Ab initio exact diagonalization simulation of the Nagaoka transition in quantum dots

Recent progress of quantum simulators provides insight into the fundamental problems of strongly correlated systems. To adequately assess the accuracy of these simulators, the precise modeling of the many-body physics, with accurate model parameters, is crucially important. In this paper, we employed an ab initio exact diagonalization framework to compute the correlated physics of a few electrons in artificial potentials. We apply this approach to a quantum-dot system and study the magnetism of the correlated electrons, obtaining good agreement with recent experimental measurements in a plaquette. Through control of dot potentials and separation, including geometric manipulation of tunneling, we examine the Nagaoka transition and determine the robustness of the ferromagnetic state. While the Nagaoka theorem considers only a single-band Hubbard model, in this work we perform extensive ab initio calculations that include realistic multiorbital conditions in which the level splitting is smaller than the interactions. This simulation complements the experiments and provides insight into the formation of ferromagnetism in correlated systems. More generally, our calculation sets the stage for further theoretical analysis of analog quantum simulators at a quantitative level.

Figure 1

Solution of single-well wave functions for $V_0=100$ and $\mathrm{\Sigma }=1$. (a) The Gaussian quantum well in two dimensions. (b) Eigenenergy solutions for all bound states in the quantum well of (a), with the colors denoting different angular quantum numbers. (c) Sample eigenstate wave functions for $(n,m)=(0,0)$, (4, 0), (1, 1), (4, 1), (1, 5), and (2, 7), respectively.

Figure 2

The interaction terms that are ignored in the tight-binding Hamiltonian: (a) the density-dependent hopping and (b) the scattering terms involving more than two orbitals with different site energies.

Figure 3

On-site interactions within one quantum dot: Hubbard $U$, interorbital Hubbard $U^{\prime }$ (and its spin-antiparallel form ${\overline{U}}^{\prime }$), and Hund's exchange $J$ (and its spin-antiparallel form $\overline{J}$).

Figure 4

Long-range interactions between two quantum dots: direct Coulomb interaction $V$, long-range Hund's exchange $K$, correlated on-site exchange $V^{\prime }$, and correlated off-site exchange $K^{\prime }$.

Figure 5

Cartoon for multiorbital Nagaoka transition in a four-dot system. For moderate effective interaction, the multiplets in each quantum well form an overall low-spin state, with total spin $S=1/2$ (left). In contrast, a large interaction relative to the tunneling gives a Nagaoka FM state (right). The shaded surfaces denote the potential wells, while the white dots denote the single-well energy levels, which are slightly shifted according to different angular quantum numbers. The spin configuration is a conceptual sketch instead of a realistic solution.

Figure 6

(a) Ground first excited-state level spacing, (b) ground-state interaction $U$, and (c) effective interaction as a function of the depth $V_0$ and the width $\mathrm{\Sigma }$ of quantum well. The calculation is obtained on a single quantum well without hybridization.

Figure 7

The effective hopping $t$ estimated by a quarter of the single-particle bandwidth calculated for various distances in a $2\times 2$ plaquette.

Figure 8

(a) The Nagaoka gap and (b) the ground-state energy of three electrons in four quantum dots, as a function of the distance $d$. The red open circles denote the low-spin ground states, while the blue dots denote the high-spin ground states. The size of the data points reflects the energy difference between the lowest low-spin and high-spin states in a logarithmic scale.

Figure 9

(a) Schematic illustrating the potential detuning applied on one of the four quantum dot potential wells. (b) The ground-state energy for the entire system as a function of the potential detuning $dV$, calculated for various distances $d$. The gray lines denote the energy drop with slope 1.

Figure 10

(a) The first three excited-state energies in high-spin (blue) and low-spin (red) sectors. The arrow denote the region of Nagaoka phase. (b) The Nagaoka gap and (c) an enlarged energy evolution for the dashed boxed region in (a).

Figure 11

(a) Schematic illustrating the bond rotation in a $2\times 2$ system. (b) The ground-state energy for the entire system as a function of the rotating angle $\theta$, calculated for various distances $d$. (c) The Nagaoka gap and (d) the first three excited-state energies in high-spin (blue) and low-spin (red) sectors for $d=210\mathrm{nm}$. The arrow denotes the region of Nagaoka phase.