Lieb-Lattice Altermagnets: Symmetry-Driven Spin Splitting

• Lieb-lattice altermagnets are two-dimensional magnetic systems characterized by compensated collinear order and d-wave spin splitting derived from crystal symmetries.
• Microscopic models reveal that specific symmetry operations, such as C4 rotations and mirror reflections, enable momentum-dependent spin splitting without net magnetization.
• These systems offer tunable routes to novel topological phases, unconventional superconductivity, and enhanced functional responses like piezomagnetism and optical selectivity.

Searching arXiv for the cited Lieb-lattice altermagnet papers and closely related work. Lieb-lattice altermagnets are altermagnetic systems realized on the two-dimensional Lieb lattice and on closely related inverse, modified, decorated, and Lieb-$5$ lattices, where compensated collinear order on symmetry-related magnetic sublattices produces momentum-dependent spin splitting without net magnetization. In the standard minimal setting, the Lieb lattice has three sites per unit cell, with two magnetic sublattices related by crystal rotation or mirror symmetry rather than by translation or inversion, and the resulting spin splitting commonly has $d_{x^2-y^2}$-wave character. Across recent model studies and materials proposals, this platform has become a central setting for interaction-driven altermagnetism, spin-biased quantum spin Hall states, field- or strain-controlled Hall responses, phonon-mediated equal-spin superconductivity, and higher-order topology (Leraand et al., 12 Feb 2025, Kaushal et al., 2024, Wang et al., 7 Apr 2026).

1. Lattice families and symmetry logic

The unifying feature of Lieb-lattice altermagnets is not a single crystallographic realization but a symmetry principle. In the standard Lieb lattice, the three-site basis is commonly described as a nonmagnetic site and two symmetry-related magnetic sites, often labeled $A$, $B$, and $C$, with the two magnetic sublattices exchanged by $C_4$ rotations and mirror operations. In the modified Lieb-lattice Hubbard model for anti-CuO$_2$ oxychalcogenides, the nonmagnetic site is oxygen-like and the two transition-metal sites are related by $C_4$ but not by inversion or lattice translation. In the inverse Lieb lattice, the relevant operation exchanging compensated spin sublattices is a $(110)$ mirror. These relations are precisely what distinguish altermagnets from ordinary collinear antiferromagnets, where opposite-spin sublattices are typically related by translation or inversion and thus do not generically yield nonrelativistic spin splitting in momentum space (Leraand et al., 12 Feb 2025, Kaushal et al., 2024, Chang et al., 6 Aug 2025).

The same logic extends to more elaborate Lieb-derived geometries. The Lieb-$5$ lattice contains five sites per unit cell grouped into three sublattice types, with the altermagnetic symmetry encoded by spin flip between $d_{x^2-y^2}$0 and $d_{x^2-y^2}$1 sublattices followed by a $d_{x^2-y^2}$2 rotation around the central $d_{x^2-y^2}$3 site. Decorated and Janus decorated Lieb lattices retain the same basic idea but enlarge the local environment and symmetry content, allowing additional control through mirror breaking, Néel-vector orientation, or higher-order topology (Biswas et al., 20 Jan 2026, Chen et al., 20 Oct 2025, Huo et al., 19 Dec 2025).

Lattice family	Unit-cell motif	Symmetry relation central to AM
Standard / minimal Lieb lattice	Three-site basis with two magnetic and one nonmagnetic site	$d_{x^2-y^2}$4, mirrors, or $d_{x^2-y^2}$5 relating opposite spins
Modified Lieb lattice	Oxygen-like nonmagnetic site plus two magnetic transition-metal sites	$d_{x^2-y^2}$6 rotation between magnetic sites, not inversion/translation
Inverse Lieb lattice	Square-net antiperovskite arrangement with two magnetic sublattices	$d_{x^2-y^2}$7 mirror exchanging spin sublattices
Decorated / Lieb-$d_{x^2-y^2}$8 lattices	Enlarged Lieb-derived basis with symmetry-related magnetic clusters	Combined spin-lattice rotation symmetries

For the minimal three-site altermagnet considered in phonon-mediated superconductivity, the surviving magnetic-space-group operations include $d_{x^2-y^2}$9, $A$0, $A$1, $A$2, $A$3, $A$4, $A$5, and $A$6, so ordinary $A$7 is broken by magnetic order while a spin-flipping $A$8 remains. This enforces the relation

$A$9

which is the canonical statement that the spin-up and spin-down spectra are related by a $B$0 rotation rather than by a uniform exchange shift (Leraand et al., 12 Feb 2025).

2. Microscopic models and routes to altermagnetic order

Several complementary routes to altermagnetism have been established on Lieb-derived lattices. In the itinerant “Lieb metal,” a single-orbital Hubbard model near a $B$1-polarized van Hove singularity undergoes a direct instability into a translation-preserving collinear altermagnetic state. The order is identified as a $B$2-irrep spin Pomeranchuk instability with

$B$3

and the microscopic mechanism is explicitly traced to sublattice interference rather than orbital ordering or preformed local moments (Dürrnagel et al., 2024).

In the two-dimensional Lieb-lattice Hubbard model at filling $B$4, a self-consistent mean-field treatment finds three phases without SOC: a nonmagnetic metal, an altermagnetic metal, and an altermagnetic insulator. Altermagnetic order appears once $B$5 exceeds roughly $B$6, and the phase boundaries are approximately fitted by

$B$7

between the nonmagnetic metal and the altermagnetic metal, and

$B$8

between the altermagnetic metal and the altermagnetic insulator (Wang et al., 7 Apr 2026).

A different interaction-driven route appears in the modified Lieb-lattice Hubbard model motivated by anti-CuO$B$9 oxychalcogenides. There the Hubbard interaction acts only on the two transition-metal sublattices, and unrestricted Hartree-Fock together with exact diagonalization establishes spin-$C$0 altermagnetic Mott insulating ground states at average electron densities $C$1 and $C$2 per unit cell. In both cases the low-energy magnetic degrees of freedom live on the two magnetic sublattices, and the resulting bands exhibit $C$3-wave spin splitting with symmetry-protected nodes on $C$4 (Kaushal et al., 2024).

Stronger-correlation physics has been addressed by slave-rotor theory. At half filling, the anisotropic Lieb-lattice Hubbard model exhibits the sequence

$C$5

followed at still larger $C$6 by a strongly localized $C$7 regime. The main conclusion is that altermagnetic order survives into the Mott regime, but the observable spin splitting in the electronic spectral function is progressively reduced as the quasiparticle weight $C$8 collapses (Carvalho et al., 28 May 2026).

The Lieb-$C$9 Hubbard model provides an additional minimal route. With only nearest-neighbor hoppings, it supports a transition from a normal metal to an altermagnetic isolated band metal; with intracell diagonal hopping, the sequence becomes normal metal $C_4$0 altermagnetic metal $C_4$1 altermagnetic isolated band metal. The order parameter is

$C_4$2

and the key point is that this decorated Lieb-type geometry realizes altermagnetism without requiring longer-range hopping as a prerequisite (Biswas et al., 20 Jan 2026).

Not every magnetic Lieb-lattice state is altermagnetic. Hartree-Fock studies of the repulsive Hubbard model on the standard Lieb lattice mainly establish ferrimagnetic, ferromagnetic, and spiral spin-density-wave states, with the half-filled $C_4$3 state identified as the Lieb-theorem ferrimagnet rather than a compensated altermagnet. Likewise, distorted covalent-organic-framework realizations of the Lieb lattice were shown to host paramagnetic, ferromagnetic, and Néel antiferromagnetic phases under hole doping, but not the defining momentum-dependent spin splitting of altermagnetism (Gouveia et al., 2015, Cui et al., 2019).

3. Electronic structure and spectroscopic signatures

The defining electronic signature of a Lieb-lattice altermagnet is anisotropic spin splitting with sign changes in momentum space. In the minimal three-site model, the spin-up and spin-down Fermi contours are rotated copies of one another rather than translated copies, and ordinary zero-momentum spin-singlet pairing is excluded in the regime where opposite-spin Fermi surfaces do not overlap. The normal-state bands show the characteristic $C_4$4-type altermagnetic splitting, and the bands become increasingly flat near the crossing at $C_4$5, particularly along the $C_4$6 and $C_4$7 directions (Leraand et al., 12 Feb 2025).

For the modified Lieb lattice, the $C_4$8 and $C_4$9 altermagnetic Mott states display spin splitting over most of the Brillouin zone, vanishing along the nodal lines $_2$0. In the Lieb-metal instability picture, the ordered phase has finite splitting along $_2$1-$_2$2 and $_2$3-$_2$4, but symmetry-protected degeneracy along $_2$5-$_2$6, reflecting the $_2$7 form factor and the combined spin-space operations $_2$8, $_2$9, and $C_4$0 (Kaushal et al., 2024, Dürrnagel et al., 2024).

The $C_4$1 and $C_4$2 valleys play a recurrent role. In monolayer Mn$C_4$3WS$C_4$4, the top valence and bottom conduction states at $C_4$5 and $C_4$6 are fully spin polarized with opposite spin assignment, yielding spin-valley locking without net magnetization. In the field-driven topological model on a Lieb lattice, the $C_4$7 and $C_4$8 valleys acquire opposite mass tendencies through the $C_4$9-wave form factor $(110)$0, which is why valley-resolved topology emerges so naturally in this setting (Wang et al., 16 Sep 2025, Tagani et al., 2 Apr 2026).

Many-body broadening does not erase these signatures in the calculated regimes. For a $(110)$1-wave Lieb-lattice altermagnet, electron self-energies from magnons, phonons, and hybridized magnon-phonon modes still leave the spin splitting spectroscopically resolvable. The paper further finds a distinct difference between spectral broadening for up and down spins close to the Fermi surface in the electron-magnon case, tracing it to the spin splitting of magnon modes; no analogous asymmetry appears for electron-phonon coupling (Leraand et al., 17 Dec 2025).

4. Topology on Lieb and decorated Lieb lattices

Spin-orbit coupling reshapes Lieb-lattice altermagnets into several topological phases rather than simply perturbing an otherwise fixed magnetic spectrum. In the interacting Hubbard model on the two-dimensional Lieb lattice at filling $(110)$2, Kane-Mele SOC converts a broad region of the altermagnetic phase diagram into a quantum spin Hall state with

$(110)$3

while the total Chern number remains $(110)$4. Because time-reversal symmetry is already broken by altermagnetic order, this is a time-reversal-symmetry-broken QSHE. Its edge states are nondegenerate in energy, differently localized, and differently dispersing for the two spin channels, which is why the authors describe it as a spin-biased QSHE rather than an ordinary Kane-Mele state (Wang et al., 7 Apr 2026).

A distinct topological route appears in the minimal Lieb-lattice $(110)$5-wave altermagnet under applied magnetic field. There, zero field supports a normal insulator, a spin Chern insulator, and an accidental Dirac semimetal or nodal-line regime, while the field breaks the $(110)$6 valley equivalence and yields quantum anomalous Hall phases with $(110)$7. The mechanism is valley selective: one valley can host $(110)$8 or $(110)$9, and the other $5$0 or $5$1, with Berry-curvature hotspots centered near shifted Dirac points rather than only at the valley centers (Tagani et al., 2 Apr 2026).

Higher-order topology emerges when altermagnetism reconstructs a topological Lieb-lattice parent state. Using a spin-cluster construction, both $5$2- and $5$3-wave altermagnetic patterns were designed on the Lieb lattice. The altermagnetic unit cell reconstructs strip-geometry edge states and produces Dirac points, while in-plane magnetic moments open gaps at these points. In an open square geometry, corner modes appear inside the gaps, realizing higher-order topological states. The same induction of higher-order topology was verified for all altermagnetic configurations constructed there and was found to be most pronounced for altermagnetism as compared with ferromagnetism and ferrimagnetism (Huo et al., 19 Dec 2025).

Topological response theory has also been developed for insulating Lieb-lattice altermagnets. In the spinful Lieb-lattice model with collinear Néel order, the half-filled SOC-gapped phase is a mirror Chern insulator, and the orbital piezomagnetic response contains a topological contribution inherited from a parent Dirac quadrupole semimetal. For the $5$4-type strain channel, the response is

$5$5

with

$5$6

The essential picture is that strain converts the Dirac quadrupole into a Dirac dipole by shifting valleys of opposite Berry curvature in opposite energies (Radhakrishnan et al., 5 Feb 2026).

Decorated Lieb altermagnets add an orientation degree of freedom. In monolayer Ta$5$7TeSeO, described as a Janus decorated Lieb lattice, each spin block hosts a pair of mirror-protected spin-polarized Weyl points in the SOC-free limit. Aligning the Néel vector along the crystallographic $5$8 or $5$9 direction selectively preserves one unitary mirror and breaks the other, so one spin sector remains gapless while the opposite spin sector is gapped. Because unitary $d_{x^2-y^2}$00 is also broken for in-plane Néel vectors, the surviving Weyl points become inequivalent in energy, converting a half semimetal into an intrinsic half-metallic state (Chen et al., 20 Oct 2025).

5. Collective modes and superconductivity

The Lieb lattice is not only a host for altermagnetic band structures but also a setting in which collective modes and pairing kernels are strongly shaped by sublattice form factors. In the weak-coupling phonon problem on a minimal Lieb-lattice altermagnet, expanding the nearest- and next-nearest-neighbor hoppings to first order in displacement yields an electron-phonon vertex that depends on both incoming and outgoing momenta rather than only on the transfer $d_{x^2-y^2}$01. After a Schrieffer-Wolff transformation, the resulting same-spin interaction supports an odd-momentum equal-spin instability with

$d_{x^2-y^2}$02

The dominant state is an unconventional zero-momentum superconducting phase of fully spin-polarized Cooper pairs, described as odd in momentum and even in spin, with a nodal, $d_{x^2-y^2}$03-wave-like gap and a predicted V-shaped quasiparticle density of states (Leraand et al., 12 Feb 2025).

Magnons on Lieb-derived altermagnets inherit the same symmetry anisotropy. In the inverse Lieb family, linear spin-wave calculations show that chiral magnon splitting is directly correlated with anisotropy between the inequivalent second-neighbor exchanges $d_{x^2-y^2}$04 and $d_{x^2-y^2}$05. When $d_{x^2-y^2}$06, the $d_{x^2-y^2}$07 sector is effectively isotropic and the magnon splitting collapses. The flagship case is Sr$d_{x^2-y^2}$08CrO$d_{x^2-y^2}$09Cr$d_{x^2-y^2}$10OAs$d_{x^2-y^2}$11, where the splitting exceeds half the total magnon bandwidth at the $d_{x^2-y^2}$12 points and reaches about $d_{x^2-y^2}$13 of the full bandwidth according to the text (Chang et al., 6 Aug 2025).

The same $d_{x^2-y^2}$14-wave logic appears in quasiparticle lifetimes. For the representative Lieb-lattice model used in the many-body self-energy study, the magnon spectrum itself shows $d_{x^2-y^2}$15-wave spin splitting, and the largest broadenings for the two electron spin species occur on opposite sides of $d_{x^2-y^2}$16. The hybrid magnon-phonon case is reported to be very similar to the pure magnon case, because electrons couple to the hybrid modes mainly through their magnon content (Leraand et al., 17 Dec 2025).

6. Materials platforms, strain, optics, and functional responses

The inverse Lieb family provides the clearest materials-based synthesis of the field. A classical $d_{x^2-y^2}$17-$d_{x^2-y^2}$18-$d_{x^2-y^2}$19-$d_{x^2-y^2}$20 analysis, combined with DFT and DFT+$d_{x^2-y^2}$21, identifies La$d_{x^2-y^2}$22O$d_{x^2-y^2}$23Mn$d_{x^2-y^2}$24Se$d_{x^2-y^2}$25, V$d_{x^2-y^2}$26Se$d_{x^2-y^2}$27O, KV$d_{x^2-y^2}$28Se$d_{x^2-y^2}$29O, RbV$d_{x^2-y^2}$30Te$d_{x^2-y^2}$31O, and Sr$d_{x^2-y^2}$32CrO$d_{x^2-y^2}$33Cr$d_{x^2-y^2}$34OAs$d_{x^2-y^2}$35 as altermagnetic members or candidates. The reported trend is that $d_{x^2-y^2}$36 and $d_{x^2-y^2}$37 transition-metal configurations show propensity for altermagnetic behavior, while $d_{x^2-y^2}$38 and $d_{x^2-y^2}$39 fillings tend to destabilize altermagnetism in favor of stripe, block, or orthogonal-spin phases. Sr$d_{x^2-y^2}$40CrO$d_{x^2-y^2}$41Cr$d_{x^2-y^2}$42OAs$d_{x^2-y^2}$43 is singled out as a metallic altermagnet with large exchange anisotropy and Néel temperature $d_{x^2-y^2}$44 K (Chang et al., 6 Aug 2025).

Two-dimensional ternary chalcogenides $d_{x^2-y^2}$45 with $d_{x^2-y^2}$46 Mn, Fe, Co extend the subject into electromechanical and optical functionality. In monolayer Mn$d_{x^2-y^2}$47WS$d_{x^2-y^2}$48, Fe$d_{x^2-y^2}$49WS$d_{x^2-y^2}$50, and Co$d_{x^2-y^2}$51WS$d_{x^2-y^2}$52, axial stress along $d_{x^2-y^2}$53 breaks $d_{x^2-y^2}$54 and induces a giant piezomagnetic response while residual $d_{x^2-y^2}$55 symmetry suppresses piezoelectricity; diagonal stress along $d_{x^2-y^2}$56 instead preserves compensated magnetism through mirror symmetry while enabling an out-of-plane piezoelectric response. For Mn$d_{x^2-y^2}$57WS$d_{x^2-y^2}$58, the axial piezomagnetic stress coefficient is reported as $d_{x^2-y^2}$59, and the diagonal piezoelectric stress coefficient as $d_{x^2-y^2}$60 (Xu et al., 27 Jun 2025).

The same material family supports hidden Berry-curvature phenomena. In strained Mn$d_{x^2-y^2}$61WS$d_{x^2-y^2}$62, an “axial Hall effect” was predicted by identifying the two orthogonal lattice axes as a hidden axial pseudospin degree of freedom. Dresselhaus SOC and the intrinsic piezomagnetic response lift the axial degeneracy, open a narrow pseudogap near the $d_{x^2-y^2}$63 point, and generate strong localized Berry curvature. The anomalous Hall conductivity peak is reported to be about $d_{x^2-y^2}$64, centered roughly $d_{x^2-y^2}$65 meV below the Fermi level, and its magnitude remains nearly unchanged as strain varies (Xu et al., 16 Sep 2025).

Excitonic optics in the same Lieb-lattice altermagnets are valley and spin selective in a fourfold, not hexagonal, sense. In monolayer Mn$d_{x^2-y^2}$66WS$d_{x^2-y^2}$67, $d_{x^2-y^2}$68-polarized light excites transitions mainly near the $d_{x^2-y^2}$69 valley and $d_{x^2-y^2}$70-polarized light mainly near the $d_{x^2-y^2}$71 valley, with opposite spin assignment at the two valleys. The $d_{x^2-y^2}$72 quasiparticle gap is about $d_{x^2-y^2}$73 eV, the lowest bright exciton lies at $d_{x^2-y^2}$74 eV, and the corresponding binding energy is $d_{x^2-y^2}$75 eV. Under $d_{x^2-y^2}$76 tensile uniaxial strain along $d_{x^2-y^2}$77, the $d_{x^2-y^2}$78 and $d_{x^2-y^2}$79 excitons split by about $d_{x^2-y^2}$80 meV, enabling selective preparation of one spin-polarized valley exciton species (Wang et al., 16 Sep 2025).

Taken together, these results establish Lieb-lattice altermagnets as a broad class rather than a single model: the standard and modified Lieb lattices furnish minimal electronic realizations; the inverse Lieb lattice supplies a chemically versatile materials family; decorated and Janus decorated Lieb lattices enable mirror-selective and Néel-vector-selective topology; and two-dimensional ternary chalcogenides illustrate how altermagnetism on a Lieb motif can be converted into piezomagnetic, piezoelectric, Hall, and excitonic functionality under controlled symmetry breaking (Kaushal et al., 2024, Chang et al., 6 Aug 2025, Chen et al., 20 Oct 2025).