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.

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 UK Cold Atom/Condensed Matter Network Meetings, Nottingham, 7 September 2011

2. Context: 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).

3. Problem:What will happen in 1D? Hamiltonian: Evaluate double occupancy:

4. Effect of the trap – no fluctuations

5. Effect of the trap – no fluctuations D Mott insulator Band +Mott Band insulator D

6. 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).

7. 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).

8. Ground state - harmonic trap

9. Ground state - harmonic trap Evaluate D in the local density approximation:

10. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2)

11. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2)

12. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7

13. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7

14. Ground state - harmonic trap Evaluate D in the local density approximation: D()= = j Dno trap(+½x2) U/t = 0 U/t = 4,5,6,7

15. Finite temperature – no trap Use high-temperature expansion: (must go at least to 2nd order) Double occupancy: = + + ...

16. 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:

17. Finite temperature – no trap

18. Finite temperature – no trap Very good match between high-T and low-T expansions.

19. 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)

20. 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) A local picture accounts well for the observed behaviour:

21. Quantum fluctuations + thermal fluctuations + trap

22. 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