Low-temperature properties of the Hubbard model on highly frustrated one-dimensional lattices

We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices—sawtooth chain and two kagomé chains—with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential ${\mu }_0$. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.

Figure 1

(Color online) Three one-dimensional lattices considered in this paper: (a) the sawtooth chain, (b) the kagomé chain I, and (c) the kagomé chain II. For the sawtooth Hubbard chain the hopping parameter along the zigzag path $t^{\prime }$ is $\sqrt{2}$ times larger than the hopping parameter $t>0$ along the base line. For the kagomé Hubbard chains all the hopping parameters $t>0$ are identical. Bold (red) lines denote the minimal trapping cells for localized electrons.

Figure 2

(Color online) Illustration of a multicluster state. For each cluster the total spin of the cluster can be flipped independently by applying the cluster spin-flip operator $S_{n_i}^{-}={\sum }_{j∊\text{cluster}{n}_i}{c}_{j,\downarrow }^{†}{c}_{j,\uparrow }$.

Figure 3

(Color online) Ground-state degeneracies for $n=1,\dots ,n_{\text{max}}$ for the sawtooth chain ($\mathcal{N}=7$, empty diamonds; $\mathcal{N}=8$, empty squares), the kagomé chain I ($\mathcal{N}=7$, empty diamonds; $\mathcal{N}=8$, filled squares), and the kagomé chain II ($\mathcal{N}=7$, filled diamonds; $\mathcal{N}=8$, filled squares).

Figure 4

(Color online) Canonical residual entropy $S\left(n,\mathcal{N}\right)/\mathcal{N}=\text{ln}\left[D_{\mathcal{N}}\left(n\right)\right]/\mathcal{N}$ versus $n/\mathcal{N}$ for system sizes $\mathcal{N}=32,64,128,256,512,1024$.

Figure 5

(Color online) Average number of electrons in the ground state $\overline{n}\left(0,\mu ,N\right)/\mathcal{N}$ versus chemical potential $\mu /{\mu }_0$ for the sawtooth chain [$N=12$ (solid) and $N=20$ (long dashed)], the kagomé chain I [$N=18$ (short dashed)], the kagomé chain II [$N=20$ (dotted)] with $U\to \infty$. The result for $\overline{n}\left(0,\mu ,N\right)/\mathcal{N}$ which follows from Eqs. (5.3, 5.5) is given by $\theta \left(1-\mu /{\mu }_0\right)$ and $\left[\left(\mathcal{N}+1\right)/\mathcal{N}\right]\theta \left(1-\mu /{\mu }_0\right)$, respectively.

Figure 6

(Color online) Entropy $S\left(T,\mu ,N\right)/\mathcal{N}$ and grand-canonical specific heat $C\left(T,\mu ,N\right)/\mathcal{N}$ for the sawtooth chain ($N=12$, triangles) and the kagomé chains I ($N=9$, diamonds) with $t=1$, $U\to \infty$. (a) $S\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ at $\mu =0.98{\mu }_0,{\mu }_0,1.02{\mu }_0$ (from top to bottom). (b) $C\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ at $\mu =0.98{\mu }_0,{\mu }_0,1.02{\mu }_0$ (from top to bottom). We also show the hard-dimer result for $N\to \infty$ as it follows from Eq. (5.6) (thin solid lines) as well as the results which follow from Eq. (5.3) for $\mathcal{N}=3$ (dotted lines) and $\mathcal{N}=6$ (dashed lines). Note that for these systems no additional leg states exist. Note further that for $\mu ={\mu }_0$ and $\mu =1.02{\mu }_0$ the hard-dimer results for $\mathcal{N}=3$, 6, and $\infty$ are indistinguishable.

Figure 7

(Color online) Entropy $S\left(T,\mu ,N\right)/\mathcal{N}$ and grand-canonical specific heat $C\left(T,\mu ,N\right)/\mathcal{N}$ for the kagomé chain I ($N=12$, diamonds) and the kagomé chain II ($N=15$, the sectors with up to seven electrons were taken into account, pentagons) with $t=1$, $U\to \infty$. (a) $S\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ at $\mu =0.98{\mu }_0,{\mu }_0,1.02{\mu }_0$ (from top to bottom). (b) $C\left(T,\mu ,N\right)/\mathcal{N}$ versus temperature $T/{\mu }_0$ at $\mu =0.98{\mu }_0,{\mu }_0,1.02{\mu }_0$ (from top to bottom). We also show the hard-dimer result for $N\to \infty$ as it follows from Eq. (5.6) (thin solid lines) as well as the finite-size results for $\mathcal{N}=3$ (dash-dotted lines) and $\mathcal{N}=4$ (dotted lines) which follow from Eq. (5.5) and include the contribution of the leg states.

Figure 8

(Color online) $C\left(T,\mu ,N\right)/\mathcal{N}$ versus $T/{\mu }_0$ for the kagomé chain I with $N=12$ sites and $t=1$ for different values of $U$ [$U=1$ (empty circles), $U=10$ (filled diamonds), and $U\to \infty$ (empty diamonds)]. The results for finite $U$ were obtained taking into account the sectors with up to eight electrons. We also show the result which follows from Eq. (5.5) for $\mathcal{N}=4$ (lines).

Figure 9

(Color online) Canonical specific heat $C\left(T,n,N\right)/n$ versus $T$ for ideal $\left(t_d=t_L=1\right)$ and distorted ($t_d=1$ and $t_L=1.01$) kagomé chains I with $n=N/6$ electrons; $U\to \infty$.

Figure 10

(Color online) Average ground-state magnetic moment of the sawtooth-Hubbard chain: ${⟨{\mathit{S}}^2⟩}_n/{\mathcal{N}}^2$ versus $n/\mathcal{N}$ for $\mathcal{N}=8,16,32,64,128,256$. For saturated ferromagnetic ground states ${⟨{\mathit{S}}^2⟩}_n/{\mathcal{N}}^2$ would equal ${\left(n/\mathcal{N}\right)}^2/4$ (thin line) in the thermodynamic limit $\mathcal{N}\to \infty$.

Figure 11

(Color online) Average ground-state magnetic moment of the kagomé-Hubbard chains: ${⟨{\mathit{S}}^2⟩}_n/{\mathcal{N}}^2$ versus $n/\mathcal{N}$ for $\mathcal{N}=8,16,32,64,128,256$. For saturated ferromagnetic ground states ${⟨{\mathit{S}}^2⟩}_n/{\mathcal{N}}^2$ would equal ${\left(n/\mathcal{N}\right)}^2/4$ (thin line) in the thermodynamic limit $\mathcal{N}\to \infty$.

Figure 12

(Color online) Uniform magnetic susceptibility $3T\chi \left(T,n,N\right)/C_n$ for the sawtooth chain $\left(t=1\right)$ with $n=4$ electrons and $N=16$ sites with $U=4$ (filled triangles and circles) as well as $N=24$ sites with $U\to \infty$ (empty triangles and circles). Triangles correspond to $t^{\prime }=\sqrt{2}$ and circles correspond to $t^{\prime }=1$. Here we use for the normalization of the vertical axis $C_n=4$ for $\mathcal{N}=8$ and $C_n=60/17$ for $\mathcal{N}=12$.