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Numerical Methods as Exploration Tools for Theoretical Condensed Matter Physics

Numerical Methods as Exploration Tools for Theoretical Condensed Matter Physics. Siew-Ann CHEONG Division of Physics and Applied Physics, School of Physical and Mathematical Sciences. Life in Cornell. Nanyang Technological University. Nanyang Technological Institute （ 1981–1990 ）.

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Numerical Methods as Exploration Tools for Theoretical Condensed Matter Physics

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  1. Numerical Methods as Exploration Tools for Theoretical Condensed Matter Physics Siew-Ann CHEONG Division of Physics and Applied Physics, School of Physical and Mathematical Sciences

  2. Life in Cornell

  3. Nanyang Technological University Nanyang Technological Institute （1981–1990） Nanyang Technological University （1991–） Nanyang University （1956–1980） School of Physical and Mathematical Sciences Nanyang Technological University Singapore University of Technology and Design National University of Singapore Singapore Management University SINGAPORE Former Central Library Presently Chinese Heritage Centre

  4. teacher for a day 师 一日为 teacher/ mentor/ master 父 终生为 father for life

  5. My Current Research

  6. Density Matrix of a Chain of Fermions

  7. Projection Operators & Conjecture Define Example: For written in terms of = Green function matrix of chain = Green function matrix of block

  8. Grand Canonical Density Matrix grand canonical ensemble matrix elements Fermi-Dirac distribution

  9. Grassmannian Trace Formulas fermionic coherent state overcomplete basis 1 2 3

  10. Exact Formula for ρ block environment Cheong & Henley, PRB 69, 075111 (2004)

  11. ρ as a Diagnostic Tool model Hamiltonian long-range order? Hamiltonian matrix non-trivial local structure? exact diagonalization (ED) ρ of two blocks r apart ED ground state Cheong & Henley, PRB 79, 212402 (2009)

  12. Physical Intuition & Mapping

  13. Exact Ground State different number of configurations Bloch state Bloch state exact in the limit

  14. Exact Correlations corresponding local observables corresponding nonlocal observables intervening particle expansion p = specific occupation of sites between j and j + r

  15. Asymptotic Behavior of Correlations Cheong & Henley, PRB 80, 165124 (2009)

  16. Conclusions • Numerical explorations • Matrix computation and diagonalization of ρ • Exact diagonalization of Ht,V=∞ • Analytical insights • Connection between ρ and G • Connection between Ht,V=∞ and Ht,V=0 • Exact formulas

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