LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II
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1 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II
2 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops
3 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops *Fermions too but sign problem makes it useless
4 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops Worm Algorithm description. *Fermions too but sign problem makes it useless
5 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops Worm Algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. *Fermions too but sign problem makes it useless
6 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops Worm Algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. Examples and illustrations: WIBG & Helium-4 *Fermions too but sign problem makes it useless
7 LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops Worm Algorithm description. Estimators: energy, density, superfluid stiffness, density matrix, gaps, etc. Examples and illustrations: WIBG & Helium-4 A can of worms. Multicomponent systems / U(1) symmetry breaking fields *Fermions too but sign problem makes it useless
8 Path-integrals in continuous space (see Werner Krauth s lectures) P 2 1
9 Path-integrals in continuous space (see Werner Krauth s lectures) P 2 1 P
10
11
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15
16
17 One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better)
18 One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better)
19 One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better) Here (PBC in time)
20 One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better) Here (PBC in time) In the limit we write it as
21 One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better) Here (PBC in time) In the limit we write it as Kinetic Potential
22 Exchange cycles exist only for identical particles For fermions every pair-wise exchange changes trajectory sign Exchange cycles is the ONLY difference between Bosons, fermions, and distinguishable particles
23 Exchange cycles exist only for identical particles For fermions every pair-wise exchange changes trajectory sign Exchange cycles is the ONLY difference between Bosons, fermions, and distinguishable particles Ground states for bosons and distinguishable particles are the same (Feynman).
24 P 2 1
25 P 2 1 P
26
27 Ira Masha
28 Ira Masha
29
30 (open/close update)
31 - select any bead at random; this is proposed Ira bead (open/close update)
32 - select any bead at random; this is proposed Ira bead - select at random any bead ahead of Ira but not further than M slices away (open/close update)
33 - select any bead at random; this is proposed Ira bead - select at random any bead ahead of Ira but not further than M slices away - select at random any bead ahead of Ira but not further than M slices away; this is proposed Masha bead. Quit if (open/close update)
34 - select any bead at random; this is proposed Ira bead - select at random any bead ahead of Ira but not further than M slices away - select at random any bead ahead of Ira but not further than M slices away; this is proposed Masha bead. Quit if - the proposal is to remove all beads between I and M (open/close update) Accept it with probability
35
36 (open/close update)
37 - check whether and proceed if true (open/close update)
38 - check whether and proceed if true - seed positions for all beads between Ira and Masha from normalized Gaussian (open/close update)
39 - check whether and proceed if true - seed positions for all beads between Ira and Masha from normalized Gaussian (open/close update)
40 - check whether and proceed if true - seed positions for all beads between Ira and Masha from normalized Gaussian (open/close update) - the proposal is to insert new beads between I and M and go to Z-sector. Accept it with probability
41
42 (insert/remove update)
43 - select any at random any space point and any time slice; this is proposed Masha-bead position (insert/remove update)
44 - select any at random any space point and any time slice; this is proposed Masha-bead position - select at random Ira-slice ahead of Masha but not further than M slices away (insert/remove update)
45 - select any at random any space point and any time slice; this is proposed Masha-bead position - select at random Ira-slice ahead of Masha but not further than M slices away - seed positions for all new beads between Ira and Masha (including Ira from normalized Gaussian (insert/remove update)
46 - select any at random any space point and any time slice; this is proposed Masha-bead position - select at random Ira-slice ahead of Masha but not further than M slices away - seed positions for all new beads between Ira and Masha (including Ira from normalized Gaussian - the proposal is to insert a new piece of worldline and go to G-sector. (insert/remove update) Accept it with probability
47 - select any at random any space point and any time slice; this is proposed Masha-bead position - select at random Ira-slice ahead of Masha but not further than M slices away - seed positions for all new beads between Ira and Masha (including Ira from normalized Gaussian - the proposal is to insert a new piece of worldline and go to G-sector. (insert/remove update) Accept it with probability OK, use to control relative statistics between Z- and G-sectors
48
49 (insert/remove update)
50
51 - If Ira and Masha at a distance of M slices or less, propose to eliminate the worldline between them and go to Z-sector Accept it with probability
52 (advance/recede update)
53 (advance/recede update)
54 (advance/recede update)
55 - select at random number of slices (between 1 and M) to move Ira forward (advance/recede update)
56 - select at random number of slices (between 1 and M) to move Ira forward - seed positions for all beads between Ira and Ira from normalized Gaussian (advance/recede update)
57 - select at random number of slices (between 1 and M) to move Ira forward - seed positions for all beads between Ira and Ira from normalized Gaussian (advance/recede update)
58 - select at random number of slices (between 1 and M) to move Ira forward - seed positions for all beads between Ira and Ira from normalized Gaussian - the proposal is to insert new beads between I and I (including I ) and move I to I Accept it with probability (advance/recede update)
59 (advance/recede update)
60 (advance/recede update)
61 (advance/recede update)
62 - select at random number of slices (between 1 and M) to move Ira backward (advance/recede update)
63 - select at random number of slices (between 1 and M) to move Ira backward - the proposal is to erase beads between I and I (including I) and move I to I Accept it with probability (advance/recede update)
64 A B (swap update)
65 A B (swap update)
66 A B (swap update)
67 A B The proposal is too long to bother you with complete description! The idea is to move Ira to one of the nearby lines, say to bead B, and reconnect the ends starting from bead A which is M slices away. [see Phys. Rev. E, 74, (2006)] (swap update)
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
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 P 2 1
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 P name; slice(name), r(dim,name), type(name) 2 1
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 P name1=next(name) name; slice(name), r(dim,name), type(name) 2 1
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 P name1=next(name) name; slice(name), r(dim,name), type(name) name2=prev(name) 2 1
73 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 P name1=next(name) name; slice(name), r(dim,name), type(name) name2=prev(name) 2 + Ira & Masha, if present 1
74 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 P name1=next(name) name; slice(name), r(dim,name), type(name) name2=prev(name) 2 + Ira & Masha, if present 1 Hash table on each time slice for fast space search
75 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 P name1=next(name) name; slice(name), r(dim,name), type(name) name2=prev(name) 2 + Ira & Masha, if present 1 List of all beads in this space/time bin Hash table on each time slice for fast space search
76 Estimators:
77 Estimators: kinetic potential
78 Estimators: kinetic potential + higher-order schemes, virial estimator, etc.
79 Estimators: kinetic potential + higher-order schemes, virial estimator, etc.
80 Estimators: kinetic potential + higher-order schemes, virial estimator, etc.
81 Estimators: kinetic potential + higher-order schemes, virial estimator, etc.
82 Estimators: kinetic potential + higher-order schemes, virial estimator, etc. + measuring at specified points, not bins
83 Density matrix, gaps, quasiparticle dispersion and residue
84 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function:
85 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Monte Carlo estimator for a given point is 1
86 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function:
87 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and
88 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density
89 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
90 Density matrix, gaps, quasiparticle dispersion and residue Well,we have full Green s function: Density matrix: correlation length, coherence, and condensate density
91 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
92 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,
93 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
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 Now,
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, hole excitation particle excitation
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, hole excitation particle excitation Quai-particle residue Quai-particle energy
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, hole excitation particle excitation Quai-particle residue Quai-particle energy If there is a gap then is finite is defines the gap.
98 Some examples: Weakly interacting Bose gas:
99 Plots for determining critical temperature at Old scheme Small base of small system sizes (from L=3 to L=6 only) 2. Large error bars leading to multiple crossings 3. Intersection at the level of 1.1 (close to Gaussian universality class) instead of U(1) value of 1.548
100 Plots for determining critical temperature at Old scheme 97 Old scheme Small base of small system sizes (from L=3 to L=6 only) 2. Large error bars leading to multiple crossings 3. Intersection at the level of 1.1 (close to Gaussian universality class) instead of U(1) value of 1.548
101 Plots for determining critical temperature at Old scheme 04 Worm algorithm 100,000 Imperfect crossing due to corrections to scaling
102 Some examples: Weakly interacting Bose gas: Worm algorithm: Pilati, Giorgini, NP
103 Some examples: Weakly interacting Bose gas: Worm algorithm: Pilati, Giorgini, NP
104 Some examples: Weakly interacting Bose gas: Worm algorithm: Pilati, Giorgini, NP Very strong speckle potentials barely suppress superfluidity
105 Some examples: in liquid helium Accuracy of about 0.5% [Aziz pairwise potential]
106 Some examples: in liquid helium Accuracy of about 0.5% [Aziz pairwise potential]
107 Some examples: Vacancy and interstitial gaps in helium-4
108 Some examples: Vacancy and interstitial gaps in helium-4 Now, hole excitation particle excitation
109 Some examples: Vacancy and interstitial gaps in helium-4 Now, hole excitation particle excitation melting density
110 Some examples: Screw dislocation in soli helium-4
111 Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis
112 Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis winding number map Initial atomic positions
113 Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis
114 Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis
115 Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis Screw core = pipe filled with superfluid
116 A can of worms: Symmetry breaking fields:
117 A can of worms: Symmetry breaking fields: Transverse-field Ising model:
118 A can of worms: Symmetry breaking fields: Transverse-field Ising model:
119 A can of worms: Multi-component systems / paring
120 A can of worms: Multi-component systems / paring Two interacting (weakly) superfluids
121 A can of worms: Multi-component systems / paring Two interacting (weakly) superfluids
122 A can of worms: Multi-component systems / paring Two interacting (weakly) superfluids
123 A can of worms: Multi-component systems / paring Two interacting (weakly) superfluids
124 A can of worms: Multi-component systems / paring Paired superfluid
125 A can of worms: Multi-component systems / paring Paired superfluid loosing paring energy
126 A can of worms: Multi-component systems / paring Paired superfluid
127 A can of worms: Multi-component systems / paring Paired superfluid
128 A can of worms: Multi-component systems / paring Paired superfluid Use an appropriate correlator:
129 Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function
130 Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function
131 Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function The rest is worm algorithm in this configuration space: draw and erase interaction vertexes using exclusively Ira and Masha
132 Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function The rest is worm algorithm in this configuration space: draw and erase interaction vertexes using exclusively Ira and Masha
133 Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function The rest is worm algorithm in this configuration space: draw and erase interaction vertexes using exclusively Ira and Masha
134 Why bother with mappings and algorithms?
135 Why bother with mappings and algorithms?
136 Why bother with mappings and algorithms? Efficiency
137 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams
138 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams PhD while still young
139 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics PhD while still young
140 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young
141 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
142 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
143 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
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