Goldstone Modes Preview

A Goldstone mode is a collective excitation whose energy tends to zero at long wavelength because it moves along a continuously degenerate family of ordered states. The relevant symmetry is a physical continuous global symmetry, spontaneously broken in the appropriate infinite-system limit. A low energy alone is insufficient: gapless fermions or a critical excitation need not be Goldstone modes. Here the emphasis is the physical mechanism; Goldstone modes in many-body systems develops the counting rules and effective dynamics.

Helpful background. Spontaneous symmetry breaking explains the limiting ordered state, generators describes infinitesimal transformations, and approximate symmetry explains how weak explicit breaking changes the conclusions.

Order-parameter rotations and stiffness

Suppose an exact global group $G$ leaves the Hamiltonian invariant, while an ordered state preserves only a subgroup $H$. Symmetry-related orientations are locally described by the coset $G/H$. A Goldstone field describes a slowly varying orientation, rather than a new microscopic particle label.

For a phase angle $\theta$, the symmetry acts as $\theta\mapsto\theta+\alpha$ with constant $\alpha$. A uniform rotation costs no energy. In a local, stable medium, a leading long-wavelength energy can be

$$E[\theta]-E_0=\frac{\rho_s}{2}\int d^d x\,(\nabla\theta)^2, \qquad \rho_s>0.$$

The coefficient $\rho_s$ is the stiffness: it measures resistance to neighboring regions choosing different orientations. For a sinusoidal distortion $\theta=A\cos(\mathbf k\cdot\mathbf x)$ in a periodic volume $V$, the spatially averaged energy is $\rho_sVA^2k^2/4$. Its cost at fixed amplitude vanishes as $k\to0$.

This is a static statement. To find an excitation frequency one must also specify the dynamics. With a positive temporal coefficient $\chi$,

$$\mathcal L=\frac\chi2(\partial_t\theta)^2 -\frac{\rho_s}{2}(\nabla\theta)^2,$$

the equation $\chi\partial_t^2\theta-\rho_s\nabla^2\theta=0$ gives $\omega=c|\mathbf k|$, where $c=\sqrt{\rho_s/\chi}$. A quantized excitation has energy $\hbar\omega$. The $k^2$ in the static energy therefore does not imply quadratic dispersion.

Broken directions need not count independent modes

For broken internal symmetries in the usual local nonrelativistic setting, two broken directions can form a conjugate pair. The charge-commutator density detects this possibility. Its precise counting formula and assumptions are given in the many-body treatment and Watanabe–Murayama’s work.

An isotropic ferromagnet ordered along $z$ illustrates the mechanism. Rotations generated by $S_x$ and $S_y$ are broken, while $[S_x,S_y]=i\hbar S_z$ has nonzero expectation per volume. The two transverse fluctuations act as a canonical pair and give one gapless magnon branch, usually quadratic at small momentum. In a simple collinear antiferromagnet with zero uniform spin density, the corresponding commutator density vanishes and the usual ordered-phase description has two linearly dispersing transverse modes. These examples assume dimensions and temperatures where the stated long-range order exists.

The familiar relativistic theorem gives one mode per broken internal generator under its assumptions. Applying that count to every magnet, or to broken spacetime symmetries, would be incorrect. The field-theory bridge explains the relativistic current argument.

Physical examples and their limits

In an interacting neutral superfluid, slow phase fluctuations yield sound. A phase order parameter such as $\langle\Psi\rangle=|\Psi_0|e^{i\theta}$ describes a selected broken-symmetry state; in a finite fixed-particle-number state this expectation is zero, although correlations can reveal the ordering tendency. Bose condensation and superfluidity should not be identified automatically. In particular, the uniform ideal Bose gas does not supply the usual interaction-generated linear sound mode.

An unpinned crystal formed from a continuum system breaks continuous translations to a lattice. Its long-wavelength acoustic phonons describe displacement fluctuations. An imposed substrate can explicitly break translations and alter or pin those modes. The crystal also involves rotations, but broken rotation generators do not each require extra independent phonon branches: spacetime symmetry counting differs from the internal-symmetry rule.

The existence of the ordered phase must be checked. For example, short-range systems with continuous internal symmetry cannot have the corresponding conventional long-range order at nonzero temperature in one or two spatial dimensions under the Mermin–Wagner assumptions. A two-dimensional superfluid can nevertheless have quasi-long-range order and a Berezinskii–Kosterlitz–Thouless regime. Long-range interactions, finite size, and zero temperature require their own analysis.

Finite size, pinning, and pseudo-Goldstone gaps

A unique finite-system ground state of a symmetric Hamiltonian is invariant as a ray; its symmetry-odd order parameter vanishes. Ordered states become sharply distinct in the thermodynamic limit, typically after a weak selecting field is removed in the proper order. Finite-size low-lying levels and long-distance correlations can anticipate this behavior, but a small two-level splitting alone does not establish a Goldstone branch.

Weak explicit breaking can select a preferred orientation. Near a stable minimum, write

$$\mathcal L=\frac\chi2(\partial_t\theta)^2 -\frac{\rho_s}{2}(\nabla\theta)^2-\frac\kappa2\theta^2, \qquad \kappa>0.$$

Then $\omega^2=(\rho_s k^2+\kappa)/\chi$, and the energy gap is $\hbar\sqrt{\kappa/\chi}$. The quadratic potential is a local expansion of a periodic pinning potential, not a globally periodic description of the angle. This is a pseudo-Goldstone mode when the small gap results from weak breaking of the original continuous symmetry; a residual exact continuous symmetry can still protect other gapless directions.

Breaking a discrete symmetry produces distinct alternatives but no continuous orientation direction. It may yield domain walls or finite-size doublets, not a Goldstone mode on that basis alone.

Gauge coupling changes the phase dynamics

Gauge transformations relate descriptions of the same physical state. The global Goldstone theorem therefore cannot be applied by simply counting “broken gauge generators.” Coupling a condensate to a dynamical electromagnetic field changes its gauge-invariant collective response. In a three-dimensional bulk charged condensate with the usual Coulomb interaction, the phase and charge dynamics lead to a plasma scale rather than the neutral acoustic branch. Other dimensions and electromagnetic environments can have different dispersions. Anderson’s analysis explains this connection to the Higgs mechanism.

Exercises

1. For the quadratic phase Lagrangian with $\kappa\geq0$, derive the dispersion and find the lowest nonzero-wavevector frequency in a periodic cube of side $L$.

Solution

The field equation is $\chi\partial_t^2\theta-\rho_s\nabla^2\theta+\kappa\theta=0$. Substituting $e^{i(\mathbf k\cdot\mathbf x-\omega t)}$ yields $\omega^2=(\rho_s k^2+\kappa)/\chi$. The smallest nonzero wave number is $2\pi/L$, giving

$$\omega_{\min,\,k\ne0} =\sqrt{\frac{\rho_s}{\chi}\left(\frac{2\pi}{L}\right)^2+\frac\kappa\chi}.$$

With $\kappa=0$ this scales as $1/L$. The uniform collective coordinate needs its own finite-volume quantization; this formula is for the nonzero-wavevector modes.

2. A phase has a pinning energy density $u(1-\cos n\theta)$, with $u>0$ and positive integer $n$. What are the small-angle curvature and pseudo-Goldstone frequency near $\theta=0$?

Solution

Expanding the cosine gives $un^2\theta^2/2$, so $\kappa=un^2$ and $\omega(0)=n\sqrt{u/\chi}$. There are $n$ discrete minima over a full $2\pi$ interval, but discrete degeneracy does not remove the local restoring force. The gap tends to zero as the continuous shift symmetry is restored at $u\to0$.

References

• P. W. Anderson, “Plasmons, gauge invariance, and mass,” Physical Review 130, 439–442 (1963). doi:10.1103/PhysRev.130.439.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.
• J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Physical Review 127, 965–970 (1962). doi:10.1103/PhysRev.127.965.
• N. D. Mermin 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.
• H. Watanabe and H. Murayama, “Unified description of Nambu–Goldstone bosons without Lorentz invariance,” Physical Review Letters 108, 251602 (2012). doi:10.1103/PhysRevLett.108.251602.