Engineering the Kitaev Spin Liquid in a Quantum Dot System

The Kitaev model on a honeycomb lattice may provide a robust topological quantum memory platform, but finding a material that realizes the unique spin-liquid phase remains a considerable challenge. We demonstrate that an effective Kitaev Hamiltonian can arise from a half-filled Fermi-Hubbard Hamiltonian where each site can experience a magnetic field in a different direction. As such, we provide a method for realizing the Kitaev spin liquid on a single hexagonal plaquette made up of 12 quantum dots. Despite the small system size, there are clear signatures of the Kitaev spin-liquid ground state, and there is a range of parameters where these signatures are predicted, allowing a potential platform where Kitaev spin-liquid physics can be explored experimentally in quantum dot plaquettes.

Figure 1

Our system consists of 12 Fermi-Hubbard sites with interaction strength $U$ arranged on a hexagon as shown. We separate the sites into the odd sites that live on the vertices, $V$, and the even sites that live on the bonds, $B$. There is hopping $t_1$ between all adjacent sites and hopping $t_2$ between the vertices. Even though a next-nearest-neighbor hopping $t_3$ between two bond sites should have roughly the same magnitude as $t_2$, we show in the Supplemental Material (SM) [67] that its effect is $\mathcal{O}(U^{-3})$ and does not alter our construction. The six edges of the hexagon are each given a label, $x$, $y$, or $z$ in the pattern indicated by color. The direction of the magnetic field for the bond (vertex) sites points in the direction of the bond label (sum of the two adjacent bond’s labels), respectively. For example, ${\mathit{h}}_4=-h_B\widehat{y}$ and ${\mathit{h}}_3=h_V(\widehat{z}+\widehat{y})$. If $|t_1|,|h_B|\ll U$, $|h_B|\gg |t_1{|}^2/U$, and $|t_2|$ and $|h_V|$ are related to $|t_1|$, $|h_B|$, and $U$ as shown, then the six vertex spins will interact with an effective Kitaev interaction of strength $K=2|t_1{|}^4/(U^2|h_B|)$.

Figure 2

We perform DMRG on twelve Fermi-Hubbard sites at half filling (scatter plot points) as well as exact diagonalization (ED) on six spin-$1/2$ sites using the effective Hamiltonian, Eq. (3). We set $t_1=h_B=1$, and the values of $t_2$ and $h_V$ are set so as to give a specified value of $J/K$ and $h_{\mathrm{eff}}$ via Eq. (5). In (a), (b) $|h_{\mathrm{eff}}|/K=0$ and in (c),(d) $|h_{\mathrm{eff}}|/K=0.6$ and the value of $J/K$ is indicated on the horizontal axis. In all plots, the value of $U$ used for DMRG is $U=10$ ($U=33$) for the circle ($\times$) points, respectively. By $U=33$, the DMRG results are almost entirely on top of the ED curves showing that the effective Hamiltonian is a Heisenberg-Kitaev Hamiltonian in a magnetic field. We compare two observables: in (a), (c), we plot the plaquette operator Eq. (4) and in (c),(d) we plot the correlator $⟨S_i^z{S}_j^z⟩$ for a $z$ bond $(i=1,j=3)$, an $x$ bond $(i=1,j=11)$, and the farthest spins $(i=1,j=7)$. In (b) and (d), the color of the points and dashed line indicates which spin correlator is being plotted. A Kitaev plaquette would have $W_P=1$ and the $x$ bond and farthest spin correlators will be zero. Clearly $J/K=0$ and $|h_{\mathrm{eff}}|/K=0$ satisfy these requirements, but when $|h_{\mathrm{eff}}|/K\ne 0$, there is a small range of $J/K$ where these are approximately true as well.