Nagaoka Ferromagnetism in $3\times 3$ Arrays and Beyond

Nagaoka ferromagnetism (NF) is a well-known result of the Hubbard model, where Nagaoka [1] proved rigorously the existence and uniqueness of a saturated, ferromagnetic ground state in a single-band Hubbard model on a lattice. In the case of one hole in a half-filled band and infinite, repulsive, same-site Coulomb interaction $U$ between opposite spins, the ground state has the maximum total spin $S$. This comes from a non-trivial interplay between quantum dynamics and Coulomb interaction, where the kinetic and repulsive energy terms compete resulting in a ferromagnetic ground state. This competition is determined by how the single hole can move around the lattice.

NF is one of the exact results for itinerant ferromagnetism (IF) and has been studied theoretically for years in various conditions [2, 3]. However, the specific conditions proved by Nagaoka do not arise in natural materials and was observed experimentally for the first time in an engineered $2\times 2$ quantum dot plaquette [4]. It remains an open question when NF can be extended to larger, finite-systems that have internal as well as edge sites, how NF would behave in realistic conditions where $U$ is large but not infinite, and how NF is affected by long-range interactions or disorder.

Following Nagaoka’s original work, many have tried to generalize Nagaoka’s result. Tasaki provided a proof of a generalized version of Nagaoka’s theorem [2]. Mielke and others found a new class of ferromagnetic states in the Hubbard model which they called ”flat-band ferromagnetism” [5]. The possibility of extending NF to finite $U$ with finite densities of holes has also been studied for different infinite lattices [6, 7], for large 2D lattices [8], and for square lattices [9]. also see [10] for Nagaoka polaron regime for finite hole density. There have also been many theoretical works done on the instability of NF, that show when NF does not take place [11, 12, 13]. Additional theoretical work has elucidated the polaronic correlation around holes in NF [10, 14, 15].

In recent years, as computational techniques have advanced, there has been more discussion of small, finite systems that exhibit NF. For example, finite lattices with up to 24 sites with periodic boundary conditions were studied using quantum Monte Carlo [16, 17]. Different geometries of finite systems were studied using exact diagonalization techniques (ED) [18, 19]. These studies revealed the importance of geometry for realizing NF in finite systems, which could be achievable with current experiments using gated quantum dot or dopant arrays.

Quantum dots, also known as artificial atoms, can be used to perform simulations beyond what classical computers can do [20, 21, 22, 23]. Among various experimental platforms, dopant-based quantum dots have many advantages, such as the precise placement of dopant atoms and tunable site-to-site hopping, making them a strong candidate for simulating strongly correlated systems, like the Hubbard model in the large $U$ limit. Silver recently fabricated $3\times 3$ arrays of single/few-dopant quantum dots and demonstrated analog quantum simulation of a 2D Fermi-Hubbard model with tunable parameters [24]. This was the first two-dimensional quantum-dot simulation with internal sites. This provides an experimental pathway to probe NF in $3\times 3$ arrays and in many other finite geometries. Gated semiconductor quantum dots show similar promise. The original effort to study NF in a $2\times 2$ plaquette [4] has been extended to study exciton and spin dynamics in $2\times 4$ ladders. [25, 26] Linear chains formed from multidopant quantum dot clusters has been used to study topological states in the Su-Schrieffer-Heeger model. [27]

In this paper, we simulate $3\times 3$ dopant arrays in the conditions required for NF, i.e. one less electron than a half-filled band and in the large $U$ limit. We predict that NF will exist in a $3\times 3$ array, as in a $2\times 2$ array, as well as in $N\times N$ arrays, for a finite region of $t/U$, where $t$ is the nearest-neighbor hopping. For $t/U$ below the transition point, the ground state of the system has maximum total spin $S$, i.e. the ground state is ferromagnetic. Since the direction of the ferromagnetism is arbitrary, the system energy only depends on total spin $S$ but not the spin in the $z$-direction ($s_{z}$). The ferromagnetic state with maximum $s_{z}=S$ with all of the spins aligned in the same $z$ direction is degenerate with all other ferromagnetic states with smaller $s_{z}$ as long as they have the same total $S$. The ground state is $(2S+1)$-fold degenerate when there is Nagaoka ferromagnetism. Above the transition point, the ground state is no longer the maximum-total-spin state and is no longer ferromagnetic. We also simulate variations of the perfect $3\times 3$ array to determine when NF is robust to experimental impurities or disorder.

The quantum nature of magnetism plays an essential role in determining how and when ferromagnetism occurs. The small-scale simulators that we consider in this paper provide an excellent testbed for investigating quantum effects. We find that the connectivity of the many-body wavefunction plays a key role in determining when NF can occur. Wavefunction connectivity also determines when forms of itinerant ferromagnetism different from NF at fillings other than one hole in a half-filled band can occur. Band filling is another key condition that determines the type of ferromagnetism. We will show that very different fillings, related to the number of holes in a half-filled band or the number of electrons in an empty band, will define the possibility of ferromagnetism in similar, related structures. The competition between kinetic energy and coulomb repulsion effects determines where in $t/U$ space the transition to ferromagnetism occurs. The $t/U$ ratio at the transition can increase or decrease with system size, depending on the array geometry (and the related wavefunction connectivity), and the band filling. We develop simple scaling arguments to explain both types of size-dependence and the critical role that the kinetic energy plays, through its dependence on band filling, in determining the form of the scaling. The role of these quantum effects will be highlighted as we discuss NF in different small arrays to show how quantum simulation with small arrays could reveal these critical effects. It should be emphasized that dopant arrays, once they can be made perfectly [28] would be a versatile solid-state platform to explore the diversity of arrays and the range of effects that we discuss. Gated quantum dots will also be a versatile solid-state platform as techniques to control dots in complex geometries are developed.

The paper is structured as follows: In Section II, we discuss the 2D Hubbard model, the signatures of finding NF in finite systems, and how the systems are modelled in our simulations. In Section III, we present our simulation results for the $3\times 3$ and $N\times N$ square arrays, which exhibit NF up to a finite $t/U$ transition point that decreases with increasing $N$. In Section IV, we describe the robustness of NF in a $3\times 3$ array with disordered sites, that is sites that have been removed. Most importantly, we show that a loop array, i.e. an $N\times N$ with the internal sites removed, possesses a ferromagnetic ground state for three-electron filling with a $t/U$ transition that scales linearly with loop length $N$, in stark contrast to NF in perfect, square arrays. In this section, we also present results to show how the $t/U$ ratio at the transition to ferromagnetism scales with system size $N$. Scaling arguments explain this dependence on $N$ and elucidate the competition between kinetic energy and repulsive interaction affects to determines when the transition occurs. Finally in Section V, we summarize and discuss essential results about quantum magnetism that can be revealed by simulations done on small arrays of dopants or quantum dots.

NF occurs for electrons described by the Hubbard model, which takes the form as follows

where $t$ is the hopping between nearest neighbor sites, $c^{{\dagger}}$ and $c$ are the creation and annihilation operators, $n$ is the particle number operator, and $U$ is the on-site interaction between spin-up and spin-down electrons. NF is determined by a competition between hopping energy and repulsion, hence the parameter that matters is $t/U$. Some use the hopping term $t$ with a minus sign, but here we use the convention without a minus sign and allow the value of $t$ to be positive or negative. The sign of $t$ is important because in some systems the Nagaoka theorem can be proved for one sign but not the other. The sign of t can often be changed with a basis change. In particular, the transformation between the signs of $t$ can be done for a bipartite system. In these cases, NF, if it occurs, will occur for both signs of $t$. The $N\times N$ arrays we study are bipartite lattices.

To find out if a system has NF, it is necessary to determine whether the one-hole ground state has maximum total spin. For a finite system like the $2\times 2$ array with 3 electrons, the Hamiltonian is small and the energy spectrum can be calculated directly, as shown in [4]. The energies for the two different total spins, $S=3/2$ and $S=1/2$ are plotted against the parameter $t/U$ in Fig. 1a. Both energies are shifted with respect to the energy of the maximum-total-spin state $E_{S=3/2}=-2t$. Below $|t/U|\approx 0.053$, the ground state of the system is the maximum-total-spin $S_{max}=3/2$ state. In the ferromagnetic ground state, the states with $S=S_{max}$ have a degeneracy of $2S_{max}+1$. In the $2\times 2$ array with 3 electrons, there are four degenerate $s_{z}$ states with $s_{z}=-3/2,-1/2,1/2,3/2$. Because the Hubbard model does not include spin-mixing, ED can be done for each $s_{z}$ subspace, thereby reducing the size of each diagonalization. In our simulation, instead of plotting the ground state energy for each total spin $S$, we plot the ground state energy for the different spin-$z$ subspaces as shown in Fig. 1b, and shift them by the energy of the state with maximum $s_{z}$ $E_{s_{z}=3/2}=-2t$. Below the transition when the system is in the NF regime, the ground state is $4$-fold degenerate. The signature of NF is when the ground states of the spin-$z$ subspaces become degenerate with each other. This guarantees that the ground state of the system has maximum total spin.

Fig. 1 shows how energy behaves as $|t/U|$ varies. The lattice is bipartite so the sign of $t$ does not matter here, and $t$ and $-t$ give the same energies. This is not always the case as seen later when the system is no longer bipartite, and the $\pm t$ cases can behave very differently. In Fig. 1b, the $s_{z}=\pm 1/2$ state is always a ground state, but the ground state after the transition has a different total $S$ and degeneracy compared to before the transition. This change in total spin can be verified by direct calculation of the total spin for each spin $z$.

The existence of NF in $2\times 2$ array has been verified experimentally in [4] for a $2\times 2$ gated quantum dot plaquette. An ab initio calculation of NF in quantum dots was done in [29]. Its robustness was discussed in [29] and theoretically in [18].

The sites in a semiconductor array need not be identical. The site energies can be controlled by bias. In dopant systems, disorder in dopant position and dopant number can affect the tunnelling and on-site energies. For example, there can be more than one dopant clustered near a site. The properties of that disordered site will be determined by the multidopant cluster which will have a different on-site energy. It is harder for electrons to hop on and off a disordered site with a shifted on-site energy, lowering the participation of that site in the array states. When the energy of the disordered site is sufficiently far off-resonance from the other sites, the disordered site is effectively removed from the array. To investigate the robustness of NF in fabricated $3\times 3$ arrays with disordered sites, we consider $3\times 3$ arrays with one site missing to see if ferromagnetism survives or disappears, and, if ferromagnetism survives, what electron fillings display the ferromagnetism. This will provide us insight into how and when NF exists in small finite systems with variations in on-site energies. In this paper we look at the limit when the on-site energy difference is large enough to remove the site. In an upcoming paper, we will show how the ferromagnetism evolves as on-site energy differences vary.

For the $3\times 3$ array, there are three different types of site, the corner, edge, and center sites, corresponding to having 2, 3, or 4 hoppings connecting it to other other sites. Removing one site of each type of atom is shown in Fig. 6. We discuss these three cases in the next three sections. We also discuss ferromagnetism in other finite structures that can be made by removing similar sets of sites in larger $N\times N$ arrays. We summarize the results for removing sites from the $3\times 3$ array and other extended structures in Table 2.

In this work we have presented theoretical simulations done to probe Nagaoka ferromagnetism in arrays with different geometries. Starting from the $2\times 2$ plaquette, which was experimentally realized in [4], we have shown that perfect $3\times 3$ and $N\times N$ arrays up to $8\times 8$ exhibit NF for one-hole in a half-filled band, with a $t/U$ ratio for the onset of NF that decreases as the array size N increases, with the transition scaling as $1/n^{4}$. So far, 2D dopant arrays have only been studied for the $3\times 3$ configuration. However, recent advances in fabrication of dopant structures give significant promise for making highly engineered dopant arrays that will be an excellent testbed as an analog quantum simulator for this quantum magnetism .[28] Extending coupled quantum dot systems to large, complex and extended arrays shows similar promise.

The existence of Nagaoka ferromagnetism or other forms of itinerant ferromagnetism in small arrays crucially depends on the array geometry. One of the geometrical, necessary criteria is connectivity, not only in physical space, but also in the many-body spin-configuration space, which requires that any configuration of spins in the array can be transformed to any other configuration by a series of nearest neighbor hoppings without double occupancy. In $N\times N$ arrays, at least one hole is needed in a half-filled band for wavefunction connectivity at that filling. The $N\times N$ array has saturated ferromagnetism at this filling. A chain of $2\times 2$ square plaquettes with adjacent plaquettes connected at a common corner needs at least two holes in the half-filled band to support wavefunction connectivity and supports saturated ferromagnetism for small $t/U$. In contrast, $N$-site loops have wavefunction connectivity only for three-electron filling and saturated ferromagnetism for three-electron filling.

The robustness of the saturated ferromagnetism to disorder in $N\times N$ arrays has been tested by considering $3\times 3$ arrays with one site removed. When a corner site is removed, saturated ferromagnetism is still possible for one hole in a half-filled band (which now has one less electron). When one of the edge sites is removed from the $3\times 3$ array, saturated ferromagnetism is no longer possible because wavefunction connectivity is broken. When the center site is removed, the $3\times 3$ array becomes a loop. Saturated ferromagnetism is possible but only for three-electron filling. Higher electron fillings don’t have wavefunction connectivity. The transition between different realizations of ferromagnetism when a site is removed can be mapped out by varying the on-site energy of the site to be removed. The phase diagrams associated with these transitions will be discussed in a future publication.

Electron filling plays an essential role in determining the type of ferromagnetism that can occur and the scaling of the transition $|t/U|$ that marks the crossover to ferromagnetism. Loop ferromagnetism occurs for the same, fixed, small electron filling for all loop sizes $N$. The ferromagnetic state occurs for three electron filling but not for more electrons. Nagaoka ferromagnetism in $N\times N$ occurs when there is one hole in a half-filled band. As one consequence the scaling of the transition $|t/U|$ is linear in $N$ for loops and $1/N^{4}$ in arrays. Loop ferromagnetism occurs for a broader range of $U$ as $N$ increases, while for NF, the ferromagnetic state occurs for a smaller range of $U$ as $N$ increases. Scaling arguments support and elucidate the scaling obtained from the computational studies. Three-electron loop ferromagnetism occurs when the kinetic energy costs of becoming ferromagnetic is compensated by the elimination of repulsive interaction in the ferromagnetic state. For NF, the transition out of the ferromagnetic state occurs when the hopping energy cost of the hole, partially localized by scattering from the minority spins is compensated by the exchange energy that is gained when hopping to create doubly occupied states is allowed at finite $U$.

Interestingly, the first experiment to demonstrate Nagaoka ferromagnetism was for a $2\times 2$ plaquette. For $2\times 2$ plaquette, one hole in a half-filled band requires three electrons. However the $2\times 2$ array is also a loop that displays the three-electron ferromagnetism expected for loops. Ferromagnetism in $2\times 2$ plaquettes could be described as Nagaoka ferromagnetism or as three-electron loop ferromagnetism. These are very different, but it is not clear if one is a better characterization of the ferromagnetism of a $2\times 2$ plaquette.

Our results show that many forms of ferromagnetism should be possible in arrays with various geometries. Both small-scale and extended dopant and quantum dot arrays should be a rich playground for quantum simulators to explore quantum magnetism.