Extended Checkerboard Hubbard Model

• Extended Checkerboard Hubbard Model is a family of Hubbard-type Hamiltonians that incorporate checkerboard patterns through charge order, lattice geometry, or plaquette hopping modulation.
• It reveals diverse phenomena including second-order charge-density-wave transitions, frustration-induced dimensional reduction, and unconventional, odd-frequency superconducting pairing.
• Nonlocal interactions and plaquette inhomogeneity are central to tuning phase transitions, exact ordered states, and emergent edge modes across various lattice constructions.

The extended checkerboard Hubbard model denotes a family of Hubbard-type lattice Hamiltonians in which a checkerboard structure enters either as a real-space charge pattern, as the geometry of the underlying lattice, or as an imposed plaquette modulation of the hopping network. In the fermionic literature covered here, the term includes at least four distinct settings: the half-filled square-lattice extended Hubbard model with nearest-neighbor interaction $V$ and checkerboard charge-density-wave order at $\mathbf q=(\pi,\pi)$; the quarter-filled checkerboard-lattice Hubbard model with nearest- and next-nearest-neighbor hoppings $t_1$ and $t_2$; checkerboard-modulated two-leg Hubbard ladders; and generalized checkerboard-lattice Hamiltonians engineered to admit exact plaquette-ordered ground states (Loon et al., 2017, Yanagi et al., 2012, Karakonstantakis et al., 2010, Huang et al., 29 Sep 2025, Nakatsuji et al., 2020). The common theme is that off-site interactions, frustration, or plaquette inhomogeneity couple strongly to bipartite or plaquette ordering tendencies, but the resulting phases and even the meaning of “checkerboard” differ substantially across these constructions.

1. Definitions and Hamiltonian families

The simplest extended checkerboard problem is the half-filled square-lattice extended Hubbard model with nearest-neighbor hopping $t$, on-site interaction $U$, and nearest-neighbor density interaction $V$,

$$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$

With $t\equiv 1$, this model is used to study the competition between local and nonlocal interactions and, for $V>0$, the onset of checkerboard charge order at half filling (Loon et al., 2017).

A second class is the checkerboard-lattice Hubbard model at quarter filling,

$\mathbf q=(\pi,\pi)$0

defined on a two-sublattice checkerboard lattice with nearest-neighbor $\mathbf q=(\pi,\pi)$1-$\mathbf q=(\pi,\pi)$2 hopping $\mathbf q=(\pi,\pi)$3 and next-nearest-neighbor $\mathbf q=(\pi,\pi)$4-$\mathbf q=(\pi,\pi)$5 or $\mathbf q=(\pi,\pi)$6-$\mathbf q=(\pi,\pi)$7 hopping $\mathbf q=(\pi,\pi)$8. Here the checkerboard character is geometric rather than an emergent $\mathbf q=(\pi,\pi)$9 density pattern (Yanagi et al., 2012).

A third class consists of checkerboard ladders, where the hopping amplitudes are modulated in a period-2 plaquette pattern. In the two-leg model of Karakonstantakis et al., the along-chain hopping alternates between $t_1$0 and $t_1$1, while the rung hopping remains $t_1$2; in the later extended variant, an intraplaquette nearest-neighbor attraction $t_1$3 is added on the four bonds inside each $t_1$4 plaquette (Karakonstantakis et al., 2010, Huang et al., 29 Sep 2025).

Finally, generalized checkerboard-lattice Hamiltonians include not only edge and diagonal hoppings but also density-density, pair-hopping, correlated-hopping, and mixed interaction terms. In that setting, the Hamiltonian is tuned into a sum of positive-semidefinite plaquette projectors, which yields exact plaquette-ordered states and exact edge modes (Nakatsuji et al., 2020).

2. Half-filled square lattice: checkerboard charge order

On the half-filled square lattice at $t_1$5, $t_1$6 and moderate to large $t_1$7 drive a second-order transition from a uniform Fermi liquid into a checkerboard charge-density wave characterized by alternating high and low charge densities on the two sublattices. A natural order parameter is the staggered density

$t_1$8

and the equivalent susceptibility criterion is

$t_1$9

Within the dual boson calculation, the checkerboard phase boundary $t_2$0 grows roughly linearly with $t_2$1 (Loon et al., 2017).

$t_2$2	$t_2$3 from DB
1	$t_2$4
4	$t_2$5
8	$t_2$6

At small $t_2$7, single-site EDMFT overestimates the critical repulsion. The explicit comparison at $t_2$8 gives $t_2$9 in EDMFT and $t$0 in DB, so the nonlocal corrections reduce the transition scale by roughly $t$1. The same study emphasized an asymmetry between repulsive and attractive nonlocal interactions: in the attractive $t$2 regime, where phase separation occurs, EDMFT and DB give almost identical results, whereas in the repulsive regime they yield very different checkerboard phase boundaries, especially at small $t$3.

The ordering wavevector is fixed by the lattice Fourier transform of the nearest-neighbor interaction,

$t$4

whose most relevant instability at half filling occurs at $t$5. In this sense, the checkerboard ordered phase of the extended square-lattice Hubbard model is the canonical fermionic realization of “checkerboard” charge order in the present literature.

3. Diagrammatic descriptions and the character of the transition

The checkerboard instability has been studied with several nonperturbative or partially resummed frameworks, and these do not produce identical descriptions of the transition. In EDMFT, the nonlocal interaction enters the impurity problem only through a local retarded interaction $t$6, with lattice susceptibility

$t$7

In the dual boson method, one supplements this with a nonlocal dual polarization $t$8 in the ladder approximation,

$t$9

or, at zero bosonic frequency,

$U$0

At $U$1, the numerical estimate

$U$2

shows that the nonlocal ladder corrections selectively enhance the checkerboard response (Loon et al., 2017).

A distinct picture emerges in the strong coupling diagram technique. For the two-dimensional extended Hubbard model at half filling, infinite ladder series are summed in both charge and spin channels, with explicit short-range antiferromagnetic fluctuations and the actual short-range antiferromagnetic order at nonzero temperatures. In that treatment, a first-order phase transition in the charge subsystem occurs at $U$3. The transition is signaled by an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin zone, and the resulting states exhibit alternating deviations of site occupancies from the mean value on neighboring sites; these deviations possess only short-range order for the considered parameters (Sherman, 2023).

For $U$4, the SCDT estimates are

$U$5

Thus, for $U$6, a first-order charge transition is found, whereas for $U$7 no charge-order transition appears in $U$8; instead, the system remains a Mott insulator and the Mott gap grows continuously with $U$9. At $V$0, $V$1, the reported values are $V$2, $V$3, and $V$4 for $V$5, $V$6, and $V$7, respectively. The methodological contrast is therefore sharp: EDMFT/DB identify a checkerboard susceptibility divergence with a second-order onset, whereas SCDT finds a first-order charge-sector transition accompanied by finite, sign-changing $V$8 and short-range checkerboard correlations.

4. Checkerboard lattice at quarter filling: frustration, dimensional reduction, and odd-frequency pairing

In the quarter-filled checkerboard-lattice Hubbard model, the dominant organizing principle is geometrical frustration rather than a nearest-neighbor density repulsion. When $V$9, the lattice decomposes into two sets of orthogonal one-dimensional chains, and each chain carries a quarter-filled 1D band with Fermi points at $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$0 on the $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$1 chains or $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$2 on the $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$3 chains. Turning on $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$4 hybridizes the chains, but the hybridization vanishes exactly on the zone-boundary lines $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$5. As a result, the Fermi surface remains quasi-1D as long as $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$6, and the noninteracting susceptibility develops ridge-like peaks along $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$7 and $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$8; this one-dimensional character survives up to $$H = -t\sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} + V\sum_{\langle ij\rangle} n_i n_j .$$9 (Yanagi et al., 2012).

At the Mielke point $t\equiv 1$0, the completely saturated ferromagnet is exact for any $t\equiv 1$1. Away from that line, the mean-field and RPA phase diagram in the $t\equiv 1$2 plane contains a ferromagnetic region, a sequence of antiferromagnetic or incommensurate phases, and superconducting instabilities obtained from the linearized Eliashberg equation,

$t\equiv 1$3

Within RPA,

$t\equiv 1$4

with $t\equiv 1$5 for spin singlets and $t\equiv 1$6 for spin triplets.

The pairing states depend strongly on the frustration ratio. For $t\equiv 1$7 near the antiferromagnetic boundary at $t\equiv 1$8, the leading instability is an even-frequency spin-singlet extended-$t\equiv 1$9 state. For $V>0$0, the pronounced 1D spin fluctuations favor an odd-frequency spin-singlet $V>0$1-wave state over a wide interval of $V>0$2. Very close to $V>0$3, where the spin-fluctuation momentum dependence becomes almost flat, an odd-frequency spin-triplet $V>0$4-wave state appears at intermediate $V>0$5. The odd-frequency character follows from Fermi statistics: for a spin-singlet state,

$V>0$6

so a spatially odd $V>0$7-wave gap must be odd in frequency.

This checkerboard-lattice problem is therefore conceptually separate from the half-filled square-lattice charge-order problem. Its checkerboard physics is rooted in flat-band ferromagnetism, frustration-driven dimensional reduction, and unconventional pairing, not in a nearest-neighbor-repulsion-driven $V>0$8 CDW.

5. Checkerboard ladders and superconductivity

Checkerboard two-leg ladders provide a controlled quasi-one-dimensional setting in which plaquette inhomogeneity can be varied continuously. In the period-2 checkerboard ladder without explicit off-site interaction, the along-chain hopping alternates between $V>0$9 and $\mathbf q=(\pi,\pi)$00 while the rung hopping remains $\mathbf q=(\pi,\pi)$01. DMRG calculations on lengths $\mathbf q=(\pi,\pi)$02 with open boundary conditions and up to $\mathbf q=(\pi,\pi)$03 kept states found the largest pairing scales near density $\mathbf q=(\pi,\pi)$04, $\mathbf q=(\pi,\pi)$05, and $\mathbf q=(\pi,\pi)$06. After thermodynamic extrapolation, the reported values are $\mathbf q=(\pi,\pi)$07 and $\mathbf q=(\pi,\pi)$08. Period-1 and period-3 modulations also enhance pairing relative to the uniform ladder, but less strongly; the checkerboard period-2 ladder shows up to a $\mathbf q=(\pi,\pi)$09 enhancement of $\mathbf q=(\pi,\pi)$10 (Karakonstantakis et al., 2010).

The extended checkerboard ladder adds an intraplaquette nearest-neighbor attraction $\mathbf q=(\pi,\pi)$11,

$\mathbf q=(\pi,\pi)$12

and was studied by DMRG on a $\mathbf q=(\pi,\pi)$13 ladder at density $\mathbf q=(\pi,\pi)$14, with up to $\mathbf q=(\pi,\pi)$15 states and truncation error $\mathbf q=(\pi,\pi)$16. Pairing is diagnosed through singlet pair fields $\mathbf q=(\pi,\pi)$17, $\mathbf q=(\pi,\pi)$18 and correlations $\mathbf q=(\pi,\pi)$19, where $\mathbf q=(\pi,\pi)$20 signals dominant superconducting correlations (Huang et al., 29 Sep 2025).

$\mathbf q=(\pi,\pi)$21	$\mathbf q=(\pi,\pi)$22 at $\mathbf q=(\pi,\pi)$23	$\mathbf q=(\pi,\pi)$24 at $\mathbf q=(\pi,\pi)$25
1.0	$\mathbf q=(\pi,\pi)$26	$\mathbf q=(\pi,\pi)$27
0.4	$\mathbf q=(\pi,\pi)$28	—
0.2	$\mathbf q=(\pi,\pi)$29	—
0.1	$\mathbf q=(\pi,\pi)$30	—
0.05	$\mathbf q=(\pi,\pi)$31	$\mathbf q=(\pi,\pi)$32–$\mathbf q=(\pi,\pi)$33

The main numerical trend is that inhomogeneity, $\mathbf q=(\pi,\pi)$34, drastically reduces the magnitude of the attraction required to enter the superconducting phase. In the non-superconducting regime, both spin and single-particle correlations decay algebraically, indicating gapless modes. Once $\mathbf q=(\pi,\pi)$35, they cross over to exponential decay, indicating simultaneous spin and single-particle gaps. The pairing symmetry is identified from the signs of three correlation functions: $\mathbf q=(\pi,\pi)$36-wave behavior corresponds to $\mathbf q=(\pi,\pi)$37, $\mathbf q=(\pi,\pi)$38, and $\mathbf q=(\pi,\pi)$39. In the homogeneous case, the pairing is strongly anisotropic, $\mathbf q=(\pi,\pi)$40; in the inhomogeneous regime, $\mathbf q=(\pi,\pi)$41 within numerical error, implying an approximate $\mathbf q=(\pi,\pi)$42 symmetry of the $\mathbf q=(\pi,\pi)$43-wave pair wavefunction.

The ladder literature thus isolates two related mechanisms. Plaquette-scale hopping modulation alone enhances pairing relative to the uniform ladder, and adding a moderate intraplaquette attraction can convert that enhancement into a bona fide $\mathbf q=(\pi,\pi)$44-wave superconducting ground state with a reduced critical interaction scale.

6. Generalized checkerboard models with exact plaquette order

A more algebraic extension of the checkerboard Hubbard model is obtained by enlarging the local interaction content on each four-site plaquette $\mathbf q=(\pi,\pi)$45. The local term $\mathbf q=(\pi,\pi)$46 may contain edge hopping $\mathbf q=(\pi,\pi)$47, diagonal hopping $\mathbf q=(\pi,\pi)$48, on-site $\mathbf q=(\pi,\pi)$49, edge density-density couplings $\mathbf q=(\pi,\pi)$50 and $\mathbf q=(\pi,\pi)$51, diagonal density-density couplings $\mathbf q=(\pi,\pi)$52 and $\mathbf q=(\pi,\pi)$53, bond-bond interactions $\mathbf q=(\pi,\pi)$54 and $\mathbf q=(\pi,\pi)$55, and three-site correlated-hopping and mixed terms $\mathbf q=(\pi,\pi)$56. Under the constraints

$\mathbf q=(\pi,\pi)$57

the plaquette Hamiltonian can be rewritten as a sum of positive-semidefinite products of plaquette projectors (Nakatsuji et al., 2020).

This construction yields exact “plaquette-Néel” ground states on the checkerboard lattice. Introducing the plaquette molecular orbitals $\mathbf q=(\pi,\pi)$58, $\mathbf q=(\pi,\pi)$59, $\mathbf q=(\pi,\pi)$60, and $\mathbf q=(\pi,\pi)$61, one obtains exact states at $\mathbf q=(\pi,\pi)$62, $\mathbf q=(\pi,\pi)$63, and $\mathbf q=(\pi,\pi)$64 filling. At $\mathbf q=(\pi,\pi)$65 filling, there is one electron per plaquette occupying the symmetric $\mathbf q=(\pi,\pi)$66 mode on one plaquette sublattice for spin up and on the other for spin down. At $\mathbf q=(\pi,\pi)$67 filling, two degenerate patterns appear, denoted $\mathbf q=(\pi,\pi)$68 and $\mathbf q=(\pi,\pi)$69. At $\mathbf q=(\pi,\pi)$70 filling, two further Néel-like patterns, $\mathbf q=(\pi,\pi)$71 and $\mathbf q=(\pi,\pi)$72, are obtained.

Open boundaries generate exact edge states. At $\mathbf q=(\pi,\pi)$73 filling, each broken plaquette contributes one localized edge electron with energy shift

$\mathbf q=(\pi,\pi)$74

and the total ground-state energy becomes

$\mathbf q=(\pi,\pi)$75

The same framework also gives an exact area-law bipartite entanglement entropy,

$\mathbf q=(\pi,\pi)$76

This generalized model is not equivalent to the ordinary checkerboard Hubbard Hamiltonian with only $\mathbf q=(\pi,\pi)$77 hopping and on-site $\mathbf q=(\pi,\pi)$78. Its significance is instead constructive: by adding diagonal hopping, off-site density interactions, pair-hopping, and correlated-hopping terms, one can tune the local checkerboard plaquette problem into an exactly solvable projector form and thereby obtain nonperturbative results for ordered bulk states, edge modes, and entanglement.

7. Terminology, related models, and recurring themes

A persistent source of ambiguity is that “checkerboard” does not designate a single object. In the half-filled extended square-lattice Hubbard model, it refers to a $\mathbf q=(\pi,\pi)$79 charge pattern. In the quarter-filled checkerboard-lattice Hubbard model, it refers to the lattice geometry and the associated frustration. In checkerboard ladders, it refers to a plaquette-periodic hopping modulation. In generalized projector models, it refers simultaneously to the corner-sharing plaquette geometry and to exact plaquette order. The literature summarized here therefore treats a family of related but non-identical models rather than a unique canonical Hamiltonian.

A second recurring theme is the central role of nonlocality. In the half-filled square-lattice problem, nonlocal density interactions and nonlocal polarization corrections determine the checkerboard boundary (Loon et al., 2017). In the SCDT treatment, short-range antiferromagnetic fluctuations feed back into the charge sector and change the apparent nature of the transition (Sherman, 2023). In the checkerboard lattice, nonlocal hoppings $\mathbf q=(\pi,\pi)$80 and $\mathbf q=(\pi,\pi)$81 generate dimensional reduction and unconventional odd-frequency pairing (Yanagi et al., 2012). In checkerboard ladders, plaquette inhomogeneity and intraplaquette attraction together reduce the critical scale for superconductivity (Karakonstantakis et al., 2010, Huang et al., 29 Sep 2025). In generalized checkerboard models, explicit off-site interactions are what make exact plaquette ordering possible (Nakatsuji et al., 2020).

Closely related checkerboard phenomena also occur in Bose-Hubbard systems, although these are distinct from the fermionic Hubbard models proper. The bosonic literature includes checkerboard superlattices in artificial gauge fields, extended Bose-Hubbard models realized with dipolar excitons, hard-core models with checkerboard supersolids, and three-body-constrained attractive models with a dimer checkerboard solid (Iskin, 2011, Lagoin et al., 2022, Chen et al., 2017, Chen et al., 2011). This broader context reinforces a narrower conclusion for fermionic systems: whenever checkerboard geometry, bipartite density order, or plaquette modulation is combined with off-site couplings or frustration, the resulting Hubbard-type model tends to exhibit phases that are highly sensitive to momentum-dependent correlations, short-range ordering tendencies, and the precise spatial structure of the nonlocal terms.