Bilayer graphene quantum dots as a quantum simulator of Haldane topological quantum matter

Abstract

We demonstrate here that a chain of Bilayer Graphene Quantum Dots (BLGQDs) can realize topological quantum matter by effectively simulating a spin-1 chain that hosts the Haldane phase within a specific range of parameters. We describe a chain of BLGQDs with two electrons per dot using an atomistic tight-binding model combined with exact diagonalization to solve the interacting few-electron problem. Coulomb interactions and valley-mixing effects are treated within a single microscopic framework, allowing us to systematically investigate spin and valley polarization transitions as functions of interaction strength and external tuning parameters. We calculate the low energy states for single and double QDs as a function of the number of electrons, identifying regimes of highly correlated multi-electron states. We confirm the presence of a spin-one ground state for two electrons. Then, we explore two coupled QDs with 4 electrons and extend the analysis to QD arrays. Using a mapping of the BLGQD chain to an effective bilinear-biquadratic (BLBQ) spin model, we demonstrate that BLGQD arrays can work as a quantum simulator for one-dimensional spin chains with emergent many-body topological phases.

Figures (7)

• Figure 1: (a) Top view of bilayer graphene, showing the four atoms within the unit cell: $A_1$, $B_1$, $A_2$, $B_2$, distinguished by colour. (b) Side view (zoomed in), highlighting the two primary hopping parameters: $\gamma_0$ for intralayer coupling and $\gamma_1$ for interlayer coupling. The applied perpendicular electric field is shown schematically; note that the diagram depicts the resulting potential energy profile rather than the electrostatic potential. (c) Schematic illustration of the confining potential for a double quantum dot. The parameters $V_0$, $R_{\rm{QD}}$, and $d$ denote the potential depth, dot radius, and center-to-center distance between the dots, respectively.
• Figure 2: (a) Low-energy levels of a double quantum dot. Each state shown is doubly degenerate by spin; only the valley-degenerate states are explicitly labeled. Insets show the corresponding electron densities for each shell, illustrating the increased coupling between interdot states with higher energy, as a consequence of the greater spatial extension of the wavefunctions. (b) Low-energy spectrum for four electrons in a double quantum dot as a function of interaction strength. For weak interactions, the ground state is a valley-unpolarized singlet, while for strong interactions, it becomes a valley-polarized quintuplet. There is an artificial horizontal shift for the valley-polarized quintuplets to show both valley cases. In the regime where the ground state is a singlet, the first and second excited states are a triplet and a quintuplet, respectively, closely resembling the spectrum of a spin-1 Heisenberg antiferromagnetic chain with $L = 2$ sites.
• Figure 3: (a) Triplet and quintuplet energies obtained from the fitted BLBQ and Heisenberg models as a function of interaction strength. The blue dashed lines show the energies of the double quantum dot. As discussed in the main text, the BLBQ Hamiltonian accurately reproduces the low-energy spectrum of two coupled quantum dots, each containing two electrons. (b) Fitted values of the BLBQ parameters $\beta$ and $J$ as functions of the electron-electron interaction strength.
• Figure 4: (a) Low energies of the $L=100$ BLBQ spin chain as a function of the $\beta$. (b) Spin-spin correlation function $g(n) = (-1)^n \langle S_1^z S_n^z \rangle$ plotted for the first half of the chain. The inset shows the correlation length $\xi$ extracted from exponential fits to $g(n)$, illustrating its dependence on $\beta$. The correlation length $\xi$ characterizes the spatial decay of spin correlations in the system.
• Figure S1: (a) Low-energy levels of a single quantum dot. Each state shown is doubly degenerate by spin; only the valley-degenerate states are explicitly labeled. Insets display the corresponding electron densities for each shell, illustrating the increasing spatial extension of the states with energy. (b) Low-energy spectrum for two electrons as a function of interaction strength. The ground state remains a triplet across the entire range of interaction strengths.