Numerical Methods as Exploration Tools for Theoretical Condensed

Matter Physics

Siew-Ann CHEONGDivision of Physics and Applied Physics, School of Physical and Mathematical Sciences

Life in Cornell

Cornell University | Sep 12, 2014 2

Nanyang Technological University

SINGAPORE

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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师

父一日为

终生为

teacher/mentor/master

teacher for a day

father for life

My Current Research

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Density Matrix of a Chain of Fermions

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Projection Operators & Conjecture

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Example:

written in terms of

= Green function matrix of chain= Green function matrix of block

Define

For

Grand Canonical Density Matrix

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grand canonical ensemble

matrix elements Fermi-Dirac distribution

Grassmannian Trace Formulas

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fermionic coherent state

overcomplete basis

1

2

3

Exact Formula for ρ

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block

environment

Cheong & Henley, PRB 69, 075111 (2004)

ρ as a Diagnostic Tool

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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)

Physical Intuition & Mapping

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Exact Ground State

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different number of configurations

Bloch state

Bloch state

exact in the limit

Exact Correlations

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corresponding local

observables

corresponding nonlocal observables

intervening particle expansionp = specific occupation of sites between j and j + r

Asymptotic Behavior of Correlations

Cornell University | Sep 12, 2014 15Cheong & Henley, PRB 80, 165124 (2009)

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– –

Cornell University | Sep 12, 2014 16