Finite-temperature dynamical properties of SU($N$) fermionic Hubbard models in the spin-incoherent regime

We study strongly correlated Hubbard systems extended to symmetric $N$-component fermions. We focus on the intermediate-temperature regime between magnetic superexchange and interaction energy, which is relevant to current ultracold fermionic atom experiments. The $N$-component fermions are represented by slave particles, and, by using a diagrammatic technique based on the atomic limit, spectral functions are analytically obtained as a function of temperature, filling factor, and the component number $N$. We also apply this analytical technique to the calculation of lattice modulation experiments. We compute the production rate of double occupancy induced by modulation of an optical lattice potential. Furthermore, we extend the analysis to take into account the trapping potential by use of the local density approximation. We find an excellent agreement with recent experiments on ${}^{173}$Yb atoms.

Figure 1

The possible NCA self-energy diagrams for a holon [(a) and (b)] and for doublons [(c)–(g)]. The solid, double-solid, and dashed lines denote the full propagators of the doublon, the holon, and the spinon, respectively. The indices $\eta$ and ${\eta }^{\prime }$ appearing in the doublon and spinon propagators are dummy spin ones, in which the summation over possible spin values is taken. The holon self-energies ${\Sigma }^{\mathrm{h}\left(1\right)}$ and ${\Sigma }^{\mathrm{h}\left(2\right)}$ correspond to (a) and (b). One of the two parts of the doublon self-energy ${\Sigma }_{{\sigma }_1{\sigma }_2}^{\mathrm{d}\left(1\right)}$ includes (c)–(f), and the other ${\Sigma }_{{\sigma }_1\sigma 2}^{\mathrm{d}\left(2\right)}$ is illustrated by (g). In the simple NCA idea, each propagator should be dealt with as the full ones, but in the approximation shown in this paper, the spinon propagators are replaced by the bare, namely, atomic-limit, ones. In addition, diagrams (b) and (g) are off-shell, because they must not be relevant in the energy regime $\sim U$ since $U$ is very large. Thus in this paper diagrams (a) and (c)–(f) are taken into consideration.

Figure 2

Two examples of diagrams which are not included in our NCA: (a) an example of crossing diagrams and (b) one of the off-shell diagrams which contains intermediate processes at higher energies than the main energy scale, i.e., creation of doublons in the holon propagation. The solid, double-solid, and dashed lines denote the propagators of the holon, the doublon, and the spinon, respectively. Note that the doublon and holon propagators shown here mean a bare propagator, while all lines denote full propagators in Fig. 1.

Figure 3

The holon bandwidth $W_{\mathrm{h}}/\sqrt{4z}J$ for different chemical potentials as a function of temperature and chemical potential: (a) for $N=2$, (b) for $N=6$, and (c) for $N=10$, respectively. The lines from upper to lower denote $\mu /U=0.5$, $0.3$, $0.1$, $0.0$, and $-0.1$, respectively.

Figure 4

The spectral functions of the fermionic atoms for $N=2$ as a function of the chemical potential for different temperatures: (a) $k_{B}T<J$ ($k_{B}T/U=0.025$, $J/U=0.05$), (b) $k_{B}T=J$ ($k_{B}T/U=J/U=0.05$), and (c) $k_{B}T>J$ ($k_{B}T/U=0.075$, $J/U=0.05$). The chemical potential runs from $\mu /U=0$ to $0.7$.

Figure 5

The spectral functions of the fermionic atoms for $N=6$ as a function of the chemical potential for different temperatures: (a) $k_{B}T<J$ ($k_{B}T/U=0.025$, $J/U=0.05$), (b) $k_{B}T=J$ ($k_{B}T/U=J/U=0.05$), and (c) $k_{B}T>J$ ($k_{B}T/U=0.075$, $J/U=0.05$). The chemical potential runs from $\mu /U=0$ to $0.7$.

Figure 6

The spectral functions of the fermionic atoms for $N=10$ as a function of the chemical potential for different temperatures: (a) $k_{B}T<J$ ($k_{B}T/U=0.025$, $J/U=0.05$), (b) $k_{B}T=J$ ($k_{B}T/U=J/U=0.05$), and (c) $k_{B}T>J$ ($k_{B}T/U=0.075$, $J/U=0.05$). The chemical potential runs from $\mu /U=0$ to $0.7$.

Figure 7

A sketch of optical lattice modulation and double occupancy. Due to the modulation perturbation, the system is excited, and doubly occupied sites are created. In experiments, the number of formed atom pairs is measured as a function of the modulation time duration, and the production rate is estimated.

Figure 8

Diagrams corresponding to the kinetic-energy correlation functions: (a) ${\chi }_{\mathrm{K}}^{\mathrm{h}}\left(\tau \right)$, (b) ${\chi }_{\mathrm{K}}^{\mathrm{d}}\left(\tau \right)$, and (c) ${\chi }_{\mathrm{K}}^{\mathrm{hd}}\left(\tau \right)$. The solid, double-solid, and dashed lines, respectively, denote the propagators of the doublon, the holon, and the spinon. The shaded rectangle includes a vertex correction which is not considered in this paper.

Figure 9

A sketch of the contributions to the kinetic-energy correlation function. There are four possible nearest-neighboring pairs in an equilibrium state: (a) singly occupied and empty site, (b) doubly and singly occupied site, (c) singly occupied sites, and (d) doubly occupied and empty site. Based on these configurations of the pair of nearest-neighboring sites, we can categorize the three types of contributions, ${\chi }_{\mathrm{K}}^{\mathrm{h}}\left(\tau \right)$ for pair (a) in the left panel, ${\chi }_{\mathrm{K}}^{\mathrm{d}}\left(\tau \right)$ for pair (b) in the middle panel, and ${\chi }_{\mathrm{K}}^{\mathrm{hd}}\left(\tau \right)$ for pairs (c) and (d) in the right panel. The $N=2$ case is taken here for the sake of simplicity, but it can easily be extended to general $N$ cases.

Figure 10

The DPR spectra per site, $P_{\mathrm{D}}\left(\omega \right)/[{\left(\delta F\right)}^{2}J/\hslash ]$, as a function of modulation frequency for $N=2$ in a cubic lattice ($z=6$): (a) $k_{B}T/U=0.025$ ($k_{B}T<J$), (b) $k_{B}T/U=0.05$ ($k_{B}T=J$), and (c) $k_{B}T/U=0.075$ ($k_{B}T>J$). The hopping parameter is taken to be $J/U=0.05$. The solid, dashed, and dotted lines denote $\mu /U=0.5$, $0.3$, and $0.1$, respectively.

Figure 11

The DPR spectra per site, $P_{\mathrm{D}}\left(\omega \right)/[{\left(\delta F\right)}^{2}J/\hslash ]$, as a function of modulation frequency for $N=6$ in a cubic lattice ($z=6$): (a) $k_{B}T/U=0.025$ ($k_{B}T<J$), (b) $k_{B}T/U=0.05$ ($k_{B}T=J$), and (c) $k_{B}T/U=0.075$ ($k_{B}T>J$). The hopping parameter is taken to be $J/U=0.05$. The solid, dashed, and dotted lines denote $\mu /U=0.5$, $0.3$, and $0.1$, respectively.

Figure 12

The DPR spectra per site, $P_{\mathrm{D}}\left(\omega \right)/[{\left(\delta F\right)}^{2}J/\hslash ]$, as a function of modulation frequency for $N=10$ in a cubic lattice ($z=6$): (a) $k_{B}T/U=0.025$ ($k_{B}T<J$), (b) $k_{B}T/U=0.05$ ($k_{B}T=J$), and (c) $k_{B}T/U=0.075$ ($k_{B}T>J$). The hopping parameter is taken to be $J/U=0.05$. The solid, dashed, and dotted lines denote $\mu /U=0.5$, $0.3$, and $0.1$, respectively.

Figure 13

The dimensionless DPR spectra $P_{\mathrm{d}}\left(\omega \right)$ scaled by $J/\hslash$ as a function of modulation frequency $\hslash \omega /U$ for an $N=6$ trapped system with different parameters: (a) $J/U=0.0089$, (b) $J/U=0.016$, (c) $J/U=0.030$, (d) $J/U=0.061$, and (e) $J/U=0.091$. In all cases, $1.87\times {10}^4$ trapped atoms and a modulation amplitude $\delta F=0.08$ are taken. The solid line and points with an error bar, respectively, denote the theoretical result and the ${}^{173}$Yb experiments [50]. The temperatures are determined by the least-square fit to the experiment, and the values are (a) $k_{B}T/U=0.0719$, (b) $k_{B}T/U=0.0577$, (c) $k_{B}T/U=0.0909$, (d) $k_{B}T/U=0.0963$, and (e) $k_{B}T/U=0.179$.