Exchange Interactions and Effective Spin Hamiltonians

An effective spin Hamiltonian is justified when charge excitations are separated from a low-energy manifold of localized moments. Virtual hopping, direct Coulomb exchange, spin–orbit coupling, and lattice symmetry then determine the allowed couplings. The sign and scale of exchange must be derived for that manifold; “superexchange is antiferromagnetic” is a controlled result only in its stated limit.

Required background. Superexchange and the $t$–$J$ Projection supplies the strong-coupling projection from the Hubbard model. Helpful background. Internal, Spacetime, Discrete, and Antiunitary Symmetries supplies the symmetry constraints on anisotropic exchange.

Superexchange from virtual charge fluctuations

Consider the repulsive one-band Hubbard model at one electron per site,

$$H=-\sum_{\langle ij\rangle,\sigma} (t_{ij}c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.}) +U\sum_i n_{i\uparrow}n_{i\downarrow}, \qquad U\gg |t_{ij}|.$$

Let $P$ project onto singly occupied sites. Since $PHP=0$ for the hopping part, second-order degenerate perturbation theory gives

$$H_{\mathrm{eff}}^{(2)}=-PH_tQ\frac1{QH_UQ-E_0}QH_tP =\sum_{\langle ij\rangle}\frac{4|t_{ij}|^2}{U} \left(\mathbf S_i\cdot\mathbf S_j-\frac14n_in_j\right).$$

At fixed unit filling the density term is constant, leaving antiferromagnetic Heisenberg exchange $J_{ij}=4|t_{ij}|^2/U>0$. The physical reason is Pauli blocking: the triplet cannot access the same virtual double occupancy as the singlet, so the singlet is lowered. Anderson’s superexchange construction makes this virtual-state origin explicit Anderson 1950, pp. 350–356.

Corrections begin at higher powers of $t/U$. Fourth order changes pair exchange and produces multispin ring exchange on loops. Near charge degeneracy, with multiple orbitals, or when ligand charge-transfer energies compete with $U$, the denominator and Hund coupling can reverse or reshape the interaction. A spin-only reduction is controlled only when all omitted charge states remain gapped relative to exchange and the processes of interest.

The symmetry-allowed exchange tensor

The most general bilinear coupling between two localized moments is

$$H_{ij}=J_{ij}\mathbf S_i\cdot\mathbf S_j +\mathbf D_{ij}\cdot(\mathbf S_i\times\mathbf S_j) +S_i^a\Gamma_{ij}^{ab}S_j^b,$$

where $\Gamma^{ab}$ may be chosen symmetric and traceless after its isotropic trace is absorbed into $J$. The Dzyaloshinskii–Moriya vector $\mathbf D_{ij}$ is odd under exchanging $i$ and $j$. An inversion center at the bond midpoint forbids it; lower point-group symmetry constrains its direction. Moriya derived these rules from spin–orbit-assisted virtual hopping Moriya 1960, pp. 91–98.

Single-ion terms such as $D(S_i^z)^2$ exist for $S\ge1$ but are constant for spin $1/2$. Zeeman coupling is $-\sum_i\mathbf h\cdot\mathbf S_i$ when $\mathbf h$ includes the $g$ tensor and magnetic field. Further-neighbor exchange, biquadratic terms, scalar chirality $\mathbf S_i\cdot(\mathbf S_j\times\mathbf S_k)$, and ring exchange must be included when symmetry and perturbation theory permit them.

Matching and validity checks

A proposed model should state the local spin representation, lattice and bond orientation, sign convention ($J>0$ antiferromagnetic here), hierarchy of exchange scales, and which charge or orbital states were removed. Spectroscopic fits can determine exchange parameters, but a good dispersion fit need not uniquely identify the microscopic virtual process. Thermodynamics, polarization dependence, field response, and ab initio or cluster estimates supply independent constraints.

The expansion also distinguishes a Heisenberg model from the doped $t$–$J$ model: away from unit filling, projected carriers remain dynamical. Setting $n_i=1$ after doping would discard the very low-energy charge degrees of freedom that invalidate a spin-only theory. A systematic fourth-order strong-coupling expansion is given by MacDonald, Girvin, and Yoshioka 1988, pp. 9753–9756.

Exercises

1. For two Hubbard sites with two electrons, compute the singlet–triplet splitting to order $t^2/U$.

Solution

The effective bond Hamiltonian is $J(\mathbf S_1\cdot\mathbf S_2-1/4)$ with $J=4t^2/U$. The singlet has $\mathbf S_1\cdot\mathbf S_2=-3/4$ and energy $-J$; the triplet has $1/4$ and energy $0$. Hence $E_T-E_S=4t^2/U$.

2. Show that bond-center inversion forbids Dzyaloshinskii–Moriya exchange.

Solution

Inversion interchanges sites $i$ and $j$ while axial spins themselves are unchanged. Thus $\mathbf S_i\times\mathbf S_j\mapsto\mathbf S_j\times\mathbf S_i=-\mathbf S_i\times\mathbf S_j$. If the bond is inversion symmetric, the Hamiltonian must be invariant, forcing $\mathbf D_{ij}=0$.

References

• Anderson, P. W. “Antiferromagnetism. Theory of Superexchange Interaction.” Physical Review 79 (1950): 350–356. DOI.
• MacDonald, A. H., S. M. Girvin, and D. Yoshioka. “$t/U$ Expansion for the Hubbard Model.” Physical Review B 37 (1988): 9753–9756. DOI.
• Moriya, T. “Anisotropic Superexchange Interaction and Weak Ferromagnetism.” Physical Review 120 (1960): 91–98. DOI.