Weak Coupling: Mean-Field Theory and the Slater Insulator

Starting from the non-interacting limit, a mean-field decoupling of the Hubbard interaction with staggered magnetization gives an antiferromagnetic two-band insulator. Solving the self-consistent gap equation reveals a Slater gap that opens exponentially at weak coupling and crosses over to the Mott gap Δ → U/2 at strong coupling.

Methods Mean-field theory

A standard first approximation treats one term of the Hubbard model as a perturbation around the solvable limit of the other. Here we begin from the non-interacting limit , with but strong enough to qualitatively change the free Fermi gas, and apply a mean-field approximation. Writing ,

The constant is dropped and is redefined as , entering .

Mean-field decoupling

Assuming small fluctuations of the local occupation, introduce deviation operators

so that, dropping the term quadratic in the (small) deviations,

Discarding the constant shift, the mean-field Hubbard Hamiltonian is

Staggered magnetization and zone folding

At half-filling (), antiferromagnetic order is observed experimentally. Introduce a staggered magnetization with ,

where is the sublattice magnetization amplitude. This assumes magnetic order at wave vector — the nesting vector met earlier — since

with on the two sublattices. Parameterizing the occupations,

At both equal (paramagnet); at the other extreme the site is fully spin-up on one sublattice and fully spin-down on the other — perfect AFM order. This doubles the unit cell and rotates the reciprocal lattice by . The old primitive and reciprocal vectors,

become, in the -rotated lattice,

so the new Brillouin zone is half the old one. Substituting into ,

The last term cancels the shift, so and , giving

In momentum space the factor nests the vectors,

The third term scatters electrons from to without flipping spin. Useful identities:

Diagonalizing the mean-field Hamiltonian

Use the Heisenberg equations of motion, , with

which give (after a shift , at half-filling ),

In matrix form, with ,

Equivalently, splitting the full-BZ Hamiltonian into the reduced zone and its complement,

Diagonalize with a unitary so that , , where

and , built from the eigenvectors of , is

with

The diagonalized Hamiltonian is then a two-band insulator,

with an energy gap at half-filling — exactly the insulating behaviour of the parent cuprates.

Self-consistency: the gap equation

We assumed AFM order with amplitude ; now we close the loop, computing from the diagonalized and equating it to . Starting from the real-space occupation,

and inverting ,

Using and ,

Splitting the full-BZ sum into two over (and using ),

At half-filling only the lower band () is filled, so the only surviving expectation is , giving and hence

so that, with , the self-consistent gap equation is

Going to polar coordinates, changing variables and introducing the density of states (with ),

Weak vs. strong coupling: Slater meets Mott

In the weak-coupling limit (), the 2D density of states diverges logarithmically,

so the gap integral is dominated by :

Solving for the gap,

(the prefactor is cutoff-dependent, hence non-universal). As the gap collapses, but the exponent makes the approach far slower than the usual of metals with finite : even weak interactions favour antiferromagnetic order on the nested square lattice. This is a Slater insulator — driven by Fermi-surface nesting.

In the opposite strong-coupling limit (, ), dominates the denominator,

and since , we get ; as grows further , recovering the Mott gap of the single-site limit.