Numerical Methods in Quantum Many-body Theory. Gun Sang Jeon Pyeong-chang Summer Institute 2014

Transcription

1 Numerical Methods in Quantum Many-body Theory Gun Sang Jeon Pyeong-chang Summer Institute 2014

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3 Contents Introduction to Computational Physics Monte Carlo Methods: Basics and Applications Numerical Calculations in Quantum Theory of Many Particles Example: Dynamical Mean-Field Theory

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5 Computational Physics What does atomic physics do? Atoms, Molecules What does nuclear physics do? Nuclei What does computational physics do? Computers(?), Computation(?) Physics

6 Let s play with computers. Let s play with physics. Be Friends with computers

7 Steps in numerical works 1. Setting up a problem 2. Coding and running a program 3. Analyzing and visualizing results

8 1. Setting up a problem State and understand a problem Find relevant variables and how to solve it Convert it to a computer-friendly form

9 2. Coding and running a program Design a global structure Writing a program (Coding) Use pseudocode first verified existing routine separated logic groups comments Avoid complex logic tricky technique computer-dependent language Leave a backup-copy of programs before significant changes are made

10 2. Coding and running a program Testing and debugging a program Grammatical error debugging at the compiler stage Terminated without results infinite loops, memory access error,... Running but wrong results???

11 How to debug Working like a detective!!! Check global logic Check local logic by simulating a program personally Check line-by-line values of variables

12 How to test results Check the result of the current program in the known special cases Check whether the current program reproduce other previous results Check whether the results are physically sensible

13 3. Analyzing & visualizing results Analysis of results are very important since it lays physical meaning on the results. Significance of the results is not given by how difficult the computation is but by how important its physical meaning is. Visualization helps us to understand results. It can emphasize the significance of results.

14 3. Analyzing & visualizing results Example

15 3. Analyzing & visualizing results Example

16 Remark Think in a way that computers do. Computers can perform numerical calculation very fast. Computers do not know physics, nor even have common senses. Understand how computer works as much as possible. Let s play with computers.

17 Example: Throwing a ball through air How do we know if the result is correct?

18 How to test results Check special cases Check whether other previous results are reproduced Check whether the results are physically sensible

19 Compare previous results v x 0 0, vy0 0

20 Let s play with physics variation of b

21 Let s play with physics variation of angle

22 Let s play with physics

23 Let s play with physics n=2 n=4

24 Let s play with physics

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26 Area of a circle 1 1 Area 4 (5/7) 2.9

27 Area of a circle 1 1

28 Area of a circle 1 N tot area ± ± ± ± ± ±0.002 Area =

29 Integration by Monte Carlo

30 Simple Sampling

31 Simple Sampling N tot Method I Simple Sampling ± ± ± ± ± ± ± ± ± ± ± ± 0.001

32 Importance Sampling

33 Simple versus Importance Sampling N tot Simple Sampling Importance Sampling ± ± ± ± ± ± ± ± ± ± ± ±

34 Metropolis Method

35 Metropolis Method (Cont d)

36 Example: Metropolis Method

37 Alternative Way for Metropolis Method

38 Metropolis Method for Multi-Dimension Integral

39 Statistical Mechanics of Thermodynamic Systems

40 Metropolis method

41 Ensemble average

42 Metropolis importance sampling with Ising spins

43 Simple example

44 Simple example (cont d)

45 Metropolis importance sampling (simple example)

46 Some Useful Tips

47 Some Useful Tips

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49 RMn 2 O 5 (R=Tb,Y,Dy,Ho,Bi, ) Mn 4+ : octahedrally coordinated (S=3/2) Mn 3+ : tetrahedrally coordinated (S=2)

50 BiMn 2 O 5 Application of high magnetic fields along the a axis Sharp symmetric peak in dm/dh and ε at low temperatures Abrupt sign change in P Magnetic-field-induced metaelectric transition Kim,Haam,Oh,Park,Cheong,Sharma,Jaime,Harrison,Han,Jeon,Coleman,Kiim, PNAS(2009)

51 Exchange interactions J3 J4,J5

52 Spin-lattice coupling via exchange striction

53 Single-ion anisotropy + magnetic field

54 Polarization and Magnetization T > T * : continuous crossover T < T * : first-order metaelectric transition Jeon, Park, Kim, Kim, Han, PRB (2009)

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56 Skyrmion Topological excitation in twodimensional Heisenberg spin model Indirect NMR observation of skyrmions in two-dimensional quantum Hall ferromagnetic state

57 Skyrmion Crystal Monte Carlo simulations of the Hamiltonian zero magnetic field weak magnetic field Yu, Onose, Kanazawa, Park, Han, Matsui, Nagaosa Tokura, Nature (2010)

58 Real-space observation of Skyrmion Crystal zero magnetic field weak magnetic field Yu, Onoze, Kanazawa, Park, Han, Matsui, Nagaosa Tokura, Nature (2010)

59 Phase Diagram Yu, Onose, Kanazawa, Park, Han, Matsui, Nagaosa Tokura, Nature (2010)

60 Example: Dynamical Mean-Field Theory

61 Numerical Methods in Quantum Theory of Many Particles Quantum Monte Carlo Method Density Matrix Renormalization Group Numerical Renormalization Group Exact Diagonalization Slave-Boson Theory Hartree-Fock Approximation Decoupling Method Dynamical Mean-Field Theory

62 Hubbard model Tight-binding electrons with on-site Coulomb repulsion U

63 Mott-Hubbard Transtion Weakly correlated Intermediate regime : hard to describe quantitatively Strongly correlated

64 Dynamical Mean-Field Theory

65 Weiss Mean-Field Theory: classical Ising model STEP 1: Construct an effective single-site model STEP 2: Solve a single-site model STEP 3: Impose a self-consistent condition

66 Weiss Mean-Field Theory: classical Ising model Weiss mean-field equation Or m h eff

67 Dynamical Mean-Field Theory Hubbard model STEP 1: Construct an effective single-site (impurity) model

68 Dynamical Mean-Field Theory Hubbard model STEP 2: Solve a single-site (impurity) model

69 Dynamical Mean-Field Theory Hubbard model STEP 3: Impose a self-consistent condition

70 Mean-Field Theory Classical vs. Quantum

71 Why DMFT? Non-perturbative and time-resolved treatment of local interactions Unified description of metals and Mott insulators

72 How to Solve Impurity Problem Exact Diagonalization (ED) Quantum Monte Carlo (QMC) continous time QMC Hirsch-Fye QMC Projective QMC Numerical renormalization group Density matrix renormalization group Non-Crossing Approximation Iterated Perturbation Theory Hubbard Approximations

73 DMFT + ED + Self-consistent equations for bath parameters

74 DMFT + ED (1) Prepare initial sets of {, ε }

75 DMFT + ED (2) Lanczos method

76 DMFT + ED (3) Minimize d to yield new sets of {, ε }

77 Bath Geometry (DMFT+ED)

78 DMFT + CTQMC by AJK

79 DMFT + CTQMC by AJK

80 DMFT+ED (IPT) Metal U Insulator Caffarel, Krauth (1994) Georges, Kotliar, Krauth, Rozenberg (1996)

81 DMFT+QMC (Hirsch-Fye) Metal U Insulator Georges, Kotliar, Krauth, Rozenberg (1996)

82 DMFT + NRG Metal-insulator transition at Uc2=1.47W Uc1=1.25W Bulla, PRL (1999)

83 Quasiparticle weight (DMFT+NRG) Bulla, PRL (1999)

84 Phase Diagram Existence of coexistence region First-order transition line ends at finite-temperature critical point Crossover region above a critical point Bulla,Costi,Vollhardt, PRB(2001) Capone,Medici,Georges (2007)

85 Phase Diagram of V2O3 McWhan et al. PRL (1971) Limelette,Georges,Jerome,Wzietek,Metcal f, Honig, Science (2003)

86 Phase Diagram of V2O3 DMFT : first unified framework for various phases

87 Concluding Remark