Extracting density information from finite Hamiltonian matrices

1. Extracting density information from finite Hamiltonian matrices We demonstrate how to extract approximate, yet highly accurate, density-of-state information over a continuous range of energies from a finite Hamiltonian matrix. The approximation schemes which we present make use of the theory of orthogonal polynomials associated with tridiagonal matrices. However, the methods work as well with non-tridiagonal matrices. We demonstrate the merits of the methods by applying them to problems with single, double, and multiple density bands, as well as to a problem with infinite spectrum.

2. With every Hamiltonian (hermitian matrix), there is an associated positive definite density of states function (in energy space). • Simple arguments could easily be under-stood when the Hamiltonian matrix is tridiagonal. • We exploit the intimate connection and interplay between tridiagonal matrices and the theory of orthogonal polynomials.

4. Solutions of the three-term recursion relation are orthogonal polynomials. • Regular pn(x) and irregular qn(x) solutions. • Homogeneous and inhomogeneous initial relations, respectively.

5. pn(x) is a polynomial of the “first kind” of degree n in x. • qn(x) is a polynomial of the “second kind” of degree (n1) in x. • The set of n zeros of pn(x) are the eigenvalues of the finite nn matrix H. • The set of (n1) zeros of qn(x) are the eigenvalues of the abbreviated version of this matrix obtained by deleting the first raw and first column.

6. They satisfy the Wronskian-like relation: • The density (weight) function associated with these polynomials: • The density function associated with the Hamiltonian H:

7. y G00(x+iy) x     Discrete spectrum of H Continuous band spectrum of H

8. Connection:

9. For a single limit: The density is single-band with no gaps and with the boundary For some large enough integer N

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

11. One-band density example

12. Two-Band Density giving again a quadratic equation for T(z) The boundaries of the two bands are obtained from the reality of T as

13. Two-band density example Three-band density example Infinite-band density example

14. Asymptotic limits not known? • Analytic continuation method • Dispersion correction method • Stieltjes Imaging method

15. Non-tridiagonal Hamiltonian matrices? Solution will be formulated in terms of the matrix eigenvalues instead of the coefficients .

16. Analytic Continuation         

18. One-Band Density Two-Band Density Infinite-Band Density

19. Dispersion Correction Gauss quadrature: Numerical weights:

21. n 4  () 3  2   1   0 1 2 3 4 0

22. One-Band Density Two-Band Density Infinite-Band Density

23. Stieltjes Imaging

24. Stieltjes Imaging

25. One-Band Density Two-Band Density Infinite-Band Density

26. الحمد لله والصلاة والسلام على رسول اللهوالسلام عليكم ورحمة الله وبركاته Thank you