Extracting density information from finite Hamiltonian matrices

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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.

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• 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.

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

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• 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.

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

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y
G00(xiy)
x
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Discrete spectrum of H
Continuous band spectrum of H
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Connection
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For a single limit The density is single-band
with no gaps and with the boundary
For some large enough integer N
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Note the reality limit of the root and its
relation to the boundary of the density band
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One-band density example
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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
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Two-band density example
Three-band density example
Infinite-band density example
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Asymptotic limits not known?

• Analytic continuation method
• Dispersion correction method
• Stieltjes Imaging method

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Non-tridiagonal Hamiltonian matrices?
Solution will be formulated in terms of the
matrix eigenvalues instead of the
coefficients .
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Analytic Continuation
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One-Band Density
Two-Band Density
Infinite-Band Density
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Dispersion Correction
Gauss quadrature
Numerical weights
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n
4
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3
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2
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1
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0
?1
?2
?3
?4
?0
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One-Band Density
Two-Band Density
Infinite-Band Density
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Stieltjes Imaging
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Stieltjes Imaging
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One-Band Density
Two-Band Density
Infinite-Band Density
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Thank you