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The two-state calculation is not a general formula for a ferromagnet. Exchange interactions, magnetic domains, magnetic domain-wall pinning, magnetic anisotropy, and hysteresis can determine which response is accessible. Both spin angular momentum and orbital angular momentum can contribute; a single constant gyromagnetic ratio may be only a restricted effective description. Rotation sets an internal bias, while the material and its preparation determine magnetization. The Barkhausen effect illustrates why a smooth bias can produce jumps instead of the toy smooth curve.

For a limited, reversible, isotropic linear model, suppose the same intrinsic volume magnetic susceptibility ${\chi }_v$ responds to the applied bias and to $B_{\Omega }/{\mu }_0$. For a uniformly magnetized ellipsoid along a principal axis, include its demagnetizing factor $N_d$:Solving this linear equation yieldsHere $H_{\mathrm{int}}$ and $H_{\mathrm{app}}$ are actual magnetic field intensities; the added rotational bias is an equivalent drive in the constitutive response. In SI, dimensionless ${\chi }_v$ multiplies an intensity in $A/m$, so $M_z={\chi }_v\Omega /\gamma$ would omit ${\mu }_0$ and have the wrong units. A susceptibility defined instead as $\partial M/\partial B$ has different units. State the convention before comparing formulas.

At zero applied magnetic field intensity, a stipulated ${\chi }_v=3$ gives prefactor $3/2$ in a sphere, with $N_d=1/3$, and a prefactor approaching $3$ in a long axial ellipsoid. This is a geometry check of the assumed linear model. A real cylinder does not have an exactly uniform demagnetizing field, and an irreversible ferromagnet cannot in general be represented by this single linear equation. Inferring an intrinsic gyromagnetic ratio requires an independently calibrated material response and a geometry-dependent conversion of the measured stray magnetic field.

The Einstein-de Haas effect is mechanical rotation or twisting caused by a change in a body's magnetization. The microscopic carriers of magnetic dipole moment also carry angular momentum: changing their net orientation can transfer angular momentum to the body. A freely suspended ferromagnet can therefore turn when its magnetization reverses. A suspension instead stores part of the response as an elastic twist. This makes a microscopic property accessible through a macroscopic angle.

The central measurement compares a change of magnetic dipole moment with transferred angular momentum, giving a gyromagnetic ratio. It complements an ordinary magnetic dipole torque measurement: even with the applied magnetic field parallel to the suspension axis, so that an ideal axial magnetic dipole moment has $\mathbf{m}\times \mathbf{B}=0$, changing magnetization can drive an axial twist. The external drive supplies energy; conservation of angular momentum still requires accounting for the suspension and any external axial torque.

A ferromagnet is a material that exhibits ferromagnetism. Its magnetic domains can cancel the net magnetization even while each domain remains ordered.

A physicist associated with the Barkhausen effect: the magnetization of a ferromagnet can change in discrete jumps even when the applied magnetic field varies smoothly. A pickup coil detects the accompanying changes in magnetic flux; amplification makes the irregular signal audible. The experiment connects a macroscopic noise measurement with sudden rearrangements of magnetic domains and the motion of domain walls.^[ref] His work at Dresden also developed the experimental teaching of electrical engineering.^[ref]

Consider an explicitly stipulated dilute independent spin-one-half two-level system with zero applied magnetic field. Each constituent has $J_z=\pm \hslash /2$ and $m_z=\gamma J_z$, with constant signed gyromagnetic ratio $\gamma$. Ignore interactions and the specimen's own magnetic field. At positive temperature $T$, thermal equilibrium with the rotating lattice gives energy levels $E_{\pm }=\mp \hslash \Omega /2$. Define the dimensionless variable $x=\hslash \Omega /(2k_{B}T)$, where $\hslash$ is the reduced Planck constant and $k_B$ is the Boltzmann constant. The partition function and Boltzmann distribution giveTaking the expected value giveswhere $n$ is number density. This is the rotational version of the independent-moment magnetic response. The angular momentum aligns with positive rotation even when the magnetic moment points oppositely. The probability ratio $p_{+}/p_{-}=e^{2x}$ provides a direct population check of the sign.

For $|x|\ll 1$, expand the hyperbolic tangent to obtainThe slope against $B_{\Omega }$ is positive for either sign of $\gamma$, while the slope against $\Omega$ has the sign of $\gamma$. At fixed $T>0$, reversing $\Omega$ reverses both means, and $\Omega =0$ gives zero. At fixed nonzero $\Omega$, the formal $T\to 0^{+}$ limit saturates $|⟨J_z⟩|$ at $\hslash /2$; the linear approximation must not be extrapolated there. At $T\to \infty$, the response vanishes. These limits describe the toy canonical ensemble, whose equilibration is assumed.

For a numerical example, stipulate $n=10^{28}{\mathrm{m}}^{-3}$, $T=300\mathrm{K}$, $\gamma =-1.76\times 10^{11}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, and rotation frequency $f=1000\mathrm{Hz}$. With $\Omega =2\pi f$, $\hslash =1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$, and $k_B=1.380649\times 10^{-23}J/K$, the model givesThese are teaching inputs and calculated outputs, not measured material values. The corresponding internal angular momentum density is $M_z/\gamma =+4.22\times 10^{-17}\mathrm{J}s/{\mathrm{m}}^3$. Keeping a rotor at prescribed $\Omega$ while its internal angular momentum changes requires the drive to supply the appropriate angular impulse. In an isolated body the mechanical part instead changes oppositely, as in magnetomechanical angular momentum balance. There is no creation of net angular momentum by choosing a rotating frame.

Heinrich Barkhausen's 1919 report used iron, a 300-turn pickup coil, moving magnet, amplified telephone readout and galvanometer. Removing iron corrected drive pickup. ^[ref]

A modern primary experiment by Gianfranco Durin and Stefano Zapperi used annealed iron–silicon strips and amorphous alloys. A 60 cm solenoid swept a triangular magnetic field intensity; a 50-turn pickup coil, wound within 1 mm, sensed the specimen. A preamplifier with a 3–20 kHz low-pass filter preceded sampling. Measurements selected the central hysteresis region dominated by magnetic domain wall motion. Above-background threshold crossings defined event durations; integrating between crossings defined event sizes. These are reported procedures, rather than data generated for the diagrams.^[ref]

For this stationary implementation, secure a thin ferromagnet and its pickup coil inside a longer drive solenoid, keeping their relative positions fixed. Drive the solenoid with a slow triangular electric current, monitor that electric current, and record the amplified, filtered induced electromotive force with a digitizer. A speaker may make suitable frequencies audible, but the voltage record is needed for quantitative analysis. Slow variation of the drive does not require smooth magnetization. A triangular drive supplies an approximately constant applied-field sweep rate along each ramp; it does not guarantee constant instantaneous $dM/dt$.

A proposed control sequence makes the inference testable. First record the electronics background with no drive. Repeat the same sweep with the specimen removed, retaining coil geometry: mutual induction from the drive can itself induce a voltage. A matching empty pickup coil wired in opposition can suppress common drive pickup, after checking its matching. Compensation of air flux by an opposed coil is an established method.^[ref] Then restore the specimen, repeat both sweep directions, reduce sweep rate, and compare the same hysteresis region with fixed amplifier gain, filter and threshold. Monitor drive irregularities and mechanical vibration; either can generate spurious pulses. These controls are recommendations for the schematic, rather than undocumented controls attributed to the modern paper.

A genuine material contribution should differ from the blank-coil record and change reproducibly with the magnetic preparation and sweep region. Quieting the drive tests drive-correlated pickup, but a held field need not eliminate all delayed material motion. Reversing the sweep is a consistency check on signed changes, not a demand that individual events retrace. A changing pulse count under a different filter or threshold cannot by itself establish changed microscopic dynamics. The useful inference is intermittent collective magnetization change within the instrument's signal bandwidth, with spatial detail left unresolved.

Authored lecture notes, dated 9 July 2015 on the title page. Section 4.2 covers the vector ferromagnet; §4.3, printed pp.10–13, develops the nematic tensor, rotational invariants and discontinuous transition. The tensor and coefficient normalization differs from $S=Q(3\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I)$ and must be translated explicitly.

Magnetostriction is a change of a material's shape or dimensions associated with its magnetization. A magnetic field can therefore produce strain without a contact force. The resulting magnetoelastic coupling lets a ferromagnet convert changes of magnetic state into mechanical motion; a constraint can instead turn the preferred deformation into stress. James Prescott Joule published his study of iron and steel bars in 1847.^[ref]

The surprising point is that reversing the magnetization need not reverse the deformation. A material can prefer a longer axis along either $+\mathbf{m}$ or $-\mathbf{m}$: an axis has no arrow. Isotropic linear magnetostriction vanishes excludes one particular linear constitutive term, while saturated isotropic magnetostrictive strain supplies a nonzero response with a quadratic dependence on the magnetic direction. The pulsed-field optical magnetostriction experiment shows how an actual length change can be resolved, and quadratic magnetostriction and frequency doubling separates an illustrative response law from a measured material curve.

An order parameter is a quantity chosen to distinguish phases and measure their degree of order. For a conventional spontaneous symmetry breaking transition, it vanishes in the symmetric phase and acquires a nonzero value in an ordered phase once a state or symmetry-breaking limit is selected.

The magnetization of a ferromagnet is a vector example. In a nematic liquid crystal, an ordinary average molecular axis can vanish because opposite directions are equivalent, so the nematic order parameter is a tensor instead. Choosing the correct object is therefore part of identifying the order. In a finite symmetric ensemble, averaging over differently oriented ordered states can also give zero; correlations or an infinitesimal conjugate field reveal the underlying order. Some phases require nonlocal diagnostics beyond this conventional local-order-parameter description.

A material exhibiting paramagnetism. Its induced magnetization points along the applied magnetic field in the linear-response regime, giving positive magnetic susceptibility. Unlike a ferromagnet, it has no spontaneous ordered moment in that phase. The inverse-temperature response of independent localized moments is one model, rather than a definition that includes all microscopic mechanisms of paramagnetism.

A bulk ferromagnet generally separates into magnetic domains. Within each domain many moments align and the local magnetization can be large, but different domain orientations partially or almost completely cancel the vector sum measured for the entire specimen. This reduces its external stray field and magnetostatic energy. Formation of magnetic domain walls costs exchange and anisotropy energy, so the actual pattern balances these costs with the reduction in stray-field energy.

A strong applied field can move domain walls and rotate moments toward a nearly saturation magnetization. When the field is removed, pinning and the sample's magnetic history determine the remanent moment. Thus a small net moment does not show that local ferromagnetic order has disappeared, or that moments within each domain are all randomly oriented. At finite temperature, thermal spin fluctuations also reduce the local ordered moment below its zero-temperature maximum even in a single-domain specimen. These two reductions—between-domain cancellation and within-domain thermal reduction—should be distinguished.

Within the stipulated localized, one-moment-per-atom spin model, the number density follows directly from mass density and molar mass:The face-centered cubic lattice is consistent with four atoms per conventional unit cell; no additional factor four is needed once $n$ has been obtained from the density. The Curie constant is $C=0.71549\mathrm{K}$. Identifying the model's $\theta =\lambda C$ with the specified Curie temperature givesAt very low temperature, a fully ordered single domain has $M_s=n{\mu }_B=8.4764\times 10^5\mathrm{A}{\mathrm{m}}^{-1}$. Its Weiss molecular field isThe cancellation of $n$ in the middle expression provides a useful check. The ordinary magnetostatic scale is only ${\mu }_0{M}_s\simeq 1.065\mathrm{T}$, smaller by the factor $\lambda$. Thus $B_{\mathrm{eff}}$ is not a plausible ordinary self-generated dipolar magnetic field of the nickel moments. It represents the exchange interaction energy in field units, not an additional Maxwell field of that magnitude. This does not assert that such large externally generated fields are forbidden. Nickel is an itinerant ferromagnet; the imposed one-${\mu }_B$-per-atom model is an estimate rather than its measured microscopic moment.

A material with paramagnetism develops a magnetization parallel to a weak applied magnetic field intensity, with positive magnetic susceptibility and no spontaneous magnetization in its paramagnetic phase. Molecular oxygen is an example; it has unpaired electron spins.

A material with diamagnetism develops an induced magnetization opposing the applied magnetic field, with negative magnetic susceptibility. A closed-shell atom such as helium is an example; its orbital response is diamagnetic.

A ferromagnet has a spontaneously ordered magnetization below its Curie temperature, even in the limit of vanishing applied magnetic field. Iron below its Curie temperature is an example. Different magnetic domains can cancel the net moment of an unmagnetized bulk sample, despite nonzero local spontaneous magnetization.

A material showing paramagnetism has an induced magnetization in the direction of a weak applied field, giving positive magnetic susceptibility, and no spontaneous magnetization once the field is removed in the paramagnetic phase. Alignment of permanent atomic or nuclear moments is one mechanism.

A material showing diamagnetism develops an induced moment opposing the applied field, giving negative magnetic susceptibility. Closed electronic shells supply a familiar orbital contribution; the response need not require permanent moments.

A ferromagnet has a phase with a spontaneous ordered magnetization below its Curie temperature, produced by cooperative alignment of moments. Its magnetic domains can cancel in a macroscopic zero-field sample, so a measured zero net moment alone does not show that the material lacks ferromagnetism.

Let $\ell =2\pi r-d$ be the mean path length in the core. Neglect leakage and fringing across the narrow air gap, and assume a cross-section small compared with $r$. Continuity of normal magnetic flux density then gives the same tangential magnetic field magnitude $B$ in the core and gap. Ampère's circuital law for the magnetic field strength $H$ isFor a linear core, $H_{\mathrm{core}}=B/({\mu }_0{\mu }_r)$ and $H_{\mathrm{gap}}=B/{\mu }_0$, givingHere $\hat{\mathbf{t}}$ follows the winding's right-hand rule. The gap has much larger $H$ than the core when ${\mu }_r\gg 1$, although $B$ is approximately continuous. Equivalently, the core and gap magnetic reluctances add in series.

On a magnetic hysteresis loop, remanence is the nonzero $B_r$ remaining at $H=0$ after magnetizing the sample. The normal magnetic coercivity $H_{cB}$ is the magnitude of reverse $H$ at which $B=0$ on that branch. The intrinsic magnetic coercivity instead refers to zero magnetization; these two conventions should not be confused. A comparison of coercive $H$ with remanent $B$ requires a unit conversion or a specified constitutive scale, because their SI units differ. A soft ferromagnet is understood here to have a narrow loop and small coercive field.

Use the stated single-valued constitutive equation for the calculation. Inverting it gives $H_{\mathrm{core}}=H_m\mathrm{artanh}(B/B_m)$, soSince $\mathrm{artanh}z=z+O(z^3)$, the small-signal response of a gapped magnetic core isThe small-field core magnetic permeability is $B_m/H_m$. Strictly, $B=B_m\tanh (H/H_m)$ is a reversible approximation: at $H=0$ it has $B=0$, so it cannot itself model nonzero remanence. The displayed loop and this constitutive approximation serve different purposes. Keeping a genuinely remanent branch would generally add a history-dependent offset; it would not imply $B=\kappa I$ through the origin.

A spin valve contains a magnetically soft free ferromagnet, a thin nonmagnetic metallic spacer and a reference ferromagnet pinned by exchange bias from an antiferromagnet. The recording track's stray field rotates or reverses the free-layer moment while the reference remains approximately fixed. Spin-dependent scattering makes the resistance lower for parallel moments and higher for antiparallel moments. A sense current converts this into a voltage signal, or a fixed voltage into a current signal.

A useful angular approximation is $R(\theta )=R_P+(R_{AP}-R_P)(1-\cos \theta )/2$. Biasing the equilibrium moments near $90^{\circ }$ gives an approximately linear response to small track fields. Pinning stability, magnetic shielding and the sensor's spatial extent matter for reading adjacent narrow bits.^[ref]

Heat the bilayer into a window $T_N<T<T_c$, apply an in-plane magnetic field sufficient to align the ferromagnet, and cool through the antiferromagnet's ordering/blocking range while retaining the field. The ferromagnet supplies a preferred interfacial spin direction as the antiferromagnet orders. Sufficient antiferromagnetic anisotropy then pins part of the interfacial spin configuration, producing unidirectional magnetic anisotropy rather than merely uniaxial magnetic anisotropy in the ferromagnet.

Measure a full major hysteresis loop after removing the cooling field, and compare with the isolated ferromagnet, a zero-field-cooled bilayer and the result of reversing the cooling-field direction. Those controls separate a true loop-centre shift from a changed coercive width, minor-loop artefacts or trapped applied field. The effect requires appropriate interfacial exchange and stable pinning; an ideally compensated or freely rotating interface need not produce a large bias. Exchange anisotropy in coupled ferromagnetic/antiferromagnetic material was demonstrated in the classic experiment below.^[ref]

For an isolated easy-axis ferromagnet undergoing coherent reversal, the ideal low-temperature loop is approximately rectangular, switching between $\pm M_s$ at $\pm H_c$. It has remanent Magnetization and a large response near switching. An ideal compensated antiferromagnet has essentially no remanence and a small, approximately reversible linear net Magnetization $M={\chi }_{\mathrm{AF}}H$ at fields far below its exchange/spin-flop scales. Strong antiferromagnetic exchange makes substantial canting costly, so its low-field magnetic susceptibility is small relative to the ferromagnet's unsaturated switching response.

This comparison does not say the ferromagnet has a larger differential slope at every field: on its saturated branches $dM/dH$ can also be small. Nor does it rule out spin-flop transitions at much larger fields, outside the sketched range. The original plot uses arbitrary normalized scales.

Exchange-split bands in a ferromagnet generally give a different spin-resolved density of states for each spin branch, $D_{\uparrow }(E_F)$ and $D_{\downarrow }(E_F)$. If spin is approximately conserved across a multilayer, its two spin channels conduct in parallel. A spin valve combines a comparatively fixed reference-layer magnetization with a soft free layer that rotates in a small applied field. Its readout depends on the relative angle, not simply the absolute spin population. Biasing the layers near a perpendicular alignment gives a useful linear response instead of operating at the flat parallel/antiparallel extrema.

In giant magnetoresistance, a metallic spacer separates ferromagnetic layers. Spin-dependent bulk and interface scattering make their channel resistances unequal. The density of available final $d$ states influences scattering of more mobile states; however, a large total density of states does not by itself guarantee a large conductivity. The scattering-weighted spin-channel asymmetry also involves velocity and scattering matrix elements. In the elementary identical-layer two-current resistor model, let each magnetic layer have resistances $a,b$ in the two spin channels and neglect spacer resistance. Parallel moments give channels $2a$ and $2b$, while antiparallel moments give $a+b$ in each channel. ThereforeOne favorable spin channel traverses both aligned layers efficiently; reversal removes that common easy path. A conducting spacer adds resistance, and spin mixing weakens the separate-channel approximation. The channel asymmetry is material/interface dependent, so positive magnetoresistance is not a theorem for every pair of different ferromagnets.^[ref]

In tunnelling magnetoresistance, the spacer is an ultrathin insulator and current flows across a magnetic tunnel junction. In the simple spin-conserving density-of-states tunnelling magnetoresistance model, assume spin-independent matrix elements and low voltage. Available initial and final states giveWriting $P_i=(D_{i\uparrow }-D_{i\downarrow })/(D_{i\uparrow }+D_{i\downarrow })$ and $G_0=C{D}_1{D}_2/2$, these become $G_P=G_0(1+P_1{P}_2)$ and $G_{AP}=G_0(1-P_1{P}_2)$. ThusThe effective tunnelling polarization can depend on the barrier and orbital symmetry, not just a bulk density-of-states sketch. Crystalline MgO junctions exploit symmetry-selective transmission, so the elementary product model is not a quantitative universal formula for them.^[ref]

Both devices convert a free-layer rotation into an electrical change. A useful approximate GMR interpolation is $R(\theta )=R_P+\Delta R(1-\cos \theta )/2$; the elementary tunnelling model instead gives $G(\theta )=G_0(1+P_1{P}_2\cos \theta )$. The spin-valve small-field sensitivity is a chain rule,magnetic anisotropy, biasing and pinning set the rotation susceptibility. Conducting-spacer GMR usually offers low resistance and supports current-in-plane or current-perpendicular geometries. TMR uses perpendicular tunnelling, often offers a larger fractional signal and a larger resistance-area product, but demands uniform, stable barriers and has bias dependence and breakdown limits. Neither is automatically the better small-field sensor: heating, Johnson and low-frequency noise, domain switching, hysteresis and stability set the detectable field. Reverse sweep direction, vary current and temperature, and independently test reference-layer pinning to distinguish a reproducible magnetic signal from drift or heating.

The Holstein–Primakoff transformation converts a spin algebra into bosonic operators while retaining the finite physical spin Hilbert space. With $[a_n,a_m^{†}]={\delta }_{nm}$, writeThe Holstein–Primakoff occupation constraint is $0\le n_n\le 2S$. Acting on $|n⟩$, the raising matrix element is $\sqrt{n(2S-n+1)}$ and the lowering element is $\sqrt{(n+1)(2S-n)}$. Their squared difference is $2(S-n)$, reproducing $[S^{+},S^{-}]=2S^z$. Operator ordering inside the square roots is essential. For low boson occupation relative to $S$, expand them to obtain $S^{+}\simeq \sqrt{2S}a$ and $S^{-}\simeq \sqrt{2S}{a}^{†}$. Keeping quadratic terms gives the linear spin-wave approximation; higher terms describe magnon interactions. Large $S$ helps control the expansion, but a low-dimensional infrared divergence can invalidate it even at large finite $S$.

For the ferromagnet choose $H_F=-J{\sum }_n{\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$, $J>0$, with unit lattice spacing and $\hslash =1$. Expand around the all-up state. A longitudinal bond contributes $-J{S}^2+JS(n_n+n_{n+1})$, and the transverse exchange contributes $-JS(a_n^{†}{a}_{n+1}+a_{n+1}^{†}{a}_n)$. After a Fourier transform,These are ferromagnetic magnons. The polarized vacuum is an exact ground state of the full ferromagnetic Hamiltonian: every bond is in its maximum total-spin sector, minimizing that bond energy. It therefore has no quantum zero-point reduction of its magnetization. The one-magnon dispersion is also exact, although interacting many-magnon states need not be free bosons. Rotating the polarization generates the maximum-total-spin ground-state multiplet. Selecting a definite direction in a thermodynamic symmetry-breaking description breaks $SU(2)$ to $U(1)$; the two transverse spin components form a canonical pair, producing a single quadratic type-B Goldstone boson. A small aligning field gaps it by the Zeeman energy.

At nonzero temperature the one-dimensional ferromagnet is different: its thermal magnon number contains $\int dk[e^{\beta JS{k}^2}-1{]}^{-1}$, with infrared behaviour $\int dkT/k^2$. The divergent magnetization reduction signals the absence of finite-temperature long-range order in an infinite short-range isotropic chain, consistent with the Mermin-Wagner theorem. It does not disallow the polarized zero-temperature ground states just constructed.^[ref]

For the antiferromagnet use $H_{AF}=+J{\sum }_n{\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$, still $J>0$, on an even bipartite chain. The classical reference state alternates up and down. Rotate one sublattice through $\pi$ so that both local quantization axes point along their reference spins. In unrotated physical components, to leading order $S_A^z=S-a^{†}a$, $S_B^z=-S+b^{†}b$, $S_A^{+}\simeq \sqrt{2S}a$ and $S_B^{+}\simeq \sqrt{2S}{b}^{†}$. Choosing Fourier phases using the actual site positions givesThe reduced Brillouin zone has $M=N/2$ momenta and length $\pi$. The pair creation and annihilation terms express the fact that an antiparallel reference product state is not an eigenstate. A bosonic Bogoliubov transformation with $u_k=\cosh {\theta }_k$, $v_k=\sinh {\theta }_k$, $\tanh 2{\theta }_k=\cos k$ diagonalizes each two-mode block. With the sign convention $a_k=u_k{\alpha }_k-v_k{\beta }_{-k}^{†}$ and $b_{-k}=u_k{\beta }_{-k}-v_k{\alpha }_k^{†}$, the result isThus the harmonic antiferromagnetic modes have linear, rather than quadratic, long-wave dispersion. In an unfolded Brillouin-zone description the same result is represented by zeros at both zero and the antiferromagnetic wavevector; these are not extra independent degrees of freedom beyond the two-sublattice accounting.

The Bogoliubov vacuum has no $\alpha$ or $\beta$ excitations but contains original spin bosons, with $v_k^2=\frac{1}{2}(|\sin k{|}^{-1}-1)$. Its staggered magnetization is $m_s=S-M^{-1}{\sum }_{k\in \mathrm{RBZ}}{v}_k^2$. This quantum depletion of Néel order diverges logarithmically at small $k$ in one dimension. Linear spin-wave theory therefore does not establish a stable ordered antiferromagnetic ground state for this chain. Its nominal gapless dispersion is a result of expanding around an invalid ordered reference, not a general proof of the exact excitation spectrum. For the nearest-neighbour isotropic chain, integer-spin systems have a Haldane gap, whereas half-integer-spin systems are gapless; the spin-one-half chain in particular has fractional spin excitations rather than a simple dilute gas of ordered-state magnons. This distinction requires physics beyond the harmonic expansion, including the topological term in the continuum theory.^[ref]