Filling-driven Mott transition in $\text{SU}\left(N\right)$ Hubbard models

We study the filling-driven Mott transition involving the metallic and paramagnetic insulating phases in $\mathrm{SU}\left(N\right)$ Fermi-Hubbard models, using the dynamical mean-field theory and the numerical renormalization group as its impurity solver. The compressibility shows a striking temperature dependence: near the critical end-point temperature, it is strongly enhanced in the metallic phase close to the insulating phase. We demonstrate that this compressibility enhancement is associated with the thermal suppression of the quasiparticle peak in the local spectral functions. We also explain that the asymmetric shape of the quasiparticle peak originates from the asymmetry in the dynamics of the generalized doublons and holons.

Figure 1

Particle number per site $n\equiv \langle {\widehat{n}}_i\rangle$ vs chemical potential $\mu$ along the homogeneous, paramagnetic phases of the $\mathrm{SU}\left(N\right)$ Hubbard models [cf. Eq. (1)] at temperature $T=0$ (thick lines). Due to particle-hole symmetry at $\mu =0$, the curves for $\mu >0$ can be deduced by $n\left(\mu \right)=N-n(-\mu )$. (a) For small $U$, the systems remain compressible, i.e., metallic. (b) For intermediate $U=3$, plateaus start to develop at integer $n$. (c) For large $U=7$, wide plateaus demonstrate the incompressibility of the Mott insulating phase. As the flat plateaus for the insulating phase connect to the slanting lines for the metallic phase at the Mott transition, weak hysteretic behavior occurs, which thus indicates coexistence. Insets: Zoom-in to individual hysteresis loops for $N=4$ and 5, as examples for the left and right ends of the Mott plateaus, respectively. Each thin solid line connects two data points (crosses) across the Mott transition: For the insulator-to-metal transition (IMT) at ${\mu }_{c1}$ (dashed vertical lines), it connects the last data point in a plateau with the subsequent next point in the metallic phase. Conversely, for the metal-to-insulator transition (MIT) at ${\mu }_{c2}$ (dash-dotted vertical lines) it connects the last point in the metallic phase (slanted line) with the next data point in the insulating plateau (here at $T=0$, these are very short line segments visible only in the insets).

Figure 2

Compressibility $\tilde{\kappa }$ of Eq. (2), obtained as numerical derivative from the curves in Fig. 1 at $T=0$, as function of filling fraction $n/N$ (thick solid lines). In (b) and (c), thin lines with arrows indicate discontinuous jumps between finite $\tilde{\kappa }$ in the metallic phase and $\tilde{\kappa }=0$ in the insulating phases. Thus the boundary of each shaded area represents a hysteresis loop, where the upper solid line part of the boundary indicates the metallic solution in the coexistence region. Note that the shading just below $n=1$ for $(N,U)=(2,7)$ is not visible, since the coexistence region $[{\mu }_{c1},{\mu }_{c2}]$ in the curve $n\left(\mu \right)$ is narrower than the grid size $\mathrm{\Delta }\mu =0.05$. To indicate the second-order nature of MIT at $T=0$, we have extrapolated the shaded areas to integer $n$, while the lines with downward arrows with finite slopes connect actual data points. The curves for $n>N/2$ can be deduced by particle-hole symmetry $\tilde{\kappa }\left(n\right)=\tilde{\kappa }(N-n)$.

Figure 3

Compressibility $\tilde{\kappa }$ vs filling fraction $n/N$ for fixed $U=7$ and varying $T$. Below the critical end point temperature $T^{*}(N,U,n)$, there are Mott transitions appearing as discontinuous jumps in $\tilde{\kappa }$. Thick horizontal bars above the curves $\tilde{\kappa }\left(n\right)$ (solid lines) near integer $n$ (vertical gray lines) indicate the range of $n$ for the metallic phase in the coexistence regime, color matched with the curves $\tilde{\kappa }$ for the same $T$. As $T$ increases from 0 to $T^{*}$, the coexistence regions get narrower in $n$. For some curves, e.g., for $N=2$, the coexistence region is narrower than the numerical grid size $\mathrm{\Delta }\mu =0.05$. For $T\sim T^{*}$, local maxima of $\tilde{\kappa }\left(n\right)$ appear near integer $n$, which we call compressibility enhancement. At $T>T^{*}$, the discontinuity in the curves $\tilde{\kappa }\left(n\right)$ disappears, which indicates that a crossover (rather than a phase transition) occurs between metallic and insulating behavior.

Figure 4

Particle number per site $n$ vs chemical potential $\mu$ zooming into the Mott transitions just above $n=\lfloor N/2\rfloor$, i.e., particle-doped regime, for $U=7$ and $2\le N\le 5$. Thick solid lines and symbols represent $n\left(\mu \right)$. Thin lines connect data points in different phases, where the arrow specifies the direction of phase transition. For the values of $\mu$ indicated by vertical dotted lines, we show the local spectral functions in Fig. 5 below. In Fig. 6, we illustrate the local spectral functions for $N=4$, with the $\mu$ values marked by vertical dashed lines in (c).

Figure 5

Local spectral functions $A\left(\omega \right)$ in the metallic phase with slight particle doping, for large interaction $U=7$, with varying parameters $(N,T,\mu )$, where $\mu$ corresponds to the values marked by the vertical dotted lines in Fig. 4. Left insets zoom into the low-frequency regime containing the quasiparticle peak. Middle insets show how the occupation number $n$ (filled symbols) together with the spectral weight of the lower Hubbard band, $n_{\mathrm{LHB}}\equiv N{\int }_{\mathrm{LHB}}d\omega A\left(\omega \right)$ (empty symbols), change with $T$, where for the large value of $U$ here we delineate the range of the LHB by $\omega <-2$. For reference, the dark blue symbols on top of the left axis in the middle insets give $n$ and $n_{\mathrm{LHB}}$ for $T=0$, not for $T={10}^{-3}$.

Figure 6

Evolution of the local spectral functions $A\left(\omega \right)$ for $N=4,U=7$, and a set of five fixed temperatures, as $\mu$ is decreased (top to bottom) towards the MIT at ${\mu }_{c2}$. The three $\mu$ values shown here all lie in the vicinity of the value $\mu =2.3$ of Fig. 5. Insets zoom into low-frequency regime in which the quasiparticle peak or the Mott gap appears. Note that the curve for $T=0.01$ has no quasiparticle peak in (c), since $\mu =2.2$ lies below ${\mu }_{c2}(T=0.01)=2.275\left(25\right)$, i.e., $\mu$ has already been lowered past the MIT transition point.

Figure 7

The correlation functions of generalized doublons and holons at $T=0$. The creation operators for the generalized doublons and holons, $d_{i\nu }^{†}$ and $h_{i\nu }^{†}$, are defined in Eq. (3). Here we choose the occupation numbers $n$ to satisfy $\left[n\right]=\lfloor N/2\rfloor$, where $\left[n\right]$ means the nearest integer to $n$. The first column of (a)–(d) show the correlation functions for the weakly particle-doped cases shown in Fig. 5. The second column [(e)–(h)] presents data for increased particle doping, whereas in the last column [(i)–(l)], the interaction strength was significantly reduced down to $U=3$ while keeping the particle doping comparable to (a)–(d). Insets zoom into low-frequency regime containing the quasiparticle peak.