Scientists are increasingly recognising Van Hove singularities as key drivers of novel phase transitions in materials. Chen Lu and Lun-Hui Hu, from their respective institutions, alongside colleagues, demonstrate that these singularities within multi-orbital systems can stabilise diverse and competing magnetic orders, including a previously unobserved form of intrinsic altermagnetism arising from spontaneous orbital antiferromagnetism. This research is significant because it reveals that intrinsic altermagnetism, where antiparallel spins occupy separate orbitals, can be realised across all two-dimensional crystal structures, offering a generic route to this phenomenon in correlated materials and potentially paving the way for new spintronic technologies. They map several magnetic transitions via Hubbard model phase diagrams, linking these behaviours to specific electronic conditions near Van Hove singularities.
This work demonstrates that VHSs within multi-orbital systems can stabilize a variety of competing magnetic orders, notably including an intrinsic altermagnetism originating from spontaneous orbital antiferromagnetism.
This intrinsic phase, characterized by antiparallel spins residing on distinct orbitals, is remarkably realized across all four two-dimensional Bravais lattices. The emergence of this phase is driven by orbital-resolved spin fluctuations, enhanced by inter-orbital hopping, and favored by suppressed Hund’s coupling, strong inter-orbital hybridization, and electronic filling near a VHS arising from quadratic band touching.
Researchers mapped several magnetic phase transitions through Hubbard-U, JH phase diagrams, specifically observing transitions from ferrimagnet to d-wave extrinsic altermagnet, from d-wave intrinsic altermagnet to ferromagnet, and from g-wave extrinsic altermagnet to either d-wave extrinsic altermagnet or ferromagnet. These findings establish a clear connection between VHSs and the stabilization of altermagnetism in correlated materials, offering a generic route to achieve this exotic magnetic state.
The study began with a symmetry analysis establishing orbital antiferromagnetism as a mechanism for altermagnetism in two-dimensional systems. Considering a 2D Bravais lattice with locally degenerate orbitals, the research identified how point-group symmetries, combined with time-reversal symmetry, govern the symmetry-breaking patterns of orbital antiferromagnetic order.
This analysis revealed that the combination of P · T (parity-time reversal) and t · T (translation-time reversal) symmetries are broken in all cases, a hallmark of altermagnetism. Furthermore, the work details how specific lattice types influence the realization of altermagnetism, with oblique lattices proving unsuitable due to a lack of symmetry constraints needed for zero net magnetization.
Conversely, rectangular, centered rectangular, and square lattices support dx2−y2-wave altermagnetism, while square lattices also exhibit the possibility of a distinct altermagnetic phase. These detailed investigations provide a comprehensive understanding of the conditions under which VHSs can induce and control altermagnetic behavior in correlated materials, potentially paving the way for novel spintronic devices.
Renormalised susceptibility calculations define magnetic phase diagrams and critical behaviour
Random Phase Approximation (RPA) calculations underpinned the investigation of magnetic phase transitions in multi-orbital systems. Specifically, the research determined the RPA-renormalized static susceptibility for the order parameter Oα, expressed as χRPA α (k) = 1 2X l1l2l3l4 [ Oα]l1l2[ Oα]l3l4[χRPA spin ]l1l2 l3l4.
The spin susceptibility tensor, χRPA spin, was obtained via a Dyson equation, enabling comparison of susceptibility channels for competing magnetic orders. Critical interaction strength, Uc, and ordering wavevector, Q, were identified as key parameters defining the magnetic/non-magnetic phase boundary and spatial periodicity of the dominant magnetic phase.
To systematically map magnetic transitions, a three-step procedure was implemented. First, parameter sets maximizing χRPA α (k) at the Γ-point for a target order Oα were identified. Next, the critical interaction strength, Uc, was determined.
Finally, the dominant magnetic orders near criticality were established. Interaction phase diagrams were constructed in the U, JH plane, exploring systems exhibiting intrinsic or extrinsic altermagnetism. Calculations began with band parameters detailed in Figure 1(c), setting U = 1 and JH = 0.2 to examine the momentum-resolved RPA susceptibility χRPA α (k).
Analysis revealed a pronounced peak in the dominant susceptibility channel, χRPA 4 (k), at the Γ point, indicating that the O4 order, extrinsic altermagnetism, became the leading instability. A computationally simpler approach, relying on the susceptibility difference χRPA Γ,3 −χRPA Γ,4, confirmed these findings. Intrinsic altermagnetism, emerging from spontaneous orbital antiferromagnetism, is realized across all four two-dimensional Bravais lattices within this study. This phase is driven by orbital-resolved spin fluctuations enhanced by inter-orbital hopping and favors suppressed Hund’s coupling, strong inter-orbital hybridization, and filling near a VHS from quadratic band touching.
Hubbard-U, JH phase diagrams map several magnetic phase transitions including a ferrimagnet to extrinsic altermagnet transition, an intrinsic altermagnet to ferromagnet transition, and an extrinsic altermagnet transitioning to either another extrinsic altermagnet or a ferromagnet. Transitions among four distinct phases, designated O1 through O5, are mediated by variations in hopping parameters.
Specifically, a transition from the d-wave intrinsic altermagnet O1 to the g-wave extrinsic altermagnet O5 was observed by decreasing t1,2 while increasing t3,5. The research identifies that the Q = 0 orders emerge precisely as the chemical potential approaches the VHS, highlighting its crucial role in stabilizing these magnetic phases.
At low temperatures, a divergence at Q = 0 requires three simultaneous conditions: a finite numerator indicating opposite band occupancy, vanishing band energies approaching the Fermi level, and a sufficiently high density of states at k1-points. The condition for a high density of states is automatically satisfied when the Fermi level is tuned near a VHS at high-symmetry points, implying the band structure exhibits Dirac semi-metal character.
High-DOS “hot spots” emerge in momentum space as the Fermi level approaches a VHS, with Q = 0 identified as the dominant nesting vector linking these hot spots when the VHS is located near a single high-symmetry point. Intra-VHS contributions to the Lindhard function scale as 1/(2t1k2) + 1/(2t2k2), exceeding the inter-VHS rate of 2/(t1k2 + t2k2) by the harmonic-arithmetic mean inequality as k approaches zero.
Consequently, the susceptibility consistently peaks at Q = 0 in semi-metal systems with Fermi levels near VHSs. Slight shifts in filling away from the VHSs cause the random-phase-approximation susceptibilities to rapidly shift their divergence to nonzero Q. Investigations into Hubbard models reveal several magnetic phase transitions, including transitions between ferrimagnetic, extrinsic and intrinsic altermagnetic, and ferromagnetic states.
These transitions are highly tunable, governed by both kinetic energy, represented by hopping parameters, and interaction strengths, specifically the Coulomb interaction and Hund’s coupling. Future work could focus on mapping out the full parameter space to provide a comprehensive understanding of these transitions and their potential for material design.