Re Tκκ is not the level shift

Solid State Communications,

Printed in Great Britain

Pergamon Press.

Vol. 22, pp. 63-65,1977.

Re Tkb;IS NOT THE LEVEL SHIFT E.W. Fenton Physics Division, National Research Council of Canada, Ottawa KIA 0R6, Canada (Received 12 November 1976; in revised form 23 December 1976)

In a time-independent theory or in a Greens function formalism, the shift in energy of a k-wave electron state due to an impurity in a metal is not equal to the real part of the forward-scattering T-matrix, Re Tkk, as often assumed.

IN A NUMBER OF theories concerning dilute alloys [l--S], the shift in the energy level of an electron k-wave state in a metal caused by a single impurity, from the corresponding state in the pure metal identified by turning off the impurity potential, has been taken as the real part of the forward-scattering T-matrix, Re T,. We will show that this assumption or derivation is not correct, and in at least one case leads to errors which become arbitrarily large in the strong-coupling limit [6]. The statement in the above title is known to a large number of physicists. However identification of the level shift as Re T, occurs frequently in the literature, and it has been suggested [S] that the results obtained in reference [6] which contradict this were in fact an aberration of the Green’s function formalism which was used. We will therefore discuss directly the derivation of this identification which is sometimes referenced or used. The following sequence of equations has been used by Stern [2] and by Jones and March [7]:

T-matrix, to first-order in (vol.)-’ or in c, the concentration of impurities near the dilute limit. However we do not think that this physical argument is correct, as will be discussed below. We investigate equation (3) by using the same Lippmann-Schwinger scattering equations on which the T-matrix is based [8,9] to obtain an exact formal expression for I $k). This investigation will show that with the Stern definition of the T-matrix [2], or with a definition in reference 9 appropriate to the LippmannSchwinger equations when a level shift occurs, equation (3) does not mean that the level shift is Re T,+ We will discuss an example which shows that the arguments made are not merely academic, because in this example the level shift is vastly different from Re T, [6]. When the perturbing potential causes a level shift, the “in” and “out” scattering states are given exactly by equation 4-200 of reference [9]:

I$:) = I$$+

=

Ill/E)

lim VOE~-(@

+ liio 1 kp

Equation (1) is the Schrodinger time-independent equation for the pure metal. Equation (2) is the Schrodinger equation for the metal with an impurity which changes the hamiltonian to Ho + V. The quantity ($EIVI$k) in equation (3) is identified as CJI~ITlJl~), or T,, by Stem [2], and as the “forward scattering amplitude” by Jones and March [7]. In the denominator on the right-hand-side of equation (3) Stern and Jones and March argue that physically the function l$k) should differ from the pure-metal function I$$ only at and near the impurity and so the denominator in equation (3) is just [ 1 + O( 1/Vol.)]. If this argument is correct, then near the infinite-volume limit the level shift is the real part of the forward-scattering

(V-A) +A)+in

Tl?k Ek - Ekl f ip

Itif)

(4)

Iti:+

(5)

Id&.

(6)

Here A is formally a level-shift operator given by: Al@

=

c (

I$$&,

-@)($:*I

=” (Ek - E;)I$&

(7)

= Akl@. In equation (6) we have used the T-matrix definition 4-58a combined with equation 4-198, in reference [9]. In the macroscopic but formally-finite crystal volume, either the in or out states are eigenstates of H and may be used in equation (3). Using equation (3) and the scalar product of ($:I with equation (6) we obtain: 63

Re Tfi IS NOT THE LEVEL SHIFT

64

(8) The denominator in the first term on the right-handside diverges and so this term contributes zero. Equation (3) is reduced to equation (8) which does not involve the T-matrix in the limit and which simply states that the level shift is equal to the level shift. Formally, ($:I$$ is divergent and so, even taking Stern’s definition of the T-matrix [2] as ($:I I’l$i), equation (3) does not state that the level shift is equal to Re Tkh- The scalar product (9) does not diverge because, as shown for example in reference [9] (p. 298) the divergence of ($J:[$:) formally cancels out in C$~l$~). The divergent mixed scalar product ($:I$;) occurs only for the itinerant-electron k-wave representation. For bound impurity states or in the lattice-site (Wannier) representation, an equation analogous to equation (3) does define a non-zero level shift. However to summarize our arguments to this point, the k-wave level shift cannot be taken as Re Tkk as a result from equation (3).

Equation (8) but with the Stern definition of Tti which eliminates the second term on the right-handside, is analogous to one obtainable in the Greens function formalism [6]. We start with a well-known equation [6, 10, 111: G,,(z)

= G,O(z)&,p +

G:(z)T,,fz)G:,(z). (10)

z is a complex frequency. Usually n is written in this equation as k [ 10, 1 l] however we have changed the notation [6] to emphasize that G:(z) is the “free” propagator for the electron in a fevel-shifted reference system: G;(z)

= (z-.@-I;

E,O = Ek” + Ak.

(11)

Using equation (10) and a Dyson equation for G,,(z), the electron self-energy in the full propagator, G$, = GO-’ n - C,, is given by [6]:

E”(Z) =

T,,(z) 1 + TJz)(z

-E,O)-’

’

(12)

X,(z) is zero at the pole of G:(z), which means that the pole position of the exact propagator G,,(z) is exactly the same as the pole position of the referencesystem propagator G:(z) (although the pole-strengths are different). This result is just equivalent to an old argument by Bench-Bruevich and Tyablikov [ 121, that

Vol. 22, No. 1

a different pole position for G and shifted energy levels of the perturbed system cannot be obtained in any perturbation theory which uses an expansion in the reference propagator Go which stops at finite order. This argument applies even when the finite-order expansion is formally exact, as in equation (lo), and shows that the T-matrix quantity in equation (10) cannot be identified with the level shift as in for example reference [ 11. The T-matrix must be defined in terms of referencesystem states in which any level shift due to the impurity potential Vis already incorporated [6,9] and, as above, the level shift is not given by the real part of the forwardscattering T-matrix. The problem of actually calculating the level shift E, - Ez still remains. There are at least two possible approaches. For the general case the level shift can be calculated using a formal expression with the timedevelopment operator which was orginially derived by Cell-Mann and Low [ 131. In a second approach we have used equation (10) where time-ordering is automatically included [6]. This procedure yields an exact result for a model Kondo system described by a Kondo Hamiltonian with exchange interaction J and scattering potential V due to a magnetic impurity with both interactions delta-functions in r (measured from the impurity site). The electron density of states curve is taken as flat in this model, which along with the delta-function impurity potentials means that the model itself (and not any approximation procedure) has rigid-band character. For this model, the exact result for the level shift is not equal to Re Tti [6]. No exact theory exists as yet for Re Tkk in this model Kondo system. However when Re T, reaches a maximum value at or near the Kondo temperature, then Re T, is proportional to J-’ times the exact result for the level shift (when V = 0). The error in using the incorrect assumption that the level shift is Re TM is therefore a divergent error in the J + 0 limit for the Kondo system [6, 141. Although lessspectacular errors would occur in general, nevertheless the general point is clear, that for k-wave states the level shift is nor equal to the real part of the forwardscattering T-matrix. The situation for the imaginary part of the k-state self-energy due to the impurity is different. For an impurity potential which is a delta-function in r, we have shown that for energies near to the Fermi level, Im .&(a + is) is equal to Im Tnn(w + i6) with Ez = Ei -t Ak [6]. Of course this is expected, because the lifetime of an electron k-state has a direct physical interpretation in terms of the imaginary part of the forward scattering T-matrix, whereas no similar physical relation exists between the level shift and Re Tkk. In conclusion, we stress that the null result for level shifts between a suitable reference-state spectrum in the

Re T,

Vol. 22, No. 1

65

IS NOT THE LEVEL SHIFT porated into the reference spectrum.) This does not change the fact that the level shift of a k-wave electron state in a metal is not equal to Re T,.

Lippmann-Schwinger equations and the exact scattering states occurs when the reference state and the exact scattering state both extend through the entire volume of the crystal. For bound impurity states, or if a latticesite representation is used [ 151, non-zero level shifts can be determined using an equation similar to equation (3). (Of course, in this case the level shift need not be incor-

Acknow/e&emcnr,s - I am grateful to P.T. Cole&lge, C.R. Leavens, and R. Taylor for useful discussions.

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P.T., J. Phys. F:Mefal Phys. 5,13 17 (1975). Equations

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ROMAN P., Advanced Quantum Theory, pp. 286-344. Addison-Wesley, Reading (1965) especially Section 4-4~. As the author comments on p. 344, the mixed scalar product relation of equation 4-204 when level shifts occurs is incompatible with equation 4-53 (modified for level shifts as discussed on p. 343) which is equivalent to equations (6) and (8) herein. However in the scalar product of Gk with equation 4-202 a ratio _+irllf. iv becomes + iv/(- E t iv) if the perturbing potential H’ is defined as lim (H-H” + E) when level shifts occur.

Theor. Phys. (Kyoto)

In this case equations

50,1797

6,363

3.2, 3.3, and 3.11, and compare 3.2 with

(1973) Equations

2.4 and 3.2. 4 and 5.

(1976).

to Relativistic

Wiley Interscience,

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4-203 and 4-204 do not follow from equatios-202

London

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is resolved.

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See Figure 2 of reference [6] for a dramatic contrast between values of the level shift and Re Te for a realistic value of J in a Kondo system.

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