Double occupancy as a probe of the Mott transition for fermions in one-dimensional optical lattices<br />VIVALDO L. CAMPO,...
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 />
Bulk 1D Hubbard model (no trap)<br />U / t<br />f<br />Mott insulator:<br />0<br />1<br />2<br />Luttinger Liquid<br />Ell...
Finite temperature<br />Use high-temperature expansion:<br />	(must go at least to 2nd order)<br />Double occupancy:<br />...
Finite temperature<br />Match to low-T expansion from quantum transfer method [Klümper and Bariev 1996]<br />Obtain<br />C...
Finite temperature<br />Very good match between high-T and low-T expansions.<br />dvsT is non-monotonic (suggests cooling ...
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 />
Add quantum fluctuations<br />Evaluate D in the local density approximation:<br />D()= = jDno trap(+½x2)<br />U/t = 0<...
Quantum + thermal fluctuations<br />
In summary...<br />Fermionic Hubbard model in one dimension.<br />Mott phase has inherent double occupancy fluctuations.<b...
Upcoming SlideShare
Loading in …5
×

Double occupancy as a probe of the Mott transition for fermions in one- dimensional optical lattices

833 views

Published on

Contributed talk at SCES 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, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
833
On SlideShare
0
From Embeds
0
Number of Embeds
5
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • Thanks for organising this.I am going to talk about double occupancy as a probe of the Mott transition for fermions in one dimension.I am Jorge Quintanilla and my collaborators are Vivaldo Campo, Vito Scarola, Chris Hooley and Klaus Capelle.
  • Double occupancy has been used to detect the Mott transition for the harmonically-trapped, three-dimensional, fermionicHubbard model.These graphs show the suppression of double occupancy as a function of atom number as we go from non-interacting (U=0) to strongly interacting fermions.
  • Will the same happen in one dimension?Our hamiltonian features hopping, on-site repulsion and a trapping potential.We have evaluated the double occupancy as the ratio of the expectation value of this operator, which counts the number of doubly-occupied sites, mutiplied by two because there are two atoms at each doubly-occupied sites, to the total number of atoms. This can be evaluated as a derivative of the energy with respect to the strength of the interaction.
  • 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.
  • To describe the regime dominates by thermal fluctuations we use a series expansion of the free energy which starts from the atomic limit and includes perturbatively hopping to nearest neighbour sites.At high particle number, temperature introduces additional fluctuations that reduce double occupancy even further.At low particle numbers, on the other hand, temperature intially suppresses double occupancy as it destroyes quantum fluctuations and then it enhances it, i.e. The behaviour is non-monotonic. Further characterisation of this behaviour is needed to guide future experiments.
  • To describe the regime dominates by thermal fluctuations we use a series expansion of the free energy which starts from the atomic limit and includes perturbatively hopping to nearest neighbour sites.At high particle number, temperature introduces additional fluctuations that reduce double occupancy even further.At low particle numbers, on the other hand, temperature intially suppresses double occupancy as it destroyes quantum fluctuations and then it enhances it, i.e. The behaviour is non-monotonic. Further characterisation of this behaviour is needed to guide future experiments.
  • To describe the regime dominates by thermal fluctuations we use a series expansion of the free energy which starts from the atomic limit and includes perturbatively hopping to nearest neighbour sites.At high particle number, temperature introduces additional fluctuations that reduce double occupancy even further.At low particle numbers, on the other hand, temperature intially suppresses double occupancy as it destroyes quantum fluctuations and then it enhances it, i.e. The behaviour is non-monotonic. Further characterisation of this behaviour is needed to guide future experiments.
  • 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
  • It is a simple but pleasant exercise to work out the phase diagram as a function of the number of particles and the relative strengths of the trap and the interaction.Note that in the Mott insulating region the double occupancy is, of course, zero. It then increases in the region of coexistence until it reaches 1 in the band insulator region.
  • Putting back the hopping term introduces quantum fluctuations.Although in the presence of the trap we do not have an exact solution, we can construct an approximate one using a local density approximation.This is all right for sufficiently weak trapping potentials.In particular we will look at the effect of these quantum fluctuations on the double occupancy, which we know exactly in the absence of the trap.As in the 3D case, interactions induce a significantreduction of doubleoccupancy. ThestrongestcontributiontothisreductionismadebytheMottinsulatingregion.This can be seen by looking at the density profiles. We see here the U=4 case and we note that the strongest suppression of double occupancy, compared to the non-interacting case, occurs when the Mott insulating region is largest.
  • 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 transition 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 />SCES 2011, Cambridge, 1 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. Bulk 1D Hubbard model (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 />
    5. 5. Finite temperature<br />Use high-temperature expansion:<br /> (must go at least to 2nd order)<br />Double occupancy:<br />= + + ...<br />
    6. 6. Finite temperature<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 />
    7. 7. Finite temperature<br />Very good match between high-T and low-T expansions.<br />dvsT is non-monotonic (suggests cooling mechanism with 1D system as reference state)<br />A local picture accounts well for the observed behaviour:<br />
    8. 8. Effect of the trap – no fluctuations<br />
    9. 9. Effect of the trap – no fluctuations<br />D<br />Mott insulator<br />Band<br />+Mott<br />Band insulator<br />D<br />
    10. 10. Add quantum fluctuations<br />Evaluate D in the local density approximation:<br />D()= = jDno trap(+½x2)<br />U/t = 0<br />U/t = 4,5,6,7<br />
    11. 11. Quantum + thermal fluctuations<br />
    12. 12. 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 – may be used for cooling.<br />THANKS!<br />arxiv.org:1107.4349<br />

    ×