Quantum Monte Carlo Study of vortices in cold atoms

1. Quantum Monte Carlo Study of vortices in cold atoms Etienne Thibierge Project supervised by Prof. Olav Syljuåsen FysiskInstitutt, Universitetet i Oslo, Norway May – July 2010

2. 1. Introduction • Cold atomic systems in optical lattices: high interest! • Very well understood at the microscopic level • Extremely tunable experimentally • Model systems of Condensed Matter type • In particular, the 3d quantum XY model can be achieved: • Model largely studied: • Phase transition at finite temperature • Vortices at high temperature Etienne Thibierge

3. 1. Introduction • Aim of this project: • study vortices and their properties • in particular establish a link with the atoms in an atomic eigenstate • Outline: • 1. Introduction • 2. Effective model • 3. The multi-loop algorithm • 4. Vortices and BKT transition • 5. A link between vortices and eigenstates? • 6. Conclusion Etienne Thibierge

4. 2. Effective model Etienne Thibierge 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion

5. 2.1. Optical lattices 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Action of laser light on atoms can be described by an optical potential (AC-Stark effect): • Periodic potential achieved by using stationary waves, created by counter-propagating lasers • If the potential barriers are high enough, the atoms are trapped in an optical lattice • Lots of experimental possibilities: • Lattice geometry (in our case square lattice) • Exactly one atom per site • Two-state systems, by using atomic hyperfine states Etienne Thibierge

6. 2.2. A Hamiltonian for trapped atoms 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Starting point: second quantized Hamiltonian for a weakly interacting Bose gas in an external potential • By expanding it on the basis of the Wannier functions and after simplification, it reads: • Then we assume:  The first term of can be treated as a perturbation of the second one. Etienne Thibierge

7. 2.2. A Hamiltonian for trapped atoms 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Schwinger pseudo-spin operators: • Finally the Hamiltonian reads: • By a clever choice: Etienne Thibierge

8. 3. The multi-loop algorithm Etienne Thibierge 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion

9. 3.1. Introduction 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Monte Carlo methods: • Class of various algorithms • Compute thermodynamic quantities • First classical and then quantum systems E. Fermi S. Ulam • Basic idea: sampling of the configuration space by choosing most often the most probable configurations. Etienne Thibierge

10. 3.2. Space-time plaquettes 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Partition function: • Graphical representations: spin representation Spin Spin Etienne Thibierge

11. 3.2. Space-time plaquettes 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Graphical representations: vertex representation Spin Spin Etienne Thibierge

12. 3.3. Updating procedure 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion Etienne Thibierge

13. 3.4. Algorithm outline 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Global subdivision into several Monte Carlo Steps • One single Monte Carlo Step: • Starts from the final configuration of the previous MCS • Breakup specification for all the plaquettes, with suitable probabilities • Loops construction, randomly flipped with probability ½ • Measurements in the new configuration Etienne Thibierge

14. 4. Vortices and BKT transition Etienne Thibierge 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion

15. 4.1. BKT phase transition 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • BKT = Berezinskii – Kosterlitz – Thouless (1972-1973) • Transition of infinite order occurring in the XY model • Description of the phases: • Low temperature: quasi-ordered phase • Around the critical temperature: tightly bound vortex-antivortex pairs • High temperature: separate vortices and antivortices • Signature of the transition: universal jump in the spin stiffness at the critical temperature Etienne Thibierge

16. 4.1. BKT phase transition 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Stiffness and temperature: in units of in units of Etienne Thibierge

17. 4.2. Vortices and vorticity 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Vortex and anti-vortex: • Vortex density operator: +1 -1 0 4 3 1 2 Etienne Thibierge

18. 4.3. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Vortex density: • In the graphical representation: • An operator plaquette flips the spin, exactly as a Pauli matrix does  even number of operator plaquettes in the same loop in order the configuration to contribute one Pauli matrix one operator plaquette = Etienne Thibierge

19. 4.3. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • On the algorithmic level: • Scan over all lattice cells • At each cell, check if it contributes or not to the bilinear and quadrilinear terms • Bilinear term: • If the sites n. 1 and 3 (of the cell) belong to the same loop, the contribution is +1 • Otherwise it is 0 4 3 1 2 Etienne Thibierge

20. 4.3. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Quadrilinear term: • Two possible contributions: • (1) The operator plaquettes belong to two different loops, by pairs • (2) All the operator plaquettes belong to the same loop • (1) Disconnected contribution 1 (1) -i (+i) -i (+i) -i (+i) 1 (1) +i (-i) 1 (1) 1 (1) +1 Etienne Thibierge 0

21. 4.3. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • (2) Connected contribution -i (+i) -i (+i) +i (-i) 1 (1) -i (+i) 1 (1) 1 (1) 1 (1) +1 -1 Etienne Thibierge

22. 4.4. Vortex density and temperature 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion in units of Etienne Thibierge

23. 5. A link between vortices and eigenstates? Etienne Thibierge 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion

24. 5.1. A physical picture 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Idea: near vortices, the spins exhibit a larger value of their z component • Far from any vortex: • Near a vortex: • Atomic language: Etienne Thibierge

25. 5.2. Eigenstate and vortex densities 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Eigenstate density: • Vortex density: in units of Etienne Thibierge

26. 5.2. Eigenstate and vortex densities 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Derivative of the vortex density: • Derivative of the eigenstate density: in units of in units of Etienne Thibierge

27. 5.3. Single configuration analysis 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • At that point: link between vortex and eigenstate densities • But: is it true that one eigenstate = one vortex? • Idea: study correlations between eigenstates, keeping in mind that at low temperature vortices are tightly bound in pairs vortex-antivortex • Instead of implementing the correlators, we study single configuration snapshots Etienne Thibierge

28. 5.3. Single configuration analysis 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Strongcorrelations in vortices, but do not appearthere  tends to indicatethat eigenstates and vortices are not equivalent • Nonetheless: how doesit look likewhen a vortex and an antivortex are closerthan one latticespacing? Etienne Thibierge

29. 6. Conclusion 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • The atomicHamiltonian in opticallatticescanbemappedinto a spin-like XY Hamiltonian • By the way of a multi-loop QMC algorithm, we have studiedvorticesappearing in this model in the framework of the BKT phase transition • Vortex and eigenstate densities have been compared and theirderivativesalso, showingsomesimilarity • A strict equivalencebetweenvortices and eigenstates does not seemobviousaccording to the study of correlations in a single configuration Etienne Thibierge

30. Værså god før forsiktiget Etienne Thibierge

31. Worldline representation 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Graphical representations: worldline representation Spin Spin Etienne Thibierge

32. Plaquette configurations 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Possible plaquette configurations: Etienne Thibierge

33. Spin stiffness 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • Signature of the transition: universal jump in the spin stiffness at the critical temperature • Spin stiffness: • Winding number in direction : related to geometrical properties of the spin configuration Etienne Thibierge

34. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • The operators do not belong to the same loop: Etienne Thibierge

35. QMC estimator of the vortex density 1. Introduction 2. Effective model 3. The multi-loop algorithm 4. Vortices and BKT transition 5. A link between vortices and eigenstates? 6. Conclusion • The operators belong to the same loop: Etienne Thibierge