Cluster Mott Insulator Model

• The Cluster Mott Insulator Model describes strongly correlated systems where electrons localize on atomic clusters, leading to unconventional charge and spin dynamics.
• The model employs an extended Hubbard Hamiltonian that balances kinetic energy with strong intra- and inter-cluster Coulomb repulsions to stabilize unique localized states.
• Effective low-energy theories reveal emergent gauge fields and complex spin-orbital interactions, which underpin experimental signatures in materials like lacunar spinels and Kagome magnets.

A cluster Mott insulator (CMI) is a distinct class of strongly correlated electron systems in which electrons are localized not on individual atomic sites but on clusters or molecular units comprising several atoms. This "cluster Mottness" arises from the interplay between kinetic energy, on-site and intersite Coulomb interactions, and, in many cases, crystal geometry or underlying orbital structure. CMIs differ fundamentally from conventional atomic-site Mott insulators; they exhibit exotic ground states, emergent degrees of freedom, and often support unconventional magnetic, orbital, and topological phenomena. Realizations span several materials classes, including trimerized and breathing Kagome magnets, lacunar spinels, pyrochlore oxides, and layered dichalcogenides.

1. Cluster Mott Physics: Fundamentals and Model Hamiltonians

In the prototypical CMI scenario, electrons are tightly constrained by strong on-site repulsion $U$ and strong intersite repulsions $V_{ij}$, often at partial electronic filling. The minimal model is the extended Hubbard Hamiltonian incorporating both types of interactions: $$H = -\sum_{\langle ij \rangle, \sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + \sum_{\langle ij \rangle} V_{ij} n_i n_j + U\sum_i n_{i\uparrow} n_{i\downarrow}$$ where $t_{ij}$ is the (possibly cluster-dependent) hopping amplitude, and $n_{i}$ is the electron (or boson) number operator. At certain commensurate fillings, CMIs form when $V_{ij} \gg t_{ij}$, stabilizing charge localization on clusters (e.g., triangles, tetrahedra, or larger units) rather than single sites (Chen et al., 2015, Lv et al., 2015).

CMI states feature emergent low-energy degrees of freedom associated with the collective quantum numbers of the clusters—such as total cluster spin, orbital, or charge multiplets—rather than those of atomic sites (Jayakumar et al., 2023, Magnaterra et al., 2023). The intracluster Hamiltonian often takes the Kanamori form to account for intra-orbital and inter-orbital interactions as well as Hund's couplings.

2. Cluster Localization Mechanisms and Phase Structure

The localization into clusters arises when intersite repulsions dominate over hopping, generally in systems with partial band filling (e.g., $1/6$ filling on a Kagome lattice) such that on-site $U$ alone cannot provide a charge gap. This distinguishes CMIs from standard atomic-site Mott phases. Two archetypal cluster localization regimes, extensively explored in trimerized and breathing Kagome systems, include:

• Type-I CMI: Electrons localize on one sublattice of clusters (e.g., up-triangles), with full site-to-site delocalization within each cluster but immobility between clusters. The system retains "local metallic" behavior within clusters and typically supports a gapless U(1) spin liquid with a spinon Fermi surface (Yao et al., 2020, Chen et al., 2015).
• Type-II CMI: Localization on all clusters (e.g., up and down triangles), leading to a full charge-ordered state. Low-energy dynamics are governed by higher-order ring-exchange (plaquette) processes, generating quantum "resonances" between distinct cluster occupations (Chen et al., 2017, Chen et al., 2015).

The resultant phase diagrams can be understood in terms of the competition between $V_{1}/t_{2}$ and $V_{2}/t_{1}$, where $V_{1,2}$ and $t_{1,2}$ denote intersite repulsions and hoppings on distinct cluster types (such as up/down triangles) (Yao et al., 2020). Transitions between these phases are typically continuous Higgs transitions of the order parameter representing cluster charge delocalization.

3. Effective Low-Energy Theories and Emergent Gauge Structures

CMIs exhibit emergent gauge structures in their low-energy descriptions. In the strong-coupling limit, ring-exchange (higher-order virtual hopping around a closed loop of sites) becomes the leading process, yielding effective quantum dimer or compact U(1) gauge models. For example, in the breathing Kagome or pyrochlore lattices: $H_{\rm ring} = -J_{\rm ring} \sum_{\hexagon} (L_1^+ L_2^- L_3^+ L_4^- L_5^+ L_6^- + {\rm h.c.}),\qquad J_{\rm ring} \sim \frac{t^3}{V^2}$ creates resonance between different cluster charge configurations (Yao et al., 2020, Chen et al., 2017, Lv et al., 2015).

Charge sector constraints (such as "one per triangle") are recast as Gauss's-law conditions for the emergent gauge field, and low-energy charge fluctuations map to divergence-free electric field configurations. In three dimensions, as for the pyrochlore lattice, this leads to a Coulomb liquid phase with photon-like emergent excitations and characteristic $T^3$ specific heat (Lv et al., 2015).

The spin sector is typically described by effective spin or spin-orbital models on the emergent cluster lattice, often combined with gauge fields if charge and spin degrees of freedom fractionalize. In strong cluster Mottness, Kugel–Khomskii-type interactions naturally arise: $$H_{\rm KK} = \sum_{\langle RR'\rangle} \big[(\mathbf{s}_R \cdot \mathbf{s}_{R'}) (1 + 4 T^{\mu}_{R}) (1 - 2 T^{\mu}_{R'})\big]$$ coupling spin and emergent orbital (pseudospin) degrees of freedom on clusters (Chen et al., 2017, Chen et al., 2015).

4. Physical Properties, Emergent Degrees of Freedom, and Experimental Signatures

CMIs realize a variety of unconventional physical behaviors distinct from single-site Mott insulators:

• Spinon Fermi Surface States: In many CMIs, a gapless U(1) spin liquid ground state is stabilized with a spinon Fermi surface. This is reflected in thermodynamic (specific heat), transport (finite residual susceptibility), and neutron-scattering (continuum excitations, 2$k_F$ features) measurements (Chen et al., 2015, He et al., 2018).
• Plaquette Charge Order (PCO): Quantum ring-exchange selects translational-symmetry-breaking plaquette charge order, resonating clusters that triple the unit cell and lead to observable modulations in X-ray or neutron scattering (Yao et al., 2020, Chen et al., 2017).
• Partial Magnetic Moment Freezing: In the PCO phase, only a fraction of cluster spins remain magnetically active (often precisely 1/3), resulting in multiple Curie-Weiss regimes in spin susceptibility—accounting for the 2/3 reduction of Curie constant observed in LiZn$_2$Mo$_3$O$_8$ (Yao et al., 2020, Chen et al., 2015).
• Emergent Orbitals: Resonating hexagons or other clusters develop low-lying orbital-like degrees of freedom protected by point-group symmetry, leading to novel spin-orbital multiplets and possible orbital order under external perturbations (Chen et al., 2017, Jayakumar et al., 2023).
• Cluster Hund's Physics: The hierarchy of intra-cluster spin, orbital, and charge orders is distinct from single-site Hund rules; a "cluster Hund's rule" emerges, minimizing local charge fluctuations and maximizing cluster-wide spin/orbital quantum numbers (Jayakumar et al., 2023).

Experimental signatures are diverse: diffuse or superlattice peaks in scattering (evidence for PCO or cluster localization); double Curie regimes in susceptibility; specific-heat power laws ($T^{2/3}$ in 2D U(1); $T^3$ in 3D Coulomb liquids); direct observation of internal cluster excitations in Raman scattering for spin cluster CMIs such as Cu$_2$OSeO$_3$ (1908.10279).

5. Representative Material Systems and Ab Initio Connections

Several material systems are now established as archetypes of the cluster Mott paradigm:

• Nb$_3$Cl$_8$: Monolayer and bulk forms realize a single-orbital cluster Hubbard model (one symmetric $\psi_0$ orbital per Nb$_3$ trimer). Ab initio cRPA and Wannier analysis yields $U^*_{{\rm mono}} \approx 1.13$ eV, bandwidth $W \approx 0.20$ eV, and robust Mott gaps $\Delta \sim 1$–1.2 eV (Grytsiuk et al., 2023).
• LiZn$_2$Mo$_3$O$_8$, Li$_2$InMo$_3$O$_8$: Kagome/trimerized lattices at 1/6-filling, with experimentally confirmed multiple Curie regimes, PCO, and cluster Mott phases; supported by both theoretical modelling (Yao et al., 2020, Chen et al., 2017, Nikolaev et al., 2020) and ab initio calculations.
• GaTa$_4$Se$_8$ and Lacunar Spinels: Electron delocalization over Ta$_4$ tetrahedra forms cluster $J_{\rm tet}=3/2$ moments, with the cluster wavefunction and effective SOC renormalized and tunable by intracluster hopping (Magnaterra et al., 2023).
• Cu$_2$OSeO$_3$: Magnetism is dominated by rigid Cu$_4$ tetrahedral spin clusters, evidenced by clear separation of intra- and inter-cluster excitation scales and Raman-active internal magnon modes (1908.10279).

Ab initio methods such as constrained RPA, density functional theory, and explicit Wannier orbital construction are essential for parameter extraction and quantitative comparison to experiment (Grytsiuk et al., 2023).

6. Variations: Bosonic and Multi-Orbital Cluster Mott Insulators

Cluster localization physics is not restricted to fermionic electrons. In the hard-core boson Hubbard model on the pyrochlore lattice, robust cluster Mott phases form at fractional fillings ($\rho=1/4,\;1/2,\;3/4$), resulting in Coulomb quantum liquids described by a compact U(1) gauge theory (Lv et al., 2015). These phases exhibit characteristic dipolar correlations in electric field correlators, $T^3$ specific heat due to emergent photons, and entropy plateaus associated with ice-rule constraints.

Multi-orbital and multi-site cluster models—analyzed with exact diagonalization—reveal a taxonomy of symmetry-protected cluster multiplets (spin, orbital, and non-Kramers degeneracies), their susceptibility to symmetry breaking, and the phase competition between uniform and clumped charge, all crucial for many candidate CMIs (Jayakumar et al., 2023).

7. Outlook, Open Questions, and Experimental Probes

CMIs provide a versatile platform for strong-correlation physics beyond the canonical Mott–Hubbard paradigm, including unconventional spin-liquids, spin-orbital entanglement, emergent gauge excitations, and tunable quantum magnetism. Open directions include:

• The nature and robustness of cluster spin liquids and the precise conditions for stabilization of spinon Fermi surfaces.
• The role of disorder, stoichiometric deviations, and external fields in melting or modifying cluster orders (Chen et al., 2017).
• The interplay of spin-orbit coupling, spin-phonon coupling, and inter-cluster hopping in modifying the collective behavior (Magnaterra et al., 2023, 1908.10279).
• Extensions to itinerant or multiorbital systems, and potential for topologically nontrivial or heavy-fermion–like behavior as observed in certain transition-metal cluster compounds.

State-of-the-art probes—such as resonant X-ray and neutron scattering, Raman spectroscopy for internal cluster modes, muon spin rotation, thermodynamic measurements, and advanced ab initio simulation—are indispensable tools for disentangling the complex phenomenology and revealing the full organizing principles of cluster Mott physics.