Chiral entanglement in triangular lattice models

We consider the low-energy spectrum of spin-$\frac{1}{2}$ two-dimensional triangular lattice models subject to a ferromagnetic Heisenberg interaction and a three spin chiral interaction of variable strength. Initially, we consider quasi-one-dimensional ladder systems of various geometries. Analytical results are derived that yield the behavior of the ground states, their energies, and the transition points. The entanglement properties of the ground state of these models are examined and we find that the entanglement depends on the lattice geometry due to frustration effects. To this end, the chirality of a given quantum state is used as a witness of tripartite entanglement. Finally, the two-dimensional model is investigated numerically by means of exact diagonalization and indications are presented that the low energy sector is a chiral spin liquid.

Figure 1

(Color online) Quasi-1D geometries. The spins reside on the vertices and spin-spin interactions are represented by the edges. At each triangle there is also a chiral three-spin interaction which has the same sense everywhere, as indicated by the arrows.

Figure 2

(Color online) Absolute value of the chirality vs relative strength of chiral interaction for the different geometries of Fig. 1 (for $N=9$ spins with periodic boundary conditions). We see that the chirality of lattice types A, B, C, and D becomes nonzero at the points ${\lambda }_{\text{A}}\approx 1.1$, ${\lambda }_{\text{B}}\approx 1.7$, ${\lambda }_{\text{C}}\approx 1.7$, and ${\lambda }_{\text{D}}\approx 1.1$, respectively. The horizontal lines (from top to bottom) correspond to the maximum chirality $\left(=2\sqrt{3}\right)$, and the entanglement witnesses for $W$-like $\left(=3\sqrt{3}/2\right)$, GHZ-like $\left(=2\right)$, and bipartite entangled $\left(=1\right)$ states, as demonstrated in Sec. 2.

Figure 3

(Color online) Ground-state energy per lattice site vs relative strength of chiral interaction for a type-A ladder [Fig. 1a]. The solid line is the analytical estimate computed using the fermionization in Sec. 3B, while the dashed and dashed-dotted lines come from an exact diagonalization of the model for $N=7$ and $N=15$ spins, respectively.

Figure 4

Spin-spin correlations $C_{0,\Delta }$ [Eq. (22)] for a periodic type-A ladder with $N=24$ spins. The dashed-dot line corresponds to $\lambda =0.2$ and the solid line to $\lambda =100$. In the chiral regime the correlations decay exponentially (on either direction of the periodic ladder).

Figure 5

(Color online) (a) Ground-state mean chirality vs relative strength of chiral interaction for different lattice sizes. (b) Energy gap between the ground state and the first excited state.

Figure 6

A geometrical configuration of the $4\times 4$ periodic lattice. The position $\left(n,m\right)$ of the site on the lattice is given in terms of the integers $n,m=1,\dots ,4$. A certain bond is depicted as $\left(n,m\right)–\left(k,l\right)$ that connects the corresponding two neighboring sites. (a) Chiral correlation between the plaquette with sites (2, 2), (2, 3), and (3, 3) and the remaining ones. Note the difference between the value of $⟨\mathcal{X}\mathcal{X}⟩$ for the same plaquette (7.042) and neighboring ones. (b) Dimer correlations between the bond (2, 2)–(2, 3) and the remaining bonds. The simulations were performed deep in the chiral regime $\left|\lambda \right|=100$.

Figure 7

(Color online) The triangular configuration. Three-spin interaction terms appear between sites 1, 2, and 3 since tunneling between two of the sites, such as 1 and 2, can happen directly or through the third site. The latter case results in an exchange interaction between 1 and 2 that is influenced by the state of the spin at site 3.