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 Fixing the number of particles update  Self-consistency is reached Done NoYes

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

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

Power of Path Integral Formulation quantum particle at TP classical particles at PT

Natural Variables in a Path Integral Approach PP PP

Universal Functional

Evaluation of D(  P,  P ) PP PP

Universal Potential F ferm (  P,  P  P  PP PP

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