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Computational Solid State Physics 計算物性学特論　第８回

Explore the use of Quantum Monte Carlo methods, such as Diffusion Monte Carlo, to calculate ground state properties in solid state physics. Topics covered include the Pauli principle, importance sampling, Schrödinger equation, time-dependent and imaginary time methods, as well as branching processes. Application to hydrogen atoms and Fermi fluids.

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Computational Solid State Physics 計算物性学特論　第８回

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  1. Computational Solid State Physics 計算物性学特論　第８回 8. Many-body effect II: Quantum Monte Carlo method

  2. Diffusion Monte Carlo method to calculate the ground state Importance sampling method How to treat the Pauli principle: fixed node approximation Quantum diffusion Monte Carlo method

  3. Schrödinger equation in atomic unit How to solve the Schrödingerequation for many electrons?

  4. Time-dependent Schrödinger equation Imaginary Time The ground state wave function can be obtained in the limit of infinite time.

  5. Diffusion equation with branching process for the ground state wave function diffusion branching Diffusion equation holds for

  6. Diffusion equation for particles Flux: diffusion flux drift flux D : diffusion constant, vd: drift velocity Conservation of number of particles: Diffusion equation

  7. Rate equation R>0 : growth rate R<0: decay rate

  8. Time-dependent Green’s function : Boundary Condition

  9. Time evolution of wave function

  10. Short time approximation

  11. Green’s function of the classical diffusion equation The transition probability from x to ycan be simulated by random walk in 3N dimensions for N electron system.

  12. Green’s function of the rate equation The branching process can be simulated by the creation or destruction of walkers with probability GB

  13. Importance sampling : analytical trial fn. Diffusion equation with branching process Diffusion Branching Drift : Local energy : Quantum force

  14. Biased diffusion Green’s function Kinetic energy operator Drift term The transition probability from x to ycan be simulated by biased random walk with quantum force F in 3N dimensions for N electron system.

  15. Detailed balance condition To guarantee equilibrium Acceptance ratio of move of the walker from x to y

  16. DMC and Importance-sampled DMC for the hydrogen atom Branching process: suppression of branching process DMC Importance-sampled DMC

  17. Walker 1 Biased diffusion Walker 2 Walker 3 Branching Walker 4 Schematic of the Green’s function Monte Carlo calculation with importance sampling for 3 electrons

  18. Evaluation of the ground state energy

  19. How to remove the condition ? • Fixed node approximation to treat wave functions with nodes • Fixed phase approximation to treat complex wave functions

  20. Fixed node approximation Importance sampling on condition Wave function φ is assumed to have the same nodes with ΨＤ.

  21. Pauli principle for n like-spin electrons Slater determinant Slater determinant has nodes.

  22. Fixed phase approximation Importance sampling on condition Wave function φ is assumed to have the same phase with

  23. Ground states of free electrons D.M.Ceperley, B.J.Alder: PRL 45(1980)566

  24. Transition of the ground state of free electrons • Unpolarized Fermi fluid • Polarized Fermi fluid • Wigner crystal n: concentration of free electrons

  25. Problems 8 • Calculate the ground state wave function of a hydrogen atom, using the diffusion Monte Carlo method. Consider how to calculate the ground state energy. • Calculate the ground state of a hydrogen atom, using the diffusion Monte Carlo method with importance sampling method. Assume the trial function as follows. • Derive the diffusion equation for in importance sampling method.

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