Goldstone Modes Preview

Goldstone modes are low-energy collective excitations associated with spontaneously broken continuous global symmetries. They are the dynamical fluctuations that move slowly along the family of symmetry-related ordered states.

The compressed symmetry pattern is

$$G \longrightarrow H,$$

where $G$ is the original continuous symmetry group and $H$ is the subgroup preserved by the ordered state. The broken directions in $G/H$ are not ordinary microscopic labels. They are collective coordinates of the ordered medium.

This page is a physical preview. Goldstone Modes in Many-Body Systems gives the full nonrelativistic counting rule, effective dynamics, and condensed-matter caveats. The reference-card theorem statement is Goldstone Theorem Preview, while current algebra and the field-theory proof strategy are developed in From Symmetry Breaking to Goldstone Theorem.

The Basic Picture

Suppose an order parameter $\Phi$ transforms nontrivially under a continuous symmetry. In a symmetry-broken phase, the system chooses one value

$$\langle\Phi\rangle = \Phi_0.$$

Acting with the original symmetry produces a continuous family of equally good ordered states:

$$\Phi_0 \longmapsto g\Phi_0, \qquad g\in G.$$

The elements of $H$ leave $\Phi_0$ unchanged, so physically distinct orientations are labeled by the coset $G/H$. A Goldstone field is a slowly varying coordinate on that coset. It describes how the ordered state is locally rotated along the symmetry-broken manifold.

The word “mode” matters: a Goldstone mode is not just the existence of many symmetry-related ground states. It is a low-energy excitation in which the order parameter varies gently in space and time.

Why the Energy Is Small

A continuous broken symmetry gives a flat direction for a uniform change of the order parameter. If every constant value of a phase $\theta$ is symmetry-related, then the energy cannot depend on the absolute value of $\theta$:

$$E[\theta+\alpha] = E[\theta].$$

For a slowly varying configuration, the leading energy cost is usually controlled by gradients:

$$E[\theta] = \frac{\rho_s}{2} \int d^d x\, (\nabla\theta)^2 + \cdots,$$

where $\rho_s$ is a stiffness. A uniform shift $\theta\mapsto\theta+\alpha$ costs no energy, while a long-wavelength wave with wave number $k$ costs energy that vanishes as $k\to0$.

The dispersion depends on the dynamics. A relativistic or sound-like mode often has

$$\omega(k) \sim c|k|,$$

while some nonrelativistic systems have quadratic Goldstone modes,

$$\omega(k) \sim Dk^2.$$

The gapless conclusion is robust under the appropriate assumptions, but the exact number of modes and their dispersions require more care.

Finite Systems and Nearly Broken Symmetry

In a finite quantum system, exact spontaneous symmetry breaking is subtle. If the ground state is unique and the Hamiltonian has an exact symmetry, the ground state is usually symmetric. The broken-symmetry picture becomes sharp in a thermodynamic or infinite-volume limit.

Before that limit, one often sees:

• a set of low-lying states whose splittings shrink with system size;
• order-parameter correlations that look ordered over long distances;
• a small symmetry-breaking field selecting one branch;
• collective excitations whose lowest momenta are set by the finite system size.

Thus the Goldstone mode is not an isolated two-level splitting. It is a many-degree-of-freedom, long-wavelength excitation. The finite-size caveat is developed in Spontaneous Symmetry Breaking Preview.

Examples

A neutral superfluid or Bose condensate has an approximate order parameter

$$\langle\Psi\rangle = |\Psi|e^{i\theta}.$$

The phase $\theta$ is the broken $U(1)$ direction, and its long-wavelength fluctuations are the sound mode of the neutral superfluid. The reference-model entry is Ideal Bose Gas, with interactions and true superfluidity belonging to later many-body pages.

A magnet with broken spin-rotation symmetry has spin-wave excitations. The local magnetization direction varies slowly in space, and the resulting collective modes are magnons. Depending on the magnetic order and commutators of broken charges, the dispersion can be linear or quadratic.

A crystal breaks continuous translation symmetry down to a discrete lattice. Acoustic phonons are Goldstone modes of the broken translations in the long-wavelength elastic description. The microscopic atoms still obey translation-invariant laws; the ordered phase does not display the full continuous symmetry.

In relativistic field theory, idealized chiral symmetry breaking produces pion-like Goldstone bosons. In real QCD the pions are light but not exactly massless because the corresponding symmetry is approximate rather than exact. That is a pseudo-Goldstone situation, not an exact theorem conclusion.

Discrete Symmetries Do Not Produce Goldstone Modes

Breaking a discrete symmetry can produce multiple ordered phases, domain walls, and nearly degenerate finite-size structures. It does not provide a continuous flat direction.

For example, an Ising-like spin-flip symmetry has two alternatives rather than a circle of alternatives:

$$m \longrightarrow -m.$$

There is no small continuous rotation of $m$ that stays in the space of equally good ordered states. Domain walls can be low-energy in some regimes, but they are not Goldstone modes merely because a discrete symmetry is broken.

Pseudo-Goldstone Modes

If the continuous symmetry is explicitly but weakly broken, the would-be flat direction is slightly tilted. A simple effective energy for a phase-like coordinate is

$$E[\theta] = \int d^d x \left[ \frac{\rho_s}{2}(\nabla\theta)^2 + \frac{\kappa}{2}\theta^2 \right].$$

The $\kappa$ term selects a preferred value of $\theta$. With a kinetic susceptibility $\chi$, the small oscillations obey a schematic dispersion

$$\omega^2(k) = \frac{\rho_s}{\chi}k^2 + \frac{\kappa}{\chi}.$$

The mode has a small gap when $\kappa>0$. This is the effective-theory meaning of a pseudo-Goldstone mode. The symmetry is not exact, but the breaking is small enough that a light collective excitation remains.

This language should be tied to Approximate Symmetry and Explicit Symmetry Breaking, not used as a synonym for every low-energy mode.

Gauge Symmetry Is Different

Goldstone modes come from broken physical global symmetries. Gauge symmetry is a redundancy of description, not an ordinary symmetry that maps one physical state to another distinct physical state.

In superconductors and Higgs-like systems, the would-be Goldstone mode can combine with a gauge field, changing the spectrum. In particle-physics language this is the Higgs mechanism; in condensed-matter language it is related to the electromagnetic response of a charged condensate. The important caution here is simple:

$$\text{do not apply the global Goldstone theorem to gauge redundancy.}$$

This does not mean superconductors lack low-energy physics. It means the correct degrees of freedom and observable spectrum are not obtained by naively counting broken gauge generators.

Counting Caution

In many relativistic systems, the number of Goldstone modes equals the number of broken generators. In nonrelativistic many-body systems, this equality can fail.

One reason is that two broken generators can form a conjugate pair in the ordered state. A common schematic diagnostic uses the expectation value of charge commutators:

$$\rho_{ab} = -\frac{i}{V} \langle[Q_a,Q_b]\rangle.$$

When this matrix has nonzero rank in the thermodynamic limit, the number of Goldstone modes can be smaller than the number of broken generators. This page only flags the issue. Detailed counting belongs to the theorem and many-body pages.

How They Appear in Practice

To identify a Goldstone mode, look for more than a low energy:

• an underlying continuous global symmetry;
• a symmetry-broken state or phase in the appropriate limit;
• an order parameter that transforms under the broken symmetry;
• a long-wavelength excitation that changes the order-parameter orientation;
• a gap that vanishes as momentum goes to zero, unless explicit breaking or gauge coupling changes the conclusion.

The observable may be a pole in a response function, a low-energy branch in neutron scattering, a sound mode in a fluid, a spin wave in a magnet, or a phonon branch in a crystal. The common structure is collective motion along a broken continuous direction.

Common Mistakes

• Calling every gapless excitation a Goldstone mode.
• Applying Goldstone reasoning to a broken discrete symmetry.
• Ignoring the thermodynamic-limit assumption behind spontaneous symmetry breaking.
• Counting broken generators as modes in nonrelativistic systems without checking charge commutators.
• Treating gauge redundancy as an ordinary global symmetry.
• Forgetting that weak explicit breaking gaps the mode.
• Using the word “phase” without distinguishing a phase angle from a phase of matter.

Related Pages

• Spontaneous Symmetry Breaking Preview
• Goldstone Theorem Preview
• From Symmetry Breaking to Goldstone Theorem
• Approximate Symmetry
• Explicit Symmetry Breaking
• Emergent Symmetry
• Why Symmetry Becomes Central in QFT
• Entanglement in Many-Body Physics

References

• J. Goldstone, “Field theories with superconductor solutions,” Il Nuovo Cimento 19, 154-164, 1961.
• J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Physical Review 127, 965-970, 1962.
• Y. Nambu, “Quasi-particles and gauge invariance in the theory of superconductivity,” Physical Review 117, 648-663, 1960.
• P. W. Anderson, “Plasmons, gauge invariance, and mass,” Physical Review 130, 439-442, 1963.
• H. Watanabe and H. Murayama, “Unified description of Nambu–Goldstone bosons without Lorentz invariance,” Physical Review Letters 108, 251602, 2012.
• S. Weinberg, The Quantum Theory of Fields, Volume II, Cambridge University Press, 1996.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.

Exercises

1. Phase stiffness and gaplessness.

Consider

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

Explain why a uniform shift $\theta(x)=\theta_0$ costs no energy and why a plane wave $\theta(x)=A\cos(\mathbf k\cdot\mathbf x)$ has energy proportional to $k^2$.

Solution

For a uniform shift, $\nabla\theta=0$, so $E[\theta]=0$ relative to the chosen ordered state. For a plane wave,

$$\nabla\theta = -A\mathbf k\sin(\mathbf k\cdot\mathbf x),$$

so the energy density is proportional to $A^2k^2$ after spatial averaging. The energy cost vanishes as $k\to0$, which is the static stiffness version of gaplessness.

2. Discrete versus continuous breaking.

Why does an Ising-like order parameter with $m\mapsto -m$ not imply a Goldstone mode?

Solution

The two ordered states are separated alternatives, not a continuous family. There is no infinitesimal symmetry direction along which the order parameter can vary while remaining on an exactly degenerate manifold. Domain walls or finite-size doublets may occur, but those are not Goldstone modes.

3. Pseudo-Goldstone gap.

For the effective quadratic dynamics

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

derive the dispersion relation for plane waves.

Solution

The Euler–Lagrange equation is

$$\chi\,\partial_t^2\theta - \rho_s\nabla^2\theta + \kappa\theta = 0.$$

For $\theta\propto e^{i(\mathbf k\cdot\mathbf x-\omega t)}$, this gives

$$-\chi\omega^2 + \rho_s k^2 + \kappa = 0,$$

so

$$\omega^2(k) = \frac{\rho_s}{\chi}k^2 + \frac{\kappa}{\chi}.$$

When $\kappa=0$, the mode is gapless. When $\kappa>0$, the gap is $\omega(0)=\sqrt{\kappa/\chi}$.

4. Gauge caution.

A charged condensate has a phase-like order parameter and couples to electromagnetism. Why is it not enough to count the phase as an ordinary global Goldstone mode?

Solution

The electromagnetic phase transformation is tied to gauge redundancy and a dynamical gauge field. The would-be phase mode is not counted like a broken physical global symmetry. Coupling to the gauge field changes the observable spectrum, as in the Anderson–Higgs mechanism. One must analyze gauge-invariant degrees of freedom and response functions rather than naively applying the global Goldstone theorem.