LOW ENERGY DENSITY OF STATES FOR DISORDERED CHAINS

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LOW ENERGY DENSITY OF STATES FOR

DISORDERED CHAINS

S. Alexander, J. Bernasconi, R. Orbach

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, suppf6ment au no 8 , Tome 39, aolit

1978,

page

_{~ 6 - 7 0 6}

LOW ENERGY DENSITY OF STATES FOR DISORDERED

CHAINS

f

S. Alexander, J. Bernasconi and R. Orbachf*

Physics Department, University of California, Los Angeles, Ca. 90024, U.S.A. (Permanent address : The Racah T n s t i t u t e , Hebrew University, JerusaZem,IsmeZ)

"

Brown Boveri Research Center, CH-5405 Baden-Datt~il, Switzerland

XX

Physics Department, University of California, Los Ange l e s , Ca. 90024, U. S. A.

Rdsum6.- Nous calculons exactement la limite basse 6nergie pour la densitd d'dtats de chaines lind- aires B couplage aldatoire entre premiers voisins. Nos rdsultats peuvent Stre appliquds aux vibrations du rdseau d'une chazne

B

constantes dlastiques aldatoires ; B la diffusion de particules clas- siques avec des taux de transfert algatoires ; et aux bandes dlectroniques liaison forte avec ddsor- dre corrdld diagonal et non-diagonal.

Abstract.- We calculate exactly the low energy limit for the density of states of linear chains with random near-neighbor coupling. Our results are applicable to lattice vibrations of a chain with random force constants ; diffusion of classical particles with random transfer rates ; low lying spin wave states of the random Heisenberg chain ; and tight binding electron bands with correlated diagonal and off-diagonal disorder.

E 1. INTRODUCTION.- The purpose of this paper is to

xoE

= (E + g1 + present exact results for the low energy limit of

where the density of states for random one dimensional

systems. Our results will be useful for the deter- 'nE = 'n-i ,ncl

-

( ~ n ~ / ~ n - : ) ~

mination of low temperature specific heats ; auto- with g E obeying the iterative equations correlation functions at long times~and estimates

g n E

-

D I I ~ ~ , ~ + ~ ) + 1 1 ( g ~ + ~ ~ + '-])E (4) of the localization of excitations for low energy

From Eq. 4, the probability distribution for g E

states. Weexhibit exact results for these quantities n '

f E(g), obeys the integral equation for random one dimensional systems which generate

f eigenvalue equations of the TEplitz type :

E

-W n,n-1 X n-i + (E+Wn,n-l+Wn,n+l)Xn E

-

We shall be interested in situations where the W are random variables distributed independently ac- cording to a probability distribution p(W). Physi- cal systems which fall into this category are :

a) lattice vibrations of a linear chain with random force constants /l/ ; b) diffusion of classical p a r ticles with random transfer rates 121 ; c) low lying spin wave states for the random linear Hiesenberg chain 131; and d) tight binding bands for electrons with correlated diagonal and off-diagonal disorder 141.

2. MATHEMATICAL FORMALITIES.- The detailed mathematical techniques will be described in detail else- where 151. Sunrmarizing, we calculate the statistical

E

average of XO

.

One has, for a definite distribution of W, the result

This allows us to write the statistical average of

For a large class of distributions p(W), and for small E, the leading contribution to integrals such as equation 6 is given correctly by replacing by a delta function :

fE(g)

=

6(g

-

<g>")

,

E -+ 0 ( 7 )

where <g>E is the solution of

E P (cg>'

+

E)

I

dW P(W) (W + <g>€ + E)-' ₍₈₎

Clearly, the small E behavior of equation I is do- minated by the behavior of p(W) near W = 0. We con-

sider three cases.

3 . DENSITY OF STATES.- (a) li% p(W) = 0. One can

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define

so that, from equation 8, to leading order

<g>E = €112 <l /w>-1/2 (10)

(b) l i m ~ , ~ p(W) constant. To leading order, this results in

<g>E=

(€1

l

an,

l

)

'l2

( 1 1 )

(C) P(W) diverges as W , but is integrable. We set

p(~) =

,

a

< I (1 2)

with a suitable cutoff for normalization purposes. This distribution leads to

One notes that for all three cases one has <g>E >> E as E -+ 0, so that

xoE

' (14)

E

In all three cases cited above, X. is the diagonal element of a Green's function. The density of states is the inverse Laplace transform of the inverse Laplace transform of X E. For lattice vibrations, for example, w2 plays the role of E, so that one obtains for the "phonon" density of states for case (b)

while for case (c) N(w) a W -a/(2-a)

In other cases, the relevant density of states is proportional to We see that the low e n e r gy density of states is enhanced compared to the usual one dimensional behavior. This must reflect the localization of the small E eigenstates on ac- count of the presence of small transfer rates W.

4. LOCALIZATION OF EXCITATI0NS.- Because of the appearance of W in the definition of g E (equation 3).

n

one cannot determine easily the spatial extent of the eigenfunctions. Consider, for example, case (c) above. For this situation, X E is the Laplace transform of the occupation probability of site n at time

t when site 0 was occupied at t = 0. One finds 151

X,(t)

=

t -(l-a)/(z-a) (17)

Moreover, the X (t) are monotonic at all times :

n

cupations must obey the inequality

2

t (l-a)/(z-a) (19) This sets a lower limit on the extention of the eigenfunctions for E Q I/t. That is, we find

C,

In1 $E2(n)

rl

E -(l-a)/(2-a) (20) where $ (n) is the eigenfunction at site n for

E energy E.

5. CONCLUSION.- We have shown how the low energy density of states and the spatial localization of eigenfunctions in random linear chains governed by equation 1 can be calculated for essentially arbri- trary probability distributions. We have (briefly) discussed the associated time development of the excitation amplitude at the initial and spatially separated sites at subsequent times. One dimensional systems with random site-site couplings are known /6/ so that our conclusions may by direct relevance to experiment.

The authors adknowledge very helpful con- versations with Prof. T. Holstein, and Drs.

W.R. Schneider and H.J. Wiesmann. This research was supported in part by the U.S. National Science Foun- dation and by the U.S. Office of Naval Research.

References

/l/ Dyson, F.J., Phys. Rev.

92

(1953) 1331 ;

Schmidt. H., Phys. Rev.

105

(1957) 425 ;

Domb, C., Maradudin, A.A., Montroll, E.W., and Weisse, G.H., Phys. Rev.

115

(1959) 24

/2/ Alexander, S., Bernarsconi. J., and Orbach, R., Phys. Rev., accepted for publication (1978) /3/ Alexander, S.. and Holstein, T., Phys. Rev.

accepted for publication (1978) /4/ Governed b the Hamiltonian

H m

-

'n'6n-1

-

,n+ 'n,n+>

CA

n' + 'n,n+~ k + c +h.c.]}

where the c2:" on site Fermion operators, and the Vn are Pandom.

/S/ Reference 2 and to be published by the same authors.

/6/ An example is quinoliniom (TCNQ)z. See Azevedo, L.J., and Clark, W.G.. Phys. Rev.

B16

(1977) 3252