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Trace map properties of variations of Fibonacci chains
IOURNaL or ~ ' ~ / ~ d ~ i~}IlI~
Journal of Non-Crystalline Solids 156-158 (1993) 944-948 North-Holland
Trace map properties of variations of Fibonacci chains Tadashi Takemori a, Masahiro Inoue a and Hiroshi Miyazaki b a Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan b Department of Applied Physics, Tohoku University, Sendai, Miyagi 980, Japan
The electronic trace-map for one-dimensional tight-binding binary chains with a constant nearest-neighbour interaction, whose atomic sequence is generated by the stacking rule C(1)= B, C(2)= A and C(n + 1)= C ( n ) P C ( n - 1) q, have been examined. Here A and B are the two types of atoms with different site energies, and p and q are integers > 0. The invariant of the map for q = 1 becomes a 'quasi-invariant' for q > 1. Vanishing of the quasi-invariant defines a manifold in the trace space, on which the wave function is essentially Bloch-like. The wavefunction is therefore extended at a dense subset of the entire spectrum. The study of cyclic orbits on this manifold suggests that an invariant exists only when q = 1 or q = p + 1. The bandwidth distribution has fractal structure only when q -- 1. For q > 1, most of the states are extended, but are interspersed with localized states whose distribution appears to be fractal.
1. Introduction
There has now been a considerable amount of research on the electronic states on self-similar structures (see, for example, ref. [1]), the simplest of which is the one-dimensional Fibonacci chain of two types of atom A and B: ABAABABAABAABABAABABA... There are a number of ways of producing this sequence, among which are (1) projection from two-dimensional square lattice, (2) inflation according to the replacement rule B + A and A AB starting with B, and (3) recursive stacking according to the rule D(1)= B, D(2)= A and D(n) = D(n - 1)D(n - 2). Many properties of the Fibonacci lattice have been shown to be related to its quasi-periodicity [1]; i.e., that the lattice can be constructed by the projection from a regular two-dimensional lattice. On the other hand, the self-similarity is largely reflected in the fractal (i.e., renormalization-group) structure, which must be closely related to the inflation/stacking rules. In the hope of shedding some light on the Correspondence to: Dr T. Takemori, Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan. Tel: +81-298 53 5312. Telefax: +81-298 53 5205. Telex: 3652580 untuku j.
role of the stacking rule, we have investigated certain variations of stacking lattices constructed by the rule D(1)=B,
D(2)=A
and
D(n) = D(n - 1)PD(n - 2) q,
(1)
where p and q are positive integers. (Similar lines of research are being pursued also by a number of other authors. For earlier work, see, for example, refs. [2,3].)
2. Trace map and the quasi-invariant surface
The nature of electronic states on a tight-binding chain with the nearest-neighbour interaction is analyzed most conveniently by examining the behaviour of the trace of the transfer matrix [4,5]. Here we consider a tight-binding chain with a constant nearest-neighbour interaction, t, and with two different site energies, EA and EB. Taking t to be the unit of energy, and defining M(1)= M(2)=
(0 -1
1) e--eB '
_1
E--E A '
(o 1)
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
(2)
T. Takemori et al. / Trace map properties of variations of Fibonacci chains
the (unimodular) transfer matrix for energy e at nth generation is determined recursively by M ( n ) =M(n1 ) P M ( n - 2) q. Using the H a m i l t o n Cayley theorem, one can reduce the powers of a 2 X 2 matrix M to a linear combination of M and the identity matrix I, M k = Ck(Tr(M))M-
Ck_I(Tr(M))I
(3)
,
where C k is the Chebyshev polynomial of the second type. Putting T r ( M ) = 2 c o s ( 0 ) , Ck(Tr (M)) is expressed as Ck(Tr ( M ) ) -- s i n ( k O ) / s i n ( 0 ) .
(4)
For a general construction of the trace map [6], one needs in addition to T ( n ) - Tr(M(n)) another variable defined by S(n) = Tr{M(n)M(n
- 1)}.
(5)
In terms of these, the recursive relation is given by T ( n + 1) = C p ( T ( n ) ) C q ( T ( n
- 1))S(n)
- Cp+l(T(n))Cq_l(T(n
- 1))
-- C p _ l ( T ( n ) ) C q + l ( T ( n
- 1))
(6)
and S ( n + 1) = C p + l ( T ( n ) ) C q ( T ( n
- 1))S(n)
-- C p + 2 ( T ( n ) ) C q _ l ( T ( n - Cp(T(n))Cq+l(T(n
- 1))
- 1)).
(7)
This defines the trace map ( T ( n - 1), T ( n ) , S ( n ) ) ( T ( n ) , T ( n + 1), S ( n + 1)) in the three-dimensional trace space where the initial point is T(1) = e a - e, T(2) = e A - e and S(2) = T(1)T(2) - 2. It has been shown [7] that the trace map has a quasi-invariant F ( n + 1) = T r ( n + 1 ) 2 + T r ( n ) 2 + S ( n
of dynamics on the manifold F = 0 can be seen by reparameterizing as T ( n ) = 2 cos On, T ( n - 1) = 2 c o s On_ 1 and S ( n ) = 2 cos(0n + On_l). Then the trace map translates into 0n+ 1 = gO n + qOn_ 1 (mod2"rr).
(9)
This shows that the part of the manifold that falls within - 2 < T(n), T ( n - 1), S ( n ) < 2 is closed under the trace map. We shall call this the quasiinvariant surface (QIS).
3. States on the QIS It is immediately clear that, if the trace-map trajectory falls on QIS, then it corresponds to an extended state of the semi-infinite lattice. A closer examination of the condition for the trajectory to fall on QIS reveals that the corresponding electronic states are essentially Bloch-like. If the trajectory falls on QIS first at (n + 1)st generation, then from eq. (8) C q ( T ( n - 1)) = 0. From eq. (4), this means 0n _ ~ = m ~ r / q ( r e = l , 2 . . . . , q - l ) . Then the two eigenvalues of M ( n - 1 ) are exp(_+im'rr/q) so that M ( n - 1)q = + I (identity matrix). Since M ( n - 1) appears only as M ( n 1) q in generations later than n, it may be dropped altogether in calculating the wavefunctions except for the possible negative sign. Therefore, the wavefunction can be obtained from the wavefunction on the lattice that is a periodic repetition of D ( n ) . At such energies where the trace-map trajectory falls on the QIS, the system behaves effectively as a periodic lattice whose unit cell is D ( n ) .
4. Stability of cyclic orbits on QIS
+ 1) 2
- T r ( n + 1) T r ( n ) S ( n + 1) - 4 = Cq(T(n - 1))2F(n).
945
(8)
The quasi-invariant F has the properties that: (i) its sign does not change with generation (in fact, it is always positive for the initial condition given above); (ii) once it has vanished at a generation, it is zero for all future generations; and (iii) it is an invariant for q = 1 (because C l ( x ) = 1). Simplicity
The type of dynamical flow around the recurrent fixed points (cyclic orbits) of trace map on QIS will be restricted if there exists a conserved quantity of sufficiently regular nature. The quantity J = T r ( M ( n ) - ° M ( n + 1)) = T(n)Cq+l(T(n
- 1)) - S ( n ) C q ( T ( n
- 1))
(lO)
946
T. Takemori et a L / Trace map properties of variations of Fibonacci chains
has been found to be invariant under the tracemap for the case q ---p + 1 [8]. If such a regular invariant exists, and if its first derivative is finite at the fixed point, then one expects the marginal behaviour (an eigenvalue + 1) on linearizing the trace map around the fixed point, because at least one local coordinate J must remain unchanged. In the angle variables of eq. (9), the dynamics on the QIS are already linear, and the fixed-point stability is given by ~0n
1
p
j"
(11)
One of the eigenvalues A1 = ( p + p ~ + 4q ) / 2 is always larger than l, while the other A 2 - - ( P - ~ + rq)/2 is negative. The marginal behaviour is possible only when q = p + 1. It has been proven that there are an infinite number of recurrent orbits on the QIS for any p, q > 0 (see appendix B of ref. [8]). Therefore, if q 4~p + 1, the only possibility for a non-singular invariant to exist on QIS is for it to have vanishing first derivatives at all such cyclic points. Since the
1.0
recurrent fixed points are distributed densely all over the QIS, the invariant would have to be constant. Barring the slight logical possibility of an invariant that just happens to be constant on the QIS, we can therefore conclude that, for the case q > 1, no regular (analytical) conserved quantity exists other than that already found for q = p + 1. (A regular invariant would be ruled out for q > p + 1 if there were only a finite number of fixed points. The invariant must be a function of /3=(log x)(loglA zl)-(log
y ) ( l o g l A 11)
(12)
in the immediate neighbourhood of the fixed point, where x and y are the local coordinates along the two eigenvectors of the trace map with the origin at the fixed point. For q > p + 1, the fixed point is unstable with both I hll and I Az I greater than 1. Then /3 can take on arbitrary values near the origin. Therefore the invariant would have to be either singular there or trivial, i.e., independent of/3. For q < p + 1 on the other hand, 1~21 < 1 and the invariant can be analytical at x = y = 0 o n l y i f t h e ratio log]h e I / l o g l A l l
(a)
(p,q)=(3,1), eB-eA=2.0
"~o.5
generation=3-6 0.0 ~ -eA=0.2 l.O (b) "~0.5
~
0.00.0
1.0
~ B 5 / 6 / 7 /
1...7
2.0
3.0
4.0
5.0
6.0
Fig. 1. Developmentof 'fractal dimension',f(a), through generationsas a function of the exponent a calculatedfor (a) the case (p, q) ~ (3, 1) and (b) the case(2, 2).
T. Takemori et al. / Trace map properties of variations of Fibonacci chains
1), suggesting the fractal nature of the band-width distribution. We have observed the same behaviour for (p, q) = (1, 1), (2, 1), (3, 1) and (4, 1) which are all quasi-periodic lattices. The localization is thus critical (or singular-continuous) for all these cases, although the direct examination of the spatial distribution of wavefunctions did not yield a definite result for the system of such small generations. On the other hand, fig. l(b) shows no sign of convergence with the curve tailing off to larger values of a while approaching the point (a, f(a)) = (1, 1) on the left with increasing system size. This is indicative of the distribution such that majority of A E are proportional to l (i.e., extended states, or nearly absolutely continuous spectrum) while strongly localized states exist as a minority decreasing in proportion with the system size. Although differing in the degree of divergence, the behaviour is essentially the same forl_
is rational, which is possible only for very limited combinations of integers p and q.)
5. Distribution of bandwidths and localization
Localization of an energy eigenfunction can be examined by looking at the sensitivity of the energy eigenvalue to the change of the boundary condition. For this purpose, one calculates the energy 'bandwidth', AE, through which Tr(M(n)) traverses the region - 2 < Tr(M) < 2 [9]. If AE is expected to behave as l ", where l is some measure and the exponent a is distributed over some range with the distribution function behaving also as l -r~"), then the fractal analysis by Hasey et al. [10] can be applied. The natural measure here is the inverse of the system size (the total number of atoms). If the calculated fractal dimension, f(a), converges with increasing system size, then one may conclude the fractal nature of the distribution. Figure 1 shows the calculated values of f(a) with increasing generation for the cases of (p, q) = (3, 1) and (p, q ) = (2, 2). The curve (fig. l(a)) is seen to quickly converge for (p, q ) = (3,
20.0
I
'
'
'
I
'
'
'
947
'
I
'
'
'
'
'
I
(p,q)=(2,3) 10.0
b~
0.0
- 1 0 . 0
,
I
-2.0
,
,
~
~
I
-1.0
i
i
i
i
I
L
~
0.0 e n e r g y (arb. u n i t )
i
~
I
1.0
I
2.0
Fig. 2. A typical recurrent trace curve for (p, q ) = ( 2 , 3). The arrow indicates the region to be enlarged to ( - 1, 1) in the next generation.
948
T Takemori et a L / Trace map properties of variations of Fibonacci chains
number of atoms in the chain. However, if the energy scale is magnified at each generation step so that the level spacing is renormalized to 1, it is often found that a similar pattern repeats itself locally generation after generation. A typical pattern for (p, q ) = (2, 3) is shown in fig. 2. The same pattern is repeated if the energy region indicated by the arrow is enlarged to ( - 1 , 1) at each generation step. Such a recurrent pattern depends on (p, q) but does not depend on EB e A. If a similar region to the left of the indicated one is enlarged every time, then the pattern quickly converges to a sinusoidal oscillation between T(M)=-2 and T(M)---2, i.e., to the absolute continuum. On the other hand, if a region to the right is enlarged, then the amplitude of oscillation blows up, giving small bandwidths, i.e., localized states. A localized region can never develop out of an absolute continuum, but continuous regions have been seen to develop out of regions of large amplitudes after some generations. Therefore, we conclude that, for q > 1, the spectrum is largely continuous with strongly localized states (or regions of such states) thinly interspersed between continuous regions. From the way continuous regions develop and divide the localized regions, we conjecture that the distribution of such localized states/regions is fractal. In terms of the oscillation amplitude of Tr(M), fractal structure of q = 1 manifests itself in the fact that the oscillation amplitude neither blows up nor dies down with generation in the vicinity of energy eigenvalues when the level spacing is renormalized to 1 at each generation.
6. Conclusions
We have examined the electronic trace map on certain variations of stacking lattices indexed by (p, q). The quasi-invariant surface traps the trace-map trajectories and gives essentially Bloch-like extended states, which constitute a dense subset of the entire spectrum. From the analysis of flow around the fixed points on the QIS, it is very likely that no conserved quantity exists other than those known for q = p + 1 and q = 1. The energy band distribution exhibits fractal structure only for the quasi-periodic case q = 1. For q > 1, most states are extended with small number of strongly localized states between the extended regions.
References [1] P.J. Steinhardt and D.P. DiVincenzo, eds., Quasicrystals; the State of the Art (World Scientific, Singapore, 1991). [2] G. Gumps and M.K. Ali, Phys. Rev. Lett. 60 (1988) 1081. [3] M. Holzer, Phys. Rev. B38 (1988) 1709; 5756. [4] M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett. 50 (1983) 1870. [5] S. Ostlund, R. Pandit, D. Rand, H.J. Schellhuber and E.D. Siggia, Phys. Rev. Lett. 50 (1983) 1873. [6] F. Axel and J. Peyri~re, J. Stat. Phys. 57 (1989) 1013. [7] M. Kolar and M.K. Ali, Phys. Rev. A39 (1988) 6538. [8] M. Inoue, T. Takemori and H. Miyazaki, J. Phys. Soc. Jpn. 60 (1991) 3460. [9] M. Kohmoto, B. Sutherland and C. Tang, Phys. Rev. B35 (1987) 1020. [10] T.C. Hasey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev. A33 (1986) 1141.
Journal of Non-Crystalline Solids 156-158 (1993) 944-948 North-Holland
Trace map properties of variations of Fibonacci chains Tadashi Takemori a, Masahiro Inoue a and Hiroshi Miyazaki b a Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan b Department of Applied Physics, Tohoku University, Sendai, Miyagi 980, Japan
The electronic trace-map for one-dimensional tight-binding binary chains with a constant nearest-neighbour interaction, whose atomic sequence is generated by the stacking rule C(1)= B, C(2)= A and C(n + 1)= C ( n ) P C ( n - 1) q, have been examined. Here A and B are the two types of atoms with different site energies, and p and q are integers > 0. The invariant of the map for q = 1 becomes a 'quasi-invariant' for q > 1. Vanishing of the quasi-invariant defines a manifold in the trace space, on which the wave function is essentially Bloch-like. The wavefunction is therefore extended at a dense subset of the entire spectrum. The study of cyclic orbits on this manifold suggests that an invariant exists only when q = 1 or q = p + 1. The bandwidth distribution has fractal structure only when q -- 1. For q > 1, most of the states are extended, but are interspersed with localized states whose distribution appears to be fractal.
1. Introduction
There has now been a considerable amount of research on the electronic states on self-similar structures (see, for example, ref. [1]), the simplest of which is the one-dimensional Fibonacci chain of two types of atom A and B: ABAABABAABAABABAABABA... There are a number of ways of producing this sequence, among which are (1) projection from two-dimensional square lattice, (2) inflation according to the replacement rule B + A and A AB starting with B, and (3) recursive stacking according to the rule D(1)= B, D(2)= A and D(n) = D(n - 1)D(n - 2). Many properties of the Fibonacci lattice have been shown to be related to its quasi-periodicity [1]; i.e., that the lattice can be constructed by the projection from a regular two-dimensional lattice. On the other hand, the self-similarity is largely reflected in the fractal (i.e., renormalization-group) structure, which must be closely related to the inflation/stacking rules. In the hope of shedding some light on the Correspondence to: Dr T. Takemori, Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan. Tel: +81-298 53 5312. Telefax: +81-298 53 5205. Telex: 3652580 untuku j.
role of the stacking rule, we have investigated certain variations of stacking lattices constructed by the rule D(1)=B,
D(2)=A
and
D(n) = D(n - 1)PD(n - 2) q,
(1)
where p and q are positive integers. (Similar lines of research are being pursued also by a number of other authors. For earlier work, see, for example, refs. [2,3].)
2. Trace map and the quasi-invariant surface
The nature of electronic states on a tight-binding chain with the nearest-neighbour interaction is analyzed most conveniently by examining the behaviour of the trace of the transfer matrix [4,5]. Here we consider a tight-binding chain with a constant nearest-neighbour interaction, t, and with two different site energies, EA and EB. Taking t to be the unit of energy, and defining M(1)= M(2)=
(0 -1
1) e--eB '
_1
E--E A '
(o 1)
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
(2)
T. Takemori et al. / Trace map properties of variations of Fibonacci chains
the (unimodular) transfer matrix for energy e at nth generation is determined recursively by M ( n ) =M(n1 ) P M ( n - 2) q. Using the H a m i l t o n Cayley theorem, one can reduce the powers of a 2 X 2 matrix M to a linear combination of M and the identity matrix I, M k = Ck(Tr(M))M-
Ck_I(Tr(M))I
(3)
,
where C k is the Chebyshev polynomial of the second type. Putting T r ( M ) = 2 c o s ( 0 ) , Ck(Tr (M)) is expressed as Ck(Tr ( M ) ) -- s i n ( k O ) / s i n ( 0 ) .
(4)
For a general construction of the trace map [6], one needs in addition to T ( n ) - Tr(M(n)) another variable defined by S(n) = Tr{M(n)M(n
- 1)}.
(5)
In terms of these, the recursive relation is given by T ( n + 1) = C p ( T ( n ) ) C q ( T ( n
- 1))S(n)
- Cp+l(T(n))Cq_l(T(n
- 1))
-- C p _ l ( T ( n ) ) C q + l ( T ( n
- 1))
(6)
and S ( n + 1) = C p + l ( T ( n ) ) C q ( T ( n
- 1))S(n)
-- C p + 2 ( T ( n ) ) C q _ l ( T ( n - Cp(T(n))Cq+l(T(n
- 1))
- 1)).
(7)
This defines the trace map ( T ( n - 1), T ( n ) , S ( n ) ) ( T ( n ) , T ( n + 1), S ( n + 1)) in the three-dimensional trace space where the initial point is T(1) = e a - e, T(2) = e A - e and S(2) = T(1)T(2) - 2. It has been shown [7] that the trace map has a quasi-invariant F ( n + 1) = T r ( n + 1 ) 2 + T r ( n ) 2 + S ( n
of dynamics on the manifold F = 0 can be seen by reparameterizing as T ( n ) = 2 cos On, T ( n - 1) = 2 c o s On_ 1 and S ( n ) = 2 cos(0n + On_l). Then the trace map translates into 0n+ 1 = gO n + qOn_ 1 (mod2"rr).
(9)
This shows that the part of the manifold that falls within - 2 < T(n), T ( n - 1), S ( n ) < 2 is closed under the trace map. We shall call this the quasiinvariant surface (QIS).
3. States on the QIS It is immediately clear that, if the trace-map trajectory falls on QIS, then it corresponds to an extended state of the semi-infinite lattice. A closer examination of the condition for the trajectory to fall on QIS reveals that the corresponding electronic states are essentially Bloch-like. If the trajectory falls on QIS first at (n + 1)st generation, then from eq. (8) C q ( T ( n - 1)) = 0. From eq. (4), this means 0n _ ~ = m ~ r / q ( r e = l , 2 . . . . , q - l ) . Then the two eigenvalues of M ( n - 1 ) are exp(_+im'rr/q) so that M ( n - 1)q = + I (identity matrix). Since M ( n - 1) appears only as M ( n 1) q in generations later than n, it may be dropped altogether in calculating the wavefunctions except for the possible negative sign. Therefore, the wavefunction can be obtained from the wavefunction on the lattice that is a periodic repetition of D ( n ) . At such energies where the trace-map trajectory falls on the QIS, the system behaves effectively as a periodic lattice whose unit cell is D ( n ) .
4. Stability of cyclic orbits on QIS
+ 1) 2
- T r ( n + 1) T r ( n ) S ( n + 1) - 4 = Cq(T(n - 1))2F(n).
945
(8)
The quasi-invariant F has the properties that: (i) its sign does not change with generation (in fact, it is always positive for the initial condition given above); (ii) once it has vanished at a generation, it is zero for all future generations; and (iii) it is an invariant for q = 1 (because C l ( x ) = 1). Simplicity
The type of dynamical flow around the recurrent fixed points (cyclic orbits) of trace map on QIS will be restricted if there exists a conserved quantity of sufficiently regular nature. The quantity J = T r ( M ( n ) - ° M ( n + 1)) = T(n)Cq+l(T(n
- 1)) - S ( n ) C q ( T ( n
- 1))
(lO)
946
T. Takemori et a L / Trace map properties of variations of Fibonacci chains
has been found to be invariant under the tracemap for the case q ---p + 1 [8]. If such a regular invariant exists, and if its first derivative is finite at the fixed point, then one expects the marginal behaviour (an eigenvalue + 1) on linearizing the trace map around the fixed point, because at least one local coordinate J must remain unchanged. In the angle variables of eq. (9), the dynamics on the QIS are already linear, and the fixed-point stability is given by ~0n
1
p
j"
(11)
One of the eigenvalues A1 = ( p + p ~ + 4q ) / 2 is always larger than l, while the other A 2 - - ( P - ~ + rq)/2 is negative. The marginal behaviour is possible only when q = p + 1. It has been proven that there are an infinite number of recurrent orbits on the QIS for any p, q > 0 (see appendix B of ref. [8]). Therefore, if q 4~p + 1, the only possibility for a non-singular invariant to exist on QIS is for it to have vanishing first derivatives at all such cyclic points. Since the
1.0
recurrent fixed points are distributed densely all over the QIS, the invariant would have to be constant. Barring the slight logical possibility of an invariant that just happens to be constant on the QIS, we can therefore conclude that, for the case q > 1, no regular (analytical) conserved quantity exists other than that already found for q = p + 1. (A regular invariant would be ruled out for q > p + 1 if there were only a finite number of fixed points. The invariant must be a function of /3=(log x)(loglA zl)-(log
y ) ( l o g l A 11)
(12)
in the immediate neighbourhood of the fixed point, where x and y are the local coordinates along the two eigenvectors of the trace map with the origin at the fixed point. For q > p + 1, the fixed point is unstable with both I hll and I Az I greater than 1. Then /3 can take on arbitrary values near the origin. Therefore the invariant would have to be either singular there or trivial, i.e., independent of/3. For q < p + 1 on the other hand, 1~21 < 1 and the invariant can be analytical at x = y = 0 o n l y i f t h e ratio log]h e I / l o g l A l l
(a)
(p,q)=(3,1), eB-eA=2.0
"~o.5
generation=3-6 0.0 ~ -eA=0.2 l.O (b) "~0.5
~
0.00.0
1.0
~ B 5 / 6 / 7 /
1...7
2.0
3.0
4.0
5.0
6.0
Fig. 1. Developmentof 'fractal dimension',f(a), through generationsas a function of the exponent a calculatedfor (a) the case (p, q) ~ (3, 1) and (b) the case(2, 2).
T. Takemori et al. / Trace map properties of variations of Fibonacci chains
1), suggesting the fractal nature of the band-width distribution. We have observed the same behaviour for (p, q) = (1, 1), (2, 1), (3, 1) and (4, 1) which are all quasi-periodic lattices. The localization is thus critical (or singular-continuous) for all these cases, although the direct examination of the spatial distribution of wavefunctions did not yield a definite result for the system of such small generations. On the other hand, fig. l(b) shows no sign of convergence with the curve tailing off to larger values of a while approaching the point (a, f(a)) = (1, 1) on the left with increasing system size. This is indicative of the distribution such that majority of A E are proportional to l (i.e., extended states, or nearly absolutely continuous spectrum) while strongly localized states exist as a minority decreasing in proportion with the system size. Although differing in the degree of divergence, the behaviour is essentially the same forl_
is rational, which is possible only for very limited combinations of integers p and q.)
5. Distribution of bandwidths and localization
Localization of an energy eigenfunction can be examined by looking at the sensitivity of the energy eigenvalue to the change of the boundary condition. For this purpose, one calculates the energy 'bandwidth', AE, through which Tr(M(n)) traverses the region - 2 < Tr(M) < 2 [9]. If AE is expected to behave as l ", where l is some measure and the exponent a is distributed over some range with the distribution function behaving also as l -r~"), then the fractal analysis by Hasey et al. [10] can be applied. The natural measure here is the inverse of the system size (the total number of atoms). If the calculated fractal dimension, f(a), converges with increasing system size, then one may conclude the fractal nature of the distribution. Figure 1 shows the calculated values of f(a) with increasing generation for the cases of (p, q) = (3, 1) and (p, q ) = (2, 2). The curve (fig. l(a)) is seen to quickly converge for (p, q ) = (3,
20.0
I
'
'
'
I
'
'
'
947
'
I
'
'
'
'
'
I
(p,q)=(2,3) 10.0
b~
0.0
- 1 0 . 0
,
I
-2.0
,
,
~
~
I
-1.0
i
i
i
i
I
L
~
0.0 e n e r g y (arb. u n i t )
i
~
I
1.0
I
2.0
Fig. 2. A typical recurrent trace curve for (p, q ) = ( 2 , 3). The arrow indicates the region to be enlarged to ( - 1, 1) in the next generation.
948
T Takemori et a L / Trace map properties of variations of Fibonacci chains
number of atoms in the chain. However, if the energy scale is magnified at each generation step so that the level spacing is renormalized to 1, it is often found that a similar pattern repeats itself locally generation after generation. A typical pattern for (p, q ) = (2, 3) is shown in fig. 2. The same pattern is repeated if the energy region indicated by the arrow is enlarged to ( - 1 , 1) at each generation step. Such a recurrent pattern depends on (p, q) but does not depend on EB e A. If a similar region to the left of the indicated one is enlarged every time, then the pattern quickly converges to a sinusoidal oscillation between T(M)=-2 and T(M)---2, i.e., to the absolute continuum. On the other hand, if a region to the right is enlarged, then the amplitude of oscillation blows up, giving small bandwidths, i.e., localized states. A localized region can never develop out of an absolute continuum, but continuous regions have been seen to develop out of regions of large amplitudes after some generations. Therefore, we conclude that, for q > 1, the spectrum is largely continuous with strongly localized states (or regions of such states) thinly interspersed between continuous regions. From the way continuous regions develop and divide the localized regions, we conjecture that the distribution of such localized states/regions is fractal. In terms of the oscillation amplitude of Tr(M), fractal structure of q = 1 manifests itself in the fact that the oscillation amplitude neither blows up nor dies down with generation in the vicinity of energy eigenvalues when the level spacing is renormalized to 1 at each generation.
6. Conclusions
We have examined the electronic trace map on certain variations of stacking lattices indexed by (p, q). The quasi-invariant surface traps the trace-map trajectories and gives essentially Bloch-like extended states, which constitute a dense subset of the entire spectrum. From the analysis of flow around the fixed points on the QIS, it is very likely that no conserved quantity exists other than those known for q = p + 1 and q = 1. The energy band distribution exhibits fractal structure only for the quasi-periodic case q = 1. For q > 1, most states are extended with small number of strongly localized states between the extended regions.
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