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

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Presentation on theme: "Numerical Methods as Exploration Tools for Theoretical Condensed Matter Physics Siew-Ann CHEONG Division of Physics and Applied Physics, School of Physical."— Presentation transcript:

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 Cornell University | Sep 12, 20142

3 Nanyang Technological University National University of Singapore Singapore Management University Singapore University of Technology and Design Nanyang Technological University Former Central Library Presently Chinese Heritage Centre Nanyang University ( 1956–1980 ) Nanyang Technological Institute ( 1981–1990 ) Nanyang Technological University ( 1991– ) Cornell University | Sep 12, 2014 School of Physical and Mathematical Sciences 3

4 Cornell University | Sep 12, 师 父 一日为 终生为 teacher/ mentor/ master teacher for a day father for life

5 My Current Research Cornell University | Sep 12, 20145

6 Density Matrix of a Chain of Fermions Cornell University | Sep 12, 20146

7 Projection Operators & Conjecture Cornell University | Sep 12, Example: written in terms of = Green function matrix of chain = Green function matrix of block Define For

8 Grand Canonical Density Matrix Cornell University | Sep 12, grand canonical ensemble matrix elementsFermi-Dirac distribution

9 Grassmannian Trace Formulas Cornell University | Sep 12, fermionic coherent state overcomplete basis

10 Exact Formula for ρ Cornell University | Sep 12, block environment Cheong & Henley, PRB 69, (2004)

11 ρ as a Diagnostic Tool Cornell University | Sep 12, model Hamiltonian Hamiltonian matrix exact diagonalization (ED) ED ground state long-range order? non-trivial local structure? ρ of two blocks r apart Cheong & Henley, PRB 79, (2009)

12 Physical Intuition & Mapping Cornell University | Sep 12,

13 Exact Ground State Cornell University | Sep 12, different number of configurations Bloch state exact in the limit

14 Exact Correlations Cornell University | Sep 12, corresponding local observables corresponding nonlocal observables intervening particle expansion p = specific occupation of sites between j and j + r

15 Asymptotic Behavior of Correlations Cornell University | Sep 12, Cheong & Henley, PRB 80, (2009)

16 Conclusions Numerical explorations – Matrix computation and diagonalization of ρ – Exact diagonalization of H t,V=∞ Analytical insights – Connection between ρ and G – Connection between H t,V=∞ and H t,V=0 Exact formulas – – Cornell University | Sep 12,


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