Ferromagnetism and Borromean Binding in Three-Fermion Clusters

A three-particle spin-$\frac{1}{2}$ fermion problem with on-site repulsion and nearest-neighbor attraction is solved on the two-dimensional square lattice by discretizing a Schrödinger equation in momentum space. Energies of bound complexes (trions) and their binding conditions are obtained. For total spin $S=1/2$, a wide region of trion instability toward decaying into a stable singlet pair plus a free fermion is identified. The instability is attributed to the formation of a wave function node upon addition of the third fermion. In the $S=3/2$ sector, trions are found to form in the absence of bound pairs indicating Borromean binding. In the strong coupling limit the system transitions from an $S=1/2$ ground state to a ferromagnetic $S=3/2$ ground state in agreement with the Nagaoka theorem for a four-site plaquette.

Figure 1

Total spin $S=3/2$. (a) Six basic particle configurations in the strong and medium coupling regimes. (b) A typical six-band spectrum of $S=3/2$ trions for $V=10t$ (solid lines). Also shown are the lowest three free fermion energy $E_{\{111\}}=-6t\cos (P_x/3)-6t\cos (P_y/3)$ (dotted-dashed line), and the lowest energies of one pair plus one free fermion (dashed lines) for the same total $\mathbf{P}$. (c) The same trion spectrum at a larger scale. Only five lowest bands are shown. (d) $\mathbf{P}=(0,0)$ trion energies compared with the lowest $\{2p1\}$ energies. The two middle trion states are double degenerate.

Figure 2

Total spin $S=1/2$. (a) The twelve basic “corner” $S_z=1/2$ configurations of the strong coupling regime. (b) The trion spectrum for $V=U=10t$. (c) $\mathbf{P}=(0,0)$ trion energies vs $V$ for $U=10t$. The first, fifth, and ninth states are double degenerate. For $\frac{10}{9}<V<3.425$ singlet $s$ pairs are stable, whereas trions are unstable.

Figure 3

Main panel: the Nagaoka boundary line for $\mathbf{P}=(0,0)$. The solid line is Eq. (5). Inset: the difference between the lowest $S=1/2$ energy and the lowest $S=3/2$ energy as a function of $V$ for $U=30t$. The zero crossing at $V/t=11.285$ indicates a change in the ground state spin from $S=1/2$ to $S=3/2$.

Figure 4

Phase boundaries of two-fermion states [for $\mathbf{K}=(0,0)$] and three-fermion states [for $\mathbf{P}=(0,0)$] in the two-dimensional $UV$ model. Bound pairs and trions form above the respective lines. The singlet pair line is $V_s$ from Eq. (2). The triplet pair line is $V_p$ from Eq. (3). The $S=3/2$ line is $V=4.05t$ (independent of $U$). The Nagaoka line is a fragment of Fig. 3.