Half-filled Hubbard model on a Bethe lattice with next-nearest-neighbor hopping

We study the interplay between Néel antiferromagnetism and the paramagnetic metal-insulator transition (PMIT) on a Bethe lattice with nearest- and next-nearest-neighbor hoppings $t_1$ and $t_2$. We concentrate in this paper on the situation at half-filling. For $t_2/t_1\to 1$ the PMIT outgrows the antiferromagnetic phase and shows a scenario similar to ${\text{V}}_2{\text{O}}_3$. In this parameter regime we also observe another magnetic phase.

Figure 1

(Color online) DOS for increasing NNN hopping $t_2$. Observe the van Hove singularity at the lower band edge. $\rho \left(\omega \right)$ and $\omega$ are scaled with the bandwidth $W=4t_2+2t_1+t_1^2/\left(4t_2\right)$ $\left(t_2>1/4t_1\right)$.

Figure 2

(Color online) The transition lines for the PMIT for different frustrations as a function of temperature and interaction strength. For each frustration the right line represents the transition from the metal to the insulator while the left line represents the transition from the insulator to the metal. Symbols mark the calculated data points, in which the lines are fits meant as guide for the eyes.

Figure 3

(Color online) The left (right) panel shows the $T-U$ phase diagram for $t_2/t_1=0.6$ (0.8). The shaded area represents the antiferromagnetic phase while the white area represents the paramagnetic phase. The lines show where the PMIT in the paramagnetic phase would occur. The points denote the parameter values, where DMFT calculations were done. From these points the shaded area was constructed as guide for the eyes. Additional calculations were performed to find the PMIT lines.

Figure 4

(Color online) Staggered magnetization versus interaction $U/W$ for two different temperatures and $t_2/t_1=0.8$. In the upper panel there are for each temperature two transition lines, representing either increasing or decreasing interaction strength. The region between both lines embodies a hysteresis region. The lower panel shows the transition for large interaction strengths. Here no hysteresis region could be found but a smooth transition.

Figure 5

(Color online) Phase diagram for $t_2/t_1=0.8$ including the temperature depending hysteresis region (between dashed and full lines). The meaning of symbols is the same as in Fig. 3. The shaded area is the same as before meant as guide for the eyes.

Figure 6

(Color online) Ground state $\left(T=0\right)$ phase diagram for different strengths of frustration as a function of the interaction. The brindled regions are hysteresis regions for increasing or decreasing interaction. The phase boundaries of the IC phase are only qualitative (for explanation, see text).

Figure 7

(Color online) Phase diagram $T-U$ for $t_2/t_1=0.98$ including the PMIT. The brindled area represents again the hysteresis region.

Figure 8

Example of a nonconvergent DMFT calculation. The figure shows the staggered polarization over the DMFT iteration number for $t_2/t_1=0.98$, $U/W=1.35$, and $T\approx 0$. The lines are meant as guide for the eyes.

Figure 9

(Color online) Left: Bethe lattice $Z=3$ with sites numbered according to the vector spins on the right. The nearest neighbors of site 1 must lie on a circle so there is an angle $\theta$ between 1 and 2. Similarly the nearest neighbors of one of the two spins must lie on a circle including the three spins and the one spin.