Double occupancy as a probe of the Mott state for fermions in 1D lattices
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1) This document proposes measuring double occupancy as a probe of the Mott transition in a one-dimensional fermionic Hubbard model with an optical lattice. 2) It finds that the Mott phase exhibits inherent fluctuations in double occupancy that can be used to detect the Mott phase. 3) The double occupancy in the bulk can be determined from measurements in a trapped system using the local density approximation.
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Double occupancy as a probe of the Mott state for fermions in 1D lattices
- 1. Double occupancy as a probe of the Mott state for fermions in one-dimensional optical lattices VIVALDO L. CAMPO, JR (1), KLAUS CAPELLE (2), CHRIS HOOLEY (3), JORGE QUINTANILLA (4,5), and VITO W. SCAROLA (6) (1) UFSCar, Brazil, (2) UFABC, Brazil, (3) SUPA and University of St Andrews, UK, (4) SEPnet and Hubbard Theory Consortium, University of Kent, (5) ISIS Facility, Rutherford Appleton Laboratory, and (6) Virginia Tech, USA arxiv.org:1107.4349 QuAMP, Oxford, 20 September 2011
- 3. Why double occupancy? Experiments on 3D Hubbard model - Experimental evidence for the Mott transition: U. Schneider, L. Hackermuller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch, A. Rosch, Science322, 1520-1525 (2008). Robert Jordens, NielsStrohmaier, Kenneth Gunter, Henning Moritz & TilmanEsslinger, Nature 455, 204-208 (2008).
- 4. Problem:What will happen in 1D?
- 5. Problem:What will happen in 1D? “A theory of correlations [...] will be mainly concerned with understanding [...] the balance between band-like and atomic-like behaviour.” Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...] John Hubbad, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 276, 238-257 (1963)
- 6. Problem:What will happen in 1D? Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...] Without the trap, we have an exact solution [Lieb & Wu 1968]
- 7. Problem:What will happen in 1D? Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...] Without the trap, we have an exact solution [Lieb & Wu 1968] We know how to deal with the trapping potential [ C. Hooley & JQ, PRL (2004); V.L. Campo, K. Capelle, JQ & C. Hooley, PRL (2007) ]
- 8. Problem:What will happen in 1D? Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...] Without the trap, we have an exact solution [Lieb & Wu 1968] We know how to deal with the trapping potential [ C. Hooley & JQ, PRL (2004); V.L. Campo, K. Capelle, JQ & C. Hooley, PRL (2007) ] Straight-forward to evaluate double occupancy:
- 9. Effect of the trap – no fluctuations
- 10. Effect of the trap – no fluctuations
- 11. Effect of the trap – no fluctuations D Mott insulator Band +Mott Band insulator D
- 12. Ground state – no trap U / t f Mott insulator: 0 1 2 Luttinger Liquid Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
- 13. Ground state – no trap U / t f Mott insulator: 0 1 2 Luttinger Liquid Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
- 14. Ground state – no trap U / t f Mott insulator: 0 1 2 Luttinger Liquid Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
- 15. Ground state – no trap U / t f Mott insulator: 0 1 2 Luttinger Liquid Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
- 16. Ground state - harmonic trap
- 17. Ground state - harmonic trap Evaluate D in the local density approximation:
- 18. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2)
- 19. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2)
- 20. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7
- 21. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7
- 22. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7
- 23. Finite temperature – no trap Use high-temperature expansion: (must go at least to 2nd order) Double occupancy: = + + ...
- 24. Finite temperature – no trap Match to low-T expansion from quantum transfer method [Klümper and Bariev 1996] Obtain C(x) is the unity central charge from CFT for the Hesienberg universality class:
- 25. Finite temperature – no trap
- 26. Finite temperature – no trap Very good match between high-T and low-T expansions.
- 27. Finite temperature – no trap Very good match between high-T and low-T expansions. d vs T is non-monotonic (suggests cooling mechanism with 1D system as reference state) [c.f. 3D case: F. Werner, O. Parcollet, A. Georges & S.R. Hassan, PRL (2005)]
- 28. The non-monotonic beahviour is a strictly local phenomenon!
- 29. The non-monotonic beahviour is a strictly local phenomenon!
- 30. The non-monotonic beahviour is a strictly local phenomenon!
- 31. The non-monotonic beahviour is a strictly local phenomenon!
- 32. Quantum fluctuations + thermal fluctuations + trap
- 33. In summary... Fermionic Hubbard model in one dimension. Mott phase has inherent double occupancy fluctuations. Mott phase detectable via double occupancy. Can read out double occupancy in the bulk from the trapped data. Non-monotonic temperature dependence a universal, local feature. THANKS! arxiv.org:1107.4349
Hinweis der Redaktion
- In the presence of the trap, we can still solve problem analytically if we ignore the hopping term. In this limit the system is always an insulator.We have three regimes: -for weak interactions, the system forms a band insulator with two atoms per site-For strong interaction, we have a Mott insulator with one atom per site-in the intermediate regime we have coexistence of band insulator and mott insulator regions
- In the presence of the trap, we can still solve problem analytically if we ignore the hopping term. In this limit the system is always an insulator.We have three regimes: -for weak interactions, the system forms a band insulator with two atoms per site-For strong interaction, we have a Mott insulator with one atom per site-in the intermediate regime we have coexistence of band insulator and mott insulator regions
- Without the trap, exact results are available [Lieb & Wu 1968]The groundstate phase diagram features Luttinger liquid and Mott insulator phases.Can compute D exactly.Interestingly, find large D even deep in the Mott insulating region: for U as large as the bandwidth one in five atoms are in a doubly-occupied site. This is due to strong quantum fluctuations inherent to the Mott insulating state in 1D.
- Without the trap, exact results are available [Lieb & Wu 1968]The groundstate phase diagram features Luttinger liquid and Mott insulator phases.Can compute D exactly.Interestingly, find large D even deep in the Mott insulating region: for U as large as the bandwidth one in five atoms are in a doubly-occupied site. This is due to strong quantum fluctuations inherent to the Mott insulating state in 1D.
- Without the trap, exact results are available [Lieb & Wu 1968]The groundstate phase diagram features Luttinger liquid and Mott insulator phases.Can compute D exactly.Interestingly, find large D even deep in the Mott insulating region: for U as large as the bandwidth one in five atoms are in a doubly-occupied site. This is due to strong quantum fluctuations inherent to the Mott insulating state in 1D.
- Without the trap, exact results are available [Lieb & Wu 1968]The groundstate phase diagram features Luttinger liquid and Mott insulator phases.Can compute D exactly.Interestingly, find large D even deep in the Mott insulating region: for U as large as the bandwidth one in five atoms are in a doubly-occupied site. This is due to strong quantum fluctuations inherent to the Mott insulating state in 1D.