ROLE OF LONG RANGE COULOMB REPULSION IN METAL-INSULATOR TRANSITIONS

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ROLE OF LONG RANGE COULOMB REPULSION

IN METAL-INSULATOR TRANSITIONS

T. Izuyama

To cite this version:

T. Izuyama. ROLE OF LONG RANGE COULOMB REPULSION IN METAL-INSULATOR

TRANSITIONS. Journal de Physique Colloques, 1976, 37 (C4), pp.C4-227-C4-229.

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JOURNAL DE PHYSIQUE Colloque C4, supple'ment au no 10, Tome 37, Octobre 1976, page C4-227

ROLE OF LONG RANGE COULOMB REPULSION

IN METAL-INSULATOR TRANSITIONS

T. IZUYAMA

Institute of Physics, College of General Education, University of Tokyo, Komaba, Meguro-ku, Tokyo, Japan

Resumi:. - On admet d'ordinaire que les transitions de phase du second ordre sont influencees par l'instabiliti: de Peierls. Bien que les idi:es originales de Mott donnaient une grande importance

a l'interaction coulombienne,

A

grande distance les theories les plus rkentes font appel a

un modkle qui tient compte de l'interaction

A

courte distance seulement. En consequence, on decrit habituellement les transitions de Mott ou de Peierls comme des transitions du deuxikme ordre, alors que Mott predisait une transition du premier ordre. Nous montrons dans cette Btude prbliml- naire comment la nature longue distance de l'interaction electronique peut donner aux transitions de Mott ou de Peierls un caractkre du premier ordre.

Abstract. - The Peierls instability is usually supposed to induce the second order phase transition. Althought the long range nature of the Coulomb interaction was important in Mott's original argument, the advanced theories have resort to a model with the short range part of the interaction only. As a result, the Mott or Peierls transition has usually been described as the second order transition, though Mott obtained the first order one. In this preliminary report we show how the long range nature of the electron interaction may endow the Mott or Peierls transition with the first order character.

R. E. Peierls [l] pointed out that the one-dimensional 2) when the coupling strength reaches a critical metal is always unstable against the formation of a value, the CDW amplitude suddenly becomes finite. standing wave with wave number 2 k, due to any This implies that the Peierls transition becomes the mechanism which couples the state k, with

-

k,.

In first order one.

three dimensional case, the standing wave formation is not always possible but may be possible on certain occasions as can be seen in Slater's band theory of antiferromagnetism and Overhauser's SDW theory.

A more illuminating argument was given by

N. F. Mott [2] on the correlation-induced transition of a many-electron system from its metallic phase to an insulating state. In his original argument, the long range part of the Coulomb interaction was very important and led him to conclude that the transition would be of the first order. Much more refined theories [3, 41 developed later have revealed that the Mott transition is caused primarily by the short range part of the Coulomb interaction. The Mott transition described by these theories is of the second order. Since the long range part is neglected by these theories, consideration on this part is still important for achie- ving ultimate character of the Mott transition.

If the long range nature is so important, then it will affect the phonon-coupled Peierls transition as well. This problem was considered in our previous paper [5]. Our argument was based upon RPA, which is far from complete. With this reservation the conclusions derived are :

1) The CDW is not possible even in one-dimensional metal in its ground state if the coupling between k, and

-

k, is weak enough, and

So far as the present problem is concerned, there is no distinction between the phonon-coupled CDW and the correlation-induced SDW formations. Presu- mably the paramagnetic insulating phase formation would also be disturbed by the long range Coulomb force in the same way, because the physics seems to be similar to that in the case of CDW or SDW.

For simplicity we consider here again the one-dimensional half-filled band. We will deal with the same model as that considered in our previous paper 151.

Strictly speaking, purely one-dimensional lattice is always collapsed by the zero-point vibrations. Such break-down does not occur in the crystal of regular array of mutually parallel linear chains, where acoustic phonon modes propagating in the direction perpendicular to the chains possess non-vanishing frequencies. To avoid all these complexities, we consider the single mode model, where a single phonon mode with wave number 2 k, is retained and all the others are neglected. In the case of SDW, all the spin density fluctuations are neglected except for the single considered mode with wave number 2 k,. The fluctuation of long range order is regarded as a different problem, though it is important, and will not be considered here. Here we restrict ourselves to the long range part of the density fluctuation, which is mostly affected by Mott's screening mechanism.

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C4-228 T. IZUYAMA

We adopt again RPA. In this sense our analysis is still preliminary. However, we will consider also the parquet diagrams. It is found that they do not affect our conclusions. ,Thus basic points of our previous assertions are reconfirmed, though the present result is slightly different from the previous one.

We start with the Hamiltonian X = X,

+

X,,

where

X, =

C

(- 2 cos k) a:, a,,

+

o b t b -t

k,a

and

The notations are given in ref. [5]. Only the long range part of the Coulomb interaction is considered by adop- ting the following model interaction :

For small q we have J(q) = 2 e2 In (l/a

I

q

I).

See refs [5] and [6] for, implications of this J(q).

The case of SDW is achieved by

X, =

C

(- 2 cos k) a:, a,,

-

IM+(- n) M-(n)

.

(l') Where I is essentially Hubbard's I, and

In the mean field theory, where (M,

-

<

M.,

>)

(M-

-

<

M-

>)

is neglected, the following one-to- one correspondense is found between (l) and (1') :

We first diagonalize X,. Then

where ~ ( k ) is defined in I for the CDW and the replacement 6' sin2 k + 6' in this definition gives ~ ( k ) for the SDW. In the above,

where the double sign takes

+

for the CDW and - for the SDW, y, is defined in I for the CDW and the SDW case is established by the replacement 6' sin2 k -+ 6,. The difference between sin2 k and 1 is trivial, because important region of k is in the vicinity of 4 2 .

The first step of our reductio ad absurdurn is to assume that the ground state is derived from the filled cl-band by adiabatic switching on of X,. This is in fact the usual assumption, when the CDW (or SDW) is believed to exist. Under the adiabatic hypothesis the rigorous screened Coulomb potential is %(g, o = J(q)/{ 1

+

J(9) m q , o )

),

where m) is the sum of all such polarization diagrams that receive momentum q and frequency o from the J(q) interaction and can not fall into two separate parts by elimi- nating any single Coulomb line. This type of diagrams is labelled as proper.

The correlation energy U per atom is also expressed in terms n(q, m). We differentiate such expression with respect to 6'. Then it follows that

where a finite temperature T is considered first and

o = 2 n in/T with n = 0, t_ 1, $. 2,

....

Later on we set T -+ 0 and (Ti2 xi) d o

.

0

Our first approximation is to replace %(g, m) by %(g, 0) E %(g) in the rigorous expression (3). This

was called the static approximation. Notice also that

C

W

m) =

<

P, P - , > p -= S(q),

where the suffix P indicates that only the proper diagrams should be counted. Then

Let us consider the ground state. In the lowest order approximation, S(q) is given by the unperturbed one, and Z(q) by the RPA result. They are listed in table I of I. The present procedure seems to be a little more natural than the previous one.

It is noticed that

and in the small q region %(g) LX 17(q)-'. We notice also

that 9, c q c g, with q,

=

const. 6 is the most important region. From Table I of I, it follows that

(for small 6).

Therefore, the correlation energy loss caused by the CDW (SDW) is

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ROLE OF LONG RANGE COULOMB REPULSION IN METAL-INSULATOR TRANSITIONS C4-229

In (116). For small S the loss overcomes the gain due to figure 1 are cancelled out. This is seen by removing a

X,, which is vertex @ from the electron (or hole) line to the hole

The CDW is not possible for small v. This conclusion agrees with the previous one.

In view of renormalization group theories [7, 101 on the BGD model [l 1, 121 one might think that the appearance of ln3 (116) is simply a symptom of the serious divergences inherent in one-dimensional metals and after adequate treatments the correlation would strengthen (rather than resist against) the CDW. This point was already argued in I, bur there we overlooked the fact that in BGD theories the coupling g, is mis- sing [13]. This coupling is crucial in our analysis. It is straightforward to see that many of the parquet diagrams (zero-sound and Cooper channels) cancel each other in our case. Due to the electron-hole sym- metry the cancellation is even more spectacular in our case than in the BGD model. First, the

f(k)

ladders considered in a single closed loop which could contribute to

(electron) line within the same loop. The third order AL diagrams as shown in figure a can contribute to

a

S(q)/a(62). It is seen, however, that their contribu- tion is at most ln3 (116). A more detailed report requires space and will be presented elsewhere.

Finally we mention an implication of the present result. Suppose an (( external D field with amplitude 6

is applied to the system, giving rise to the Hamilto- nian (1). a(E,

+

U)/a(S2) is the induced CDW amplitude due to this external field. If the derivative is minus, the system fitts itself to this field so as to achieve vanish after the @-integration. Consider next the AL

lower energy. In our case the derivative is plus. Then diagrams for S(q)- The second order ones as shown in

the system.s energy is rather raised, though the field couples to the system linearly. This seems to contradict with the physics. We do not claim, however, that the state obtained from the ground state of X, through the adiabatic hypothesis is the real ground state. Such state would be an eigenstate of X but can not be the lowest energy state when the field exists. The long range Coulomb force is so important that the weak external field is incapable of producing the standing CDW in the ground state. The result would rather encourage the

FIG. 1. renormalization group approach.

References

[l] PEIERLS, R. F., Quantum Theory of Solids (Oxford Univer- [8] SOLYOM, J., J. LOW Temp. Phys. 12 (1973) 547.

sity Press) 1955. _{[9] CHUI, S. T., RICE, T. M. and VARMA,}_C.M., Solid State [21 MOTT, N. F., PYOC. Phys. Soc. (London) 62 (1949) 416. Commun. 15 (1974) 155.

[3] HUBBARD, J., P ~ o c . R. SOC. A 281 (1964) 401. _{[l01 FUKUYAMA,}_{H., RICE, T. M,, VARMA,}_{C. M. and HALPE-} [4] BRINKMAN, W. F. and RICE, T. M., Phys. Rev. B 2 (1970) _RIN,_{B. I., Phys. Rev. B}₁₀_{(1974) 3775.}

4302.

[5] IZUYAMA, T. and SAITOH, M., Solid State Commun. 16 (1975), [l1] BYCHKOV' YU'

*"

L' and DZYALOSHINSK1ly I' 549 referred to as I. E., Sov. Phys.-JETP 23 (1966) 489.

[6] IZUYAMA, T., Prog. Theov. Phys. 22 (1959) 681. [l21 DZYALOSHINSKII, I. E. and LARKIN, A. I., SOV. Phys.-JETP

[7] MENYHARD, N. and SOLYOM, J., J. LOW Temp. Phys. 12 34 (1972) 422.