Singularities in spectra of disordered systems

Physica A 167 (1990) 43-65 North-Holland

SINGULARITIES IN SPECTRA OF DISORDERED SYSTEMS

Th.M. NIEUWENI-IUIZEN Natuurkundig Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

This paper starts with a review of work on singularities in the spectral density of random harmonic chains. Detailed descriptions of singular behavior near the maximal frequency, near special frequencies and near island frequencies are discussed. Next the site-disordered Anderson model is considered. Here the discussion is limited to the singularity in the density of states near the low-energy band edge for systems with arbitrary disorder in dimensions d = 1, 2, 3. In a specific example it is shown that renormalization improves the prediction for the band tail quantitatively.

1. Introduction

A simple object such as the densityof states of a random system can contain many intriguing aspects. When the model is defined on a lattice, this density may exhibit complicated singular behavior, where " t h e influence of each random parameter in the system is felt". These singularities are most pronounced in one'dimensional random systems, where they may appear in pure form or as leading singular contributions. In higher dimensions, however, they are small singularities on top of a smooth background. Only in the case of Lifshitz tails at the band edge, this smooth background is absent. For historical reasons, we shall' mainly discuss two classes of random systems. The first is the harmonic random mass chain, described by the equation of motion for the amplitudes ar at site r ~nd frequency to, rn~to2ar = 2at - ar-I

- ar+1 •

(1.1)

Here the force constant has been normalized to unity. It is assumed that the random masses m r are independent random variables with common distribution v ( m r ) . For convenience, we shall set the smallest value of the random masses equal to unity. This implies that the maximal eigenfrequency that can occur in the system (1.1) is equal to 2; it actually only shows up if the chain has an infinity of consecutive masses equal to unity. This is very unlikely to occur, and implies already that the density of states is very small near the maximal 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

44

Th.M. Nieuwenhuizen / Singularities in spectra o f disordered systems

frequency. Indeed, there is an exponential tail in the (integrated) spectral density, which will be discussed in great detail here. The second system to be considered is the site-disordered Anderson model on a hyper-cubic lattice in d = 1, 2, 3 dimensions. It is defined by the time-independent Schr6dinger equation (~0r - ~br+o) + VA0r = E~Or.

(1.2)

0

Here p denotes a nearest-neighbor vector. The Vr are independent random site-potentials with common distribution v(Vr). If they are bounded from below, a shift in E will place this lowest value at Vr = 0; if they have a Gaussian distribution, the average can be taken equal to zero. The system (1.2) also shows up for diffusion of independent particles on a lattice with random trap rates Vr I--0. The probability p , ( t ) to find a particle at site r at time t is governed by the master equation d dt pr(t) = ~" (Pr+o - P~) - V~pr . P

(1.3)

The eigenvalues E then are the decay rates of eigenmodes p r ( t ) = ~O~e x p ( - E t ) . The long-time behavior of this system is determined by the low-energy tail of the density of states p ( E ) connected t o (1.2). The harmonic system near the maximal frequency is closely related to the Anderson model near the lower band edge E = 0 , In one dimension the connections are E = 4 - to 2, ~Or = (_ 1)ra r and Vr = to 2( m r - 1) ~ 4 ( m ~ - 1 ) . All results to be discussed later on hold for both systems, with slightly different parametrization. Let us first recall the Lifshitz [1] argument for the exponential tail in the density of states of the three-dimensional Anderson model with binary disorder,

ro a i,i,y V~ =

,

probability c = 1 - p .

(1.4 )

Lifshitz noted that a mode with E small but positive may occur if it is localized on a large enough cluster with all sites having zero random potentials. The optimal cluster will be approximately spherical, and have radius R. In a continuum approximation, the solution of ( 1 . 2 ) i n s i d e this sphere will be proportional to the spherical Bessel function s i n ( r V ~ ) / r . Lifshitz further notes that the wave function will decay rapidly outside the cluster, and can essentially be assumed to vanish at the surface of the hypersphere. As a result one finds that the energy .of the mode is inversely proportional to the square of the

Th.M. Nieuwenhuizen / Singularitiesin spectra of disordered systems

45

radius of the cluster, viz. E = ~r2/R 2. The probability to find this cluster and thus this m o d e is of the order p4~,R313, which leads to an exponential tail in the density of states p(E) at E~0. In arbitrary dimension, it has the form

p(E) - e x p [ - l n ( 1 / p ) v a E -d'2]

(E$0),

(1.5)

where u 1 ~--"i'C, /')2= 'IT~L~2with /.t2 = 2.40483 being the first zero of the Bessel function J0, and where v 3 = 4~4/3. In a harmonic system, a state with eigenfrequency close to the highfrequency band edge occurs on a large enough cluster of light masses. In one dimension the spectral density will have a related Lifshitz tail, p =lv4-o,2

(1.6)

(to1'2),

where p is the probability to find a light mass. The Lifshitz argument can simply be extended to cases where the distribution of site-energies or masses has no delta-peak at V = 0 or m = 1. One assumes [2] that the relevant eigenfunction is localized on a cluster with all site energies less than or equal to some effective value Veff. From dimensional considerations, Ve~f has to be of the order of E. The probability to find a value 0 < V r < V e f f is given by the integrated distribution of disorder h ( V ) = f v do v(o) at V = Vole- E. Hence the factor p in the original Lifshitz argument is replaced by h(const, x E). As a result, one finds for a power-law distribution of disorder

v(V) ,-~ AotV ':'-1

(V,I,O)

(1.7)

that the factor In p in (1.5) is replaced by a factor a In E for very small E. As discussed in section 3, the full result has the form

p( E ) - e x p { - a v aE-a/ZFa(ln( Eo/ E ) ) } ,

(1.8)

where E 0 is a scale and Fd(A ) is a universal function, which behaves as A for large A. In case the distribution of disorder has an exponential tail itself, viz.

v(V) ~ e x p ( - B V - 8 )

(1.9)

(V$0),

the tail deviates from the Lifshitz result (1.5):

p(E) - e x p [ - B C a ( f l ) v a E

-d/2-t3 ]

(E~O).

(1.10)

46

Th.M. Nieuwenhuizen / Singularities in spectra o f disordered systems

This result states that the density of states is much smaller than in the case of binary disorder, because it is much harder to find large regions with small site potentials. A more pronounced p h e n o m e n o n occurs near so-called island frequencies [3]. Consider the localized mode of one light (L) mass, which is embedded in a sea of heavy ( H ) masses (H=LH=). Let its eigenfrequency be too. In the random mass chain, a succession H U L H u will have an eigenfrequency to, which deviates from too by an amount of order e x p ( - N / x ) , where /x is the inverse decay length of the eigenfulaction inside the region of heavy masses. The probability of occurrence of 2 N heavy masses and one light mass is p(1 -p)ZU. This probability to find such a region is also the probability to find the corresponding eigenfrequency. On eliminating N in favor of to - too one finds a power-law contribution to the integrated spectral density: H(to 2) - H(to0:) --I to2 - to2012"

(to $1' too),

(1.11)

where ( ~ = l n ( 1 - p ) / l n / ~ . It may be less than 1/2, and then leads to a divergent spectral density at too. Many papers have been devoted to singular behavior in the density of states of random systems, for instance involving a rigorotls mathematical approach [4, 5]. It is the purpose of the present paper to discuss detailed calculations confirming, and widely extending, the above rough estffnates. In section 2, we review results on the spectral density in one dimension, including exact solutions, special frequencies and island frequencies. In section 3, we review a recent approach to the calculation of the Lifshitz tail at the band edge in arbitrary dimension for arbitrary disorder. In section 4, we propose a new method with which the Lifshitz results can be improved in order to describe an energy- or frequency-interval of more practical interest. A model with Gaussian disorder in one dimension is analyzed in detail. Section 5, finally, consists of a discussion.

2. One-dimensional harmonic systems The equation of motion for a harmonic chain with random masses may be cast in the form o f a recurrence equation for z r = a r + i / a r , 1 Zr :

2 - mrto 2 -

Zr_ 1

(2.1)

Given the boundary conditions a 0 = 0, a 1 = 1 (z 0 = ~), this fixes the 2'r recursively. In the limit r---> ~, the random variables z r have a limiting distribution

47

Th.M. Nieuwenhuizen I Singularities in spectra of disordered systems

(2.2)

W ( u ) = !irn [Prob(z~ < u) - Prob(z r < 0)1.

According to (2.1) it satisfies (2.3)

W ( u ) = f v ( m ) d m W(2 - mto 2 - 1 / u ) - O ( - u ) - W ( - o o ) ,

where 0 is the Heaviside step function (O(x) = 1 for x > 0 and zero elsewhere). The integrated density of states H(to 2) is by definition the fraction of eigenfrequencies % less than to. Due to the Sturm-Liouville theorem, it equals the fraction of sign changes in the vector (a,} (0 ~< r < oo), which is the same as the fraction of negative z~ values. Since the integrated density of states is selfaveraging, it holds that H(to 2) = - W ( - o o ) .

(2.4)

Eq. (2.3) is called the Dyson-Schmidt [6] integral equation, and W is called the Schmidt function. Given the distribution of disorder and the value of frequency, W yields the integrated spectral density by (2.4). Eq. (2.3) was generalized for complex frequency values by the present author [7]. One introduces the function (2.5)

D ( u ) = !im= ( l n ( 1 / z r - 1 / u ) ) .

It satisfies D ( u ) = f v ( m ) d m [D(2 - mto 2 - 1 / u ) + I n ( 2 - into 2 - l/u)]

D(oo), (2.6)

and defines the "characteristic function" or complex Lyapunov coefficient

(1

12(_to2) = !irn r In = ~

a r

) ! [Xl.(ar at, -~ at-1 a t - 2 =

.

.

.

.

.

a2

a 1

)]

a 1

(2.7)

lim (In z r ) = D(oo). r--~

(Here the chain average has been replaced by an ensemble average; this is allowed since 12 is self-averaging,) Forreal frequencies (real positive to2)12 has the decompositioa 12(_ to2 ___i0) = y(to2) +__i~rH(to2),

,

(2.8)

48

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

where y is the inverse localization length or Lyapunov coefficient and H is the integrated density of states. They are related by a K r a m e r s - K r o n i g equation y(to 2) = f In[1 - to2/x[ d H ( x ) ,

(2.9)

which is usually called the H e r b e r t - J o n e s - T h o u l e s s formula [8].

2.1. Exact solutions A class of exactly soluble models was found by the present author [9-11]. For the harmonic problem, one assumes that the value of the smallest mass has been normalized to unity, and that the random masses are equal to m r = 1 + MXr, where M > 0 is the strength of disorder and where the x r have the c o m m o n distribution

v(x)=p6(x)+(1 -p)exp(-x)

O(x).

(2.10)

This says that a fraction p of the sites has m, = 1, whereas the other masses are drawn from an exponential distribution. The solution for t h e characteristic function at complex frquencies is given by an infinite continued fraction with non-random coefficients, g2=/.~+(1-p)r/

1-

1 Pl-

1 P2-

1 P3-

"-" ) .

"

(2.11)

H e r e / x ( R e / x t> 0) and 7/are defined by 2cosh/x = 2-to

2

-Mr°2

,

r/ = 2- s i-n h, / ,

(2.12)

and the coefficients Pk read 1 - exp(-2k/x)

(2.13)

Ok = 2 + rlk[1 - p e x p ( - 2 k / x ) ] " The real part of 12 yields the Lyapunov coefficient, and the imaginary part gives the integrated density of states. Similar exact solutions exist for many related systems [9, 10]. For instance, in the one-dimensional Anderson xnodel, one can also solve exactly for diluted exponential distributions of site disorder ( V r = W 0 -~- W l X r , where ;the sign of W 1 is arbitrary and where the x r are distributed by (2.10)). In this model, the definitions o f / z and 7/read

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems 2cosh ~ = 2 + W 0 - E ,

7/=

2

W1 sinh/.,

49 (2.14)

"

Other solved cases are Anderson models with power-law-distributed offdiagonal disorder [10] and various random Kronig-Penney models [9]. For these models, the average one-particle Greens function can also be solved exactly from three-term recurrence equations with non-random coefficients [10].

2.2. Lifshitz tails for diluted distributions We call disorder diluted when the lowest mass (m r - 1) has a finite probability of occurrence. This is always the case for binary disorder; in the above section it means that p is strictly positive. The Lifshitz tail for the diluted exponential distributions has been extracted from the continued fraction (2.11). In the harmonic problem, it occurs near the maximal frequency. The result reads [12] 1 - H(to 2 = 4 - e 2) = p~/'Q(~/e)

(e$O),

(2.15)

where the first factor is the Lifshitz prediction (1.6). The second factor is a periodic amplitude with unit period and having the Fourier decomposition Q(y)= (1_p)2 ~ f z exp(2rriny), p ,=-~ - x n

(2.16)

with xn = In p + 2"rrin. This form of Q is valid for arbitrary diluted distributions; the coefficients f~ have to be solved from the Dyson-Schmidt equation at the band edge to 2 = 4. In the special case of diluted exponential disorder this reduces to solving the second order differential equation d2 1 - exp(-x) dx 2 f(x) = 4Mx[1 - p e x p ( - x ) ]

f(x),

(2.17)

with boundary conditions riO) = 1, f ( + ~ ) = 0. The coefficients f, are then given by the values of the function f(x) at the singular points x = x n. In fig. 1, the function Q(y) has been plotted for a typical case ( p = 0 . 2 , M = 5). It was calculated from the continued fraction (2.11) for H(to2), and agrees with the Fourier series defined by (2.16), (2.17). In fig. 2 the leading coefficient f0 is plotted as a function of M / ( M + 1) for different values of p. In the limit M---> ~, the masses either take the value unity or infinity. This case can be solved exactly, because the infinite masses do not vibrate and

50

Th.M. Nieuwenhuizen / Singularities in spectra o f disordered systems

5.~

,~

'

t

I

I

I

;

I

~

I

'

/,.8

"1-

'

/

/,.2

3.6

3

J

I

I

5.2

I

a

6./,

I

z

7.6

I

,

8.8

10

"n;/E Fig. 1. Plot of ( 1 - H ) p -~/E versus Ir/e in a typical case of diluted exponentially distributed disorder ( p = 0.2, M = 5), illustrating the periodic amplitude.

10

'°\o.~,\

Io.'\'

'\'

'

'

I

I

'

'

I

,

,,...=

0

I

0

I

0.2

I

I

J

0./,

0.6

0.8

M M+I Fig. 2. Plot of the leading Fourier coefficient f0 of the scaling function Q for diluted exponential disorder, versus M / ( M + 1). Values of the dilution fraction p are indicated on the curves.

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

51

hence divide the infinite chain into independent finite segments [13]. In this limit, the solution of (2.17) goes to f(x) = 1 at finite x, implying fn = 1 and Q(y) = ( 1 - p)pi,t(y)-y, where int(y) denotes the integer part of y. The spectral density at fixed wavevector q, p(q, to2), also has Lifshitz tails. They have been discussed in ref. [12]. Near oJ = 2, q = ~r there is a scaling form

p(~ + ek, 4 - e 2) =

I d H ( 4 - e 2) S(k) -~e--2

(2.18)

with scaling function 4 1 + cos('nk) S(k) = "rre ( 1 - k2) 2

(2.19)

The function S is normalized such that f ~ (dq/2~r)S ~ f~_=(e dk/2"rr) S(k) = 1. In the Anderson problem, such a scaling behavior is present near E = 0, q -- 0.

2.3. Binary disorder: Lifshitz tails near the band edge and near the special frequencies The integrated spectral density of binary random harmonic chains has a very rich structure in a certain high frequency interval. This is the region where the heavy masses can only oscillate due to the presence of light masses, thereby damping these excitations. Near the high-frequency band edge, the integrated spectral density behaves as (2.15). The periodic amplitude Q has been derived in the limit of small concentration p of light masses, and leads to [14] 1 - H ( 4 - e z) = (1 -p)p,nte,/,+2a),

(2.20)

where 6 = 1 _ ½[M/(M - 1)] ~/z for a binary chain with masses equal to unity or t o M ( M > l ) . The leading Fourier coefficient f0 is the only Fourier coefficient showing up in the long-time behavior of the return probability in the diffusion model (1.3) with random traps. It expresses the influence of non-perfect trapping [15] (a trap is called perfect if its rate Vr equals + oo). In fig. 3, we present a plot of f0 for binary disorder, for different values of the probability p to find a light mass m r = 1 or vanishing trap rate. The variable along the horizontal axis is a/(a + 1), where a denotes the non-vanishing trap rate in the diffusion model, and a -- 4(M - 1) in a binary mass chain with masses equal to unity or to M. Binary harmonic chains may also have so-called special frequencies [16]. These are values of the frequency kvhich are not an eigenfrequency of any chain in the ensemble. Consider a given frequency value t o = to~2 s i n [ ' r r ( l - k ) / 2 l ] , which corresponds to a rational wavelength in the pure

Th.M. Nieuwenhuizen / Singularities in spectra o f disordered systems

52

l

14 12 a

1.0

8

fo 6 4 C

2 d I

0

0.0

i

l

i

0.~

l

l

I

,

t

,~'

0.4

I 0.8

t

,

~

I

,

0.8

1.0

a/(a+1) Fig. 3. T h e leading Fourier coefficient fo for binary disorder, as function of a/(a + 1) for different values of p. A fraction p of the masses is equal to unity, and a fraction 1 - p has the value M -~ 1 + a/4. (a) p = 0.5; (b) p = 0.7; (c) p = 0.9; (d) p = 0.95.

syfftem with only light masses. It becomes a special frequency in the binary random chain when the mass ratio M is large enough. This happens for the first time when M = 2; then an infinity of isolated frequencies (k = 1, 1 = 1, 2 , . . . ) , accumulating at the band edge, becomes special. Another set of special frequencies shows up for M > 3, etc. The integrated spectral density is known exactly at special frequencies [16]. This is due to their nature. The point is that no eigenfrequency of any (finite) chain can pass through the special frequency, when the mass ratio is enlarged. Hence the integrated spectral density at a special frequency is the same as in the case of infinite mass ratio; the latter quality can be calculated explicitly [13, 16]. The Lyapunov coefficient has been evaluated at the special frequency in the limit of infinite mass ratio [5]. Extension of this result for finite mass ratio would be interesting. The nature of special frequencies is the same as the nature of the band-edge frequency. Hence one also expects Lifshitz tails near special frequencies. This has been shown to be the case [17, 18]. However, the problem is rather complicated. There are three different situations:

1. Frequencies approaching the special frequency from the right Here, large enough repetitions of units consisting of one heavy mass and a k-independent number ( l - 1) of light masses, will be able to oscillate. This

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

53

leads to a Lifshitz picture, where the role of each light mass is replaced by the above-mentioned unit. The integrated spectral density behaves as

H(to z) - H(to~) = ( qt)'~'~'Qz('rr/lz)

(toStos) ,

(2.21)

where qt = (1 - p)pt-~ is the probability of occurrence of one heavy and l - 1 light masses, ~ is proportional to V-~ - tos, and Q2 is a bounded amplitude, which becomes periodic in the limit p---> 0; see ref. [14] for the full result.

2. Frequencies approaching the special frequency from the left: critical mass ratio This situation is close to the above one. Large enough repetitions of units consisting of one heavy and a certain number a of light masses, will be able to oscillate (a = 1 when k = l - 1 ) . The integrated density of states H(to 2) H(to 2) - A H has a similar form as in (2.21). A typical case is the special frequency with k = 2, 1 = 7 in a random chain with p = 0.1, and mass ratio equal to the critical value connected to this special frequency, i.e. M = 3.110. For this system we present a plot of const, x Iln AHI in fig. 4, as function of a variable which is proportional to 1 / x / - ~ - to. The form of the curve is consistent with an exponential times a periodic amplitude, as in (2.21). For the definition of the variables, and for an explanation of the structures in the curve, see ref. [14].

'

I

'

I

~

I

'

I

'

Y _= i _~

/

3

( ,

I

2

i

I

3

,

l

,

I 5

i 6

Fig. 4. Plot of const, x Iln AH[ for a binary mass chain with p = 0.1, M = 3.110, to the left of the special frequency to, =2sin(5~r/14). The variable along the horizontal axis is proportional to 1/~Vff~:~. This behavior is consistent with an exponential times a periodic amplitude.

54

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

3. Frequencies approaching the special frequency from the left: general mass ratio Here the relevant units contain l light masses, but no heavy mass. In both above cases, the relevant units were such that a periodic chain built up from an infinite repetition of these units, always had the special frequency as an endpoint of one of its vibration bands. Since dispersion laws are quadratic at band edges, there appears a square root of the frequency difference in the exponents of eqs. (1.6), (2.15), (2.20) and (2.21). In the present case, however, the special frequency lies inside the spectrum of the relevant periodic chain (which consists only of light masses). As a result, the relevant dispersion relation is linear near the special frequency, and one obtains an exponential decay of the form H ( o J ~ ) - H ( w ~ - e)~ exp(-const/e). The full result, derived in ref. [14], involves an additional powerlaw prefactor of e, and a complicated bounded amplitude. Its form was discussed in detail.

2.4. Binary disorder: generalized special frequencies and island frequencies We expect that similar exponential tails will show up near frequencies corresponding to band edges related to infinite repetitions of more complicated unit cells. An example is the case where the unit cell consists of two heavy masses and one light mass. In these situations, the exponential singularity will occur on top of a "smooth" background (special frequencies are, by definition, frequencies where this background is strictly zero). In higher dimensions, there are probably only generalized special frequencies, and no special frequencies. To our knowledge, a study of the integrated spectral density near generalized special frequencies in one dimension has not been performed. Next we consider the behavior of H near the "island" frequency of a light mass in a sea of heavy masses. The rough form of the powerlaw singularity was derived by considering one light mass in a large array of heavy masses and is already given in (1.11). The full behavior is governed by the boundary conditions at the left and right ends of this array. This is described by the Schmidt function, which may be approximated by its value at the island frequency [19]. This leads to

H(o)~ _+ E ) - H(c.D~) = + e 2aR+ ( I n E

- \ln/.t/

(e>0)

(2.22)

where a = -In(1 - p ) / I n / z . R+ are periodic functions with unit period, which can probably be evaluated in some limits. Also the evaluation of H(wZo) and y(to02) has not been performed, to our knowledge. A behavior like in (2.22) occurs near the eigenfrequencies of any small cluster (containing, for instance, two light masses, or one light, one heavy and

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

55

one light mass) embedded in a sea of heavy masses. As a result, there is an infinity of island frequencies in the interval 4 / M < to 2 < 4 (the lower bound corresponds to the maximal frequency of a pure chain consisting only of heavy masses). Under mild conditions there is a high-frequency interval where the exponent 2a in (2.22) is less than unity. In this interval the density of states diverges on a dense set of points, and thus is not a well-defined object. It was shown in ref. [19] that the Lyapunov coefficient y also has singularities as in (2.22), involving periodic amplitudes too. This effect may lead to a considerably larger localization length than expected naively. 2.5. Lifshitz tails for arbitrary disorder It was argued in section 1, that when the probability density of random masses near the lower bound m = 1 has a continuous behavior (i.e. it has no delta peak), the form of this behavior will show up in the leading exponential of the Lifshitz tail. For the one-dimensional case, this point has been investigated with use of the Dyson-Schmidt integral equation [2]. One class of distributions involves the power-law density (1.7) of the deviations Vr=-4 ( m , - 1). This class contains an exactly solvable case, see eq. (2.10) with p = 0, corresponding to a = 1 and A = 1/4M. Both for the exactly solvable case and for distributions with general a, it was found that -tr - cbo

1 - H(4 - e 2) - exp - -~-

(2.23) 4'0

where Y satisfies the implicit equation Y(~b)- In Y(~b) = In ( 4 E ° sin2(~b) ) ,

(2.24)

E

and where th0 and ~r - ~b0 are the points where Y vanishes. The quantity E 0 is the only independent parameter in the integral occurring in (2.23), and given by E o = a [ A F ( a + 1)] -~/~ .

(2.25)

Eq. (2.23) defines the function F 1 entering eq. (1.8), with slightly different parametrization. The leading expansion of (2.23) is 1-H(4-eZ)-exp

/ ----~-ol In

e

+lnln

(1) ( 1 ~ +6

)]}

(2.26)

56

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

and involves a universial log-log correction to the In(l/e) behavior expected from the physical argument given above (1.8). Another class of distributions starts as the exponential (1.9). Here it was found that the integrated spectral density behaves as (1.10), with the coefficient C I given C l ( j ~ ) = 4fl

F(j~ + 1 / 2 )

(2.27)

In conclusion, also the modified Lifshitz predictions are confirmed by exact calculations in the one-dimensional situation.

3. Lifshitz tails in higher dimensions for arbitrary disorder In dimensions larger than one, the powerful approach of the integral equation is no longer present. We follow here a field-theoretic method, described in ref. [20]. The left-hand side of eq. (1.2) defines a matrix H - - A + V, where A is the lattice Laplacian and where V is a random potential. The density of states p ( E ) of H is expressed in terms of a two-point Green's function,

1 Im[G(E + i0)l

p(e)=

G(E) ~a =

1 " Tr E------H

(3.1)

N is the number of lattice points; it will be sent to infinity at an appropriate moment. The Green's function is written as a path integral [21], G(E)=

f Dqr D ~ e x p [ - ~ ( U -

1Z*~b~ ~b~,

E)~] ~

(3.2)

where ~r = (~b,, q'r) is a two-component field defined at each lattice site. Here ~br is a complex "bosonic" variable and ~br a Grassmann ("fermionic") variable. A factor i, absorbed in ~r, makes the integrals convergent provided Im(E) > 0. In (3.2), the average over disorder can be performed exactly and leads to G(E)=

f D ~ D ~ e x p ( - A ) ~1Zr ~br*~br,

(3.3)

where the "action" A = ~: [ - ~ r ( A ~ ) r - E ~ r qt + U(~r ~r)] r

(3.4)

57

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

involves a "potential" U ( x ) = - I n ( dV v ( V ) e -vx .

(3.5)

d

It is site-independent because the distribution of disorder is translationally invariant. For binary disorder (1.4) it reads U ( x ) = - l n ( 1 - c + c e-WX) .

(3.6)

The Lifshitz tail in the density of states is obtained by assuming that the path integral (3.3) is governed by a saddle-point solution (also called classical solution or instanton) [22-24]. One thus inserts the* = ~br = (1 - ½'r0)fr, ~, = ~fr, ~ = ~Tf~, where ,~ and 7/ are fixed Grassmann variables, and where the instanton form function f~ is fixed by the requirement that the classical action A c = ~ [-f Af-

Ef 2 + U(f2)]

(3.7)

r

be minimal. This leads to the classical equation of motion -Af - Ef + U'(f2)f

(3.8)

= O.

It has a localized solution with finite action in the region E < U'(0). The action is minimal when its center a coincides with a lattice point. The localized instanton mimics the localized eigenfunction in the Lifshitz argument. In order to have the full contribution of the instanton to the Green's function one must integrate over the fluctuations around it. This is usually done in Gaussian approximation. There are also zero modes, which have to be treated separately, and one has to sum over the possible locations of the center of the instanton. The result for the density of states in d dimensions is 1 e_Ac ( d e t ' M T "IT) - d e t M L E~ f 2

1/2

p(e) = ~

~r f ~ .

(3.9)

Here, there appear determinants of the longitudinal and transversal fluctuation matrices, M L = - A - E + U ' ( f 2) + 2 f 2 U " ( f 2 ) ,

MT

=

--A

--

E

+

U'(f2)

.

(3.10) The instanton f is a zero mode of the latter matrix, c.f. (3.8). This eigenvalue must be excluded from the determinant, which is indicated by the prime in eq. (3.9). Its zero mode brings the second factor within the square root. The matrix M L has one negative eigenvalue, and for this reason the instanton has

58

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

contributed to the imaginary part of G(E), i.e., to the density of states. If space is continuous, M L also has d translational zero modes generated by shifting the origin a of the instanton, i.e., g / = (O/Oaj)fr]a=O, j = 1, 2 , . . . , d. These eigenvalues have to be excluded from the determinant and their norm also enters eq. (3.9) by a factor (f g~ dr/~r) die. In ref. [20] it was pointed out that the leading form p - e x p ( ' A c ) is very general and covers all different Lifshitz tails discussed already. In particular, after making the continuum approximation, all leading results in the onedimensional case can also be obtained from the instanton approach. The general result reads p - e x p ( - A c ) with Y+

At(E) = ~

2

f dy ~ / - 1 - 0

1 Y

(3.11)

ln[f ~W)dVexp(--~)].

Prefactors and periodic amplitudes are harder to obtain in this way. In dimensions two and three, also the universal function F a, defined in eq. (1.8), has been determined. It is convenient to define a function Y by (3.12)

Fd(A ) = A + In Y(A).

This function behaves asymptotically as Y(A)= KA + . . . for large A, thus yielding the log-log term in (2.26). A plot of Y(A) is presented in fig. 5 for I

I

1.0

0.5

o

I

I

o.1

0.2

I/A

0.3

Fig. 5. T h e f u n c t i o n 1/Y(A), d e f i n e d in eq. (3.12), as f u n c t i o n of 1/A ( d a s h e d curves) in d i m e n s i o n s d = 1 a n d 3. Full c u r v e s i n d i c a t e the a s y m p t o t i c b e h a v i o r s . D a t a in d = 2 w o u l d b e close to t h o s e in d = 3.

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems I

I

I

I

I

t

I

O. 5

1.0

1.5

2.0

59

I

6

2

fl

2.5

Fig. 6. The coefficient Ca, defined in (1.10), as function of/3 in dimensions d = 1, 2, 3.

dimensions d = 1 and d = 3. Data for d = 2 would be very close to those of d=3. Also, the constants Cd(~), defined in eq. (1.9), have been derived [20]. They are plotted in fig. 6 for dimensions d = 1, 2, 3. For Gaussian disorder in continuum space, H a l p e r i n - L a x tails [25] of the form p ~ e x p ( - c o n s t . × [El e-d~2) for E - - - > - ~ are recovered.

4. Renormalized Lifshitz tails

The practical importance of the description of band tails by the Lifshitz formula is essentially limited to the one-dimensional situation. In higher dimensions there are important corrections to the classical action [23, 24], which change the e x p ( - E -d/2) behavior of p(E) into a e x p ( - - E - d / 2 + E-(d-1)/2+ E-(d-2)/2+ "" ") behavior at small E. In one dimension, such corrections only show up in the prefactor of the exponent (which may involve a periodic amplitude). The three-dimensional Anderson model with binary disorder (1.4) was recently studied by the present author in the limit of small concentration c [26]. It was shown that non-Gaussian fluctuations around the instanton are large and already enter the first c o r r e c t i o n exp(E-(d-1)/2). Thus the Lifshitz formula (1.5), (3.9) has only a meaning for very small E, where the density of states is

60

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

extremely small. It was discussed that the leading contributions from nonGaussian fluctuations can be summed by renormalizing the action (3.4) before performing the instanton calculation. There are a number of different reasons for doing so: (1) in the continuum limit the ratio of determinants in eq. (3.9) diverges for dimensions d 1>2; (2) binary systems have an ill-defined potential U (c.f. eq. (3.6)) in the limit W--*~; (3) one always has to renormalize a theory. In the limit considered in ref. [26] this led to a renormalized potential strength WR. This resulted in a scaling behavior of the density of states at small concentrations for energies scaling with concentration: 1

(4.1)

where F is a given scaling function. This result incorporates a prediction on a seemingly unrelated problem, namely the distribution of the number of distinct sites visited in a random walk on a lattice without disorder [26]. Here we wish to generalize this approach. We shall apply the method to the one-dimensional model of a particle diffusing in continuous space in the presence of a Gaussian white-noise potential. It is described by (4.2)

- ½q/'(x) + V ( x ) ¢,(x) = E ¢ , ( x ) ,

where V has average zero and is delta-correlated, (4.3)

( V ( x ) V ( x ' ) ) = 0-2 (x - x ' ) .

It was shown by Halperin [27] that the integrated density of states can be solved exactly and is given by

H(E) =

(u3 v~

~

exp

12

(2o-2) 2/3

(4.4)

0

By saddle-point methods, this yields the leading asymptotic behavior pas(E)~ 41-~E21e x p ( "fro-

4X/2]EI3/2] 30.2 ]

(E--->-oo),

(4.5)

which is often called the Halperin-Lax tail [25]. This result can also be derived with the instanton approach of previous section. The potential in eq. (3.5) is equal to U(x)-- ½0-2x2. The equivalent of eq. (3.8) leads to the classical equation of motion

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

~f"(x)-Ef(x)-~2f3(x)=O.

61 (4.6)

For E < 0 it has a solution

f = tr c o s h ( x ~ )

'

(4.7)

with action equal to the exponent in eq. (4.5). The fluctuation operators are d2 dx 2

ML-

MT=

E-30r2f2(x),

d2 dx 2

E-o'2f2(x)

(4.8)

Their determinants can be calculated using the formula ( d2 2 det - ~ y 2 + / z

A(A + 1)~ ~-os~-2y ]

r(~) r ( ~ + 1)

( det _ - __dy2 + ].2)

r ( ~ - x) r ( ~ + a + 1)"

d 2

(4.9)

\

Both operators have a zero eigenmode. The zero eigenvalues have to be excluded from the determinants. Inserting all factors in (3.9), one recovers eq. (4.5) from the instanton calculation. The renormalized theory is defined by performing a one-loop "mass renormalization" [28]. Let us call the squared mass z. Its bare value z = - E , as can be read off from (3.4) with U ( x ) = - ltr2x 2. It has a leading correction from the one-loop Feynman diagram, z = -E-

tr2g(-E),

(4.10)

where oo

g(z) =-

dq 1 - X/~ _J~ 2~r(1q2 + z)

"

(4.11)

If this renormalization is done self-consistently, one has E= -z - g2g(z).

(4.12)

This equation has a solution for E ~< E _ - - 3 t r 4 / 3 / 2 . Next, the instanton calculation is repeated with this renormalization squared mass z replacing the bare value - E . All steps remain the same. The only difference is that contributions to the determinants involving g(z) should be subtracted. These

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

62

have been taken into account already. They can be found simply by expanding 1n ~{det'MT'~ ,~,} = Tr'ln ( M ~ ) = T r '

1

(4.13)

1 - (-0x2/2 + z)-'3crZf 2 '

in powers of f2. The first term involves Tr _02/21 + z 2cr2f 2 =

- ( - 0 x2/ 2 + z)-lor2f 2

g(z) and becomes

g(z) 2o"z f dx fZ(x) = 4 •

(4.14)

As a result, the one-loop subtraction reduces the ratio of determinants by a factor exp(4). This leads to the renormalized prediction for the tail of the density of states

PR(E ) = e -2Pas(Z(E)),

(4.15)

where z(E) is defined by (4.12). It is readily checked that this expression satisfies the asymptotic behavior (4.5). In fig. 7 we present a plot containing

0.1

P

O.Ol

0,001

-1.8

-lj.6

-ln.4

-ll.2

_lJ0 I E

E_

Fig. 7. Density of states for diffusion in a continuous, one-dimensional m e d i u m in the presence of Gaussian disorder with strength ~r = 1/X/2. Full curve: exact result by Halperin. D a s h e d curve: leading asymptotic behavior for E---~-o0. Boxes: renormalized asymptote.

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

63

the exact values (full curve) of p as function of E, obtained from (4.4), the leading asymptote (4.5) (dashed curve), and the renormalized asymptote (4.15) (boxes), for the case or2 = 1/2. It is seen that the renormalized asymptote describes the data very accurately up to E = -0.9449. At this point the renormalized prediction stops. The integrated density of states is here of the order of one percent, and the relative error of the leading asymptote (4.5) is some 30 percent. At E_, the error of the renormalized asymptote (4.15) is of the order of a few percent; for smaller energies the error is much smaller. This shows that fluctuations around the renormalized instanton are indeed small, at least in the situation considered.

5. Discussion

This paper first contains a review of various precise results on the spectral density of a harmonic chain with random masses. It is discussed that singularities near the maximal frequency, near special frequencies and near island frequencies usually have periodic prefactors. These singularities are caused by coherent regions of the chain (near the high-frequency band edge one needs a large succession of light massess). If such a relevant segment of light masses is enlarged by one extra unit, it contributes in a closely related fashion to the integrated spectral density at a frequency closer to the singular value. This effect leads to the periodicity of the prefactors. Their form is determined by the non-coherent surroundings of the coherent regions. Since there are many possible surroundings, the periodic amplitudes may be very detailed. As a consequence, they are hard to calculate, except in some limits. The distribution of boundary effects from possible surroundings is coded in the Schmidt function W(u). Also, this function shows interesting effects at special frequencies and at island frequencies [14, 19]. Due to the presence of island frequencies, the spectral density may diverge in the one-dimensional situation (if this happens, it will diverge on a dense set of points in a whole interval). In higher dimensions, island frequencies will also occur. In d dimensions, the frequency w0 will be connected to the localized vibration mode of one light mass in a large hypersphere of heavy masses. Let its radius be R. The shift in frequency will be exponentially small in R, whereas the probability to find such a region is roughly exp(--Rd). As a result one will find the behavior

p(o% +- e) ~ exp ( - [ I n (1/e)] a}

~

~ ['n(1/e)]d-1

,

(5.1)

64

Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems

Hence, for dimensions larger than one, island frequencies only produce an exponential singularity. Nevertheless, they bring an important contribution in practical situations, such as in numerical simulations on large systems. It is thus seen that the only mechanism which brings divergencies in the spectral density in one dimension, fails to do so in higher dimensions. We therefore expect that the spectral density of random harmonic systems and the density of states of random Anderson models is C ~, even for binary disorder. Only in percolation limits (infinite masses, infinite site-potentials, vanishing hopping rates), divergences may occur. Next, we have reviewed an instanton calculation in a field theoretic approach for calculating low-energy band tails of the density of states in disordered Anderson models. This problem is closely connected to the singular behavior near the maximal frequency in harmonic systems. The method discussed here works for arbitrary distribution of disorder on arbitrary lattices in arbitrary dimension. The outcome covers all known results. Finally we have presented new results on the renormalization of these band tails, which is needed for energies in regions of practical interest. We have restricted ourselves to Gaussian disorder in one dimension. It was found that renormalization indeed gives a better description of the tails. The generalization of these ideas to general disorder on lattices in arbitrary dimension is a subject of current research.

Acknowledgements The author's work discussed here was performed when he was member of the Instituut voor Theoretische Fysica at the Rijksuniversiteit Utrecht, of the Service de Physique Th6orique of the CEN in Saclay, and of the Institut f/ir Theoretische Physik of the RWTH in Aachen. It is a pleasure to thank many colleagues for stimulating interactions. Special thanks are due to Jean Marc Luck for the warm atmosphere in which our common research has taken place.

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Th.M. Nieuwenhuizen / Singularities in spectra of disordered systems [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

65

D.C. Herbert and R. Jones, J. Phys. C 4 (1971) 1145; D.J. Thouless, J. Phys. C 5 (1972) 53. Th.M. Nieuwenhuizen, Physica A 120 (1983) 468. Th.M. Nieuwenhuizen, Physica A 125 (1984) 197. Th.M, Nieuwenhuizen, Phys. Lett. A 103 (1984) 333. Th.M. Nieuwenhuizen and J.M. Luck, Physica A 145 (1987) 161. C. Domb, A.A. Maradudin, E.W. Montrol and G.W. Weiss, Phys. Rev. 115 (1959) 24. Th.M. Nieuwenhuizen and J.M. Luck, J. Stat. Phys. 48 (1987) 393. Th.M. Nieuwenhuizen and H. Brand, J. Stat. Phys. 59 (1990) 53. J. Hori, Spectral Properties of Disordered Chains and Lattices, D. ter Haar, ed. (Pergamon, Oxford, 1968). M. Endrullis and H. Englisch, Commun. Math. Phys. 108 (1987) 591. Th.M. Nieuwenhuizen, J.M. Luck, J. Canisius, J.L. van Hemmen and W.J. Ventevogel, J. Stat. Phys. 45 (1986) 395. Th.M. Nieuwenhuizen and J.M. Luck, J. Stat. Phys. 41 (1985) 745. Th.M. Nieuwenhuizen and J.M. Luck, Europhys. Lett. 9 (1989) 407. K.B. Efetov, Adv. Phys. 32 (1983) 53. J.L. Cardy, J. Phys. C 11 (1978) L321. J.M. Luttinger and R. Tao, Ann. Phys. (NY) 145 (1983) 185. T.C. Lubensky, Phys. Rev. A 30 (1984) 2657. B.I. Halperin and M. Lax, Phys. Rev. 148 (1966) 722. Th.M. Nieuwenhuizen, Phys. Rev. Lett. 62 (1989) 357. B.I. Halperin, Phys. Rev. A 139 (1965) 104. E. Brezin and G. Parisi, J. Phys. C 13 (1980) L307.