Hubbard Models: Symmetries and Controlled Limits

The Hubbard model isolates kinetic delocalization and onsite repulsion. Its reliable anchors are exact symmetries, the noninteracting and atomic limits, and the strong-coupling expansion. None of these alone proves a Mott phase in a finite-dimensional model; the charge gap, compressibility, order, and thermodynamic limits must also be tested.

Required background. Use second-quantized fermions, Dyson equations, and emergent effective degrees of freedom. Helpful background. Tensor-network ansätze and lattice error budgets govern numerical claims.

Hubbard Hamiltonian and symmetries

For one orbital per site,

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

It has charge $U(1)$ and spin $SU(2)$ when hopping is spin independent. For real nearest-neighbor hopping on a bipartite lattice, the particle–hole transformation $c_{i\sigma}\mapsto\eta_i c_{i\sigma}^\dagger$, with $\eta_i=+1$ on one sublattice and $-1$ on the other, fixes half filling at $\mu=U/2$. Frustrating hoppings or an inequivalent unit cell remove this simple statement.

Three controlled limits

At $U=0$, the exact state is the filled band sea; whether it is metallic or a band insulator depends on dispersion and filling. At $t=0$, one site has energies

$$E_0=0,\qquad E_1=-\mu,\qquad E_2=U-2\mu.$$

For $0<\mu<U$, singly occupied states minimize the energy and charge addition/removal costs are positive. The degeneracy of their spins means that the atomic charge gap does not determine magnetic order.

At half filling and $t/U\ll1$, virtual doublon–holon states cost $U$ and generate an antiferromagnetic exchange scale $J=4t^2/U$. Corrections begin at higher orders in $t/U$ and, with doping, include projected hopping and three-site processes. Hubbard’s original formulation and atomic-band logic appear in Hubbard 1963, pp. 238–257.

What “Mott regime” requires

At minimum, specify commensurate filling, a charge gap from addition energies or a causal spectrum, vanishing thermodynamic compressibility, and whether the gap survives when conventional order is suppressed. Large $U/W$ makes a strong-coupling description plausible but is not an observable. Finite temperature can show bad metallicity above a coherence scale without a zero-temperature insulating state Imada, Fujimori, and Tokura 1998, §§ II–III, pp. 1047–1106.

The superexchange page derives the low-energy spin Hamiltonian. Spectral-weight transfer and the insulator diagnostic supply the missing observable tests.

Exercises

Apply the particle–hole transformation to the interaction and chemical-potential terms and find the symmetric chemical potential.

Solution

$n_{i\sigma}\mapsto1-n_{i\sigma}$, so $Un_{i\uparrow}n_{i\downarrow}-\mu n_i$ becomes the same operator expression plus constants when $\mu=U/2$. Bipartite signs preserve nearest-neighbor hopping. Thus the transformed model has the same form at half filling.

References

• John Hubbard, “Electron Correlations in Narrow Energy Bands,” Proceedings of the Royal Society A 276 (1963) 238–257, doi:10.1098/rspa.1963.0204.
• Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal–Insulator Transitions,” Reviews of Modern Physics 70 (1998) 1039–1263, §§ II–III, doi:10.1103/RevModPhys.70.1039.