---
title: Single-Band Hubbard Model
url: https://www.emergentmind.com/topics/single-band-hubbard-model
type: topic
---

# Single-Band Hubbard Model

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 [1508.04357] [2112.07531] [2211.14019] [2502.19483].

## 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 \(t'\) and next-next-nearest \(t''\), giving
$$
\varepsilon_{\mathbf{k}}=
-2t(\cos k_x+\cos k_y)
-4t'\cos k_x\cos k_y
-2t''(\cos 2k_x+\cos 2k_y),
$$
while in the hypercubic \(d\to\infty\) limit one scales \(t=t^*/\sqrt d\) and uses the normalized energy scale \(D=2\sqrt d\,t\), with a Gaussian noninteracting density of states \(\rho_0(\epsilon)=\frac{1}{\sqrt{\pi}D}\exp[-(\epsilon/D)^2]\) [2112.07531] [1912.05053].

The filling conventions are model- and context-dependent but are consistently expressed through the site density \(n\). In the nickelate square-lattice study, \(n=1-\delta\) with \(\delta\) the hole doping of the \(d_{x^2-y^2}\) band; in electron-doped cuprate work, \(x_e=n-1\) and hole doping is \(x_h=1-n\); in exact-diagonalization studies on \(N=16\) clusters, hole doping is fixed by particle number \(N_e\) through \(x=1-N_e/N\) [2112.07531] [2210.14930] [1012.4013].

The canonical observables are likewise standard but context-sensitive. Magnetic response is often extracted from the uniform susceptibility \(\chi\), 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
$$
A(\mathbf{k},\omega)=-\frac{1}{\pi}\,\mathrm{Im}\,G(\mathbf{k},\omega+i0^+),
$$
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 \(T^*\) is identified from the maximum of \(\chi(T)\), and the magnetic correlation length \(\xi\) is extracted by an Ornstein–Zernike fit to \(\chi_m(\mathbf{q},0)\) [2112.07531].

## 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
$$
\tilde t = \frac{t_{pd}^2}{\Delta},\qquad
\frac{1}{\tilde U}=\frac{1}{U_d}+\frac{1}{\Delta+U_p/2},
$$
so that the superexchange satisfies \(J=4\tilde t^2/\tilde U\). Exact diagonalization of the Cu\(_2\)O 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 [2211.14019].

The same reduction is more conditional in nickelates. For LaNiO\(_2\) and closely related NdNiO\(_2\), controlled downfolding finds that the Ni \(d_{x^2-y^2}\) orbital dominates near the Fermi level, motivating a single-band square-lattice model with material-realistic parameters \(t=395\) meV, \(t'=-0.25t\), \(t''=0.12t\), and \(U=8t=3.16\) eV. The same study emphasizes, however, that O \(p\) and Ni \(d_{z^2}\) contributions are not absent, so the one-band reduction is a low-energy approximation rather than an exact microscopic identity [2112.07531].

Recent moiré proposals make the single-band reduction itself tunable. Twisted homobilayer C\(_{568}\) yields an isolated square-lattice moiré band per spin, with an effective dispersion of the same \(t\)–\(t'\)–\(t''\) form as the cuprate square lattice and a broadly tunable \(t'/t\) ratio; at \(\theta=1.8^\circ\), the gap to the next moiré band is \(\approx 4.2\) meV and the top-band bandwidth is \(\approx 0.014\) meV, enabling a strictly single-band description in a sizable window [2502.19483]. Twisted diamond homobilayers analogously realize a single-band anisotropic triangular-lattice Hubbard model, with intrachain \(t\), interchain \(t'\), and a displacement field \(D_z\) that tunes the system from quasi-one-dimensional weakly coupled chains to a two-dimensional frustrated lattice [2503.19829]. A separate materials-design program identified LiCuF\(_3\) in the hexagonal manganite structure as a “bespoke” single-band triangular-lattice realization, with an isolated Cu \(d_{z^2}\) band of bandwidth \(W\approx 1.0\) eV and a DMFT Mott transition for \(U_{\mathrm{DMFT}}\gtrsim 1.25\) eV at \(T=290\) K [1508.04357].

## 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 \(\Delta=t^2G_{\mathrm{loc}}\) [1912.05053] [1904.01724]. 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 [2112.07531].

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 [2112.07531] [2210.14930]. Ladder D\(\Gamma\)A starts from the local DMFT irreducible vertex, resums nonlocal spin ladders, applies Moriya-like \(\lambda\)-corrections, and respects the Mermin–Wagner theorem in two dimensions, making it particularly suitable for pseudogap and susceptibility studies on the square lattice [2112.07531].

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
$$
g(\Delta,\delta\Delta)=\frac{2}{N}\,\frac{1-F(\Delta,\Delta+\delta\Delta)}{(\delta\Delta)^2},
$$
with \(F=|\langle\Psi_0(\Delta)|\Psi_0(\Delta+\delta\Delta)\rangle|\) [2409.18833] [1012.4013]. 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 [1502.05839] [1912.07631].

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 [1605.09620]. In the atomic \(t=0\) limit with nearest- and next-nearest-neighbor density interactions, an extended transfer-matrix method gives an exact \(T=0\) phase diagram across all fillings [1308.5196]. On the algorithmic side, exact diagonalization can exploit the factorized structure
$$
H_t = \mathcal{I}_\downarrow\otimes \mathbf{H}_\uparrow + \mathbf{H}_\downarrow\otimes \mathcal{I}_\uparrow,
$$
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 [1307.7542]. 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 [2205.01161].

## 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 \(\mathbf{Q}=(Q,Q)\), \((Q,\pi)\), \((Q,Q,Q)\), \((Q,\pi,\pi)\), 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 [1502.05839] [1912.07631].

A distinct regime occurs away from half-filling in large-\(U\) DMFT on the hypercubic lattice. There the uniform susceptibility develops a maximum around \(n\sim 0.95\), and at \(U/D=1000\) a ferromagnetically ordered metal appears for \(0.85<n<0.99\). The same instability is absent on the Bethe lattice, and comparison with a Student-\(t\) density of states shows that slowly decaying high-energy tails strongly stabilize ferromagnetism. This shifts the interpretation away from a purely Stoner-like \(N(0)\) criterion toward a kinetic-energy mechanism controlled by the full density of states [1904.01724] [1912.05053].

In two-dimensional square-lattice nickelate modeling, the salient magnetic feature is instead a non-Curie–Weiss susceptibility. D\(\Gamma\)A produces a broad maximum in \(\chi(T)\) at \(T^*\) for all considered hole dopings, with \(T^*\) 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\(\Gamma\)A does not order at finite \(T\). The magnetic correlation length at pseudogap onset is only \(\xi(T^*)\approx 1.2\)–\(2.5\) lattice spacings, which identifies the pseudogap as strong-coupling rather than weak-coupling in origin [2112.07531].

There is also a rigorous route to metallic ferromagnetism. A decorated multi-band Hubbard construction can be taken to \(U\to\infty\) 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 \(N_h\le \mathrm{const.}\,|\mathcal{M}|^{2/(d+2)}\), and nonsaturated but extensive ferromagnetism for \(N_h\le \mathrm{const.}\,|\mathcal{M}|^{(d+1)/(d+2)}\), on closely packed lattices in \(d\ge 2\). This suggests that kinetic ferromagnetism can be established rigorously for a single conducting band, albeit with subextensive hole count in the thermodynamic limit [1601.04265].

## 5. Spectral, charge, and pairing phenomena

The model’s one-particle spectrum displays strong momentum selectivity once nonlocal correlations are included. In the LaNiO\(_2\) square-lattice parameter set, lowering temperature at \(\delta=7.5\%\) suppresses antinodal spectral weight while nodal weight continues to grow, producing Fermi-arc formation below \(T^*\). At \(T\approx 920\) K, decreasing the doping from \(7.5\%\) to \(2.5\%\) reduces both nodal and antinodal intensity and drives a tendency from hole-like to electron-like Fermi-surface topology near \((\pi,0)\) [2112.07531].

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 \((Q,0)\) and \((0,Q)\), while the spin structure factor remains centered near \((\pi,\pi)\) 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 \(Q\) argues against a simple weak-coupling nesting interpretation [2210.14930].

On wide diagonal square cylinders with \(U/t=12\) and \(t'/t\in[-0.1,-0.3]\), DMRG identifies a more intertwined regime on the hole-doped side. At \(\delta\sim 14\%\), the ground state exhibits infinite-length period-3 charge stripes with ordering vector \(Q_1=(0,2\pi/3)\), antiferromagnetic stripes with anti-phase domain walls, and dominant incommensurate pair-density-wave correlations along the stripes with
$$
\Phi(r)\sim r^{-K_{sc}}\cos(Q_{\mathrm{PDW}}r+\theta),
$$
where \(K_{sc}\approx 1.6\) and \(Q_{\mathrm{PDW}}\approx 0.55\pi\). At \(\delta\sim 10\%\), 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 [2409.18833].

Exact diagonalization with an added infinite-range \(d\)-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 \(d\)-wave pair density matrix show an antiferromagnetic ground state at half-filling, a \(d\)-wave superconducting ground state at \(25\%\) hole doping, and a weak short-range checkerboard charge-ordered state at larger hole doping. Negative \(t'\) reduces the crossover scale \(\Delta^*\) required to precipitate the superconducting state and therefore facilitates \(d\)-wave superconductivity in this setting [1012.4013].

In one dimension, exact spectral analysis is even sharper. At finite magnetic field, the one-electron spectral function is organized by charge \(c^\pm\) and spin \(s1\) 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 [1605.09620].

## 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 \(t=0\) and added nearest- and next-nearest-neighbor density interactions, the transfer-matrix solution yields a complete \(T=0\) phase diagram with commensurate and incommensurate charge-ordered phases across \(n\in[0,2]\), including explicit energies such as \(E_0=U/4\), \(V_2/2\), or \(V_1/4\) at \(n=1/2\), depending on whether doublon crystallization or NN/NNN charge order is favored [1308.5196]. 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 \(I_\uparrow\) or \(I_\downarrow\) 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 [1307.7542]. 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
$$
\mathbf{G}_{\mathbf{k}\sigma}(\omega)
=
\mathbf{M}_\sigma(\omega)\,
\big[\mathbf{I}-\mathbf{W}_{\mathbf{k}\sigma}\mathbf{M}_\sigma(\omega)\big]^{-1},
$$
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 [2205.01161].

The model is now also being realized or closely approximated in designed materials and moiré systems. LiCuF\(_3\) provides a triangular-lattice \(d_{z^2}\) realization with \(W\approx 1.0\) eV and a DMFT metal–insulator transition at \(U_{\mathrm{DMFT}}\gtrsim 1.25\) eV; DCA\(+\) on the corresponding triangular-lattice model finds two divergent \(d\)-wave pairing channels, with \(d_{xy}\) favored and \(T_c\approx 0.08\,t\) at \(U=8t\) [1508.04357]. Twisted C\(_{568}\) offers a square-lattice single-band extended Hubbard platform with broad in situ control of \(t'/t\) and access to the regime \(t\ll k_BT\ll U\), while twisted diamond homobilayers realize an anisotropic triangular-lattice model whose half-filled \(U/t=10\) phase diagram on YC4 cylinders contains a chiral spin liquid for \(0.9\lesssim t'/t\lesssim 1.1\), flanked by nonmagnetic and Néel regimes [2502.19483] [2503.19829].

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 [2211.14019] [2112.07531].

Source: https://www.emergentmind.com/topics/single-band-hubbard-model