Lieb–Kagome Interconversion

• Lieb/Kagome interconversion is a family of geometric and Hamiltonian deformations that transforms a bipartite flat-band Lieb lattice into a frustrated kagome network using parameters like t'/t, λ, θ, shear, and strain.
• It reveals model-specific evolution of band structures where flat bands and Dirac cones shift, merge, or vanish, impacting magnetic order, transport properties, and topological phases.
• This interconversion framework enables programmable platforms across electronic, photonic, and quantum systems, offering insights into correlated magnetism, quantum phase transitions, and flat-band engineering.

Lieb/Kagome interconversion denotes a family of geometric and Hamiltonian deformations that connect the three-site Lieb lattice to the three-site kagome lattice. In the recent literature, this connection is realized through several one-parameter constructions: by turning on a bond $t'$ between the edge sites $B$ and $C$, by varying a coupling ratio $\lambda=J'/J$, by changing a morphological angle $\theta\in[\pi/2,2\pi/3]$, by applying a shear parameter $s\in[0,\pi/6]$, or by introducing a strain parameter $\eta\in[0,1]$. Across these formulations, the central theme is the controlled transformation of a bipartite flat-band lattice into a frustrated network of corner-sharing triangles, with consequences for flat bands, Dirac cones, magnetic order, collective excitations, topology, and transport (Ying et al., 12 Jan 2026, Lara et al., 17 Jun 2025, Lang et al., 2022).

1. Geometric definitions and interpolation schemes

The most direct interpolation is the half-filled Hubbard model on a two-dimensional Bravais lattice with three orbitals per unit cell, labeled $A$ (corner site) and $B,C$ (edge sites), with nearest-neighbor hoppings $A\!-\!B$ and $B$0 of amplitude $B$1, and an intra-cell $B$2 hopping $B$3, where $B$4. In this construction, $B$5 is the Lieb lattice and $B$6 is the kagome-graph limit (Ying et al., 12 Jan 2026).

A closely related classical-spin interpolation uses a square unit cell with three spin-$B$7 sites at $B$8, $B$9, and $C$0, and defines a single parameter

$C$1

where $C$2 couples the original square edges and $C$3 the square diagonal $C$4. Here $C$5 is the pure Lieb lattice and $C$6 the perfect kagome lattice (Lopez-Bezanilla et al., 24 Jul 2025).

A second major family of models uses a morphological angle $C$7. In the monolayer and multilayer tight-binding framework, $C$8 gives the Lieb lattice and $C$9 the kagome lattice, with intermediate $\lambda=J'/J$0 defining transition lattices (Lara et al., 17 Jun 2025). The same angular interval appears in the line-graph Hubbard model used for quantum thermodynamics (Sur et al., 2024) and in the intrinsic-spin-orbit model for topological phase transitions (Lima et al., 16 Jun 2025).

In photonic realizations, the interpolation is implemented as a shear transformation $\lambda=J'/J$1 acting on the primitive vectors of the square Lieb lattice, with $\lambda=J'/J$2 corresponding to Lieb and $\lambda=J'/J$3 to kagome (Lang et al., 2022). In strain-driven Hubbard models, an analogous role is played by $\lambda=J'/J$4, which turns on a second set of bonds with amplitude $\lambda=J'/J$5, so that $\lambda=J'/J$6 is Lieb and $\lambda=J'/J$7 kagome (Kunwar et al., 11 Aug 2025).

Parameter	Endpoints	Representative use
$\lambda=J'/J$8	$\lambda=J'/J$9	Hubbard and magnon spectra (Ying et al., 12 Jan 2026)
$\theta\in[\pi/2,2\pi/3]$0	$\theta\in[\pi/2,2\pi/3]$1	Ising frustration on a quantum annealer (Lopez-Bezanilla et al., 24 Jul 2025)
$\theta\in[\pi/2,2\pi/3]$2	$\theta\in[\pi/2,2\pi/3]$3	Monolayer, multilayer, topological, and thermodynamic models (Lara et al., 17 Jun 2025)
$\theta\in[\pi/2,2\pi/3]$4	$\theta\in[\pi/2,2\pi/3]$5	Sheared photonic Lieb–kagome lattice (Lang et al., 2022)
$\theta\in[\pi/2,2\pi/3]$6	$\theta\in[\pi/2,2\pi/3]$7	Strain-driven Hubbard interpolation (Kunwar et al., 11 Aug 2025)

Taken together, these constructions indicate that “interconversion” is not a single universal mapping but a class of controlled deformations. The choice of parameter determines which structural relations, symmetries, and effective couplings are preserved.

2. Band-structure evolution, flat bands, and Dirac-point migration

At the tight-binding level, the Lieb limit and kagome limit share a three-band structure with one flat band, but the location and evolution of that flat band are model dependent. In the $\theta\in[\pi/2,2\pi/3]$8-interpolated Hubbard model, the Lieb limit has point group $\theta\in[\pi/2,2\pi/3]$9 and a flat band at $s\in[0,\pi/6]$0, exactly at the Fermi energy at half filling, while the kagome-graph limit retains a flat band at the top of the spectrum and two dispersive bands below; the Brillouin zone remains square with point group $s\in[0,\pi/6]$1 because the original unit cell is kept fixed (Ying et al., 12 Jan 2026).

In the $s\in[0,\pi/6]$2-controlled monolayer tight-binding model, diagonalizing the $s\in[0,\pi/6]$3 Bloch Hamiltonian yields three bands $s\in[0,\pi/6]$4. The flat band lies at zero energy in the Lieb limit and “slowly ‘bends’ downward” as $s\in[0,\pi/6]$5, reaching the kagome flat-band energy $s\in[0,\pi/6]$6 in the $s\in[0,\pi/6]$7 convention. The two dispersive bands touch at the $s\in[0,\pi/6]$8-point in the Lieb limit and, as $s\in[0,\pi/6]$9 increases, the Dirac cones migrate so that for $\eta\in[0,1]$0 they are the upper and lower kagome Dirac bands touching at the $\eta\in[0,1]$1-point (Lara et al., 17 Jun 2025).

The photonic shear model reaches a different conclusion about the flat band. There, the Lieb and kagome lattices can be continuously converted into each other by a shearing transformation, but “during this transformation, the flat band is destroyed, while the Dirac cones remain and become tilted,” with type I, II, and III cones occurring for different parameters (Lang et al., 2022). By contrast, the line-graph Hubbard model used for quantum thermal machines states that one of the three bands is “strictly flat at $\eta\in[0,1]$2 for all $\eta\in[0,1]$3” (Sur et al., 2024). A related ribbon study finds that in the pure nearest-neighbor approximation the middle band is exactly flat for all $\eta\in[0,1]$4, while next-nearest-neighbor couplings split it into quasi-flat subbands (Uchôa et al., 2024). These results show that flat-band survival under interconversion is a model-specific property rather than a universal one.

The reverse direction, kagome $\eta\in[0,1]$5 Lieb, can be realized by Floquet engineering. Under off-resonant, linearly polarized light, the effective hopping along each kagome bond is renormalized by a Bessel factor,

$\eta\in[0,1]$6

and for purely $\eta\in[0,1]$7-polarized light the first zero of $\eta\in[0,1]$8 occurs at $\eta\in[0,1]$9, where one hopping vanishes while the other two remain $A$0. At that point the Dirac cones migrate and merge at the $A$1 point, and the resulting quasienergy spectrum matches the nearest-neighbor Lieb model with $A$2 (Kumar et al., 10 Jan 2025).

A distinct, non-geometric notion of interconversion appears in flat-band embedding. By adding a direct hopping $A$3, a fourth orbital $A$4, and a coupling $A$5, the extended Hamiltonian retains a perfectly flat band provided

$A$6

Under this condition, a unitary transformation block-diagonalizes the enlarged model into the original Lieb or kagome Hamiltonian plus a decoupled adatom level (Lee et al., 2019). This construction generalizes interconversion from geometric morphing to flat-band-preserving embedding.

3. Correlated-electron magnetism and collective spin excitations

In the half-filled multiorbital Hubbard model, the interconversion parameter $A$7 controls a competition between flat-band ferrimagnetism and frustration-driven antiferromagnetism. The Hamiltonian is

$A$8

with $A$9 and half filling enforced by $B,C$0 (Ying et al., 12 Jan 2026).

At strong coupling $B,C$1, virtual hopping generates the spin Hamiltonian

$B,C$2

with $B,C$3 on $B,C$4 and $B,C$5 links and $B,C$6 on $B,C$7 links. Increasing $B,C$8 therefore turns on antiferromagnetic $B,C$9 exchange on corner-sharing triangles and drives a transition from ferri- to antiferromagnetism in the large-$A\!-\!B$0 regime (Ying et al., 12 Jan 2026).

The phase diagram obtained by self-consistent Hartree–Fock combined with real-time two-particle response functions from the Bethe–Salpeter equation in the random phase approximation has three regions at $A\!-\!B$1. There is a paramagnet for $A\!-\!B$2, with $A\!-\!B$3 at $A\!-\!B$4 and $A\!-\!B$5 near $A\!-\!B$6, and the PM–magnetic transition is first-order. For $A\!-\!B$7 and $A\!-\!B$8, the system is ferrimagnetic with net moment per cell $A\!-\!B$9, decreasing continuously with $B$00 and vanishing at $B$01. For large $B$02 and $B$03, it is antiferromagnetic with $B$04 but oppositely polarized sublattices; the FI–AFM transition is continuous. A small-staggered-field Hartree–Fock solution also reveals a metastable altermagnetic phase at intermediate $B$05 (Ying et al., 12 Jan 2026).

The transverse susceptibility

$B$06

gives the magnon spectrum directly from $B$07, without analytical continuation. In both ferrimagnetic and antiferromagnetic phases, the spectrum contains gapless Goldstone modes and gapped Higgs magnon bands. Near $B$08, the Goldstone dispersion is linear, $B$09, in the AFM phase and quadratic, $B$10, for the net-ferromagnetic branch of the FI phase. The Higgs gaps satisfy $B$11 and $B$12; at $B$13, the numerical values are $B$14 and $B$15, in units of $B$16. At $B$17, an additional Higgs branch appears for $B$18, reflecting enhanced frustration near the AFM boundary (Ying et al., 12 Jan 2026).

4. Topological, strain, Floquet, and photonic control

With intrinsic spin–orbit coupling, Lieb, kagome, and intermediate transition lattices support topological phase transitions under interconversion and strain. In the general tight-binding Hamiltonian

$B$19

nearest-neighbor hopping amplitudes $B$20 and intrinsic spin–orbit amplitudes $B$21 depend on the morphological angle $B$22 and on strain-modified distances. The spin-up sector reduces to a $B$23 Bloch Hamiltonian with ordinary hopping matrix elements $B$24 and ISO matrix elements $B$25 (Lima et al., 16 Jun 2025).

Without ISO, the Lieb lattice has a threefold degeneracy at $B$26, transition lattices host two twofold Dirac nodes, and kagome has a Dirac point at $B$27 with a flat bottom band. Nearest-neighbor ISO alone gaps transition and kagome, while next-nearest-neighbor ISO is needed to gap Lieb (Lima et al., 16 Jun 2025). Berry curvature and Chern numbers confirm the resulting topological phases. For $B$28 and $B$29, representative spin-up Chern numbers are $B$30 for Lieb and $B$31 for kagome (Lima et al., 16 Jun 2025).

The unstrained hybrid-ISO model has a unique topological phase transition at $B$32, where $B$33 jumps from $B$34 to $B$35. Under strain, critical lines appear in the $B$36 plane. Examples given in the paper are UX strain in Lieb, with $B$37 for $B$38, BI strain in kagome, with $B$39, and PS strain in Lieb, with successive critical values $B$40 and $B$41 (Lima et al., 16 Jun 2025). The same study also shows, by hypothetical calculations with intentionally unchanged hopping and ISO parameters, that the strain-induced phase transitions arise from changes in the hopping and ISO coupling parameters.

Periodic driving provides another route through the Lieb–kagome manifold. In the off-resonant regime $B$42 bandwidth, bond-selective renormalization of kagome hoppings can switch off a single bond and force the migration and merging of Dirac cones at a high-symmetry $B$43 point, producing a Lieb-like quasienergy spectrum with a flat midband and two linearly dispersing bands (Kumar et al., 10 Jan 2025).

Photonic lattices realize the same geometry in real space. In laser-written fused silica $B$44, the sheared tight-binding Hamiltonian yields tilted Dirac cones whose tilt ratio $B$45 classifies them as type I, II, or III. Split-step beam propagation simulations and experiments show asymmetric conical diffraction: when only cones tilted upward are excited, the output shifts upward; when only downward cones are launched, it shifts downward; and symmetric excitation splits the ring into two lobes along $B$46 (Lang et al., 2022).

5. Finite-size, multilayer, and edge-dependent interconversion

Interconversion acquires additional structure in multilayers and nanoribbons. In the multilayer tight-binding framework, a system with $B$47 stacked layers is described by a $B$48 Hamiltonian composed of monolayer blocks $B$49 and interlayer couplings $B$50. For AA stacking, $B$51; for AB stacking, only two of the three sublattices form vertical dimers while the third is coupled by skew-diagonal $B$52 terms (Lara et al., 17 Jun 2025).

For the AA bilayer, the spectrum is exactly

$B$53

so each monolayer band splits into bonding and antibonding copies separated by $B$54. Under a perpendicular electric field, the bilayer splitting becomes

$B$55

that is, $B$56. For $B$57, the bands must be diagonalized numerically, but AA stacking retains a standing-wave-like layer dependence, whereas AB stacking supports richer symmetry-protected structures (Lara et al., 17 Jun 2025).

In the Lieb limit of AB stacking, the half-unit-cell shift makes the space group nonsymmorphic and enforces twofold degeneracies along the $B$58 line in $B$59-space. For even $B$60, the spectrum consists of $B$61 bilayer-like bands; for odd $B$62, it contains $B$63 bilayer copies plus one monolayer copy. A perpendicular electric field breaks the nonsymmorphic symmetry, splits the $B$64 degeneracies, and opens local gaps (Lara et al., 17 Jun 2025).

Nanoribbons add strong edge dependence. In the ribbon interpolation with $B$65, straight edges remain metallic for all $B$66 because a Dirac-like crossing survives at $B$67. Bearded edges are more delicate: in the kagome limit, a non-zero $B$68 persists only for the two smallest widths $B$69, while for $B$70 $B$71 vanishes although the indirect gap may survive pointwise. Asymmetric edges keep a full gap for all $B$72, but the gap closes as $B$73 (Uchôa et al., 2024).

Edge states also evolve nontrivially. Bearded kagome ribbons host four edge modes: two around $B$74, decaying exponentially into the ribbon with

$B$75

and two pinned near $B$76, double-degenerate at each $B$77, localized on dangling-bond $B$78-sites. Asymmetric kagome ribbons host two un-degenerate edge modes, one in the lower and one in the upper gap (Uchôa et al., 2024).

6. Programmable platforms, transport responses, and functional uses

The interconversion has been used as a programmable testbed for frustration and design. In the quantum-annealer study, each logical spin is encoded as a ferromagnetically coupled chain of three physical qubits with penalty coupling $B$79, enabling embedding on the D-Wave Advantage Pegasus topology. The observables are the magnetization per spin $B$80, the static structure factor $B$81, and real-space correlators $B$82 (Lopez-Bezanilla et al., 24 Jul 2025).

At zero field, the annealer model evolves from AFM-ordered Lieb, with Bragg peaks at $B$83 in $B$84, to a maximally frustrated “disordered” regime at $B$85, where $B$86 is diffuse and the average magnetization dip occurs precisely at $B$87. At the kagome limit, even a small field lifts the macroscopic degeneracy and produces sharp peaks in $B$88; numerically, the threshold for restoring order is $B$89 in units where $B$90, beyond which $B$91 becomes almost $B$92-independent (Lopez-Bezanilla et al., 24 Jul 2025). The same work proposes phthalocyanine assemblies as a structurally constrained prototype and frames the annealer as a surrogate for closed-loop design.

In the strain-driven SPA–Monte Carlo Hubbard model, low-temperature transport across the interconversion exhibits a re-entrant sequence of phases: a gapped magnetic insulator for $B$93, a non-Fermi-liquid metal dome for $B$94, and a gapless flat-band-localized insulator for $B$95. The low-$B$96 resistivity follows

$B$97

with $B$98 at $B$99, $C$00 at $C$01, $C$02 for $C$03, and $C$04 at $C$05. The low-frequency optical conductivity follows $C$06, with $C$07 at $C$08, $C$09 at $C$10, and $C$11 at $C$12 (Kunwar et al., 11 Aug 2025).

The same geometric control can be used thermodynamically. In the many-body quantum thermal machine, the line-graph Lieb–kagome Hubbard model serves as the working medium of a quantum Stirling cycle in which $C$13 is the external “piston.” In both interacting and non-interacting regimes, the heat-engine function dominates when the strain is induced from the kagome to the Lieb limit, while the reverse deformation favors refrigeration. The efficiency and coefficient of performance are maximized when the bath-temperature difference is small, and the Carnot limit is approached in the quasi-low-$C$14 regime $C$15 (Sur et al., 2024).

The literature therefore portrays Lieb/Kagome interconversion as a unified but model-dependent control principle. Depending on the microscopic realization, it tunes flat-band placement, frustration, magnetic order, Goldstone and Higgs magnons, Berry curvature, Chern numbers, Dirac-cone tilt, ribbon edge modes, multilayer degeneracies, and transport exponents. A plausible implication is that the most robust aspect of the subject is not a single invariant spectral feature, but the existence of a reproducible geometric pathway between flat-band bipartite physics and frustrated triangular-network physics across electronic, spin, photonic, and programmable platforms.