Low temperature properties of the Anderson lattice

Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 65-69 North-Holland, Amsterdam

65

I N V I T E D PAPER L O W T E M P E R A T U R E P R O P E R T I E S OF T H E A N D E R S O N LATYICE H. K E I T E R lnstitut fur Physik der Universitgit Dortmund, Postfach 500 500, D-4600 Dortmund 50, Fed. Rep. Germany Dedicated to Prof. U. Bonse on the occasion of his 60th birthday New developments in infinite order perturbation expansions in the hybridization for the Anderson lattice are reviewed, which allow for the exact incorporation of the large U Coulomb interaction between local electrons at the same lattice site. The resulting dressing of quasiparticles is treated within the frame of established approximations, like the self-consistent non-crossing approximation for the lattice. Results for the one-particle properties are presented. The coherence gap is shown to be formed parallel to a many body resonance. Finally quasiparticle scattering is briefly investigated for instabilities and criteria for superconducting or magnetic ground states.

1. Introduction Infinite order perturbation expansions in terms of the hybridization for the single impurity Anderson model [1] are known for a long time [2] and have been repeatedly reviewed [3]. Approximations like the non-crossing one (NCA), which is a finite temperature version of part of the Gunnarsson-Sch/Snhammer approach, reviewed in ref. [4], showed excellent agreement with measured thermodynamic and transport properties of alloys containing rare earths [5] as well as with temperature dependent photoemission spectra [6]. Furthermore, the 1/N-expansion ( N = degeneracy of the 4f~-level) excellently agreed [7] with the Bethe-ansatz results for the f-valence and the magnetic susceptibility at zero temperature. In all these applications, the limit of infinite U in the Anderson model was taken. The expansion in terms of the hybridization can also be set up with auxiliary bosons [8], and the results are in one-to-one correspondence to those of the direct expansion, provided, that the restrictions on the boson states are treated exactly. At present, both the 1 / N - e x p a n sion and the N C A are studied for the case of finite U [9]. The enhanced numerical difficulties are less than those in the NCA-improvements [10] for infinite U. An extension of the N C A to the Anderson lattice [11,12], called LNCA, will be reviewed in the following two sections. It formally results in the single-site N C A equations with an effective band-electron density of states, which contains the lattice effects, depends on temperature, and which

has to be determined self-consistently. Up to this point it represents the first step in the general renormalization scheme for the local and non-local interaction processes suggested in ref. [12]. This scheme yields new quasiparticles instead of bare f- and band-electrons, and allows to study their remaining interactions. So, if an improvement of the single-site N C A can be found, which avoids the well-known N C A difficulties at the Fermi-level at zero temperature [13], this scheme could provide a microscopic derivation of Fermiliquid parameters. The first attempts to find effective quasiparticle interactions are sketched in the final section. 2. From renormalization of hybridization processes to the LNCA

Starting from the Anderson model for infinite U, the grand partition function or Green's functions are expanded in the hybridization interaction; the resulting terms are most conveniently represented diagrammatically as hybridization processes [14]. In the following, for clarity and simplicity, only those will be pictured, which occur between empty 4f °-states at each lattice site as initial states (at the bottom) and final states (at the top in the diagrams of fig. 1). Due to the infinite U condition, one may then, on an imaginary vertical time axis, view the f-shell processes as flip-flops from 4f 0 to 4f I to 4f 0, and so on. Each flip from 4f 0 is accompanied by an absorption of a band electron (full line with arrow pointing

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66

H. Keiter / Low temperature properties of the A ndermm lattice

I

4fo~ 4fi

I

I

I

I

I

I

I

,

4fo'

i I

I I

i

R1 R2 {b)

R1 R2 (a)

R1 R2

R1 R2

(c',

{d',.

I

I

I

I

I

t

A

+

I

I

ized band electron line (full double line). Up to this point, the interaction processes in the lattice have been used for a renormalization of the on-site contributions. Neglecting the remaining interactions between the new quasiparticles for the moment, it is obvious, that in this approximation the grand partition function of the system is a product of the (unrenormalized) band part Zh~,n " d and N independent (renormalized) site contributions Z~

I ÷

•

z = z~:..,~ ( z, ) " .

I

I

I

I I

t I

6 I

I

RI {e)

(fl

(1)

{g)

Ih}

R2

i

R1

R2 (i}

Fig. 1. Simplest hybridization processes for two sites of an Anderson lattice, starting from a 4f°-level at each site. (a) (c) contain independent processes, (d) the simplest interaction process, which, together with processes like (e) result in a renormalization of f- and band-electrons, as shown in (f). Diagrams with crossings of band electron lines at a particular site like (g) are neglected. Parts of diagrams (h) and (i) contribute to the quasiparticle interaction.

towards the "flip-vertex"), each flop by an emission of a band electron. The simplest processes involving two sites are pictured in fig. l(a)-(d). In (a) and (b) one of the two sites remains empty, in (c) the simplest excitations of the f-shells occur independently, while in (d) the simplest interaction process between two sites is pictured. The basic idea of renormalizing the on-site processes by interaction processes with other sites, first correctly described by Grewe [12], may be seen as follows: For the two elementary on-site processes of fig. l(a) and (b) there is only one interaction process between two sites (fig. l(d)). So one has to take half of the numerical contribution of (d) for renormalizing (a) and the other half for renormalizing (b). Continuing this redistribution with three and more sites, one finally arrives at effective band electron lines in processes (a), (b) and (c). More complicated on-site processes, like the one pictured in (e) can also be renormalized. With the renormalization, diagrams (a) and (e) in fig. 1 form the first two terms of a geometric series for an effective 4fLline. It is shown as a double dotted line in fig. 1(13 together with the renormal-

Here Z~ is related to the 4f-resolvents (or ionic propagators) in the same way as for the Anderson impurity model

The sum in (2) contains the 4f°-contribution and the 4f 1 one (including degeneracy by spin or angular momentum). The contour C encircles the singularities of the resolvent in a counterclockwise direction. The resolvents are related to their selfenergies

P,,(:)=[:-E,,

'

(3)

of which Z'0(z) is the quantity, for which the simplest contribution is shown in fig. l(f). E M denotes the f-energies. If contributions with crossings of band electrons are neglected (like the one in fig. l(g)), the system of equations for the (renormalized) LNCA reads:

(4) o

xo(:)=

v2fdoaoo(~o)(1-f(~0))P0(:-~0)

(5)

(V is the hybridization coupling constant, N~ denotes the number of lattice sites, and f ( ~ ) is Fermi's function). In (4) and (5) the density of states of the band electrons in the NCA is re-

H. K e i t e r / L o w t e m p e r a t u r e p r o p e r t i e s o f the A n d e r s o n lattice

placed by the spectral function corporates the lattice effects:

po(oa), which

po(w) = - (1/,~) Im R,(~0 + i8),

in(6)

Ro(:) 1 ,+1

= ~

1 N,

[(VZ/N')F°(')(z)I

t

l=0

~

Q'+'

kl -.. k/- i

QI+ I =

E' exp{i(k2-kl)R~,

_}_ . . .

P l " ' " Pl

+i(k,+l-

=(Z,)

--I

(s)

z" d z ' Jc~exp(-flz

,

,

)Po(z )Po(z' + z). (9)

The effective band propagator starts from and ends at site R , , = 0 . When toucing / + 1 other sites, the unperturbed Green's functions for the band (z - %0) 1 are coupled to the corresponding Green's functions for the f-electrons Ff)(z), and the process requires a weight factor 1 / ( / + 1), which was discussed before for the special case l = 1. The complicated coherence factor Q~+I involves a sum on all sets of site induces Pl---Pl, which are different from each other and from v0 = O. Up to now it could only be calculated with the additional approximation, that only consecutive sites differ from each other [15]. In this approximation one obtains a closed expression for

67

3. Single particle properties within the LNCA From the numerical solution of the L N C A [11] one first obtains the effective band density of states (DOS) via eq. (6), see fig. 2, and the local DOS for an effective site from the corresponding equation with Ro(a~ + i8) replaced by F~l)(~0 + i6), see fig. 3. Compared to the unperturbed band, the effective band DOS shows moderate enhancement at the (renormalized) position of the f-level, and a peak slightly above the Fermi-level e F = 0, which increases at lowered temperature. The increased DOS is compensated for by a decrease towards the band edges. The peak is seen to reflect the m a n y b o d y r e s o n a n c e h la Abrikosov-Suhl in the effective 4f-DOS of fig. 3. Compared to p(4f)(w) (the DOS without lattice contributions), ~(o4r)(o0) shows an increased peak, slightly shifted to higher energies. But there is no new energy scale within the effective site approximation. As a second result of the numerical solution of the LNCA-eqs., one can obtain Green's functions for the one-particle propagation in the lattice. Green's function for the band electrons Gko(z ) is related exactly to the one for f-electrons Fk, ( z ) via

Gko(z)=(z-%o)-l[l

+ V2Fko(z)(z-%o)

(11) In Fko(z ) on-site as well as intersite f-exitations are summed,

Fko(z) = Fo°)(z)N.fiRo(z)/3[( z -

%o)-'],

02)

Ro(z):

-1+ E

'

l=1

x

therefore, in (12), in the quantity Ro(z) from (10), the functional differentiation should not be carried out in the 2nd, site-excluding term in the square bracket of (10). The DOS following from (11) and (12), e.g.

_1] l+1 }

1

N, E ( z •

'].

p~'°~"')(w) = - ( ' ~ U s ) -1 I m Y ] F k o ( ~ + i3 )

%'o)

k*

(13)

k

(lO) Eqs. (2)-(6), (9) and (10) form a closed system, from which the thermodynamics and the one-particle properties of the Anderson lattice in the effective site approximation are obtained. This system is called LNCA.

are shown in fig. 4. The coherence gap widening with decreasing temperature seems to support the hybridization band picture at zero temperature. A two peak structure in the local DOS was also obtained by Grewe [15], and within the X N C A [16], which is an expansion in the reciprocal num-

H. Keiter / Low temperature properties oj the Anderson lattice

68 -0 4

200

O

-0 2

02

0L,

Po

F

175

010

15 cL=. (b)

125

OOg

1•0

150

OO6

125

?

~ (e)

1.00 075 -5

Li

i

i

4

.

-3

i

i

-2

p(tOcl a :

[

-1

00L,

0

100

0 25

i

I

2

w

Fig. 2. Effective band D O S in the L N C A for thermal energies k BT = 0.2 (dotted line), 0.03 (full) and 0.07 (dashed). Fermi level at ~o = 0, edges of the unperturbed band at + 10, unperturbed f-level at - 3 , impurity Kondo energy at 0.02. Energy unit is the A n d e r s o n width A = ,~V2NF , and unit for the DOS is N v. (b) shows the magnification of the peak region in (a).

0 20 015

r' ~\

ii

0 l0

005 __L. . . . .

ber of effective neighbours. The temperature dependence of the two peaks differs from the one in fig. 4, however. The coherence gap in the band D O S has been seen in Gd 3 + -ESR on Ce,.La 1 ,.Os 2 [17]. 4. Two particle properties and the ground state For an investigation of the ground state one has to know the remaining interactions between the quasiparticles. The simplest processes, which partly survive the renormalization to effective site-contributions are those of fig. l(h) and (i). Note that (i) differs from (h) by an exchange of the two lines running from site R~ to R=. Sche-

030

{b)

025

(a)

]A

/

.~

020

O0

06

-0L~

Fig. 4. Lattice DOS

-02 for

the

0 local

02 part

0~

from

06

eqs. (12) and (13)

and for the band part from eq. (11). Units and different cases as in fig. 2.

matically one obtains from these processes the following two-site correction AZ 2 for the partition function:

AZ= = - ( 4 # ) - '

E all loop variables

× [11 + H e~ch - #2F¢I~F ¢t~] •, V402ph [/7 + H exch - fl2g'I)g('~]

R 2"

(14)

Here k "¢~ is given in (9), Qp~ denotes the particle-hole propagator, involving two Green's functions for the band, running between R 1 and R=, /7 is the local two particle vertex

":e,,eoP,,eo

o_

L#

<15)

015 c 'c~ 010

0 05

001

-5

I

-4

I

-3

L

-2

I

-1

I

0

L

1

L

2 co

Fig. 3. Local DOS of the L N C A (full line) and the NCA (dotted) for kBT = 0.03. Units as in fig. 2.

which together with its exchange part ~x~h is corrected in the square bracket in (14) by the product of two F {1) which were needed for the renormalization to effective sites before. From a mathematical point of view, the square brackets contain cumulant vertices. These have only been treated approximately [11,12] with respect to their dependence on three external energy variables,

H. Keiter / Low temperature properties of the Anderson lattice

relating the cumulant vertices to measurable quantities like charge- or spin-susceptibilities. Also elementary particle-hole or particle-particle propagators (essentially spatial Fourier-transforms of the O ph in (14)) could only be calculated approximately, guessing the effects of the cumulant vertices as a tendency towards a hybridized quasi-particle band-structure. Apparently the difference between the two kinds of propagators, known from the free particle case, dominates their behaviour for small energy- and momentum transfer. Within these approximations one finds a local, essentially wave-vector independent repulsion of quasiparticles with opposite spins (singlet-channel), while for equal spins (triplet-channel) the quasiparticle interaction is strongly wave-vector dependent and may show a local attraction for certain combinations of them. A magnetic instability, showing up in the Stoner-like susceptibility, could occur in the singlet channel at finite wave-vector, while in the triplet channel a possible magnetic instability would lead to a more complicated spin structure. An inspection of the particle-particle propagation should furnish criteria for superconductivity. Within the same approximation scheme as used before for detecting magnetic instabilities, singlet superconductivity with a purely electronic mechanism is unlikely. The chances are better for triplet superconductivity, but more detailed knowledge of the momentum dependence of the quasiparticle interaction is required. The calculations supporting the conclusions of the present section, are found in ref. [11]. The author is indebted to N. Grewe for almost a decade of exchanging ideas on highly correlated

69

electron systems, and to G. Czycholl and T. Pruschke for many discussions.

References [1] P.W. Anderson, Phys. Rev. 124 (1961) 41. [2] H. Keiter and J.C. Kimball, Int. J. Magn. 1 (1971) 233. [3] a) H. Keiter and G. Morandi, Phys. Rep. 109 (1984) 227. b) G. Czycholl, Phys. Rep. 143 (1986) 277. c) P.A. Lee, T.M. Rice, J.W. Serene, L.J. Sham and J.W. Wilkins, Comments Condensed Matter Phys. 12 (1986) 99. d) N.E. Bickers, Rev. Mod. Phys. 59 (1987) 845. e) P. Fulde, J. Keller and G. Zwicknagel, to appear in: Solid State Physics vol. 41, eds. F. Turnbull and H. Ehrenreich (Academic Press, New York, 1988). [4] O. Gunnarsson and K. Sch/Snhammer, in: Handbook on the Physics of Chemistry of Rare Earths, vol. 10: eds. K.A. Gschneider Jr., L. Eyring and S. Htifner (North-Holland, Amsterdam, 1987) p. 64. [5] N.E. Bickers, D.L. Cox and J.W. Wilkins, Phys. Rev. B36 (1987) 2036. [6] a) Y. Baer and W.-D. Schneider, in ref. [4] p. 1. b) Y. Baer, F. Patthey, J.M. liner, W.-D. Schneider and B. Delly, paper presented at the 8th meeting of the Condensed Matter Division of the EPS at Budapest, Hungary 1988 (to appear in Physica Scripta). [7] J.W. Rasul and A.C. Hewson, J. Phys. C 17 (1984) 3332. [8] P. Coleman, Phys. Rev. B 29 (1984) 3035. [9] J. Holm and K. Sche,nhammer (preprint); H. Keiter, C. Mielsch and Q. Qin: 8th meeting of the Condensed Matter Division of the EPS; T. Pruschke (private communication). [10] H. Keiter, Z. Phys. B 60 (1985) 337. [11] N. Grewe, T. Pruschke and H. Keiter, Z. Phys. B 71 (1988) 75. [12] N. Grewe, Z. Phys. B 67 (1987) 323. [13] E. Miiller-Hartmann, Z. Phys. B 57 (1984) 281. [14] N. Grewe and H. Keiter, Phys. Rev. B 24 (1981) 4420. [15] N. Grewe, Solid State Commun. 50 (1984) 19. [16] Y. Kuramoto, C.-I. Kim and Y. Kasuya, Proceedings of the LT 18 (Kyoto). [17] M. Schlott, H. Schaeffer and B. Elschner, Z. Phys. B 63 (1986) 427.