Single-Band Hubbard Model

• The single-band Hubbard model is a simplified lattice framework that captures electron interactions via hopping terms and on-site Coulomb repulsion.
• It provides a versatile tool for studying Mott physics, magnetism, charge ordering, pseudogap behavior, and unconventional superconductivity in various lattice geometries.
• The model underpins multiple analytical and numerical techniques, offering concrete insights into low-energy phenomena and effective multiorbital reductions.

The single-band Hubbard model is a lattice model for interacting electrons in which a single electronic orbital per site is retained, nearest- and longer-range hopping set the kinetic energy, and an on-site Coulomb term penalizes double occupancy. In its standard form,

$$H = -\sum_{\langle ij\rangle,\sigma} t_{ij}\, c^\dagger_{i\sigma} c_{j\sigma} + U \sum_i n_{i\uparrow}n_{i\downarrow} -\mu \sum_{i,\sigma} n_{i\sigma},$$

with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$, it conserves $N$, $N_\uparrow$, $N_\downarrow$, and hence $S_z$ in the absence of spin-flip terms. Across square, triangular, hypercubic, and related lattices, it functions as a minimal framework for Mott physics, magnetism, charge ordering, pseudogap formation, and unconventional pairing, and it also arises as an effective low-energy model by downfolding multiorbital Hamiltonians or isolated moiré minibands (Griffin et al., 2015, Klett et al., 2021, Sheshadri et al., 2022, Luo et al., 26 Feb 2025).

1. Hamiltonian, parameter space, and observables

The model is specified by the hopping graph, the amplitudes $t_{ij}$, the on-site repulsion $U$, and the filling controlled by $\mu$. On the two-dimensional square lattice, a common parametrization supplements nearest-neighbor $t$ by next-nearest $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$0 and next-next-nearest $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$1, giving

$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$2

while in the hypercubic $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$3 limit one scales $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$4 and uses the normalized energy scale $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$5, with a Gaussian noninteracting density of states $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$6 (Klett et al., 2021, Koga et al., 2019).

The filling conventions are model- and context-dependent but are consistently expressed through the site density $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$7. In the nickelate square-lattice study, $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$8 with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$9 the hole doping of the $N$0 band; in electron-doped cuprate work, $N$1 and hole doping is $N$2; in exact-diagonalization studies on $N$3 clusters, hole doping is fixed by particle number $N$4 through $N$5 (Klett et al., 2021, Mai et al., 2022, Jia et al., 2010).

The canonical observables are likewise standard but context-sensitive. Magnetic response is often extracted from the uniform susceptibility $N$6, either from linear response to a small Zeeman or ferromagnetic field or from a two-particle Bethe–Salpeter equation. Single-particle properties are encoded in

$N$7

while density, spin, and pairing tendencies are diagnosed through equal-time structure factors, density matrices, and correlation functions. In the nickelate application, the pseudogap onset scale $N$8 is identified from the maximum of $N$9, and the magnetic correlation length $N_\uparrow$0 is extracted by an Ornstein–Zernike fit to $N_\uparrow$1 (Klett et al., 2021).

2. Effective reductions and lattice-specific incarnations

A central question is when a many-orbital electronic structure can be reduced to a one-band Hubbard description. Near half-filling in cuprates, the three-band Cu–O model can be downfolded into an effective one-band form with

$N_\uparrow$2

so that the superexchange satisfies $N_\uparrow$3. Exact diagonalization of the Cu$N_\uparrow$4O cluster shows that the ground state has substantial Zhang–Rice singlet weight, which clarifies why the one-band and three-band parameterizations can be quantitatively connected near half-filling (Sheshadri et al., 2022).

The same reduction is more conditional in nickelates. For LaNiO$N_\uparrow$5 and closely related NdNiO$N_\uparrow$6, controlled downfolding finds that the Ni $N_\uparrow$7 orbital dominates near the Fermi level, motivating a single-band square-lattice model with material-realistic parameters $N_\uparrow$8 meV, $N_\uparrow$9, $N_\downarrow$0, and $N_\downarrow$1 eV. The same study emphasizes, however, that O $N_\downarrow$2 and Ni $N_\downarrow$3 contributions are not absent, so the one-band reduction is a low-energy approximation rather than an exact microscopic identity (Klett et al., 2021).

Recent moiré proposals make the single-band reduction itself tunable. Twisted homobilayer C$N_\downarrow$4 yields an isolated square-lattice moiré band per spin, with an effective dispersion of the same $N_\downarrow$5–$N_\downarrow$6–$N_\downarrow$7 form as the cuprate square lattice and a broadly tunable $N_\downarrow$8 ratio; at $N_\downarrow$9, the gap to the next moiré band is $S_z$0 meV and the top-band bandwidth is $S_z$1 meV, enabling a strictly single-band description in a sizable window (Luo et al., 26 Feb 2025). Twisted diamond homobilayers analogously realize a single-band anisotropic triangular-lattice Hubbard model, with intrachain $S_z$2, interchain $S_z$3, and a displacement field $S_z$4 that tunes the system from quasi-one-dimensional weakly coupled chains to a two-dimensional frustrated lattice (Sun et al., 25 Mar 2025). A separate materials-design program identified LiCuF$S_z$5 in the hexagonal manganite structure as a “bespoke” single-band triangular-lattice realization, with an isolated Cu $S_z$6 band of bandwidth $S_z$7 eV and a DMFT Mott transition for $S_z$8 eV at $S_z$9 K (Griffin et al., 2015).

3. Analytical and numerical formulations

Because the model is nontrivial outside special limits, its literature is methodologically plural. In infinite dimensions, DMFT assumes a purely local self-energy, maps the lattice problem onto a self-consistent impurity model, and is exact in that limit; on the hypercubic lattice this yields a Gaussian-DOS Hilbert transform, while on the Bethe lattice the self-consistency closes as $t_{ij}$0 (Koga et al., 2019, Kamogawa et al., 2019). In two dimensions, single-site DMFT misses nonlocal correlations and, in magnetic applications, can generate spurious finite-temperature antiferromagnetic order because it violates the Mermin–Wagner constraint (Klett et al., 2021).

Cluster and diagrammatic extensions address that deficiency in distinct ways. CDMFT embeds a finite real-space cluster and captures short-range nonlocal correlations up to the cluster size, while DCA coarse-grains momentum space into cluster patches and accesses equal-time spin and charge structure factors at finite temperature (Klett et al., 2021, Mai et al., 2022). Ladder D$t_{ij}$1A starts from the local DMFT irreducible vertex, resums nonlocal spin ladders, applies Moriya-like $t_{ij}$2-corrections, and respects the Mermin–Wagner theorem in two dimensions, making it particularly suitable for pseudogap and susceptibility studies on the square lattice (Klett et al., 2021).

Ground-state tensor-network and exact-diagonalization approaches emphasize complementary observables. DMRG on wide diagonal square cylinders resolves stripe, pair-density-wave, and charge-crystal textures in the doped square-lattice model, while exact diagonalization on 16-site periodic clusters can track changes in the many-body ground state through the fidelity metric

$t_{ij}$3

with $t_{ij}$4 (Xu et al., 2024, Jia et al., 2010). At the mean-field level, Hartree–Fock and Kotliar–Ruckenstein or spin-rotation-invariant slave-boson formalisms describe commensurate and spiral magnetism, quasiparticle renormalization, and response functions in a computationally tractable but nonexact manner (Igoshev et al., 2015, Riegler et al., 2019).

Special limits admit more controlled formulations. The one-dimensional model at finite magnetic field is solved by Bethe ansatz and pseudofermion dynamical theory, which identify branch-line and border-line singularities in the one-electron spectral function and define lower and upper Hubbard bands away from half-filling through rotated electrons (Carmelo et al., 2016). In the atomic $t_{ij}$5 limit with nearest- and next-nearest-neighbor density interactions, an extended transfer-matrix method gives an exact $t_{ij}$6 phase diagram across all fillings (Mancini et al., 2013). On the algorithmic side, exact diagonalization can exploit the factorized structure

$t_{ij}$7

which allows one-spin-at-a-time basis construction and reduces non-diagonal Hamiltonian-generation time by about an order of magnitude relative to two-spin basis schemes (Sharma et al., 2013). A different cluster-based route is the cumulant Green’s functions method, which diagonalizes a finite correlated seed, builds cumulants, and embeds them into the lattice Green’s function without self-consistency (Lira et al., 2022).

4. Magnetic order, spin response, and ferromagnetic instability

Near half-filling on bipartite lattices, antiferromagnetism and its incommensurate descendants dominate much of the phase diagram. Hartree–Fock and slave-boson analyses on square and cubic lattices find commensurate antiferromagnetic and ferromagnetic phases together with spiral states characterized by $t_{ij}$8, $t_{ij}$9, $U$0, $U$1, and related vectors. Correlation effects included through slave bosons strongly suppress both spirals and especially ferromagnetism unless Fermi-surface nesting is present or van Hove singularities lie close to the paramagnetic Fermi level; phase separation is particularly wide near half-filling (Igoshev et al., 2015, Riegler et al., 2019).

A distinct regime occurs away from half-filling in large-$U$2 DMFT on the hypercubic lattice. There the uniform susceptibility develops a maximum around $U$3, and at $U$4 a ferromagnetically ordered metal appears for $U$5. The same instability is absent on the Bethe lattice, and comparison with a Student-$U$6 density of states shows that slowly decaying high-energy tails strongly stabilize ferromagnetism. This shifts the interpretation away from a purely Stoner-like $U$7 criterion toward a kinetic-energy mechanism controlled by the full density of states (Kamogawa et al., 2019, Koga et al., 2019).

In two-dimensional square-lattice nickelate modeling, the salient magnetic feature is instead a non-Curie–Weiss susceptibility. D$U$8A produces a broad maximum in $U$9 at $\mu$0 for all considered hole dopings, with $\mu$1 highest near half-filling and decreasing monotonically with doping. CDMFT yields a similar trend but retains a finite-temperature ordering line because of finite-cluster mean-field tendencies, whereas D$\mu$2A does not order at finite $\mu$3. The magnetic correlation length at pseudogap onset is only $\mu$4–$\mu$5 lattice spacings, which identifies the pseudogap as strong-coupling rather than weak-coupling in origin (Klett et al., 2021).

There is also a rigorous route to metallic ferromagnetism. A decorated multi-band Hubbard construction can be taken to $\mu$6 and infinite band gap so that only a single finite-energy dispersive band remains. In that effective one-band sector, saturated ferromagnetism is proven for $\mu$7, and nonsaturated but extensive ferromagnetism for $\mu$8, on closely packed lattices in $\mu$9. This suggests that kinetic ferromagnetism can be established rigorously for a single conducting band, albeit with subextensive hole count in the thermodynamic limit (Tanaka et al., 2016).

5. Spectral, charge, and pairing phenomena

The model’s one-particle spectrum displays strong momentum selectivity once nonlocal correlations are included. In the LaNiO$t$0 square-lattice parameter set, lowering temperature at $t$1 suppresses antinodal spectral weight while nodal weight continues to grow, producing Fermi-arc formation below $t$2. At $t$3 K, decreasing the doping from $t$4 to $t$5 reduces both nodal and antinodal intensity and drives a tendency from hole-like to electron-like Fermi-surface topology near $t$6 (Klett et al., 2021).

Charge correlations need not track spin correlations. Finite-temperature QMC-DCA for the electron-doped square-lattice model finds robust charge-density-wave correlations with simultaneous checkerboard and unidirectional components at wave vectors $t$7 and $t$8, while the spin structure factor remains centered near $t$9 without matching spin-density peaks at the CDW wave vector. The charge-order wave vector grows with electron doping, and the absence of concomitant spin modulation at that $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$00 argues against a simple weak-coupling nesting interpretation (Mai et al., 2022).

On wide diagonal square cylinders with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$01 and $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$02, DMRG identifies a more intertwined regime on the hole-doped side. At $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$03, the ground state exhibits infinite-length period-3 charge stripes with ordering vector $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$04, antiferromagnetic stripes with anti-phase domain walls, and dominant incommensurate pair-density-wave correlations along the stripes with

$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$05

where $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$06 and $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$07. At $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$08, an additional intra-stripe CDW aligns across stripes to form a holon Wigner crystal, and the emergent charge-order momentum is close to the PDW center-of-mass momentum (Xu et al., 2024).

Exact diagonalization with an added infinite-range $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$09-wave pair field produces a complementary finite-cluster diagnosis. On a 16-site square cluster, the fidelity metric and the ratio of the two largest eigenvalues of the $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$10-wave pair density matrix show an antiferromagnetic ground state at half-filling, a $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$11-wave superconducting ground state at $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$12 hole doping, and a weak short-range checkerboard charge-ordered state at larger hole doping. Negative $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$13 reduces the crossover scale $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$14 required to precipitate the superconducting state and therefore facilitates $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$15-wave superconductivity in this setting (Jia et al., 2010).

In one dimension, exact spectral analysis is even sharper. At finite magnetic field, the one-electron spectral function is organized by charge $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$16 and spin $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$17 branch lines, plus border lines where charge and spin velocities coincide. Rotated electrons provide a nonperturbative definition of lower and upper Hubbard bands for all densities and spin polarizations, extending the half-filled Mott vocabulary into the doped regime (Carmelo et al., 2016).

6. Exact limits, representations, and physical realizations

The single-band Hubbard model has several exact or nearly exact boundary cases that clarify which sectors of the full problem are intrinsically difficult. In one dimension with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$18 and added nearest- and next-nearest-neighbor density interactions, the transfer-matrix solution yields a complete $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$19 phase diagram with commensurate and incommensurate charge-ordered phases across $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$20, including explicit energies such as $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$21, $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$22, or $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$23 at $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$24, depending on whether doublon crystallization or NN/NNN charge order is favored (Mancini et al., 2013). This exactness disappears once itinerancy is restored, but it makes clear that even before kinetic exchange enters, the charge sector alone supports a large variety of competing orders.

Computational representations of the Hilbert space are correspondingly important. Spin-separated basis organization for exact diagonalization, in which only $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$25 or $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$26 configurations are stored at a time and the full hopping Hamiltonian is assembled as tensor products, reduces non-diagonal matrix-generation time by about an order of magnitude and is inherently parallelizable; it is also directly useful in ED impurity solvers for DMFT (Sharma et al., 2013). The cumulant Green’s functions method shows a different strategy: diagonalize a finite correlated seed, compute its Lehmann-representation cumulants, and reconstruct the lattice Green’s function as

$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$27

without a self-consistency loop. In one dimension, this method converges systematically toward exact results for the gap, ground-state energy, and double occupancy as the seed size increases (Lira et al., 2022).

The model is now also being realized or closely approximated in designed materials and moiré systems. LiCuF$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$28 provides a triangular-lattice $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$29 realization with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$30 eV and a DMFT metal–insulator transition at $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$31 eV; DCA$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$32 on the corresponding triangular-lattice model finds two divergent $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$33-wave pairing channels, with $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$34 favored and $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$35 at $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$36 (Griffin et al., 2015). Twisted C$n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$37 offers a square-lattice single-band extended Hubbard platform with broad in situ control of $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$38 and access to the regime $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$39, while twisted diamond homobilayers realize an anisotropic triangular-lattice model whose half-filled $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$40 phase diagram on YC4 cylinders contains a chiral spin liquid for $n_{i\sigma}=c^\dagger_{i\sigma}c_{i\sigma}$41, flanked by nonmagnetic and Néel regimes (Luo et al., 26 Feb 2025, Sun et al., 25 Mar 2025).

Taken together, these results support a technically precise but limited conclusion. The single-band Hubbard model is genuinely minimal only after a nontrivial projection, and its predictive success depends on lattice geometry, density of states, dimensionality, and the observables under consideration. In some settings—electron-doped cuprates, square-lattice nickelate pseudogap phenomenology, triangular and square moiré bands, and selected designed materials—the reduction is quantitatively useful. In others, especially where charge-transfer, multiorbital, or long-range interactions are central, the one-band description is best regarded as a controlled low-energy truncation rather than a complete microscopic theory (Sheshadri et al., 2022, Klett et al., 2021).