Paramagnetic Phases of Strongly Correlated Ultracold Fermions Coupled to an Optical Cavity

Abstract

We numerically study a gas of two-component fermions coupled to a transversely pumped optical cavity and confined to a two-dimensional static square optical lattice. In the dispersive regime, the steady state of the system is described by an extended Hubbard Hamiltonian with cavity-mediated long-range interactions. Using real-space dynamical mean-field theory (RDMFT), we investigate the formation of the (superradiant) checkerboard density-wave phase both at quarter and half filling. We focus on the paramagnetic phase assuming sufficiently high temperatures such that no magnetic long-range order develops. At quarter filling, we find a reentrant homogeneous Fermi liquid to density wave phase transition with increasing temperature, which is due to the higher entropy of the ordered phase. At half filling, in addition to the Fermi liquid to Mott insulator phase transition, marked by a vanishing quasiparticle residue at the Fermi level, we identify the transition into a density-wave phase. Due to perfect Fermi surface nesting at half filling, we find that arbitrarily small long-range interactions destabilize the system towards the density-wave phase in the absence of short-range interactions. By varying short- and long-range interactions at a fixed low temperature, we obtain the full phase diagram and identify a region of coexistence between the homogeneous Fermi liquid and Mott insulating phase with the density-wave phase. In this region, we determine the thermodynamic phase transition by comparing the energies of the different RDMFT solutions.

Figures (6)

• Figure 1: a) Schematic representation of a two-component Fermi gas (with the two components (hyperfine states) denoted as $\ket{{\uparrow}}$, $\ket{{\downarrow}}$) confined to a static square optical lattice inside a single mode optical cavity. The atoms are transversely driven by a coherent pump-laser field. b) Schematic illustration of the atomic configuration of the different quantum phases we observe in the system at half filling. For weak long-range interactions $U_l$, we observe a phase transition from a Fermi-liquid phase (FL) to a Mott-insulating phase (MI) as the short-range interaction $U_s$ is increased. For fixed $U_s$, increasing $U_l$ drives the system into a density-wave insulating phase (DW), in which the atoms self-organize, occupying either the even or the odd sublattice. In this phase, the expectation value of the occupation imbalance operator $\langle\hat{\Theta}\rangle$ (Eq. \ref{['eq7']}) asymptotically approaches $+1$ for a checkerboard pattern with occupied even sites and $-1$ for occupied odd sites.
• Figure 2: Phase diagram of the extended Fermi-Hubbard model with cavity-mediated long-range interactions at quarter and half filling in the $T$-$U_l$-plane. The distinct phases are denoted as: FL (homogeneous Fermi liquid), MI (Mott insulator), DW (checkerboard density-wave). For both a) and the c) the short-range interaction strength is fixed at $U_s=16$, while for b) its value is $U_s=3$. At quarter filling a) we observe reentrant behavior at low temperatures. In contrast, no such behavior is present at half filling, i.e. in b) and c). The inset in a) shows the $U_l$ dependence of the occupation imbalance $\langle\hat{\Theta}\rangle$ at temperature $T=0.8$ where the green dot-dashed line marks the FL-DW transition at $N_sU_l=4.77$.
• Figure 3: a) Occupation imbalance $\langle\hat{\Theta}\rangle$ for increasing long-range interaction strength $U_l$ at half filling, for $U_s=0$ and $T=0.02$. Due to the perfect nesting of Fermi surface, the imbalance becomes finite as soon as the long-range interactions are non zero signaling the transition to the density-wave phase. b) Quasiparticle weight $Z_{\vb{e},\vb{o}}$ as a function of the short-range interaction strength $U_s$ at half filling, for $U_l=0$ and $T=0.02$. The paramagnetic Mott transition is characterized by a linear decrease of $Z_{\vb{e},\vb{o}}$ for increasing $U_s$ until it becomes zero at $U_s^c=10.3$ (dot-dashed green line) marking the vanishing of the Fermi liquid solution obtained using RDFMT.
• Figure 4: a) Phase diagram of the extended Fermi-Hubbard model with cavity-mediated long-range interactions at half filling and for $T=0.02$. The distinct phases are denoted as: FL (homogeneous Fermi liquid), MI (Mott insulator), DW (checkerboard density-wave). The FL–MI transition is marked by red squares. The purple circle dots correspond to the border of the region of metastability of the DW solution obtained within RDMFT, while the orange empty circles represent the border of the region of metastability of the homogeneous FL and MI solutions. The region where the MI and DW solutions coexist is depicted in shaded pink, while the region where the FL and DW solutions coexist is represented by the yellow shaded area. The inset shows a zoom into the FL-DW coexistence region. The diamond blue markers represent the thermodynamic phase transition obtained by comparing the energies of the homogeneous FL and MI solutions with the DW solution obtained within RDMFT. These points follow closely the dotted black line $N_sU_l=U_s$ obtained by analyzing the model in the atomic limit. Plots b) and c) show the hysteretic behavior of the occupation imbalance $\left\lvert*\right\rvert{\langle\hat{\Theta}\rangle}$ and the expectation value of the mean-field Hamiltonian $\ev*{\hat{H}_{\rm MF}}$ (Eq. \ref{['efH']}) for $U_s=12$ and $T=0.02$, respectively. In b) and c) the dashed purple line indicates the DW solution while the continuous orange line corresponds to the MI solution. The thermodynamic phase transition is indicated in c) by the vertical dot-dashed blue line at $N_sU_l=11.6$.
• Figure 5: Expectation value of the energy in the atomic limit Eq. \ref{['eqA1']} for different values of the ratio $N_sU_l/U_s$.