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Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach

Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach. M. Skorobogatiy and J.D. Joannopoulos MIT. Non Zero Temperature DFT. Self-Consistent Algorithm for Finding Density Matrix. Given density matrix calculate. Given H KS calculate.

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Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach

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  1. Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach M. Skorobogatiy and J.D. Joannopoulos MIT

  2. Non Zero Temperature DFT

  3. Self-Consistent Algorithm for Finding Density Matrix Given density matrix calculate Given HKS calculate Fixing the number of particles update m Done Self-consistency is reached No Yes

  4. Ways of evaluating 1) Explicit Hamiltonian diagonalization – expansion into the eigen modes ψof a Hamiltonian 2) Implicit Hamiltonian diagonalization – expansion into any complete set of modes ψ

  5. Ways of evaluating 1) Expansion into polynomial series 2) Integral representation 3) Path integral representation – 2) assumes |ψ>=|r>, evaluates 2) by introducing intermediate states and separating kinetic and potential energies

  6. Power of Path Integral Formulation quantum particle at T P classical particles at PT

  7. Natural Variables in a Path Integral Approach hP xP

  8. Universal Functional

  9. Evaluation of D(xP,hP ) xP hP

  10. Universal Potential Fferm(1,P,xP,hP) xP hP

  11. Conclusions • Path Integral Formalism allows an elegant separation of “universal” and “system-dependent” properties for a particular problem • Universal potential can be calculated in advance and used with different systems • Advanced Monte-Carlo techniques can be used to evaluate a “system-dependent” distribution of natural variables

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