Hubbard Model

The single-band Hubbard model describes spin-$1/2$ fermions that hop between lattice sites and interact when opposite-spin particles occupy the same site. In its standard nearest-neighbor form,

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

where $n_{i\sigma}=c_{i\sigma}^\dagger c_{i\sigma}$. The hopping amplitude $t$ favors delocalization. Repulsive $U>0$ penalizes double occupancy, while attractive $U<0$ favors onsite pairs.

The formula is short, but it poses an interacting many-body problem. The kinetic term is simple in momentum space, the interaction is simple in real space, and the two terms generally cannot be diagonalized in the same one-particle basis. Their competition produces correlated metals, local moments, interaction-driven charge localization, magnetic exchange, pairing tendencies, and difficult numerical benchmarks.

Helpful background. Fermionic mode operators explains hopping signs; occupation representation explains the basis used for the finite examples.

Hopping and four local occupation states

Choose a lattice or, more generally, a graph with $L$ sites. Each site has two fermionic modes labeled by

$$\sigma\in\{\uparrow,\downarrow\}.$$

The canonical anticommutation relations are

$$\begin{aligned} \{c_{i\sigma},c_{j\sigma'}^\dagger\} &= \delta_{ij}\delta_{\sigma\sigma'}, \\ \{c_{i\sigma},c_{j\sigma'}\} &= \{c_{i\sigma}^\dagger,c_{j\sigma'}^\dagger\} =0. \end{aligned}$$

A single site has four occupation states,

$$\lvert 0\rangle_i, \qquad \lvert\uparrow\rangle_i, \qquad \lvert\downarrow\rangle_i, \qquad \lvert\uparrow\downarrow\rangle_i.$$

The first three have zero onsite interaction energy in the unshifted Hamiltonian above. The doubly occupied state has interaction energy $U$.

The Hubbard model combines intersite hopping with a local four-state Hilbert space. In the unshifted Hamiltonian, the onsite interaction changes only the energy of $\lvert\uparrow\downarrow\rangle$; a chemical potential shifts states according to their particle number.

The site label is a mode label, not a distinguishable-particle label. Fermionic signs therefore depend on a declared global ordering of the $2L$ modes even when the Hamiltonian is local.

Hilbert Space and Symmetry Sectors

After fixing a mode ordering, the Fock space is isomorphic as a vector space to a product of local four-dimensional spaces:

$$\mathcal F \simeq \bigotimes_{i=1}^{L} \mathbb C^4, \qquad \dim\mathcal F=4^L.$$

This ordinary tensor-product notation is useful for counting. The locality of fermionic operators still carries the parity and ordering structure explained on the occupation-number pages.

For spin-conserving hopping, the particle numbers

$$N_\uparrow = \sum_i n_{i\uparrow}, \qquad N_\downarrow = \sum_i n_{i\downarrow}$$

are separately conserved. The corresponding sector has dimension

$$\dim\mathcal H_{N_\uparrow,N_\downarrow} = \binom{L}{N_\uparrow} \binom{L}{N_\downarrow}.$$

At fixed total particle number $N=N_\uparrow+N_\downarrow$ but unrestricted spin projection,

$$\dim\mathcal H_N = \binom{2L}{N} = \sum_{N_\uparrow+N_\downarrow=N} \binom{L}{N_\uparrow} \binom{L}{N_\downarrow}.$$

These formulas explain why small clusters already become difficult. At half filling with $N_\uparrow=N_\downarrow=L/2$, the sector dimension grows approximately exponentially with $L$ even after both particle numbers are fixed.

General Single-Band Hamiltonian

On an arbitrary lattice, a useful number-conserving form is

$$H ={} - \sum_{i,j,\sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + U \sum_i n_{i\uparrow}n_{i\downarrow} + \sum_i v_i n_i,$$

with

$$n_i=n_{i\uparrow}+n_{i\downarrow}, \qquad t_{ji}=t_{ij}^*.$$

The matrix $t_{ij}$ specifies hopping amplitudes, Peierls phases, and possibly longer-range motion. The scalar $v_i$ may represent a trap, disorder, or a site-dependent orbital energy. A chemical potential belongs to the grand Hamiltonian

$$K = H-\mu\hat N, \qquad \hat N=\sum_i n_i,$$

rather than to the isolated fixed-$N$ Hamiltonian unless a convention explicitly folds it into $H$.

For real uniform nearest-neighbor hopping, one commonly writes

$$H_t = -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + \mathrm{h.c.} \right),$$

where $\langle i,j\rangle$ denotes each unordered bond once. If oriented bonds are summed instead, the Hermitian-conjugate term and numerical normalization must be adjusted.

Parameters and Conventions

Quantity	Meaning	Important qualification
$t$	nearest-neighbor hopping energy	its sign may or may not be gauge-removable
$U$	onsite interaction energy	$U>0$ repulsive; $U<0$ attractive
$\mu$	chemical potential	convention-dependent zero at half filling
$L$	number of lattice sites	distinct from the number of fermionic modes $2L$
$N$	total particle number	ranges from $0$ to $2L$
$n=N/L$	filling per site	half filling is $n=1$, not $n=1/2$
$W$	noninteracting bandwidth	depends on lattice and hopping network
$T$	temperature	compare $k_{\mathrm B}T$ with $t$, $U$, and emergent scales

The ratio $U/t$ is common for a fixed lattice convention, while $U/W$ is often more portable across lattices. Neither ratio alone specifies the model: dimension, geometry, filling, boundary conditions, flux, and hopping range also matter.

For hole doping away from half filling, a common convention is

$$\delta=1-n.$$

Then $\delta>0$ denotes hole doping and $\delta<0$ electron doping. Other communities use different signs, so the definition should always accompany the symbol.

The Sign of the Hopping

On a bipartite lattice with only real hopping between opposite sublattices, the transformation

$$c_{i\sigma} \longmapsto \eta_i c_{i\sigma}, \qquad \eta_i= \begin{cases} +1,&i\in A,\\ -1,&i\in B, \end{cases}$$

changes $t\to -t$. The overall hopping sign is therefore gauge-equivalent in this bipartite setting. The same staggered rephasing reverses every intersublattice hopping even if those hoppings carry phases; it preserves each even-loop flux.

On a general graph, products of hopping phases around closed loops are invariant. Same-sublattice links or odd loops can obstruct reversal of every bond sign; independent changes of a flux cannot be removed by rephasing sites. The sign and phase pattern can then affect the spectrum and cannot be dismissed as a convention.

Noninteracting Limit

When $U=0$, the model reduces to the spinful Tight-Binding Model. For a translationally invariant $d$-dimensional hypercubic lattice with spacing $a$ and nearest-neighbor hopping,

$$c_{j\sigma} = \frac{1}{\sqrt L} \sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf r_j} c_{\mathbf k\sigma},$$

and

$$H_t = \sum_{\mathbf k,\sigma} \epsilon_{\mathbf k} c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma},$$

with dispersion

$$\epsilon_{\mathbf k} = -2t \sum_{\alpha=1}^{d} \cos(k_\alpha a).$$

The continuous bulk band runs from $-2d\lvert t\rvert$ to $2d\lvert t\rvert$, so

$$W=4d\lvert t\rvert = 2z\lvert t\rvert,$$

where $z=2d$ is the coordination number. This identity is specific to the nearest-neighbor hypercubic band. A finite periodic momentum grid may miss a band edge, so its actual spectral width can be smaller.

At zero temperature, each spin species fills the lowest one-particle energies up to the Fermi level. Turning on $U$ couples momenta and destroys the reduction to independent occupation numbers, even though translation symmetry may remain exact.

Atomic Limit

When $t=0$, sites decouple. For one site in the grand-canonical convention,

$$K_i = U n_{i\uparrow}n_{i\downarrow} - \mu(n_{i\uparrow}+n_{i\downarrow}),$$

the local energies are

$$\begin{array}{c|cccc} \text{state} &\lvert0\rangle &\lvert\uparrow\rangle &\lvert\downarrow\rangle &\lvert\uparrow\downarrow\rangle \\ \hline K_i &0 &-\mu &-\mu &U-2\mu \end{array}$$

and the one-site partition function is

$$Z_i = 1 + 2e^{\beta\mu} + e^{2\beta\mu-\beta U}.$$

The mean occupation and double occupancy are

$$\begin{aligned} \langle n_i\rangle &= \frac{ 2e^{\beta\mu} + 2e^{2\beta\mu-\beta U} }{Z_i}, \\ \langle D_i\rangle &= \langle n_{i\uparrow}n_{i\downarrow}\rangle = \frac{e^{2\beta\mu-\beta U}}{Z_i}. \end{aligned}$$

For repulsive $U$ at the particle-hole-symmetric value $\mu=U/2$, one finds $\langle n_i\rangle=1$ at every temperature. As $\beta U\to\infty$, empty and doubly occupied states are suppressed relative to the two singly occupied states. Charge is frozen locally while a spin-$1/2$ degree of freedom remains.

The atomic limit is not by itself a proof of a thermodynamic Mott phase. It is an exact reference point that reveals the local charge scale and the emergent spin degeneracy.

Particle-Hole-Symmetric Convention

For real hopping only between opposite sublattices, with no additional site potentials, the particle-hole-symmetric grand Hamiltonian is

$$K_{\mathrm{ph}} = H_t + U \sum_i \left(n_{i\uparrow}-\frac12\right) \left(n_{i\downarrow}-\frac12\right).$$

Expanding the interaction gives

$$U \left(n_{i\uparrow}-\frac12\right) \left(n_{i\downarrow}-\frac12\right) ={} U n_{i\uparrow}n_{i\downarrow} - \frac U2 n_i + \frac U4.$$

Thus $K_{\mathrm{ph}}$ is the standard grand Hamiltonian at

$$\mu=\frac U2$$

plus the additive constant $UL/4$. In the centered convention, half filling is described by zero shifted chemical potential. Statements such as “$\mu=0$ at half filling” and “$\mu=U/2$ at half filling” can therefore both be correct under different conventions. Chemical Potential explains the underlying energy-zero and particle-number conventions.

Let $\eta_i=+1$ on sublattice $A$ and $-1$ on sublattice $B$. The particle-hole transformation

$$c_{i\sigma} \longmapsto \eta_i c_{i\sigma}^\dagger$$

sends $n_{i\sigma}-1/2$ to its negative and leaves this centered Hamiltonian invariant. As a unitary particle-hole transformation it conjugates the hopping amplitudes, so an arbitrary complex-flux hopping pattern need not be invariant. Bipartiteness alone is insufficient without the hopping and potential conditions. This symmetry pins the half-filled density in the grand-canonical ensemble under the stated assumptions.

Continuous and Lattice Symmetries

For spin-independent hopping and no spin-dependent fields, total particle number is conserved:

$$[H,\hat N]=0.$$

The Hamiltonian is also invariant under global spin rotations. Define dimensionless local spin operators by

$$\mathbf s_i = \frac12 \sum_{\alpha,\beta} c_{i\alpha}^\dagger \boldsymbol\sigma_{\alpha\beta} c_{i\beta}.$$

Then

$$[H,\mathbf S_{\mathrm{tot}}]=0, \qquad \mathbf S_{\mathrm{tot}}=\sum_i\mathbf s_i.$$

The separate conservation of $N_\uparrow$ and $N_\downarrow$ is compatible with this spin $SU(2)$ symmetry: $N_\uparrow-N_\downarrow=2S_{\mathrm{tot}}^z$ is one generator, while the raising and lowering generators connect sectors with different $N_\uparrow$ and $N_\downarrow$ at fixed $N$.

Additional symmetries depend on the lattice and parameters:

• translations for uniform periodic lattices;
• point-group operations preserved by the hopping graph;
• time reversal for real spin-independent hopping without magnetic flux;
• particle-hole symmetry for the centered model on an appropriate bipartite lattice;
• an additional pseudospin structure in special bipartite forms of the model.

Symmetry labels reduce finite-cluster calculations and constrain correlation functions. They do not, by themselves, solve the interacting spectrum.

Exact Hubbard Dimer Spectrum

The two-site, two-fermion problem is the smallest cluster that contains hopping, double occupancy, spin symmetry, and interaction-generated exchange. Let

$$H = -t \sum_\sigma \left( c_{1\sigma}^\dagger c_{2\sigma} + c_{2\sigma}^\dagger c_{1\sigma} \right) + U\sum_{i=1}^{2} n_{i\uparrow}n_{i\downarrow}.$$

Choose the mode order $(1\uparrow,1\downarrow,2\uparrow,2\downarrow)$. In the spin-singlet sector, define

$$\begin{aligned} \lvert S\rangle &= \frac{ \lvert\uparrow,\downarrow\rangle - \lvert\downarrow,\uparrow\rangle }{\sqrt2}, \\ \lvert D_+\rangle &= \frac{ \lvert\uparrow\downarrow,0\rangle + \lvert0,\uparrow\downarrow\rangle }{\sqrt2}, \\ \lvert D_-\rangle &= \frac{ \lvert\uparrow\downarrow,0\rangle - \lvert0,\uparrow\downarrow\rangle }{\sqrt2}. \end{aligned}$$

With this phase convention,

$$H_{S,D_+,D_-} = \begin{pmatrix} 0&-2t&0\\ -2t&U&0\\ 0&0&U \end{pmatrix}.$$

Changing the phase of $\lvert D_+\rangle$ changes both off-diagonal signs but no observable. The three triplet states have energy zero because Pauli exclusion prevents an onsite same-orbital triplet.

The two coupled singlet energies are

$$E_\pm = \frac12 \left( U \pm \sqrt{U^2+16t^2} \right),$$

while the decoupled odd doublon has energy $U$. For $t\ne0$, the lower singlet is the unique two-particle ground state for either sign of $U$. At $t=0$ additional degeneracies appear. The Hubbard dimer gives the other particle-number sectors and their comparison.

The Hellmann–Feynman theorem gives its total double occupancy:

$$\begin{aligned} D_{\mathrm{tot}} &= \left\langle \sum_{i=1}^{2} n_{i\uparrow}n_{i\downarrow} \right\rangle \\ &= \frac{\partial E_-}{\partial U} = \frac12 \left( 1- \frac{U}{\sqrt{U^2+16t^2}} \right). \end{aligned}$$

At $U=0$, $D_{\mathrm{tot}}=1/2$. At large positive $U$,

$$D_{\mathrm{tot}} = \frac{4t^2}{U^2} + O\!\left(\frac{t^4}{U^4}\right).$$

The exact singlet-triplet gap is

$$\Delta_{ST} = -E_- = \frac{ \sqrt{U^2+16t^2}-U }{2}.$$

Its large-$U$ limit is $4t^2/U$. The controlled effective-Hamiltonian derivation explains why this scale is an antiferromagnetic exchange rather than an ordinary first-order hopping energy.

Strong-Coupling Spin Physics

For repulsive $U$ at half filling and

$$\lvert t\rvert\ll U,$$

the low-energy states have approximately one fermion per site. A single hop creates a virtual doublon-hole pair with energy cost of order $U$; a second hop can remove it. To second order, the resulting low-energy Hamiltonian is

$$H_{\mathrm{eff}} = J \sum_{\langle i,j\rangle} \left( \mathbf s_i\cdot\mathbf s_j - \frac14 n_i n_j \right), \qquad J=\frac{4t^2}{U},$$

in the half-filled low-energy subspace, to the stated second order. At exactly one fermion per site, $n_i n_j=1$ and the density term is an additive constant, leaving the antiferromagnetic Heisenberg model.

The hierarchy

$$J\sim\frac{t^2}{U} \ll \lvert t\rvert \ll U$$

separates spin dynamics from charge excitations. A system can have well-formed local moments while its spins remain thermally disordered when

$$J \ll k_{\mathrm B}T \ll U.$$

Away from half filling, real hopping survives inside the projected low-energy space, leading to the t–J model rather than a pure spin Hamiltonian. At finite doping, second-order elimination also generates three-site correlated hopping; the simplest t–J Hamiltonian drops those terms as a further approximation. Complex bond hopping gives $J_{ij}=4|t_{ij}|^2/U$ at this order. Longer-range hopping, multiple orbitals and higher orders generate additional terms. The coefficient $4t^2/U$ is therefore a controlled result for a specific limit, not a universal formula for every antiferromagnet.

Mott Physics Preview

A band can be partially filled and therefore metallic in the noninteracting description, yet become charge insulating when repulsion suppresses charge fluctuations. This interaction-driven mechanism is the central Mott-physics lesson of the Hubbard model.

For a finite cluster, the single-fermion addition/removal gap is

$$\Delta_c(L) = E_0(N+1,L) + E_0(N-1,L) - 2E_0(N,L).$$

Its interpretation requires a controlled $L\to\infty$ limit and an examination of all relevant charge sectors. In the ordinary repulsive Mott regime it measures the charge gap. In a paired attractive system, a positive single-fermion gap can instead measure pair breaking while pairs remain compressible. A finite gap to odd particle number does not by itself establish a density plateau; compare pair addition/removal and the global minimum of $E_0(N)-\mu N$ as well. On a finite lattice, avoided crossings and nonzero level spacings are not by themselves phase transitions.

Several distinctions matter:

• suppressed double occupancy indicates local correlation, not a complete definition of a Mott insulator;
• a charge gap does not require a spin gap;
• antiferromagnetic order can coexist with interaction-driven charge localization;
• at half filling in one dimension, the repulsive nearest-neighbor model has a charge gap for every $U>0$, while spin excitations remain gapless;
• higher-dimensional behavior depends on lattice, frustration, temperature, and the distinction between symmetry breaking and paramagnetic localization;
• the doped two-dimensional model contains important open questions, so no single universal phase diagram should be presented as settled.

This page introduces those diagnostics without replacing the dedicated treatment of Mott phases and correlated materials.

Local Moments and Magnetism

Define the local double occupancy

$$D_i = \langle n_{i\uparrow}n_{i\downarrow}\rangle.$$

Because $n_{i\sigma}^2=n_{i\sigma}$,

$$\begin{aligned} \left\langle (n_{i\uparrow}-n_{i\downarrow})^2 \right\rangle &= \langle n_i\rangle - 2D_i. \end{aligned}$$

Repulsive interaction at fixed filling tends to reduce $D_i$ and increase this local-moment diagnostic. Long-range magnetic order is a separate question requiring intersite correlations and a thermodynamic limit.

The spin structure factor is commonly defined by

$$S_s(\mathbf q) = \frac1L \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \langle \mathbf s_i\cdot\mathbf s_j \rangle.$$

On a bipartite square lattice, enhanced weight near $\mathbf Q=(\pi/a,\pi/a)$ signals antiferromagnetic correlations. Demonstrating spontaneous order requires finite-size scaling rather than inspection of one small cluster.

Structure factors explains connected versus full correlations, peak scaling and scattering normalization for these channels.

Attractive Interaction

For $U<0$, onsite double occupation lowers the energy. The local low-energy tendency is toward empty or doubly occupied sites rather than isolated spins. This favors onsite spin-singlet pairing and charge correlations.

The onsite pair operator is

$$\Delta_i = c_{i\downarrow}c_{i\uparrow},$$

with equal-time pair correlation

$$P_{ij} = \langle \Delta_i^\dagger\Delta_j \rangle.$$

With real intersublattice hopping, a particle-hole transformation applied to one spin species maps the repulsive and attractive models into one another while exchanging aspects of spin and charge order. This exact correspondence is useful, but boundary conditions, chemical-potential terms, spin imbalance, and extra hopping can modify or break it.

Core Observables

No single observable characterizes all Hubbard regimes. A reliable analysis usually combines several of the following.

Filling and compressibility

$$n = \frac1L \sum_i \langle n_i\rangle, \qquad \kappa_c = \frac{\partial n}{\partial\mu}.$$

The normalization of compressibility varies across subfields; some definitions include factors of $n^{-2}$ or volume.

Susceptibilities separates $\partial n/\partial\mu$, isothermal compressibility, local response, and the static-versus-uniform limit conventions.

Double occupancy

$$D = \frac1L \sum_i \langle n_{i\uparrow}n_{i\downarrow}\rangle = \frac1L \frac{\partial F}{\partial U}$$

in the canonical ensemble at fixed $T,N,t$ and other couplings. In the grand-canonical ensemble use $(\partial\Omega/\partial U)_{T,\mu}/L$ instead. Differentiating along a path such as $\mu(U)=U/2$ introduces an additional chain-rule term. The free-energy derivative makes double occupancy a direct measure of interaction response.

Kinetic energy

$$E_{\mathrm{kin}} = \langle H_t\rangle.$$

This measures delocalization but is convention-dependent through the hopping network and energy zero.

Momentum distribution

$$n_\sigma(\mathbf k) = \langle c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma} \rangle.$$

For an interacting system this is not generally restricted to zero or one even at zero temperature, although the total occupation per fermionic mode remains bounded by one.

Charge and spin correlations

With $\delta n_i=n_i-\langle n_i\rangle$,

$$C_c(i,j) = \langle\delta n_i\delta n_j\rangle, \qquad C_s(i,j) = \langle\mathbf s_i\cdot\mathbf s_j\rangle.$$

Their Fourier transforms diagnose characteristic wavevectors and correlation lengths.

One-particle spectrum

The retarded Green function and spectral function reveal addition and removal energies, coherent quasiparticle features, incoherent continua, and interaction-induced gaps. Their definitions belong on the correlation and response pages; here the key point is that the interacting spectrum is not determined by the noninteracting dispersion alone.

From Continuum or Orbitals to the Model

The Hubbard Hamiltonian can arise by projecting a continuum or multiorbital problem onto orthonormal localized orbitals $w_i(\mathbf r)$. Plain projection gives matrix elements within that subspace; eliminating virtual higher-band excitations can further renormalize them and generate additional interactions. A schematic hopping matrix element is

$$t_{ij} = - \int d^dr\, w_i^*(\mathbf r) h_0 w_j(\mathbf r).$$

In a weak-coupling contact approximation for opposite spins, the leading onsite matrix element is

$$U = g \int d^dr\, \lvert w_i(\mathbf r)\rvert^4.$$

For electronic orbitals with an effective interaction $V$,

$$U ={} \int d^dr\,d^dr'\, \lvert w_i(\mathbf r)\rvert^2 \times V(\mathbf r-\mathbf r') \lvert w_i(\mathbf r')\rvert^2.$$

Here $g$ is the physical low-energy coupling in the appropriate dimensional and confinement regime, not an arbitrary bare delta-potential coefficient. Near resonance or when interactions mix higher bands, the overlap formula is insufficient and the effective parameters need scattering or few-body matching. A one-band truncation is controlled only when eliminated excitation energies are large and their coupling to retained states is perturbative on that scale. Real materials may require several orbitals, longer-range Coulomb terms, spin-orbit coupling, lattice vibrations, disorder, or frequency-dependent effective interactions.

In Optical Lattices, the periodic potential and short-range atomic collisions can provide a comparatively direct realization of the same structure. The mapping is still an effective one: trap inhomogeneity, finite temperature, higher bands, and calibration of $t$ and $U$ remain part of the experimental problem.

Exact Solution Status

The Hubbard model is not generally exactly solvable. Its status depends sharply on geometry and parameters.

Regime	What is controlled
$U=0$	free-fermion band problem
$t=0$	independent atomic sites
two sites	complete finite-dimensional diagonalization
one-dimensional nearest-neighbor chain	Bethe-ansatz solution for the standard uniform model
$U/\lvert t\rvert\gg1$ near half filling	controlled strong-coupling expansion for low energies
general two-dimensional doped model	no complete exact solution; multiple numerical and analytical methods are required
standard infinite-dimensional or large-coordination lattice limit, with $t\propto z^{-1/2}$ and local interaction	single-site dynamical mean-field theory becomes exact

An exact solution in one dimension does not determine the two-dimensional phase diagram. Likewise, a strong-coupling expansion controls a scale-separated regime, not arbitrary intermediate $U/t$.

Numerical Representation

For exact diagonalization, choose a fixed ordering such as

$$(1\uparrow,2\uparrow,\ldots,L\uparrow, 1\downarrow,2\downarrow,\ldots,L\downarrow)$$

or an interleaved site ordering, and use it everywhere. A basis state can be represented by two $L$-bit strings, one for each spin. The interaction is diagonal:

$$H_U\lvert\boldsymbol n_\uparrow, \boldsymbol n_\downarrow\rangle = U \left( \sum_i n_{i\uparrow}n_{i\downarrow} \right) \lvert\boldsymbol n_\uparrow, \boldsymbol n_\downarrow\rangle.$$

Each hopping term flips one occupied and one empty bit in the same spin string. Its matrix element includes the fermionic parity accumulated between the two modes in the declared ordering. The resulting Hamiltonian is sparse.

Useful method families include exact diagonalization for small clusters, tensor networks in one dimension and selected quasi-one-dimensional geometries, determinant and auxiliary-field Monte Carlo where sign structure permits, dynamical mean-field methods, diagrammatic expansions, and controlled weak- or strong-coupling approximations. Variational Many-Body States uses the Hubbard dimer to show how a Gutzwiller factor variationally suppresses, but need not eliminate, finite-$U$ doublons.

The repulsive model with real hopping on a bipartite lattice at particle-hole-symmetric half filling is a standard sign-problem-free setting for determinant quantum Monte Carlo. Generic doping, frustration, or additional hopping can reintroduce a severe sign problem. The Sign Problem Preview explains the determinant pairing and why this is an algorithm-specific structural statement, not a claim that every half-filled Hubbard calculation is easy.

Exercises

Particle-hole transformation

For real hopping on a bipartite nearest-neighbor lattice, show that $c_{i\sigma}\mapsto\eta_i c_{i\sigma}^\dagger$ sends $n_{i\sigma}-1/2$ to its negative and leaves $K_{\mathrm{ph}}$ invariant.

Solution

The number operator transforms as

$$n_{i\sigma} = c_{i\sigma}^\dagger c_{i\sigma} \longmapsto c_{i\sigma}c_{i\sigma}^\dagger = 1-n_{i\sigma}.$$

Hence

$$n_{i\sigma}-\frac12 \longmapsto -\left(n_{i\sigma}-\frac12\right),$$

and the product of the two centered spin densities is invariant.

For a nearest-neighbor bond, $\eta_i\eta_j=-1$. Anticommuting the transformed operators back into normal order supplies the second minus sign required to preserve the hopping term. Constants do not appear in the centered interaction, so the full $K_{\mathrm{ph}}$ is invariant.

Local moment versus magnetic order

Show that

$$(n_{i\uparrow}-n_{i\downarrow})^2 = n_i-2n_{i\uparrow}n_{i\downarrow}.$$

Explain why a large expectation value of this operator does not establish antiferromagnetic long-range order.

Solution

Using $n_{i\sigma}^2=n_{i\sigma}$,

$$\begin{aligned} (n_{i\uparrow}-n_{i\downarrow})^2 &= n_{i\uparrow}^2 + n_{i\downarrow}^2 - 2n_{i\uparrow}n_{i\downarrow} \\ &= n_i - 2n_{i\uparrow}n_{i\downarrow}. \end{aligned}$$

This is a one-site diagnostic. It measures whether a site tends to carry an uncompensated spin, but it contains no information about alignment between distant sites. Antiferromagnetic order requires long-distance spin correlations or scaling of $S_s(\mathbf Q)$ with system size.

A single-fermion gap without a pair plateau

Take two decoupled sites, $t=0$, with $U<0$. Find the minimum energies $E_0(N)$ for $N=0,\ldots,4$. At $N=2$, compare the single-fermion gap with $E_0(4)+E_0(0)-2E_0(2)$. Does $N=2$ minimize $E_0(N)-\mu N$ over an open interval of chemical potential?

Solution

Attraction maximizes the number of doublons, giving

$$(E_0(0),E_0(1),E_0(2),E_0(3),E_0(4))=(0,0,U,U,2U).$$

The odd-sector addition/removal combination at $N=2$ is $-U>0$, whereas the pair combination is zero. At $\mu=U/2$, the $N=0,2,4$ ground energies of $H-\mu N$ all coincide. Below this value the empty state wins; above it the four-particle state wins. There is no open $N=2$ plateau. This atomic example separates pair breaking from incompressibility; by itself it makes no claim of transport, since $t=0$ forbids motion.

References

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