Exact low-temperature properties of a class of highly frustrated Hubbard models

We study the repulsive Hubbard model both analytically and numerically on a family of highly frustrated lattices which have one-electron states localized on isolated trapping cells. We construct and count exact many-electron ground states for a wide range of electron densities and obtain closed-form expressions for the low-temperature thermodynamic quantities which are universal for all lattices of the family. Furthermore, we find that saturated ferromagnetism is obtained only for sufficiently high electron densities and large Hubbard repulsion $U$ while there is no finite average moment in the ground states at lower densities.

Figure 1

(Color online) (a) Diamond chain, (b) frustrated ladder, (c) square-kagome, and (d) checkerboard lattices. The circles indicate the lattice sites, the thick (thin) lines indicate the hopping paths along the trap (connecting) bonds. For the ideal diamond chain with $t_1=t_3$ we will use the notation $t=t_1=t_3$, $t^{\prime }=t_2$.

Figure 2

(Color online) Electron density $\overline{n}/\mathcal{N}$ versus chemical potential $\mu /{\mu }_0$ for the diamond chain $\left(N=18,t^{\prime }=3\right)$, the frustrated ladder $\left(N=16,t^{\prime }=3\right)$, the square-kagome lattice $\left(N=24,t^{\prime }=2\right)$, and the checkerboard lattice $\left(N=16,t^{\prime }=2\right)$ for $U\to \infty$ and $T=0$. The universal dependence [Eq. (6)] at $T=0$ is given by $\theta \left(1-\mu /{\mu }_0\right)$. Inset: charge gap $\Delta \mu /{\mu }_0$ at $\overline{n}/\mathcal{N}=1$ versus $U$ for the diamond chain [$N=12$ (triangles), $N=18$ (line), $t^{\prime }=3$].

Figure 3

(Color online) $C\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ in the limit $U\to \infty$ for the diamond chain (triangles), the frustrated ladder (squares), the square-kagome lattice (diamonds), and the checkerboard lattice (circles). We also show the universal dependence [Eq. (8)] (lines).

Figure 4

(Color online) $C\left(T,\mu ,N\right)/\mathcal{N}$ for the frustrated ladder with $N=12$ and $t^{\prime }=3$ for different values of $U$. We also show the universal dependence [Eq. (8)] (lines).

Figure 5

(Color online) Entropy $S\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ in the limit $U\to \infty$ for the diamond chain (triangles), the frustrated ladder (squares), the square-kagome lattice (diamonds), and the checkerboard lattice (circles). We also show the universal dependence [Eq. (7)] (lines).

Figure 6

(Color online) $C\left(T,\mu ,N\right)/\mathcal{N}$ for the ideal diamond chain $\left(t^{\prime }=4,t=1\right)$ and the distorted diamond chain $\left(t_1=1.1,t_2=4,t_3=0.9\right)$. The system size is $N=12$ in all cases.

Figure 7

(Color online) Specific heat $C\left(T,n,N\right)/\mathcal{N}$ for the ideal diamond chain $\left(t^{\prime }=4,t=1\right)$ and the distorted diamond chain $\left(t_1=1.1,t_2=4,t_3=0.9\right)$ with $n=N/3$ electrons for $N=9,12,15,18$. The Hubbard repulsion is $U=10$.