LT 21 Proceedings of the 21st International Conference on Low Temperature Physics Prague, August 8-14, 1996

Part $5 - LT Properties of Solids 2: Correlated electrons

D y n a m i c a l l o c a l i z a t i o n a n d e l e c t r o n c o r r e l a t i o n

Talmshi Hotta and Yasutami Takada

Institute for Solid State Physics, University of Tokyo 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan

By investigating the divergence in the electronic self-energy in an electron-phonon system with the on-site Coulomb repulsion U, we study the localization due to U and/or the phonon-mediated attractive interaction V. In addition to the generalization of the Mott transition in the Hubbard model by including V, particular attention is paid to the localization due to the confinement by local phonons occurring at finite energies which may be called as the dynamical localization.*

1. I N T R O D U C T I O N

Competition between the attractive interaction between electrons mediated by local phonons and the electron-electron short-range repulsion leads to sev- eral interesting phenomena including the interaction- induced electron localization. In order to make a quantitative discussion on such a localization, we consider the Hubbard-Holstein (HH) model, de- scribed by [I]

x-" tlj , + U E # Z H = - f " ~ c i ~ c j ~ niTnil -- ni~ i,j,a - - i i,~

n,.(a, + + (1) i,a i

where ci~ is an operator to annihilate a spin-~ elec- tron at site i, ni~ -- c~oci~, tij the hopping integral between i- and j-sites, d the spatial dimensionality, p the chemical potential, U the on-site Coulomb inter- action, ai the phonon annihilation operator at site i, ~ the non-dimensional electron-phonon couphng

�9 constant, and w0 the phonon energy which will be taken as units for energies hereafter.

From the physical meaning of the single-particle Green's function G, electron localization is signified by the vanishment of G. In terms of the self-energy E, the condition of G = 0 at w = w* amounts to the divergence in V.(w) as

- (2)

In the HH model with a sufficiently small bare band- width W, the localization is expected to occur at w* -- 0 for U ~ 2a. This will be discussed in Sec. 2.

On the other hand, for U = 2~, such a divergence does not appear at w = 0 even at W = 0. Instead, we can find the behavior of the form of Eq. (2) at nonzero w*, i.e., the dynamical localization, which will be argued quantitatively in Sec. 3.

2. L O C A L I Z A T I O N AT U ~ 2a

In the atomic limit, an analytic expression for can be obtained with use of the Lang-Firsov canon- ical transformation, irrespective of U or ~ [2]. At half-filling, a* in Eq. (2) is given exactly by

~, = eO + ([U/2 - ~l + t ) 2 / e = 0 t! [~,.2 _ ( I U / 2 - ~[ + t)2] 2' (3)

for arbitrary U and ~. In Fig. 1, we plot a* corre- sponding to w* = 0 as a function of U for c~ = 0, I, and 2. A nonzero value of a* for U p 2a leads to the finite charge-excitation gap at the Fermi level, sug- gesting the interaction-induced localization. Quanti- tatively, magnitude of the gap is equal to ]U - 2a[ in the present case. At ~ = 0, this is nothing but the Mott-Hubbard gap.

Now we consider the situation in which electrons begin to itinerate. Due to the polaron nature of an electron motion, the kinetic energy is not simply rep- resented by W but We-% Thus, the interaction- induced localization exists if the charge gap iU - 2~ I is large compared to We-% ,~t [U - 2~] ~ We-% the metal-insulator transition is expected to occur,

* This work has been partially supported by Grant-in-Aids for Scientific Research given by the Ministry of Education, Science, Sports and Culture of Japan.

Czechoslovak Journal of Physics, Vol. 46 (1996), Suppl. $5 2625

just as the case in the Mott transition. Quantita- tively, the transition occurs at U ~. 1.7W for a = 0 in the limit of infinite spatial dimensionality (d --, oo) in which E is known to be a function of only the energy variable w [3].

3. D Y N A M I C A L L O C A L I Z A T I O N

At U = 2a, we obtain a metallic ground state, irrespective of W; E does not diverge at uJ = 0 and thus no charge gap exists. (Note the or* in Fig. 1 vanishes for IU - 2tx[ --* 0.) This different situation calls for an alternative approach to this case. In order to make a quantitative evaluation of E when W is small compared to w0, we consider the d = oo case at which the itineracy of electrons is expressed in terms of the so-called effective dynamical field A. We have developed the expansion theory with respect to this A in the d = oo HH model [4]. In an actual calculation for E as well as .A, we adopt the Bethe lattice for which the density of states is given by (8/ w v ' ( w l 2 Y -

As the result of such calculations, we find the dynamical localization for a sufficiently small W. Namely, ~(w) diverges at w = w* ~ 0 and at the same time ~4(w*) vanishes. With the increase of W, w* shifts to a higher value, but a* remains the same as the value in the atomic limit. Those val- ues of ~* at ~ = 1 and 2 are given in solid cir- cles in Fig. 1. As W increases further, ~* shifts so much that it touches the tail of the phonon side- band around w = w0, resulting in ImA(w*) ~ 0 even though ReA(w*) = 0. The nonzero ImA(w*) leads to the suppression of the divergence in ~, indicating the absence of the dynamical localization. This critical value of the bandwidth, We, is important physically, because it determines the transition point for the dy- namical localization-delocalization. For example, We is given by Wc = 0.75 for ~ = 1 and the temperature T = 0.01. The values of We depend strongly on but only weakly on T.

By evaluating We with the change of ~ at U = 2e, we obtain the phase diagram for the dynamical localization-delocalization transition in the (W, e)- plane [4]. Because of U = 2~, this may also be re- garded as the phase diagram for the transition in the (W, U)-plane. In this view, this diagram agrees qualitatively with that of the Mott transition at

= 0. We find that the dynamical transition oc- curs at U ~ 2.8W, universal in the accuracy of our numerical calculation.

4. D I S C U S S I O N

We have discussed the interaction-induced local- ization as well as the dynamical localization in the HH model as a generalization of the Mott transition. For the localization at w* = 0, we obtain a symmetric behavior in a* with respect to U - 2~ at half-filling. Actually, two distinct insulating phases appear, de- pending on the sign of U - 2~: For U < 2a, the insu- lating phase is the charge-density-wave (CDW) one or the array of the immobile bipolaron state. On the other hand, the spin-density-wave (SDW) phase, i.e., the antiferromagnetic state, appears for U > 2tx. Be- cause of the retardation effects of phonons, a quanti- tative difference can be expected between the SDW- metal transition and the CDW-metal one. Clarifi- cation of this difference is a direction of future re- searches in this problem.

a/w- o , ' / i'/ L T ~ t / a

o= o i / " ~ t / I S , , / - - ' ) ' , / /

o : ._4\/~ : ' ( a " ~ - - ~ . ",L~. .' /

�9 e ... " x / ., / = ~ - \ - - - - ~ \ . W , . _ . , ". / " ~ ( . I

o ' , / . /',, :

\4 " , 0 2 4 6

U ( units of ta o)

Figure 1: Weight ~* of the divergence in the self- energy at w = w*. The solid, dashed, and long- dashed curves indicate, respectively, the results for a = 0, 1, and 2 at w* = 0 for U ~ 2a. For U = 2t~, a*'s corresponding to the dynamical localization (w* ~ 0) are shown as solid circles.

R E F E R E N C E S

[1] We employ such units as h = kB = 1. [2] Y. Takada and T. Higuchi, Phys. Rev. B52,

12720 (1995). [3] A review on d = oo electronic systems is given

in: A. Georges, G. Kotliar, W. Kranth, and M. J. Rozenberg, Rev. Mod. Phys. 68, No.1 (1996).

[4] T. Hotta and Y. Takada, Phys. Rev. Lett. 76, 3180 (1996).

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