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Quantum magnetism studies collective magnetic behavior in which quantum spin states, noncommuting observables and quantum fluctuations matter. A lattice of local spins can be modeled by a Heisenberg model, whose exchange interaction favors particular relations between neighbours. Its states need not resemble a set of classical arrows pointing in definite directions.

A two-spin spin singlet state illustrates the difference: it has zero total spin while retaining nontrivial spin correlations, rather than two independently known opposite arrows. Large ordered magnets can support spin waves, whereas strong quantum fluctuations can invalidate an expansion around a chosen ordered state. Measurements of magnetic susceptibility, heat capacity and excitation spectra probe different aspects of the same interacting system. The subject is therefore about both collective order and correlated states that resist a simple ordered reference picture.

A time-dependent unitary operator $R(t)$ defines transformed quantum states by $|{\psi }_{\mathrm{r}}⟩=R^{†}|\psi ⟩$. Substituting $|\psi ⟩=R|{\psi }_{\mathrm{r}}⟩$ into the Schrödinger equation gives the displayed transformed Hamiltonian operator, in units $\hslash =1$. If $R=e^{-it{H}_0}$ for constant $H_0$, this becomes $H_{\mathrm{r}}=R^{†}HR-H_0$. Removing a common nuclear Zeeman energy with $H_0=\omega (ZI+IZ)/2$ leaves an isotropic Heisenberg model coupling unchanged, because it commutes with the generator of the common rotation. Removing two unequal nuclear frequencies instead makes their exchange terms oscillate at the difference frequency.

A spin lattice assigns a classical or quantum spin degree of freedom to each site of a specified crystal lattice. The lattice specifies spatial organization, while the interaction graph specifies which spins couple. Nearest-neighbor models couple adjacent sites, but longer-range interactions are also possible.

For an Ising model, each site carries a two-valued spin variable. For a quantum Heisenberg model, spin operators have noncommuting components, and a typical interaction is $J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. The sign and geometry of the couplings determine whether neighboring preferences can be satisfied together or produce geometric frustration.

A spin rotation acts on a quantum state through the unitary representation generated by its angular momentum operators. With $\hslash =1$, rotation by angle $\theta$ about axis $\hat{\mathbf{n}}$ is $e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{S}}$. Simultaneously rotating every site is a global symmetry of an isotropic Heisenberg model.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.