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A lattice boson Hamiltonian operator combining hopping and on-site repulsion. The on-site interaction counts distinct particle pairs through $n_i(n_i-1)/2$. It is the bosonic counterpart of the Hubbard model; projecting a continuum contact interaction onto localized orbitals gives $U=g\int |{\phi }_i{|}^4$ when intersite overlaps are negligible.

A direct intersite Coulomb exchange integral acts without an intermediate doubly occupied state. In a repulsive one-band Hubbard model, by contrast, virtual hopping into a doubly occupied state lowers the singlet energy by $4t^2/U$ relative to the triplet. The latter is antiferromagnetic kinetic exchange, even when hopping is directly between neighbouring magnetic orbitals. Ligand-mediated superexchange introduces an intermediate ligand orbital as well.

The first term assigns energy ${ϵ}_i$ to occupying the localized orbital at site $i$. The hopping term moves an electron between neighbouring sites, producing kinetic delocalization and band dispersion; Hermitian conjugate hops make the Hamiltonian Hermitian. The final term costs $U>0$ only when both spin states occupy the same local orbital. This is a Hubbard model with an on-site Coulomb penalty, not a classical interaction between magnetic dipoles.

At one electron per site and large $U$, a real hop is energetically costly, but a virtual hop into a doubly occupied intermediate state is possible. Opposite-spin states can exploit these virtual processes and lower their energy, while parallel spins cannot occupy the same one-orbital site. The resulting singlet–triplet splitting produces antiferromagnetic Hubbard kinetic exchange.

There is a terminology qualification. The paper calls this overlap-mediated process “direct exchange” because the hopping is directly between neighbouring magnetic orbitals. In the stricter distinction between direct Coulomb exchange versus kinetic exchange, the displayed Hamiltonian omits an explicit intersite Coulomb exchange integral. The antiferromagnetic $t^2/U$ mechanism derived below is kinetic exchange; a genuine intersite direct exchange term is separate and can compete with it. This distinction also separates the model from ligand-mediated superexchange in part f.

The microscopic electric interaction is spin independent to leading order, but the Pauli exclusion principle ties the spatial symmetry of an electronic state to its spin symmetry. Different spin combinations therefore have different Coulomb and kinetic energies. Projecting those energies onto localized low-energy moments often gives an approximately rotationally invariant bilinear exchange Hamiltonian operator $-J{\mathbf{S}}_1\cdot {\mathbf{S}}_2$. For spin one half, the spin singlet state and triplet state energies are distinguished by ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=-3/4$ and $1/4$, so any isotropic two-level spin singlet state–triplet state splitting can be written in this form plus a constant. More general spins or orbital degeneracy can require extra interactions.

An antiferromagnetic oxide commonly has superexchange through a ligand such as oxygen. Virtual hopping between half-filled metal orbitals lowers the energy of antiparallel spins more strongly because the corresponding intermediate doubly occupied configurations are allowed by the Pauli exclusion principle. The simple one-orbital Hubbard model limit gives $+4t^2{\mathbf{S}}_1\cdot {\mathbf{S}}_2/U$, equivalent to negative $J$ in the question's minus-sign convention. Bond angles and orbital occupancy can change the sign in more complicated oxides, so superexchange is not universally antiferromagnetic irrespective of geometry.

Let $D_n={\hat{n}}_{n\uparrow }{\hat{n}}_{n\downarrow }$. On a site, empty and doubly occupied states have spin zero, while either singly occupied state has spin $1/2$. ThusThe on-site spin-square normal-ordering identity rewrites the interaction as $3U{\sum }_n{D}_n$. Its contact coefficient is $3U$ in this parameterization, rather than the $U$ convention often used for the Hubbard model.

Using a fermionic coherent-state trace and time slicing, replace each normal-ordered monomial by its Grassmann field symbol. With ${\xi }_{\mathbf{k}}={ϵ}_{\mathbf{k}}-\mu$,The independent fields have antiperiodic thermal boundary conditions. Normal ordering is crucial: the explicit $3\hat{n}/4$ subtraction removes the spin-square contraction, so there is no extra chemical-potential shift from it.

The Hubbard model combines band motion and local repulsion:Hopping favors itinerant states, while positive $U$ penalizes simultaneous opposite-spin occupation on one site. In a uniform mean-field treatment, a spin imbalance reduces the interaction cost but raises the band-filling energy. The Stoner criterion for instability of the unpolarized metal is $U{g}_{\uparrow }(E_F)>1$ when $g_{\uparrow }$ is the density of states per site, per energy, for one spin. Using a total two-spin density of states changes the displayed numerical convention.

The ordered mean-field state has exchange-split spin bands and a spontaneous magnetization, with the kinetic and interaction costs balancing at equilibrium. This is a model of itinerant ferromagnetism, not a guarantee that every repulsive Hubbard system becomes ferromagnetic. Filling, lattice structure and competing order matter: near half filling on a bipartite lattice, antiferromagnetism and a Mott insulator can be favored instead. The approximation also omits important spin fluctuations and strong-correlation effects.

The Hubbard model combines hopping with a local repulsion,Hopping favours a broad itinerant band; the interaction penalizes opposite-spin double occupancy. For uniform Stoner decoupling of the Hubbard interaction, let $s=n_{\uparrow }-n_{\downarrow }$ be the spin imbalance per site. The interaction is $U(n^2-s^2)/4$, so polarization lowers it by $U{s}^2/4$, at the expense of kinetic energy.

If $D(E_F)$ is the noninteracting density of states per spin per site, small polarization costs $s^2/[4D(E_F)]$. Thereforeis the mean-field ferromagnetic instability criterion. The exchange splitting is $\Delta =Us$, and the paramagnetic response is enhanced as $\chi ={\chi }_0/[1-UD(E_F)]$. A density of states including both spins changes the written factor in the criterion.

The model explains how itinerant bands can polarize without imposing pre-existing localized atomic spins. It is not a claim that every large-$U$ metal becomes ferromagnetic: near half filling the Hubbard model can instead form a Mott insulator with antiferromagnetic superexchange of scale $4t^2/U$. Band filling, bandwidth and fluctuations matter.^[ref] The interaction-energy and kinetic-cost expansion above is the stated mean-field derivation of the enhancement.^[ref]

Specify the spin counting first. Let $g_s=g_s(E_F)$ denote the unpolarized single-spin density of states per atom per energy, and put $\delta n=n_{\uparrow }-n_{\downarrow }$. At fixed total occupation $n$, the two Hubbard model mean-field energies can be writtenThe common shift $Un/2$ can be absorbed into the chemical potential. Lowering the up-spin band by $\Delta$ adds $g_s\Delta$ carriers, while raising the down-spin band removes the same number. ThusIf $n_a$ is the atom number density, the magnetization is $M=n_a{\mu }_B\delta n$. The Stoner-enhanced magnetic susceptibility in SI volume units is thereforeHere ${\chi }_P$ is the noninteracting Pauli paramagnetism result. If the paper's $g(E_F)$ counts both spins, $g=2g_s$ and the same result isA susceptibility per atom omits $n_a$. Stating the spin and volume conventions avoids an otherwise spurious factor of two or a dimensional mismatch.

The Stoner criterion is $U{g}_s(E_F)>1$, equivalently $Ug(E_F)>2$ with spin-summed $g$. To see the instability, transfer $\eta$ electrons per atom from one spin band to the other. The band-filling cost is ${\eta }^2/g_s$, whereas the interaction energy changes by $-U{\eta }^2$. Their sum $(1/g_s-U){\eta }^2$ becomes negative at the threshold. The unpolarized state then ceases to be a local minimum, and nonlinear band filling determines the spontaneous ferromagnetism. The negative continuation of the displayed linear susceptibility beyond the threshold is not a stable paramagnetic response.

Use $\hslash =k_{\mathrm{B}}=1$ and dimensionless spin operators. At one site, the canonical anticommutation relations imply the on-site spin-square normal-ordering identityIndeed the empty and doubly occupied states have spin zero, while either singly occupied state has ${\mathbf{S}}^2=3/4$. Thus the interaction in normal-ordered form is $3U{\sum }_n{c}_{n\uparrow }^{†}{c}_{n\downarrow }^{†}{c}_{n\downarrow }{c}_{n\uparrow }$. There is no residual $3Un/2$ chemical-potential shift: that contraction has cancelled.

Insert fermionic coherent state resolutions between imaginary-time evolution factors. With ${\xi }_k={ϵ}_k-\mu$ and $D_n={\bar{\psi }}_{n\uparrow }{\bar{\psi }}_{n\downarrow }{\psi }_{n\downarrow }{\psi }_{n\uparrow }$, the grand-canonical coherent-state action isThe independent barred and unbarred Grassmann variables satisfy antiperiodic boundary conditions at $0,\beta$. Adjacent time-slice labels in the normal-ordered symbol fix the continuum prescription and thermal normalization. The interaction coefficient is $3U$, not the $U$ of a differently normalized Hubbard model.

The t-J model approximates the low-energy sector of a repulsive Hubbard model with $U\gg t$ near one electron per site. Large on-site repulsion suppresses double occupancy. Virtual hopping out of and back into that low-energy sector produces antiferromagnetic superexchange with $J=4t^2/U$. One conventional effective Hamiltonian iswith projected electron operator ${\tilde{c}}_{i\sigma }=c_{i\sigma }(1-n_{i\bar{\sigma }})$. At half filling, nearest-neighbor hopping cannot move an electron without entering a forbidden doubly occupied state, leaving the leading spin-exchange model. Doped holes can move and couple the charge dynamics to these spin correlations. Additional terms, including three-site processes at order $t^2/U$, are omitted in this simplified Hamiltonian.

The exchange term energetically favors a nearest-neighbor spin singlet. With $b_{ij}^{†}=(c_{i\uparrow }^{†}{c}_{j\downarrow }^{†}-c_{i\downarrow }^{†}{c}_{j\uparrow }^{†})/\sqrt{2}$ in the no-double-occupancy subspace, ${\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4=-b_{ij}^{†}{b}_{ij}$. A square-lattice pairing ansatz can therefore use bond singlets with opposite signs along the two axes. This d-wave bond order hasIt changes sign under a quarter-turn and vanishes along the zone diagonals. It is an even-parity, spin-singlet pairing pattern, not a statement of triplet pairing or necessarily broken time reversal. Phase-sensitive interference experiments distinguish this sign structure from a merely anisotropic nonnegative gap in cuprate superconductors.^[ref]

The model captures strong local repulsion, antiferromagnetism and mobile doped carriers. Its interacting phase diagram is not solved by writing down the Hamiltonian: a mean-field d-wave ansatz is a proposed approximation, not a proof that every parameter choice has that exact ground state or a complete microscopic explanation of every high-temperature superconductor.

Let $g_s(E_F)$ be the single-spin density of states per site per energy, and let the on-site Hubbard model repulsion have energy $U$. For a weak uniform magnetic field $H$, the half-splitting of the two spin bands is $\Delta =U\delta n/2+{\mu }_0{\mu }_{B}H$, where $\delta n=n_{\uparrow }-n_{\downarrow }$. Constant density of states gives $\delta n=2g_s\Delta$. ThereforeThis is the response per site; multiplying by the site number density gives the dimensionless SI volume magnetic susceptibility. With spin-summed density of states $g=2g_s$, the denominator is $1-Ug/2$. Its vanishing is the mean-field Stoner criterion. The unpolarized linear-response expression is stable only for $U{g}_s<1$.

An effective low-energy lattice model obtained from a strongly repulsive Hubbard model near one electron per site. The retained Hilbert space excludes double occupancy. Its usual nearest-neighbor Hamiltonian has projected hopping $-t{\sum }_{⟨ij⟩,\sigma }({\tilde{c}}_{i\sigma }^{†}{\tilde{c}}_{j\sigma }+h.c.)$ and antiferromagnetic superexchange $J{\sum }_{⟨ij⟩}({\mathbf{S}}_i\cdot {\mathbf{S}}_j-n_i{n}_j/4)$. Virtual intermediate states with one double occupancy cost energy $U$, giving $J=4t^2/U$ to leading order. Three-site hopping terms of the same expansion order are omitted in this simplified version.

At half filling the projected hopping is blocked. Doped holes restore charge motion coupled to spin correlations. Nearest-neighbor singlet pairing can support a d-wave bond order ansatz on a square lattice, but the Hamiltonian alone is not a proof of a unique superconducting ground state for all parameters.