Vibrational spectrum on an exact fractal lattice

Physica 128A (19&1) 307-317 North-Holland, Amsterdam

VIBRATIONAL

SPECTRUM

ON AN EXACT Kin-Wah

PBACTAL

YU

Department of Physics, University of California, Los Angeles,

Received

LATTICE

CA 90024, USA

6 March 1984

The vibrational properties were studied on the two-dimensional Sierpinski gasket. If one allows the mass on each site to move in two orthogonal directions, one needs to introduce a transverse coupling K in addition to the longitudinal couplings Kt and K2. Using the standard decimation procedures, one derives the resealing properties of the couplings Kr, Kr and K. One finds, if K is the geometric mean of KI, KS, then upon resealing, the new coupling K’ remains as the geometric mean of Ki and Ki. By using the lattice Green’s function and its resealing properties, one is able to obtain the density of vibrational states as well as the correlation between two given sites. According to the correlation function, one finds there exists an energy dependent localization length &, such that when jr, - r21> 5, the ratios K/K, and K/K* go to constants upon resealing. Also, in the limit of K = 0, the two orthogonal motions are completely decoupled and our results reduce to the case of one degree of freedom.

1. Introduction

Fractal is a self-similar structure embedded in the Euclidean space of dimension d. In these structures, the mass of a certain size goes as rd, where r is the linear scale of the volume occupied. d is known as the fractal dimension. This idea was first introduced by Mandelbrot’), and it is now under very intensive studies. Fractal structures serve as model of disorder, describing crosslink polymers2), and percolating backbone at its critical concentration3). One believes that the fractal character of disordered systems dominates and the disorder is irrelevant. These structures do not have translational invariance but they rather possess dilatation symmetry, thus allowing one to study their properties by scaling arguments4*5); by decimation technique&j) and by real-space renormalization

methods’).

It should

be emphasized

that

in general

a large

class of problems can be solved exactly in any dimension on such lattices. It is well known that various classes of spectral problems map onto the master equation’). Examples are the tight-binding electronic problem, the vibrational problem, the diffusion problem and the problem of spin waves in a ferromagnet. Thus by solving one problem, one is able to transform the solutions to the others. In particular, the vibrational problem was recently analyzed by Alexander and Orbach4.5) on the fractal structures. They stressed that for spectral properties on fractal lattices, one should distinguish the 037%4371/84/!$03.00 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

308

K.-W.

spectral

dimension

2, the Euclidean

YU

d, and the fractal

dimension

dimension

d

For a percolating network at p,, one has d = 4/3 according to the AlexanderOrbach conjecture4). More recently, the vibrational problems on a percolating network near p, were studied by Derrida et al.9) and the dispersion relation of the vibrational spectrum by Entin-Wohlman et al.“). Their results strongly support

the fractal

nature

employed a decimation fractal lattices. Tremblay

of some disorder

solids*). Domany

procedure to solve the vibrational and Southern7) used a generating

et al.6) (DABK) problem on some function technique

to calculate the density of states. In their analysis, only one degree of freedom per site is considered. In the present paper, one wants to generalize the analysis to two or more degrees of freedom. It turns out that, if one allows the mass on each site to move in two orthogonal directions, one needs to introduce a transverse coupling K in addition to the longitudinal couplings K, and K,. In the following, we would like to use the recent DABK method to derive the resealing properties of such elastic coupling constants. Then we use the lattice Green’s function to obtain the density of states and the correlation between two given Sierpinski

sites. All of the calculations gasket.

are carried

out on the two-dimensional

2. Theory The Sierpinski gasket is constructed as shown in fig. 1. Starting from the zero stage n = 0, one has a largest triangle with sites 1, 2, 3. With periodic boundary conditions, another identical triangle is placed opposite to site 1. At each stage, one inserts three sites in each of the triangles and joins them by bonds to obtain the next stage. Fig. 1 shows this construction procedure to 2nd stage.

2

3

2

3

1

1-

2 _:

3

n=O

2x

3

n=l

n=2

Fig. 1. The construction of the two-dimensional Sierpinski gasket. Starting from the zeroth stage, one inserts three sites in each of the triangles and joins them by bonds to obtain the first stage. The higher stages are obtained in the same way.

VIBRATIONAL

The decimation

SPECTRUM

procedure

ON AN EXACT

is done

by elimination

FRACTAL

LATTICE

309

of the sites on the smallest

scale of the nth stage, one obtains the (n - 1)th stage. In the DABK method, one puts identical masses each of mass m on the sites of such a fractal structure. Then one joins nearest neighboring masses by springs

of force constant

m 2

K. One then writes down the equation

of motion

(1)

= -K C (xi - xi+8), s

where i + S denote the nearest neighboring sites of i. In two dimensions, there are four of them. Since one looks for the eigen modes, one puts xi - e-‘“‘, and thus d*xjdt* = -w2xi, and one obtains:

mo2xi = K C (xi - x~+~).

(2)

6 Now we allow motions in the y-direction. Suppose the force constant associated with displacements in the x-direction to be K, and the force constant associated with displacements in the y-direction to be K2. These are longitudinal couplings. We also introduce the transverse coupling K associated with the displacement in the x-direction interacting with a nearest neighbor with displacement in the y-direction. Then the equations of motion are

mo*xi =K, C (XI- xi+6)+ K C (Yi - Yi+s) mw*yi =K2 2

(Yi- Yi+s) + K

C (3 - xi+61*

(3b)

s

6

For mathematical

@a)

3

6

6

simplicity,

one defines

dimensionless

parameters:

A, = K,lmw* , A, = K,lmw* ,

(4)

h = Klmw* . If A = 0, then x and y are completely decoupled. We reduce to the case of one degree of freedom. One can easily obtain (3a), (3b) in a matrix form

310

K.-W. YU

Eq. (2) can be written as (1 - 4A)Xi + A c (xi+8) = 0. 6

(6)

Now one can see that eqs. (5) and (6) have the same structure if one allows A to be a matrix. One also wants to write down the tight-binding Hamiltonian which leads to equations of motion of the form given by (5) and (6)

H =- 2

[tijli>(il+

tjilj>(il]

(7)

9

i.i

with i, j being nearest neighbors, and tii = 1 if i, j are nearest neighbors, tij = 0 otherwise. Using H/i) = Eli), one gets the equations of motion of the form (5) (6) with the requirement that

These conclude the equations of motion. One can easily generalize eq. (6) to any degree of freedom by introducing more coupling constants. In DABK, one writes the eigenvalue equation HI+) = :I$), where H is given by eq. (7) and I#)is an eigenstate. One now divides the lattice at nth stage into two sublattices denoted by 2, which contains the sites at the smallest scale, and by 1 which contains all other sites. The projection of I+) onto these two sublattices are lJ/i) and I&); then the eigenvalue equation can be written as:

(9) From the second line of this equation, we express

and, substituting into the first line, we get

One can thus introduce

an effective

Hamiltonian

VIBRATIONAL SPECTRUM ON AN EXACT FRACTAL LATTICE

311

such that

For the Sierpinski gasket, one has (HI,), = 0. (HrJij = (Hz,), = (H22>ij= -1+ 6, (i, Jo= 1,2,3). Using (lla), one has (I&),

= 4Z[(Z - ?)(Z + 27)]_l, = (T-Zci)[(Z-

-cI

i = j = 1,2,3 )

I)(& +29-r,

i#j.

(12)

Using (llb), and forcing the (n - 1)th stage into the same form, namely, (I-I’), = -1 + 6, and H’(I,!I,)= 2\&), one has immediately: 2=-;(;:+3?).

(13)

Using (8) eq. (13) becomes

One should be aware of the fact that, in the present case, the DABK method does not introduce any new couplings, as this is not true in general”). Consequently, if one requires K’ = K for the two successive stages, one finds the transformation of the frequency for one degree of freedom w ‘* =

d(5 - w’) .

(15)

At low frequencies, one has 0” = 5”‘. Thus by reducing the length scale of this system by a factor of 2, one increases the square of the frequency by a factor of 5, or equivalently one has L

=w

-2v

)

log 2 y=log* We are now in a position to investigate the resealing properties of eq. (5) on the Sierpinski gasket. One writes

I=

A, A [ A

A,

1 -

(17)

312

VIBRATIONAL SPECTRUM ON AN EXACT FRACTAL LATTICE

Using eq. (14) one obtains the following transformations:

%A-(A:+A2)

*,=

’

A,

254 -5(A,+A,)+l’

WA - (A;+ A21

=

(184

’

25A-5(A,+A,)+l’

Wb)

A,=

A[5A-(A,+A2)1 25A-5(A,+AJ+l’

(18~)

where A =AIA2-A2.

(19)

These equations, though complicated, can be reduced to the well-known case when the transverse coupling is identically zero. Thus, putting A = 0 into eq. (18) one obtains, A’=O, A;-’ = A,‘(5 - A;‘), A;-’ = A;‘(5 - A,‘),

in accordance with Rammal and Toulouse’). Thus in this case, the motions are completely decoupled. Furthermore, if A is the geometric mean of A, and A,, i.e., A,A,- A2= 0, then one can show by explicit use of eq. (18) that A’A’-A”=0 1 2 Thus, upon transformation A’ remains as the geometric mean of A; and A;. Also in this case, we have Al/A; = A/A, and Al/A; = A/A,, or, equivalently, K’IK; = K/K, and K’IK; = K/K,. That means the ratios of the transverse coupling to the longitudinal couplings are invariant under the resealing transformation. Also, it is generally true that A’

AlA

7=5-(A,+A&A

’

which goes to zero very rapidly and thus A/A, and A/A, always transform constants.

(20)

to

VIBRATIONAL SPECTRUM ON AN EXACT FRACTAL LATTICE

313

Further analysis of the density of states and localization properties fractal lattices require the definition of the Green’s function,

on these

G;;‘(z) =

[(: - H)-‘], .

(21)

One can show that G;-l’(g) =

[(z - He,)-l]ij,

(22)

for the same two sites in the (n - 1)th and nth stage. Using eq. (12) and (13) one obtains,

where f(Z) = (.Z + 2 ?)(Z - 7)(2 Ii - ;)-I )

(23b)

2=-Z(Z+37).

(23~)

One thus has, from eq. (23), the relationship zeroth stage and Gt’(z,) in the nth stage: G’“‘(r, E” ) =

f(z n )f(:n-1),. . .

)

f(:,)G;'(;,)

between

Gt’(&)

defined in the

.

(24)

The sequence z,,, . . . , ‘& is given by eq. (13). In the following, one is interested in calculating the density of states by N(w’) = i Im[Tr G$)(w* - iO’)] ,

(25)

where ‘Tr’ denotes taking trace of a 2 X 2 matrix. One should be aware of the fact that the density of states obtained by eq. (25) is in some sense different from that obtained by DABK because according to eq. (25) one only gets the modes supported by the sites 1, 2, and 3 in the zeroth stage. One can calculate G!‘(z) = [(z - HJ’],, where (H,), = 2(- 1 + S,), i, j = 1,2,3. One obtains Gl”l(‘E)= -2(;

- 2:)7:

+ 4?)-‘ ,

Gl”l(:) = (‘E+ 2 ?)[(; - 2 :)(;

+ 4if)]-’ .

(26a) (26b)

314

K.-W. YU

One calculates the density of states for different values of K,, K2 and K. The results shall be discussed in the next section. On the other hand, one also wants G%)(z) as a function of n. Upon resealing the transformation, one increases the distance between the two sites 1 and 2 by a factor of 2, thus from G$(z), one can calculate the localization properties of the vibrational modes. These are discussed in the next section.

3. Numerical calculations Using eq. (25) in section 2, one calculates N(o*) for different values of K,, K2 and K. For the purpose of convergence, one puts in a finite width 71.In fig. 2a, one puts in K = 0,K, = K2 = 1.0. Since there is no transverse coupling, the two orthogonal modes are decoupled from each other but degenerate. The results are in very good agreement with those by Tremblay and Southern’). In Fig. 2b, one still has K = 0 but K, = 3K2 = 1. In this case, the two modes are again decoupled but now nondegenerate. It shows very serious overlapping in the low energy region. The higher frequency mode belonging to K, is well separated. Now in fig. 2c, we consider K = 0.5and this spectrum is calculated for the first time. One can see there exists a higher frequency mode at W*= 9; this is the result of interaction between the two orthogonal motions. Also the mode at W*= 6 disappears as a result of this transverse coupling. In fig. 3, one where IGI plots log,,( G$)(w*)/ as a function of 2” for K, = K2 = 1.0 and K = 0.5, denotes the absolute value of the determinant of G. As K/K, and K/K, rescale to constants only after a few iterations, the calculation of Go) is essentially independent of the initial values of K,,K2 and K. One sees that this correlation function decreases exponentially with increasing separation between two sites. This signifies a localization length which is the negative of the inverse of the slope. Also this exponential decay becomes weaker as one goes to lower frequency, signifying a frequency dependence of the localization length. Thus one can write 2” log,,lGl”,)(w*)[ = constant - 5(w’) ’ where ((0’) is the frequency-dependent plot that log, 5(w2) = 3 - log,($)

localization length. One finds from the

t

where W; = 0.69 is the square of the frequency

of one of the normal modes.

VIBRATIONAL

SPECTRUM

ON AN EXACT

FRACTAL

LATTICE

315

K,=K,=l.O K=O 7)= 0.10

t

n

2

,.,2

4

6

4

6

K,=l.O K,= 113=0.33.. K=O

b 2

0 c

K,=K,= 1.0

W2

Kz0.5 7)=0.10

C

c-l

0

2

6

8

Fig. 2. The density of states N(o*) as a function of w* for different values of K1, KZ and K, (a) K,= K2 = 1.0, K = 0; (b)K, = 3K2 = 1.0, K = 0; (c)Kl= K2 = 1.0, K = 0.5.In all cases, 11= 0.10.

316

K.-W. YU

nz67

8

\

\

\

9

10

11

Fig. 3. Correlation between two sites Gt’(o*) as a function of the separation between at the nth stage. One plots the correlation for different values of oz. The localization is determined from the negative of the inverse of the slope (see text).

Thus, t(w2) = constant

x w-‘“,

sites 1 and 2 length t(o’)

and

log 2 v=log5

(29)

This gives us a quantitative support for eq. (16) in the last section. In conclusion, we have analyzed the vibrational properties on the twodimensional Sierpinski gasket. We also studied the localization nature of the normal modes in this structure. Finally, the generalization of these results to d dimensions is easily done by replacing the number 5 by d + 3 (which takes a value 5 for d = 2) in the various equations.

Acknowledgements

The author wishes to thank Professor S. Alexander for giving his attention to this problem and to thank Professor R. Orbach for guidance during the course of this work. This research was supported by the National Science Foundation, under contract number DMR 81-15542.

VIBRATIONAL

SPECTRUM

ON AN EXACT

FRACT’AL

LATTICE

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References 1) B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982). Also, Fractals: Form, Chance and Dimension (Freeman, San Francisco, 1977). 2) S. Alexander, C. Laermans, R. Orbach and H.M. Rosenberg, Phys. Rev. B 28 (1983) 4615. 3) Y. Gefen, A. Aharony, B.B. Mandelbrot and S. Kirkpatrick, Phys. Rev. Lett. 47 (1981) 1771. 4) S. Alexander and R. Orbach, J. Physique Lett. 43 (1982) L625. 5) R. Rammal and G. Toulouse, J. Physique Lett. 44 (1983) L-13. 6) E. Domany, S. Alexander, D. Bensimon and L.P. Kadanoff, Phys. Rev. B 28 (1983) 3110. 7) A.M.S. Tremblay and B.W. Southern, J. Physique Lett. 44 (1983) L843. 8) S. Alexander, J. Bernasconi, W.R. Schneider and R. Orbach, Rev. Mod. Phys. 53 (1981) 175. 9) B. Derrida, R. Orbach and K.-W. Yu, Phys. Rev. B 29 (1984) 4588. 10) 0. Entin-Wohlman, S. Alexander, R. Orbach and K.-W. Yu, Phys. Rev. B 29 (1984) 6645. 11) P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena (Wiley, New York, 1977).