Finite-temperature phase transitions in the ionic Hubbard model

We investigate paramagnetic metal-insulator transitions in the infinite-dimensional ionic Hubbard model at finite temperatures. By means of the dynamical mean-field theory with an impurity solver of the continuous-time quantum Monte Carlo method, we show that an increase in the interaction strength brings about a crossover from a band insulating phase to a metallic one, followed by a first-order transition to a Mott insulating phase. The first-order transition turns into a crossover above a certain critical temperature, which becomes higher as the staggered lattice potential is increased. Further, analysis of the temperature dependence of the energy density discloses that the intermediate metallic phase is a Fermi liquid. It is also found that the metallic phase is stable against strong staggered potentials even at very low temperatures.

Figure 1

Fermi-level spectral weight ${\tilde{A}}_{\alpha }$ for $\Delta =0.5$. (a) The colored plot displays ${\tilde{A}}_{\alpha }$ on the plane of the interaction strength $U$ and temperature $T$, obtained via increasing $U$. (b) ${\tilde{A}}_{\alpha }$ at temperatures $T=1/32$ [(red) squares] and $1/128$ [(blue) circles]. Filled and open symbols for $T=1/128$ represent data obtained via increasing and decreasing $U$, respectively.

Figure 2

Spectral function $A\left(\omega \right)$ for $\Delta =0.5$ and $T=1/128$. Corresponding interaction strengths are (a) $1.0$, (b) $2.5$, (c) $2.8$ (increasing $U$), (d) $2.8$ (decreasing $U$), and (e) $4.0$. Solid (red) and dotted (blue) lines represent the spectral function at sublattices $A$ and $B$, respectively.

Figure 3

Double-occupancy $d_O$ as a function of $U$: (a) for staggered lattice potential $\Delta =0.5$ at temperatures $T=1/32$ [(red) squares] and $T=1/128$ [(blue) circles] and (b) at temperature $T=1/128$ for various values of the staggered lattice potential—from top to bottom, $\Delta =1.0$, 0.5, and $0.0$. Inset in (a): Detailed behavior in the coexistence region. Data for $T=1/64$, (green) diamonds; $T=1/128$, (blue) circles. Data obtained via increasing $U$, filled symbols; via decreasing $U$, open symbols.

Figure 4

Staggered charge density $n_B-n_A$ as a function of $U$: (a) for staggered lattice potential $\Delta =0.5$ at temperatures $T=1/32$ [(red) squares] and $T=1/128$ [(blue) circles] and (b) at $T=1/128$ for $\Delta =0.5$ [(green) circles] and $\Delta =1.0$ [(blue) triangles]. Inset in (a): Details in the coexistence region. Data for $T=1/64$, (green) diamonds; $T=1/128$, (blue) circles. Data obtained via increasing $U$ filled symbols; via decreasing $U$, open symbols.

Figure 5

Energy densities as functions of $U$ at temperature $T=1/128$ and $\Delta =0.5$. (a) Interaction energy ${\varepsilon }_U$, (b) kinetic energy ${\varepsilon }_k$, (c) staggered lattice potential energy ${\varepsilon }_{\Delta }$, and (d) total energy $\varepsilon$ (see text for definitions). The (blue) circles and (red) squares represent data for the IHM and the HM, respectively. Inset in (d): Total energy density in the coexisting region; DMFT solutions obtained via increasing U (filled symbols) and decreasing $U$ (open symbols).

Figure 6

Phase diagram for $\Delta =0.5$ on the plane of $T$ and $U$. Filled (red) circles and surrounding horizontal bars indicate the crossover strength $U_{\mathrm{co}}$ and estimated crossover regions, respectively. Two transition points for the Mott transition, $U_{c1}$ and $U_{c2}$, are plotted by squares for various temperatures. The critical point of the Mott transition is represented by diamonds, along with the first-order transition line. Regions of the band insulator (BI), metal (M), and Mott insulator (MI) phases. The three open (blue) circles on the horizontal axis correspond to $U_{\mathrm{co}}$, $U_{c1}$, and $U_{c2}$, respectively, obtained from NRG-DMFT at zero temperature [47].

Figure 7

Total energy density $\varepsilon$ as a function of temperature $T$. From top to bottom, the staggered lattice potential and interaction strength are given by $(\Delta ,U)=(0,2.2)$ (squares), $(\Delta ,U)=(0.5,2.5)$ (circles), $(\Delta ,U)=(1,3.2)$ (triangles), and $(\Delta ,U)=(3,6.9)$ (inverted triangles). The horizontal axis is drawn on the scale of $T^2$.

Figure 8

Inverse susceptibility ${\chi }^{-1}$ versus temperature $T$ for the staggered lattice potential $\Delta =0.5$ [(red) squares], $\Delta =1.0$ [(orange) triangles], $\Delta =3.0$ [(green) inverted triangles], and $\Delta =5.0$ [(blue) circles].

Figure 9

Coexistence region for $\Delta =0.5$ (diamonds), $1.0$ (triangles), and $3.0$ (inverted triangles). For clear comparison with the Hubbard model [represented by (red) squares], data for $\Delta =0.5$, 1.0, and $3.0$ are shifted to the left by the amount $\delta U=0.3$, $1.0$, and $4.69$, respectively.

Figure 10

Phase diagram on the plane of $\Delta$ and $U$ at temperature $T=1/128$. Band insulator (BI), metal (M), and Mott insulator (MI) phases. The (blue) squares represent $U_{c1}$ and $U_{c2}$ of the Mott transition. The (red) circles and vertical bars describe $U_{\mathrm{co}}$ and the crossover region between the BI and the metal.