Double occupancy as a probe of the Mott state for fermions in one-dimensional optical lattices<br />VIVALDO L. CAMPO, JR (...
Why double occupancy?<br />
Why double occupancy?<br />Experiments on 3D Hubbard model - Experimental evidence for the Mott transition:<br />U. Schnei...
Problem:What will happen in 1D?<br />
Problem:What will happen in 1D?<br />“A theory of correlations [...] will be mainly concerned with understanding [...] the...
Problem:What will happen in 1D?<br />Hamiltonian: 	[Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an ...
Problem:What will happen in 1D?<br />Hamiltonian: 	[Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an ...
Problem:What will happen in 1D?<br />Hamiltonian: 	[Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an ...
Effect of the trap – no fluctuations<br />
Effect of the trap – no fluctuations<br />
Effect of the trap – no fluctuations<br />D<br />Mott insulator<br />Band<br />+Mott<br />Band insulator<br />D<br />
Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. L...
Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. L...
Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. L...
Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. L...
Ground state - harmonic trap<br />
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t...
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t...
Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t...
Finite temperature – no trap<br />Use high-temperature expansion:<br />	(must go at least to 2nd order)<br />Double occupa...
Finite temperature – no trap<br />Match to low-T expansion from quantum transfer method [Klümper and Bariev 1996]<br />Obt...
Finite temperature – no trap<br />
Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />
Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />d vs T is non-monotonic (sugge...
The non-monotonic beahviour is a strictly local phenomenon!<br />
The non-monotonic beahviour is a strictly local phenomenon!<br />
The non-monotonic beahviour is a strictly local phenomenon!<br />
The non-monotonic beahviour is a strictly local phenomenon!<br />
Quantum fluctuations + thermal fluctuations + trap<br />
In summary...<br />Fermionic Hubbard model in one dimension.<br />Mott phase has inherent double occupancy fluctuations.<b...
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Double Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices

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Contributed talk at QuAMP 2011, Oxford.

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Double Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices

  1. 1. Double occupancy as a probe of the Mott state for fermions in one-dimensional optical lattices<br />VIVALDO L. CAMPO, JR (1), KLAUS CAPELLE (2), CHRIS HOOLEY (3), JORGE QUINTANILLA (4,5), and VITO W. SCAROLA (6)<br />(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<br />arxiv.org:1107.4349<br />QuAMP, Oxford, 20 September 2011<br />
  2. 2. Why double occupancy?<br />
  3. 3. Why double occupancy?<br />Experiments on 3D Hubbard model - Experimental evidence for the Mott transition:<br />U. Schneider, L. Hackermuller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch, A. Rosch, Science322, 1520-1525 (2008).<br />Robert Jordens, NielsStrohmaier, Kenneth Gunter, Henning Moritz & TilmanEsslinger, Nature 455, 204-208 (2008).<br />
  4. 4. Problem:What will happen in 1D?<br />
  5. 5. Problem:What will happen in 1D?<br />“A theory of correlations [...] will be mainly concerned with understanding [...] the balance between band-like and atomic-like behaviour.”<br />Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...]<br />John Hubbad,<br />Proceedings of the Royal Society of London. Series A, Mathematical and <br />Physical Sciences, 276, 238-257 (1963)<br />
  6. 6. Problem:What will happen in 1D?<br />Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an exact solution <br /> [Lieb & Wu 1968]<br />
  7. 7. Problem:What will happen in 1D?<br />Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an exact solution <br /> [Lieb & Wu 1968]<br />We know how to deal with the trapping potential <br /> [ C. Hooley & JQ, PRL (2004); <br /> V.L. Campo, K. Capelle, JQ & C. Hooley, PRL (2007) ]<br />
  8. 8. Problem:What will happen in 1D?<br />Hamiltonian: [Hubbard 1963; Gutzwiller 1963; ...]<br />Without the trap, we have an exact solution <br /> [Lieb & Wu 1968]<br />We know how to deal with the trapping potential <br /> [ C. Hooley & JQ, PRL (2004); <br /> V.L. Campo, K. Capelle, JQ & C. Hooley, PRL (2007) ]<br />Straight-forward<br /> to evaluate <br /> double occupancy:<br />
  9. 9. Effect of the trap – no fluctuations<br />
  10. 10. Effect of the trap – no fluctuations<br />
  11. 11. Effect of the trap – no fluctuations<br />D<br />Mott insulator<br />Band<br />+Mott<br />Band insulator<br />D<br />
  12. 12. Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. Lieb<br />and F. Y. Wu, <br />Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).<br />
  13. 13. Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. Lieb<br />and F. Y. Wu, <br />Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).<br />
  14. 14. Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. Lieb<br />and F. Y. Wu, <br />Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).<br />
  15. 15. Ground state – no trap<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Elliott H. Lieb<br />and F. Y. Wu, <br />Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).<br />
  16. 16. Ground state - harmonic trap<br />
  17. 17. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />
  18. 18. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
  19. 19. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
  20. 20. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t = 0<br />U/t = 4,5,6,7<br />
  21. 21. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t = 0<br />U/t = 4,5,6,7<br />
  22. 22. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />U/t = 0<br />U/t = 4,5,6,7<br />
  23. 23. Finite temperature – no trap<br />Use high-temperature expansion:<br /> (must go at least to 2nd order)<br />Double occupancy:<br />= + + ...<br />
  24. 24. Finite temperature – no trap<br />Match to low-T expansion from quantum transfer method [Klümper and Bariev 1996]<br />Obtain<br />C(x) is the unity central charge from CFT for the Hesienberg universality class:<br />
  25. 25. Finite temperature – no trap<br />
  26. 26. Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />
  27. 27. Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />d vs T is non-monotonic (suggests cooling mechanism with 1D system as reference state)<br /> [c.f. 3D case: <br /> F. Werner, O. Parcollet, A. Georges & S.R. Hassan, PRL (2005)]<br />
  28. 28. The non-monotonic beahviour is a strictly local phenomenon!<br />
  29. 29. The non-monotonic beahviour is a strictly local phenomenon!<br />
  30. 30. The non-monotonic beahviour is a strictly local phenomenon!<br />
  31. 31. The non-monotonic beahviour is a strictly local phenomenon!<br />
  32. 32. Quantum fluctuations + thermal fluctuations + trap<br />
  33. 33. In summary...<br />Fermionic Hubbard model in one dimension.<br />Mott phase has inherent double occupancy fluctuations.<br />Mott phase detectable via double occupancy.<br />Can read out double occupancy in the bulk from the trapped data. <br />Non-monotonic temperature dependence a universal, local feature. <br />THANKS!<br />arxiv.org:1107.4349<br />

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