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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 and Mathematical Sciences

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Life in Cornell Cornell University | Sep 12, 20142

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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

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Cornell University | Sep 12, 20144 师 父 一日为 终生为 teacher/ mentor/ master teacher for a day father for life

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My Current Research Cornell University | Sep 12, 20145

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

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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

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Grand Canonical Density Matrix Cornell University | Sep 12, 20148 grand canonical ensemble matrix elementsFermi-Dirac distribution

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

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

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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)

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Physical Intuition & Mapping Cornell University | Sep 12, 201412

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

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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

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Asymptotic Behavior of Correlations Cornell University | Sep 12, 201415 Cheong & Henley, PRB 80, 165124 (2009)

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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

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