LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS

Transcription

1 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS

2 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course

3 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course *Fermions too but sign problem makes it useless

4 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course Worm Algorithm. Incomplete algorithm description. *Fermions too but sign problem makes it useless

5 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course Worm Algorithm. Incomplete algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. *Fermions too but sign problem makes it useless

6 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course Worm Algorithm. Incomplete algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. Examples and illustrations. *Fermions too but sign problem makes it useless

7 LECTURE 3 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS Path-integral for lattice bosons*: oriented closed loops, of course Worm Algorithm. Incomplete algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. Examples and illustrations. Ising model in transverse field. *Fermions too but sign problem makes it useless

8 Interacting particles on a lattice:

9 Interacting particles on a lattice: diagonal off-diagonal

10 Interacting particles on a lattice:

11 Interacting particles on a lattice:

12 Interacting particles on a lattice: by definition of the time-ordered exponent

13 Interacting particles on a lattice:

14 Interacting particles on a lattice:

15 Interacting particles on a lattice:

16 Interacting particles on a lattice: In the diagonal basis set (occupation number, or Fock, representation):

17 Interacting particles on a lattice: In the diagonal basis set (occupation number, or Fock, representation):

18 Interacting particles on a lattice: In the diagonal basis set (occupation number, or Fock, representation): Each term describes a particular evolution of as imaginary time increases

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23 0-order term one of the 2-order terms imaginary time + + in this example

24 0-order term one of the 2-order terms imaginary time + + in this example 0-order term

25 0-order term one of the 2-order terms imaginary time + + in this example 0-order term 2-d order term; in this example matrix elements of equal to for bosons

26 0-order term one of the 2-order terms imaginary time + + in this example 0-order term

27 0-order term one of the 2-order terms imaginary time + + in this example 0-order term 2-d order term; in this example matrix elements of equal to for bosons

28 high-order term for off-diagonal matrix elements for the trajectory with K kinks at times (ordered sequence on the -cylinder) all possible trajectories for N particles with K hopping transitions potential energy contribution

29 high-order term for off-diagonal matrix elements for the trajectory with K kinks at times (ordered sequence on the -cylinder) all possible trajectories for N particles with K hopping transitions potential energy contribution (easy to decompose into one-, two-, etc. body piece-wise terms)

30 high-order term for closed oriented loops (PBC in time) off-diagonal matrix elements for the trajectory with K kinks at times (ordered sequence on the -cylinder) all possible trajectories for N particles with K hopping transitions potential energy contribution (easy to decompose into one-, two-, etc. body piece-wise terms)

31 Green s function

32 Green s function Similar expansion in hopping terms for + two special points for Ira and Masha

33 Green s function Similar expansion in hopping terms for + two special points for Ira and Masha

34 Green s function Similar expansion in hopping terms for + two special points for Ira and Masha Worm Algorithm in this configuration space is as before: draw and erase lines using exclusively Ira and Masha

35 Green s function Similar expansion in hopping terms for + two special points for Ira and Masha graph element = interval Worm Algorithm in this configuration space is as before: draw and erase lines using exclusively Ira and Masha

36 Ergodic set of local updates

37 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha

38 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha space shift ( particle type): j i j i

39 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha j j i i space shift ( particle type): j space shift ( hole type): j i i

40 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha j j i i space shift ( particle type): j space shift ( hole type): j i i Insert/delete Ira and Masha:

41 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha j j i i space shift ( particle type): j space shift ( hole type): j i i Insert/delete Ira and Masha:

42 Ergodic set of local updates another kink (or off-diagonal) event time shift: Ira or Masha j j i i space shift ( particle type): j space shift ( hole type): j i i Insert/delete Ira and Masha: connects and configuration spaces

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44 M I

45 M I

46 I M

47 I M

48 Data structure and implementation of updates

49 Data structure and implementation of updates standard updates (integration using MC):

50 Data structure and implementation of updates standard updates (integration using MC): time shift:

51 Data structure and implementation of updates standard updates (integration using MC): time shift: -Collect information about the neighborhood of the Worm : and

52 Data structure and implementation of updates standard updates (integration using MC): time shift: -Collect information about the neighborhood of the Worm : and - Propose from the normalized probability density defined on, for example,

53 Data structure and implementation of updates standard updates (integration using MC): time shift: -Collect information about the neighborhood of the Worm : and - Propose from the normalized probability density defined on, for example, - Accept with probability (unity for on-site interactions).

54 j space shift ( particle type): i j i

55 space shift ( particle type): j i - Collect information about the neighborhood of the Worm j i

56 space shift ( particle type): j i - Collect information about the neighborhood of the Worm - Propose from the normalized probability density defined on j i

57 space shift ( particle type): j i - Collect information about the neighborhood of the Worm - Propose from the normalized probability density defined on j i - Accept with probability

58 space shift ( particle type): j i - Collect information about the neighborhood of the Worm - Propose from the normalized probability density defined on j i - Accept with probability Diagrammatic MC style (new continuous variables) Will be described in detail by Boris Svistunov later

59 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions

60 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions

61 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions a

62 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions a b=next(a)

63 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions c=prev(a) a b=next(a)

64 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions c=prev(a) a=prev(b) b=next(a)

65 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions site(a) c=prev(a) a=prev(b) b=next(a)

66 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions time(a) site(a) c=prev(a) a=prev(b) b=next(a)

67 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions time(a) tau(a), occ(a) site(a) c=prev(a) a=prev(b) b=next(a)

68 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions time(a) tau(a), occ(a) site(a) c=prev(a) a=prev(b) b=next(a) d=kink(a)

69 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions time(a) tau(a), occ(a) site(a) c=prev(a) a=prev(b) a=kink(d) b=next(a) d=kink(a)

70 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions e=link(a,dir) time(a) tau(a), occ(a) site(a) c=prev(a) a=prev(b) a=kink(d) b=next(a) d=kink(a)

71 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions e=link(a,dir) site(a) c=prev(a) time(a) tau(a), occ(a) a=prev(b) a=kink(d) Ira b=next(a) d=kink(a)

72 Data structure Locality: Any local spot on the path can be reconstructed without knowing the rest all updates require O(1) operations irrespective of T and L for short range interactions e=link(a,dir) site(a) c=prev(a) time(a) tau(a), occ(a) a=prev(b) a=kink(d) Ira b=next(a) d=kink(a) + management of name allocation: (i) all names are unique, (ii) not ordered and drawn from a finite pool

73 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

74 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

75 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

76 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

77 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

78 Energy, density, compressibility, superfluid stiffness, density matrix, gaps,

79 Recall, winding numbers and superfluid density z y x

80 Recall, winding numbers and superfluid density (cross-section independent in Z-sector) z y x

81 Recall, winding numbers and superfluid density z y x

82 Recall, winding numbers and superfluid density z y x

83 Recall, winding numbers and superfluid density z y x

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86 Density matrix, gaps, quasiparticle dispersion and residue

87 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function:

88 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Monte Carlo estimator for a given point is 1

89 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function:

90 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and

91 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density

92 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density Monte Carlo estimator 1

93 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density

94 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution

95 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now,

96 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now, Meaning: >> microscopic time scales

97 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now,

98 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now, hole excitation particle excitation

99 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now, hole excitation particle excitation Quai-particle residue Quai-particle energy

100 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density and, more generally, the entire momentum distribution Now, hole excitation particle excitation Quai-particle residue Quai-particle energy If there is a gap then is finite is defines the gap.

101 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF

102 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF determine gaps for adding/removing particles from the MI state with

103 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF gaps control the exponential decay of the Green s function in time determine gaps for adding/removing particles from the MI state with

104 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF gaps control the exponential decay of the Green s function in time determine gaps for adding/removing particles from the MI state with

105 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF Otherwise, good luck in calculating energy differences determine gaps for adding/removing particles from the MI state with

106 Mott insulator superfluid T=0 phase diagram: plane, 3D case MI SF determine gaps for adding/removing particles from the MI state with

107 Quantum spin chains gaps, spin wave spectra, magnetization curves Energy gap: One dimensional S=1 chain with Spin gap Z -factor

108 Grand canonical ensemble (a must for disorder problems!) V(x) R 1 R 2

109 Grand canonical ensemble (a must for disorder problems!) V(x) R 1 R 2 R 1 R 2 exponentially rare event

110 Grand canonical ensemble (a must for disorder problems!) V(x) R 1 R 2

111 Grand canonical ensemble (a must for disorder problems!) R 1 R2 V(x) R 1 R 2

112 Grand canonical ensemble (a must for disorder problems!) R 1 R2 V(x) R 1 R 2 Exponential slowing done is solved (no need to tunnel to explore.and. ) R 1 R 2

113 uniform distribution on

114 uniform distribution on

115 Added particle wavefunction: mobility thresholds, participation ratio, etc. Current standard for simulations of bosons in optical lattices and in traps: all experimental parameters as is, including particle number

116 Added particle wavefunction: Not fully explored yet mobility thresholds, participation ratio, etc. Current standard for simulations of bosons in optical lattices and in traps: all experimental parameters as is, including particle number

117 Transverse-field Ising model:

118 Transverse-field Ising model:

119 Transverse-field Ising model:

120 Transverse-field Ising model:

121 Transverse-field Ising model: Closed non-oriented loops similar to the classical Ising model Worm Algorithm is obtained by considering

122 Transverse-field Ising model: Closed non-oriented loops similar to the classical Ising model Worm Algorithm is obtained by considering