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 $N=2$.

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

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.

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.

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.