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The Berezinskii–Kosterlitz–Thouless transition in the two-dimensional XY model separates a low-temperature phase with algebraically decaying correlations from a high-temperature phase with unbound vortices and exponentially decaying correlations. It does not produce conventional long-range order.

How can something resist a twist when its orientation never settles down? The distinction is between long-range order, a nonzero limiting correlation function, and phase stiffness, a free energy cost for a slow spatial twist. In a uniform two-dimensional XY model at positive temperature, smooth spin waves can erase the first while leaving the second finite. Vortex-antivortex binding then decides whether that resistance survives on arbitrarily large scales.

Vadim Lvovich Berezinskii developed an earlier description of two-dimensional order. John Michael Kosterlitz and D. J. Thouless independently developed the defect mechanism; Kosterlitz's 2016 lecture explicitly acknowledges their initial ignorance of Berezinskii's work. The central step was to include rare vortices alongside smooth spin waves.^[ref]

The Mermin-Wagner theorem concerns continuous symmetry and sufficiently short-range interactions in the thermodynamic limit. It does not say that every two-dimensional system is disordered in every sense, nor that every continuous symmetry gives a Berezinskii–Kosterlitz–Thouless transition. Here the order parameter has a circular phase, permitting integer winding numbers. Finite-scale vortex renormalization explains why a finite sample can look coherent even when its infinite counterpart lacks long-range order.

An XY model orientation takes values on a circle, so a circuit around an excluded defect core can carry integer winding number. Smooth single-valued spin waves cannot change that integer without a core event or boundary crossing. The locally quadratic order-parameter phase action therefore describes a sector, rather than automatically summing every compact-field configuration. Vortex-antivortex unbinding changes the large-distance correlation function and renormalized phase stiffness, even though ordinary long-range order is already absent in two dimensions at positive temperature.

On a finite simply connected planar graph, fix the height of the exterior face to zero. Every divergence-free integer bond current is uniquely the oriented difference of the two neighboring face heights. The positive graph Laplacian $L$ on the remaining faces gives its energy. Applying the Poisson summation formula to these heights and completing the square givesThe Gaussian prefactor $Z_{\mathrm{sw}}$ equals the spin wave determinant of the original graph, up to angle zero-mode normalization. One way to verify this is that the matrix-tree theorem equates the spanning-tree counts of a planar graph and its dual. Boundaries can admit nonzero total vortex charge; neutrality is an additional periodic or finite-energy condition.

The 1988 study grew alternating iron/chromium layers by molecular beam epitaxy on gallium arsenide. Magnetization and torque measurements tested the magnetic configuration. An AC electrical resistance measurement passed electric current along the layers while the applied magnetic field changed. Comparing field directions and spacer thicknesses linked the resistance decrease to magnetic saturation. Electron microscopy and surface analysis checked layer formation, while leaving atomic-scale interface roughness unresolved.^[ref]

The independent Jülich study used four-terminal sensing on epitaxial Fe/Cr/Fe strips. A single iron film supplied an anisotropic magnetoresistance comparison. Magneto-optic Kerr effect signals and Brillouin light scattering from spin waves identified the layer alignments, rather than inferring them from resistance alone. Field scans along different magnetic axes connected the optical signatures to the resistance changes.^[ref]

Peter Andreas Grünberg's retrospective explains why the preceding interlayer exchange coupling measurements mattered: coupled spin wave modes supplied signatures of magnetic alignment before that alignment became an electrical signal.^[ref]

A resistance curve alone does not reveal the microscopic direction of every magnetic layer. The optical measurements add a different observable, while the reference film tests an orientation-dependent response that can occur without a multilayer. Sample preparation and cross-checks therefore belong to the physical explanation.

On a rectangular torus, an integer height for the integer-current representation of the Villain model may have twists $w_x,w_y\in \mathbb{Z}$. Subtract $w_{x}x/L_x+w_{y}y/L_y$ to get a periodic real height. Its Gaussian energy splits into the ordinary gradient energy and $(L_y{w}_x^2/L_x+L_x{w}_y^2/L_y)/(2\beta )$. Height Poisson summation supplies a phase with dipole moments $P_x=\sum q_{x}x$ and $P_y=\sum q_{x}y$. A further Poisson summation of the twists gives, up to a charge-independent factor,Here $G$ is the inverse lattice Laplacian on the mean-zero subspace. This keeps the smooth spin wave determinant separate, while correctly retaining the coupling between vortex dipole moments and global winding sectors at finite volume. Dropping those sectors without stating boundary conditions is not an exact torus duality.

Inelastic neutron scattering resolves $E_i-E_f\ne 0$. Through conservation of energy, a peak at positive $\hslash \omega$ identifies an allowed excitation energy; through momentum conservation, its location in $\mathbf{Q}$ constrains the spatial variation of that excitation. A phonon is a quantum of lattice vibration; a magnon is a quantum of a spin wave. Their different couplings make energy transfer alone insufficient to identify which was created.

Bertram Neville Brockhouse's aluminium experiment rotated a large single crystal, scanning analyzer angle for neutron groups. His vanadium comparison exposed the elastic resolution. Later magnetite measurements used a magnetic field intensity test to identify magnons.^[ref]

Successive orientations sample different wave vectors; analyzer settings distinguish outgoing energies. This supplies points on a dispersion relation, rather than only an angular diffraction pattern. Finite resolution can merge neighboring excitations; background, multiple scattering and processes involving several phonons can complicate interpretation. A lattice model is a testable explanation of selected features, not a guarantee that every spectral peak has the same origin.

For the Gaussian functional integral with dimensionless phase stiffness $K$ and a microscopic cutoff $a$, the Gaussian phase averaging identity gives the displayed correlation function. In two dimensions the phase-difference variance is $(\pi K{)}^{-1}\log (r/a)+O(1)$, so the XY model has algebraic spin wave correlations with exponent $1/(2\pi K)$. In one dimension the variance is $r/K$, giving exponential decay. Above two dimensions the variance has a finite large-distance limit within the low-temperature spin wave approximation. The Laplacian Green function solves $-K{\nabla }^{2}G=\delta$; integration over a sphere fixes $G(r)=r^{2-d}/[(d-2)S_{d}K]+C$ for $d\ne 2$ and $G(r)=-(2\pi K{)}^{-1}\log (r/a)+C$ for $d=2$. A fixed uniform phase and ultraviolet regularization are required to define $G$ itself; phase differences eliminate the constant.

A magnon is a quantum of a spin wave.

For a two-component order parameter of nonzero magnitude, write $m_x+i{m}_y=\bar{m}{e}^{i\theta }$. Its phase is the orientation angle modulo $2\pi$, not a thermodynamic phase such as an ordered or disordered state. A global continuous symmetry rotates this angle uniformly; spin waves are smooth spatial changes, while a phase vortex has nonzero winding around an excluded core. At a zero of the magnitude the angle is undefined, allowing the topology of an orientation field to change.

For a finite-range isotropic Heisenberg model, the Mermin-Wagner theorem excludes spontaneous magnetic order in one dimension at every nonzero temperature. Long-wavelength rotations cost very little energy: a smooth orientational fluctuation has $\delta F=({\rho }_s/2)\int dx({\partial }_x\theta {)}^2$, so its thermal variance contains $(k_{\mathrm{B}}T/{\rho }_s)\int dq/q^2$, which diverges in the infrared. In a Heisenberg ferromagnet, the same obstruction appears as a divergent thermal occupation of quadratic long-wave spin waves. This explains why a finite magnetization cannot be supported, although correlations can be strong over a long finite distance. The rigorous statement requires short-range interactions and the stated continuous symmetry; interchain coupling and anisotropy change the problem. Symmetry forcing the mean to zero in a finite chain alone would not prove absence of thermodynamic order. ^[ref]

For the Ising model, use dimensionless $s_i=2S_i^z=\pm 1$, $h=g{\mu }_{\mathrm{B}}B$, $K=\beta J/2$ and $b=\beta h/2$. Its pair Boltzmann factor isMultiplying N such factors around the periodic ring counts each bond once and each field term twice with weight one half. Summing the closed spin-index sequence therefore gives the transfer matrix for the one-dimensional Ising modelThe two eigenvalues areAt finite temperature every entry of P is positive, so the Perron–Frobenius theorem gives ${\lambda }_{+}>|{\lambda }_{-}|$. ConsequentlyThe exact finite-N correction to the extensive expression is $-k_{\mathrm{B}}T\ln [1+({\lambda }_{-}/{\lambda }_{+}{)}^N]$. No assumption that ${\lambda }_{-}$ is positive is needed.

Differentiate at fixed temperature. For $D=\sqrt{{\sinh }^{2}b+e^{-2\beta J}}$, ${\partial }_b\ln (\cosh b+D)=\sinh b/D$. The Ising-chain magnetization with spin-half normalization gives total magnetic momentThis is the thermodynamic-limit result; the moment per site is M/N. For every finite $T>0$, ${\lim }_{B\to 0}M/N=0$, even with the thermodynamic limit taken first. The zero-field magnetic susceptibility per site is $(g{\mu }_{\mathrm{B}}{)}^2{e}^{\beta J}/(4k_{\mathrm{B}}T)$, large but finite.

The Ising-chain domain-wall energy convention explains the result for $J>0$: changing a parallel bond into an antiparallel bond costs J in this spin-half normalization. Domain walls, occurring in pairs on a ring, acquire a nonzero density at any nonzero temperature. Indeed the zero-field correlation length is $\xi =-a/\ln [\tanh (\beta J/2)]\sim (a/2)e^{\beta J}$ at low temperature. The Ising symmetry is discrete, so this is a separate one-dimensional domain-wall mechanism rather than an application of the continuous-symmetry theorem. At zero temperature a ferromagnetic chain can select a fully polarized state; the $B\to 0$ and $T\to 0$ limits do not commute.

The dot product of the two magnetizations is ${\bar{m}}^2\cos [\theta (x)-\theta (0)]$. For the Gaussian spin wave field, the Gaussian phase averaging identity $⟨e^{iX}⟩=e^{-⟨X^2⟩/2}$ for a centered Gaussian variable givesKeeping this exponential is essential: expanding the cosine to quadratic order would fail when phase differences grow with separation. Equivalently, the phase-difference variance isFor $d>2$, the integral defining the absolute phase variance is finite at small momentum after fixing the uniform mode; its microscopic ultraviolet cutoff still matters. Its nonconstant Green function decays as $r^{2-d}$, so $D(r)$ tends to a finite constant $D_{\infty }$ andIn two dimensions, the logarithmic Green function gives $D(r)=(\pi \bar{K}{)}^{-1}\log (r/a)+O(1)$. ConsequentlyFor $1\le d<2$, $D(r)$ grows as $2r^{2-d}/[(2-d)S_d\bar{K}]$, up to bounded microscopic terms, and the correlation function again vanishes. In particular, in one dimension $S_1=2$ gives $D(r)=r/\bar{K}$ and exponential decay ${\bar{m}}^2{e}^{-r/(2\bar{K})}$. Therefore there is no conventional long-range order for $d\le 2$ at positive temperature in this short-range continuous-symmetry model. In two dimensions, algebraic decay can still support quasi-long-range order.

These infrared spin-wave correlations of the XY model explain the mechanism behind the Mermin-Wagner theorem. They are a test of the low-temperature spin wave approximation, not a claim that every $d>2$ XY model is ordered at every temperature. Large local fluctuations and defects destroy the ordered phase at sufficiently high temperature. Also, the assumed small angle is a local smooth-field approximation; phase differences at arbitrarily large separation need not remain small.

The physical angle takes values on a circle, so it is defined modulo $2\pi$. Around a closed curve enclosing a phase vortex,The integer winding number cannot change through a smooth single-valued deformation unless a defect crosses the curve. Thus a globally smooth Gaussian spin wave field omits whole topological sectors of the XY model.

Outside a vortex core of radius $a$, an isolated charge-$n$ phase vortex has $\theta =n\phi$ and $|\nabla \theta |=|n|/r$. Its dimensionless energy in a sample of size $L$ isThere are of order $(L/a{)}^2$ positions for its core, so the positional entropy contributes $2\log (L/a)$. The competition for unit charge is therefore $(\pi \bar{K}-2)\log (L/a)$, apart from core terms. Opposite charges have a logarithmic attraction and form a vortex Coulomb gas.

At low temperature, vortex-antivortex binding leaves no free charges on large scales. Spin waves still destroy true long-range order, but the correlation function decays algebraically with the renormalized phase stiffness. At the Berezinskii–Kosterlitz–Thouless transition, pairs unbind; free vortices screen the interaction and produce a finite correlation length and exponential decay above the transition. The energy-entropy estimate identifies the threshold stiffness $K_R=2/\pi$, where $K_R$ is the long-distance dimensionless stiffness, not necessarily the bare $\bar{K}$. It agrees with the universal stiffness jump and the limiting algebraic exponent $1/(2\pi K_R)=1/4$. The two-dimensional transition is vortex unbinding between quasi-long-range order and disorder, rather than the onset of nonzero spontaneous magnetization.

Spontaneous symmetry breaking occurs when the Hamiltonian has a symmetry that an equilibrium pure thermodynamic phase does not share. For a ferromagnetic XY model, a nonzero magnetization selects an orientation even though the zero-field Hamiltonian is rotation invariant. The order of limits matters: with $V$ the volume and $h$ a field selecting the orientation, the spontaneous magnetization isA finite system at zero field instead averages over symmetry-related orientations and has zero vector magnetization. Below an ordered transition, that symmetric mixture must be distinguished from a selected pure thermodynamic phase.

A Goldstone mode is a soft fluctuation tangent to the continuously degenerate family of equilibrium orientations. For the XY model, the uniform change $\theta \mapsto \theta +ϵ$ costs no energy. Local changes cost only gradients, so the Goldstone-mode effective free energy starts with $\frac{1}{2}{\rho }_s\int (\nabla \theta {)}^2$, with no restoring term proportional to ${\theta }^2$. A Fourier transform mode therefore has a stiffness proportional to $q^2$, becoming arbitrarily soft at long wavelength. The dimensionless phase stiffness here is $\bar{K}=\beta {\rho }_s$. These are the spin waves of the angle field. Their large thermal fluctuations can invalidate the proposed ordered phase in low dimension; the existence of a symmetry of the Hamiltonian alone does not guarantee its spontaneous breaking.

The Berezinskii–Kosterlitz–Thouless transition is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range XY model. At long wavelengths, smooth spin waves have energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the Mermin-Wagner theorem. The low-temperature phase instead has quasi-long-range order, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.

The phase is periodic modulo $2\pi$, so it also permits topological defects with integer winding $n$. A phase vortex has $|\nabla \theta |=|n|/r$, and integration givesA singly charged phase vortex can be placed in roughly $(L/a{)}^2$ locations, giving entropy $2k_B\log (L/a)$. This energy-entropy balance suggests that isolated vortices become favorable when the effective stiffness falls to $2k_{B}T/\pi$. Below the transition, vortex-antivortex binding predominates; above it their unbinding screens the logarithmic interaction and produces a finite correlation length. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.

The dilute-vortex vortex-pair renormalization flow incorporates this screening. With $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe first equation says polarizable phase vortex pairs reduce stiffness. The second compares the phase vortex's position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave renormalization-group fixed points with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.

Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.

At the transition the limiting stiffness from below has the universal stiffness jumpAbove $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order ${\xi }^{-2}$ and is exponentially small in $1/\sqrt{\tau }$, so this infinite-order transition is unlike a Landau transition with a discontinuous order parameter or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in Kosterlitz's original vortex analysis. The transition separates bound vortices and algebraic order from unbound vortices and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.

Angle integration gives a dual representation in any lattice dimension. Discrete summation by parts rewrites the phase asEach normalized angle integral is a Kronecker delta. This proves the integer-current representation of the Villain modelThe PDF does not specify dimension or boundaries. The point-vortex vortex Coulomb gas interpretation below is two-dimensional. In three dimensions the defects are vortex lines, and the dual constrained-current representation still holds but is not a gas of point charges with logarithmic interactions.

First take a connected simply connected planar lattice, with free outer angles, so no torus winding sectors are silently discarded. Assign an integer height $\ell$ to each dual face, fixing the exterior face height to zero. A divergence-free integer current is exactly the oriented height difference on the two faces adjoining its bond. To see existence, integrate the rotated current along a dual path: the change around a closed dual path is the total divergence inside, hence zero. Integer currents give integer heights, and fixing the exterior removes their constant-shift ambiguity. Thereforewhere $L$ is the positive dual graph Laplacian with the exterior height fixed. This gives the dual integer-height representation of the planar Villain model.

Apply the supplied Poisson summation formula separately to every interior height. For $N_f$ interior faces,Complete the Gaussian square, or use its characteristic function, to evaluate the integral asThus the smooth fluctuations and integer defects separate:The Gaussian factor is $Z_{\mathrm{sw}}=(2\pi \beta {)}^{-(N_b-N_f)/2}/\sqrt{\det L}$. It really is the spin wave factor of the original lattice: the planar Euler identity gives $N_b-N_f=N_x-1$, and the matrix-tree theorem identifies $\det L$ with the spanning-tree count of the original graph. Its reduced primal Laplacian has the same determinant, so the noncompact Gaussian angle integral with one angle fixed has exactly this prefactor. The compact common-angle mode has been normalized to one. The remaining $q$ variables are integer vortex windings on the faces, giving the vortex Coulomb gas.

For periodic boundaries, $L$ has a zero mode, and there are also two noncontractible current sectors. Neutral charges use the inverse $G$ on the mean-zero subspace. An integer dual height may have twists $w_x,w_y\in \mathbb{Z}$ across periods $L_x,L_y$. Subtracting $w_{x}x/L_x+w_{y}y/L_y$ makes the remaining height periodic; the gradient energy gains $(L_y{w}_x^2/L_x+L_x{w}_y^2/L_y)/(2\beta )$. Its height Poisson summation supplies the phase $\exp [2\pi i(w_x{P}_x/L_x+w_y{P}_y/L_y)]$, where $P_x=\sum q_{x}x$ and $P_y=\sum q_{x}y$. A second Poisson summation in $w_x,w_y$ yields the positive harmonic-sector factor in periodic Villain dualityUp to charge-independent normalization, the exact periodic result is $Z_V=Z_{\mathrm{sw}}{\sum }_{\sum q=0}{e}^{-2{\pi }^2\beta q^{T}Gq}\mathcal{W}(q)$. Smooth spin waves still factor out, but the finite-volume global winding sectors couple to vortex dipole moments. The familiar neutral bulk formula suppresses this boundary information; it must not be asserted as an exact finite-torus formula without specifying those sectors.

When $m_1=m_2={\mu }^2$, the free energy has continuous symmetry $O(2)$. Its ordered minima satisfyContinuous phase fluctuations make the lower critical dimension $d_{\mathrm{l}}=2$, in agreement with the Mermin-Wagner theorem. In two dimensions write the complex order parameter as $\psi ={\rho }_0{e}^{i\theta }$. Neglecting the massive amplitude mode gives the Goldstone-mode effective free energyThe phase-difference variance isand thereforewithThus spin waves replace true long-range order by quasi-long-range order.

A vortex-antivortex pair of separation $R$ has the logarithmic energy $E_{\mathrm{pair}}\simeq 2\pi {\rho }_0^2\log (R/a)$. The number of pair separations below $R$ grows as $R^2$, giving the coarse entropy $S_{\mathrm{pair}}\simeq 2\log (R/a)$. Thusand widely separated pairs become favorable at ${\rho }_0^2\simeq 1/\pi$. Substituting the mean-field stiffness givessuggesting a Berezinskii–Kosterlitz–Thouless transition. The numerical location is only a bare-stiffness estimate: vortex-core fluctuations renormalize the stiffness near the transition.

For an infrared-regulated phase covariance, fix the uniform order-parameter phase zero mode and introduce ultraviolet and infrared cutoffs before taking limits. With the Fourier conventionthe dimensionless quadratic Hamiltonian diagonalizes asCompleting the square with a source gives $Z[J]/Z[0]=\exp [\frac{1}{2}\int JGJ]$, with inverse kernelThere is no additional $k_{B}T$ factor: it is already included in $\bar{K}$ because the supplied Hamiltonian is $\beta H$.

It is simplest to evaluate the real-space dependence by its defining Green equation, $-\bar{K}{\nabla }^{2}G={\delta }^{(d)}$. Rotational invariance gives $r^{d-1}{G}^{\prime }(r)=$ constant for $r>0$. Integrating over a small sphere fixes $-\bar{K}{S}_d{r}^{d-1}{G}^{\prime }(r)=1$, where $S_d=2{\pi }^{d/2}/\Gamma (d/2)$. Integrating once more,In two dimensions, take the limit after absorbing the divergent $r$-independent piece into $C$:For $d\le 2$, $G$ itself has an infrared-divergent constant in the infinite system; it is not a finite thermodynamic covariance without a regulator. The physical phase-difference variance is finite at fixed separation and cancels that constant. These are smooth spin wave correlations; compact-field vortex sectors are considered separately below.

The magnetization correlator is an average of a phase difference, not merely its lowest Taylor expansion. Gaussian phase averaging giveswhereThe coincident value is ultraviolet-regularized at a microscopic scale $a$. At large $r$:

Thus two is the lower critical dimension for conventional positive-temperature continuous-symmetry order under the short-range assumptions. Two dimensions can still have quasi-long-range order and a thermodynamic transition; zero limiting magnetization does not forbid them. The $d>2$ spin wave result describes an admissible low-temperature ordered thermodynamic phase, not order at arbitrarily high temperature. The Mermin-Wagner theorem supplies the more general no-order statement below or at two dimensions.^[ref]

For the even periodic chain, every classical Néel state ${\mathbf{S}}_m=(-1{)}^{m}S\mathbf{n}$ has energy $-JN{S}^2$, with arbitrary unit vector $\mathbf{n}$. Thus there is a continuous sphere of classical orientations, not merely two Ising-like states. Opposite staggered directions are connected by a global spin rotation.

A quantum Néel product is not an eigenstate: transverse exchange flips opposite neighbouring spins. The ground state therefore contains quantum superpositions and zero-point spin waves, reducing the classical staggered moment. A finite isotropic balanced chain has a symmetry-invariant ground state; a symmetry-broken reference is an approximation for analysing fluctuations, not its exact finite-system eigenstate.

Quanta of collective motion. An elementary excitation is a low-energy departure from the many-body ground state that can often be treated as a quasiparticle. Near an ordered equilibrium, small collective motions form normal modes. Quantization turns a harmonic mode of frequency ${\omega }_{\mathbf{k}}$ into bosonic excitations with energy $\hslash {\omega }_{\mathbf{k}}$. Their number is not an independently conserved thermodynamic variable in equilibrium, so their chemical potential is zero andHere $\nu$ labels branches, and $U_0$ contains the ground-state contribution, including phonon zero-point energy where appropriate. The approximation is useful while interactions and damping of the excitations are weak enough that the modes remain identifiable.

Phonons. A phonon is a quantum of an elastic lattice-vibration mode. Long-wavelength acoustic phonon branches have $\omega \simeq c_{\nu }k$, with one longitudinal and two transverse polarizations in a three-dimensional monatomic solid. A multi-atom primitive cell can also support optical phonon branches, generally with nonzero frequency at $k=0$. Linear acoustic dispersion gives $D(ϵ)\propto {ϵ}^2$, so the energy integral has $U-U_0\propto T^4$ and the heat capacity obeys the Debye $T^3$ law. In the Debye model,At temperatures high compared with all phonon energies but below structural changes, the classical harmonic limit gives approximately $C_V=3N{k}_B$. Optical modes are frozen out at temperatures well below their gaps.

Spin waves. A spin wave is a spatially varying collective precession of ordered spins; its quantum is a magnon. In a simple isotropic ferromagnet, a long-wavelength gapless branch has energy ${ϵ}_{\mathbf{k}}\simeq D_s{k}^2$, where the spin stiffness $D_s$ has units $\mathrm{J}{\mathrm{m}}^2$. State counting gives $D(ϵ)\propto {ϵ}^{1/2}$. For one isotropic branch in three dimensions,This exponent is specific to a gapless quadratic ferromagnetic branch. Long-wavelength antiferromagnetic spin waves can be linear and instead give a $T^3$ heat-capacity law in three dimensions. Magnetic anisotropy or an applied field can open a gap, causing activated behaviour at still lower temperatures. Near a magnetic ordering transition, a dilute noninteracting spin-wave description is no longer adequate.

These examples illustrate the low-temperature energy of gapless bosonic excitations: for $ϵ\propto k^s$ in $d$ dimensions, $D(ϵ)\propto {ϵ}^{d/s-1}$, hence $U-U_0\propto T^{1+d/s}$ and $C_V\propto T^{d/s}$. The exponent records both the spatial dimension and the dispersion, making calorimetry a probe of elementary excitations.