In Chaps. 2 and 3, we saw examples of how spontaneous symmetry breaking (SSB) gives rise to massless particles in the spectrum. Our operational understanding of these Nambu–Goldstone (NG) bosons was based on a classical (tree-level) analysis of certain scalar field potentials. The purpose of this chapter is to develop a general understanding of NG bosons without the limitations of a particular model or approximation. The basic intuition is built in Sect. 6.1. This explains both where NG bosons come from, and how their spectrum is related to the nature of broken symmetry. The subsequent sections put this intuition on a solid footing. In Sect. 6.2 I prove the so-called Goldstone theorem, which asserts the very existence of a NG boson as a consequence of SSB. Section 6.3 then delves into the question how many NG bosons there are in a given system, and how their number relates to their dispersion relations. This generalizes the observations made in Chap. 3; see Sect. 3.3 for a quick summary. Altogether, this chapter completes the background needed in Parts III and IV, where the effective field theory (EFT) formalism for SSB is developed in detail.

1 Intuitive Picture

The intuitive understanding of NG bosons relies heavily on the concept of order parameter. Consider for simplicity a system whose ground state is spatially uniform, and imagine that we disturb it by a weak, local perturbation. Locally, it is energetically favorable for the system to remain in one of the degenerate ground states. The order parameter will therefore respond to the perturbation by developing spatial variation while remaining on the vacuum manifold everywhere. The energy cost of creating such a spatially varying excited state must be proportional to gradients of the order parameter. This is because changing the order parameter uniformly just amounts to a different choice of ground state. We can then imagine order parameter configurations that vary over progressively longer and longer length scales. In the limit that the scale of spatial variation goes to infinity, the gradient of the order parameter vanishes, and the energy cost must go to zero.

This is a completely general observation, valid regardless of the choice of field variables and dynamics behind SSB. The only assumption that is really necessary is that the spontaneously broken symmetry is continuous. This makes smooth spatial variation of the order parameter possible while keeping it on the vacuum manifold everywhere. We conclude that NG bosons are local, propagating fluctuations of the order parameter whose energy goes to zero in the infinite-wavelength limit.

The identification of NG bosons with fluctuations of the order parameter is not necessarily one-to-one. Eventually, we would like to know the precise spectrum of NG bosons; this among others governs the low-temperature thermodynamics of systems with SSB. What we really need in order to establish a dictionary between broken symmetries and NG bosons is to answer the following questions:

  • How many different types of order parameter fluctuations (NG fields) are there in a given system?

  • What is the correspondence between the various order parameter fluctuations (NG fields) and NG modes in the spectrum?

I will deal with these two questions in the given order in the next two subsections.

1.1 Redundancy of Order Parameter Fluctuations

I will build upon the intuitive picture of a NG boson in terms of a fluctuation of the order parameter that locally remains on the vacuum manifold. One can imagine such a fluctuation as being generated from the ground state by a broken symmetry transformation with a coordinate-dependent parameter. This leads immediately to the important observation that fluctuations induced by different broken symmetry generators may coincide.

Example 6.1

The Lagrangian of a free massless relativistic scalar field \(\phi \) is \(\mathcal {L}[\phi ]=(1/2)(\partial _\mu \phi )^2\). The action of this theory is invariant under the polynomial shift transformation \(\phi (x)\to \phi (x)+\epsilon _1+\epsilon _{2\mu }x^\mu \), where \(\epsilon _1,\epsilon _{2\mu }\) are constant parameters. This symmetry is necessarily spontaneously broken, and the order parameter can be chosen as \(\langle {\phi (x)}\rangle \). One might expect to obtain \(D+1\) different fluctuations of the order parameter by applying a polynomial shift with coordinate-dependent parameters \(\epsilon _1(x)\), \(\epsilon _{2\mu }(x)\). However, shifting \(\phi (x)\) by is identical to shifting it by \(\epsilon _1(x)\) if we set . The set of \(D+1\) broken generators associated with our polynomial shift therefore corresponds to a single independent fluctuation, induced by \(\phi (x)\to \phi (x)+\epsilon _1(x)\). The only independent NG field in the theory is \(\phi \) itself.

Symmetries that become equivalent once their parameters are made coordinate-dependent are called redundant. Such a redundancy is typical for spacetime symmetries. As Example 6.1 shows, however, it is also possible for point symmetries that do not affect spacetime coordinates. Here is a less trivial example, relevant for any crystalline phase of matter.

Example 6.2

In relativistic theories of scalar fields, the generators of spacetime rotations \(J^{\mu \nu }\) and translations \(P^\mu \) are known to satisfy the relation \(J^{\mu \nu }=x^\mu P^\nu -x^\nu P^\mu \). A local rotation \(\exp [(\mathrm{i} /2)\epsilon _{2\mu \nu }(x)J^{\mu \nu }]\) with antisymmetric matrix parameter \(\epsilon _{2\mu \nu }\) is thus equivalent to a local translation \(\exp [\mathrm{i} \epsilon _{1\mu }(x)P^\mu ]\) with \(\epsilon _{1\mu }(x)\equiv -\epsilon _{2\mu \nu }(x)x^\nu \). Imagine now a system where spacetime rotations and translations (or a subset thereof) are both spontaneously broken. For example, in crystalline solids, all the continuous spatial rotations and translations are spontaneously broken. The rotations are clearly redundant; the only independent NG fields are those associated with broken translations. These parameterize the vibrations of the crystal lattice.

The basic moral to remember is that the number of independent order parameter fluctuations may be lower than the number of broken symmetry generators. I will further refine this observation in Sect. 6.3.1. The discussion of redundancy of local symmetry transformations will be of great importance for the development of EFT for broken spacetime symmetries in Part IV.

1.2 Canonical Conjugation of Nambu–Goldstone Fields

The number of NG modes in the spectrum may be lower than the number of independent NG fields in case some of the latter are canonically conjugated. We saw this in Sect. 3.2. To make statements independent of a specific model, we need to relate the possibility of canonical conjugation of NG fields directly to the broken symmetry.

Suppose we have two NG fields, \(\pi ^1\) and \(\pi ^2\), associated with two broken symmetry generators \(Q_{1,2}\). We can assume without loss of generality that under the respective transformations generated by \(Q_{1,2}\), these fields transform as

$$\displaystyle \begin{aligned} \pi^a\to\pi^a+\epsilon^b[\delta^a_b+\mathcal{O}(\pi)]\;,\qquad a=1,2\;. {} \end{aligned} $$
(6.1)

Indeed, one can always choose the fields so that the ground state corresponds to \(\pi ^a=0\). That the leading, constant piece of the Taylor expansion of the transformation rule for \(\pi ^a\) around this ground state is nonzero, follows from the assumption of broken symmetry. The bases of NG fields and broken generators can then be aligned so that this constant piece is simply \(\pi ^a\to \pi ^a+\epsilon ^a\).

Suppose in addition that we can construct a low-energy EFT for the NG fields whose Lagrangian contains a term with a single time derivative,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\varrho_{12}\pi^1\partial_0\pi^2+\dotsb\;. {} \end{aligned} $$
(6.2)

The ellipsis stands for terms with more than one derivative or more than two NG fields. Recall from Sect. 4.2.1 that the Noether currents corresponding to (6.1) can be identified by making the parameters \(\epsilon ^a\) coordinate-dependent. From (6.2) alone, one can then extract the leading contributions to the Noether charge densities,

$$\displaystyle \begin{aligned} J^0_1[\pi]=-\varrho_{12}\pi^2+\dotsb\;,\qquad J^0_2[\pi]=+\varrho_{12}\pi^1+\dotsb\;. {} \end{aligned} $$
(6.3)

The ellipses now represent terms with derivatives or more than one NG field.

Here comes the key step: I will evaluate the transformation of \(J^0_2[\pi ]\) under the symmetry generated by \(Q_1\) in two different ways. On the one hand, it is obvious that under (6.1), \(J_2^0[\pi ]\to J_2^0[\pi ]+\varrho _{12}\epsilon ^1+\dotsb \). On the other hand, one may think of the currents as quantum operators and represent the same transformation as

$$\displaystyle \begin{aligned} \exp(\mathrm{i}\epsilon^1Q_1)J_2^0\exp(-\mathrm{i}\epsilon^1Q_1)=J_2^0+\mathrm{i}\epsilon^1[Q_1,J_2^0]+\mathcal{O}((\epsilon^1)^2)\;. \end{aligned} $$
(6.4)

The NG fields vanish in the ground state. Hence, upon taking the vacuum expectation value (VEV), a comparison of our two little calculations leads to

$$\displaystyle \begin{aligned} \varrho_{12}=\mathrm{i}\langle{[Q_1,J_2^0]}\rangle =-\mathrm{i}\langle{[Q_2,J_1^0]}\rangle \;, {} \end{aligned} $$
(6.5)

where the second equality follows by running the same argument on \(J_1^0[\pi ]\) and \(Q_2\).

This is a remarkable result that is as model-independent as it gets. In any theory that realizes the same symmetry-breaking pattern, one can construct the Noether currents via Noether’s theorem. Evaluating the VEV of the commutator (6.5) then gives us a simple criterion for when to expect two NG fields to be canonically conjugated: whenever (6.5) is nonzero.

Example 6.3

We already saw the relation between canonical conjugation and the charge commutator (6.5) at work in Sect. 5.3.1. Indeed, one can think of the free Schrödinger field \(\psi \) therein as corresponding to two real NG fields. These are in turn associated with the invariance of the free Schrödinger theory under independent shifts of the real and imaginary parts of \(\psi \). The commutator of the generators of these shifts, \(Q_{\mathrm {R}}\) and \(Q_{\mathrm {I}}\), turns out to be nonzero upon quantization. It is easy to check that the normalization of the commutator as shown in (5.36) agrees with (6.5). The factor of spatial volume V  in (5.36) arises from the latter displaying a commutator of two charges rather than a charge and a charge density.

Although noninteracting, the free Schrödinger theory is nontrivial in that it makes the VEV (6.5) arise from a central charge in the symmetry algebra. Let us have a look at one more example, which is mathematically simpler and physically familiar.

Example 6.4

In the ideal (that is spatially uniform and isotropic) approximation, both ferromagnets and antiferromagnets possess a symmetry \(G\simeq \mathrm {SU}(2)\), generated by the operator \(\boldsymbol S\) of total spin. In the ground state of both, this is spontaneously broken down to \(H\simeq \mathrm {U}(1)\), consisting of spin rotations around the axis along which the spins are aligned. Suppose that one chooses the ground state oriented along the third direction in the spin space. Then H is generated by \(S_3\), whereas the two independent broken generators can be chosen as \(S_{1,2}\).

The difference between ferro- and antiferromagnets is that the former feature a nonzero net magnetization. This amounts to a nonzero VEV \(\langle {S_3}\rangle =-\mathrm{i} \langle {[S_1,S_2]}\rangle \). As a consequence, the two NG fields in a ferromagnet are canonically conjugated. The spectrum only contains a single NG mode: the ferromagnetic magnon. Its dispersion relation at long wavelengths is known to be quadratic in momentum. In antiferromagnets, on the other hand, \(\langle {\boldsymbol S}\rangle =\mathbf {0}\). As a consequence, the spectrum of antiferromagnets features two different magnon branches. Their dispersion relation is known to be linear in momentum in the long-wavelength limit.

1.3 The Big Picture

Let me summarize what we have learned so far. Spontaneous breaking of a continuous symmetry implies the existence of a gapless mode in the spectrum: the NG boson. This can be viewed as a propagating fluctuation of the order parameter. The number of such NG modes may be lower than the number of broken symmetry generators. There are two basic mechanisms how such a reduction may occur.

First, different broken symmetry generators may induce fluctuations of the order parameter that are indistinguishable from each other. I will refer to this mechanism as “geometric” to keep in mind that it represents a restriction on the configuration space of off-shell NG fields. Second, there may not be a one-to-one correspondence between the independent NG fields and the independent NG modes in the spectrum. This happens whenever a pair of NG fields is canonically conjugated, which in turn requires a nonzero VEV of the charge commutator (6.5). This mechanism I will refer to as “dynamical” since it restricts the space of on-shell fields, or eigenstates of the Hamiltonian. For the reader’s convenience, an outline of the correspondence between broken symmetry, NG fields and NG modes is shown graphically in Fig. 6.1. This scheme is intuitively simple yet somewhat imprecise. I will return to the question of how many NG modes there are in a given system in Sect. 6.3.

Fig. 6.1
figure 1

Basic mechanisms that may cause the number of independent NG modes in the spectrum to be lower than the number of broken symmetry generators. The geometric reduction arises whenever fluctuations of the order parameter, induced by different generators, are indistinguishable from each other. The dynamical reduction is signaled by a nonzero VEV of the charge commutator (6.5), or by a term in the effective Lagrangian for NG fields with a single time derivative

Full size image

2 Goldstone Theorem

The Goldstone theorem is one of the most profound exact results in quantum field theory that deserves a more careful justification than the intuitive but hand-waving argument of Sect. 6.1. In the next two subsections, I will give two different proofs, following in spirit the original work of Goldstone, Salam and Weinberg [1, 2]. Along the way, I will clarify the technical assumptions on which the theorem is based. A reader seeking a higher level of mathematical rigor than what I can offer here is advised to consult the early review [3] or the book [4].

2.1 Operator Proof

I will start with a proof using the operator formalism of quantum field theory in the Heisenberg picture. Before proceeding to the details of the proof, however, I need to spend some time listing its technical assumptions.

First, I assume that in the limit of infinite spatial volume, the system possesses unbroken continuous (spacetime) translation invariance. This ensures the existence of a basis of the Hilbert space consisting of eigenstates of the energy–momentum operator \(P^\mu \). It is always possible, and I will implicitly do so, to define this operator so that the energy and momentum of the ground state are both zero. The assumption of continuous spatial translation invariance is, in fact, unnecessarily strong. The minimum requirement is unbroken discrete translation invariance, which is necessary to have a well-defined notion of quasiparticles. The proof then proceeds along the same steps at the cost of a somewhat cluttered notation [5].

Second, I assume the existence of a local operator \(J^0(x)\), interpreted as the density of a conserved charge. The charge density should not depend explicitly on spacetime coordinates; this is needed to ensure the translation property

$$\displaystyle \begin{aligned} J^0(x)=\mathrm{e}^{\mathrm{i} P\cdot x}J^0(0)\mathrm{e}^{-\mathrm{i} P\cdot x}\;. {} \end{aligned} $$
(6.6)

Now define the total charge contained in a finite spatial domain \(\Omega \),

$$\displaystyle \begin{aligned} Q_\Omega(t)\equiv\int_\Omega\mathrm{d}^d\!\boldsymbol x\,J^0(\boldsymbol x,t)\;. {} \end{aligned} $$
(6.7)

The last assumption we need is the existence of a local, time-independent operator \(\Phi \) such that the VEV in the chosen ground state \(\left \lvert {0}\right \rangle \),

$$\displaystyle \begin{aligned} \left\langle{0}\right\rvert [Q_\Omega(t),\Phi]\left\lvert{0}\right\rangle \;, {} \end{aligned} $$
(6.8)

is nonzero and time-independent in the limit \(\Omega \to \infty \).

It is this last assumption that captures the essence of SSB. Think of \(Q_\Omega \) as the Noether charge associated with a continuous symmetry. One would be tempted to say that in the limit \(\Omega \to \infty \), \(Q_\Omega \) itself becomes time-independent. But we saw in Sect. 5.3 that in the infinite-volume limit, the operators representing spontaneously broken symmetry may be ill-defined. The limit \(\Omega \to \infty \) is only safe if performed on the commutator in (6.8), not on \(Q_\Omega \) itself.

Usually, the time-independence of the integral charge is a consequence of a local conservation law, \(\partial _\mu J^\mu =0\). That is however strictly speaking not necessary. Taking this additional step requires the vanishing of the surface integral \(\int _\Omega \mathrm{d} ^d\!\boldsymbol x\,\partial _rJ^r(\boldsymbol x,t)\) as \(\Omega \to \infty \). This condition, or even its weaker form imposed only on the VEV \(\left \langle {0}\right \rvert [\partial _rJ^r(\boldsymbol x,t),\Phi ]\left \lvert {0}\right \rangle \), may be compromised in systems with long-range interactions [3]. Here I avoid having to deal with this issue simply by assuming right away the time-independence of (6.8).

Think of \(\left \langle {0}\right \rvert \Phi \left \lvert {0}\right \rangle \) as an order parameter. Then (6.8) is, up to a factor, its infinitesimal variation under the transformation \(\Phi \to \exp (\mathrm{i} \epsilon Q_\Omega )\Phi \exp (-\mathrm{i} \epsilon Q_\Omega )\). The assumption that (6.8) is nonzero guarantees that \(\left \lvert {0}\right \rangle \) cannot be an eigenstate of \(Q_\Omega \). Hence the symmetry generated by \(Q_\Omega \) must be spontaneously broken.

We are now ready to prove the existence of a NG mode in the system. I will initially assume that the system is enclosed in a finite volume V  with a periodic boundary condition. The Hilbert space then admits a basis of momentum eigenstates \(\left \lvert {n,\boldsymbol p}\right \rangle \), where \(\boldsymbol p\) stands for a set of discrete momenta consistent with the boundary condition. In a finite volume, the basis states can be normalized as \(\left \langle {m,\boldsymbol p}\middle \vert {n,\boldsymbol q}\right \rangle =\delta _{mn}\delta _{\boldsymbol p\boldsymbol q}\). The label n indicates all other quantum numbers the states may possess, such as relative momenta in multiparticle states, or internal degrees of freedom.

Upon inserting the partition of unity in terms of \(\left \lvert {n,\boldsymbol p}\right \rangle \) and using the translation property (6.6), the VEV (6.8) can be rewritten as

$$\displaystyle \begin{aligned} \begin{aligned} \left\langle{0}\right\rvert [Q_\Omega(t),\Phi]\left\lvert{0}\right\rangle =\sum_{n,\boldsymbol p}\int_\Omega&\mathrm{d}^d\!\boldsymbol x\,\Big[\exp(-\mathrm{i} p_n\cdot x)\left\langle{0}\right\rvert J^0(0)\left\lvert{n,\boldsymbol p}\right\rangle \left\langle{n,\boldsymbol p}\right\rvert \Phi\left\lvert{0}\right\rangle \\ &-\exp(+\mathrm{i} p_n\cdot x)\left\langle{0}\right\rvert \Phi\left\lvert{n,\boldsymbol p}\right\rangle \left\langle{n,\boldsymbol p}\right\rvert J^0(0)\left\lvert{0}\right\rangle \Big]\;. \end{aligned} {} \end{aligned} $$
(6.9)

Here \(p_n\) is a shorthand notation for the energy–momentum of \(\left \lvert {n,\boldsymbol p}\right \rangle \). Now recall that SSB implies the existence of degenerate ground states. Even if one picks a unique vacuum \(\left \lvert {0}\right \rangle \), the other, alternative ground states do not disappear. They contribute to (6.9) among the states \(\left \lvert {n,\mathbf {0}}\right \rangle \) with \(\boldsymbol p=\mathbf {0}\). Is this something to worry about?

In the free Schrödinger theory, worked out in Sect. 5.3.1, \(\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\mathbf {0}}\right \rangle \) scales as \(1/\sqrt {V}\), cf. (5.35). This observation turns out to be general. With the periodic boundary condition, the integral charge \(Q_V(t)\) is translationally invariant. A classic argument [6] then shows that the norm of \(Q_V(t)\left \lvert {0}\right \rangle \) scales as \(\sqrt {V}\),

$$\displaystyle \begin{aligned} \big\|Q_V(t)\left\lvert{0}\right\rangle \big\|{}^2=\int_V\mathrm{d}^d\!\boldsymbol x\,\left\langle{0}\right\rvert J^0(\boldsymbol x,t)Q_V(t)\left\lvert{0}\right\rangle =V\left\langle{0}\right\rvert J^0(0)Q_V(0)\left\lvert{0}\right\rangle \;. \end{aligned} $$
(6.10)

This means, roughly, that the norm of \(J^0(0)\left \lvert {0}\right \rangle \) scales as \(1/\sqrt {V}\), and so does therefore \(\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\mathbf {0}}\right \rangle \). Altogether, the contribution of the alternative ground states to (6.9) at fixed \(\Omega \) and increasing V  is suppressed as \(1/\sqrt {V}\). Should (6.8) be nonzero in the infinite-volume limit, it must be dominated by excitations with nonzero momentum.

We can now switch to infinite volume. This requires changing the normalization of the basis states to \(\left \langle {m,\boldsymbol p}\middle \vert {n,\boldsymbol q}\right \rangle =(2\pi )^d\delta _{mn}\delta ^d(\boldsymbol p-\boldsymbol q)\) and replacing the sum over \(\boldsymbol p\) in (6.9) with an integral. Keeping \(\Omega \) still finite, the integration over \(\boldsymbol x\) then amounts to a Fourier transform of unity, . The notation underlines that \(\delta ^d_\Omega (\boldsymbol p)\) is a finite-volume approximation of the Dirac \(\delta \)-function. We then have

$$\displaystyle \begin{aligned} {} \left\langle{0}\right\rvert [&Q_\Omega(t),\Phi]\left\lvert{0}\right\rangle =\sum_n\int\mathrm{d}^d\!\boldsymbol p\,\big\{\exp[-\mathrm{i} E_n(\boldsymbol p)t]\delta^d_\Omega(\boldsymbol p)\left\langle{0}\right\rvert J^0(0)\left\lvert{n,\boldsymbol p}\right\rangle \\ \notag &\times\left\langle{n,\boldsymbol p}\right\rvert \Phi\left\lvert{0}\right\rangle -\exp[+\mathrm{i} E_n(\boldsymbol p)t]\delta^d_\Omega(-\boldsymbol p)\left\langle{0}\right\rvert \Phi\left\lvert{n,\boldsymbol p}\right\rangle \left\langle{n,\boldsymbol p}\right\rvert J^0(0)\left\lvert{0}\right\rangle \big\}\;, \end{aligned} $$
(6.11)

where \(E_n(\boldsymbol p)=p_n^0\) is the dispersion relation of the state \(\left \lvert {n,\boldsymbol p}\right \rangle \).

By our assumptions, (6.8) should be nonzero and time-independent in the limit \(\Omega \to \infty \). For large \(\Omega \), the function \(\delta ^d_\Omega (\boldsymbol p)\) is smooth but sharply peaked around \(\boldsymbol p=\mathbf {0}\). The time-independence thus requires that only states such that

$$\displaystyle \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}E_n(\boldsymbol p)=0 {} \end{aligned} $$
(6.12)

contribute to (6.11). The assumption that (6.8) should be nonzero guarantees that such states exist. These must be one-particle states, for if the one-particle spectrum had a gap, then so would the multiparticle one. We have thus established the existence of one-particle states \(\left \lvert {n,\boldsymbol p}\right \rangle \) with the property (6.12), for which both \(\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\boldsymbol p}\right \rangle \) and \(\left \langle {0}\right \rvert \Phi \left \lvert {n,\boldsymbol p}\right \rangle \) are nonzero. This concludes the proof of Goldstone’s theorem.

There is a simple, intuitive way to understand why NG modes must satisfy (6.12). Suppose we know beforehand that there is a one-particle state \(\left \lvert {n,\boldsymbol p}\right \rangle \) created by \(J^0\)and that the latter descends from a local conservation law, \(\partial _\mu J^\mu =0\). Extending the translation property (6.6) to \(J^\mu \), we get at once \(\left \langle {0}\right \rvert J^\mu (x)\left \lvert {n,\boldsymbol p}\right \rangle =\exp (-\mathrm{i} p_n\cdot x)\left \langle {0}\right \rvert J^\mu (0)\left \lvert {n,\boldsymbol p}\right \rangle \). Upon using current conservation and the fact that \(\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\boldsymbol p}\right \rangle \) is nonzero, it then follows that \(E_n(\boldsymbol p)={\boldsymbol p\cdot \left \langle {0}\right \rvert \boldsymbol J(0)\left \lvert {n,\boldsymbol p}\right \rangle }/{\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\boldsymbol p}\right \rangle }\). This shows that the vanishing of the energy of NG modes in the long-wavelength limit is an immediate consequence of symmetry. The nontrivial part of Goldstone’s theorem is that such NG states exist at all.

2.2 Effective Action Proof

I will now present an entirely different proof of Goldstone’s theorem which will prepare the ground for the discussion of redundancies among NG fields in the next section. In spirit, this proof is not based on current conservation but rather directly on the symmetry of the action. I will follow closely the appendix of [7].

The starting assumption is the existence of an action \(\Gamma \) as a functional of a set of fields \(\psi ^i\). The notation indicates that \(\Gamma \) is the quantum effective action, since we want to make exact statements about a quantum field theory. The argument below applies in principle equally well to the classical action S. However, the fields \(\psi ^i\) need not be the “elementary” fields that appear in S. They may be composite operators, and are chosen so that the VEV \(\langle {\psi ^i}\rangle \) is a suitable order parameter for SSB. The action is assumed to be invariant under the infinitesimal transformation

$$\displaystyle \begin{aligned} \updelta\psi^i(x)=\epsilon F^i[\psi,x](x)\;. {} \end{aligned} $$
(6.13)

Unlike in the previous, operator proof of Goldstone’s theorem, the symmetry transformation, and thus the Noether current, is allowed to depend explicitly on coordinates.

It is not guaranteed a priori that the quantum effective action \(\Gamma \) inherits verbatim all the symmetries of the classical action S. It turns out that the symmetry transformations on the classical and quantum actions are the same at least if \(F^i[\psi ,x]\) is linear in the fields; see Sect. 16.4 of [8]. This is something I assumed already back in Sect. 5.2 when I introduced the very concept of order parameter. The only restriction involved in the transition to the quantum effective action therefore is that the symmetry should preserve the functional integral measure. This is needed to exclude quantum anomalies, whose presence would invalidate Goldstone’s theorem.

The invariance of the action is encoded in the condition

$$\displaystyle \begin{aligned} \int\mathrm{d}^D\!y\,\frac{\updelta\Gamma}{\updelta\psi^j(y)}F^j[\psi,y](y)=0\;. \end{aligned} $$
(6.14)

Taking one more variational derivative and setting the fields equal to their VEVs, \(\langle {\psi ^i(x)}\rangle \equiv \varphi ^i(x)\), one gets

$$\displaystyle \begin{aligned} \int\mathrm{d}^D\!y\,\left.{\frac{\updelta^2\Gamma}{\updelta\psi^i(x)\updelta\psi^j(y)}}\right\rvert_{\psi=\varphi}F^j[\varphi,y](y)=0\;. {} \end{aligned} $$
(6.15)

Now we need the assumption of continuous translation invariance of the action and of the ground state. The second variational derivative of \(\Gamma \) can thus be traded for its Fourier transform, which is the inverse propagator of the theory, . Multiplying (6.15) with \(\mathrm{e} ^{\mathrm{i} p\cdot x}\) and integrating over x then yields the condition

$$\displaystyle \begin{aligned} G^{-1}_{ij}[\varphi](p)\int\mathrm{d}^D\!y\,\mathrm{e}^{\mathrm{i} p\cdot y}F^j[\varphi,y](y)=0\;. {} \end{aligned} $$
(6.16)

Hence, the propagator \(G_{ij}[\varphi ](p)\) has a singularity at energy–momentum \(p^\mu \) whenever the integral in (6.16) is nonzero. The spectral representation ensures that such singularities arise solely from states in the spectrum of the Hamiltonian.

By the assumption of broken symmetry, \(F^j[\varphi ,y]\), which represents the symmetry transformation of the order parameter, is nonzero. This alone implies that the integral in (6.16) must be nonzero for some\(p^\mu \). For coordinate-independent symmetries, the integral is proportional to \(\delta ^D(p)\). Moreover, for known examples of coordinate-dependent symmetries, \(F^j[\varphi ,y]\) is polynomial in coordinates. In such cases, integration over y leads to a linear combination of derivatives of \(\delta ^D(p)\). One way or another, we conclude that SSB implies the existence of a mode whose energy vanishes in the limit of zero momentum, as expected.

3 Classification and Counting

None of the above two proofs of the Goldstone theorem addresses the question how many NG bosons there are. To that end, we have to put in some extra effort. I will follow closely the scheme outlined in Sect. 6.1.3. This means that as the first step, we need to know, given the symmetry-breaking pattern, how many independent NG fields, or order parameter fluctuations, there are.

3.1 Independent Order Parameter Fluctuations

We add to (6.13) an index A, distinguishing the action of different symmetry generators \(Q_A\), \(\updelta \psi ^i(x)=\epsilon ^AF^i_A[\psi ,x](x)\). Recall the view of NG fields as local fluctuations of the order parameter. The easiest (albeit not the only; cf. Sect. 4.2.1) way to generate such fluctuations is by replacing \(\epsilon ^A\to \epsilon ^A(x)\). Redundancies among order parameter fluctuations are then detected by the existence of nonzero functions \(\epsilon ^A(x)\) such that

$$\displaystyle \begin{aligned} \epsilon^A(x)F^i_A[\varphi,x](x)=0\;. {} \end{aligned} $$
(6.17)

This relation between symmetry transformations implies an analogous dependence between the zero modes of the inverse propagator of the quantum theory. Namely, trading the coordinate inside \(\epsilon ^A(x)\) for a derivative with respect to the momentum variable \(p^\mu \), denoted as \(\nabla _\mu \), a combination of (6.16) and (6.17) gives

$$\displaystyle \begin{aligned} \epsilon^A(-\mathrm{i}\nabla)\int\mathrm{d}^D\!y\,\mathrm{e}^{\mathrm{i} p\cdot y}F^j_A[\varphi,y](y)=0\;. \end{aligned} $$
(6.18)

The conclusion is that the number of independent NG fields equals the dimension of the symmetry group G minus the number of linearly independent solutions of (6.17). Note that the linear independence is meant in the functional sense: multiplying all the \(\epsilon ^A(x)\) by the same function \(f(x)\) does not count as a new solution.

The rule for counting the number of independent NG fields does not have an established form in the literature. My formulation is close in spirit to [9] where, however, the functions \(\epsilon ^A(x)\) were only allowed to depend on coordinates corresponding to unbroken continuous translations. That seems too restrictive for instance in case of crystalline solids where only a discrete translation invariance survives in the ground state.

Another condition for redundancy similar to (6.17) was put forward in [10]; see also the review [11]. Their criterion is however phrased in terms of the vacuum ket-vector and charge density operators. As such, it also includes the redundancy of NG states under canonical conjugation, which I treat separately in Sect. 6.3.2. Finally, in [12] a rule counting independent NG fields is formulated directly in terms of the zero modes of the inverse propagator.

In case of broken spacetime symmetries, the number of independent would-be NG fields identified with the help of (6.17) turns out to depend on the precise choice of order parameter(s). Remarkably, some of these fields may couple to gapped states in the spectrum. This subtlety is missed by the classical counting rule for NG fields based on (6.17). It is best addressed having the full EFT machinery at hand. I will do so in Chap. 13.

As a basic sanity check, consider spontaneously broken internal symmetries in systems with unbroken translation invariance. In that case, \(F^i_A[\varphi ,x]\) does not depend on coordinates, either explicitly or implicitly through the order parameter. Also, it is an ordinary local function of \(\varphi ^i\), hence the condition (6.17) reduces to \(\epsilon ^A(x)F^i_A(\varphi )=0\). The only nontrivial solutions for \(\epsilon ^A(x)\) are then those corresponding to generators of the unbroken subgroup H. The number of independent NG fields therefore equals \(\dim G-\dim H\). For spontaneously broken internal symmetries, there is a one-to-one correspondence between NG fields and broken symmetry generators. This is one of the moral lessons we reached in Chap. 3 by analyzing a set of specific models.

Oftentimes, the solutions to (6.17) descend from redundancy among local symmetry transformations acting on an arbitrary field configuration. This was certainly the case for Example 6.1 and Example 6.2. The condition (6.17) is however much weaker, as it only demands redundancy of local symmetry transformations of the ground state. Here is a nontrivial example of this latter type.

Example 6.5

Helimagnets are magnetic materials in which the alignment axis of spins gradually changes with position; see [13] for basic phenomenology. The simplest type of helimagnetic state can be modeled by the unit-vector order parameter

$$\displaystyle \begin{aligned} \langle{\boldsymbol n(\boldsymbol x,t)}\rangle =(\cos kz,\sin kz,0)\;, {} \end{aligned} $$
(6.19)

where k is a positive constant. The local axis of alignment lies in the xy plane but rotates around the z axis, forming a helix with the pitch \(2\pi /k\).

Suppose for simplicity that the material in which the helical order develops has a cubic crystal lattice. This is the case for instance for the inorganic compounds MnSi and FeGe; see Sect. 2 of [14] for a more comprehensive list of known helimagnetic materials. Then the low-energy physics of the material exhibits the following approximate continuous symmetries (in addition to time translations):

  • Spatial translations (T): \(\boldsymbol n'(\boldsymbol x',t)=\boldsymbol n(\boldsymbol x,t)\) with \(\boldsymbol x'=\boldsymbol x+\boldsymbol \epsilon \), \(\boldsymbol \epsilon \in \mathbb {R}^3\).

  • Space–spin rotations (R): \(\boldsymbol n'(\boldsymbol x',t)=R\boldsymbol n(\boldsymbol x,t)\) with \(\boldsymbol x'=R\boldsymbol x\), \(R\in \mathrm {SO}(3)\).

Unlike in ordinary (anti)ferromagnets, spin rotations cannot be considered separately from spatial rotations as a consequence of the spin-orbit coupling. This induces the so-called Dzyaloshinskii–Moriya interaction, which is eventually responsible for the development of the helical order (6.19).

In their evolutionary form, the above transformations correspond respectively to the functions and . Denoting the corresponding infinitesimal parameters as \(\epsilon ^r(\boldsymbol x,t)\) (translation in the r-th direction) and \(\omega ^r(\boldsymbol x,t)\) (rotation around the r-th axis), (6.17) turns into

(6.20)

This boils down to the following constraints on the parameter functions,

$$\displaystyle \begin{aligned} \begin{aligned} \omega^1(\boldsymbol x,t)\sin kz&=\omega^2(\boldsymbol x,t)\cos kz\;,\\ \omega^3(\boldsymbol x,t)&=k[\epsilon^3(\boldsymbol x,t)+y\omega^1(\boldsymbol x,t)-x\omega^2(\boldsymbol x,t)]\;. \end{aligned} {} \end{aligned} $$
(6.21)

Some of the solutions of (6.21) are trivial. First of all, \(\epsilon ^1\) and \(\epsilon ^2\) can be chosen arbitrarily since they do not appear in (6.21) at all. These represent unbroken translations in the \(x,y\) directions. Furthermore, there is another unbroken symmetry, corresponding to simultaneous translation in the z-direction and rotation around the z-axis. This extends to the solution of (6.21) with \(\omega ^3(\boldsymbol x,t)=k\epsilon ^3(\boldsymbol x,t)\) and \(\omega ^{1,2}(\boldsymbol x,t)=0\).

In addition to the three trivial solutions that identify unbroken symmetries, there is one solution of (6.21) that represents a genuine redundancy of order parameter fluctuations. This can be thought of for instance as picking \(\omega ^1\) arbitrarily and adjusting \(\omega ^2\) and \(\omega ^3\) so that (6.21) is satisfied. Altogether, this makes four independent solutions of (6.17) in case of the order parameter (6.19). With six different symmetry generators, three translations and three rotations, this leaves us with mere two independent NG fields. That is a result we could have guessed: the unit vector field \(\boldsymbol n(\boldsymbol x,t)\) contains exactly two degrees of freedom.

This example demonstrates that while possibly tedious, finding the solutions to (6.17) as a set of linear equations for \(\epsilon ^A(x)\) is completely straightforward. Even though (6.17) does not give an explicit expression for the number of independent NG fields, it therefore offers a streamlined algorithmic procedure how to find them.

3.2 Type-A and Type-B Nambu–Goldstone Bosons

Now we finally get to the question how many NG bosons there are in the spectrum. I will start with the simpler case of spontaneously broken internal symmetry in systems with unbroken translation invariance. In this case, there are guaranteed to be \(\dim G-\dim H\) NG fields, in a one-to-one correspondence to broken symmetry generators. Finding the number of NG modes requires just a mild generalization of the argument in Sect. 6.1.2.

As the first step, one introduces the commutator matrix

$$\displaystyle \begin{aligned} \varrho_{AB}\equiv\mathrm{i}\langle{[Q_A,J^0_B]}\rangle \;. {} \end{aligned} $$
(6.22)

Translation invariance allows to rewrite this as \(\varrho _{AB}=\mathrm{i} \lim _{V\to \infty }\langle {[Q_A,Q_B]}\rangle /V\). This makes it clear that \(\varrho _{AB}=0\) whenever \(Q_A\) or \(Q_B\) is unbroken. The nontrivial part of the commutator matrix therefore resides in its restriction to broken generators \(Q_{a,b,\dotsc }\), that is \(\varrho _{ab}\). Moreover, \(\varrho _{AB}\) is real antisymmetric, and as such can be block-diagonalized by an orthogonal change of basis of symmetry generators. With the shorthand notation \(r_\varrho \equiv \operatorname {\mathrm {rank}}\varrho \), this implies that the part of the effective Lagrangian for NG fields \(\pi ^a\), linear in time derivatives, takes the form

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\varrho_{12}\pi^1\partial_0\pi^2+\dotsb+\varrho_{r_\varrho-1,r_\varrho}\pi^{r_\varrho-1}\partial_0\pi^{r_\varrho}+\dotsb\;. {} \end{aligned} $$
(6.23)

The ellipsis indicates operators with more than one derivative or two NG fields.

This allows a unique categorization of NG fields. The \(r_\varrho \) NG fields \(\pi ^a\) with \(a=1,\dotsc ,r_\varrho \) are canonically conjugated into \(r_\varrho /2\) pairs. Each such pair corresponds to one type-B NG boson. Barring accidental cancellations in the spatial gradient part of the effective Lagrangian, the dispersion relation of type-B NG bosons is typically quadratic in momentum. The remaining NG fields \(\pi ^a\) with \(a=r_\varrho +1,\dotsc ,\dim G-\dim H\) do not appear in any bilinear operator with a single time derivative. They correspond to one type-A NG boson each. As a rule of thumb, free type-A bosons are described by bilinear operators in the Lagrangian with two temporal or two spatial derivatives. Hence, their dispersion relation is linear in momentum.

We have arrived at a simple counting rule for the respective numbers of type-A and type-B NG bosons [15, 16],

$$\displaystyle \begin{aligned} n_{\mathrm{A}}=\dim G-\dim H-\operatorname{\mathrm{rank}}\varrho\;,\qquad n_{\mathrm{B}}=\frac{1}{2}\operatorname{\mathrm{rank}}\varrho\;. {} \end{aligned} $$
(6.24)

This characterizes the spectrum of NG bosons based solely on the pattern of symmetry breaking and a bit of additional information about the ground state, encoded in the matrix \(\varrho _{AB}\). The possible values of \(\varrho _{AB}\) are strongly restricted by the unbroken symmetry. To start with, in systems with unbroken Lorentz invariance, terms in the Lagrangian with a single time derivative are forbidden, hence \(\varrho _{AB}=0\). This means that \(n_{\mathrm {A}}=\dim G-\dim H\) and \(n_{\mathrm {B}}=0\). All relativistic NG bosons are massless particles with a strictly linear dispersion relation.

Let us now relax the requirement of Lorentz invariance, but keep the assumption that only internal symmetry is spontaneously broken and that the system possesses continuous translation invariance. If in addition the internal symmetry group G is compact and semisimple, we can write

$$\displaystyle \begin{aligned} \varrho_{AB}=\mathrm{i}\lim_{V\to\infty}\frac{\langle{[Q_A,Q_B]}\rangle }V=-f^C_{AB}\lim_{V\to\infty}\frac{\langle{Q_C}\rangle }V\;, {} \end{aligned} $$
(6.25)

where \(f^C_{AB}\) are the structure constants of G. This shows that the VEV of the conserved charges themselves can serve as an order parameter for SSB. The set of VEVs \(\langle {Q_C}\rangle \) furnishes a vector in the adjoint representation of G. At the same time, it must remain unchanged under the action of H. As a consequence, the vector \(\langle {Q_C}\rangle \) must correspond to a trivial one-dimensional representation (singlet) of H. Finding such singlets in the decomposition of the Lie algebra \(\mathfrak {g}\) of G into irreducible multiplets of H is a quick way to isolate candidate order parameters.

Focusing on singlets of H in the decomposition of the adjoint representation of G may not be restrictive enough in case H is small or even trivial. We can however do much better than that. Namely, it is always possible to choose the ground state and basis of generators \(Q_A\) such that all the charges that have a nonzero VEV mutually commute. A detailed proof of this statement is given in the appendix of [7]. Thus, to identify conserved charge order parameters, it is sufficient to restrict to a Cartan subalgebra of \(\mathfrak {g}\) and find singlets of H therein. By the usual root decomposition of Lie algebras, all the generators of G are organized into pairs whose commutator lies in the Cartan subalgebra of \(\mathfrak {g}\). This maps type-B NG bosons uniquely to pairs of mutually conjugate roots of \(\mathfrak {g}\).

Example 6.6

Consider the symmetry-breaking pattern \(\mathrm {SU}(n)\to \mathrm {U}(n-1)\) with integer \(n\geq 2\). The unbroken subgroup \(H\simeq \mathrm {U}(n-1)\) can be realized explicitly for instance as the set of unitary block-diagonal matrices with blocks of sizes 1 and \(n-1\), respectively. The Lie algebra \(\mathfrak {g}\simeq \mathfrak {su}(n)\) contains a unique singlet of H, namely the generator of the \(\mathrm {U}(1)\) factor of H, \(Q\equiv \operatorname {\mathrm {diag}}(-n+1,1,\dotsc ,1)\).

The \(2n-2\) broken generators can be chosen as matrices with nonzero elements located in the first row and column. These span a complex \((n-1)\)-plet of \(\mathrm {SU}(n-1)\subset H\). Since all the broken generators reside in a single multiplet of H, we expect all the corresponding NG bosons to be either type-A or type-B. Which of these two options is realized depends on the VEV \(\langle {Q}\rangle \). If \(\langle {Q}\rangle =0\), we find that \(n_{\mathrm {A}}=2n-2\) and \(n_{\mathrm {B}}=0\). This happens in Lorentz-invariant theories, but also for instance in antiferromagnets where \(n=2\). If, on the other hand, \(\langle {Q}\rangle \neq 0\) then the broken generators contribute pairwise to \(\varrho _{AB}\). As a result, \( \operatorname {\mathrm {rank}}\varrho =2n-2\). According to (6.24), we then find \(n_{\mathrm {A}}=0\) and \(n_{\mathrm {B}}=n-1\). The special case of \(n=2\) corresponds to ferromagnets.

Example 6.7

For an example of a system where a type-B NG boson arises from the VEV of a broken symmetry generator, consider the symmetry-breaking pattern \(\mathrm {SU}(2)\to \{e\}\). Here the unbroken subgroup is trivial and any generator may develop a nonzero VEV. However, the rank of antisymmetric matrices is always even. The only possibilities are therefore \( \operatorname {\mathrm {rank}}\varrho =0\) and \( \operatorname {\mathrm {rank}}\varrho =2\). In the latter case, we find \(n_{\mathrm {A}}=n_{\mathrm {B}}=1\). This is realized in so-called canted (anti)ferromagnets, schematically shown in Fig. 6.2.

Fig. 6.2
figure 2

Schematic illustration of spin order in canted (anti)ferromagnets. In ferromagnets, the individual spins are slightly tilted away from perfect alignment. Antiferromagnets, on the other hand, exhibit overall antialignment of spins, tilted so that a nonzero net magnetization arises. In both cases, the whole \(\mathrm {SU}(2)\) group of spin rotations is spontaneously broken

Full size image

So far I have assumed that the internal symmetry group G is compact and semisimple so that (6.25) holds. When this assumption is not satisfied, the commutator matrix (6.22) may receive contributions from central charges of the Lie algebra \(\mathfrak {g}\). This is the case for instance for the free Schrödinger field. Without going into detail, let me just mention that such central extensions are classified by the so-called second Lie algebra cohomology of \(\mathfrak {g}\). See Chap. 6 of [17] for further information.

How does the counting of NG bosons change for symmetries that are not internal, such as spacetime symmetries? In this case, there are no established general results in the literature. However, the simplicity of the argument given in Sect. 6.1.2 suggests that the reduction of independent NG modes based on the charge commutator (6.22) is robust. Suppose that we have already identified a set of independent NG fields \(\pi ^a\). Let us further assume that we can map these to a set of broken generators \(Q_a\) that act on the fields by a shift such as (6.1). We can restrict the definition of the commutator matrix to the generators \(Q_a\). The VEV in (6.22) may now in principle be coordinate-dependent, and it may no longer be appropriate to replace \(J^0_B\) with the spatial average of \(Q_B\). However, the rest of the argument will still go through. We will then get the same identification of type-B NG bosons with pairs of broken generators whose commutator has a nonzero VEV. Nontrivial examples of systems where the counting rule (6.24) for type-B NG bosons was still found to be valid include the centrally extended algebra of spatial translations in an external magnetic field [15], and central extensions of commutators of spatial and internal symmetries in presence of topological defects such as a domain wall [18] or a line defect [19].

4 Nambu–Goldstone-Like Modes

The defining property of a NG boson is its relation to symmetry and its spontaneous breaking. In the operator proof of Goldstone’s theorem (Sect. 6.2.1), this manifests through the coupling of the NG state to the broken symmetry current, \(\left \langle {0}\right \rvert J^0(0)\left \lvert {n,\boldsymbol p}\right \rangle \). In the effective action proof (Sect. 6.2.2), the NG boson appears through a flat direction of the effective potential or action. Many physical systems possess excitations with the above properties that however lack the main attribute of a NG boson, that is being exactly gapless. Due to their similarity to genuine NG modes, I will call such excitations NG-like bosons.

The existence of SSB-related modes that are not exactly gapless is of course only possible if some of the assumptions of Goldstone’s theorem are violated. The usual suspect is the symmetry. There are very few exact symmetries in nature; such usually only exist in our models, valid to certain, possibly high, accuracy. In this last section of the chapter, I will give a brief survey of different mechanisms how a NG-like boson may arise from a symmetry that is only approximate.

4.1 Pseudo-Nambu–Goldstone Bosons

Consider a theory whose Lagrangian can be split into a part invariant under a group G, and a small perturbation invariant only under some proper subgroup of G,

$$\displaystyle \begin{aligned} \mathcal{L}=\mathcal{L}_{\mathrm{inv}}+\epsilon\mathcal{L}_{\mathrm{pert}}\;. \end{aligned} $$
(6.26)

This is a setup that I used previously to isolate a unique ground state. The difference is that now I want to keep the parameter \(\epsilon \) small but nonzero. This is called explicit symmetry breaking.

Suppose that in the limit \(\epsilon \to 0\), the symmetry under G is broken spontaneously. It is sensible to expect that by restoring small but nonzero \(\epsilon \), the spectrum will evolve adiabatically. The NG mode predicted by Goldstone’s theorem will then survive, but its mass (gap) will no longer be exactly zero. Such “approximate” NG bosons are usually called pseudo-NG bosons.

Let us try to make an educated guess at how the gap of a pseudo-NG boson depends parametrically on \(\epsilon \). Imagine that we have already derived a low-energy EFT for the (pseudo-)NG bosons in our system. The contributions of the perturbation \(\mathcal {L}_{\mathrm {pert}}\) to the effective Lagrangian can be organized by powers of \(\epsilon \). The leading contribution to the bilinear part of the effective Lagrangian typically comes from a nonderivative operator linear in \(\epsilon \). I will justify this claim more carefully in Chap. 8, where I actually construct the effective Lagrangian.

The rest of the argument depends on the type of the NG boson in the limit of vanishing perturbation. This type is still defined by the presence or absence of a term with a single time derivative, containing the given NG field. For type-A NG bosons, the bilinear Lagrangian contains two time derivatives which translate to squared energy in the Fourier space. For type-B NG bosons, on the other hand, a term with a single time derivative is present and dominates in the low-energy limit. This leads to the following parametric dependence of energy at zero momentum on \(\epsilon \),

$$\displaystyle \begin{aligned} \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}E(\boldsymbol p)&\propto\sqrt{\epsilon}\quad &&\text{(``type-A'' pseudo-NG boson)}\;,\\ \lim_{\boldsymbol p\to\mathbf{0}}E(\boldsymbol p)&\propto\epsilon\quad &&\text{(``type-B'' pseudo-NG boson)}\;. \end{aligned} {} \end{aligned} $$
(6.27)

I have just used quotation marks for a reason. There is no established, unambiguous classification of pseudo-NG bosons. All I did was to rely on the continuity of the spectrum in the limit \(\epsilon \to 0\). Moreover, the parametric scaling in (6.27) is based on the assumption of a rather specific form of perturbation in the effective Lagrangian. It is imaginable that even for an originally type-A NG boson, \(\mathcal {L}_{\mathrm {pert}}\) will induce an operator in the effective Lagrangian with a single time derivative and a coefficient proportional to some power of \(\epsilon \). That may change the way the gap of the pseudo-NG mode scales with \(\epsilon \). This is exactly what happens for the “massive” NG bosons, introduced in Sect. 6.4.3.

It is a simple application of the discussion in Sect. 6.3.2 to find how many pseudo-NG bosons one should expect in a given system. I will denote as \(G_\epsilon \) and \(H_\epsilon \) the symmetries of the action and of the ground state in presence of the perturbation. In analogy with (6.22), I will define the commutator matrix \(\varrho _\epsilon \) by

$$\displaystyle \begin{aligned} \varrho_{\epsilon,AB}\equiv\mathrm{i}\langle{[Q_A,J^0_B]}\rangle _\epsilon\;. \end{aligned} $$
(6.28)

The \(\epsilon \) in the subscripts reminds us that the indices \(A,B\) run over the generators of \(G_\epsilon \), and that the VEV is taken in the perturbed vacuum. According to (6.24), there are altogether \(\dim G-\dim H-(1/2) \operatorname {\mathrm {rank}}\varrho \) NG modes in the limit \(\epsilon \to 0\). By the same formula, \(\dim G_\epsilon -\dim H_\epsilon -(1/2) \operatorname {\mathrm {rank}}\varrho _\epsilon \) genuine NG bosons survive upon switching on the perturbation. On the assumption of continuity of the spectrum, the number of pseudo-NG bosons therefore equals

$$\displaystyle \begin{aligned} n_{\mathrm{pseudoNG}}=(\dim G-\dim G_\epsilon)-(\dim H-\dim H_\epsilon)-\frac{1}{2}(\operatorname{\mathrm{rank}}\varrho-\operatorname{\mathrm{rank}}\varrho_\epsilon)\;. {} \end{aligned} $$
(6.29)

We saw an example of a pseudo-NG boson already in Chap. 2. In that case, \(G\simeq \mathrm {U}(1)_{\mathrm {V}}\times \mathrm {U}(1)_{\mathrm {A}}\) and \(H\simeq \mathrm {U}(1)_{\mathrm {V}}\), whereas \(G_\epsilon \simeq H_\epsilon \simeq \mathrm {U}(1)_{\mathrm {V}}\). The counting rule (6.29) is trivially satisfied and the prediction (6.27) for the scaling of the gap agrees with our explicit calculation of the mass, cf. (2.12). Next, let us have a look at a couple of less trivial examples.

Example 6.8

We already met quantum chromodynamics (QCD) in Chap. 5, see Example 5.5. In this case, the quark mass plays the role of the perturbation \(\epsilon \). With \(n_{\mathrm {f}}\) flavors of massless quarks, QCD possesses a \(G\simeq \mathrm {SU}(n_{\mathrm {f}})_{\mathrm {L}}\times \mathrm {SU}(n_{\mathrm {f}})_{\mathrm {R}}\) symmetry, spontaneously broken down to the “vector” subgroup \(H\simeq \mathrm {SU}(n_{\mathrm {f}})_{\mathrm {V}}\). In presence of the perturbation (with equal masses of all quarks), this reduces to \(G_\epsilon \simeq H_\epsilon \simeq \mathrm {SU}(n_{\mathrm {f}})_{\mathrm {V}}\). The vacuum of QCD is Lorentz-invariant, so the matrices \(\varrho \) and \(\varrho _\epsilon \) are both identically zero. According to the counting rule (6.29), this leaves us with \(n_{\mathrm {f}}^2-1\) pseudo-NG bosons, which can be identified with a multiplet of pseudoscalar mesons. The fact that their mass scales with the square root of the quark mass, as predicted by (6.27), has been verified in numerical lattice simulations of QCD.

Example 6.9

As discussed before, ideal (isotropic) antiferromagnets possess a \(G\simeq \mathrm {SU}(2)\) spin symmetry, spontaneously broken down to \(H\simeq \mathrm {U}(1)\). The low-energy EFT for antiferromagnetic spin waves (magnons) is a nonrelativistic version of the nonlinear sigma model, introduced in Example 3.3,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac{\varrho_{\mathrm{s}}}{2v^2}\big(\delta_{ij}\partial_0n^i\partial_0n^j-v^2\delta_{ij}\boldsymbol\nabla n^i\cdot\boldsymbol\nabla n^j\big)+\epsilon(n^3)^2\;. \end{aligned} $$
(6.30)

Here \(\varrho _{\mathrm {s}}\) is the so-called spin stiffness, \(\boldsymbol n\) a unit-vector variable encoding the two NG fields, and v the magnon phase velocity in the symmetric limit \(\epsilon =0\).

For \(\epsilon >0\), the perturbation proportional to \((n^3)^2\) describes a crystal anisotropy, corresponding to “easy-axis” antiferromagnets; see [20] for a mild introduction to antiferromagnetic magnons. The perturbation breaks the continuous spin symmetry explicitly down to \(G_\epsilon \simeq \mathrm {U}(1)\), consisting of rotations around the third spin axis. At the same time, it leaves only two possible ground states, \(\langle {\boldsymbol n}\rangle _\epsilon =(0,0,\pm 1)\), both of which preserve \(H_\epsilon \simeq \mathrm {U}(1)\). By the counting formula (6.29), both magnons become pseudo-NG bosons. Their gap is proportional to \(\sqrt {\epsilon }\) in accord with (6.27).

4.2 Quasi-Nambu–Goldstone Bosons

The above general discussion of pseudo-NG bosons applies in principle to any NG-like mode, associated with an approximate symmetry. However, the strength of the perturbation, and hence the size of the gap in the spectrum, may depend on the concrete mechanism of explicit symmetry breaking. It is in particular possible to have systems with a well-defined classical limit whose symmetry is explicitly broken only by quantum corrections. The ensuing pseudo-NG modes have been dubbed quasi-NG bosons [21]. I refer the reader to [22, 23] for a detailed discussion of the spectrum of quasi-NG modes.

To be more concrete, suppose we are given a Lorentz-invariant Lagrangian whose potential has a higher symmetry than the kinetic term. At the classical level, the spectrum of NG bosons is determined by the zero modes of the Hessian of the potential in the ground state. But some of these zero modes may correspond to symmetries of the potential that do not preserve the kinetic term. The corresponding fluctuations of the order parameter will receive a mass generated by quantum corrections.

The lifting of the mass of the would-be NG modes does not have to be induced by the kinetic term. What matters is that the classical ground state is determined entirely by a part of the Lagrangian that has a higher symmetry than the whole. The reduction of the symmetry may also be caused for instance by other contributions to the classical potential, or by a coupling to gauge fields.

4.3 Massive Nambu–Goldstone Bosons

What the pseudo-NG bosons and their subclass, quasi-NG bosons, have in common is that their gap can rarely be determined from first principles. In general, it depends on all parameters a given theory might have. One says that the gap is not protected by symmetry, in contrast to genuine NG bosons whose gap is guaranteed to vanish by SSB. There is however a remarkable special class of pseudo-NG bosons whose gap can be calculated exactly using symmetry alone. Their existence was first pointed out about a decade ago [24] and I will call them massive NG bosons following [25].

Consider a system with a “microscopic” Hamiltonian H, possessing an internal symmetry group G. Pick one generator, Q, of G. In statistical physics, the many-body ground state of the system with a fixed average value of Q is determined by minimizing the grandcanonical Hamiltonian \(H_\mu \equiv H-\mu Q\). Here \(\mu \) is the corresponding chemical potential, which is fixed by external conditions imposed on the system. If needed, one can always shift \(H_\mu \) by a constant so that the many-body ground state \(\left \lvert {0}\right \rangle _\mu \) satisfies \(H_\mu \left \lvert {0}\right \rangle _\mu =0\). In this ground state, the symmetry under G may be spontaneously broken. We are particularly interested in the fate of those generators of G that are spontaneously broken but do not commute with Q.

As long as G is compact, we can always choose Q to be part of a Cartan subalgebra of the Lie algebra \(\mathfrak {g}\) of G. Those generators of G that do not commute with Q then organize themselves into pairs \(Q_\pm \) such that \(Q_-={Q}^{\dagger }_+\) and

$$\displaystyle \begin{aligned} {[}Q,Q_\pm{]}=\pm qQ_\pm\;. {} \end{aligned} $$
(6.31)

The factor q is determined by the corresponding root vector, and is thus fixed by symmetry. We can always choose the generators so that both \(\mu \) and q are positive. I will now rerun the operator proof of Goldstone’s theorem detailed in Sect. 6.2.1 on the VEV \({ }^{}{\mu }{\left \langle {0}\right \rvert }[Q_{+\Omega }(t),J^0_-(0)]\left \lvert {0}\right \rangle _\mu \).

Of course, (6.31) is only formal since we know that generators of spontaneously broken symmetry are ill-defined as operators on the Hilbert space. I therefore assume that (6.31) also holds for the finite-volume charges \(Q_{\pm \Omega }\), possibly up to a correction that vanishes in the limit \(\Omega \to \infty \). This can be ensured for instance if the charge densities satisfy the local commutation relation \([Q,J^0_\pm (x)]=\pm qJ^0_\pm (x)\). One can then define a related pair of charges via “time evolution” with respect to \(H_\mu \),

$$\displaystyle \begin{aligned} \begin{aligned} {}^{\mu}{}Q_{\pm\Omega}(t)&\equiv\int_\Omega\mathrm{d}^d\!\boldsymbol x\,\exp(\mathrm{i} H_\mu t-\mathrm{i}\boldsymbol{P}\cdot\boldsymbol{x})J^0_\pm(0)\exp(-\mathrm{i} H_\mu t+\mathrm{i}\boldsymbol{P}\cdot\boldsymbol{x})\\ &=\mathrm{e}^{-\mathrm{i}\mu Qt}Q_{\pm\Omega}(t)\mathrm{e}^{+\mathrm{i}\mu Qt}=\mathrm{e}^{\mp\mathrm{i}\mu qt}Q_{\pm\Omega}(t)\;. \end{aligned} \end{aligned} $$
(6.32)

The point of this redefinition is that the spectrum of the many-body system consists of eigenstates of \(H_\mu \) rather than H. I will keep the notation \(E_n(\boldsymbol p)\) for the eigenvalue of \(H_\mu \) (excitation energy) in the eigenstate \(\left \lvert {n,\boldsymbol p}\right \rangle \). Following the same steps as in Sect. 6.2.1 then leads to an analog of (6.11),

(6.33)

With positive \(\mu \) and q, the assumed time independence of VEVs of commutators of \(Q_{+\Omega }(t)\) in the \(\Omega \to \infty \) limit guarantees that \(\lim _{\boldsymbol p\to \mathbf {0}}{ }^{}{\mu }{\left \langle {0}\right \rvert }J^0_-(0)\left \lvert {n,\boldsymbol p}\right \rangle =0\) for any eigenstate \(\left \lvert {n,\boldsymbol p}\right \rangle \). Assuming that the VEV we started with is nonzero, on the other hand, ensures the existence of a state for which \({ }^{}{\mu }{\left \langle {0}\right \rvert }J^0_+(0)\left \lvert {n,\boldsymbol p}\right \rangle \neq 0\) and

$$\displaystyle \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}E_n(\boldsymbol p)=\mu q\;. {} \end{aligned} $$
(6.34)

This is our massive NG boson. With some extra effort, one can also derive a counting rule for the number of massive NG modes in a system. To formulate such a rule, we adapt (6.22) for the present problem by introducing and . The matrix \(\tilde \varrho \) is constructed using the generators of the subgroup of G that commutes with Q. The number of massive NG modes then is

$$\displaystyle \begin{aligned} n_{\mathrm{massiveNG}}=\frac{1}{2}(\operatorname{\mathrm{rank}}\varrho_\mu-\operatorname{\mathrm{rank}}\tilde\varrho_\mu)\;. {} \end{aligned} $$
(6.35)

See [25] for a detailed proof.

The counting rule (6.35) may resemble the last term in (6.29) that counts pseudo-NG bosons. It may thus be worthwhile to stress the difference. Namely, the matrices \(\varrho \) and \(\varrho _\epsilon \) in (6.29) are evaluated in different ground states, corresponding respectively to the absence and presence of the perturbation. On the contrary, both matrices \(\varrho _\mu \) and \(\tilde \varrho _\mu \) in (6.35) are evaluated in the same ground state, “perturbed” by the presence of the chemical potential. The difference is best illustrated on an example.

Example 6.10

Let us have one more look at antiferromagnets, this time assuming perfect isotropy and spin symmetry, that is \(G\simeq \mathrm {SU}(2)\) and \(H\simeq \mathrm {U}(1)\). In an external magnetic field \(\boldsymbol B\), the Hamiltonian of the antiferromagnet receives a contribution \(-\mu _{\mathrm {m}}\boldsymbol {B}\cdot \boldsymbol {S}\), where \(\mu _{\mathrm {m}}\) is the magnetic moment and \(\boldsymbol S\) the operator of total spin. The combination \(\mu _{\mathrm {m}}\left \lvert {\boldsymbol B}\right \rvert \) plays the role of a chemical potential for the projection of \(\boldsymbol S\) into the direction of \(\boldsymbol B\). The latter can without loss of generality be chosen as, say, \(S_3\).

Let us first treat the effect of the magnetic field as a perturbation in the sense of Sect. 6.4.1. The field breaks the spin symmetry explicitly down to \(G_\epsilon \simeq \mathrm {U}(1)\). This necessarily implies \( \operatorname {\mathrm {rank}}\varrho _\epsilon =0\). Somewhat against the intuition, the antiferromagnet responds to the magnetic field by aligning its spins largely in a direction perpendicular to \(\boldsymbol B\). This ground state breaks the residual exact symmetry so that \(H_\epsilon \simeq \{e\}\). At the same time, in the unperturbed antiferromagnetic ground state, there is no net magnetization so that \( \operatorname {\mathrm {rank}}\varrho =0\). Equations (6.22) and (6.29) then tell us that there is one genuine, type-A NG boson and one pseudo-NG boson. Unlike in the case of an easy-axis anisotropy, one of the two magnons therefore remains gapless.

The fact that the magnetic field acts as a chemical potential allows us to make a stronger statement about the spectrum. The magnetic field does induce net magnetization in the perturbed ground state, given by nonzero \(\langle {S_3}\rangle _\mu =-\mathrm{i} \langle {[S_1,S_2]}\rangle _\mu \). This implies \( \operatorname {\mathrm {rank}}\varrho _\mu =2\). At the same time, \( \operatorname {\mathrm {rank}}\tilde \varrho _\mu =0\) simply because there are not enough symmetry generators left intact by the magnetic field. In accord with (6.35), we then expect one massive NG mode in the spectrum. In other words, the previously predicted pseudo-NG magnon has a gap exactly fixed by (6.34). The algebraic factor q equals one thanks to the commutator of \(S_3\) with the ladder operators \(S_\pm \equiv S_1\pm \mathrm{i} S_2\), \([S_3,S_\pm ]=\pm S_\pm \). Hence the gap of the magnon equals \(\mu _{\mathrm {m}}\left \lvert {\boldsymbol B}\right \rvert \).

An interested reader will find further examples of massive NG bosons in [25]. Let me just stress that the exact expression (6.34) for the gap is not the only special property of massive NG bosons. The peculiar perturbation by a chemical potential coupled to a conserved charge leaves us with a set of exact, albeit modified, conservation laws. These in turn impose exact constraints on the interactions of massive NG bosons. As a result, the Adler zero principle (cf. Sect. 2.2) remains valid for massive NG modes, although it is generally violated for pseudo-NG bosons [26].

As an aside, note that not every NG boson that acquires a gap due to a chemical potential is a massive NG boson [27]. The simplest example of such an exception is provided by choosing \(G\simeq \mathrm {SU}(2)\) and breaking it completely in the ground state. Provided none of the three generators develops a VEV, the spectrum contains three type-A NG bosons. Upon adding a chemical potential \(\mu \) for one of the generators, two of the NG modes receive a gap, proportional to \(\mu \). Only one of these is however a massive NG boson, as is easily checked using (6.35).