Emergent Matter

Strong Coupling: Perturbation Theory and Superexchange

In the opposite limit U ≫ t, treating hopping as a perturbation and projecting out doubly occupied sites with a Schrieffer–Wolff transformation maps the half-filled Hubbard model onto the antiferromagnetic Heisenberg model, with superexchange J = 4t²/U.

By Abdulrahman Anter Université Paris-Saclay · LPS, Orsay
graduate evergreen Theory updated 2026-06-21

Methods Perturbation theory

Now take the opposite limit, , treating the kinetic energy as a perturbation to the dominant interaction. At half-filling (),

To keep things tractable we restrict to the two-site Hubbard model, which already shows the key physics:

Hubbard subbands

Consider an -site lattice at : copies of the single-site levels, read as single-electron levels. One electron occupies ; adding a second to the same site places it at . With electrons, fill the lower levels and the extra one sits in an level. Turning on hopping lets that electron delocalize across its neighbours, broadening the upper level into a band of width ; likewise a hole broadens the lower level. The result is two Hubbard subbands.

Single-electron level picture: the second electron on a site costs an extra U.
Single-electron level picture: the second electron on a site costs an extra U.
Hopping broadens the levels into two Hubbard subbands of width ~2zt.
Hopping broadens the levels into two Hubbard subbands of width ~2zt.
Formation of the Hubbard subbands: hopping broadens the discrete atomic levels into two bands of width ~2zt separated by ~U (Fazekas, 1999).

Effective Hamiltonian: projecting out double occupancy

We seek an effective Hamiltonian for the lowest subband — no doubly occupied sites. Split the hopping by how it changes the number of doublons, using the single-site projection operators

For example, with an on site 2 and a on site 1, the operator singles out the process that creates a doublon. Collecting terms by their effect on the doublon number,

so that , where raise/lower the doublon number and conserves it.

Projection-operator identities (rewriting H± in number operators)

Using , , , :

We eliminate transitions between the low- and high-energy subbands to leading order in with a Schrieffer–Wolff unitary transformation,

Schrieffer–Wolff bookkeeping (why this S, and which terms survive)

Expanding the transformation,

A single commutator hints at the right :

i.e. and . With , the orders are

which we drop, along with the contributions.

Since and , choosing

makes the doublon-changing pieces cancel at first order:

Dropping the terms,

At half-filling the lowest subband has one electron per site: no double occupancy, so and give nothing, and no hopping conserves the (zero) doublon number, so drops too. We are left with

The exchange: a Heisenberg Hamiltonian

The half-filled two-site Hilbert space is spanned by

(notation ). After the transformation the singly-occupied subband decouples from the doublon states; encodes virtual hops into a doublon and back — effective spin exchange. With the convention , the matrix elements are

The factor of 2 comes from the two indistinguishable orderings (either electron may hop first), adding coherently; the signs follow from the state conventions. In the basis,

Computing a matrix element, and two routes to the spin form

Explicitly, expands over into four projector strings,

which the projector identities (, etc.) collapse to two,

each contributing a factor of one in the expectation, so .

From here there are two equivalent routes to the spin form. The second-quantization one uses

or one introduces the Hubbard operators ,

We take the first-quantization route below.

First-quantization derivation: from H_eff to S₁·S₂

Writing with outer products,

With the spin operators , , , this is

where . Using

and ,

Collecting the spin terms gives the antiferromagnetic Heisenberg Hamiltonian,

Prerequisites

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