Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices
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Transcript of Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices
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ISIS Facility, STFC Rutherford Appleton Laboratory
Functional Materials Group
Hubbard Theory Consortium
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
UK Cold Atom/Condensed Matter Network Meetings, Nottingham, 7 September 2011
Double occupancy as a probe of the Mott state for fermions in one-dimensional optical lattices
arxiv.org:1107.4349
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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, Science 322, 1520-1525 (2008).
Robert Jordens, Niels Strohmaier, Kenneth Gunter, Henning Moritz & Tilman Esslinger, Nature 455, 204-208 (2008).
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Problem:What will happen in 1D?
• Hamiltonian:
• Evaluate double occupancy:
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Effect of the trap – no fluctuations
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Effect of the trap – no fluctuations
Mott insulator
Band+Mott
Band insulator D
D
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Ground state – no trap
Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
f
0 1 2
U / t
Luttinger Liquid
Mott insulator:
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Ground state – no trap
Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968); 21, 192 (1968).
f
0 1 2
U / t
Luttinger Liquid
Mott insulator:
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Ground state - harmonic trap
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• Evaluate D in the local density approximation:
Ground state - harmonic trap
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• Evaluate D in the local density approximation:
D() = = j Dno trap(+½x2)
Ground state - harmonic trap
![Page 11: Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices](https://reader035.fdocuments.net/reader035/viewer/2022062709/558eadc51a28ab98708b474f/html5/thumbnails/11.jpg)
• Evaluate D in the local density approximation:
D() = = j Dno trap(+½x2)
Ground state - harmonic trap
![Page 12: Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices](https://reader035.fdocuments.net/reader035/viewer/2022062709/558eadc51a28ab98708b474f/html5/thumbnails/12.jpg)
• Evaluate D in the local density approximation:
D() = = j Dno trap(+½x2) U/t = 4,5,6,7
U/t = 0
Ground state - harmonic trap
![Page 13: Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices](https://reader035.fdocuments.net/reader035/viewer/2022062709/558eadc51a28ab98708b474f/html5/thumbnails/13.jpg)
• Evaluate D in the local density approximation:
D() = = j Dno trap(+½x2) U/t = 4,5,6,7
U/t = 0
Ground state - harmonic trap
![Page 14: Doulbe Occupancy as a Probe of the Mott Transition for Fermions in One-dimensional Optical Lattices](https://reader035.fdocuments.net/reader035/viewer/2022062709/558eadc51a28ab98708b474f/html5/thumbnails/14.jpg)
• Evaluate D in the local density approximation:
D() = = j Dno trap(+½x2) U/t = 4,5,6,7
U/t = 0
Ground state - harmonic trap
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Finite temperature – no trap• Use high-temperature expansion:
(must go at least to 2nd order)• Double
occupancy:
= + + ...
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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:
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Finite temperature – no trap
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Finite temperature – no trap• Very good match between
high-T and low-T expansions.
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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)
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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:
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Quantum fluctuations + thermal fluctuations + trap
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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