LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II

LECTURE 4 WORM ALGORITHM FOR QUANTUM STATISTICAL MODELS II Path-integral for continuous systems: oriented closed loops

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

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

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

Path-integrals in continuous space (see Werner Krauth s lectures) P 2 1

Path-integrals in continuous space (see Werner Krauth s lectures) P 2 1 P

One of the simplest approximations: (there are schemes which are accurate up to, e.g. Chin s formula, or even better)

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)

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

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

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

P 2 1

P 2 1 P

Ira Masha

(open/close update)

- select any bead at random; this is proposed Ira bead (open/close update)

- 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)

- 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)

(open/close update)

- check whether and proceed if true (open/close update)

- check whether and proceed if true - seed positions for all beads between Ira and Masha from normalized Gaussian (open/close update)

- 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

(insert/remove update)

- select any at random any space point and any time slice; this is proposed Masha-bead position (insert/remove update)

- 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)

- 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

(insert/remove update)

- 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

(advance/recede update)

- select at random number of slices (between 1 and M) to move Ira forward (advance/recede update)

- 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)

- 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)

(advance/recede update)

- select at random number of slices (between 1 and M) to move Ira backward (advance/recede update)

- 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)

A B (swap update)

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, 036701 (2006)] (swap update)

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

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

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

Estimators:

Estimators: kinetic potential

Estimators: kinetic potential + higher-order schemes, virial estimator, etc.

Estimators: kinetic potential + higher-order schemes, virial estimator, etc. + measuring at specified points, not bins

Density matrix, gaps, quasiparticle dispersion and residue

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

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

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

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

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

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

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

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

Some examples: Weakly interacting Bose gas:

Plots for determining critical temperature at Old scheme 97 1. 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

Plots for determining critical temperature at Old scheme 97 Old scheme 04 1. 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

Plots for determining critical temperature at Old scheme 04 Worm algorithm 100,000 Imperfect crossing due to corrections to scaling

Some examples: Weakly interacting Bose gas: Worm algorithm: Pilati, Giorgini, NP

Some examples: Weakly interacting Bose gas: Worm algorithm: Pilati, Giorgini, NP Very strong speckle potentials barely suppress superfluidity

Some examples: in liquid helium-4 64 2048 Accuracy of about 0.5% [Aziz pairwise potential]

Some examples: Vacancy and interstitial gaps in helium-4

Some examples: Vacancy and interstitial gaps in helium-4 Now, hole excitation particle excitation

Some examples: Vacancy and interstitial gaps in helium-4 Now, hole excitation particle excitation melting density

Some examples: Screw dislocation in soli helium-4

Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis

Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis winding number map Initial atomic positions

Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis

Some examples: Screw dislocation in soli helium-4 About 1000 atoms, with BC favoring screw along z-axis Screw core = pipe filled with superfluid

A can of worms: Symmetry breaking fields:

A can of worms: Symmetry breaking fields: Transverse-field Ising model:

A can of worms: Multi-component systems / paring

A can of worms: Multi-component systems / paring Two interacting (weakly) superfluids

A can of worms: Multi-component systems / paring Paired superfluid

A can of worms: Multi-component systems / paring Paired superfluid loosing paring energy

A can of worms: Multi-component systems / paring Paired superfluid

A can of worms: Multi-component systems / paring Paired superfluid Use an appropriate correlator:

Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function

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

Why bother with mappings and algorithms?

Why bother with mappings and algorithms? Efficiency

Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams

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

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