Equation of State of the Two-Dimensional Hubbard Model

The subtle interplay between kinetic energy, interactions, and dimensionality challenges our comprehension of strongly correlated physics observed, for example, in the solid state. In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics. Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions $0\lesssim U/t\lesssim 20$ and temperatures, down to $k_{B}T/t=0.63(2)$ using high-resolution imaging of ultracold fermionic atoms in optical lattices. We show density profiles, compressibilities, and double occupancies over the whole doping range, and, hence, our results constitute benchmarks for state-of-the-art theoretical approaches.

Figure 1

Setup and in situ density profiles. (a) Experimental setup showing the arrangement of laser beams to create a stack of two-dimensional Hubbard models with ultracold atoms. Singles and doubles within a single layer are detected by combining radio-frequency spectroscopy and absorption imaging. (b)–(d) In situ density profiles of singles and doubles in the two-dimensional Hubbard model for different interaction strengths. The images are averaged over $\sim 35$ repetitions of the experiment. The density profiles cross over from a metallic phase at weak interactions, $U/t=1.6$, to a flat-top Mott insulator without doubles at strong interaction $U/t=12.0$. (e)–(g) Singles, doubles, and total density data averaged over $-10a\le x\le 10a$ [see dashed lines in (b)] as well as evaluated along isopotential contours of the trapping potential.

Figure 2

Equation of state of the two-dimensional repulsive Hubbard model. Purple, $U/t=-0.2(3)$ for $k_{B}T/t=1.35(4)$; blue, $U/t=1.6(2)$ for $k_{B}T/t=1.19(4)$; green, $U/t=8.2(5)$ for $k_{B}T/t=0.63(2)$; yellow, $U/t=12.0(7)$ for $k_{B}T/t=0.92(6)$; red, $U/t=19.5(1.3)$ for $k_{B}T/t=1.41(5)$. (a) Equation of state $n(\mu )$ vs interaction strength. Solid lines show fits using NLCE data [15] (purple, noninteracting Hubbard model) from which the temperature has been extracted. The inset shows the comparison with DCA data [16] for the temperature interval $0.55\le k_{B}T/t\le 0.82$ at $U/t=8$. (b) Compressibility $\kappa$ vs filling. The dashed line shows the prediction of the noninteracting Hubbard model at $k_{B}T/t=1.4$ and the gray band the prediction from DCA as in (a). (c) Doubles fraction vs filling. The theoretical predictions for the noninteracting (dashed line) and infinitely repulsive (dashed-dotted line) Hubbard model are shown. The inset shows the behavior at $n=0.5$ vs interaction strength. The error bars show the standard errors.

Figure 3

Equation of state vs temperature at $U/t=8.2$. Blue, $k_{B}T/t=0.67(3)$; green, $k_{B}T/t=1.55(6)$; red, $k_{B}T/t=3.25(7)$. (a) Equation of state $n(\mu )$ for different temperatures with fits using NLCE data. (b) Compressibility at half filling vs temperature together with NLCE data. (c) Doubles fraction $D$ vs filling for different temperatures. (d) Doubles $⟨{\hat{n}}_{\uparrow }{\hat{n}}_{\downarrow }⟩$ at half filling vs temperature. The solid line shows the coinciding predictions of NLCE [15] and DCA [16], and the dashed line the QMC prediction [33]. The error bars show the standard errors.