Goldstone boson

Each broken continuous internal-symmetry generator normally produces a massless Goldstone mode; for a symmetry breaking pattern $G\to H$, their number is $\dim G-\dim H$ in the standard relativistic setting.

A gapless mode associated with a canonically paired pair of broken symmetry generators. In an isotropic ferromagnet, the nonzero expectation of $[S^x,S^y]=i{S}^z$ makes two transverse coordinates one dynamical pair, giving one quadratic magnon branch. Broken-generator counting without this pairing would incorrectly predict two independent modes. This contrasts with the linear phase mode of a superfluid.

For a broken $O(2)$ symmetry with order-parameter magnitude $v$, the long-wavelength phase field has free energywhere ${\rho }_s$ is the stiffness, equal to $\gamma v^2$ in the simplest Landau-Ginzburg model.

The phase stiffness is the coefficient of $\frac{1}{2}\int d^{d}x(\nabla \theta {)}^2$ in the long-distance free energy. Its renormalized value determines the thermal phase fluctuations and algebraic correlation function of a two-dimensional O(N) model with $N=2$.

Operationally impose a small phase difference $\varphi$ between opposite sides of a square sample of side $L$. The smooth minimum has $\theta =\varphi x/L$, so the two-dimensional excess free energy isThe derivative concerns a prescribed twist at the stated observation scale; in $d$ dimensions its conventional size factor is $L^{d-2}$. This response definition includes thermal fluctuations and vortices after equilibration, whereas a microscopic gradient coefficient need not. A finite phase stiffness requires no nonzero spatially uniform order parameter; the Berezinskii–Kosterlitz–Thouless transition is the instructive example.

For a Gaussian phase field of stiffness ${\rho }_s$,Its infrared divergence rules out conventional long-range order in dimensions at or below two.

Fix the uniform order-parameter phase and retain finite system size $L$ before evaluating a massless Gaussian functional integral. In $d\le 2$, its coincident and separated order-parameter phase covariances each acquire a divergent additive constant as $L\to \infty$. Their difference $2[G_L(0)-G_L(r)]$ instead has a finite fixed-separation limit for $d>0$. This is why physical XY model correlations are obtained from Gaussian phase averaging of angle differences. An ultraviolet cutoff also remains necessary for coincident-point quantities; infrared cancellation does not remove it. The logarithmic two-dimensional Green function follows from the $d\to 2$ power-law expression only after subtracting its constant pole.

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 spin wave is a smooth long-wavelength fluctuation of the orientation of an ordered continuous-symmetry field. In an $O(2)$ phase it is a fluctuation of the angle $\theta$ governed at leading order by the Gaussian gradient energy.

A magnon is a quantum of a spin wave.

A mobile spin-triplet excitation above a singlet background in a quantum magnet. At low density, an appropriate triplet branch can be described as bosonic quasiparticles with an effective mass and repulsive interaction. Applied magnetic field can lower one branch toward condensation. The term does not imply that the total excitation number is conserved on thermodynamic timescales.

The linear spin-wave approximation expands a Holstein–Primakoff transformation about a classical ordered state and retains the quadratic oscillator Hamiltonian. It gives the leading order-$S$ excitation energies above the order-$S^2$ classical energy. In a Heisenberg antiferromagnet, first make a bipartite spin rotation. The approximation also predicts a zero-point reduction of the order parameter; infrared-divergent quantum depletion of Néel order signals that the assumed ordered state is not a valid thermodynamic starting point.

For the XY model with equal ferromagnetic exchange $J$ in its plane and field term $-hS{\sum }_j{S}_j^x$, rotate the local quantization axis along the Classical spin. The Holstein–Primakoff transformation gives the number coefficient ${\alpha }_k=S(|h|+2J-J\cos k)$ and anomalous coefficient ${\Delta }_k=JS\cos k$ with one transverse-axis convention. Bosonic Bogoliubov diagonalization gives ${ϵ}_k=\sqrt{{\alpha }_k^2-{\Delta }_k^2}$ and field gap $S\sqrt{|h|(|h|+2J)}$. A nonzero field stabilizes every oscillator mode.

The Bogoliubov vacuum has original-boson occupation $v_k^2=({\alpha }_k/{ϵ}_k-1)/2$, reducing the ordered spin by its average over momenta. For the zero-field planar ferromagnetic chain, $v_k^2\sim 1/(2\sqrt{2}|k|)$. Even after excluding the exact uniform zero mode, the sum grows as $(2\pi \sqrt{2}{)}^{-1}\log L$. It invalidates a thermodynamic fixed-orientation expansion with finite dilute depletion; it does not represent a negative physical magnetization.

For a polarized spin chain with dispersion relation $ϵ(k)=2S[J(1+u)-J\cos k-D\sin k]$, the first zero mode appears at the maximum over the allowed ring momenta. In the thermodynamic limit this threshold is $\sqrt{1+(D/J{)}^2}-1$, since the maximum of $J\cos k+D\sin k$ is $\sqrt{J^2+D^2}$. At finite size the minimizing wave vector need not be available, so a positive continuum threshold need not give a positive finite-ring threshold. For $L=4$ and $0<D<J$, the latter is zero.

The polarized ground state of an isotropic Heisenberg model with ferromagnetic exchange is an exact state. Its normal-ordered harmonic Hamiltonian has no pair-creation terms, no vacuum magnon occupation and no additional zero-point shift. An antiferromagnetic Néel state instead generates anomalous terms, a squeezed Bogoliubov transformation vacuum, a negative energy correction and reduced staggered magnetization. Neither statement licenses omitting normal-ordering constants from an arbitrary spin-wave model with anisotropy or a different reference state.

For a rotated nearest-neighbour chain, a full-zone sum of $JS(a_k^{†},a_{-k})M_k(a_k,a_{-k}^{†}{)}^T$ contains a normal-ordering constant $JSN$. Rewriting the classical constant as $-JNS(S+1)$ compensates it. One must not additionally halve the sum without changing mode counting.

With unit lattice spacing and $\hslash =1$, a nearest-neighbour Heisenberg model has quadratic low-momentum ferromagnetic magnons and linear antiferromagnetic modes in the linear spin-wave approximation. The ferromagnetic polarized vacuum is an exact ground state, whereas the assumed antiferromagnetic ordered vacuum has quantum depletion of Néel order. In one dimension its logarithmic infrared divergence invalidates finite ordered magnetization and prevents the linear approximation from deciding the exact gap.

For $0<\Delta <1$, choose a planar Néel reference and rotate local axes to align its longitudinal spins. One phase convention gives the displayed full-zone spectrum, linear near $k=0$ with slope $JS\sqrt{2(1+\Delta )}$, and gapped at $k=\pi$ by $2JS\sqrt{2(1-\Delta )}$. Shifting a sublattice phase can interchange the momentum locations. Folding the magnetic zone separates the gapless in-plane mode from the gapped out-of-plane polarization. An easy-axis reference in this regime is unstable and does not supply this physical expansion.

After a bipartite spin rotation, a leading Holstein–Primakoff transformation expansion gives $H_2=-J\Delta N{S}^2+JS{\sum }_k[2\Delta n_k+\cos k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})]$. The bosonic Bogoliubov diagonalization gives the displayed spectrum and vacuum energy $-J\Delta N{S}^2+JS{\sum }_k[\sqrt{{\Delta }^2-{\cos }^{2}k}-\Delta ]$. The anisotropy gap is $2JS\sqrt{{\Delta }^2-1}$. At $\Delta =1$ the finite-angle transformation is singular at its zero modes, and quantum depletion of Néel order is infrared divergent in one dimension.

The Mermin-Wagner theorem forbids spontaneous breaking of a continuous internal symmetry at positive temperature in one- and two-dimensional systems with sufficiently short-range interactions.