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Anderson localization with correlated disorder
Chemical Physics 177 (1993) 727-731 North-Holland
Anderson localization with correlated disorder M.D. Stephens and J.L. Skinner TheoreticalChemistry Institute and Department of Chemistry Universityof Wisconsin,Madison, WI 53706, USA Received 7 June 1993
We consider a tight-binding Hamiltonian on a three-dimensional simple cubic lattice with constant nearest-neighbor hopping matrix elements and short-range correlated diagonal disorder. Using numerical finite-size scaling techniques we show that the localization threshold is independent of the amount of correlation.
1. Introduction According to Bloch’s theorem, a system with a spatially periodic Hamiltonian has quantum-mechanical one-particle eigenstates that are uniformly extended (delocalized) throughout space. The presence of disorder breaks the translational invariance of the Hamiltonian and can cause eigenstates to be spatially localized. As the disorder is increased, more states become localized; the point at which the last extended state disappears is called the Anderson transition [ 1,2 1. Anderson localization has been invoked often to explain the absence of transport of electrons or excitons in disordered systems. The simplest model that exhibits Anderson localization involves a tight-binding Hamiltonian on a regular lattice, with constant nearest-neighbor off-diagonal matrix elements. The diagonal matrix elements (site energies) are taken to be uncorrelated random variables described by a certain probability distribution, typically a Gaussian. A dimensionless measure of the disorder is a&‘/J, where Z* is the variance of the probability distribution and J is the nearestneighbor matrix element. For a Gaussian distribution on a simple cubic lattice the Anderson transition occurs at [ 3-51 crc=6.00f0.17 (all error bars reported herein represent two standard deviations). The physical origin of the disorder can be very complicated and is often left unspecified. In some instances, however, the disorder will arise from relatively well-defined lattice imperfections, such as vacancies, interstitials, dislocations, or impurities. These
imperfections are typically quite dilute, but propagate their effects over long range. In these systems it is physically unreasonable that site energies should be uncorrelated random variables, since a given site may feel much the same local field as its near neighbors [a]. In this paper we generalize the usual Anderson model by considering the site energies to be correlated Gaussian random variables, with finite-range correlation. We are interested in determining the critical properties as a function of the amount of correlation. From universality considerations one expects that critical exponents will not depend on the amount of local correlation, since one could always form blocks of Cdsites (C is the range of correlation and d is the dimension of space), which would then be uncorrelated [ 7 1. For the critical threshold one can make two different arguments leading to contradictory conclusions. Consider the case with positive correlation, so that the magnitude of the site-energy difference of nearest neighbors is smaller than if there is no correlation. If one views the Anderson transition as a competition between J, which tends to delocalize the states, and the magnitude of the nearest-neighbor site-energy difference, which tends to localize them, then positive correlation would increase the degree of delocalization, and the critical threshold would increase with increasing correlation. Elyutin and Sokolov [ 8,9] came to this conclusion when studying localization on a Cayley tree. The same result is inherent in Logan
0301-0104/93/S 06.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved.
728
MD. Stephenr, J.L. Skinner/Chemical Physics177 (1993) 727-731
and Wolynes’ application [ lo] of localization to the problem of energy flow in molecules. On the other hand, since at the Anderson transition the localization length of the critical eigenstates is infinite, perhaps global rather than local considerations should be more important. Suppose that a critical eigenstate is spread over N sites. It therefore involves 0 (Nz) site-energy differences. The magnitudes of most of these differences are on the order of Z; the global disorder, while only U(N) differences are smaller because of local correlation. Therefore the amount of global disorder should determine the Anderson transition, and the threshold should be independent of the amount of local correlation. Our goal is to determine which, if either, of the above scenarios is correct. We consider a tight-binding model on a simple cubic lattice with correlated Gaussian random site energies and two different models of finite-range correlation. To determine the critical threshold as a function of correlation we use the concept of quantum connectivity together with finite-size scaling, as described earlier [ 3,11- 13 1. The model and theoretical method are discussed in section 2, and the results are presented in section 3. Before proceeding, we note that for one-dimensional systems several groups have studied the effect of correlated diagonal disorder on localization lengths and absorption spectra [ 14-l 8 ] and Wu and Phillips [ 19 ] have shown that special types of correlated disorder can produce extended states. For threedimensional systems Zhang and Chu [ 201 showed that local correlation affects mobility edge trajectories near band edges. Puri and Odagaki [ 2 1 ] and Saven et al. [ 221 have studied the effect of spatial correlation on the localization threshold for topologically disordered systems.
only. The random variables ei have zero mean and are described by a multivariate Gaussian probability distribution with covariance matrix v,= (c,e,). We consider two different models for the correlation. In model I, vU=z2 for i= j, vU=,??y for i, j nearest neighbors ( y> 0), and v,= 0 otherwise. Thus in this model only nearest neighbors are correlated. In model II, v~=~*exp[-rVln(l/~)], where rV is the distance between sites i and j divided by the lattice spacing. Thus in this model the correlation decreases exponentially with distance, and it is defined such that the nearest-neighbor correlation is the same in both models. When y=O both models reduce to the usual uncorrelated case. In a variety of localization problems the concept of quantum connectivity, generalized from classical percolation ideas, has been useful in calculating critical thresholds [ 3,1 l- 13,22 1. To implement this approach, consider a finite lattice with b3 sites and periodic boundary conditions. For a particular realization of the random variables, this Hamiltonian can be diagonalized to yield the eigenstates I p) , which can be written as linear combinations of the site states as
IP>= 7 Ci,lO *
From the expansion coefficients one can define the quantum connectivity of two sites i and j by [ 3,111 d,=Pij(PjiPjj)-“*
Our model Hamiltonian can be written as
H= 7 cilO(il+J<~> IOO’I , where i labels sites on a simple cubic lattice, Ii) are orthonormal site states, J is the hopping matrix element, and the second sum is over nearest neighbors
5
(3)
Pij= c Ic~rl*lQA*. p
(4)
Averaging over many configurations for the same disorder then gives the expression for the localization length [31&b(c) cij
2. Tight-biding Hamiltonian, quantum connectivity, and finite-size scaling
(2)
(L&j)
’
(5)
where we impose the minimum image convention for rp To determine the critical disorder for an infinite system from the results of calculations of finite-size systems, we employ the method of finite-size scaling [ 23-251. From a pair of finite sizes with b3 and b13 sites we can obtain an estimate, @&, of the critical threshold using the phenomenological renormalization group equation [ 3,24,25]
129
M.D. Stephens, J. L. Skinner / Chemical Physics I 78 (I 993) 727- 731
(6)
where [ 31 Tb(~)=2&(6)/(b2+2)1’2.
(7)
For a sequence of pairs we can extrapolate our results with the Nightingale ansatz [ 25 ] a&, =a,+A(b-‘+b’-l)Q ,
(8)
where a, is the critical threshold for the infinite system. 0.6 -
3. Results and discussion We perform calculations for four finite sizes: b= 4, 6,8, and 12. For each size b, value of the disorder a, and correlation y, we generate many configurations of site energies using an algorithm that samples from a multivariate Gaussian distribution [26,27]. For each configuration we diagonalize the Hamiltonian and calculate d, from eq. ( 3 ) . The configurational average in eq. ( 5 ) then yields &,(a). For each value of y this was done for several different values of u. For each pair of finite sizes eq. (6) was used to determine ubb.,which were then extrapolated using eq. ( 8 ) . This procedure was performed previously [ 31 for y=O (no site-energy correlation) yielding a,= 5.88 kO.25. In this paper we consider y = 0.15 for model I (for y> l/6 the covariance matrix is no longer positive definite [26]), and y=O.15, 0.3, and 0.4 for model II. Examples of the crossing of T6( a) and Ts (a), and the Nightingale extrapolation are-shown in figs. 1 and 2 for model II, y=O.3. The results for all values of y are shown in table 1, which should be compared with the best estimate of the critical disorder for the uncorrelated case of a,= 6.00 f 0.17 [ 3 1. While there is perhaps a slight increase in o, as the amount of correlation is increased, all values are well within error bars of the uncorrelated result. Thus on the basis of these results we conclude that the critical threshold is independent of the amount of correlation. We can contrast these results to what one might expect if the nearest-neighbor site-energy differences controlled the Anderson transition. First consider the case of no correlation. The Anderson transition oc-
\ \ b LA
4.5
5
5.5
6
65
7
75
d
Fig. 1. Ts(u) and T*(u) versus uformodel II, y=O.3. The open circles (simulation data) and solid line (best quadratic fit) are for T&u), and the squares and dash-dot line are for Ts(u). Eq.
0
0.1
03
0.2
l/b
+
0.4
l/b’
Fig. 2. uw versus l/b+ l/b’ for model II, y=O.3. The open circlesarethevaiuesofu~fromthecrossingsof7’~(u)andZ’~(u) for the six pairs of finite sizes, and the solid line is the best fit to eq. (8).
M.D. Stephens,J. L. Skinner / Chemical PhysicsI 77 (I 993) 72 7- 731
130
Table 1 Critical disorder, u.(y), for several correlation schemes Y
Model
O,(Y)
0.00 0.15 0.15 0.30 0.40
I, II I II II II
5.88 f0.25 6.04+ 0.08 5.88f0.17 6.16kO.09 6.15kO.13
incorrect. Also shown in fig. 3 is a,(r) =0,(O). As a final remark we note that although the critical threshold (and presumably the critical exponent as well) is independent of y, for localized states the actual localization lengths were found to increase with increasing y. This implies that the critical amplitude, A, as in e(a) =A(cr--b,)+, is nonuniversal (depends on y). Thus in experiments involving finite distances, transport should be enhanced as the amount of site-energy correlation increases.
8
Acknowledgement I.5
We thank Dr. Tsun-Mei Chang and professor Leslie Root for very helpful discussions during the preliminary stages of this work. We are grateful for support from the National Science Foundation (grants CHE90-96272 and CHE92- 19474).
./-,’
/” ..’ ..’ ./
6.5 ..’
..’
..’
2’
/
..’
..’
2’
..’
..a
References / P P [l] P.W. Anderson, Phys. Rev. 109 (1958) 1492. ____-6~1 ,I ___~_________________-[2]P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 1 f (1985) 287. ,..
2’
..’
i
5.5
i 0
0.1
0.2
0.3
0.4
Y
Fig. 3. cc(y) versus y. The open circles are the results from model II, and the triangle is from model I. The dash-dot line (with dottedlineerrorestimate)istheansatzu~,(y)=u~(0)/(1-y)‘~2,and the dashed line (with dotted line error estimate) is simply %(r)=%(O).
curs when the ratio of the disorder to the hopping matrix element reaches a critical value: C/ J=rs,(y=O). This can also be written as e,)*) ] l’*/fi J= a,(O) for i, j nearest neigh[<(etbors. Now suppose that even with site-energy correlation the above expression determines the critical threshold - that is, when the root-mean-square nearest-neighbor site-energy difference divided by &J reaches a certain critical value, ~~(0), all states become localized. Substituting ( ( Ei- ej) *) = 2.P ( 1 - y ) gives an ansatz for the critical threshold as a function of correlation: a&)=u,(O)/( 1 -y)‘/*. Using a,(O) =6.00+0.17, this ansatz is shown in fig. 3, along with our numerical results. The ansatz is clearly
[3] T.-M. Chang, J.D. Bauer and J.L. Skinner, J. Chem. Phys. 93 (1990) 8973. [4] B. Bulka, M. Schreiber and B. Kramer, Z. Physik B 66 (1987) 21. [ 5 ] E. Hofstetter and M. Scbreiber, Europhys. Letters 2 1 ( 1993) 933. [6] L. Root and J.L. Skinner, Phys. Rev. B 32 (1985) 4111. [7] S. John andM.J. Stephen, Phys. Rev. B 28 (1983) 6358. [S]P.V. E1yutinandI.M. Sokolov, J.Phys.C 16 (1983) 893. [9] I.M. Sokolov, Soviet Phys. Dokl. 29 (1984) 399. [lo] D.E. Logan and P.G. Wolynes, J. Chem. Phys. 93 (1990) 4994. [ 1 l] L.J. Root, J.D. Bauer and J.L. Skinner, Phys. Rev. B 37 (1988) 5518. [ 121 L.J. Root and J.L. Skinner, J. Chem. Phys. 89 (1988) 3279. [ 13 ] J.D. Bauer, V. Logovinsky and J.L. Skinner, J. Phys. C 2 1 (1988) L993. [ 141 R. Johnston and B. Kramer, Z. Physik B 63 (1986) 273. [ 15 ] E.N. Economou, C.M. Soukoulis and M.H. Cohen, Phys. Rev. B 37 (1988) 4399. [ 161 M. Kasner and W. Weller, Phys. Stat. Sol. (b) 134 (1986) 731. [ 171 H. Fidder, J. Knoester and D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [ 181 E.W. Knapp, Chem. Phys. 85 (1984) 73. [19]H.-L. Wu,W.GoffandP.Phillips,Phys.Rev.B45 (1992) 1623.
M.D. Stephens, J.L. Skinner/Chemical Physics 178 (1993) 727-731
731
[ 201 Z.-Q. Zhang and Q.-J. Chu, Phys. Letters A 13 1 ( 1988) 5 17.
[24] B. Denida
[21] A. Put-i and T. Odagaki, Phys. Rev. B 29 (1984) 1707. [22] J.G. Saven, J.R. Wright and J.L. Skinner, J. Chem. Phys. 94 (1991) 6153. [23] M.E. Fisher, in: Critical phenomena, Proceedings of the International School of Physics “Enrico Fermi”, ed. M.S. Green (Academic Press, New York, 1972 ) .
[25] M.P. Nightingale, PhysicaA 83 (1976) 561. [ 261 Y.L. Tong, The multivariate normal distribution (Springer, New York, 1990). [27] IMSL Stat/Library, Version 2.0 (IMSL, Houston, 1991).
and L. De Seze, J. Phys. (Paris)
43 (1982)
475.
Anderson localization with correlated disorder M.D. Stephens and J.L. Skinner TheoreticalChemistry Institute and Department of Chemistry Universityof Wisconsin,Madison, WI 53706, USA Received 7 June 1993
We consider a tight-binding Hamiltonian on a three-dimensional simple cubic lattice with constant nearest-neighbor hopping matrix elements and short-range correlated diagonal disorder. Using numerical finite-size scaling techniques we show that the localization threshold is independent of the amount of correlation.
1. Introduction According to Bloch’s theorem, a system with a spatially periodic Hamiltonian has quantum-mechanical one-particle eigenstates that are uniformly extended (delocalized) throughout space. The presence of disorder breaks the translational invariance of the Hamiltonian and can cause eigenstates to be spatially localized. As the disorder is increased, more states become localized; the point at which the last extended state disappears is called the Anderson transition [ 1,2 1. Anderson localization has been invoked often to explain the absence of transport of electrons or excitons in disordered systems. The simplest model that exhibits Anderson localization involves a tight-binding Hamiltonian on a regular lattice, with constant nearest-neighbor off-diagonal matrix elements. The diagonal matrix elements (site energies) are taken to be uncorrelated random variables described by a certain probability distribution, typically a Gaussian. A dimensionless measure of the disorder is a&‘/J, where Z* is the variance of the probability distribution and J is the nearestneighbor matrix element. For a Gaussian distribution on a simple cubic lattice the Anderson transition occurs at [ 3-51 crc=6.00f0.17 (all error bars reported herein represent two standard deviations). The physical origin of the disorder can be very complicated and is often left unspecified. In some instances, however, the disorder will arise from relatively well-defined lattice imperfections, such as vacancies, interstitials, dislocations, or impurities. These
imperfections are typically quite dilute, but propagate their effects over long range. In these systems it is physically unreasonable that site energies should be uncorrelated random variables, since a given site may feel much the same local field as its near neighbors [a]. In this paper we generalize the usual Anderson model by considering the site energies to be correlated Gaussian random variables, with finite-range correlation. We are interested in determining the critical properties as a function of the amount of correlation. From universality considerations one expects that critical exponents will not depend on the amount of local correlation, since one could always form blocks of Cdsites (C is the range of correlation and d is the dimension of space), which would then be uncorrelated [ 7 1. For the critical threshold one can make two different arguments leading to contradictory conclusions. Consider the case with positive correlation, so that the magnitude of the site-energy difference of nearest neighbors is smaller than if there is no correlation. If one views the Anderson transition as a competition between J, which tends to delocalize the states, and the magnitude of the nearest-neighbor site-energy difference, which tends to localize them, then positive correlation would increase the degree of delocalization, and the critical threshold would increase with increasing correlation. Elyutin and Sokolov [ 8,9] came to this conclusion when studying localization on a Cayley tree. The same result is inherent in Logan
0301-0104/93/S 06.00 Q 1993 Elsevier Science Publishers B.V. All rights reserved.
728
MD. Stephenr, J.L. Skinner/Chemical Physics177 (1993) 727-731
and Wolynes’ application [ lo] of localization to the problem of energy flow in molecules. On the other hand, since at the Anderson transition the localization length of the critical eigenstates is infinite, perhaps global rather than local considerations should be more important. Suppose that a critical eigenstate is spread over N sites. It therefore involves 0 (Nz) site-energy differences. The magnitudes of most of these differences are on the order of Z; the global disorder, while only U(N) differences are smaller because of local correlation. Therefore the amount of global disorder should determine the Anderson transition, and the threshold should be independent of the amount of local correlation. Our goal is to determine which, if either, of the above scenarios is correct. We consider a tight-binding model on a simple cubic lattice with correlated Gaussian random site energies and two different models of finite-range correlation. To determine the critical threshold as a function of correlation we use the concept of quantum connectivity together with finite-size scaling, as described earlier [ 3,11- 13 1. The model and theoretical method are discussed in section 2, and the results are presented in section 3. Before proceeding, we note that for one-dimensional systems several groups have studied the effect of correlated diagonal disorder on localization lengths and absorption spectra [ 14-l 8 ] and Wu and Phillips [ 19 ] have shown that special types of correlated disorder can produce extended states. For threedimensional systems Zhang and Chu [ 201 showed that local correlation affects mobility edge trajectories near band edges. Puri and Odagaki [ 2 1 ] and Saven et al. [ 221 have studied the effect of spatial correlation on the localization threshold for topologically disordered systems.
only. The random variables ei have zero mean and are described by a multivariate Gaussian probability distribution with covariance matrix v,= (c,e,). We consider two different models for the correlation. In model I, vU=z2 for i= j, vU=,??y for i, j nearest neighbors ( y> 0), and v,= 0 otherwise. Thus in this model only nearest neighbors are correlated. In model II, v~=~*exp[-rVln(l/~)], where rV is the distance between sites i and j divided by the lattice spacing. Thus in this model the correlation decreases exponentially with distance, and it is defined such that the nearest-neighbor correlation is the same in both models. When y=O both models reduce to the usual uncorrelated case. In a variety of localization problems the concept of quantum connectivity, generalized from classical percolation ideas, has been useful in calculating critical thresholds [ 3,1 l- 13,22 1. To implement this approach, consider a finite lattice with b3 sites and periodic boundary conditions. For a particular realization of the random variables, this Hamiltonian can be diagonalized to yield the eigenstates I p) , which can be written as linear combinations of the site states as
IP>= 7 Ci,lO *
From the expansion coefficients one can define the quantum connectivity of two sites i and j by [ 3,111 d,=Pij(PjiPjj)-“*
Our model Hamiltonian can be written as
H= 7 cilO(il+J<~> IOO’I , where i labels sites on a simple cubic lattice, Ii) are orthonormal site states, J is the hopping matrix element, and the second sum is over nearest neighbors
5
(3)
Pij= c Ic~rl*lQA*. p
(4)
Averaging over many configurations for the same disorder then gives the expression for the localization length [31&b(c) cij
2. Tight-biding Hamiltonian, quantum connectivity, and finite-size scaling
(2)
(L&j)
’
(5)
where we impose the minimum image convention for rp To determine the critical disorder for an infinite system from the results of calculations of finite-size systems, we employ the method of finite-size scaling [ 23-251. From a pair of finite sizes with b3 and b13 sites we can obtain an estimate, @&, of the critical threshold using the phenomenological renormalization group equation [ 3,24,25]
129
M.D. Stephens, J. L. Skinner / Chemical Physics I 78 (I 993) 727- 731
(6)
where [ 31 Tb(~)=2&(6)/(b2+2)1’2.
(7)
For a sequence of pairs we can extrapolate our results with the Nightingale ansatz [ 25 ] a&, =a,+A(b-‘+b’-l)Q ,
(8)
where a, is the critical threshold for the infinite system. 0.6 -
3. Results and discussion We perform calculations for four finite sizes: b= 4, 6,8, and 12. For each size b, value of the disorder a, and correlation y, we generate many configurations of site energies using an algorithm that samples from a multivariate Gaussian distribution [26,27]. For each configuration we diagonalize the Hamiltonian and calculate d, from eq. ( 3 ) . The configurational average in eq. ( 5 ) then yields &,(a). For each value of y this was done for several different values of u. For each pair of finite sizes eq. (6) was used to determine ubb.,which were then extrapolated using eq. ( 8 ) . This procedure was performed previously [ 31 for y=O (no site-energy correlation) yielding a,= 5.88 kO.25. In this paper we consider y = 0.15 for model I (for y> l/6 the covariance matrix is no longer positive definite [26]), and y=O.15, 0.3, and 0.4 for model II. Examples of the crossing of T6( a) and Ts (a), and the Nightingale extrapolation are-shown in figs. 1 and 2 for model II, y=O.3. The results for all values of y are shown in table 1, which should be compared with the best estimate of the critical disorder for the uncorrelated case of a,= 6.00 f 0.17 [ 3 1. While there is perhaps a slight increase in o, as the amount of correlation is increased, all values are well within error bars of the uncorrelated result. Thus on the basis of these results we conclude that the critical threshold is independent of the amount of correlation. We can contrast these results to what one might expect if the nearest-neighbor site-energy differences controlled the Anderson transition. First consider the case of no correlation. The Anderson transition oc-
\ \ b LA
4.5
5
5.5
6
65
7
75
d
Fig. 1. Ts(u) and T*(u) versus uformodel II, y=O.3. The open circles (simulation data) and solid line (best quadratic fit) are for T&u), and the squares and dash-dot line are for Ts(u). Eq.
0
0.1
03
0.2
l/b
+
0.4
l/b’
Fig. 2. uw versus l/b+ l/b’ for model II, y=O.3. The open circlesarethevaiuesofu~fromthecrossingsof7’~(u)andZ’~(u) for the six pairs of finite sizes, and the solid line is the best fit to eq. (8).
M.D. Stephens,J. L. Skinner / Chemical PhysicsI 77 (I 993) 72 7- 731
130
Table 1 Critical disorder, u.(y), for several correlation schemes Y
Model
O,(Y)
0.00 0.15 0.15 0.30 0.40
I, II I II II II
5.88 f0.25 6.04+ 0.08 5.88f0.17 6.16kO.09 6.15kO.13
incorrect. Also shown in fig. 3 is a,(r) =0,(O). As a final remark we note that although the critical threshold (and presumably the critical exponent as well) is independent of y, for localized states the actual localization lengths were found to increase with increasing y. This implies that the critical amplitude, A, as in e(a) =A(cr--b,)+, is nonuniversal (depends on y). Thus in experiments involving finite distances, transport should be enhanced as the amount of site-energy correlation increases.
8
Acknowledgement I.5
We thank Dr. Tsun-Mei Chang and professor Leslie Root for very helpful discussions during the preliminary stages of this work. We are grateful for support from the National Science Foundation (grants CHE90-96272 and CHE92- 19474).
./-,’
/” ..’ ..’ ./
6.5 ..’
..’
..’
2’
/
..’
..’
2’
..’
..a
References / P P [l] P.W. Anderson, Phys. Rev. 109 (1958) 1492. ____-6~1 ,I ___~_________________-[2]P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 1 f (1985) 287. ,..
2’
..’
i
5.5
i 0
0.1
0.2
0.3
0.4
Y
Fig. 3. cc(y) versus y. The open circles are the results from model II, and the triangle is from model I. The dash-dot line (with dottedlineerrorestimate)istheansatzu~,(y)=u~(0)/(1-y)‘~2,and the dashed line (with dotted line error estimate) is simply %(r)=%(O).
curs when the ratio of the disorder to the hopping matrix element reaches a critical value: C/ J=rs,(y=O). This can also be written as e,)*) ] l’*/fi J= a,(O) for i, j nearest neigh[<(etbors. Now suppose that even with site-energy correlation the above expression determines the critical threshold - that is, when the root-mean-square nearest-neighbor site-energy difference divided by &J reaches a certain critical value, ~~(0), all states become localized. Substituting ( ( Ei- ej) *) = 2.P ( 1 - y ) gives an ansatz for the critical threshold as a function of correlation: a&)=u,(O)/( 1 -y)‘/*. Using a,(O) =6.00+0.17, this ansatz is shown in fig. 3, along with our numerical results. The ansatz is clearly
[3] T.-M. Chang, J.D. Bauer and J.L. Skinner, J. Chem. Phys. 93 (1990) 8973. [4] B. Bulka, M. Schreiber and B. Kramer, Z. Physik B 66 (1987) 21. [ 5 ] E. Hofstetter and M. Scbreiber, Europhys. Letters 2 1 ( 1993) 933. [6] L. Root and J.L. Skinner, Phys. Rev. B 32 (1985) 4111. [7] S. John andM.J. Stephen, Phys. Rev. B 28 (1983) 6358. [S]P.V. E1yutinandI.M. Sokolov, J.Phys.C 16 (1983) 893. [9] I.M. Sokolov, Soviet Phys. Dokl. 29 (1984) 399. [lo] D.E. Logan and P.G. Wolynes, J. Chem. Phys. 93 (1990) 4994. [ 1 l] L.J. Root, J.D. Bauer and J.L. Skinner, Phys. Rev. B 37 (1988) 5518. [ 121 L.J. Root and J.L. Skinner, J. Chem. Phys. 89 (1988) 3279. [ 13 ] J.D. Bauer, V. Logovinsky and J.L. Skinner, J. Phys. C 2 1 (1988) L993. [ 141 R. Johnston and B. Kramer, Z. Physik B 63 (1986) 273. [ 15 ] E.N. Economou, C.M. Soukoulis and M.H. Cohen, Phys. Rev. B 37 (1988) 4399. [ 161 M. Kasner and W. Weller, Phys. Stat. Sol. (b) 134 (1986) 731. [ 171 H. Fidder, J. Knoester and D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [ 181 E.W. Knapp, Chem. Phys. 85 (1984) 73. [19]H.-L. Wu,W.GoffandP.Phillips,Phys.Rev.B45 (1992) 1623.
M.D. Stephens, J.L. Skinner/Chemical Physics 178 (1993) 727-731
731
[ 201 Z.-Q. Zhang and Q.-J. Chu, Phys. Letters A 13 1 ( 1988) 5 17.
[24] B. Denida
[21] A. Put-i and T. Odagaki, Phys. Rev. B 29 (1984) 1707. [22] J.G. Saven, J.R. Wright and J.L. Skinner, J. Chem. Phys. 94 (1991) 6153. [23] M.E. Fisher, in: Critical phenomena, Proceedings of the International School of Physics “Enrico Fermi”, ed. M.S. Green (Academic Press, New York, 1972 ) .
[25] M.P. Nightingale, PhysicaA 83 (1976) 561. [ 261 Y.L. Tong, The multivariate normal distribution (Springer, New York, 1990). [27] IMSL Stat/Library, Version 2.0 (IMSL, Houston, 1991).
and L. De Seze, J. Phys. (Paris)
43 (1982)
475.