Hubbard model ($H=H_{\mathrm{hop}}+U_H{\sum }_n{n}_{n\uparrow }{n}_{n\downarrow }$)

A lattice fermion model combining hopping with local density interaction, conventionally written $U_H{\sum }_n{n}_{n\uparrow }{n}_{n\downarrow }$. Its interaction parameter must be matched when a spin-square representation is used: $-2U({\mathbf{S}}^2-3n/4)$ equals $3U{n}_{\uparrow }{n}_{\downarrow }$, so $U_H=3U$ for that representation.

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.

An electron annihilation operator restricted to the no-double-occupancy sector of the t-J model. It removes a spin-$\sigma$ electron only when the opposite-spin site occupation vanishes. The corresponding creation operator cannot create a doubly occupied site. These projected operators do not obey the unrestricted canonical on-site fermion algebra on all states.

For spin imbalance $s=n_{\uparrow }-n_{\downarrow }$ per site, the uniform mean-field interaction energy is $U(n^2-s^2)/4$. With noninteracting density of states $D$ per spin per site, the kinetic cost is $s^2/(4D)$. The quadratic coefficient changes sign at $UD=1$, producing the Stoner mean-field instability. This does not describe every strong-coupling Hubbard phase.

Two electrons of opposite spin occupying the same spatial orbital. The mean occupation product measures its probability at a specified site. In the two-electron dimer, the sum over sites is also the probability of any doubly occupied site, since two sites cannot both be doubly occupied.

The ground-state probability of both electrons occupying one site is $P_{\mathrm{double}}=\frac{1}{2}[1-U/\sqrt{U^2+16t^2}]$. It equals the derivative of the relative ground energy with respect to $U$. It tends to zero for $|t|/U\to 0$ and to $1/2$ for $|t|/U\to \infty$.

Two localized orbitals coupled by hopping, with an on-site penalty for double occupancy. At two-electron filling the competition between hopping and local repulsion can be solved exactly within the finite basis. It gives a compact example of a molecular bonding crossover and electron correlation; it is not by itself a complete real-molecule binding curve.

At one electron per site and $U\gg |t|$, the low-energy singlet of a repulsive two-site Hubbard dimer mixes virtually with doubly occupied configurations. Its energy is $-4t^2/U$ to leading order, while the three triplets have no such shift. With dimensionless spin-half operators this yields the displayed effective antiferromagnetic Hamiltonian. Restoring angular-momentum operators with units of $\hslash$ requires the corresponding $1/{\hslash }^2$ in the spin product.

For real hopping $t$ and repulsion $U>0$, the covalent singlet and symmetric ionic state have relative Hamiltonian $\left(\begin{array}{cc}0& 2t\\ 2t& U\end{array}\right)$. The lower energy is $(U-\sqrt{U^2+16t^2})/2$, relative to the common one-electron energy. It approaches $-4t^2/U$ at small hopping, and its state continuously mixes covalent and ionic configurations.

A configuration with both electrons on one center, such as $aa$ or $bb$, in a two-center electronic basis. A coherent superposition of such configurations in a neutral molecule represents charge fluctuations; it does not imply a permanent classical ionic charge assignment.

For orthonormal orbitals $a,b$, the symmetric spatial state $C=(|ab⟩+|ba⟩)/\sqrt{2}$ multiplied by the antisymmetric spin singlet has one electron on each center and no double occupancy. The full state remains antisymmetric under fermion exchange.

The energy cost $U=\iint |a({\mathbf{r}}_1){|}^2|a({\mathbf{r}}_2){|}^2{e}^2/[4\pi {ϵ}_0|{\mathbf{r}}_1-{\mathbf{r}}_2|]{\mathrm{d}}^3{r}_1{\mathrm{d}}^3{r}_2$ for double occupancy of a localized orbital. It is positive for an ordinary nonzero orbital. A model retaining only this integral also neglects intersite and mixed Coulomb terms; orbital orthogonality alone does not make those terms vanish.

An isolated spin-degenerate orbital coupled to a particle reservoir has Hamiltonian $H=ϵ(n_{\uparrow }+n_{\downarrow })+U{n}_{\uparrow }{n}_{\downarrow }$. The four occupation microstates have weights $1,z,z,q{z}^2$, where $z=e^{\beta (\mu -ϵ)}$ and $q=e^{-\beta U}$. Thus $\Xi =1+2z+q{z}^2$ and $\bar{n}=(2z+2q{z}^2)/\Xi$. The factor two counts distinct spin states. At $U=0$ the grand partition function factorizes into two independent fermionic orbitals; positive U separates the two electron-addition thresholds.

For a repulsive single orbital at temperatures small compared with U, chemical potentials strictly between the first and second addition energies select a singly occupied defect. Its mean occupation is nearly one while its charge-number variance is exponentially small away from the thermally broadened step regions. The remaining spin degeneracy does not itself produce charge-number fluctuations.

With $z=e^{\beta (\mu -ϵ)}$, $q=e^{-\beta U}$ and $\Xi =1+2z+q{z}^2$, direct four-state averaging givesFor $\beta U\gg 1$ it has two separated peaks of height $1/4$ at the spin-degeneracy-shifted charge steps. At the lower bare crossing $\mu =ϵ$, three microstates share equal limiting weight, giving $\bar{n}=2/3$ and variance $2/9$. The upper bare crossing similarly gives mean $4/3$ and variance $2/9$. Between crossings occupation is pinned near one and fluctuations are suppressed. The identity is grand-canonical; fixing total particle number would change the fluctuation question.

Near the first addition threshold of a repulsive single-orbital defect, the empty and singly occupied grand weights are 1 and $2e^{\beta (\mu -ϵ)}$. Equal weights occur at $\mu =ϵ-k_{B}T\ln 2$. Near the second threshold, the single/double aggregate weights are in ratio $2:e^{\beta (\mu -ϵ-U)}$, giving half-step position $ϵ+U+k_{B}T\ln 2$. These are entropic finite-temperature shifts, which vanish in energy units as $T\to 0$. They must still be retained when evaluating statistics exactly at an unshifted degeneracy point.

For a single spinful fermion orbital, ${\mathbf{S}}^2=3(n-2n_{\uparrow }{n}_{\downarrow })/4$. The contraction term is $3n/4$, so $:{\mathbf{S}}^2:={\mathbf{S}}^2-3n/4=-3n_{\uparrow }{n}_{\downarrow }/2$. Empty and doubly occupied states are spinless while singly occupied states have spin one half, which also proves the identity directly.

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

In the homogeneous spin-vector mean-field scheme with field cost $M^2/(2U)$ and spin-summed density of states per site, the paramagnetic curvature is proportional to $U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$. It changes sign at $U{\chi }_0=1$. For a smooth low-temperature density of states, ${\chi }_0\simeq \nu (\mu )$, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

For a uniform exchange field with cost $N{M}^2/(2U)$ and spin-band energies ${ϵ}_k-\sigma M$, the grand potential is even in M. Its curvature at zero field is $N[U^{-1}-{\chi }_0]$, with ${\chi }_0=\int \nu (-n_F^{\prime })dϵ$ and spin-summed density of states per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.

For a three-dimensional zero-temperature parabolic Fermi gas, let $\zeta =(N_{\uparrow }-N_{\downarrow })/N$ and $E_F={\hslash }^2(3{\pi }^{2}N/V{)}^{2/3}/(2m)$. Suppose the interaction energy per electron is $U{N}_{\uparrow }{N}_{\downarrow }/N^2=U(1-{\zeta }^2)/4$. The total energy per electron isIts Taylor expansion has quadratic coefficient $E_F/3-U/4$ and positive quartic coefficient $E_F/81$. Hence the unpolarized state loses stability at $U_c=4E_F/3\propto (N/V{)}^{2/3}$. This $U$ has units of energy. It differs from a contact coupling $g$ defined by interaction-energy density $g{n}_{\uparrow }{n}_{\downarrow }$, for which $g=U/n$ and $g_c=4E_F/(3n)$. Normalization must be fixed before comparing critical couplings. For $U\le 4E_F/3$, convexity of $x^{-1/3}$ implies $f^{\prime \prime }(\zeta )=(E_F/3)[(1+\zeta {)}^{-1/3}+(1-\zeta {)}^{-1/3}]-U/2\ge 0$, so the unpolarized minimum is a global minimum; a first-order polarized minimum does not pre-empt this threshold in the stated model.

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 $U{g}_s<1$.

For an interaction-energy density $U{n}_{\uparrow }{n}_{\downarrow }$, let $n_{\uparrow }=n/2+\delta$, $n_{\downarrow }=n/2-\delta$ at fixed total carrier density. If $g_s(E_F)$ is the unpolarized single-spin density of states per unit volume, the kinetic-energy cost of the two occupation shifts is ${\delta }^2/g_s(E_F)$, whereas the interaction changes by $-U{\delta }^2$. ThereforeFor the three-dimensional free electron model, $g_{\mathrm{tot}}(E_F)=3n/(2E_F)$ and $U_c=4E_F/(3n)$. Both the interaction normalization and the spin counting in the density of states must be specified when quoting a Stoner criterion.

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.