Magnetism and Spin Systems

Magnetism connects microscopic magnetic moments to collective order, excitations and electrical control. A material can have large fluctuating moments yet no spontaneous magnetization, and a magnetically ordered sample can have zero total moment. Understanding these possibilities starts by separating what carries the moment, how the moments interact, and what the measurement detects.

This chapter develops those connections for solids and nanostructures. For a first pass, read magnetic moments, exchange interactions, one ordered phase, and spin waves. The itinerant, impurity and spin–orbit branches then explain cases that need more than a rigid local-spin picture.

Helpful background. Magnetic Moments and g-Factors explains moment operators and Zeeman signs. Lattice Models and Spin Systems introduces spin Hamiltonians, and Phases, Order, and Criticality explains spontaneous symmetry breaking and thermodynamic limits.

Choose a question

Question	Start here
Why do different experiments quote different moments?	Magnetic Moments in Matter
What produces the coupling between magnetic degrees of freedom?	Exchange Interactions
How can uniform order coexist with domains?	Ferromagnetism
How can a periodic ordered state have no net moment?	Antiferromagnetism
What happens when opposing, inequivalent sublattices compensate?	Ferrimagnetism
Which modes and intensities should scattering reveal?	Spin Waves and Magnons
When must the magnetism be described through mobile electrons?	Itinerant Magnetism and Stoner Criterion
How does a metal screen or couple localized moments?	Kondo Effect and RKKY Interaction
Why do spin and magnetic orientation depend on the lattice?	Spin–Orbit Coupling in Solids and Magnetic Anisotropy
How do currents generate magnetic motion or signals?	Spintronics
What stabilizes walls, vortices and skyrmions?	Skyrmions and Magnetic Textures

These are connected branches, not one compulsory sequence. For example, spin-transfer torque can occur with negligible spin–orbit coupling, while some chiral textures depend strongly on it. The volume chapter guide connects magnetism with transport, disorder, and strong correlations.

Moment, magnetization, order and response

A magnetic moment is a physical dipole operator. A projected spin or pseudospin can represent it, but the projection and its magnetic coupling must be specified. A spin label alone does not fix the measured moment, especially with crystal-field and spin–orbit effects.

Magnetization is the moment per volume. It is a signed average: equal opposite contributions can cancel even when each microscopic moment is large. Susceptibility measures how magnetization changes with an applied field, under stated temperature and boundary conditions. It is neither a moment nor an order parameter by itself.

For an isolated isotropic spin of magnitude $S$ and positive $g$ factor, the familiar moment magnitudes are

$$\mu_{\rm eff}=g\mu_B\sqrt{S(S+1)}, \qquad \mu_{\rm sat}=g\mu_B S.$$

The first follows from the Curie susceptibility and measures thermal fluctuations; the second is the maximum projection along one axis. For $S=1/2$ and $g=2$, they are $\sqrt3\mu_B$ and $\mu_B$, respectively. They differ even before any crystal-field, covalency or many-body correction. An ordered moment measured by diffraction is yet another quantity and can be reduced by fluctuations. Blundell (2001) develops these distinctions from elementary spin statistics.

Spontaneous order concerns a state selected in the thermodynamic limit, with the selecting field removed afterward. A finite symmetric equilibrium calculation may have zero one-point magnetization while its long-distance correlations reveal the tendency to order. Conversely, polarization in a nonzero field does not establish spontaneous order. The general order-parameter treatment explains this order of limits.

A magnon is a collective excitation around an appropriate ordered reference state. Its energy, intensity and decay rate answer different questions. Magnetic scattering can also reveal particle–hole excitations, multiparticle continua or short-range correlations without a sharp magnon.

Choosing magnetic degrees of freedom

A local-spin model is useful when charge and higher crystal-field excitations are sufficiently costly compared with the energies of interest. The retained moment need not be a bare electronic spin: a low-energy doublet can behave as a pseudospin with an anisotropic magnetic response. The gap to discarded states must remain large enough over the temperature, field and frequency range being studied.

For illustration, write dimensionless spins $\mathbf s_i$ and count each bond once:

$$H=J\sum_{\langle ij\rangle}\mathbf s_i\cdot\mathbf s_j -\sum_i\widehat{\boldsymbol\mu}_i\cdot\mathbf B.$$

Here $J$ has energy units. In this convention positive $J$ favors antiparallel alignment on an isolated bond. Another author can put a minus sign or a factor of two in front of the exchange sum, so numerical couplings must be translated before comparison. The Heisenberg model and common spin Hamiltonians give the detailed definitions. Anisotropic exchange, single-ion anisotropy, dipolar interactions and multispin terms may be needed in a material.

The estimate $J=4t^2/U$ is one particular example: the repulsive one-band Hubbard model at one particle per site, $U\gg|t|$, and energies below the charge gap. It does not identify every magnetic insulator. Appreciable doping leaves mobile charge in the low-energy space, motivating the t–J model or a fuller electronic description.

In an itinerant magnet, spin polarization and band occupations must be determined together. A uniform Stoner test is a local stability calculation within a mean-field approximation; it need not identify a finite-wavevector instability or a first-order transition. Local and itinerant behavior can also occur at different scales in the same metal. Dense moments coupled to mobile carriers lead onward to Kondo lattices and heavy fermions.

What does zero magnetization mean?

Suppose a sample has almost zero low-field magnetization below a thermal anomaly. Several physically different states can give that result:

Possibility	Why the net moment can vanish	A useful distinguishing observation
Antiferromagnet	The ordered magnetic basis compensates	Magnetic diffraction and the symmetry of the ordered structure
Ferrimagnet at compensation	Opposing inequivalent sublattices have equal moment magnitudes at that temperature	Site-resolved moments that remain nonzero through compensation
Multidomain ferromagnet	Different domains cancel in the sample average	Domain imaging and response to field training
Paramagnet	No spontaneous dipolar order is present	Local and dynamic correlations, and tests of a nonmagnetic origin of the anomaly

Diffraction requires the propagation vector and the magnetic basis within the cell. An antiferromagnet can have propagation vector zero when different sites in the crystallographic unit cell compensate. A nonzero absolute scattering momentum alone therefore does not distinguish these phases.

Magnetization compensation in a ferrimagnet is also different from angular-momentum compensation when the sublattices have different gyromagnetic factors. Neither is automatically the temperature at which order disappears. Coey (2010) explains the roles of sublattices, domains and sample geometry. Magnetic Susceptibility and Neutron Scattering connect these quantities to their measurements.

Why a layered magnet needs more than a dispersion fit

An ideal infinite two-dimensional isotropic Heisenberg model with sufficiently short-range interactions does not sustain spontaneous ferromagnetic or antiferromagnetic order at nonzero temperature under the Mermin–Wagner hypotheses. This does not forbid all magnetic behavior in a monolayer. Discrete anisotropy, coupling to other layers and sufficiently long-range interactions change the assumptions. An XY regime can support a Berezinskii–Kosterlitz–Thouless phase with algebraic correlations, rather than ordinary nonzero one-point order. A finite sample can show a large correlation length and a crossover without a thermodynamic phase transition.

There is a separate question about a measured excitation gap. Isotropic interlayer exchange can stabilize three-dimensional order and split branches, but it does not remove the Goldstone mode associated with a spontaneously broken continuous spin symmetry. In a two-layer ferromagnetic example, the uniform in-phase rotation costs no exchange energy, while an out-of-phase rotation costs energy because the layers tilt against one another. A gapped optical branch is therefore compatible with a gapless acoustic branch.

A gap in the lowest Goldstone branch instead requires an appropriate symmetry-breaking term or a failure of the assumptions behind that mode. Anisotropy, magnetic field and dipolar interactions are common possibilities. An avoided crossing away from the ordering wavevector does not alone establish such a gap. A weak or invisible low-energy mode can also make the observed spectrum appear gapped.

Two models can fit the same energies along one momentum path while predicting different intensities. Comparing polarization, additional momenta, field dependence and linewidths can separate them. The material spin-wave article develops these comparisons; Magnons gives the general quantization. A broad continuum by itself does not identify fractionalization, because decay, multiparticle, itinerant and disorder mechanisms can also broaden spectra.

Beyond simple ordered phases

The absence of uniform order does not select a unique alternative. Quantum Spin Liquids discusses long-range entanglement and fractionalization, while Glasses and Spin Glasses explains disorder and slow collective freezing. Impurity screening, dense-lattice coherence and fluctuating moments require different comparisons of length, energy and time scales.

For magnetic control, a bulk spin-Hall coefficient is only one ingredient. Interfaces, relaxation, spin accumulation and torques determine the device response, and spin is generally not conserved in the presence of spin–orbit coupling. For textures, topology and energetic stability are separate: an integer winding number does not guarantee a long lifetime, and a Hall hump does not uniquely establish a skyrmion. The linked spintronics and texture articles develop these physical distinctions.

Exercises

Compare two spin moments

An isotropic $S=1$ ion has $g=2$. Find its Curie effective moment and saturation moment. Would an ordered moment of $1.6\mu_B$ contradict either value?

Solution

The effective moment is $2\sqrt2\mu_B\simeq2.83\mu_B$, whereas the saturation moment is $2\mu_B$. The ordered moment of $1.6\mu_B$ is smaller than the maximum projection and is not inconsistent with either quantity. Thermal and quantum fluctuations can reduce order; crystal-field and covalency effects may also require a revised microscopic moment model. The three measurements should be compared through their definitions rather than averaged into one number.

Test a claimed uniform instability

A paramagnetic band calculation gives a spin-summed density of states $N_{\rm tot}(E_F)=2.4$ states/(eV cell) and an interaction parameter $I=0.50\,\mathrm{eV}$ defined for the per-spin Stoner criterion. Is $IN_{\rm tot}>1$ a valid argument for ferromagnetism? Suppose a correctly normalized finite-wavevector response instead loses local stability first.

Solution

Spin degeneracy gives $N_\uparrow=N_{\rm tot}/2=1.2$ states/(eV cell), so the relevant product is $IN_\uparrow=0.60$. The uniform paramagnetic state is locally stable in this scalar mean-field test. A finite-wavevector instability indicates a different fluctuation channel within the chosen approximation; its eigenvector and magnetic basis are needed to identify the candidate order. Neither calculation alone gives a transition temperature or rules out an earlier first-order transition.

Separate acoustic and optical gaps

At zero in-plane momentum, the one-magnon matrix for two identical ferromagnetic layers coupled by isotropic ferromagnetic exchange is

$$K=J_\perp S \begin{pmatrix}1&-1\\-1&1\end{pmatrix}, \qquad J_\perp>0.$$

Find its eigenvectors and excitation energies. Can a measured peak at the upper energy establish that continuous spin symmetry is explicitly broken?

Solution

The normalized vectors $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$ have energies $0$ and $2J_\perp S$. The first tilts both layers together and remains a uniform rotation; the second tilts them oppositely. The upper peak is an optical mode and does not establish a gap in the acoustic mode. Scattering can weight the two eigenvectors differently, so observing only the upper branch is insufficient to infer broken spin-rotation symmetry.

References

• Blundell, Stephen. Magnetism in Condensed Matter. Oxford University Press, 2001. Publisher record. Magnetic moments, spin statistics, exchange and ordered phases.
• Coey, J. M. D. Magnetism and Magnetic Materials. 1st ed. Cambridge University Press, 2010. Publisher record. Material moments, magnetic order, domains and applications.
• Mermin, N. D., and H. Wagner. “Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models.” Physical Review Letters 17, 1133–1136 (1966). doi:10.1103/PhysRevLett.17.1133. Finite-temperature restrictions under the theorem’s interaction and symmetry assumptions.