(N) lat-

+ c E(k+:,,C,,

,

(I) To exclude multiple occupancy of f electrons at each site we have used Hubbard operators [6] Th e only localized states at xi&3 = Iia)(jPl. site j are IjO) (no f electron) and the N states 1ja) (one f electron in channel a); the C,, are the usual Fermi operators of the conduction band. The method of functional expansion [2,3] makes possible to obtain successive approximations to the Green Functions (GF) in a systematic way. An auxiliary external potential is introduced, so that the higher-order GF that appear in the equations of motion can be expressed in terms of the functional derivatives of the original GF with respect to this auxiliary potential. Approximations are made at this stage, and in another work [7] we have used this technique to calculate, in the lowest nontrivial approximation, the Green Functions (GF) for imaginary time G(u; y, 7; y’, T’), where the ys

0921-4526/91/$03.50 0 1991- Elsevier Science Publishers B.V. (North-Holland)

M. E. Foglio

I Anderson

identify the operators Xi,O, or C,, in the GF. Here we shall only discuss the f, f GF in the frequency and wavevector domain: G(cr; fk, fk’lz) = 6,,,(z - E(ka))Q,lA(k,

z) ,

and SU(N)

lattice models

127

has a profound effect on the shape of the f spectral density, as discussed in the next section. We shall now consider a simple model to study the similarities and differences between our results and those of other treatments.

(2) where 3. Study of a simple model A(k, z) = (z - ,?)(z - E(b)) g = E(fa) - VQ,’

- Ir;-(’ ,

(3)

sTm(Xi,( .i>Cj, > 7

(4)

Q, = (&,A + MJ

PI’= (1-

c

(X”,)

SfW

7 Iv]‘.

(5)

(6)

The operator C,, destroys an electron in the Wannier state at site j, and we have considered the uniform case, in which all properties are independent of the site. In the paramagnetic case the csP% becomes N - 1 because the properties are independent of the spin component. The GF obtained [7] have poles at the solutions o,(k) (a = 1,2) of A(k, z) = 0. They correspond to the dispersion relations of a band E(ku) a_nd a lattice of localized f elec$ons with energy E hybridized with a constant V; local completeness X0, + C,X,, = I implies ]?I’ = Q,lVl’. The spectral weight of the GF in eq. (2) is the same obtained in the ‘Hubbard I’ approximation [8], i.e. Q, = 1 - no in the paramagnetic case, where nD is the number of f electrons per site and per channel, but the energy renormalization E E(f) is absent in that approximation. In all the theories we shall discuss, there is a renormalization of the hybridization which appears as a factor of V’, but their analytic expression is different from our Q, = 1 - (N - l)nU. In the MFSB and in the variational treatment of Brandow [l] one has Q, = (1 - Nn”), while employing the Gutzwiller method [9] (1 - Nn”)l (1 - n”) is obtained. All these different expressions coincide in the limit N-+ m, but for small I~Oour expression agrees to 0( InU]) with the one obtained from the Gutzwiller approximation. In the Kondo region (i.e., when Nn” - 1) our expression gives Q, - 0.5 rather than Q, - 0. This

Here we consider the problem for T = OK, taking a rectangular conduction band E(b) centered at the origin with a density of states p0 = +D per channel. Each ‘band’ w,(k) is in the interval [o,, , waM], where a 5 l(2) corresponds to the higher (lower) band, and the effective ,!? is always in the gap. The following expressions are valid when the chemical potential p is inside the lower ‘band’: E=E(f)+(N-l)IV/$“ln((~~_@~))>O,

n” = Qp”jtiI’( & (E - P)

(7)

where nc is the number of conduction electrons per channel per site. These are the expressions obtained in several ‘effective Hamiltonian’ treatments [lo] (cf. eqs. (70), (72) and (74) in ref. [5], neglecting terms of order 1 lN in our eq. (7)), except that our expression for n” has an extra factor Q. This reflects the fact that the effective H in those works describe two hybridized bands without correlation and with an integrated spectral density per channel equal to one, while in our case the spectral density integrates to Q < 1 because of the f electrons local correlation. In the effective Hamiltonian treatments, the correlation is forced on the model through extra conditions, and the resulting E is slightly above I_Lwhen E(f)

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Foglio

I Anderson

conditions to keep the total number of electrons per site less than one. In the large N limit (keeping NV2 constant) our f-electron GF tends to a delta function at E with strength Q, which is the exact solution in this limit [ll]. The variational treatment of Brandow [l] will be particularly useful in our discussion, because even in its simplest approximation, the oneparameter theory, he starts from a wavefunction that has built-in the impossibility of the double-f occupancy. This is a feature that it shares with our treatment, while it also is a variational treatment just like the several ‘effective Hamiltonian’ theories, thus giving the best possible description of the low-temperature thermodynamic properties. To compare our results with thos_e of the MFSB, we plot in fig. la the values of E - p and /1 as a function of E(f) for V= 0.3, D = 10, N = 2. T = 0 K and n’ = rzU+ nc = 0.75. The two

IO

05

50

00

-50

Fig. 1. The parameters employed are giv_en in the text. Figure l(a): The chemical potential p and E - p are plotted as a function of the unrenormalized energy E(f) for both the KB and MFSB methods. The two w curves are indistinguishable, and the two curves of E - /* are different only for E(f) below -5.0 (the dashed line corresponds to the MFSB method). Figure l(b): The total number n’= Nn” (dashed line) and the spectral density pf of the f electrons at the chemical potential Jo are plotted as a function of E(f) for the MFSB method (dotted line) and for the KB method (full line). The two pf have been multiplied by 0.005 to fit the vertical scale.

and SU(N)

lattice models

graphs of p are indistinguishable but the E - I* are different below a given energy E(f) as was discussed before. In the MFSB method, E p - 0 when E(f) is below an energy that depends on the different parameters, so that the whole Kondo region is included, while in the KB method there is only an interval of E(f) in which E - p is very close to zero. This interval coincides with that in which p increases from one plateau to a higher one, and this corresponds to the transfer of f electrons into the conduction band as E(f) increases, as can be seen in fig. lb. In this figure we also plot the spectral density p’ both for KB (full line) and MFSB (dotted line). T_he KB value of p” has large values when - 0, corresponding to Heavy-Fermion &irproperties in this region. The asymmetric peak shown by the KB spectral density is due to the jump of p from the upper band into the lower band when E(f) increases: at constant Ilf = n” + $, the 11’ increases with E(f) and Al” decreases until it is smaller than the maximum number of f electrons in the lower band. This behavior cannot be present in the MFSB treatment because p is always in the lower band to keep Nn’< 1. It is clear that in the Intermediate Valence (IV) region the two treatments give the same qualitative description of the system. In fig. 1 we have chosen a rather large Al’to emphasize the differences between the KB and the MFSB approximations, but for Al’~0.5 we find that both methods give E - p - 0 and the same qualitative properties also in the Kondo region. The same features are obtained for larger N, but the KB method gives E - p - 0 and a large p f in a smaller region, because the maximum number of f electrons per channel is 1 lN, and they are all transferred to the conduction band in a smaller interval of E(f). For the same reason, the two treatments give the same qualitative behavior of the system only when IE’< 1/N. It is interesting to compare the spectral densities p’(w) of the two methods in the Kondo region. In fig. 2 we plot this quantity as a function of w for E(f) = -8.0 and for the same parameters employed in fig. 1, except that we have used V= 1.0 to emphasize the gap between

M. E. Foglio

IO

P’ 0.5

I Anderson

(I :I ,II

.1 I: :I I

.’ I

00 ! -100

), ...’

- 80

i’

-6.0

w

Fig. 2. The f spectral density pf is represented for E(f) = -8.0 and for the same parameters of fig. 1, except that here V= 1.0. The full line corresponds to the KB method and the dashed line to the MFSB one. The dotted line is explained in the text.

the two bands in the KB method (full lines). In this case the renormalized energy, which is always inside the gap, is E = -7.64 and is not very different from E(f). The fairly large Q, = 0.502 allows a rather large hybridization between the f and c electrons, so that p’(w) is not negligible in a rather large interval of o, as shown in the figure. The MFSB method gives an E - -5.0 which is rather different from E(f). The value E - p - 0.46 x lo-l2 gives no-- 0.5 so that Q, 0 and there is negligible hybridization: the f spectral-density is then practically a delta function at E, which has been represented in fig. 2 by a dashed line. To emphasize the different effects of the value of Q, and its presence in the nU of eq. (7) for KB (or absence for MFSB) we have also plotted with a dotted line the p’(w) of the lowest band, calculated with the KB value or Q, but with the n” of the MFSB method. We see that the energy renormalization is that of the MFSB, but the large Q, allows stronfg hybridization, as shown by the width of the p . To understand the meaning of the rather different results obtained by the two methods, we shall compare with the variational calculation of Brandow [l]. As in the MFSB method, he obtains in the Kondo region a strongly renormalized peak at the chemical potential p, but with spectral weight of 1 - Nn” rather than one. This

and SU(N)

lattice models

129

peak represents the quasiparticles, that in these two cases give the best possible description of the thermodynamical properties of the system, and it has all the qualitative features of the Kondo peak. Brandow states that there is in addition a ‘nonquasiparticle contribution to the f spectral intensity’, and because of the ‘formal similarity to the single-impurity case’, he conjectures that ‘this contribution should produce a broad peak centered near the bare f level’ E(f). This last structure is just the one described by the KB method, and in our fig. 2 the complete structure of the pf conjectured by Brandow is qualitatively represented by the sum of the p f derived from the KB and the MFSB methods, with the understanding that the integrated intensity of the MFSB pf should be reduced by a factor -1 - Nn” and does not contribute practically to IZ? We conclude that the KB and the ‘effective Hamiltonian’ treatments give complementary aspects of the spectral density in the Kondo region. Also that the large spectral densities pf that give the heavy-fermion properties are present in the KB expansion only in the region of intermediate valence in which 1 - nf is small but not negligible, while in the MFSB method both E - p decreases and pf increases monotonically when E(f) decreays. It is clear that in the interval in which the E - p coincide the KB and MFSB treatments are at least qualitatively equivalent, but in the Kondo region the quasiparticles near the Fermi surface are rather different from the f electrons. As the MFSB treatment is variational in the free energy, we expect that it should give a better approximation to the thermodynamic properties than the KB, but this last gives a better overall description of the f electrons spectral density. The MFSB has a large f spectral density at the Fermi energy, but with an spectral weight that is too large. Our method does not give a Kondo resonance, but we conjecture that a higher order KB treatment would show this feature. We believe that the spectral density of the KB expansion gives a better starting point for higher-order approximations. In the present work we have not studied the stability of the paramagnetic phase against other

130

M.E.

Foglio

I Anderson

magnetic phases, nor have we calculated thermodynamic properties.

the

and SlJ(N)

[5] D.M. [6]

References [l] B.H. Brandow, Phys. Rev. B 33 (1986) 215. [2] L.P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962). [3] A.E. Ruckenstein and S. Schmitt-Rink, Phys. Rev. B 38 (1988) 7188. [4] P. Coleman, Phys. Rev. B 29 (1984) 3035.

[7] [S] [9]

[lo] [ll]

lattice models

Newns and N. Read, Adv. Phys. 36 (1987) 799; Solid State Commun. 52 (1984) 993. J. Hubbard, Proc. R. Sot. (London) A 276 (1964) 238; ibid., 281 (1965) 401, these are two of a series of six papers. M.E. Foglio, to be published. C.M. Varma and Y. Yafet, Phys. Rev. B 13 (1976) 2950. T.M. Rice and K. Ueda, Phys. Rev. B 34 (1986) 6420; A.C. Varma, W. Weber and L.J. Randall, Phys. Rev. B 33 (1986) 1015. P.A. Lee, T.M. Rice, J.W. Serene, L.J. Sham and J.W. Wilkins, Comments Conden. Matter Phys. 12 (1986) 99. G. Czycholl, Phys. Rep. 143 (1986) 277, cf. eq. (7.18).