Fractionalized long-range ordered state in a Falicov-Kimball model

A Falicov-Kimball model which thermodynamically reduces the local Coulomb interaction of particles to attraction or repulsion is studied within the dynamical mean-field theory. In the strong interaction regime a fractionalization of particles into charge and spin objects, the physical properties of which are different from the whole particles, is observed in both high- and low-temperature phases. At high temperature and strong interaction the single-particle density of states opens an excitation gap, but the charge compressibility and the spin susceptibility exhibit the features of gapless excitations. The low-temperature phase has a long-range order, and the single-particle spectra are always gapped, while the charge and spin excitations are gapless in the strong interaction regime. In the fractionalized long-range ordered phase both the charge compressibility and the spin susceptibility are universal scaling functions of temperature.

Figure 1

The critical temperature $T_c$ via the local interaction $U$. The region $T>T_c$ is the HM phase, while the region $T<T_c$ is the long-range ordered phase. As $U$ increases, the phase continuously changes from the normal to the fractionalized states at both high and low temperatures.

Figure 2

The DOS of conduction electrons in the HM phase for different values of $U$ at temperature $T=0.1$.

Figure 3

The double $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ (solid line) and triple $\langle Q{n}_{\uparrow }{n}_{\downarrow }\rangle$ (short-dotted line) occupancies via interaction $U$ in the HM phase at temperature $T=0.5$. The contributions of the repulsive ${\langle n_{\uparrow }{n}_{\downarrow }\rangle }_U$ (dashed line) and attractive ${\langle n_{\uparrow }{n}_{\downarrow }\rangle }_{-U}$ (dash-dotted line) interactions to the double occupancy are also shown.

Figure 4

Scaling of the charge compressibility $\kappa /U$ as a function of $T/T_c$. The black solid line is the fitting function, which is described in Eq. (23) with $a=7.6, b=1.945$, and $c=0.25$.

Figure 5

Scaling of the spin susceptibility ${\chi }_{\text{AF}}/U$ as a function of $T/T_c$. The black solid line is the fitting function, which is described in Eq. (23) with $a=0.0175, b=0.568$, and $c=-1.698$.

Figure 6

The electron density $n$ and $\langle Q\rangle$ via the chemical potential $\mu$ in the strong correlation regime $U=5$ at temperature $T=0.5$.

Figure 7

The DOS of conduction electrons in the charge ordered phase at temperature $T=0.01$. The solid (dotted) lines are the DOS in the sublattice $A$ ($B$).

Figure 8

The charge order parameter ${\mathrm{\Delta }}_{\text{CO}}, \langle Q\rangle$, the double and triple occupancies via the interaction $U$ at temperature $T=0.01$.

Figure 9

The total $n$ and the sublattice $n_A, n_B$ electron densities, $\langle Q_A\rangle , \langle Q_B\rangle$ via the chemical potential for the interaction $U=8$ and temperature $T=0.01$. The dotted line shows the electron density $n=1$.