Finite-Density Goldstone Counting

For spontaneously broken internal symmetries at finite density, the number of Nambu–Goldstone modes is

$$n_{\rm NG}=n_{\rm BS}-\frac12\operatorname{rank}\rho, \qquad \rho_{ab}=-i\lim_{V\to\infty}\frac{\langle[Q_a,Q_b]\rangle}{V}.$$

Each nonzero antisymmetric $2\times2$ block of $\rho$ pairs two broken generators into one type-B mode, generically with quadratic dispersion. Unpaired broken generators give type-A modes, generically linear. The formula requires the thermodynamic and zero-momentum limits, exact internal symmetries, and locality assumptions; spacetime symmetries, gauge redundancies, explicit breaking, and long-range forces need separate analysis.

Required background. Densities, Currents, and Nonrelativistic Ward Identities supplies charges and current poles. Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions supplies the general exceptions.

Helpful background. Goldstone’s Theorem: Hypotheses and Pole Argument reviews the relativistic pole argument that finite density modifies.

Broken charges and the commutator matrix

Let continuous internal charges $Q_a=\int_V\mathrm d^d x\,j_a^0$ be conserved and spontaneously broken in a translationally invariant thermodynamic state. A charge is broken when some local operator $\Phi$ has

$$\lim_{V\to\infty}\langle[Q_a,\Phi]\rangle\ne0$$

after the symmetry-breaking source is removed in the correct order. At finite $V$, an exact eigenstate usually restores the symmetry, so one must take $V\to\infty$ before sending the source to zero.

The matrix $\rho$ measures commutator density among broken charges. Because it is real antisymmetric in a Hermitian basis, its rank is even. Define

$$n_B=\frac12\operatorname{rank}\rho, \qquad n_A=n_{\rm BS}-\operatorname{rank}\rho.$$

Then $n_{\rm NG}=n_A+n_B$. Watanabe and Murayama derive the pairing from the symplectic part of the low-energy effective action Watanabe and Murayama 2012, pp. 251602-1–251602-5.

The type labels concern canonical structure, not solely dispersion. Under generic rotationally invariant analytic conditions, type A has $\omega\propto k$ and type B has $\omega\propto k^2$, but fine tuning or additional symmetries can change powers.

Effective action derivation

Let $\pi^a$ parametrize broken directions. The most general low-derivative quadratic Lagrangian can contain

$$\mathcal L_{\rm eff} =\frac12\rho_{ab}\pi^a\dot\pi^b +\frac12\bar g_{ab}\dot\pi^a\dot\pi^b -\frac12g_{ab}\nabla\pi^a\cdot\nabla\pi^b+\cdots.$$

Bring $\rho$ to canonical antisymmetric blocks. In each nonzero block, two fields form one conjugate coordinate–momentum pair; the first-order time derivative and spatial gradient term yield one quadratic branch. Directions in the kernel of $\rho$ require the second-order kinetic term and yield independent type-A branches.

This also explains why “one mode per generator” fails without Lorentz invariance. Lorentz symmetry forbids the finite-density first-order term in the usual vacuum setting, forcing $\rho=0$ under its hypotheses.

Ferromagnet and antiferromagnet

For $SU(2)$ spin symmetry broken to rotations about $z$, the broken charges are $S_x,S_y$, and

$$[S_x,S_y]=iS_z.$$

In a ferromagnet, $\langle S_z\rangle/V=s\ne0$, so

$$\rho=\begin{pmatrix}0&s\\-s&0\end{pmatrix}, \qquad \operatorname{rank}\rho=2.$$

Thus $n_{\rm BS}=2$ but $n_{\rm NG}=1$: one type-B magnon with $\omega\simeq Dk^2$. The two transverse spin components are canonically conjugate descriptions of the same excitation.

In a collinear antiferromagnet the uniform magnetization density vanishes, so $\rho=0$ for the two broken spin charges. There are two type-A transverse modes, generically linear. The symmetry-breaking pattern is the same, but the state-dependent commutator density differs.

For a neutral superfluid with one broken $U(1)$ charge, $\rho$ is a $1\times1$ antisymmetric zero matrix, giving one type-A phonon. Coulomb interactions in a charged fluid can raise that mode to a plasmon; this does not contradict the internal short-range theorem because its hypotheses have changed.

Spacetime symmetries and redundant modes

Broken spacetime generators are subtler. Their currents contain explicit coordinates, distinct generators may act identically on the low-energy fields, and inverse-Higgs constraints can remove redundant fields. A crystal breaks translations and rotations but does not possess an independent Goldstone field for every broken rotation; rotational distortions are derivatives of displacements.

Therefore one must not insert broken boosts, rotations, or translations blindly into the internal-charge rank formula. Instead construct the coset or current correlators, identify independent local fluctuations, and impose allowed inverse-Higgs relations. Watanabe’s review separates the established internal-symmetry theorem from these spacetime qualifications Watanabe 2020, §§ 2–4.

Orders of limits and explicit breaking

Three limits must be recorded:

1. take the thermodynamic limit before removing a selecting source;
2. define $\rho$ as a charge density, not an extensive divergent commutator; and
3. inspect poles as $\mathbf k\to0$ before identifying their low-energy dispersion.

If an explicit breaking parameter $h$ remains, modes are pseudo-Goldstone modes with small gaps. If a charge is gauged, the physical spectrum follows the gauge constraints and Higgs mechanism. If interactions are sufficiently long ranged, analyticity behind the derivative expansion can fail.

Common pitfalls

Counting broken generators only. At finite density, nonzero commutator density pairs generators. Compute $\rho$ before counting.

Calling every quadratic mode type B. Type B is defined by the canonical pairing encoded by $\rho$. Accidental quadratic dispersion with $\rho=0$ is not the same structure.

Applying the formula to spacetime or gauged symmetries unchanged. The theorem is strongest for internal global symmetries under locality and thermodynamic hypotheses.

Exercises

Count modes in a magnetized $SU(3)$ example

Suppose four internal generators are broken and the commutator-density matrix has rank two. Find $n_A$, $n_B$, and $n_{\rm NG}$.

Solution

The rank-two block gives $n_B=1$. The kernel has dimension $4-2=2$, so $n_A=2$. Hence $n_{\rm NG}=3=4-2/2$.

Derive a type-B dispersion

For two fields with

$$\mathcal L=\frac s2(\pi_1\dot\pi_2-\pi_2\dot\pi_1) -\frac\kappa2[(\nabla\pi_1)^2+(\nabla\pi_2)^2],$$

find the positive-frequency branch.

Solution

The equations are $s\dot\pi_2+\kappa\nabla^2\pi_1=0$ and $-s\dot\pi_1+\kappa\nabla^2\pi_2=0$. With $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$, the determinant gives $s^2\omega^2-\kappa^2k^4=0$. The positive branch is $\omega=(\kappa/|s|)k^2$.

Continue

Plasmons and Collective Charge Modes shows how long-range Coulomb forces change a density mode. The Weakly Interacting Bose Gas realizes the type-A superfluid phonon. Spin Waves and Magnons develops magnetic examples beyond counting.

References

• Watanabe, Haruki. “Counting Rules of Nambu–Goldstone Modes.” Annual Review of Condensed Matter Physics 11 (2020): 169–187. DOI. Open PDF.
• Watanabe, Haruki, and Hitoshi Murayama. “Unified Description of Nambu–Goldstone Bosons without Lorentz Invariance.” Physical Review Letters 108 (2012): 251602. DOI. Open PDF.

Further reading

• Watanabe, Haruki, and Tomáš Brauner. “Number of Nambu–Goldstone Bosons and Its Relation to Charge Densities.” Physical Review D 84 (2011): 125013. DOI. Open PDF.