Itinerant Ferromagnetism from One-Dimensional Mobility

TL;DR

This work addresses the origin of half-metallic ferromagnetism in systems with spin-independent Coulomb interactions by showing that constrained one-dimensional mobility, combined with strong on-site repulsion ($U= abla\infty$), generates only even-parity multi-spin ring exchanges that, via the Thouless rule, stabilize ferromagnetism. Using a solvable Lieb-lattice model and connections to the Emery model in the strong-coupling limit, the authors demonstrate a unique, half-metallic ground state for arbitrary doping, with a mapping to non-interacting wires that yields a gapless Fermi surface. They also prove an exact boson–fermion equivalence for these constrained-1D systems and analyze a quasi-1D vacancy scenario in a Wigner crystal, where a similar ferromagnetic metallic ground state and its low-energy excitations emerge. The results suggest a universal kinetic mechanism for itinerant half-metallic ferromagnetism across geometries and densities, with implications for Bose–Fermi analogs and potential experimental realizations.

Abstract

We propose a universal kinetic mechanism for a half-metallic ferromagnet -- a metallic state with full spin polarization -- arising from strong on-site Coulomb repulsions between particles that exhibit constrained one-dimensional (1D) dynamics. We illustrate the mechanism in the context of a solvable model on a Lieb lattice in which doped electrons have 1D mobility. Such 1D motion is shown to induce only multi-spin ring exchanges of even parity, which mediate ferromagnetism and result in a unique half-metallic ground state. In contrast to the Nagaoka mechanism of ferromagnetism, this result pertains to any doped electron density in the {\it thermodynamic} limit. We explore various microscopic routes to such (approximate) 1D dynamics, highlighting two examples: doped holes in the strong-coupling limit of the Emery model and vacancies in a two-dimensional Wigner crystal. Finally, we demonstrate an intriguing exact equivalence between the bosonic and fermionic versions of these models, which implies a novel mechanism for the conjectured Bose metallic phase.

Figures (11)

• Figure 1: (a) The single-hole problem on an $(N+1)$-site ring \ref{['eq:pinning potential']} with a pinning potential at site $i=0$. The counter-clockwise motion of the hole around the ring induces a multi-spin ring exchange ${\cal P}_N^{-1}$ in a clockwise direction \ref{['eq:N+1 perturbation']}. (b) On a bipartite lattice, single-hole motion around a loop (shown in magenta) induces a ring exchange involving an odd number of electrons.
• Figure 2: (a) A solvable $U=\infty$ model \ref{['eq:solvable model']} on a Lieb lattice under the OBC with dangling bonds on the boundary. $d$ sites are assumed to be half-filled (one spinful electron per site), whereas the filling on $p$ sites can vary. Here, blue sites are singly occupied and white sites are unoccupied. $T_{\pm x}$ ($T_{\pm y}$) are the correlated hopping processes in $\pm x$ ($\pm y$) direction. (b) A three-spin ring exchange (or an elementary 3-cycle; red box) induced by a sequence of the 1D hopping processes. The site in the middle is a $d$ site and the neighboring sites are $p$ sites.
• Figure 3: Fermi surfaces of the half-metallic ground state of the model \ref{['eq:solvable model']}. The black boundaries denote the Brillouin zone; the region interior to the magenta (blue) lines are occupied by $p_x$ ($p_y$) electrons. The bosonic version of the model also exhibits the same Fermi surfaces, as demonstrated in Sec. \ref{['sec:Exact Boson-Fermion Correspondence']}
• Figure 4: (a) Parameters in the Emery model as defined in the text around (\ref{['eq:H_0']}-\ref{['eq:H_1']}). The hopping term $t_{pp}=0$ is neglected in this work. $\tilde{n}_{\textbf{R}}^{1,1}$ denotes the number of holes on a specified triangle (dashed gray line) as defined in \ref{['eq:triangle']}. (b) Two types of effective hopping processes in the semi-classical limit of the Emery model. The first process incurs a huge intermediate energy cost of $V_{pp}+\epsilon$, resulting in a negligible hopping term $t'$ in the strong-coupling limit. The second process is the only feasible process in the strong-coupling limit, granting holes quasi-1D mobility as described by \ref{['eq:effective Hamiltonian']}.
• Figure 5: Classical (isotropic and nematic) and quantum (quantum nematic metal and isotropic half-metal) phases of the Emery model in the strong-coupling and semi-classical limit, described by $H_{\rm eff}$\ref{['eq:effective Hamiltonian']}. The black solid and dotted lines (taken from Ref. binder1980Ising) mark the classical continuous nematic-isotropic phase boundary for $t=0$, which terminates at $x_c(T\to0) \approx 0.736$. Quantum effects [$t>0$] modify the finite-temperature phase diagram, as the isotropic phase may become energetically favorable over the nematic phase due to reduced Fermi pressure; the double black line indicates this quantum-modified nematic-isotropic phase boundary, replacing the dotted line. Near $x=0$ and $x=1,$ quantum nematic phases are stabilized, becoming fully nematic at $T=0$ with complete spin degeneracy. Note that the distinction between quantum and classical nematic phases near $x=1$ is evident only at $T=0$, where any non-zero $t>0$ converts partial nematicity into full nematicity. In contrast, the nematicity near $x=0$ is purely a quantum effect. Away from these regions, the isotropic phase is expected to be the ground state and is characterized by full spin polarization at $T=0$.

Theorems & Definitions (11)

• Proposition 2.1
• Theorem 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Proposition 4.1
• Theorem 4.2
• Proposition 4.3
• Theorem 4.4
• Theorem 5.1