Staggered orbital magnetization from itinerant electrons: orbital antiferro- and ferrimagnetic phases

Abstract: Because electronic orbital angular momentum in solids is inherently non-local, its contribution to magnetism is usually cast in terms of a net orbital magnetization. Here, we show that itinerant electrons can generate orbital magnetic phases with ferromagnetic, antiferromagnetic, or ferrimagnetic orders. We demonstrate this possibility in a honeycomb lattice, using both the standard and a modified Haldane model. Employing real-space formulations, we decompose the itinerant orbital magnetization into sublattice contributions, $M_A$ and $M_B$. Their net ($M_z=M_A+M_B$) and staggered ($M_z^s=M_A-M_B$) combinations are then used to identify the orbital order. By varying the sublattice potential and the Fermi energy, we find distinct regimes: a (PT)-symmetric orbital antiferromagnet in the modified Haldane model, an orbital ferromagnet in the standard Haldane model, ferrimagnetic metallic states where net and staggered orbital magnetizations coexist, and insulating regimes in which the ferro- and antiferromagnetic orbital characters can be interchanged. These findings are explained by a low-energy theory in terms of two distinct valley mechanisms: valley-dependent Dirac masses in the standard Haldane model and valley-dependent energy shifts in its modified version.

• The paper demonstrates that itinerant electrons in honeycomb lattices can exhibit sublattice-resolved orbital order, including OFM, OAFM, and OFiM phases.
• It employs both standard and modified Haldane models along with analytical low-energy theory and Chebyshev polynomial numerical methods to probe orbital magnetization.
• The results indicate tunable orbital magnetization with potential for orbitronic device applications and advances in two-dimensional material design.

Itinerant Orbital Magnetization and Staggered Phases on the Honeycomb Lattice

Introduction

The paper "Staggered orbital magnetization from itinerant electrons: orbital antiferro- and ferrimagnetic phases" (2607.10355) systematically explores the emergence and classification of sublattice-resolved itinerant orbital magnetic orders—ferromagnetic (OFM), antiferromagnetic (OAFM), and ferrimagnetic (OFiM)—within honeycomb-lattice systems described by the standard and modified Haldane models. Through the modern theory of orbital magnetization and advanced real-space numerical techniques, the analysis reveals that orbital magnetic order, typically discussed in connection with net orbital moments or orbital currents, can manifest compensated (staggered) and partially compensated patterns, analogous to spin-based antiferro- and ferrimagnetism, in itinerant electronic systems.

Models: Standard and Modified Haldane Hamiltonians

The study utilizes two minimal two-band lattice models:

1. Standard Haldane model: Exhibits a topological phase with Chern insulating behavior due to a complex phase on next-nearest-neighbor (NNN) hopping; this mass term acts as a valley-dependent Dirac mass.
2. Modified Haldane model: Differs only in the convention for the complex hopping phase; it produces a valley-dependent energy shift and supports antichiral edge states.

The main structural distinction is in the phase factor $\nu_{ij}$ assigned to NNN hoppings: the standard Haldane convention assigns opposite phases for sublattices $A$ and $B$, whereas the modified version uses the same phase for both.

Figure 1: Convention for the complex phase $\phi$ in the $t_2$ NNN term for (a) standard and (b) modified Haldane models, and the corresponding gap structures over $\Delta/t_2$–$\phi$ parameter space.

The bulk band topology, gap closing transitions, and symmetry operations ($P$, $T$, $PT$) are fully controlled by $A$0, $A$1, and $A$2, which enables tuning between distinct orbital magnetic orders.

Real-Space and Momentum-Space Formulation of Orbital Magnetization

Itinerant orbital magnetization is inherently non-local. Unlike atomic orbital moments, its proper description leverages Berry-phase and Fermi-energy contributions via the modern theory of orbital magnetization. The real-space formulation uses sublattice-resolved operators, whose sum ($A$3) and difference ($A$4) probe net and staggered orbital order, respectively.

Numerical evaluations are performed with the Chebyshev polynomial spectral method, which allows computation of $A$5 and $A$6 for finite systems with full inclusion of bulk and edge contributions.

Phase Diagram: Sublattice-resolved Orbital Magnetization

For $A$7, the orbital order is dictated by the model's symmetry:

• Standard Haldane model: $A$8 at all Fermi energies, leading to finite $A$9 (net OFM), $B$0.
• Modified Haldane model: $B$1, so $B$2 (OAFM), $B$3.

Upon breaking sublattice equivalence ($B$4), both models exhibit $B$5, leading to nonzero $B$6 and $B$7 (OFiM). This is shown for weak sublattice potentials $B$8:

Figure 2: Variation of $B$9, $\phi$0, $\phi$1, $\phi$2, and DOS with Fermi energy for both models at $\phi$3 and $\phi$4.

In the regime $\phi$5, the orbital character reverses inside the gap:

• Standard Haldane: OAFM plateau ($\phi$6, $\phi$7) emerges despite ferromagnetic character at $\phi$8.
• Modified Haldane: OFM plateau ($\phi$9, $t_2$0) emerges despite antiferromagnetic character at $t_2$1.

  Figure 3: Sublattice-resolved orbital magnetizations and DOS at $t_2$2, $t_2$3; each plateau reveals model-dependent reversal between OFM and OAFM order within the insulating gap.

Theoretical Interpretation: Valley-Dependent Mechanisms

The analytical low-energy theory elucidates the distinct mechanisms behind the observed plateaus:

• In the standard Haldane model, the valley-dependent Dirac mass leads to net Berry curvature-induced orbital magnetization that cancels between valleys in the trivial insulator. However, sublattice projection reveals a finite staggered ($t_2$4) response.
• In the modified Haldane model, only valley-dependent energy shifts are present, producing a finite net ($t_2$5) orbital magnetization plateau even when the system is trivial and insulating.

Analytical calculations show that, for models with sublattice symmetry and $t_2$6, the magnitude of the plateau is $t_2$7, where the sign and nature (staggered or net) is dictated by the valley structure inherent to the Hamiltonian.

Symmetry Analysis

Order parameters correlate with symmetry properties:

• OFM: $t_2$8, $t_2$9; breaks $\Delta/t_2$0 and $\Delta/t_2$1, preserves $\Delta/t_2$2
• OAFM: $\Delta/t_2$3, $\Delta/t_2$4; breaks $\Delta/t_2$5 and $\Delta/t_2$6, preserves $\Delta/t_2$7
• OFiM: $\Delta/t_2$8, $\Delta/t_2$9; all symmetries broken

The symmetry classification aligns precisely with the low-energy valley analysis, and with the phase mapping of orbital and spin ferro-, antiferro-, and ferrimagnetism.

Density of States and Orbital Magnetization

The density of states (DOS) is systematically linked to the onset and nature of the orbital magnetic order. For metallic regimes with nonzero DOS at the Fermi level, all three types (OFM, OAFM, OFiM) can appear depending on the model and parameter values.

Figure 4: Sublattice and total/staggered orbital magnetization and DOS, demonstrating the interplay between sublattice-resolved responses and spectral characteristics at $\phi$0, $\phi$1 eV, $\phi$2 eV.

Implications and Outlook

These findings demonstrate that itinerant models, without localized atomic moments, can exhibit robust ferro-, antiferro-, and ferrimagnetic orbital order, which can be controlled via sublattice potentials and complex electronic hoppings. The results establish a rigorous framework for classifying and engineering orbital order in multi-sublattice and multi-valley systems, advancing the fundamental understanding of itinerant magnetism beyond the standard spin paradigm.

Practically, these insights have substantial implications for orbitronics: they suggest routes for manipulating and detecting staggered or net orbital angular momentum, impacting potential device architectures where edge currents, orbital torques, or compensated magnetization are desired. Direct applications to systems such as graphene-based heterostructures, engineered photonic lattices, and two-dimensional transition metal dichalcogenides are likely, with further relevance to novel transport and magnetoresistance phenomena.

Theoretically, the equivalence classes mapped out here could inform the design of new materials and artificial lattice systems (e.g., moiré materials, optical lattices) with targeted orbital magnetic responses and tunable symmetry breaking, potentially impacting the ongoing search for altermagnetic, multiferroic, and other unconventional magnetic phases.

Conclusion

The study experimentally and theoretically demonstrates that itinerant, sublattice-resolved orbital magnetization can be tuned to realize analogues of spin ferro-, antiferro-, and ferrimagnetism on the honeycomb lattice. These results generalize the modern theory of orbital magnetization to encompass compensated and partially compensated itinerant order and provide a foundation for future work on orbitronic functionalities in multivalley and multi-sublattice materials (2607.10355).