Square Lattice Hubbard Model

• Square lattice Hubbard model is a 2D lattice model of strongly correlated electrons with nearest-neighbor hopping t and on-site interaction U.
• It elucidates phenomena such as Mott transitions, antiferromagnetic order, charge density waves, and superconductivity through various numerical and analytical methods.
• Extensions include frustrated t-t′ variants, extended interactions, and SU(3) generalizations, enabling tunable quantum simulations and insights into competing orders.

to investigate Mott transitions, antiferromagnetism, superconductivity, and competing inhomogeneous states." That appears in various data. Good.

Need maybe include a small table for variants with Hamiltonians? But tables can be brief. Yet detailed explanations should be paragraphs. We can maybe avoid table.

Let's search arxiv. The square lattice Hubbard model is a class of lattice fermion models defined on a two-dimensional square lattice, most commonly in its one-band form with nearest-neighbor hopping $t$ and on-site interaction $U$. In the data considered here, it appears in several closely related realizations: the standard SU(2) model, extended models with nearest-neighbor interaction $V$, frustrated $t\text{-}t'\text{-}U$ variants, strong-coupling spin reductions with ring exchange, and multicomponent generalizations such as SU(3). Across these formulations, the square lattice geometry provides the common setting in which Mott physics, antiferromagnetism, charge order, superconductivity, phase separation, and inhomogeneous states are analyzed with exact diagonalization, determinant quantum Monte Carlo, variational cluster methods, density matrix embedding theory, variational Monte Carlo, and symmetry-based operator constructions (Harir et al., 2011).

1. Standard formulation and principal generalizations

In its standard one-band form on the square lattice, the Hubbard Hamiltonian is written as

$$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$

with nearest-neighbor hopping $t$, on-site repulsion $U$, and fermion operators $c_{i\sigma}^{\dagger},c_{i\sigma}$. A closely related notation used in the strong-coupling spin-expansion literature writes

$$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$

at half filling with $\mu=0$. The model is also written in a frustrated form with next-nearest-neighbor hopping,

$U$0

and in an anisotropic frustrated form with inequivalent diagonal hoppings $U$1. These extensions retain the square lattice while altering magnetic competition and the low-energy phase structure (Zheng et al., 2015).

A second major generalization is the extended Hubbard model, where a nearest-neighbor density interaction $U$2 supplements the on-site term: $U$3 At half filling, another equivalent form writes the interaction as $U$4, or as $U$5 with $U$6. Setting $U$7 reduces the model exactly to the usual square-lattice Hubbard model. The data also include an SU(3) square-lattice Fermi-Hubbard model,

$U$8

with three fermionic flavors, nearest-neighbor hopping, and onsite interactions $U$9, as well as depleted, $V$0-flux, and moiré-derived square-lattice descendants. This suggests that “square lattice Hubbard model” denotes not a single Hamiltonian, but a family of models sharing the same lattice backbone while differing in orbital content, symmetry, and range of couplings.

2. Half filling, strong coupling, and antiferromagnetic order

At half filling, the square-lattice Hubbard model is repeatedly treated as a Mott and antiferromagnetic problem. In the symmetry-based operator formulation, the repulsive model on a bipartite square lattice has exact global symmetry

$V$1

with spin, $V$2-spin, and a hidden $V$3 generated by the number of rotated-electron singly occupied sites. For $V$4, $V$5, and $V$6, long-range Néel antiferromagnetic order is argued to occur in the half-filled ground state, lowering the ground-state symmetry from $V$7 for large finite $V$8 to

$V$9

in the thermodynamic limit. In that construction, the equality of the spin effective lattice and the original square lattice is identified as a necessary condition for long-range ground-state antiferromagnetic order (Carmelo, 2010).

In the strong-coupling regime $t\text{-}t'\text{-}U$0, the half-filled square-lattice model reduces to an effective spin problem. To second order in $t\text{-}t'\text{-}U$1, one recovers the nearest-neighbor Heisenberg antiferromagnet; to fourth order, the effective Hamiltonian contains second- and third-neighbor exchanges and a four-spin ring-exchange term,

$t\text{-}t'\text{-}U$2

$t\text{-}t'\text{-}U$3

with

$t\text{-}t'\text{-}U$4

Exact diagonalization of this Hubbard-parametrized spin model finds a quantum phase transition from $t\text{-}t'\text{-}U$5 Néel order to a $t\text{-}t'\text{-}U$6 ordered state, with the critical ratio reduced by quantum fluctuations from the mean-field value $t\text{-}t'\text{-}U$7 to approximately $t\text{-}t'\text{-}U$8. Near that transition, the dynamical spin correlation function develops a pseudo-continuum at wavevectors between the competing ordering vectors (Larsen et al., 2018).

A complementary cluster exact-diagonalization study of the extended model on a $t\text{-}t'\text{-}U$9 square lattice with four electrons and periodic boundary conditions makes the low-filling many-body spectrum fully explicit. In the $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$0 limit, the first excited state energy is

$$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$1

for all $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$2, implying a purely kinetic state with no double occupancy, whereas the ground state retains double occupancy for all $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$3. This finite-cluster result is not a thermodynamic statement, but it provides an exact benchmark for how on-site repulsion alone organizes low-lying states on the square lattice (Harir et al., 2011).

3. Charge order, superconductivity, and phase separation in the extended model

Adding nearest-neighbor interaction $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$4 qualitatively restructures the half-filled square-lattice phase diagram. Large-scale determinant quantum Monte Carlo for the half-filled extended Hubbard model in the $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$5–$$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$6 plane finds quantitatively resolved boundaries among antiferromagnetic, charge-ordered, $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$7-wave superconducting, $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$8-wave superconducting, and phase-separated ground states. In the repulsive first quadrant, the antiferromagnetic–charge-density-wave boundary lies slightly above the strong-coupling line $$H = -t\sum_{\langle i,j\rangle,\sigma} \left( c^{\dagger}_{i\sigma} c_{j\sigma} + \text{h.c.}\right) + U \sum_{i} n_{i\uparrow} n_{i\downarrow},$$9; finite-size scaling gives representative critical values $t$0 at $t$1, $t$2 at $t$3, and $t$4 at $t$5. In the attractive quadrants, $t$6 supports $t$7-wave superconductivity above the line $t$8, $t$9-wave superconductivity below that line, and phase separation at sufficiently large negative $U$0. In the repulsive sector with $U$1, an intermediate $U$2-wave region appears for $U$3, flanked by antiferromagnetism and phase separation (Sousa-Júnior et al., 2023).

Exact diagonalization on a $U$4 periodic cluster yields a closely related but cluster-limited picture. At half filling and in the strong-coupling regime, the phase diagram contains spin density wave and $U$5-wave superconductivity at large positive $U$6, charge density wave and extended $U$7-wave superconductivity at large positive and negative $U$8, and $U$9-wave superconductivity at large negative $c_{i\sigma}^{\dagger},c_{i\sigma}$0 with vanishing $c_{i\sigma}^{\dagger},c_{i\sigma}$1. The energies of different particle sectors delineate the phase separation region. With hole doping, charge fluctuation produces strong competition between different orders, and at $c_{i\sigma}^{\dagger},c_{i\sigma}$2–$c_{i\sigma}^{\dagger},c_{i\sigma}$3 hole doping the $c_{i\sigma}^{\dagger},c_{i\sigma}$4-wave pairing channel is found to be enhanced, especially in an interaction range described as relevant to the cuprate superconductors (Chen et al., 2022).

A strong-coupling diagram-technique treatment of the same half-filled extended model emphasizes the charge sector. Summing infinite ladder series for spin and charge fluctuations, it finds a first-order charge transition at $c_{i\sigma}^{\dagger},c_{i\sigma}$5, visible as an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the Brillouin-zone corner $c_{i\sigma}^{\dagger},c_{i\sigma}$6. The associated states have alternating deviations of electron occupations from the mean value on neighboring sites; due to fluctuations these deviations possess short-range order. For the parameters considered, this behavior appears for $c_{i\sigma}^{\dagger},c_{i\sigma}$7, while for insulating $c_{i\sigma}^{\dagger},c_{i\sigma}$8 the transition is not observed and the Mott gap grows continuously with $c_{i\sigma}^{\dagger},c_{i\sigma}$9 (Sherman, 2023).

The finite-temperature sector is partly resolved as well. Determinant quantum Monte Carlo determines a partial $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$0 diagram in which CDW critical temperatures in the repulsive $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$1 sector rise rapidly with $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$2, and $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$3-wave superconducting critical temperatures in the attractive $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$4 sector are enhanced by nearest-neighbor attraction relative to the pure attractive Hubbard limit. By contrast, at half filling in the repulsive $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$5 sector, $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$6-wave superconductivity is found to have a much smaller $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$7 than the $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$8-wave state in the attractive sector. This suggests that nearest-neighbor interactions enrich square-lattice Hubbard physics not only by introducing new phases, but also by splitting the energy scales of ordering channels that are degenerate or absent in the on-site model.

4. Doping, frustration, and competing inhomogeneous states

Away from half filling, the square-lattice Hubbard model supports metallic, antiferromagnetic, superconducting, and inhomogeneous states whose balance depends strongly on $$H = - t\sum_{i,\tau,\sigma} \hat{c}_{i,\sigma}^\dagger \hat{c}_{i+\tau,\sigma} -\mu \sum_i n_{i,\sigma} + U \sum_i n_{i,\uparrow} n_{i,\downarrow},$$9, $\mu=0$0, and density. Density matrix embedding theory on the thermodynamic-limit square lattice, with impurity clusters up to 16 sites and an explicit error model, finds antiferromagnetism at half filling, a d-wave superconducting region at intermediate doping, a metallic regime at large doping and small $\mu=0$1, and a low-doping sector in which inhomogeneous charge, spin, and pairing states compete very closely in energy. For $\mu=0$2 and $\mu=0$3, the d-wave order parameter has a dome-like dependence on filling, peaking near $\mu=0$4 and extending roughly over $\mu=0$5. Positive $\mu=0$6 stabilizes antiferromagnetism and suppresses d-wave superconductivity, while negative $\mu=0$7 enhances d-wave superconductivity and strengthens low-doping inhomogeneity, mirroring the contrast between electron-doped and hole-doped cuprates (Zheng et al., 2015).

In the underdoped strong-coupling regime, the inhomogeneous solutions found by embedding include superconducting ribbons, classic stripes with period-4 charge modulation and period-8 spin modulation, and states combining spin-density wave, charge-density wave, and pair-density wave structure. The energies of these states are nearly degenerate with more homogeneous d-wave solutions, and the dominant uncertainty comes from cluster-size extrapolation rather than from the impurity solver. This suggests that the low-doping square-lattice Hubbard model is intrinsically a near-degenerate problem in which stripe, pair-density-wave, and uniform superconducting tendencies remain in close competition.

Frustration introduced through anisotropic next-nearest-neighbor hoppings $\mu=0$8 produces a distinct but related landscape. Variational cluster calculations at half filling on the square, crossed-square, and triangular limits of the same two-parameter lattice family find Néel, collinear, and $\mu=0$9 spiral magnetic phases, together with a nonmagnetic insulating phase separating them in broad parameter windows at strong coupling. For $U$00, along the line $U$01, the nonmagnetic insulator occupies $U$02; along $U$03, it occupies $U$04. At weaker coupling, the spiral phase shrinks and metallic phases appear. This demonstrates that geometric frustration within the square-lattice Hubbard framework can suppress conventional order and stabilize quantum-disordered insulating states (Misumi et al., 2015).

A distinct symmetry-based analysis of lightly doped square-lattice Hubbard physics argues that for very small hole concentration $U$05, the ground state has short-range incommensurate-spiral spin order rather than long-range Néel order. In that construction the spin effective lattice spacing becomes $U$06, no longer identical to the original lattice, and the order parameter $U$07 for short-range spin order decreases approximately as $U$08 for $U$09. This does not coincide methodologically with the embedding calculations, but both approaches identify low doping as the regime where antiferromagnetism gives way to more intricate spin and pairing structures.

5. Symmetry structure and multicomponent square-lattice variants

The square-lattice Hubbard model also supports nontrivial exact and emergent symmetry descriptions. In the rotated-electron formulation of the SU(2) model, the exact global symmetry $U$10 organizes the Hilbert space into charge $U$11 fermions, spinons, and $U$12-spinons. In the one- and two-electron subspace, the low-energy dynamics reduces to a two-component quantum liquid of spinless $U$13 fermions and spin-neutral $U$14 fermions. This formalism is used to connect half-filled long-range antiferromagnetism with finite-doping short-range incommensurate spin order, and to interpret spontaneous symmetry breaking in terms of the relation between the original lattice and the spin effective lattice (Carmelo, 2010).

For SU(3), the square-lattice Fermi-Hubbard model exhibits a different hierarchy of phases. Variational Monte Carlo with Slater, Jastrow, and backflow correlations on the square lattice at one particle per site finds a paramagnetic metal at small $U$15, then magnetic Mott phases labeled AF34 and AF33 at stronger coupling, while a previously predicted metallic AF32 phase with wavevector $U$16 is suppressed by backflow correlations. The principal qualitative result is a partial suppression of magnetism: the paramagnetic metal survives to larger $U$17 than in simpler variational descriptions, and the onset of magnetic order coincides with, or lies very near, the Mott transition. Upon hole doping, the strong-coupling magnetic Mott state evolves into a correlated metal, with a transition from magnetic to paramagnetic behavior (Bird et al., 10 Jul 2025).

A different multicomponent realization appears in the attractive three-color Hubbard model on a $U$18-flux square lattice at half filling. Determinant quantum Monte Carlo shows that color-dependent attractive interactions induce coexisting charge-density-wave and Néel ordered states, both tied to trion physics. The square lattice with $U$19 flux has Dirac points at $U$20, but its coordination number $U$21 produces stronger charge fluctuations than the honeycomb case, increasing the density of off-site trions and thereby substantially strengthening the Néel order. For $U$22 and $U$23, the extrapolated CDW and Néel order parameters vanish simultaneously at a melting temperature $U$24, and a Ginzburg–Landau analysis shows how color anisotropy stabilizes their coexistence (Li et al., 6 Dec 2025).

Square-lattice variants with modified geometry further broaden the scope. On the $U$25-depleted square lattice, determinant quantum Monte Carlo and random-phase approximation reveal plaquette-singlet, dimer-singlet, Néel, block-spin antiferromagnetic, and stripe phases, with the dominant pairing channel changing from d-wave in the plaquette regime to extended $U$26-wave in the Néel regime. Although this geometry is not the undepleted square lattice, it shows explicitly how bond inhomogeneity inherited from a square-lattice parent can reshape both magnetic order and pairing symmetry (Khatami et al., 2014).

6. Quantum simulation and moiré realizations of square-lattice Hubbard physics

Recent moiré proposals recast the square-lattice Hubbard model as an experimentally tunable target rather than a purely theoretical Hamiltonian. In twisted square-lattice homobilayers whose low-energy states originate at the monolayer $U$27 point, the continuum moiré problem produces an isolated flat band described by a square-lattice $U$28 model. In the low-angle limit, an emergent layer symmetry forces the nearest-neighbor hopping to vanish, so that the effective dispersion is

$U$29

with $U$30 at zero field and finite $U$31. An interlayer displacement field or in-plane magnetic field breaks that symmetry and turns on tunable $U$32, allowing access to a wide range of $U$33 ratios on a moiré square lattice (Eugenio et al., 2024).

A $U$34-valley counterpart refines this picture. Twisted square homobilayers with $U$35-valley band extrema yield two nested square sublattices and, upon projection onto one isolated top band, a single-band square-lattice Hubbard model with

$U$36

For the antibonding band, the displacement field tunes $U$37 over the range

$U$38

for $U$39–$U$40 meV, covering the cuprate-relevant value $U$41 and extending into strongly frustrated regimes. Projected screened Coulomb interactions produce a dominant onsite $U$42, with $U$43 increasing from about $U$44 at $U$45 meV to about $U$46 at $U$47 meV for a representative dielectric environment. The same study notes that $U$48–0.72 is a regime conjectured to host spin-liquid phases in the square-lattice $U$49–$U$50 model, linking moiré tunability directly to frustrated square-lattice Hubbard physics (Shi et al., 24 Mar 2026).

A material-specific implementation is proposed in twisted bilayer C$U$51. There, the monolayer has square Bravais symmetry and a valence-band maximum at the $U$52 point, yielding a top moiré band that is accurately fitted by

$U$53

At $U$54 and $U$55, the fitted parameters are $U$56 meV and $U$57 meV, while the ratio $U$58 crosses near $U$59, enabling broad $U$60 tunability without external fields. The projected onsite interaction is of order $U$61–$U$62 eV at $U$63, giving $U$64 for a quoted bandwidth $U$65 meV. The displacement field further generates anisotropic nearest-neighbor hoppings with $U$66, including a regime where one hopping nearly vanishes. This suggests that twisted square-lattice materials can access not only the isotropic square-lattice Hubbard model, but also controlled anisotropic and frustrated variants (Luo et al., 26 Feb 2025).

Taken together, these proposals imply that the square lattice Hubbard model now functions simultaneously as a benchmark many-body problem, a unifying language for correlated square-lattice materials, and a design target for moiré and cold-atom quantum simulators. A plausible implication is that the long-standing questions raised by its theoretical phase diagrams—especially the interplay of antiferromagnetism, d-wave superconductivity, charge order, stripe formation, and frustration—can increasingly be examined in settings where $U$67, $U$68, $U$69, anisotropy, and filling are tunable rather than fixed.