Breathing Kagome Lattice: Control & Phenomena

• The breathing kagome lattice is a network with alternating triangle sizes and bond strengths that tunes flat bands, Fermi surface features, and topological states.
• Its manifestations range from structural distortions in real materials to emergent orbital anisotropy in electronic systems, facilitating diverse experimental probes.
• Tuning the breathing parameter via pressure, chemical substitution, or electric fields modulates electronic correlations, magnetic frustration, and Mott-insulating behavior.

A breathing kagome lattice is a kagome network of corner-sharing triangles in which the two triangle orientations are inequivalent. Depending on context, the inequivalence is expressed as alternating long and short bonds, distinct exchange couplings or hopping amplitudes on up- and down-triangles, or an emergent orbital-network anisotropy rather than a literal structural distortion. This breathing degree of freedom is now used across condensed-matter subfields as a control parameter for flat-band formation, Fermi-surface reconstruction, Mott localization, quantum spin liquid behavior, higher-order and crystalline topology, and noncoplanar magnetic textures (Gonçalves-Faria et al., 4 Feb 2025, Jung et al., 2021).

1. Geometry, symmetry, and standard parameterizations

In real-space materials, the breathing distortion is commonly defined by alternating triangle sizes within the kagome plane. In Fe$_3$Sn$_2$, single-crystal X-ray diffraction resolves two distinct intra-layer Fe–Fe bond lengths, and the breathing amplitude is written as

$$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$

Under pressure, $\Delta_\text{breathe}$ decreases, vanishes near a regular kagome geometry, and changes sign when the distortion is reversed (Gonçalves-Faria et al., 4 Feb 2025).

In frustrated-magnet materials based on Mo clusters, a standard structural measure is the breathing parameter

$\lambda = \frac{d_\nabla}{d_\Delta},$

where $d_\Delta$ and $d_\nabla$ are the bond lengths on the two triangle types. In Li$_2$In$_{1-x}$Sc$_x$Mo$_2$0O$_2$1, $_2$2 corresponds to an ideal kagome lattice, while deviations from unity encode the lattice asymmetry that controls magnetic frustration and charge ordering tendencies (Akbari-Sharbaf et al., 2017).

Model Hamiltonians usually encode the same asymmetry through bond-selective couplings. In tight-binding descriptions this appears as unequal intra-cell and inter-cell hoppings, often denoted $_2$3 and $_2$4, or as $_2$5 and $_2$6 in the breathing kagome Hubbard model. In magnetic models, the analogous quantities are exchange constants such as $_2$7 and $_2$8 on small and large triangles, sometimes supplemented by further-neighbor terms such as $_2$9 or $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$0 (Duan et al., 9 Jun 2025, Aoyama et al., 2022).

An important generalization is that the breathing kagome lattice need not be a literal atomic lattice. In hexagonal transition metal dichalcogenides, a breathing kagome structure is hidden in the electronic Hilbert space: $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$1-like hybrid d-orbitals form an emergent kagome network with a dominant inter-site hopping $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$2 eV and a weaker intra-site “hopping” $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$3 eV arising from crystal-field splitting. In that setting, the breathing anisotropy is an orbital and hopping effect rather than a real-space rearrangement of atoms (Jung et al., 2021).

2. Electronic structure, flat bands, and Fermiology

The electronic motivation for studying breathing kagome systems is the same as for kagome metals more broadly: flat bands, Dirac features, and Van Hove singularities are especially sensitive to small symmetry-lowering distortions. In Nb$$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$4Br$$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$5, angle-resolved photoemission spectroscopy directly resolves multiple flat and weakly dispersing bands within approximately $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$6 eV below the Fermi level, and first-principles calculations attribute them to Nb d orbitals forming the breathing kagome plane (Regmi et al., 2023). In layered Nb$$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$7Cl$$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$8, Raman and DFT work identifies a van der Waals semiconductor with a breathing kagome lattice, topological flat bands in the band structure, and a low-temperature structural and magnetic transition near $$\Delta_\text{breathe} = d_\text{long} - d_\text{short}.$$9 K (Jeff et al., 2023).

Pressure provides a clean route for continuously tuning the breathing geometry without chemical disorder. In Fe$\Delta_\text{breathe}$0Sn$\Delta_\text{breathe}$1, the breathing distortion is suppressed around $\Delta_\text{breathe}$2 GPa and reversed at higher pressures. Broadband and transient optical spectroscopy, together with DFT based on the experimental structures, reveal a cascade of Lifshitz transitions: a first transition near $\Delta_\text{breathe}$3 GPa associated with the appearance of a new Fermi-surface sheet near $\Delta_\text{breathe}$4, a second near $\Delta_\text{breathe}$5 GPa involving the disappearance of Fermi-surface sheets at $\Delta_\text{breathe}$6 and merging of pockets at the Brillouin-zone boundary, and a possible further reconstruction at higher pressures (Gonçalves-Faria et al., 4 Feb 2025).

The same study shows that approaching the regular kagome geometry does not simply delocalize carriers. The optical correlation ratio,

$\Delta_\text{breathe}$7

peaks near the pressure where the kagome network becomes most regular, while the mid-infrared localization peak becomes more prominent and shifts to higher energy. This suggests that suppressing the breathing distortion can enhance electronic correlations and localization even under compression, a countertrend to the expectation that pressure generally promotes itinerancy (Gonçalves-Faria et al., 4 Feb 2025).

A closely related implication is that the sign and magnitude of breathing distortion operate as a microscopic tuning knob for kagome-band engineering. In Fe$\Delta_\text{breathe}$8Sn$\Delta_\text{breathe}$9, either sign of finite breathing distortion reduces the correlation strength and localization relative to the regular lattice, whereas the regular lattice maximizes both (Gonçalves-Faria et al., 4 Feb 2025). In Nb-halide monolayers, electric-field-driven modification of breathing can instead move the system between topologically trivial and nontrivial electronic regimes, showing that the same structural degree of freedom can control both correlation physics and band topology (Xie et al., 2024, Wang et al., 25 Feb 2025).

3. Topology, corner states, and the higher-order question

Breathing kagome models became prominent in topology because bond alternation can generate corner-localized states in finite flakes. In h-TMDs, the hidden electronic breathing kagome lattice realizes a higher-order topological insulator regime when $\lambda = \frac{d_\nabla}{d_\Delta},$0. The associated Wannier center lies at the center of a downward triangle rather than on an atomic site, producing an obstructed atomic limit, and triangular nanoflakes host triple-degenerate, spatially localized corner states that are robust in the calculated spectrum (Jung et al., 2021).

Mechanical analogues exhibit a related but not identical structure. In a spring-mass model on a breathing kagome lattice, a quantized $\lambda = \frac{d_\nabla}{d_\Delta},$1 Berry phase characterizes higher-order topological phases, and corner vibrational modes appear under fixed boundary conditions. The same work finds a phase with $\lambda = \frac{d_\nabla}{d_\Delta},$2, generated by coupling between longitudinal and transverse modes, in addition to the $\lambda = \frac{d_\nabla}{d_\Delta},$3 phase that corresponds to two copies of the tight-binding limit (Wakao et al., 2019).

The electronic nearest-neighbor breathing kagome lattice, however, is not universally accepted as a genuine second-order topological insulator. Two later analyses show that the familiar zero-energy corner modes can be moved away from zero energy or removed altogether by symmetry-respecting perturbations or by continuous deformation of nearest-neighbor hoppings without closing bulk and boundary gaps. In this view, the bulk phases are best described as obstructed atomic limits with filling anomaly rather than as phases with protected zero-energy corner states (Herrera et al., 2022, Geschner et al., 2024).

This controversy does not eliminate breathing-kagome boundary physics; it refines its classification. One consequence is that corner states in the simplest nearest-neighbor model depend sensitively on sublattice or generalized chiral symmetry, crystalline symmetry, and lattice connectivity (Herrera et al., 2022). Another is that more elaborate models can restore sharply defined bulk-corner correspondence. A breathing kagome lattice with long-range hoppings can host multiple zero-energy corner states, including topologically protected bound states in the continuum, and a momentum-space invariant

$\lambda = \frac{d_\nabla}{d_\Delta},$4

counts the number of corner states per corner and captures three distinct topological phase transitions (Zhang et al., 1 Apr 2025).

Spin-orbit coupling adds another topological layer. In the breathing kagome tight-binding model with antisymmetric nearest-neighbor SOC, both intrinsic and Rashba terms can generate nontrivial $\lambda = \frac{d_\nabla}{d_\Delta},$5 or Chern phases, and the weakly dispersing kagome band remains topological over a broad range of trimerization. Notably, Rashba SOC can produce a topological phase rather than destroy it (Bolens et al., 2018). In monolayer $\lambda = \frac{d_\nabla}{d_\Delta},$6 systems, electric-field manipulation of the breathing mode can even drive a transition from a topologically trivial insulator to a Chern insulator (Xie et al., 2024).

4. Magnetic frustration, chirality, and multi-$\lambda = \frac{d_\nabla}{d_\Delta},$7 textures

Breathing anisotropy strongly reorganizes the magnetic degeneracies of kagome spin systems. In classical Heisenberg antiferromagnets with bond alternation $\lambda = \frac{d_\nabla}{d_\Delta},$8 and a third-nearest-neighbor antiferromagnetic interaction $\lambda = \frac{d_\nabla}{d_\Delta},$9, Monte Carlo simulations find that the commensurate triple-$d_\Delta$0 state becomes noncoplanar and carries finite scalar spin chirality. The resulting state is described as a discrete miniature skyrmion crystal at zero field, with skyrmion number $d_\Delta$1 per magnetic unit cell. In the uniform kagome limit $d_\Delta$2, the same regime instead favors a collinear state selected by thermal fluctuations (Aoyama et al., 2022).

A field-driven route to skyrmion formation appears in breathing-kagome metals with competing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange. For parameters motivated by Gd$d_\Delta$3Ru$d_\Delta$4Al$d_\Delta$5, the ground state evolves from a helical phase at low field to a skyrmion phase at intermediate field and then to a polarized phase at high field. Each skyrmion spans only two unit cells, and coupling itinerant electrons to the local moments produces a strong topological Hall effect through the real-space Berry curvature of the skyrmion texture (Swain et al., 2022).

Breathing kagome magnetism also arises in heterostructures generated by “kagomerization.” In a Mn monolayer deposited on Pt(111) and capped with h-BN, the reconstructed lattice has $d_\Delta$6, nearly flat spin-spiral energy bands, and a spin-spiral minimum along the $d_\Delta$7-K line at $d_\Delta$8 in reciprocal units. Atomistic spin dynamics then yields a noncoplanar triple-$d_\Delta$9 ground state with a 36-site magnetic unit cell and a large nonzero topological charge, potentially relevant to nonlinear Hall responses (Zhou et al., 6 Feb 2025).

These examples establish a recurring pattern: in breathing kagome magnets, bond alternation does not merely perturb a preexisting frustrated manifold. It can select chirality, alter the dimensionality of the order parameter, and stabilize topological spin textures even without Dzyaloshinskii–Moriya interaction or external field, depending on the interaction set (Aoyama et al., 2022, Swain et al., 2022).

5. Correlated insulators, cluster Mott physics, and spin liquids

Breathing anisotropy also reorganizes correlation-driven insulating behavior. Determinant quantum Monte Carlo simulations of the half-filled kagome Hubbard model with intra-unit-cell hopping $d_\nabla$0 and variable inter-unit-cell hopping $d_\nabla$1 show that stronger breathing lowers the critical interaction for the metal-insulator transition. The phase diagram contains a paramagnetic metal at low $d_\nabla$2 and a Mott insulator at high $d_\nabla$3, with $d_\nabla$4 for the normal kagome limit $d_\nabla$5, and decreasing $d_\nabla$6 as $d_\nabla$7 is reduced. The same calculations find enhanced short-range antiferromagnetic correlations and a breathing-dependent sign problem (Duan et al., 9 Jun 2025).

At fractional filling, inter-site repulsion can localize electrons on clusters rather than on sites. In an extended Hubbard model on the breathing kagome lattice at $d_\nabla$8 filling, two distinct cluster Mott insulators arise. Type-I cluster Mott phases localize electrons on one triangle type and retain locally metallic motion within that cluster, while the type-II phase localizes on both triangle types and supports plaquette charge order plus an emergent compact $d_\nabla$9 gauge structure. A unified parton construction describes these phases together with the trivial Fermi liquid metal (Yao et al., 2020).

The breathing degree of freedom is equally central in spin-liquid physics. In Li$_2$0In$_2$1Sc$_2$2Mo$_2$3O$_2$4, chemical pressure from Sc substitution tunes the breathing parameter non-monotonically, and $_2$5SR measurements show a progression from antiferromagnetic long-range order to a quantum spin liquid near the most symmetric lattice. At $_2$6, where the breathing parameter is minimal, zero-field $_2$7SR detects no static internal fields down to $_2$8 mK, susceptibility shows the “1/3 anomaly,” and the specific heat is consistent with a $_2$9 quantum spin liquid (Akbari-Sharbaf et al., 2017).

A related but not identical experimental situation occurs in the vanadium oxyfluoride compound DQVOF, whose V$_{1-x}$0 ions form a breathing kagome lattice with

$_{1-x}$1

as extracted from $_{1-x}$2O NMR and series expansion. Spin-lattice relaxation yields $_{1-x}$3, indicating an essentially gapless excitation spectrum (Orain et al., 2017). This is notable because a projective-symmetry-group and variational Monte Carlo study of the spin-$_{1-x}$4 breathing kagome Heisenberg model found that breathing anisotropy stabilizes a gapped $_{1-x}$5 quantum spin liquid relative to competing $_{1-x}$6 states (Schaffer et al., 2016). A plausible implication is that additional perturbations, such as interlayer couplings or material-specific terms beyond the minimal model, are important in connecting theory and experiment.

6. Materials platforms and experimental access

Breathing kagome physics is now distributed across metals, semiconductors, correlated insulators, van der Waals magnets, and artificial platforms. Representative structural materials include Fe$_{1-x}$7Sn$_{1-x}$8, Nb$_{1-x}$9Br$_x$0, Nb$_x$1Cl$_x$2, Nb$_x$3 and $_x$4 monolayers, DQVOF, and Li$_x$5In$_x$6Sc$_x$7Mo$_x$8O$_x$9 (Gonçalves-Faria et al., 4 Feb 2025, Regmi et al., 2023, Jeff et al., 2023, Orain et al., 2017, Akbari-Sharbaf et al., 2017). There are also electronic-orbital realizations in h-TMDs and designed models in spring-mass systems (Jung et al., 2021, Wakao et al., 2019).

Experimentally, the field is unusually method-diverse. Structural breathing is resolved by single-crystal X-ray diffraction and Raman spectroscopy; bandstructure and flat-band signatures are accessed by ARPES and first-principles calculations; correlation strength, Lifshitz transitions, and carrier localization can be followed by broadband infrared and ultrafast optical spectroscopy; local magnetism and low-energy spin dynamics are probed by NMR and $_2$00SR; and real-space or nanoflake boundary states are addressed through tight-binding, DFT, and proposed STM or forced-vibration measurements (Gonçalves-Faria et al., 4 Feb 2025, Regmi et al., 2023, Jeff et al., 2023, Orain et al., 2017, Wakao et al., 2019).

A major recent development is active control of the breathing mode itself. In monolayer $_2$01, the breathing mode is coupled to ferroelectricity and can be reversed or suppressed by electric-field switching in low-barrier materials, which in turn can reverse the chirality of topological spin structures or switch the electronic state between topologically trivial and Chern-insulating regimes (Xie et al., 2024). In niobium halide monolayers, breathing ferroelectricity further enables electric-field-driven reversal of valley polarization and access to topologically nontrivial valley states, including a quantum anomalous valley Hall regime (Wang et al., 25 Feb 2025).

Taken together, these results establish the breathing kagome lattice as a unifying structural and effective-lattice motif. Its significance lies not in a single universal phase, but in the way a simple alternation between two triangle types propagates through electronic, magnetic, and topological sectors, creating a controlled route between regular kagome behavior and a wide spectrum of symmetry-broken, strongly correlated, and topological states.