Double occupancy as a probe of the Mott state for fermions in one-dimensional optical lattices<br />VIVALDO L. CAMPO, JR (...
Context: Experiments on 3D Hubbard model<br />Experimental evidence for the Mott transition:<br />U. Schneider, L. Hackerm...
Problem:What will happen in 1D?<br />Hamiltonian:<br />Evaluate double occupancy:<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 - 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...
Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />d vs T is non-monotonic (sugge...
Quantum fluctuations + thermal fluctuations + trap<br />
In summary...<br />Fermionic Hubbard model in one dimension.<br />Mott phase has inherent double occupancy fluctuations.<b...
Upcoming SlideShare
Loading in …5
×

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

904 views

Published on

Contributed talk at the Annual MSCES 2011, Cambridge. We study theoretically double occupancy D as a probe of the Mott transition for trapped fermions in
one-­‐dimensional optical lattices and compare our results to the three-­‐dimensional case. The ground
state is described using the Bethe Ansatz in a local density approximation and the behavior at finite
temperatures is modelled using a high-­‐temperature series expansion. In addition, we solve
analytically the model in the limit in which the interaction energy is the dominant energy scale.
We find that enhanced quantum fluctuations in one dimension lead to increased double occupancy
in the ground state, even deep in the Mott insulator region of the phase diagram (see figure).
Similarly, thermal fluctuations lead to high double occupancies at high temperatures. Nevertheless,
D is found to be a good indicator of the Mott transition just as in three dimensions. Moreover, unlike
other global observables, the bulk value of D in the Mott phase coincides, quantitatively, with that of
a suitably-­‐prepared trapped system. We discuss possible experiments to verify these results and
argue that the one-­‐dimensional Hubbard model could be used as a benchmark for quantitative
quantum analogue simulations.

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
904
On SlideShare
0
From Embeds
0
Number of Embeds
490
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • 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 &amp; 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 &amp; 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.
  • Doulbe 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 />UK Cold Atom/Condensed Matter Network Meetings, Nottingham, 7 September 2011<br />
    2. 2. Context: Experiments on 3D Hubbard model<br />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 />
    3. 3. Problem:What will happen in 1D?<br />Hamiltonian:<br />Evaluate double occupancy:<br />
    4. 4. Effect of the trap – no fluctuations<br />
    5. 5. Effect of the trap – no fluctuations<br />D<br />Mott insulator<br />Band<br />+Mott<br />Band insulator<br />D<br />
    6. 6. 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 />
    7. 7. 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 />
    8. 8. Ground state - harmonic trap<br />
    9. 9. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />
    10. 10. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
    11. 11. Ground state - harmonic trap<br />Evaluate D in the local density approximation:<br />D()= = j Dno trap(+½x2)<br />
    12. 12. 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 />
    13. 13. 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 />
    14. 14. 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 />
    15. 15. Finite temperature – no trap<br />Use high-temperature expansion:<br /> (must go at least to 2nd order)<br />Double occupancy:<br />= + + ...<br />
    16. 16. 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 />
    17. 17. Finite temperature – no trap<br />
    18. 18. Finite temperature – no trap<br />Very good match between high-T and low-T expansions.<br />
    19. 19. 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 />
    20. 20. 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 />A local picture accounts well for the observed behaviour:<br />
    21. 21. Quantum fluctuations + thermal fluctuations + trap<br />
    22. 22. 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 />

    ×