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The inverse of a symmetric banded toeplitz matrix
Vol. 39 (1997)
REPORTS
THE INVERSE
ON MATHEMATICAL
OF A SYMMETRIC
No. I
PHY.VI(‘S
BANDED
TOEPLITZ
MATRIX
D. A. UVIS Department
of Mathematics, King’s College, Strand, London, WCZR ZLS, England (e-mail: [email protected])
and B. W. SOUTHERN Department
of Physics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received June
18, 1996)
We describe a method for obtaining an analytic form for the inverse of a finite symmetric banded Toeplitz matrix. Explicit formulae are given for the tridiagonal and pentadiagonal cases and the results are applied to the evaluation of the Green’s function for nearest and next-nearest neighbour one-dimensional tight-binding systems.
1.
Introduction
A Toeplitz matrix A has elements A,j with the property A,, = n.(s - j). In the case of semi-infinite matrices (s, j = 0.1.. . .) necessary and sufficient conditions have long been known [l, 21 for the existence of an inverse and for finite matrices the inverse can be computed numerically using the Trench algorithm [3]. The eigenvalues and eigenvectors for the finite symmetric tridiagonal case were obtained by Streater [4]. These results provide an expression for the inverse matrix, which has also been obtained by Hu and O’Connell [5] from a calculation of the determinant and cofactors of the matrix. The inverse of the symmetric tridiagonal matrix can be used in the solution of various single-charge-tunnelling problems and of the one-dimensional Poisson equation with Dirichlet boundary conditions. It also provides the Green’s function for a one-dimensional homogeneous nearest-neighbour tight-binding system with open boundaries [6]. The resealing method for excitations in tight-binding systems [7, 81 and quantum spin chains [9, lo] uses a transfer matrix approach. This has the potential to provide a generalization of Hu and O’Connell’s result to all NxN matrices A, with elements’ ‘For any matrix X we denote the row and column vectors formed by its s-th row and ,I-th column by (s[X and Xlj) respectively and the .s.j-th element by (slX]j).
[I371
138
D. A. LAVIS
and B. W. SOUTHERN
of the form
(+lj)
=
a(js - jj)
if /s - jj < 71,
1
if 1s - jl = n,,
0
if Is - jl > 72,
n < N.
(1)
In Sections 2 and 3 we describe this procedure and give explicit formulae for bandwidths 2n + 1 = 3 and 2n + 1 = 5. In Section 4 these results are applied to first and second neighbour tight-binding models on a finite one-dimensional lattice. 2.
Method
Given that B is the inverse of A,
c
4W
+ kl%) = &.
(2)
k=a(s)
where a(s) = Max{1 - s, -n},
p(s) = Min{N - s.7~).
(3)
We define h,(j)
=
(4)
where b(s,j) = (.slBlj) if 15 s,j 5 N and zero otherwise,
and the transfer matrix
(5)
Then equation
(2) can be expressed in the form
T&(j) = bs-l(j) - &,jlll). Iterating
(6) gives
b,(j) =
if j > s,
T-"ho(j),
T-ho(j)
- T-(“-j+‘)ll)
if j 5 S.
(7)
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ MATRIX
The vector b&j) has zeros in the first 7~ elements
6(s. j) = 1
k(n,T-“/
k + ?b)b(k, j)
‘El x(r,IT-‘I
I; +
?(I,,
(7) gives
if j > s, (8)
7~)b(k,j) -
(r~IT-(~‘-~~+~)ll) if j < s.
A,=1
The vector bni(j) has zeros in the last 0 =
and so equation
139
+ ~IIT-~“IX: + 7~)b(k,j)
-
71
(7n +
elements,
so, from (7)
nIT-cAv-j+l)ll).
?fL= 1.....71.
w
k=l
From equation (8) the N elements of the j-th column of B are given as as linear combinations of the first n. elements which are in turn given as solutions of the ‘rt linear equations (9). The remaining problem is to obtain an expression for the elements of powers of the transfer matrix T. We define the functions Zn-j 4j(')=
C
CL(~~J,-~l)pl‘+j.
j =
O,I.....27?,.
(10)
r=O It is then not difficult to show that the eigenvalues k = l..... 7), of the equation @O(P) = 0. with corresponding
orthonormal
of T are the roots LLr;,IL;‘, (11)
left and right eigenvectors
(12) where (13) Thus (14) k=l
A more compact form is obtained
by setting p = exp(i8)
(1%
when we have (~(T”lj)
= TLj(m- s + 1).
s,j = I,....~R,
(16)
140
D. A. LAVIS and B. W. SOUTHERN
2n and any integer !,
where, for j = l,...,
h-j
(17) and f31,. . . ,8, are the roots of the equation
n-1 0 = F(e) E c0+e) + g(o) + C +)cOgre). r=l
(18)
It follows from (4) and (14) that
q(e)
for j = 2n,
-ul(e - 1)
=
{ -a(ln
- jl)ul(a-
1) +
uj+l(l
-
1) for 1 5 j < 2n.
(19)
Iterating the second of equations (19) and comparing with (17), we obtain the result ul(e) = --up,(e + 1) = -c
” sin{ (n + e)ek)
k=l
wok)
(20)
’
We now define the set of TZx n matrices
t&+*(4 - TL) 7&+2(-e - n - 1)
u(e) =
... ... (21)
%+1(-e
- 272+ 1)
%+n(-e
- 2n + 1)
...
and, from (8), (9), (16) and (20),
b(s.j)
=
-(ljU(s
- l)[U(N)]-iU(N
-(I] {U(s - l)[U(N)IPIU(N 3.
- j)]tz)
if j > s,
- j) - U(s - 1 -
j)} 17~)if j 5 s.
(22)
Explicit formulae
We use the procedure of the previous section to rederive the (n = 1) result of Hu and O’Connell [5] and to give the formula for the case n = 2. 3.1.
The case n = 1 Now U(e) is a 1 x 1 matrix with
u(e) = ul(e) = +(e
+ I) = ue(cOse).
2~0s~ = +o).
(23)
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ MATRIX
141
where U,(cosH) = is the Chebyshev polynomial
sin{(J! + l)fI} sin H
(24)
of the second kind. From (22) we obtain
bt~sli) = cos{(N
+ 1 - 1s - j])0} - cos{(N + 1 - s - j)H}
(25)
2 sin 0 sin{(N + 1)19) With the substitution
0= I
x
if I~L(O)(< 2,
iA
if 0(O) 5 -2,
iX + 7r
if n(O) 2 2,
for real A, this is the result obtained 3.2.
(26)
by Hu and O’Connell [5].
The case n = 2 Let
+ X1<2)
=
2(51
4).
+ (2)
=
(27)
-n(l).
We define V(e) = Uf(51) - U{(G) 2(<1
-
(28)
i2)
and, from (24), (28) and (21), we get V(-e) = -V(4! - 2). “(!)= Substituting
V(! + 1) - V(2)V(1) v(e+2)-V(2)V(e+l) (
(29) V(t) v(P+l)
PO)
1 .
(30) into (22) it can be shown, after some manipulation,
that
where F(s. j. N; 19r.Q = V(s - l)W(N - j, N + 1) - V(s)W(N - j. N)
(32)
W(.s, j) = V(s + l)V(j) - V(s)V(j + 1).
(33)
and
142
D. A. LAVIS
and B. W. SOUTHERN
From (24), (28) and (33) we obtain F(s,j,N;81,82)=fl(s-j,N;81,e2)-fl(s--j,N;B2,81)+ + s1(s
-
j, N;
Bl,f92> -
Sl(S
-
+ fi(s
+ j, N;
81,02) -
f2(s
+ j.
+
+ j, N;
Ql,e,)
g2(s
+ j, N;
g2(s
-
j, N;
82,&)
+
N; 02,6,) + e2, 01) +
N;e2, 4) + +g3(s,j,N;e1,e2)-g3(s,j,~;e2,e,) + + f3(s,iN;4, -0,) - .f3(s, i N; -e2, 0,) +
+ .f3(s, j, N;
e1 T0,)
+ g3(s,
6 7 -0,)
.A N;
-
f3(s,
-
j,
g3(s,i
N;
-e2,
e,),
(34)
where fl(k, N; el, e,) =
sin(02) c0s(e1k){2c0s(2e1
+ BIN)
sin(202 + e2N)-
- c0s(3e1 + e1N) sin(02 + O2N)-
- cOs(el + e1N) sin(302 + e2N)}, gl(k. n-: el, e2) =
sin(02) sin(81k){2sin(81) sin(B2)-2sin(2&
+ BIN) sin(202 + &N)+
+ sin(3& + B1N) sin(e2 + e2 N)+ + sin(& + O1N) sin(302 + &N)}, f2(k, N; 81, e2) =
2 sin(e2) cos(&k) cos(6$ + O1N) sin(202 + &N)x
x {cos(e,) - cos@)), g2(k, N;Ql,B2) =
2sin(82)sin(81k)sin(81
+ e1N)sin(2e2
x {cos(e,) - cos(e,)), f3(s,
j,
N; 6$, e2) = sin(&) sin(e2) cos(els - &j) x
x {cos(e,) - cOs(el) + + cOs(el + elzv - 2e2 - eziv)- c0s(e2 + e2N - 2e1 - e,N)), g3(s, j, N; 6$, 0,) = sin(&) sin(02) sin(Ols - 02j)x x {sin(O,) + sin(&)+ + sin(& + &N - 2e2 - e,N)+ + sin(02 + e2N - 2e1 - e1N)}.
+ e2N)x
(35)
THE INVERSE OF A SYMMETRIC
4.
Tight-binding
143
BANDED TOEPLITZ MATRIX
systems
The Hamiltonian of an n-th neighbour lattice of N sites is given by I% = 2
tight-binding
F
system on a one-dimensional
(S)&k(S+ I:/.
(36)
s=l kn(s)
where N(S) and /J(s) are given by (3). The Green’s function defined, for energy E, by 6(N; E){Ej - if} = i.
operator
&(N; E) is (37)
Thus, with the identification, a(0) = -27
co - E
= -
n(k) =
E7?
5.
x:= 1..
,n,.
(38)
3.0
2.5
Fig. 1. Local
density
of states
at site s = 5 plotted tight-binding
system
against
y for a one-dimensional
of N = 100 sites.
nearest-neighbour
144
D. A. LAVIS and B. W. SOUTHERN
4.0
PE2 3.0
2.0
-1.0
-0.5
Fig. 2. Local density of states at site s = 5 plotted against y for a one-dimensional neighbour tight-binding system of N = 100 sites with ( = 0.5.
next-nearest-
(39) The local density of states p(N. s; y) at site s is given by p(N, s; y) = For the nearest-neighbour are given, from (25) as
(&Y$ls)
Q:((@(N,y lim C5+0+ r
+ iQls)}
(40)
model (n = 1) the diagonal elements of the Green’s function
= - cos{(N
+ 1)0} - cos{(N + 1 - 2s)B) 2q sin 8 sin{(N + 1)0}
(41)
where, from (23) B = arccosy. As an example, a plot of p(lOO,$r) is shown in Fig. 1. In computing this curve S was chosen to have the value 0.001. Since the density of states for real y is, as indicated in (40), given by the limit 6 + 0, the effect of a small non-zero b is to replace delta function singularities by steep lorentzians.
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ
MATRIX
14s
b 1
Fig. 3. Mean density of states plotted
against y for a one-dimensional system of N = 100 sites.
A similar analysis applies to the next-nearest-neighbour (33), (35) and (36) for b(s.j), where, from (27) cos&2
= -a{<*
nearest-neighbour
tight-binding
chain (7) = 2) using formulae
<2 + 8(1 + y)}.
(42)
with < = EI/E~. A plot of p(100,5;y) for this model with < = 0.5 is shown in Fig. 2:. Again S has the value 0.001. The local density of states at site s of a semi-infinite chain would be given in the limit of large N and the result for an infinite homogeneous chain is obtained by the second limit s -+ E. Alternatively the limit as N - x of the mean density of states a(N; 7) =
$2 p(N. s; y)
(43)
SXl
will yield the result for an infinite homogeneous chain. A plot of a(100; y) for the nearest neighbour model is shown in Fig. 3. The well-known basin-shaped curve [6] for the density of states of the infinite homogeneous system is clearly seen with delta singularities, arising from the finite value of N, superimposed.
146 5.
D. A. LAVIS and B. W. SOUTHERN
Conclusions
In this paper we have extended the work of Hu and O’Connell [5] by developing a method which will give an analytic formula for inverse of any finite N x N symmetric Toeplitz matrix of band-width 2n + 1 5 2N - 1. The matrix size N enters the formula as a parameter and does not affect the complexity of the calculation. The only explicit matrix inversion that is required is of an n x n matrix. Since in most problems of interest n will be much smaller than N, the matrix inversion can be easily performed using an algebraic computing package. Acknowledgments
This work was supported by NATO Research Grant No. 0087/87 and the Natural Sciences and Engineering Research Council of Canada. REFERENCES [l] [2] [3] [4] [5] [6]
[7] [8] [9] [lo]
Calderon A., Spitzer F. and Widom H.: Ill. J. Math. 3 (1959) 490. Widom H.: Ill. J Math. 4 (1960), 88. Golub G. H. and Van Loan C. F.: Matrix Computations, Johns Hopkins University Press, 1989. Streater R. F.: Bull. London Math. Sot. 11 (1979), 354. Hu G. Y. and O’Connell R. F.: J. Phys. A: Math. Gen. 29 (1996), 1511. Economou E. N.: Green’s Functions in Quantum Physics, Springer Series in Solid-State Sciences 7, Springer, New York 1983. Southern B. W., Kumar A. A., Loly P. D. and Tremblay A-M. S.: Phys. Rev. B 27 (1983) 1405. Southern B. W., Kumar A. A. and Ashraff J. A.: Phys. Rev. B 28 (1983) 1785. Southern B. W., Liu T. S. and Lavis D. A.: Phys. Rev. B 39 (1989), 160. Cyr S. L. M., Southern B. W. and Lavis D. A.: J. Phys.: Condens. Matter 8 (1996), 4781.
REPORTS
THE INVERSE
ON MATHEMATICAL
OF A SYMMETRIC
No. I
PHY.VI(‘S
BANDED
TOEPLITZ
MATRIX
D. A. UVIS Department
of Mathematics, King’s College, Strand, London, WCZR ZLS, England (e-mail: [email protected])
and B. W. SOUTHERN Department
of Physics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received June
18, 1996)
We describe a method for obtaining an analytic form for the inverse of a finite symmetric banded Toeplitz matrix. Explicit formulae are given for the tridiagonal and pentadiagonal cases and the results are applied to the evaluation of the Green’s function for nearest and next-nearest neighbour one-dimensional tight-binding systems.
1.
Introduction
A Toeplitz matrix A has elements A,j with the property A,, = n.(s - j). In the case of semi-infinite matrices (s, j = 0.1.. . .) necessary and sufficient conditions have long been known [l, 21 for the existence of an inverse and for finite matrices the inverse can be computed numerically using the Trench algorithm [3]. The eigenvalues and eigenvectors for the finite symmetric tridiagonal case were obtained by Streater [4]. These results provide an expression for the inverse matrix, which has also been obtained by Hu and O’Connell [5] from a calculation of the determinant and cofactors of the matrix. The inverse of the symmetric tridiagonal matrix can be used in the solution of various single-charge-tunnelling problems and of the one-dimensional Poisson equation with Dirichlet boundary conditions. It also provides the Green’s function for a one-dimensional homogeneous nearest-neighbour tight-binding system with open boundaries [6]. The resealing method for excitations in tight-binding systems [7, 81 and quantum spin chains [9, lo] uses a transfer matrix approach. This has the potential to provide a generalization of Hu and O’Connell’s result to all NxN matrices A, with elements’ ‘For any matrix X we denote the row and column vectors formed by its s-th row and ,I-th column by (s[X and Xlj) respectively and the .s.j-th element by (slX]j).
[I371
138
D. A. LAVIS
and B. W. SOUTHERN
of the form
(+lj)
=
a(js - jj)
if /s - jj < 71,
1
if 1s - jl = n,,
0
if Is - jl > 72,
n < N.
(1)
In Sections 2 and 3 we describe this procedure and give explicit formulae for bandwidths 2n + 1 = 3 and 2n + 1 = 5. In Section 4 these results are applied to first and second neighbour tight-binding models on a finite one-dimensional lattice. 2.
Method
Given that B is the inverse of A,
c
4W
+ kl%) = &.
(2)
k=a(s)
where a(s) = Max{1 - s, -n},
p(s) = Min{N - s.7~).
(3)
We define h,(j)
=
(4)
where b(s,j) = (.slBlj) if 15 s,j 5 N and zero otherwise,
and the transfer matrix
(5)
Then equation
(2) can be expressed in the form
T&(j) = bs-l(j) - &,jlll). Iterating
(6) gives
b,(j) =
if j > s,
T-"ho(j),
T-ho(j)
- T-(“-j+‘)ll)
if j 5 S.
(7)
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ MATRIX
The vector b&j) has zeros in the first 7~ elements
6(s. j) = 1
k(n,T-“/
k + ?b)b(k, j)
‘El x(r,IT-‘I
I; +
?(I,,
(7) gives
if j > s, (8)
7~)b(k,j) -
(r~IT-(~‘-~~+~)ll) if j < s.
A,=1
The vector bni(j) has zeros in the last 0 =
and so equation
139
+ ~IIT-~“IX: + 7~)b(k,j)
-
71
(7n +
elements,
so, from (7)
nIT-cAv-j+l)ll).
?fL= 1.....71.
w
k=l
From equation (8) the N elements of the j-th column of B are given as as linear combinations of the first n. elements which are in turn given as solutions of the ‘rt linear equations (9). The remaining problem is to obtain an expression for the elements of powers of the transfer matrix T. We define the functions Zn-j 4j(')=
C
CL(~~J,-~l)pl‘+j.
j =
O,I.....27?,.
(10)
r=O It is then not difficult to show that the eigenvalues k = l..... 7), of the equation @O(P) = 0. with corresponding
orthonormal
of T are the roots LLr;,IL;‘, (11)
left and right eigenvectors
(12) where (13) Thus (14) k=l
A more compact form is obtained
by setting p = exp(i8)
(1%
when we have (~(T”lj)
= TLj(m- s + 1).
s,j = I,....~R,
(16)
140
D. A. LAVIS and B. W. SOUTHERN
2n and any integer !,
where, for j = l,...,
h-j
(17) and f31,. . . ,8, are the roots of the equation
n-1 0 = F(e) E c0+e) + g(o) + C +)cOgre). r=l
(18)
It follows from (4) and (14) that
q(e)
for j = 2n,
-ul(e - 1)
=
{ -a(ln
- jl)ul(a-
1) +
uj+l(l
-
1) for 1 5 j < 2n.
(19)
Iterating the second of equations (19) and comparing with (17), we obtain the result ul(e) = --up,(e + 1) = -c
” sin{ (n + e)ek)
k=l
wok)
(20)
’
We now define the set of TZx n matrices
t&+*(4 - TL) 7&+2(-e - n - 1)
u(e) =
... ... (21)
%+1(-e
- 272+ 1)
%+n(-e
- 2n + 1)
...
and, from (8), (9), (16) and (20),
b(s.j)
=
-(ljU(s
- l)[U(N)]-iU(N
-(I] {U(s - l)[U(N)IPIU(N 3.
- j)]tz)
if j > s,
- j) - U(s - 1 -
j)} 17~)if j 5 s.
(22)
Explicit formulae
We use the procedure of the previous section to rederive the (n = 1) result of Hu and O’Connell [5] and to give the formula for the case n = 2. 3.1.
The case n = 1 Now U(e) is a 1 x 1 matrix with
u(e) = ul(e) = +(e
+ I) = ue(cOse).
2~0s~ = +o).
(23)
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ MATRIX
141
where U,(cosH) = is the Chebyshev polynomial
sin{(J! + l)fI} sin H
(24)
of the second kind. From (22) we obtain
bt~sli) = cos{(N
+ 1 - 1s - j])0} - cos{(N + 1 - s - j)H}
(25)
2 sin 0 sin{(N + 1)19) With the substitution
0= I
x
if I~L(O)(< 2,
iA
if 0(O) 5 -2,
iX + 7r
if n(O) 2 2,
for real A, this is the result obtained 3.2.
(26)
by Hu and O’Connell [5].
The case n = 2 Let
+ X1<2)
=
2(51
4).
+ (2)
=
(27)
-n(l).
We define V(e) = Uf(51) - U{(G) 2(<1
-
(28)
i2)
and, from (24), (28) and (21), we get V(-e) = -V(4! - 2). “(!)= Substituting
V(! + 1) - V(2)V(1) v(e+2)-V(2)V(e+l) (
(29) V(t) v(P+l)
PO)
1 .
(30) into (22) it can be shown, after some manipulation,
that
where F(s. j. N; 19r.Q = V(s - l)W(N - j, N + 1) - V(s)W(N - j. N)
(32)
W(.s, j) = V(s + l)V(j) - V(s)V(j + 1).
(33)
and
142
D. A. LAVIS
and B. W. SOUTHERN
From (24), (28) and (33) we obtain F(s,j,N;81,82)=fl(s-j,N;81,e2)-fl(s--j,N;B2,81)+ + s1(s
-
j, N;
Bl,f92> -
Sl(S
-
+ fi(s
+ j, N;
81,02) -
f2(s
+ j.
+
+ j, N;
Ql,e,)
g2(s
+ j, N;
g2(s
-
j, N;
82,&)
+
N; 02,6,) + e2, 01) +
N;e2, 4) + +g3(s,j,N;e1,e2)-g3(s,j,~;e2,e,) + + f3(s,iN;4, -0,) - .f3(s, i N; -e2, 0,) +
+ .f3(s, j, N;
e1 T0,)
+ g3(s,
6 7 -0,)
.A N;
-
f3(s,
-
j,
g3(s,i
N;
-e2,
e,),
(34)
where fl(k, N; el, e,) =
sin(02) c0s(e1k){2c0s(2e1
+ BIN)
sin(202 + e2N)-
- c0s(3e1 + e1N) sin(02 + O2N)-
- cOs(el + e1N) sin(302 + e2N)}, gl(k. n-: el, e2) =
sin(02) sin(81k){2sin(81) sin(B2)-2sin(2&
+ BIN) sin(202 + &N)+
+ sin(3& + B1N) sin(e2 + e2 N)+ + sin(& + O1N) sin(302 + &N)}, f2(k, N; 81, e2) =
2 sin(e2) cos(&k) cos(6$ + O1N) sin(202 + &N)x
x {cos(e,) - cos@)), g2(k, N;Ql,B2) =
2sin(82)sin(81k)sin(81
+ e1N)sin(2e2
x {cos(e,) - cos(e,)), f3(s,
j,
N; 6$, e2) = sin(&) sin(e2) cos(els - &j) x
x {cos(e,) - cOs(el) + + cOs(el + elzv - 2e2 - eziv)- c0s(e2 + e2N - 2e1 - e,N)), g3(s, j, N; 6$, 0,) = sin(&) sin(02) sin(Ols - 02j)x x {sin(O,) + sin(&)+ + sin(& + &N - 2e2 - e,N)+ + sin(02 + e2N - 2e1 - e1N)}.
+ e2N)x
(35)
THE INVERSE OF A SYMMETRIC
4.
Tight-binding
143
BANDED TOEPLITZ MATRIX
systems
The Hamiltonian of an n-th neighbour lattice of N sites is given by I% = 2
tight-binding
F
system on a one-dimensional
(S)&k(S+ I:/.
(36)
s=l kn(s)
where N(S) and /J(s) are given by (3). The Green’s function defined, for energy E, by 6(N; E){Ej - if} = i.
operator
&(N; E) is (37)
Thus, with the identification, a(0) = -27
co - E
= -
n(k) =
E7?
5.
x:= 1..
,n,.
(38)
3.0
2.5
Fig. 1. Local
density
of states
at site s = 5 plotted tight-binding
system
against
y for a one-dimensional
of N = 100 sites.
nearest-neighbour
144
D. A. LAVIS and B. W. SOUTHERN
4.0
PE2 3.0
2.0
-1.0
-0.5
Fig. 2. Local density of states at site s = 5 plotted against y for a one-dimensional neighbour tight-binding system of N = 100 sites with ( = 0.5.
next-nearest-
(39) The local density of states p(N. s; y) at site s is given by p(N, s; y) = For the nearest-neighbour are given, from (25) as
(&Y$ls)
Q:((@(N,y lim C5+0+ r
+ iQls)}
(40)
model (n = 1) the diagonal elements of the Green’s function
= - cos{(N
+ 1)0} - cos{(N + 1 - 2s)B) 2q sin 8 sin{(N + 1)0}
(41)
where, from (23) B = arccosy. As an example, a plot of p(lOO,$r) is shown in Fig. 1. In computing this curve S was chosen to have the value 0.001. Since the density of states for real y is, as indicated in (40), given by the limit 6 + 0, the effect of a small non-zero b is to replace delta function singularities by steep lorentzians.
THE INVERSE OF A SYMMETRIC
BANDED TOEPLITZ
MATRIX
14s
b 1
Fig. 3. Mean density of states plotted
against y for a one-dimensional system of N = 100 sites.
A similar analysis applies to the next-nearest-neighbour (33), (35) and (36) for b(s.j), where, from (27) cos&2
= -a{<*
nearest-neighbour
tight-binding
chain (7) = 2) using formulae
<2 + 8(1 + y)}.
(42)
with < = EI/E~. A plot of p(100,5;y) for this model with < = 0.5 is shown in Fig. 2:. Again S has the value 0.001. The local density of states at site s of a semi-infinite chain would be given in the limit of large N and the result for an infinite homogeneous chain is obtained by the second limit s -+ E. Alternatively the limit as N - x of the mean density of states a(N; 7) =
$2 p(N. s; y)
(43)
SXl
will yield the result for an infinite homogeneous chain. A plot of a(100; y) for the nearest neighbour model is shown in Fig. 3. The well-known basin-shaped curve [6] for the density of states of the infinite homogeneous system is clearly seen with delta singularities, arising from the finite value of N, superimposed.
146 5.
D. A. LAVIS and B. W. SOUTHERN
Conclusions
In this paper we have extended the work of Hu and O’Connell [5] by developing a method which will give an analytic formula for inverse of any finite N x N symmetric Toeplitz matrix of band-width 2n + 1 5 2N - 1. The matrix size N enters the formula as a parameter and does not affect the complexity of the calculation. The only explicit matrix inversion that is required is of an n x n matrix. Since in most problems of interest n will be much smaller than N, the matrix inversion can be easily performed using an algebraic computing package. Acknowledgments
This work was supported by NATO Research Grant No. 0087/87 and the Natural Sciences and Engineering Research Council of Canada. REFERENCES [l] [2] [3] [4] [5] [6]
[7] [8] [9] [lo]
Calderon A., Spitzer F. and Widom H.: Ill. J. Math. 3 (1959) 490. Widom H.: Ill. J Math. 4 (1960), 88. Golub G. H. and Van Loan C. F.: Matrix Computations, Johns Hopkins University Press, 1989. Streater R. F.: Bull. London Math. Sot. 11 (1979), 354. Hu G. Y. and O’Connell R. F.: J. Phys. A: Math. Gen. 29 (1996), 1511. Economou E. N.: Green’s Functions in Quantum Physics, Springer Series in Solid-State Sciences 7, Springer, New York 1983. Southern B. W., Kumar A. A., Loly P. D. and Tremblay A-M. S.: Phys. Rev. B 27 (1983) 1405. Southern B. W., Kumar A. A. and Ashraff J. A.: Phys. Rev. B 28 (1983) 1785. Southern B. W., Liu T. S. and Lavis D. A.: Phys. Rev. B 39 (1989), 160. Cyr S. L. M., Southern B. W. and Lavis D. A.: J. Phys.: Condens. Matter 8 (1996), 4781.