Evidence of Potts-Nematic Superfluidity in a Hexagonal $s{p}^2$ Optical Lattice

As in between liquid and crystal phases lies a nematic liquid crystal, which breaks rotation with preservation of translation symmetry, there is a nematic superfluid phase bridging a superfluid and a supersolid. The nematic order also emerges in interacting electrons and has been found to largely intertwine with multiorbital correlation in high-temperature superconductivity, where Ising nematicity arises from a four-fold rotation symmetry $C_4$ broken down to $C_2$. Here, we report an observation of a three-state (${\mathbb{Z}}_3$) quantum nematic order, dubbed “Potts-nematicity”, in a system of cold atoms loaded in an excited band of a hexagonal optical lattice described by an $s{p}^2$-orbital hybridized model. This Potts-nematic quantum state spontaneously breaks a three-fold rotation symmetry of the lattice, qualitatively distinct from the Ising nematicity. Our field theory analysis shows that the Potts-nematic order is stabilized by intricate renormalization effects enabled by strong interorbital mixing present in the hexagonal lattice. This discovery paves a way to investigate quantum vestigial orders in multiorbital atomic superfluids.

Figure 1

Experimental protocol of loading atoms into the excited $s{p}^2$ band of the hexagonal optical lattice. (a) Illustrates the arrangement of the laser beams forming the hexagonal lattice. There are three laser beams in the $x\text{−}y$ plane with laser-wavelength $\lambda =1064\mathrm{nm}$ forming an optical hexagonal lattice [17]. The three angles ${\theta }_{1,2,3}$ represent the relative phases of the elliptical polarization of the light. (b) The geometry of the hexagonal lattice, having two sets of sublattices labeled by $\mathcal{A}$ and $\mathcal{B}$. The lattice is formed taking $\{{\theta }_1,{\theta }_2,{\theta }_3\}=\{4\pi /3,2\pi /3,0\}$ or $\{2\pi /3,4\pi /3,0\}$. (c) The time sequence implemented in the experiment to load atoms from the lowest to the second band. (d) The band-mapping TOF images with $T=0,0.4,5,35,45\mathrm{ms}$. When $T=5\mathrm{ms}$, three different band-mapping TOF corresponding to three nematic orientations are shown in the red dashed box. (e) The measured time evolution of the atomic population in the ground and first-excited bands normalized by their sum. Here, we average over five experimental runs for each data point, with the error bar denoting the statistical error.

Figure 2

Spontaneous Potts-nematic symmetry breaking in the hexagonal optical lattice. (a) The averaged momentum distribution. Atoms loaded in the excited band are found to spontaneously accumulate at one of the three $M$ points. We introduce a Potts-nematic contrast [PNC in Eq. (3)], where $n_{\square ,○,△}$ correspond to momentum distributions in three separate regions as marked in the middle of (a), related to each other by a three-fold lattice rotational symmetry. In the three panels, first, we classify the experimental images into three classes according to the polar angle of the nematic contrast $\arg (\mathrm{PNC})\in (-\pi /3,\pi /3)$, $(\pi /3,\pi )$, or $(\pi ,4\pi /3)$, and then take the average within each class. In the panel insets parametrized by $x(\lambda )$ and $y(\lambda )$, we show the real space density extracted from the Bloch functions at the $M$ points, which shows a bond order that breaks the lattice rotation symmetry in real space. (b) The statistical occurrence of the nematic contrast. The nematic contrast extracted from the experimental data shows the spontaneous breaking of the three-fold lattice rotation symmetry, i.e., the emergence of the Potts-nematic order in this quantum many-body system.

Figure 3

Dynamical emergence of the Potts-nematic order. (a) Time evolution of momentum distribution. In (a), we average over the experimental results having a Potts-nematic contrast with $\arg (\mathrm{PNC})\in (-\pi /3,\pi /3)$. (b) Evolution of the PNC and the coherent fraction [17]. The time point with which we quench the lattice (see Fig. 1) is set to be 0 in this plot. The phase coherence in the second band does not immediately form after the quench but, instead, appears about several milliseconds later. The emergence of the Potts-nematic order coincides with the second band phase coherence. The rise and disappearance of the Potts-nematic order define three qualitatively distinct regimes in the quantum dynamics. Here, we average over ten experimental images at each time point, with the error bar denoting the statistical error.

Figure 4

Theoretical quantum phase transitions varying the orbital Josephson coupling. The orbital Josephson coupling $J$ is introduced in Eq. (2). (a) The Gross-Pitaevskii energy $\mathcal{E}(\mathbf{k})$ for a plane-wave condensate at a lattice momentum $\mathbf{k}$. Here, we choose $t_{sp}$ as an energy unit. The chemical potentials are set at ${\mu }_s/t_{sp}=0.1$, ${\mu }_p=0$, the interaction strengths are $U_s/t_{sp}=U_{p,\parallel }/t_{sp}=0.5$, $J/t_{sp}=0.4$, and $-0.4$ in (a) and (b), respectively, and $U_{p,\perp }$ is fixed respecting the lattice rotation symmetry. The energy $\mathcal{E}(\mathbf{k})$ has minima at $K$ ($M$) points for $J>J_c$ ($J<J_c$). The ground state condensates are chiral and Potts nematic, correspondingly. (c) The sketch of the renormalization of the $p$-orbital couplings to low energy. The multiple curves correspond to different choice of bare couplings. The feature of $J$ renormalizing to the negative side is generic for the hexagonal lattice. In (c), the couplings are in arbitrary units.