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, 20144 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, 20147 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, 20148 grand canonical ensemble matrix elementsFermi-Dirac distribution

9. Grassmannian Trace Formulas Cornell University | Sep 12, 20149 fermionic coherent state overcomplete basis 1 1 2 2 3 3

10. Exact Formula for ρ Cornell University | Sep 12, 201410 block environment Cheong & Henley, PRB 69, 075111 (2004)

11. ρ as a Diagnostic Tool Cornell University | Sep 12, 201411 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, 212402 (2009)

12. Physical Intuition & Mapping Cornell University | Sep 12, 201412

13. Exact Ground State Cornell University | Sep 12, 201413 different number of configurations Bloch state exact in the limit

14. Exact Correlations Cornell University | Sep 12, 201414 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, 201415 Cheong & Henley, PRB 80, 165124 (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, 201416