Doping control of realization of an extended Nagaoka ferromagnetic state from the Mott state

Inspired by the Nagaoka ferromagnetism, we propose an itinerant model to study the transition between the Mott singlet state and a ferromagnetic state by emulating a doping process in finite lattices. In the Nagaoka ferromagnetism, the total spin of the system takes the maximum value when an electron is removed from the half-filled system. To incorporate a procedure of the electron removal, our model contains extra sites as a reservoir of electrons, and the chemical potential of the reservoir controls the distribution of electrons. As a function of the chemical potential, the system exhibits ground-state phase transitions among various values of the total spin, including a saturated ferromagnetic state due to the Nagaoka mechanism at finite hole density. We discuss the nature of the ferromagnetism by measuring various physical quantities, such as the distribution of electrons, the spin correlation functions, the magnetization process in the magnetic field, and also the entanglement entropy.

Figure 1

A lattice with five sites, which is used as a unit structure to construct an extended lattice. Here we put four electrons. Solid circles are occupied by an electron, while open circles are vacant. The electron occupation is controlled by the chemical potential at the center site $\mu$.

Figure 2

(a) The $\mu$ dependence of the total spin $S_{\mathrm{tot}}$ and the number of electrons in the subsystem $N_{\mathrm{e}}^{\mathrm{sub}}$ for the system with five sites and four electrons at $U=1000$. (b) $N_{\mathrm{e}}^{\mathrm{sub}}$ around $\mu =8$ in a magnified scale, where we clearly see a jump.

Figure 3

Extended lattices with 8, 11, 14, and 17 sites. The site numbering is presented for $N=17$ as an example. The lattice sites are labeled in the order of the sites in the lower leg in the subsystem, those in the upper leg in the subsystem, and those in the center sites.

Figure 4

The total spin $S_{\mathrm{tot}}$ and the number of electrons in the center sites $N_{\mathrm{e}}^{\mathrm{c}}$ for several system sizes: (a) $(N,N_{\mathrm{e}})=(11,8)$, (b) $(14,10)$, and (c) $(17,12)$. Here we set $U=1000$.

Figure 5

(a) The electron density in the center sites $N_{\mathrm{e}}^{\mathrm{c}}/N^{\mathrm{c}}$, and (b) that in the subsystem $N_{\mathrm{e}}^{\mathrm{sub}}/N^{\mathrm{sub}}$, for several system sizes at $U=1000$.

Figure 6

DMRG results with large system sizes up to $N=35$ at $U=1000$. (a) The total spin $S_{\mathrm{tot}}$ normalized by the maximum value $S_{\mathrm{tot}}^{\max }=N_{\mathrm{e}}/2$, i.e., $0\le S_{\mathrm{tot}}/S_{\mathrm{tot}}^{\max }\le 1$ regardless of the system size. In the shaded region near $\mu =0$, we cannot obtain well-converged data by DMRG simulations even if we keep 1000 states. (b) The size dependence of the transition points ${\mu }_{\mathrm{c}}$ denoted by arrows in (a). Here we also plot Lanczos results for $N=11$, 14, and 17 together.

Figure 7

(a) The total spin $S_{\mathrm{tot}}$ and the number of electrons in the center sites $N_{\mathrm{e}}^{\mathrm{c}}$ at $U=100$ for $N=11$. (b) The size dependence of the critical value of $U$ for the appearance of the ferromagnetism $U_{\mathrm{c}}$ with $N=11$, 14, 17, and 20.

Figure 8

The ground-state phase diagram in the coordinate $(U,\mu )$ for $N=11$. The transition points ${\mu }_{\mathrm{c}}$ are defined in the same way as those in Fig. 6. At several points denoted by crosses, we have confirmed the realization of the complete ferromagnetic state by DMRG calculations up to $N=110$.

Figure 9

The transverse spin correlation function $C_{xy}(i,j)$ for typical values of $\mu$ at $U=1000$: (a) $\mu =10$ for the Mott antiferromagnetic state with $S_{\mathrm{tot}}=0$; (b) $\mu =8$ for the partially spin-polarized state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2-2$; and (c) $\mu =5$ for the complete ferromagnetic state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2$, obtained by DMRG with $N=35$. The correlation function is measured with the middle site of the subsystem (site 6 denoted by an open circle) as starting point.

Figure 10

The longitudinal spin correlation function $C_z(i,j)$ for typical values of $\mu$ at $U=1000$: (a) $\mu =10$ for the Mott antiferromagnetic state with $S_{\mathrm{tot}}=0$; (b) $\mu =8$ for the partially spin-polarized state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2-2$; and (c) $\mu =5$ for the complete ferromagnetic state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2$, obtained by DMRG with $N=35$. The correlation function is measured with the middle site of the subsystem (site 6 denoted by an open circle) as starting point.

Figure 11

The longitudinal spin correlation function $C_z(i,j)$ for the partially spin-polarized state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2-2$ in the subspace of $M=N_{\mathrm{e}}/2-2$ at $\mu =8$ and $U=1000$, obtained by DMRG with $N=35$. The correlation function is measured with the middle site of the subsystem (site 6 denoted by an open circle) as starting point.

Figure 12

The magnetization curve for typical values of $\mu$ at $U=1000$ and several temperatures: (a) $\mu =10$ for the Mott antiferromagnetic state with $S_{\mathrm{tot}}=0$; (b) $\mu =8$ for the partially spin-polarized state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2-2$; and (c) $\mu =5$ for the complete ferromagnetic state with $S_{\mathrm{tot}}=N_{\mathrm{e}}/2$. For each value of $\mu$ we show magnetization curves at temperatures $T=0,0.001$, and $0.01$. Open circles denote numerical results with $N=11$, and the Brillouin functions $B_S$ for the corresponding temperatures are plotted by solid or dashed curves.

Figure 13

(a) The total spin of the subsystem $S_{\mathrm{sub}}$ and (b) that of the center sites $S_{\mathrm{c}}$ at $U=1000$ for $N=11$. Note that the total spin of the whole system $S_{\mathrm{tot}}$ was given in Fig. 4.

Figure 14

The total spin $S_{\mathrm{tot}}$ and the number of electrons in the center sites $N_{\mathrm{e}}^{\mathrm{c}}$ for (a) $t^{\prime }=0.5$ and (b) $t^{\prime }=0.1$ at $U=1000$ for $N=11$. Note that the plot for $t^{\prime }=1.0$ was given in Fig. 4.

Figure 15

The ground-state phase diagram in the coordinate $(t^{\prime },\mu )$ at $U=1000$ for $N=11$. The transition points ${\mu }_{\mathrm{c}}$ are defined in the same way as those in Fig. 6.

Figure 16

(a) The entanglement entropy of the subsystem $E_{\mathrm{sub}}$ for several values of $t^{\prime }$ for $N=8$. (b) $E_{\mathrm{sub}}$ at $t^{\prime }=1$ for several system sizes. Here $U=1000$.