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Phys 498CQM Lecture Notes 22
Lanczos Method
"Exact Diagonalization" and "Recursion"

Wednesday, April 11, 2001
Lecturer: Richard Martin

Reading:
Thijssen, A.8.2.2
Notes

  1. The Lanczos Method
    • Method for "Exact Diagonalization" for a few extreme eigenstates of extremely large matrices
    • Also "recursion method" that leads to many useful relations, including frequency dependent spectra and thermal expectation values
  2. Recall other iterative methods for large matrices
    • Conjugate Gradient minimization
    • RMM-DIIS (residual minimization)
    • Folded spectrum: special case with operator (H-E)2
  3. The Lanczos Method - General properties
    • Generate basis by multiplication of H times a starting vector
      (a Krylov subspace)
    • Lanczos algorithm leads to an orthonormal basis in which H is tridiagonal
    • Any problem can be converted into a "pseudo-1-dimensional" chain problem
    • Most useful for finding the few lowest (highest) eigenvectors of very large, sparse matrices
    • What is the "catch"? Why not always use Lanczos for all eigenvectors?
      Because errors accumulate. Orthogonality is guaranteed only for each successive step. For many vectors orthogonality is gradually lost, and "ghost" states may appear.
  4. Variations on way iteration is performed
    • Expansion of iterative Krylov space until lowest eigenstate(s) is converged
    • Repeated diagonalization of small m x m matrices until lowest eigenstate(s) is converged
      • Can be as small as 2 x 2
      • Very closely related to the RMM-DIIS method!
  5. Correlation functions in the ground state
    • Shown intrinsic properties of ground state
    • Also shows properties of excitations
      • Exponential decay of correlation functions implies gap in the spectrum
      • Power law decay implies no gap and special properties of excitations
      • Fermi liquid behavior requires Friedel oscillations
    • Important
    • Approach to infinite system
    • Analytic form of terminators
  6. Spectra and response functions as a function of frequency
    • Green's Function representation
    • Continued fractions
    • Approach to infinite system
    • Analytic form of terminators
  7. Thermal properties
    • Same as above but with imaginary (Matsubara) frequencies
    • Continued fraction converges quickly for high temperature
    • Correlation functions
  8. Examples of applications for independent particles
    • Tight-binding Hamiltonian
    • Recursion methods of Haydock
    • Physical interpretation of shells of neighbors as a 1-dimensional chain problem
    • Interest is in finding a few selected states or an entire spectrum
    • Useful to find total energy, etc.
  9. References
    • Koonin, p. 119
    • C. Lanczos, J. Res. Nat. Bur. STand. 45, 225 (1950).
    • G. H. Golub and C. F. Van Horn, Matrix Computations, (Johns Hopkins Press, 1989) Chapt. 9.
    • R. Haydock in Solid State Physics, vol. 35, (1980), p 215. (The recursion method for single body Lanczos type calculations and spectral functions.)
    • H. Q. Lin and J. E. Gubernatis, Computers in Physics 7, 400 (1993).
    • E. R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. 59, 2999–3002 (1987). "Dynamical properties of quantum many-body systems at zero temperature"
  10. Sources
    Example: LANZ programs in NETLIB

Last Modified April 11
Email question/comments/corrections to rmartin@uiuc.edu .