Altermagnetic Lieb-like Lattice Insights
- Altermagnetic Lieb-like lattice is a square-lattice-derived system exhibiting zero net magnetization and momentum-dependent d-wave spin splitting via sublattice interference.
- Its rich phase diagram spans metallic, Mott insulating, and higher-order topological regimes, driven by lattice geometry, electron correlations, and spin-lattice symmetry.
- Experimental realizations in materials like Mn2WS4 and V2O, along with optical lattice simulations, highlight its potential for spintronics, Hall responses, and excitonic applications.
An altermagnetic Lieb-like lattice is a square-lattice-derived geometry—ranging from the canonical three-site Lieb lattice to modified, inverse, decorated, and buckled variants—in which a collinear magnetic state with zero net magnetization produces momentum-dependent spin splitting, typically with -wave structure, through sublattice geometry and spin-lattice symmetry rather than uniform ferromagnetic exchange. In the current literature, this motif appears in weak-coupling metals, doped and undoped Mott systems, decorated Lieb-$5$ models, inverse Lieb compounds, and two-dimensional material realizations such as MnWS and VO, with consequences extending from nonrelativistic spin splitting and magnon chirality to higher-order topology, Hall responses, and excitonic selection rules (Dürrnagel et al., 2024, Biswas et al., 20 Jan 2026, Chang et al., 6 Aug 2025).
1. Lattice archetypes and symmetry setting
The canonical two-dimensional Lieb lattice is obtained by starting from a square lattice of lattice constant and removing every second site in a checkerboard pattern. It has three sites per unit cell, labeled , , and , with
and
$5$0
The $5$1 site sits at Wyckoff position $5$2, while $5$3 and $5$4 lie at Wyckoff position $5$5. Nearest-neighbor bonds connect $5$6–$5$7 and $5$8–$5$9 at distance 0, and 1–2 are next-nearest neighbors at distance 3. The point group is 4, under which 5 is invariant while 6 and 7 interchange under a 8 rotation (Dürrnagel et al., 2024).
| Archetype | Unit-cell motif | Distinctive control parameter or symmetry |
|---|---|---|
| Canonical Lieb | 9 at plaquette center; 0 at bond midpoints | 1, 2 under 3 |
| Modified Lieb | Three-site 4 lattice with magnetic 5 sector | 6 |
| Lieb-7 | 8 edge-midpoint sites and 9 corner site | 0 |
| Inverse Lieb | TM sites on the Lieb motif, anions in face centers | Inequivalent 1 |
| Buckled or materials realizations | Mn2WS3, V4O | 5, 6, buckling |
A modified Lieb lattice Hubbard model has been used as a model of quasi-2D oxychalcogenides with the so-called anti-CuO7 lattice structure. In that setting, the three-site 8 motif is retained, but the Hubbard repulsion acts only on the 9 sublattices and an on-site energy 0 controls the nonmagnetic 1 sites. A different decorated square-lattice realization is the Lieb-2 lattice, in which 3 and 4 occupy midpoints of perpendicular edges and 5 sits at the square corner, with intracell hopping 6, intercell hopping 7, and optional diagonal intracell hopping 8. The inverse Lieb lattice exchanges the crystallographic roles of transition-metal and anion sublattices relative to the usual Lieb motif, creating two inequivalent second-neighbor superexchange channels, 9 and 0 (Kaushal et al., 2024, Biswas et al., 20 Jan 2026, Chang et al., 6 Aug 2025).
Real-material realizations further broaden the category. Monolayer 1WS2 (3 Mn, Fe, Co) crystallizes in space group 4 (No. 111), point group 5, with magnetic 6 atoms at Wyckoff 7, W at 8, and S at 9, forming a Lieb-lattice-like square network. Monolayer V0O is reported in a buckled Lieb structure with space group 1 (No. 115), point group 2, and two V sublattices displaced along 3 by a buckling height 4 Å (Xu et al., 27 Jun 2025, Taşkıran et al., 7 Jun 2026).
2. Microscopic origin of altermagnetic order
In the weak-coupling metallic Lieb lattice, the central microscopic mechanism is sublattice interference rather than orbital ordering. At van-Hove filling, the noninteracting Fermi surface carries a strong sublattice texture: along 5 the Bloch states have weight solely on the 6 sublattice, while along 7 they sit purely on 8. This sublattice interference endows particle-hole scattering at the Fermi level with a nontrivial form factor and removes trivial SU(2) nesting symmetry. The dominant instability is then a 9-point magnetic fluctuation in the 0 irrep of 1,
2
which in real space gives
3
Because this order yields zero total magnetization while spin-splitting the bands, it is identified with the 4-wave, or 5, spin-Pomeranchuk instability now classified as altermagnetism (Dürrnagel et al., 2024).
In strongly correlated modified Lieb models, the mechanism can instead be traced to superexchange in a half-filled magnetic sector. At total filling 6 and large 7, the 8 sublattice is doubly occupied, 9, while 0 and 1 each host one electron, 2, yielding an antiferromagnetic superexchange
3
At 4, for 5, the 6 sites are empty and the 7 sector is again half-filled and antiferromagnetically coupled. In both cases the resulting single-particle splitting follows
8
with nodal lines 9 protected by $5$00 symmetry (Kaushal et al., 2024).
In inverse Lieb materials, a minimal collinear Heisenberg description is
$5$01
When $5$02 is antiferromagnetic, the altermagnetic phase occupies the entire quadrant with $5$03 and also a broad region around the origin when $5$04. Strong antiferromagnetic $5$05 couplings of order $5$06 instead drive geometrically frustrated stripe, orthogonal noncollinear, and block-checkerboard phases (Chang et al., 6 Aug 2025).
A common misconception is that altermagnetism on Lieb-type lattices necessarily requires orbital ordering or local-moment preselection at a higher scale. The weak-coupling Lieb-metal analysis explicitly identifies a route in which the ordering mechanism derives from sublattice interference at the Fermi level and does not involve orbital ordering (Dürrnagel et al., 2024). This suggests that, across the broader Lieb-like family, sublattice connectivity and connector symmetry are the decisive ingredients.
3. Hamiltonians, order parameters, and symmetry classification
The canonical itinerant model is a single-orbital Hubbard Hamiltonian on the Lieb lattice with nearest-neighbor hopping $5$07, next-nearest-neighbor hopping $5$08, site-dependent on-site energies $5$09, and repulsive $5$10: $5$11 Here $5$12 is the $5$13–$5$14 and $5$15–$5$16 hopping, $5$17 is the $5$18–$5$19 hopping, and $5$20 permits detuning of the central site. In the ordered phase the momentum-space mean-field structure is governed by the $5$21 form factor $5$22 and the real-space condition $5$23, $5$24 (Dürrnagel et al., 2024).
A second frequently used form is the modified Lieb-lattice Hubbard model,
$5$25
with $5$26 as the unit of energy. Its order parameters are the staggered and total moments on the magnetic sublattices,
$5$27
and the defining altermagnetic condition is $5$28 with $5$29 (Kaushal et al., 2024).
For the Lieb-$5$30 lattice, the repulsive model is written as
$5$31
with nearest-neighbor hoppings $5$32, intracell diagonal hopping $5$33, and on-site repulsion only on $5$34: $5$35 The natural Hartree-Fock order parameter is
$5$36
so that $5$37 signals the altermagnetic phase with zero net moment (Biswas et al., 20 Jan 2026).
The symmetry classification is equally important. In the canonical Lieb-metal transition, the ordered state is described as spin-group type III,
$5$38
with $5$39 a halving subgroup of the full space group $5$40. In sliding two-dimensional square-lattice bilayers, by contrast, a distinction is drawn between true altermagnets and coupled quasi-altermagnets: when the rotational connector is lost and $5$41, spin degeneracy can be lifted even at $5$42, giving $5$43, which is classified as quasi-altermagnetism rather than the usual $5$44-wave altermagnetic state (Dürrnagel et al., 2024, Dhori et al., 18 Jun 2026).
4. Phase transitions and correlation regimes
In the weak-coupling metallic Lieb lattice, the phase transition from the symmetric parent metal to collinear altermagnetism has been studied with the static truncated-unity functional renormalization group. The flow introduces a cutoff $5$45, tracks the two-particle vertex in particle-particle and particle-hole channels, and identifies an instability when one channel diverges. At van-Hove filling, the leading divergence occurs in the magnetic $5$46 channel with $5$47 form factor. For $5$48, $5$49, and $5$50, the extracted critical scale gives a second-order transition with
$5$51
A self-consistent mean-field treatment constrained to the same form factor finds continuous growth of $5$52 down to $5$53. Even for $5$54, the zero-temperature amplitude satisfies $5$55, implying a nonrelativistic spin splitting $5$56 at the Fermi surface (Dürrnagel et al., 2024).
The modified Lieb-lattice Hubbard model supports a different regime structure. Unrestricted Hartree-Fock and exact diagonalization establish spin-$5$57 altermagnetic Mott insulating ground states at average electron densities $5$58 and $5$59 per unit cell. Hole doping of the $5$60 Mott state yields an itinerant collinear antiferromagnetic metal with “cigar”-shaped hole pockets centered near $5$61 for spin down and $5$62 for spin up, while electron doping of the $5$63 Mott state places carriers mainly on $5$64 sites and produces Fermi pockets near $5$65 and $5$66, again with $5$67-wave splitting (Kaushal et al., 2024).
The Lieb-$5$68 lattice displays three half-filled regimes: normal metal (NM), altermagnetic metal (AMM), and altermagnetic isolated-band metal (AMIM). With $5$69, the transition is direct,
$5$70
and for $5$71 the threshold is $5$72. Finite $5$73 introduces an intermediate window,
$5$74
with $5$75. Increasing $5$76 lowers the onset $5$77 of altermagnetism but narrows the AMM region in $5$78 space. Moderate Kane-Mele-type SOC lowers $5$79 but does not destroy AMM or AMIM, and the finite-temperature order parameter $5$80 decays continuously to zero at $5$81, with $5$82 in a typical AMM regime and $5$83 in the AMIM regime (Biswas et al., 20 Jan 2026).
Beyond Hartree-Fock, a slave-rotor treatment at half filling predicts a cascade of correlation-driven phases in an altermagnetic Lieb model: $5$84 For $5$85, $5$86, and $5$87, the critical scales are
$5$88
Here $5$89 marks onset of staggered magnetization, $5$90 the opening of a chargon gap with $5$91, and $5$92 the complete suppression of quasiparticle weight, $5$93, in the altermagnetic Mott insulator (Carvalho et al., 28 May 2026).
A common simplification is to equate altermagnetism on Lieb-type lattices with a single metallic state. The present phase diagrams instead contain a continuous metallic transition, doped altermagnetic metals, Mott insulators with altermagnetic order, altermagnetic insulators, and isolated-band altermagnetic metals, depending on lattice decoration, filling, and correlation strength.
5. Electronic structure, spectroscopy, and magnonic signatures
In the symmetric parent state of the canonical three-band Lieb model, the spectrum contains one flat, or for $5$94 weakly dispersive, band pinned at zero energy and two symmetric dispersive bands with a quadratic band touching at the $5$95 point and a large van-Hove density of states. Once the altermagnetic order parameter condenses, time-reversal symmetry is broken but the unit-cell magnetization remains zero. The quasiparticle spectrum then shows nonrelativistic spin splitting along $5$96 and $5$97, while symmetry-protected nodal lines persist along $5$98, where $5$99 and therefore 00 (Dürrnagel et al., 2024).
In the modified Lieb model, unrestricted Hartree-Fock produces spin-resolved dispersions 01 with a Mott gap and a 02-form splitting,
03
which vanishes along 04. Exact diagonalization confirms this through the momentum-resolved hole spectral functions 05 and the squared spin-splitting
06
A tiny probe field is used in exact diagonalization to break 07 so that 08 becomes finite only when true altermagnetic order exists. The same study constructs an effective spin-09 Heisenberg model and finds two magnon branches 10 split in a 11-wave fashion between 12 and 13, with a gapless Goldstone mode at 14 and sublattice-resolved 15 away from nodal directions (Kaushal et al., 2024).
Inverse Lieb compounds exhibit an analytically transparent magnon splitting. With
16
the two-branch linear-spin-wave dispersion is
17
The splitting vanishes identically for 18. In La19O20Mn21Se22 it is only a few percent of the bandwidth, in vanadium-based inverse Lieb systems it approaches 23–24, and in metallic Sr25CrO26Cr27OAs28 it exceeds 29 of the total magnon bandwidth at 30 and 31 (Chang et al., 6 Aug 2025).
Monolayer V32O provides a first-principles electronic example of a buckled Lieb altermagnet. Without SOC, the bands along 33–34–35–36 exhibit large spin splitting, whereas along 37–38 they remain spin-degenerate. The momentum-dependent splitting
39
reaches
40
Including SOC gaps the quadratic crossings near 41–42 and 43–44, generating Berry-curvature hot spots (Taşkıran et al., 7 Jun 2026).
These band-structure characteristics define the principal experimental observables. Spin-resolved ARPES has been proposed to detect 45 in the absence of net moment, and spin-polarized scanning tunneling microscopy to image the real-space pattern 46, 47 in metallic Lieb systems (Dürrnagel et al., 2024).
6. Material realizations and symmetry-based control
Several material families have been put forward as Lieb-like altermagnets. In the modified Lieb-lattice context, candidate quasi-2D oxychalcogenides with anti-CuO48 layers include La49O50Mn51Se52, KV53Se54O, and Rb55V56Te57O. For inverse Lieb materials, density-functional calculations together with exchange extraction identify La58O59Mn60Se61, V62Se63O, KV64Se65O, RbV66Te67O, and Sr68CrO69Cr70OAs71 as altermagnetic, while La72O73Co74Se75 and La76O77Fe78Se79 fall into frustrated noncollinear sectors. A design rule proposed for inverse Lieb compounds is that nominal 80–81 and 82 configurations show propensity for altermagnetic behavior, whereas 83–84 tend toward strong antiferromagnetic 85 and loss of altermagnetism. The metallic Sr86CrO87Cr88OAs89 is highlighted as an altermagnet with highly anisotropic 90 exchange couplings and a large Néel temperature of 91 K (Kaushal et al., 2024, Chang et al., 6 Aug 2025).
Other platforms are more tunable than chemically fixed solids. The metallic Lieb-Hubbard scenario explicitly points to ultracold atoms in an optical Lieb lattice, with tunable 92 and 93, as a concrete route to the required band structure. Layered perovskite oxides and metal-organic frameworks that approximate the Lieb geometry are also proposed. In the Lieb-94 setting, Ga-doped YIG films, two-dimensional Zn-phthalocyanine polymer sheets, photonics arrays, and cold-atom optical lattices are listed as relevant geometries (Dürrnagel et al., 2024, Biswas et al., 20 Jan 2026).
The monolayer 95WS96 family illustrates how crystal-spin symmetry controls response functions. In these 97 Lieb-like altermagnets, the ground state is a Néel-type antiferromagnet with 98 and 99, and the low-order band splitting takes the form
00
with 01. Under axial stress 02 or 03, the residual 04 symmetry forbids a piezoelectric response but allows a piezomagnetic moment 05, with the only nonzero tensor component
06
Under diagonal stress 07, the in-plane mirror 08 suppresses the piezomagnetic response while a piezoelectric coefficient survives,
09
This directional alternation between piezomagnetism and piezoelectricity is the basis of the reported alter-piezoresponse (Xu et al., 27 Jun 2025).
Bilayer sliding introduces another control axis. In Mn10WS11, AA and AB stackings preserve altermagnetism with 12, whereas AC13 and AC14 stackings are classified as coupled quasi-altermagnets with
15
The two quasi-altermagnetic states are related by a symmetry that exchanges 16 and 17, enabling reversible spin-valley inversion under interlayer sliding. In Janus Mn18WS19Se20, certain stackings restore a proper rotational connector and recover a true altermagnet, while others remain quasi-altermagnetic or merely valley-polarized (Dhori et al., 18 Jun 2026).
V21O supplies a complementary buckled platform with explicit stability diagnostics. The reported formation energy is 22 eV per V23O unit, the phonon spectrum has no imaginary branches, and ab initio molecular dynamics at 24 K retains the buckled lattice. The magnetic ground state is a striped collinear antiferromagnet with local moment
25
and out-of-plane easy axis specified by 26 meV, 27 in the magnetocrystalline-anisotropy expansion (Taşkıran et al., 7 Jun 2026).
7. Topological, Hall, and excitonic phenomena
Once SOC is incorporated, Lieb-lattice altermagnets acquire a large topological response manifold. In one Hubbard-plus-intrinsic-SOC study on the three-site Lieb lattice, moderate interactions generate the altermagnetic pattern 28, and SOC gaps the crossing structure so that each conserved spin sector carries an opposite Chern number,
29
The resulting phase is a quantum spin Hall state with 30, but it is not the conventional time-reversal-invariant QSHE: the edge bands are spin-biased, nondegenerate, differently localized, and have different velocities, so an applied bias can generate simultaneous spin and charge currents (Wang et al., 7 Apr 2026).
A more general spin-cluster construction on the Lieb lattice produces multiple 31- and 32-wave altermagnetic configurations in the presence of Kane-Mele-type SOC. For out-of-plane moments, each spin block is a topological Lieb model and the mirror, or spin, Chern number satisfies
33
at 34 or 35 filling. For in-plane altermagnetic moments, the edge Dirac crossings of a 36 strip are gapped by an effective edge Hamiltonian
37
with gap
38
In an open square geometry this yields four in-gap corner modes per gap, i.e. a higher-order topological phase. The same work reports that altermagnetism is markedly more effective than comparable ferromagnetic or ferrimagnetic patterns at opening the relevant edge gaps (Huo et al., 19 Dec 2025).
Hall responses also acquire specifically altermagnetic structure. In a tight-binding model for Lieb-lattice altermagnets with Dresselhaus SOC, the twofold degeneracy between the spin-split 39- and 40-chain bands is described by an axial pseudospin 41. A symmetry-breaking perturbation
42
opens a pseudo-gap near 43, generating a sharply localized Berry curvature. For monolayer Mn44WS45 under 46 uniaxial strain, the reported first-principles values are a pseudo-gap of 47 meV at 48, a peak Berry curvature
49
and an intrinsic anomalous Hall conductivity
50
for 51 meV hole doping. In multilayers, odd and even thicknesses differ qualitatively: odd 52 retain a charge Hall response of monolayer sign, whereas even 53 cancel the charge Hall conductivity and add a pure spin Hall response linearly with 54 (Xu et al., 16 Sep 2025).
Sliding bilayer quasi-altermagnets produce a related transport effect. In AC55/AC56 stackings, SOC makes the Berry curvatures at 57 and 58 unequal in magnitude and opposite in sign, so hole doping generates a nonzero anomalous Hall conductivity that reverses sign upon sliding from AC59 to AC60. This is presented as a sliding-controlled analogue of tunnelling magnetoresistance, distinct from the valley Hall behavior of pure altermagnetic AA and AB stackings, where 61 and the anomalous Hall conductivity vanishes at charge neutrality (Dhori et al., 18 Jun 2026).
Optical many-body effects add another layer of functionality. In monolayer Mn62WS63, a Bethe-Salpeter treatment finds strongly bound excitons with lowest binding energy
64
Because the 65-valley transition has 66 but 67, 68-polarized light selectively excites the 69 valley, whereas by 70 rotation 71-polarized light selectively excites the 72 valley. Since spin and valley are locked, this becomes a valley-selective linear dichroism for spin-polarized excitons. Uniaxial strain lifts the valley degeneracy, and at 73 the reported excitonic valley splitting is 74 meV, enabling nearly complete valley and spin selectivity when the laser is tuned to one excitonic resonance (Wang et al., 16 Sep 2025).
The combined literature therefore portrays the altermagnetic Lieb-like lattice not as a single model, but as a family of square-lattice-derived systems in which sublattice interference, inequivalent exchange paths, decorated unit cells, and crystal-spin symmetries generate a common phenomenology: zero-net-moment collinear order with momentum-selective spin splitting, nodal directions set by lattice symmetry, and a broad response landscape spanning ARPES-visible nonrelativistic spin splitting, chiral magnons, alter-piezoresponse, sliding-controlled quasi-altermagnetism, quantum spin Hall transport, axial Hall conductivity, and excitonic spin-valley selection.