Lifshitz singularities in the total and the wavenumber-dependent spectral density of random harmonic chains

Physica 145A (1987) 161-189 North-Holland, Amsterdam

LIFSHITZ SINGULARITIES IN THE TOTAL AND THE WAVENUMBER-DEPENDENT SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS Th.M. NIEUWENHUIZEN

lnstitut fiir Theoretische Physik A, R.W.T.H. Aachen, Templergraben 55, 5100 Aachen, Fed. Rep. Germany and J.M. LUCK

S.Ph.T., CEN Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 18 March 1987

Lifshitz has shown that the density of states of random systems becomes exponentially small near the band edge. We perform a detailed computation in a one-dimensional system and calculate the periodic prefactor of this essential singularity. The analysis is extended to the wavenumberdependent spectral density. The periodic amplitude is derived for the exactly soluble cases, where

the distribution of masses is exponential. For binary distributions the leading behavior in the concentration of light masses is also derived.

1. Introduction

In 1964 Lifshitz 1) argued that in random systems the van Hove band-edge singularities in the density of states disappear and are replaced by exponential singularities. In the language of harmonic chains the underlying mechanism is that for having a mode with a frequency close to the maximal frequency one needs a large region containing light masses only. The probability of occurrence of such a region is exponentially small and so is the probability to find such an eigenffequency, i.e. the density of states. Several recent papers 2-4) have been devoted to the properties of such singularities near the band edge, and near so-called special frequencies of harmonic chains with binary distribution of random masses. A detailed investigation by the present authors 4) revealed that the exponential fall-off has a 0378-4371/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

162

Th.M. NIEUWENHUIZEN

A N D J.M. L U C K

periodic amplitude, which could be calculated in the limit where the probability of finding a light mass is small. The first aim of the present paper is to show that the result for the integrated density near the band edge remains valid if there is no gap in the mass distribution near the lightest mass. To do so, we start with exactly soluble cases with exponential distributions of the masses 5'6) (section 2.1, 2.2). The leading term and its first corrections are derived with a new method in section 2.3. It is shown that for the exactly soluble cases the periodic amplitude, which modulates the Lifshitz tail, can be expressed in terms of the solution of a second order differential equation. Next we extend the formalism to calculate Lifshitz singularities in the spectral density at fixed wavenumber p(q, to2). This quantity determines, for instance, the distribution of relaxation times in the present model, and the survival probability in one-dimensional trapping problems. The Lifshitz singularity in the spectral density at zero wavenumber causes the stretched-exponential decay of the survival probability for long times. In sections 3.1 and 3.2 we perform the calculations of the asymptotic 2 ~- 4 in exactly soluble cases. Again behavior of p(q, toe) for to2__~tomax periodic amplitudes, related to the solution of second order differential equations, modulate the Lifshitz tails. Section 4 is devoted to the relation with arbitrary mass distributions. In section 4.1 we discuss the connection with previously obtained results for the integrated density of states. Next, the leading correction terms are given for various forms of the mass distribution. In section 4.2 the results for the wavenumber-dependent density of states are generalized to the general situation. The special case of binary disorder is worked out in detail in the limit where the probability for finding a light mass is small.

2.

Lifshitz singularities in the density of states p(oJ 2)

2.1. Difference equations for H(oa e) We consider harmonic chains with random masses. Their equations of motion at frequency to read

- m n w 2a =a,+ l + a._ 1 - 2 a . ,

(2.1)

with boundary conditions a 0 = aN+ 1 = 0. In the present paper we take a diluted exponential distribution of the masses mn. Hereto we normalize the value of the smallest mass to be unity and make the decomposition

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS m n=l+Mx

n

(Mt>O),

163 (2.2a)

where the xn all have the distribution r(x) = p 6 ( x ) + (1 - p ) e-*O(x)

(2.2b)

and 6 and 0 are Dirac and Heaviside functions, respectively. The first quantity of interest is the integrated density of states (IDS) of the infinite chain H(0)2) = lim 1

(#0)~<0)2)

N----~~

(2.3)

where 0)i are the eigenfrequencies of the system. It happens that the IDS can be solved exactly for the mass distribution (2.2). The solution for p = 0 was discussed in ref. 5 and extended to p ~ 0 in ref. 6. We shortly recall these results. The expression for H(0) 2) can be obtained from the characteristic function g2( s~) = f log(1

+ ~/0)2) all(O)2)

(2.4)

because it decomposes along the negative real axis as 0 ( - 0 ) 2 -+ i0)

= ]/(0) 2) -i-

iTrH(0)2),

(2.5)

where 3' is the Lyapunov exponent (inverse localization length). The result from ref. 6 reads a(~:) = / , + (1 - p)rt(1 - C1) ,

(2.6)

w h e r e / , and r/ are defined in terms of ~: and M, cosh/x=l+½~

(Re/x/>0), (2.7)

7q = M ~ / ( 2 sinh ~ ) .

The coefficient C 1 in (2.6) is determined by the recursion relation Ck+l + Ck_l = pkCk

(k >l l) ,

(2.8a)

with 1 - e-2klx & =2 +

rlk(1 - p e -za~') '

(2.8b)

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Th.M. NIEUWENHUIZEN AND J.M. LUCK

subject to the boundary conditions C o = 1, C~ = 0. An explicit expression in the form of a continued fraction follows by iterating eq. (2.8a), 12(~)=/z+(1-p)~/(1-

1 Pl-

The merit of eqs. (2.6)-(2.8) masses has been carried out relations. In the more general nth order polynomial in front term recurrence relations7).

1

1

...).

(2.9)

P2-- P3--

is that the statistical average over the random at the cost of solving three-term recurrence case where the density r(x) in eq. (2.2) has an of the exponential, one ends up with (2n + 3)-

2.2. p(to 2) as an infinite series

In this section we obtain a series representation for the density of states (DOS) p(to 2) = dH(to2)/dw 2. We consider real frequencies and set ~---- --~o2 + i 0 ,

(2.10)

~2=2+2cose.

From (2.7) and (2.8), it follows that /~ = zri--ie + 0, 77= iM cotan( ½e),

(2.11)

tan(½e) (1 - e 2ik~) - iMk (1 - p e 2ike)

Oh = 2 +

According to (2.5), p(w 2) can be obained from the derivative of 12, which is given by

- p)~ ~a a ' ( ~ ) = 12 s i n h / z + (1 d~ . 2s~nh--~ = (Pk -- 2)Ck -- (1 --p)~7 dC1

(2.12)

The last factor is determined by the derivative of eq. (2.8a), dCk+l

d~

+

dCk_ 1

dC k

dp k

d----~= Pk--d-~- + Ck ~

"

(2.13)

Multiplying this equation by .Ck and summing from k = 1 to 0% one can rearrange terms and use eq. (2.8a) to cancel most of the contributions. The result is

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

dC~d~ -

k~l= C2

"

165

(2.14)

The density of s t a t e s ,0(60 2) only involves the imaginary part of O ' ,

p(w 2) =~-~1i{J2'(-w2 +i0) -J2'(-w 2 -i0)} = ---~-lIms2'(-wz +i0). (2.15) For calculating S 2 ' ( - w 2 - i0) = 0 * ( - 0 2 + i0) one has to t a k e / z = - T r i + ie + 0 instead of (2.11a). The result is that e is replaced by - e in Pk in (2.11c). D u e to the obvious relation (2.16)

p~(~) = p A - ~ ) = p - A , ) it is clear that

Ck(e ) = C_k(e ) .

(2.17)

H e n c e the combination (2.15) m a y be written as a sum from - ~ to + ~ . It happens that almost all other terms from (2.12) cancel by virtue of the recurrence relation (2.8) for k = 0. The final result is

1-_p ~1

P(°~2) - 2~- sin s

=

2

2

(2.18a)

(Ckvl'k + (Ck -- Ck)v2'k} '

where

(1 - - p ) Vl,k

--

(1 - p

e 2ike

--1

1 - - e 2ike

U2"k= 2ik sin e l - p e 2ike "

eZike) 2 '

(2.18b)

In the next section we show that the exponential Lifshitz singularity near e = 0 can be derived m o r e easily from this starting point. 2.3. Lifshitz singularity in H(to 2) near the band edge In the infinite sum (2.18) the individual terms are not small when e $ 0 ( r e m e m b e r that C o = 1). Nevertheless the resulting sum is small. In order to show this we write (2.18) as

-(1 -p)

-iA+~c

dt

2i tan ~-t -iA

-2

{C~Vl,t+(Czt-Ct)v2,t}

(A>0) (2.19)

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Th.M. NIEUWENHUIZEN AND J.M. LUCK

where C, and v t are formal analytic continuations of C k and vk, respectively. It can be verified that Ct---~0 for Im t--~ + ~ . Therefore the contour in (2.19) may be closed in the upperhalf plane. The equivalence of (2.18) and (2.19) is then clear. The exponential singularity originates from the behavior of the integrand near the poles of v r These occur at t = t~ with t, = (In p + 2~- i n ) / ( - 2 i e )

(n E 7/).

(2.20)

For small e the imaginary part of t, is large. Taking A in (2.19) large (1 ~ A ~ e -1 In 1 / p ) we may expand (2itan 7rt) -1 = ½ + e -2~it + e(e-4~rit) .

(2.21)

The constant term gives no contribution to the integral. To leading order in pl/,, we then have -iA+~

H,(to 2) = - ( 1 - p ) 2~ri

f -iA

d t -e2 r r i t ~ll_,tOl,t-~r ~2 (C~ - Ct)v2,t}

(2.22)

-~

It is then useful to introduce a scaled variable (2.23)

x = -2iet

which will be kept fixed, when the limit e--o 0 is taken. We also introduce a scaled function, which has a limit as e goes to zero, (2.24)

f ( x ) = Ctl,=ix/~2~ ) .

From eq. (2.8) it can be found that, up to terms of order e 2, f satisfies a second order differential equation, 1 - e -x f " ( x ) = 4Mx(1 - p e - x ) f ( x )

,

(2.25)

with boundary conditions f(0) = 1, f(+oo) = 0. It is clear that f has singularities at x = x n = - 2 i e t n with x,=lnp+2rrin

(n@2~).

Near these points it behaves as

(2.26)

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

f(x, + 6 ) = f , + q~,6 l n 6 + ~0,6 + 6(62 ln26).

167

(2.27)

Here f, is the limiting value f(x,) and q~ = - ( 1 - p)fJ(4Mpx,,). In terms of these quantities, eq. (2.22) takes the form -A+i~

l-p ~ p(w 2) = 4-1tie-----

f

dxe

¢~/~f (1 - p) e -x ~2. , ]. (1 _ p e_~)zf ~x)

-A-ioo

1 - e -x

+ x(~_--pe-X) (f2(x) -

f(x))}

(2.28)

where we have neglected terms of relative order e 2. We shall do so everywhere in the paper. The leading contribution in (2.28) comes when the second order pole "picks up" a derivative of exp(1rx/e). Using (2.27) we can also evaluate the first correction terms and find p(to2) - ( l--_p_ _) _2

~

2pe 2 . . . .

(pe

f, - f ~ ]'-~fn+2fnq~nlne+-Xn

21rin-,rr/ef 71" ~2

)

+ 2f,,(~°n + ~n) - 2fnq~,(ln 7r + Ye)/.1

(2.29)

Here Ye is Euler's constant. The In e term has arisen from the 6 In 6 term in (2.27). Integrating (2.29) with respect to to 2 we obtain the final result

=

¢

Ca (2.30a)

where the Qi(Y) are periodic functions with unit period. They are given by oo

Qo(Y)-(1-p)~

~] f ] e2~riny' ,=-~ - x n

(2.30b)

(1-p)~ E 2fnq~" e 2¢i"r p , =-= - rrx,

(2.30c)

p QI(y)-

Qz(y) - (1-p)~2 ~ { f ~X2fn" P , ~

2fntPn(1--1nTr--ye)

7rxn

2f,,~b,,l eZ=i,,y 7rx n ~ • (2.30d)

The parameters fn, ~o, and 46 are defined in eq. (2.27) in terms of the behavior

168

Th.M. NIEUWENHUIZEN

AND J.M. LUCK

6

'

I

i

I

I

I

'

I

J

I

'

I

t

I

'

I

i

J

5.1,t e, /'.B "rm

/'.2

3.5 3

I

5.2

6./'

7.6

8.8

10

r~/E Fig. 1. Plot of (1 - H)p -=/" versus ar/e in a typical case ( p = 0.2, M = 5), illustrating the periodic b e h a v i o r (2.30).

10

\ \

,,..=

t

I

0.2

I

I

O.&

I

M M+I

I

0.5

,

I

,

0.8

Fig. 2. Plot of the leading F o u r i e r coefficient f0 of the scaling function Q0, versus M / ( M + 1). Values of the probability p are indicated on the curves.

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

169

of the function f(x) near x n = In p + 27tin. They can be determined by solving numerically the second order differential equation (2.25). In fig. 1 we present a plot of (1 - H)p -'~/~ versus 7r/e, calculated from the infinite continued fraction (2.9) for a tyical case ( p = 0.2, M = 5.). We have also calculated the first Fourier coefficients (Inl ~ 10) of Qo(Y). The agreement with the data of fig. 1 is excellent for values of 7r/e as small as 5. In fig. 2 we show a plot of the leading coefficient f0 as a function of M/(M + 1) for different values o f p . Note that for M---*~,fo(X ) goes to unity for any value of p. This limit will be discussed in more detail in section 4.

3. Lifshitz singularities in the spectral density at fixed wavevector In this section we shall be concerned with the imaginary part of the two-point function G(q'~)=

(M~ 1) + c19 qq "

Here M denotes the diagonal random mass matrix, qb the non-random nearestneighbor force matrix, and q the value of the wavevector. Along the negative real axis this function has a cut. Its discontinuity is called the wavenumberdependent spectral density

P(q'wZ)=

27ril( G ( q , - w z + i O ) - G ( q , - w

2-i0)}

(3.2)

This is the quantity that we shall be interested in. An equivalent definition is p(q, w 2 ) = N---,~ lim -N 1 ~

~

a"(°)i2)am(Wi2) cos(nq - mq)8(~ 2 - ¢0~),

i = 1 n,rn=l

(3.3t where an(oJ~) are the normalized eigenfunctions of eq. (2.1). The average of

p(q, to 2) over wavevectors gives back the total density of states p(w2),

f ~-@q~P(q,oj2) = lim f 1i

s(o,

-

=

(3.4)

i=1

It has been shown 5'6) that whenever the IDS H(w 2) can be solved exactly, so can p(q, o92). Our program will closely follow the lines of section 2: In section 3.1 we recall and derive general expressions for G( q, ~) and then for p( q, o)2). The Lifshitz singularities in p(q, w 2) are derived in section 3.2.

170

Th,M. NIEUWENHUIZEN

AND J.M, LUCK

3.1. Recurrence relations and infinite s u m s f o r p( q, £0 2) Just like the characteristic function g2 = y + iTrH the two-point function G can be solved from three term recurrence relations if the mass distribution is the diluted exponential distribution (2.2). In eqs. (3.12) and (3.16) we give two expressions for p ( q , £02). We first quote some results from ref. 6. Let us define for k t> 1 a2k

=

1

-

p

1 = 1 -

a2k_

a2k-1

~- 1 - -

e -2kt~

,

(3.5)

p e ~+iq-2k/~

p

e/L-iq-2k/x

.

( H e r e and in the sequel a bar means that q has been relaced by - q . ) One has to solve the set of equations k(Dk+ 1 - D k ) -- ( k - 1)(D k - D k _ l )

= (*/a2~_1)-I{(1 - e

p, + i q - 2 k / x ~ / )

i q -2kt.n,[~

,~k + (1--p) c

~ck--e

2l,z

Ck_l)),

(3.6) where r / a n d / x are defined in (2.7) and where Ck is related to C k as

(3.7)

Ck=(1--p)Ck/a2k.

The boundary conditions are D k ~ 0 for k---~ ~ and regularity at k = 0. One further needs d~

~.

(3.8)

(Dg - p c- i q - 2 k l ' L [ ~~ck - e2~ck-1)}/a2~-1 •

Then G ( q , ~) can be expressed as infinite sums involving Ck, d k and d k d k ( - q ) (see ref. 5), 2sinh/xG(q, £)=-1+

~

Ck'rk= l +

k=0

Ck~k .

(3.9)

k=0

Instead of writing down the expressions for ~'k and ~k we take the sum of both. After some manipulations we obtain the "simpler" results, valid for complex ~, 4sinh t.LG(q, ~ ) = ~

dke-Zk~(--(e " +e

-iq

)c k +e2~(e -~

+e-lq)Ck_l}

k=l

+ ( q----~ - q ) .

(3.10)

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

171

W e n o w restrict ourselves to positive s q u a r e d frequencies to 2 = 2 + 2 cos e. F o r = - t o : + i0, o n e has eq. (2.11) f o r / x , ~ / a n d Ok. It was f o u n d in (2.16) that for the c o m p l e x c o n j u g a t e case, ~ = - t o a _ i0, o n e could take C~ = C_ k and in this w a y define C k for - ~ < k < ~. A similar p r o p e r t y holds b e t w e e n the dk, d k, D k a n d / ) k : d*~ =- d ~ ( q ) = - - d l _ k ( - - q ) -~ - - d l - k , a~ = --dl_k

,

(3.11) D~ = --Dl_k , D~ = -DI_ k .

It can be verified f r o m eq. (3.6), if one notes that c k = C_ k. Inserting this into (3.10) and then into (3.2) we obtain the exact result o~

8~r sin e p ( q , 0) 2) =

~

d k e 2 i k e { ( e -ie

+ ( q---) - q )

-

e-iq)c k

+

e - 2 i e ( e -iq -- e i ' ) c k _ l }

(3.12)

.

This expression should be c o m p a r e d to the infinite series (2.18) for p(w2). It will be used for extracting residues at the zeros of a2k_~ in the next section. In o r d e r to do the same at the zeroes of azk we n e e d the following expression, which is equivalent to eq. (3.10): 4 s i n h I.~G(q, ~:) = D l ( e - " + e - i q ) / a l + / ) l ( e - " + 2p(algl)-l{a2(1

+ (1 - p ) e

+ e+iq)/a 1

- e-2,cl)

-~ cos q - e -~ cos q . a 2 c l }

oo

+ ~

( 1 - p ) C k B k c~-2klzl,u2k ,

(3.13)

k=l

w h e r e a2k and a2k_ 1 are defined in (3.11) and w h e r e B k = ( a z k _ l a Z k + l ) -1 {(e - , Dk+ 1 _ e~,D~)

xk(e.Dk+l _ e-,Dk )

+ ( e-iq -- Xk eiq)(Dk+l - Dk)} + (q---~ - q ) + 2 ( 1 -- P ) ( a z k - 1 5 2 k - l a z k + l S Z k + l ) - 1 ( ( 2 C k -- e-ZUCk+ 1 -- e 2 " C k _ l ) 2

+ x k C k ( 2 -- e 2" -- e - z " ) + (2Xk cosZq -- X k ) ( C k + l + Ck_ 1 -- 2Ck)

+ COS q(1 -- xk)Z((e ~' + e - " ) C k - e-~'Ck+l -- e " C k _ l ) 2

- h'k cos q ( e " -- e - " ) ( C k + 1 -- Ck_l) } .

(3.14)

172

Th.M. NIEUWENHUIZEN AND J.M. LUCK

T h e variable X~ is a s h o r t h a n d , Xk = P e-Zk'~ = P e2ik~ •

(3.15)

if we insert eqs. (3.13), (3.14) into (3.2), using the relations (3.11) and eliminating the terms outside the sum by virtue of eq. (3.6) for k = 0 and k = 1, we are left with c~

81r sin e p ( q , w 2) = ~]

(1 - p ) C k B k e 2ike./a2k .

(3.16)

k = -oo

This expression is fully equivalent to eq. (3.12), but better b e h a v e d at the zeros of a2k. 3.2. Calculation o f Lifshitz singularity in p( q,

W 2)

In this section we extend the m e t h o d of section 2.3 to obtain the Lifshitz tail of p ( q , w 2) n e a r w e = 4. F r o m eqs. (3.12) and (3.16) we note that there are poles at zeroes of a2k and a2k_l. In terms of the scaled variable x = -2ike,

(3.17)

the zeroes o f a2~ = 1 - p e-X lie at xn = In p + 27tin

(integer n)

w h e r e a s the zeroes of a2k_ 1 x n = In p + iq + 27tin

-~

(3.18a)

1 + p e - i e - x + i q o c c u r at x = x n - ie, with

(half-integer n ) .

(3.18b)

A scaled function f ( x ) was defined in eq. (2.24) in terms of C k. Likewise we can define g(x) as a scaled limit of D k. Since eq. (3.6) b e c o m e s s y m m e t r i c in k if we m a k e the shift k ~ k + ½, we are led to define ie-iq g(x) - 2(1 - p ) sin e Dk+l/2(e)

(k = ix/2e).

(3.19)

E x p a n d i n g (3.6) in p o w e r s of e and neglecting terms of relative o r d e r e 2 we find that g o b e y s ] -}- eiq - x

xg"(x) + g ' ( x ) = 4M(1 + p e iq-x) g(x) (1 - p ) e -x f ( x ) - (1 - p e - X ) f ' ( x ) 4M (1 + p e i q - x ) ( 1 - p e-X) 2 "

(3.20)

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

173

It is seen that g is finite at the poles (3.18b), but logarithmically divergent at the poles (3.18a). Hence we also introduce the function

h(x) = g(x) + (1 +p eiq-x)-lf'(x).

(3.21)

It satisfies a similar differential equation 1 + e -x+iq X eiq( 1 - e-~)f(x) xh"(x) + h'(x) = 4M(1 + X e-iq) h(x) + 2M(1 + X eiq)2( 1 - X)

+ f ' ( x ) { XXeiq(1-Xeiq) W X e i q ( l + X e i q ) (1 + X eiq) 3

(1 ---P) e-X+iq ~ 4M(1 + X eiq) 2 J '

(3.22)

where X =- X(x) = p e -~. The function h(x) behaves near x , (integer n) as

h(x. + 3) = h. + - -

1 + cos q

(6 In 6 - 3) + fin3 + 6(3 2 1n23),

(3.23)

where q~n is proportional to f(xn) and defined below (2.27). First considering the contributions of poles at xn with half-integer n, we write (3.12) as oo

87r sin e p(q, £02) = E

~

Sk =

k =-oo

E k+l/2=-~

~-iA

Sk+l/2 = ~

i

dt ~ tan('rrt)St+l/2 .

--oe-iA

(3.24) The shift k---~ k + ½ is needed to define g(x) properly, (see (3.19)). The last expression is similar to eq. (2.19). Inserting (2.21) and introducing the varible x = -2iet one obtains -A+i~

p( q, w2) - ( l - p ) 2

f

4~'i -A

dx

e~X/~ e-X 1 + p e iq-x (1 - X) z

-i~

x ( ( 1 - x e i q ) f + ( eiq -- 1 ) ( 1 - x l f ' } + (q---~ - q ) , (3.25)

where we have omitted the argument x of f, g and X = P e-X and neglected terms of order e 2 Closing the contour downwards and only taking into account

174

Th.M. NIEUWENHUIZEN AND J.M. LUCK

the residue of the leading pole near the half-integer x, we find (1 _ p ) 2

P(q' w2) = n-1/2=-~E 4p(1 + cos q)2 x e=X°/"{ f. - 2(1 + cos

q)gn - (1 + e-iq)qJ.}

x {f. + i s i n q qJ.} + (q--* - q ) + contribution from poles with integer n .

(3.26)

Here gn g(Xn) while fn and On have been defined in (2.27). Since f is analytic near these values xn, one has of course ~0n = 0. We note that all these quantities depend on q, because x n does. In the limit q ~ 7r the denominators have to be treated a bit more carefully. Moreover, the poles xn(q) for half integer n approach those for integer n. We deduce =

p(Tr + eQ, ~oz) =

2(1-p)2c°srrQ pe4(---i---Q---?)-2

~ ....

p'~/~ 2rrZin/e~2 e

I.

+ contributions from poles with integer n.

(3.27)

In order to calculate the contributions form the poles at x n with integer n we start form eq. (3.16), with B k defined in (3.14). Writing this sum as an integral, just as it was done in (2.19), inserting (2.21) and performing the integral one finds the contribution

p( q, w 2) - 1--p 8peZ . .~. . f(x,,)B(x.)e=X./~ + contributions from poles with half-integer n ,

(3.28)

where B(Xn) is the value of B, at t, = ix,,/(2e). Expressing B(x,) in terms o f f and h we find that the logarithmic divergence in h' from (3.23) cancels the one in/~'. Adding the result to (3.26) we find the final result (1-P)Z { ~ 1( A~x,,(q)/e'-~( P(q' wz) = 2p(1 + cos q)Z .-1/2=-o~ 2~c ~zl,.~q) + (q---> - q ) )

+ ....

2,.tq)s,

(3.29a)

where x. has been defined in (3.18) for integer and half-integer n,

Ql,.(q)=(f~ +isinq~O.)(f - 2 ( l + c o s q ) g - ( l + e - i q ) o . ) ,

(3.29b)

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

175

with all quantities evaluated at x, = In p + 2 n T r i - iq for half-integer n, and Q2,,,(q) = L ( L

+ (1 + c o s q)(h,, + fz. - i s i n q(sr,, - ~,,))) ,

(3.290

with all quantities evaluated at xn = In p + 27tin with integer n. The factors h n and ~ are defined in (3.23) and h,, =-h,,(-q) etc; the quantities fn and ~0n in (2.27), where we note that q~n = 0 at the half-integer x,. We note that (3.29) can be written in the form

p( q, to 2) = p~/~R(Tr/e; q) ,

(3.30)

where R is periodic in its first argument if q / ~ is rational. More specifically, if q = (l - k)~r/l with 0 < k ~< l and k and l mutual prime, the period of R is I if k is even and 2l if k is odd. (The case k = l = 1 is included; it corresponds to q=0.) In figs. 3 and 4 we present plots o t p P t q , 0")2) as a function of 7r/e for two different rational values of q/rc. In incommensurate cases (q/rc irrational) the amplitude R is a quasi periodic function of 7r/e. In the limit q ~ ~, eq. (3.29) simplifies drastically. We find

p('n" + eQ, to 2) = ~e3(1 -

n=-~

f2n e 2~r2in/e. S ( Q )

= p(w2) • S ( Q ) ,

(3.31)

where we compared with eq. (2.29). The scaling function S(Q) is given by 4

1 + cos 7rQ

s ( a ) - roe (1 - 0 2 ) 2

(3.32)

It can be checked that S(Q) has the proper normalization 2,n-

e¢

(3.33) 0

--oo

It is clear that (3.31) has the form

p(Ir + eQ, to z) = e-ap"~Rl('rr/e, Q ) ,

(3.34)

where R~(y, Q) is periodic in y, with unit period. In fig. 5 we present a plot of p('n', to2)p-~/~e4, which nicely illustrates this behavior.

176

Th.M. NIEUWENHUIZEN

0.6

~

I

'

I

A N D J.M. L U C K

'

I

~

I

'

0.t,8

% 0.36

0 6

7.2

8.t,

9.6

10.8

12

n: / e: Fig. 3. Plot of p ( q , to2)p -~/' versus ~-/e for p = 0.5, M = 5 and q = 0, illustrating the result (3.30). T h e period 2 is clearly visible.

We can check all these expressions in one limiting case, n a m e l y w h e n M---* oo. T h e n we end up with a chain consisting of masses 1 or oo, occurring with probabilities p and 1 - p , respectively. This limit was already considered by D o m b et al.8). F r o m the differential eqs. (2.25), (3.20) and (3.22) we find that in this limit f ( x ) = 1, g(x) = h ( x ) = 0 so that

1

It

'

I

'

I

I

i

I

I

0.8

0.6

J

e 0.l,

0.2 0

i 6

7.2

8./o

9.6

10.8

"rE/E:

Fig. 4. Same as fig. 3, at wavevector q = ir/3 (period 3).

12

S P E C T R A L D E N S I T Y OF R A N D O M H A R M O N I C CHAINS

177

10

8

6 ""r(,d

6. m

2 ol

,

li

I

,

7.2

I

,

8.t,

I

9.6

I

I

10.8

i

12

"rt/E Fig. 5. Plot of p(zr, toZ)p -~+~ 6 4 v e r s u s ~r/e, for p = 0.5 and M = 5, illustrating the scaling result (3.34). Note the extra e 4 factor w.r.t, figs. 1, 3, 4.

f~=l,

q~,=~0 = g , = h , = f f , = 0

(M=oo).

(3.35)

Thus not much remains from our complicated factors Ql,n and Q2,n: they become equal to unity! Because in this limit the chain is broken up in independent segments, all eigenfrequencies occur with finite probability. Then p(q, ~o2) is an infinite sum of 6-functions. We therefore calculate its integral, which is better behaved. From (3.29) we derive, inserting (3.35) and performing the sums, for the case M = 0% 4 f o(q, x) dx = e 3p ')T / ~ '~xzl, n / / Tr/e, q) tO 2

where (1 - p) ,,[~]_~ R2(x' q) = 7r(1 + cos q)2 kX

~1 + (1 - p) cos([x + 1](7r - q ) ) - p cos([x](Tr - q)) L 1 + 2p cos q + p 2 J (3.36)

Here Ix] denotes the integer part of x. The very same result will be derived in the appendix, starting from first principles. Also this behavior scales for q ~ rr,

178

Th.M. NIEUWENHUIZEN

0.2

'

I

i

I

i

I 2.1,

I

I /~.8

A N D J.M. L U C K

I

i

I 9.6

I

I

0.16

0.12 ¢,,,,,i

rY

0.08

O.Ot,

0 0

J

I 7.2

,

12

x Fig. 6. Plot of the periodic amplitude is 3.

0.5

R~(x, q) versus x,

f o r p = 0.5 and q = ~'/3, so that the period

i

I

'

I

i

I

'

I

'

I

I

I

I

,

I

~

I

I

0.1.

0.3

0.2

0.1

0 0

2.~

~.6

7.2

9.6

X Fig. 7. Same as fig. 6, for q = ~-/2 (period 4).

12

SPECTRAL DENSITY OF R A N D O M H A R M O N I C CHAINS

179

4

f p( ~r + eQ,

x) dx = (1 - p)pI'~/~lS( Q )

(M = ~ ) ,

(3.37)

to 2

where S(Q) is defined in (3.32). Eq. (3.37) is a special case of the general scaling equation (3.31), because the first factors just consitute 1 - H(o92), cf. (4.14). Figs. 6 and 7 present plots of the amplitude R2(x, q) in eq. (3.36) for two typical rational values of q/~r.

4. Relation to general mass distributions

4.1. The integrated density of states In a separate publication4), we consider mass distributions which have a gap at m = 1. We find a limiting behavior for H(o9 2) of the form (2.30). Since no corrections are discussed, only Qo(x) appears in the result. However, instead of coefficients f , the sum involves coefficients Pn, which are related to the scaling behavior of the Schmidt function Z(u) at o92= 4 near u = 1. In the present section we aim to show that both results are the same. Next we argue that the e In e term vanishes in situations with a gap and is in general replaced by a power of e. The Schmidt function is defined 9) in terms of amplitudes an, introduced in eq. (2.1), Z(u) = l i m P r o b ( an < u ) . n

(4.1)

',an+1

It satisfies the equation 9'1°)

Z(u) = f dR(m) Z(2 - mo9 2 - 1/u) - O(-u) + H(o92),

(4.2)

where n(o) 2) = Z(0) is the integrated density of states and R(m) the mass distribution function. At the band edge o92= 4, Z(u, 4 ) = - Z s ( u ) vanishes for u < - 1 and equals unity for u > u+, where u+ ( - 1 < u+ < 0 ) is the largest root of u+ = 2 - 4mma x -- l / u + with mmax being the largest mass. If the value m = 1 occurs with probability p > 0 and there is a gap ( R ( m ) = p for 1 < m < M), then eq. (4.1) has a scaling behavior near u = - 1 (see ref. 2), z ~ ( - 1 + 1 / v ) = p Z ~ ( - 1 + 1 / ( v - 1))

(v I> v_ --- 1 + {4(M - 1 ) } - 1 ) . (4.3)

180

Th.M. N I E U W E N H U I Z E N A N D J.M. L U C K

The solution of this equation is Z~(1 + 1/v) =p°P(v)

-

P

o1-p p

(v > Vl)

~ ....

-1

2winv

(4.4)

In p + 2rrin Pn e

H e r e P(v) is a periodic function with unit period. Our result for 1 - H(o~ 2) derived in ref. 4 involves Pn rather than fn, which occur in (2.30). We now show that both objects have the same meaning. The most direct method is to investigate the definition of the coefficients c k. This will be done in section 4.2. H e r e we follow an a posteriori method. We first note that for the diluted exponential mass distribution (2.2) eq. (2.32) has the form, using H(t-o 2 = 4) = 1,

Z~(u) - p Z s ( - 2 - 1/u) + pO(- u) - 1 = ( 1 - p ) f dx e - x { Z s ( - z - 4 M x -

1/u)- O(-u)}.

(4.5)

0

In order to relate Z s to fix), defined in (2.24), we perform some manipulations with this equation. We differentiate the right-hand side with respect to u and perform a partial integration with respect to x. It happens that the new integral has the same form as the original one. This is essentially the reason why the model can be solved: in this case the integral can be replaced by the left-hand side of (4.5) and one finally obtains (1 + 4MuZO,)(Zs(u) - p Z s ( - 2 - 1/u) + p O ( - u ) - 1) = (1-p)Z~(-2-

l/u)-

(1-p)0(-u).

(4.6)

Next we define, with v_ =- (1 - u+) -1,

f ( x ) - e~X- - ~P

f

e -vx

dZs(-l+ 1/v)

(4.7)

o_

Going through the algebra we find that also this f(x) satisfies eq. (2.25) with the same boundary conditions f(0) -- 1, f ( + ~ ) = 0: both functions are the same. Inserting (4.4) into (4.7), we investigate the behavior of f near x n = In p + 2~'in,

S P E C T R A L D E N S I T Y O F R A N D O M H A R M O N I C CHAINS

181

v1

p(e ~ - 1) j Y(Xn + 6 ) - i - - p - _ e-O(x"+~)dZs(-1 + 1/v) ao

+ (e ~ - 1 ) r e - s ° Z e 2,~i(.,-n)v P., dv n' U1

--=(e ~ - 1)In(8 ) + (e ~ - 1) ~ e-~'+2~i('-n')o'e- n ' n' 8 + 2 ~ r i ( n - n')

'

(4.8)

where v 1 is defined in (2.33) and v below (2.36). For small 8 one finds

f(x,+8)=P,,+8

{

Pn'e2~ri(n-n"vll}

I n ( O ) + , ' ~¢ n T ~ ' i - - ~ n - 7 )

+

Pn "1"-~(82)

"

(4.9)

Comparing this result with eq. (2.27) one sees that fn = In, as was to be proven. But we can go one step further. It is also seen that %, defined in (2.27) vanishes. The reason is that we are studying a mass distribution with a gap, where eq. (2.33) yields exact scaling behavior. Further, the linear coefficient 0n in (2.27) is given explicitly by the term between brackets in (4.9). We conjecture that for situations with a gap the first correction to the leading asymptotic form 1 - H ( o 9 2) = p'~/~Qo(Tr/e) (see (2.30)) is given by the term Q2 in (2.30a), (2.30d), with On from (4.9). The presence of the term q~n6 In 6 in eq. (2.27) can be traced back to a subleading term pVPl(v)/v, with P1 again periodic, in the right-hand side of (4.4). This can occur because it describes a gap-less situation (M = 1); then (4.4) is only valid asymptotically. It leads to the e In e term in (2.30). We now derive the leading singular contribution, for the case where the mass distribution goes as a powerlaw near m = 1. In terms of eq. (2.2) it means that

r(x) =p6(x) + const x~-lO(x)

(x $ 0)

(4.10)

for some positive o~. Inserting the asymptotic behavior Z~(-1 + 1/v) =pVF(v)

(4.11)

into the analog of (2.35) we find for large v

F(v) = F(v - 1) + const v-2"~'(v) . The solution of this equation involves periodic amplitudes

(4.12)

182

Th.M. NIEUWENHUIZEN

AND J.M. LUCK

F(v) = P(v) + v'-2"pl(v) + " " .

(4.13)

H e r e F(v) can be expressed in terms of F and PI in terms of P. When inserted into (4.7) and (2.28), P1 leads to a non-analytic contribution eZ~-lQl(Tr/e ) in (2.30); for integer values of 2 a - 1 this becomes e 2~ - 1 In eQlOr/e ). Like in eq. (2.30) for a = 1, the Fourier coefficients of Q1 are related in a simple way to those of Q0. We thus have found correction terms of order e and of order e 2 a - 1 The latter will be leading for a < 1; the result for a = 1 has been given already in (2.30). For a t> 1 the terms of order e are the leading ones. It is easily understood that the non-analytic term e 2 ~ ' - 1 must be absent if there is a gap in the mass distribution• Finally we consider the case M--->~. Then the chain consists of masses m , = 1 (probability p) and m n = ~ (probability 1 - p). It can be checked that fix) = 1 in this case (cf. (2.25)). Inserting the values fn = 1, q~n = 0n = 0 into (2.29) we derive

1 - H(w 2) = (1 - p)p[=/~l,

(4.14)

where [~/s] denotes the integer part of rr/e. This result can also be derived in a direct way (see the appendix) from the exact solution given by D o m b et al.8). 4.2. Lifshitz singularities o f p( q, 0) 2) for arbitrary mass distributions In section 3, eq. (3.29), we have derived an expression for p(q, w 2) at w 2 close to 4 for exponential mass distributions. In section 4.1 it was discussed that the related result (2.30) f o r H(to 2) is valid for general mass distributions with an atom at the lower bound. We do not doubt that (3.29) is also valid in general, but we have not been able to prove this. In this section we first relate the coefficients occurring in (3.29) to functions introduced by Halperin 11) and Nleuwenhmzen ), which determine G(q, w 2) and p(q, w2). Next we solve them from the appropriate equations for binary mass distributions, in the limit of small p. In this way we obtain as a new result the Lifshitz singularity in p(q, w 2) for binary mass distributions. In ref. 5 the calculation of G( q, w 2) and p( q, o~2) was reduced to successively solving coupled equations for Ro(u ) and Rl(u ), •

•

5

Ro(u ) = l ( R 0 ( 2 u

mw 2 - l / u ) ) + 1 / u , (4.15)

Ra(u ) = e~q ( R 1 ( 2 - mw 2 - 1/u) + R o ( 2 - mw 2 - 1/u) ) . u

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

183

Brackets mean average respect to the mass distribution. G(q, to 2) can be expressed in terms of R 0 and R 1, but we will not need that relation here. For simplicity we consider binary mass distributions: r(m)=p6(m- 1 ) + (1 - p)6(M - m). At to z = 4 it is useful to define

So(v ) = 1/v2Ro(-1 + l/u) - 1/v , (4.16) Sa(v ) =

1~oR1(-1 + l / v ) .

These functions have a cut along the positive v-axis with discontinuities

To(v)=2~-~lfSo(v + i 0 ) - So(v - i 0 ) } , (4.17)

T,(v)

=2@i(Sl(v +i0)-

S,(v-iO)}.

In terms of the quantities U n and z n definded in ref. 5, T O and T 1 read To(v) =

(6(v - 1/(1

+ Zn))> =

TI(v ) = ((1 + z.) -1 U.6(v -

d/s(-1

-t- 1/v), (4.18)

1/(1 + z . ) ) ) ,

where Z S is the Schmidt function at to 2 = 4, defined in eq. (4.1). For binary distributions they satisfy equations following from (4.15)

To(v ) = p T o ( v - 1 ) + ( 1 - p ) ( 1 -

v-1 \-2 6 ( 6 - 1))

/

tT(v-1)

T°~6-(v-1)/(6-1))' (4.19a)

Tl(V)=_peiqTl(V_l)_(l_p)eiq( 1 X Tl(t7 -

v-1

(v5(0-- - - D / ~1)- - 1)) - (v - 1)

~-'

e'qTo(v),

(4.19b)

where t7 = ½(1 -

~/M/(M -

1)).

(4.20)

We first aim to show how these functions are related to f(x) and g(x) introduced in previous sections. The starting point is a relation between R 0 and R 1 and the coefficients Ck, Ck, dk, Dk, which was given in ref. 5. These

184

Th.M. NIEUWENHUIZEN AND J.M. LUCK

quantities are defined for arbitrary o)2. When oj2----~ 4, one has to take the s ~ 0 limit of z~

So(v ) = - 2 i e ~] ck(1 + 2ie(1 - v)) ~ .

(4.21)

k=0

Introducing the scaled variable x = - 2 i e k , the function

y(x) = Ck(e ) ,

(4.22)

replacing the sum by an integral and deforming the contour one ends up with the e = 0 limit

So(v ) = f dx y(x) e -x(1-v).

(4.23)

0 Inserting this Laplace transform and using (4.17) one gets

y(x) = e* f To(v ) do e -vx

(4.24)

1

The relation (3.7) between C k and c k yields the following equality between their scaled limits:

f(x) - 1 - p c -x y(x) - eix--p P f To(v ) d o e -vx 1-p

(4.25)

1

which, due to (4.18a), is the same as (4.7). Similarly it holds that (4.26)

S I ( U ) = f dx a(x) e -xO-v) , 1

where A(x) = lim (2ie) -~ dk+l/2 e--+O

(x

----

-2iek).

(4.27)

In analogy with (4.24) one has

A(x) = e x f Tl(v ) d v e -~x 1

(4.28)

S P E C T R A L DENSITY OF R A N D O M H A R M O N I C CHAINS

185

Finally the relation (3.8) between d k and D k yields a connection between their limits,

g(x) = - ( 1 - p )

l(e-iq + p e-X)A(x) - (1 - p )

lp e-X(y(x) _ y'(x)) (4.29)

and according to (3.21) h ( x ) -= g ( x ) -t- (1 q - p e i q - x ) - l f ' ( x )

.

(4.30)

In this way the functions f(x), g(x) and h(x) have been defined for arbitrary mass distributions. The coefficients f , , q~n, qJ,; gn; h , , ft, follow, in the same way as in previous sections, from the behavior near certain points. The Lifshitz tail in p(q, o92) is then given by (3.29) for arbitrary mass distributions. Finally we use these results to obtain the behavior of the Lifshitz tail of p(q, oJ2) for binary mass distributions. Since exact reults are not available we use a method for small values of the concentration p of light masses, introduced by us recently4). The starting point is to replace the second term in the rhs of (4.19a) by its p = 0 expression,

To(v ) = 6 ( v - 1 + 6 )

(p=0),

(4.31)

and to iterate this equation. This is done conveniently by multiplying by e x- ox, integrating over v in order to obtain y(x) and f(x) via (4.24) and (4.25). One then finds the leading behavior for small p,

f(x) = e ox .

(4.32)

A similar approach gives the expression for g(x) and h(x). Again one replaces the second term in (4.19b) by its value at p =- 0,

Tl(V)-

-u(2---1)--ei2 8(u - 1 -1--/.,7) ( p = O) 6 - 1 + 6 e iq

(4.33)

and the third term is replaced by its small-p behavior, mentioned above, and leading to f(x) given in (4.32). Multiplying by e x-vx, integrating over v and solving for g(x) one finds that first and second order poles cancel, as they should. The result for g(x), and through (4.30) for h(x), is

g(x) = e'~Xt72 eiq/(t7 - 1 + t7 e i q ) , h(x) = g(x) + eeXtT/(1 + p e-X+iq) .

(4.34)

186

Th.M. NIEUWENHUIZEN AND J.M. LUCK

It is seen that h'(x) has no logarithmic divergency at p e -x = 1 (cf. (3.23)). This is because here we have a mass distribution with a gap. See section 4.1 for a similar case. From these results we derive the coefficients in (3.29),

QL. = A( q)(p e Z r ~ i n - i q ) Q2,. = B( q)(p e Z ~ i n )

26 ,

(4.35a) 2~ ,

with A ( q ) = (1 + isin q 6 ) ( 1 -

2(1_+ cos q)tZ2i qe. v - 1 + 6 e ~q /.~2 eiq

. 4-

B(q)=l+(l+cosq) ff--i-~_~e,q 6 3 elq

. +

- i s i n q tT_ 1 + ty e, q

_ iT(1 + e_iq)) (4.35b)

/.7

~l + e , + q--+--q

~

Inserting this into (3.29a) we obtain the final result

p(q, 092)= (1--p)2prr/e+2v-1 ~n ( 2(1 + cos q)Z

)

a "n-e+ 26 - n

1 × {-~1 e-i"qA(q) + -~ ei"qA(-q) + B ( q ) } .

(4.36)

The integral of p with respect to w2 has discontinuities 4

f p(q,x) dx6o2

e3(1Z_P) 7r(l+cos

_[=/~+20]

q) 21J

× { B(q)+A(q)ei('~-q)[~/~+z':12(1 + p e -iq) + A(-q)ei(q-'°+[=/~+2~]}2(1 p e rq)

"

(4.37) Since fn = ( P e2~i") ~ at integer n the Lifshitz singularity in H(~o 2) can be extracted from eq. (2.30) 1 - H(w 2) = (1

- p ) p [rr/e+2ol .

(4.38)

This result was already derived in ref. 4 and generalizes the M = oo result (4.14).

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS

187

5. Discussion The present paper is devoted to a detailed study of the precise form of Lifshitz (exponential) singularities in the integrated density of states H(to 2) (IDS) and the wavenumber-dependent density of states p(q, toe). These singlarities occur at the upper band e d g e of, for instance, random harmonic systems. A recent review on these and related topics has been given by Simon12). In section 2, our starting point is a set of recurrence relations which determine H(to 2) in exactly soluble cases (diluted exponential mass distributions). The result of this calculation confirms the outcome of a different method4), and allows for a discussion of leading correction terms (see section 4). In section 3 the above method is generalized to obtain the Lifshitz singularities in p ( q , to2). The final result (3.29) is also valid for general mass distributions (section 4.2), and applied to the binary mass distribution. In the exactly soluble cases, the coefficients occurring in the final results are related to the solutions of second order differential equations. For general mass distributions, however, no method is known. In the binary case we have considered the limit where the probability of finding a light mass is small. Several interesting questions remain unanswered: For instance, what happens when there is no delta function at the lower bound of the mass distribution; what are the implications of the present results for trapping models with imperfect traps in one dimension? We hope to come back to these points in the future.

Acknowledgements Part of this work was done in Saclay, where Th.M.N. was supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). It is a pleasure to thank the members of the theory group for their warm hospitality.

Appendix A We calculate the Green's function G(q, ~) in the binary random model where m n = 1 with probability p and mn = ~ with probability r = 1 - p. This limit was first studied by D o m b et al.8). We consider a configuration with m 0 = ~, m 1 = m E . . . . . m n = 1, which occurs with probability rp n. In this case the amplitudes are given by an = sinh n/x,

(A. 1)

188

Th.M. N I E U W E N H U I Z E N AND J.M. LUCK

w h e r e / x is given b y e o s h / ~ = 1 + 1~___ 1 - }o) 2

(A.2)

A m e t h o d for c a l c u l a t i n g G was d e v e l o p e d in ref. 5. O n e n e e d s

U,, = ~ aj e i q ( n + l - ] )

(A.3)

.

]=1

F r o m ( A . 1 ) o n e finds U~ = einq[2(1 - e " - i q ) ( 1

- e - " + i q ) s i n h ( n + 1)/z]

1Tn,

with

T.

= e " - e - " + (e"" - e -"~') e -i("+l)q - (e ("+1)~" - e -("+1)") e -i"q

(A.4)

SPECTRAL DENSITY OF RANDOM HARMONIC CHAINS where

~0 2

189

and e are related by

o2=2+2cose.

(A.10)

T h e integral with respect to q can be carried out and gives

1 - H(~o ~) =

f f dq ~

-'rr

o)2

o ( q , x) dx = Z n=l

Z

r~p°-Xo ~ -

k~"

,

(A.II)

k~l

which can also be derived directly. In the limit o~2---~4 (e---~ 0) only the terms with k = 1 of (A.9) and (A.11) are relevant. We find back eq. (3.36) f r o m eq. (A.9) and eq. (4.14) f r o m eq. (A.11).

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