Exploring Competing Density Order in the Ionic Hubbard Model with Ultracold Fermions

Michael Messer, Rémi Desbuquois, Thomas Uehlinger, Gregor Jotzu, Sebastian Huber, Daniel Greif, and Tilman Esslinger

Phys. Rev. Lett. 115, 115303 – Published 9 September 2015

Abstract

We realize and study the ionic Hubbard model using an interacting two-component gas of fermionic atoms loaded into an optical lattice. The bipartite lattice has a honeycomb geometry with a staggered energy offset that explicitly breaks the inversion symmetry. Distinct density-ordered phases are identified using noise correlation measurements of the atomic momentum distribution. For weak interactions the geometry induces a charge density wave. For strong repulsive interactions we detect a strong suppression of doubly occupied sites, as expected for a Mott insulating state, and the externally broken inversion symmetry is not visible anymore in the density distribution. The local density distributions in different configurations are characterized by measuring the number of doubly occupied lattice sites as a function of interaction and energy offset. We further probe the excitations of the system using direction dependent modulation spectroscopy and discover a complex spectrum, which we compare with a theoretical model.

Received 18 March 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.115303

Authors & Affiliations

Michael Messer¹, Rémi Desbuquois¹, Thomas Uehlinger¹, Gregor Jotzu¹, Sebastian Huber², Daniel Greif¹, and Tilman Esslinger¹

¹Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
²Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

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Vol. 115, Iss. 11 — 11 September 2015

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Images

Figure 1
Noise correlations. (a) Schematic view of the ionic Hubbard model on a honeycomb lattice at half-filling. Circles denote lattice sites $A$ and $B$ , where larger circles indicate lower potential energy. The phase diagram exhibits two limiting cases: For $Δ ≫ U, t$ a CDW ordered state is expected with two fermions of opposite spin (red, blue) on lattice sites $B$ , and empty sites $A$ . In the other limit ( $U ≫ Δ, t$ ) a MI with one fermion on each lattice site should appear. (b) Measured noise correlation pictures obtained from absorption images of the atomic momentum distribution. Comparing panel 1 with panel 2, additional correlations appear due to broken inversion symmetry in the CDW ordered phase. When introducing strong interactions, these correlations are not observed anymore (panel 3), and the inversion symmetry of the density distribution no longer reflects the broken inversion symmetry of the lattice potential. Below each panel horizontal and diagonal cuts of the noise correlation image are shown. For the three different ratios of $Δ$ and $U$ , between 165 and 201 measurements were taken each. We show the average of $C (d_{x}, d_{y})$ and $C (d_{x}, - d_{y})$ , which reflects the symmetry of the system.
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Figure 2
Double occupancy measurement. (a) The measured double occupancy $D$ as a function of the on-site interaction $U$ for a fixed energy offset $Δ = 16.3 (4) t$ . (b) For different values of $Δ$ (different colors) we obtain the double occupancy for a range of interactions $U = [- 24.6 (13), 29.1 (7)] t$ . Hollow (full) circles represent attractive (repulsive) interactions. Vertical error bars show the standard deviation of 5 measurements and horizontal error bars the uncertainty on our lattice parameters.
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Figure 3
Modulation spectroscopy measurement. (a) Excitation spectra observed by measuring the double occupancy $D$ from amplitude modulation spectroscopy of the lattice beam in the $y$ direction for different energy offsets $Δ$ at repulsive on-site interaction $U = 24.4 (5) t$ . Solid lines are multiple Gaussian fits to the modulation spectra. (b) Schematics for the relevant energy scales $| U - Δ |$ and $U + Δ$ as a response to the lattice modulation. (c) Modulation spectroscopy of the lattice beam in the $z$ direction. The measured excitation frequencies are shown as a function of $Δ$ and compared to the value of $U = 24.4 (5) t$ (horizontal line). The inset shows the spatially dependent excitation spectrum. (d) Comparison of the measured excitation resonances (points) with the values of $| U - Δ |$ , $U + Δ$ (lines). The area of the marker indicates the strength of the response (peak height) to the lattice modulation. Full (empty) circles represent a positive (negative) response in double occupancy. Error bars as in Fig. 2, vertical error bars in (c) and (d) show the fit error for the peak position.
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Figure 4
Theoretical result for the kinetic energy response function $χ (ν)$ of the double occupancy on a modulated four site model as a function of $Δ$ at constant $U = 25 t$ . Circular (diamond) data points represent the response for the half-filled (quarter-filled) case. The area of the marker shows the relative size of the calculated response, whereas full (empty) data points have a positive (negative) response signal.
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