Direct Probing of the Mott Crossover in the $\mathrm{SU}(N)$ Fermi-Hubbard Model

We report on a detailed experimental investigation of the equation of state (EoS) of the three-dimensional Fermi-Hubbard model (FHM) in its generalized $\mathrm{SU}(N)$-symmetric form, using a degenerate ytterbium gas in an optical lattice. In its more common spin-$1/2$ form, the FHM is a central model of condensed-matter physics. The generalization to $N>2$ was first used to describe multi-orbital materials and is expected to exhibit novel many-body phases in a complex phase diagram. By realizing and locally probing the $\mathrm{SU}(N)$ FHM with ultracold atoms, we obtain model-free access to thermodynamic quantities. The measurement of the EoS and the local compressibility allows us to characterize the crossover from a compressible metal to an incompressible Mott insulator. We reach specific entropies above Néel order but below that of uncorrelated spins. Having access to the EoS of such a system represents an important step towards probing predicted novel $\mathrm{SU}(N)$ phases.

Popular Summary

Understanding the physics of strongly interacting materials is critical in condensed matter physics and may lead to new quantum technologies. In particular, fermionic many-body systems, such as the electrons in crystals, are still associated with many unanswered questions. The origin of high-temperature superconductivity, for example, is still unknown. Many of these systems can be described by a so-called Fermi Hubbard model (FHM) and exhibit properties that are notoriously hard to calculate. Here, we experimentally realize a FHM with enhanced spin symmetry and study its thermodynamic properties, shedding light on the behavior of certain fermionic many-body systems.

We focus on an ensemble of ultracold ${}^{173}\mathrm{Yb}$ atoms in their ground state situated in an optical lattice. This experimental setup allows us to realize a FHM, which was originally developed to explain interacting electrons that possess two spin components with an SU(2) symmetry. By varying the interaction strength between the atoms, we observe the transition from a conducting behavior to that of an insulator. We study the properties of this transition by both globally measuring the total entropy of the system and by locally probing inside the lattice. The details of this transition are dependent on the symmetry of the FHM, which in our system can be extended to the general $\mathrm{SU}(N)$ symmetric case. We are accordingly able to directly characterize the transition for different symmetry cases using density and compressibility as observables.

We expect that our findings will be important for advancing knowledge about $\mathrm{SU}(N)$ Fermi Hubbard physics and motivating additional studies of $\mathrm{SU}(N)$ magnetism.

Figure 1

Density profile of an SU(6) Fermi gas in an optical lattice with harmonic confinement. (a) Measured density distribution integrated along the line of sight (bottom) and local trap density obtained by performing the inverse Abel transform, displaying a plateau $n(r,y)=1$ (top) for interaction strength $U/t^{*}=6.4$ ($V=13E_r$). Three experimental realizations are averaged for the displayed data. (b) Local atomic density as a function of the distance to the trap center for different interaction parameters $U/t^{*}$.

Figure 2

EoS of the SU(6)- (blue circles) and the SU(3)- (red diamonds) spin symmetric Fermi gas in a lattice. Density as a function of the chemical potential for various interaction strengths: (a) $U/t^{*}=0.128$ (b) $U/t^{*}=0.89$ and (c) $U/t^{*}=3.6$. Solid lines are fits to the non-interacting Fermi gas EoS, with points $n{d}^3<0.5$ included in the fit. Dashed lines are the fits to the low tunneling model for SU(6) (blue) and SU(3) (red). An SU(2) model fit to the SU(6) data is shown for comparison (green), with the temperature $T$ and the central chemical potential ${\mu }_0$ as fit parameters. Error bars are the standard error of the mean (s.e.m. of the binned data).

Figure 3

Compressibility of SU(6)- (blue circles) and SU(3)- (red diamonds) spin-symmetric Fermi gases in a lattice. (a) compressibility as a function of density in the intermediate regime $U/t^{*}=0.59$. (b) compressibility as a function of density in the strongly interacting regime $U/t^{*}=3.6$. The dashed lines are obtained by deriving the fitted low-tunneling model plotted in Fig. 2. (c) minimal compressibility ${\tilde{\kappa }}_{\min }/{\tilde{\kappa }}_0$ as a function of interaction strength, where ${\tilde{\kappa }}_0$ is the compressibility of a non-interacting SU(6) Fermi gas at $T=0$ and $n=1/d^3$. The minimum of the compressibility $\tilde{\kappa }(n)$ in an interval around unit filling (shaded area in insets) is used as ${\tilde{\kappa }}_{\min }$. Error bars denote the confidence intervals of the linear regression used to obtain $\tilde{\kappa }$.

Figure 4

Entropy per particle $s$ of the SU(6)-symmetric gas before and after the lattice round-trip sequence. The blue line represents the entropy measured in the bulk before transfer into the lattice. Blue points are measurements after round-trip loading into the lattice and back to the bulk. The entropy of the gas in the lattice is constrained to the region between these two measurements. The entropy of a system with our parameters and fully random spin orientations is close to $s/k_B=\ln$ 6 (red line). Error bars and blue shaded region are the s.e.m. of the temperature measurements.

Figure 5

Density profiles of the harmonically trapped SU(6) gas for various saturation parameters without saturation correction (top) and with saturation correction of Eq. (c1) with $\alpha =3.05$ (bottom).

Figure 6

Atom loss at high densities in the center of the cloud in $V=15E_r$ deep lattice (circles). The fit function (dashed line) is a double exponential decay described by Eq. (f1).