Extracting density Extracting density information from finite information from finite Hamiltonian matrices Hamiltonian matrices

We demonstrate how to extract We demonstrate how to extract approximate, yet highly accurate, density-of-approximate, yet highly accurate, density-of-state information over a continuous range of state information over a continuous range of

energies from a finite Hamiltonian matrix. The energies from a finite Hamiltonian matrix. The approximation schemes which we present approximation schemes which we present

make use of the theory of orthogonal make use of the theory of orthogonal polynomials associated with tridiagonal polynomials associated with tridiagonal

matrices. However, the methods work as well matrices. However, the methods work as well with non-tridiagonal matrices. We demonstrate with non-tridiagonal matrices. We demonstrate the merits of the methods by applying them to the merits of the methods by applying them to

problems with single, double, and multiple problems with single, double, and multiple density bands, as well as to a problem with density bands, as well as to a problem with

infinite spectrum. infinite spectrum.

With every Hamiltonian (hermitian matrix), With every Hamiltonian (hermitian matrix), there is an associated positive definite there is an associated positive definite density of states function (in energy space).density of states function (in energy space).

Simple arguments could easily be under-Simple arguments could easily be under-stood when the Hamiltonian matrix is stood when the Hamiltonian matrix is tridiagonal.tridiagonal.

We exploit the intimate connection and We exploit the intimate connection and interplay between tridiagonal matrices interplay between tridiagonal matrices and the theory of orthogonal and the theory of orthogonal polynomials.polynomials.

0 0

0 1 1

1 2 2

2 3

0

0

a b

b a b

b a b

H b a

( ) 0xH x r

( ) ( ) ( )x n nnr f x r

1 1 1( ) ( ) ( ) ( )n n n n n n nx f x a f x b f x b f x

0n n

Solutions of the three-term recursion Solutions of the three-term recursion relation are orthogonal polynomials.relation are orthogonal polynomials.

Regular Regular ppnn((xx)) and irregular and irregular qqnn((xx)) solutions. solutions.

Homogeneous and inhomogeneous Homogeneous and inhomogeneous initial relations, respectively.initial relations, respectively.

0 0 0 1 0( ) 0 and ( ) 1a x p b p p x

0 0 0 1 0( ) 1 and ( ) 0a x q b q q x

1 1 1( ) 0 , 1n n n n n na x f b f b f n

ppnn((xx)) is a polynomial of the “first kind” is a polynomial of the “first kind” of degree of degree nn in in xx..

qqnn((xx)) is a polynomial of the “second is a polynomial of the “second kind” of degree kind” of degree ((nn1)1) in in xx..

The set of The set of nn zeros of zeros of ppnn((xx)) are the are the eigenvalues of the finite eigenvalues of the finite nnnn matrix matrix HH..

The set of The set of ((nn1)1) zeros of zeros of qqnn((xx)) are are the eigenvalues of the abbreviated the eigenvalues of the abbreviated version of this matrix obtained by version of this matrix obtained by deleting the first raw and first column.deleting the first raw and first column.

1

0

n

m m

2

0ˆ n

m m

They satisfy the Wronskian-like They satisfy the Wronskian-like relation:relation:

The density (weight) function The density (weight) function associated with these polynomials:associated with these polynomials:

The density function associated with The density function associated with the Hamiltonian the Hamiltonian HH::

1 1( ) ( ) ( ) ( ) 1n n n n nb p x q x p x q x

( ) ( ) ( )x

n m nm

x

x p x p x dx

00 00 0011

( ) ( 0) ( 0) Im ( 0)2

x G x i G x i G x ii

x

y

G00(x+iy)

Discrete spectrum of H

Continuous band spectrum of H

1( )nmnmH zG z

ConnectionConnection::

00 ( ) lim ( ) ( )n nn

G z q z p z

00 20

0 21

12

1( )

...

G zb

z ab

z az a

For a single limit:For a single limit:

The density is single-band with no gaps The density is single-band with no gaps and with the boundaryand with the boundary

lim , ,n nn

a b a b

2x a b

0 0

0 1 1

1 2

2

2 1

0

0

N

N N

a b

b a b

b a

bH

b a b

b a b

b a

For some For some large large

enough enough integer integer NN

00 20

0 21

1 22

21

1( )

...( )

N

N

G zb

z ab

z ab

z az a T z

2 2

2

2

( )( )

...

b bT z

b z a T zz a

bz a

z a

1( ) 2 2

2 2

z aT z z a b z a b

Note the reality limit of the root and its relation to the boundary of the density band

One-band density example One-band density example

Mathcad Document

Two-Band DensityTwo-Band Density2

22

11

2

( )( )

( )( )

bT z

bz a

z a T z

giving again a quadratic equation for T(z)

The boundaries of the two bands are obtained from the reality of T as

2 21 2

1 2 1 2

14

2 2

a aa a b b

2 21 2

1 2 1 2

14

2 2

a aa a b b

Two-band density exampleTwo-band density example

Three-band density Three-band density exampleexample

Infinite-band density Infinite-band density exampleexample

Mathcad Document

Mathcad Document

Mathcad Document

Asymptotic limits not Asymptotic limits not known?known?

Analytic continuation methodAnalytic continuation method

Dispersion correction methodDispersion correction method

Stieltjes Imaging methodStieltjes Imaging method

Non-tridiagonal Hamiltonian Non-tridiagonal Hamiltonian matrices?matrices?

Solution will be formulated in Solution will be formulated in terms of the matrix eigenvalues terms of the matrix eigenvalues instead of the coefficients . instead of the coefficients . ,n ma b

ˆ,n m

Analytic ContinuationAnalytic Continuation

00 00 0011

( ) ( 0) ( 0) Im ( 0)2

x G x i G x i G x ii

2

000 1

0

ˆ( )( )

( ) lim( )

( )

N

mn m

Nnn

nn

zq z

G zp z

z

One-Band DensityOne-Band Density

Two-Band DensityTwo-Band Density

Infinite-Band DensityInfinite-Band Density

Mathcad Document

Mathcad Document

Mathcad Document

Dispersion CorrectionDispersion Correction

Gauss quadrature:

1

0

( ) ( ) ( )x N

n nnx

x f x dx f

Numerical weights:

20n n

2 1

0 0

ˆN N

n n j n ij i

i n

1

0

( ) ( )x N

dn n

nx

f x dx f

1

0

( )( )( ) ( )

( ) ( )

x x Nn

nn nx x

ff xf x dx x dx

x

( ) dn n n

( )n n 1

n

dn x

d dx

()2

1

3

4

1 2 3 40

0

n

One-Band DensityOne-Band Density

Two-Band DensityTwo-Band Density

Infinite-Band DensityInfinite-Band Density

Mathcad Document

Mathcad Document

Mathcad Document

Stieltjes Imaging Stieltjes Imaging

0

( ) n

n mm

Stieltjes Imaging Stieltjes Imaging

0

12( )

n

n m nm

( )x d dx

One-Band DensityOne-Band Density

Two-Band DensityTwo-Band Density

Infinite-Band DensityInfinite-Band Density

Mathcad Document

Mathcad Document

Mathcad Document

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