• chemical potential
• Goldstone's theorem
• space-time symmetry breaking
• spontaneous symmetry breaking

Spontaneous symmetry breaking is a crucial concept in Quantum Field Theory.
When a system is invariant under the action of some symmetry group but its ground state is not, then we say that the symmetry is spontaneously broken. In a relativistic system, the breaking of a continuous group implies the appearance of gapless excitations, known as Nambu-Goldstone bosons, whose number is the same as that of the broken generators. This is the statement of Goldstone’s theorem, and is of fundamental importance since it allows to describe the low energy spectrum of the theory only by symmetry arguments, regardless of the microscopic structure of the underlying system. Moreover, these gapless excitations still have to keep track of the symmetry, which acts on them non-linearly. This condition dictates a constraint on their dynamics, as in the famous case of the pions, Goldstone bosons associated with the spontaneous breaking of the chiral group in massless QCD. In the chiral Lagrangian used to describe them, indeed, all the interactions are controlled at low energy by a single parameter, namely Fπ.
Usually, when talking about spontaneous symmetry breaking, we deal with internal symmetries of the system. Poincar´e invariance, indeed, is a property of the dynamics of every system. However, this does not have to hold for states. Indeed, it is not difficult to find cases where the Poincar´e symmetry, too, is broken by some state. Take for example a chair. Even though its action is certainly invariant under rotations, it has a definite orientation in space. We have to conclude then that a chair spontaneously breaks space-time invariance.
Spontaneous breaking of space-time symmetries yields a much richer set of possible consequences than the breaking of internal ones. For example, as opposite to what Goldstone’s theorem predicts, in this case it is not true any more that there is a single excitation for each broken generator. Moreover, some of the Goldstone’s boson can even acquire a gap.
A particularly interesting situation occurs when the ground state of a system with an internal symmetry is at finite density for some charge Q. These systems can be described by the modified Hamiltonian H˜ ≡ H − µQ where H is the original Hamiltonian and µ is the chemical potential. The ground state is then found as the eigenstate of H˜ with the lowest eigenvalue. In this work we will focus on the case where Q, and therefore time translations, i.e. H, are spontaneously broken by the ground state emerging by this procedure, together with some other charges Q_i, and see how this implies the spontaneous breaking of boosts.
Here, an adapted version of Goldstone’s theorem applies, stating that when the charge Q is associated to a non-Abelian group, some of the Nambu-Goldstone bosons acquire a gap proportional to the chemical potential µ, while some others remain gapless. More precisely, for each pair of broken charges not commuting with Q, a gapped excitation appears, whose gap is equal to µ times some factors fixed by the algebra. This value is entirely determined by symmetry and thus unaffected by quantum corrections. On the other hand, gapless excitations are associated with charges commuting with Q.
The Abelian case is realized, for example, by superfluids, where the only charge of the U(1) symmetry of the system, corresponding to the conserved particles number, is broken and is given a fixed non-zero expectation value. A realization in Nature is found in the superfluid phase of helium−4, for instance. The non-Abelian case is of physical relevance too and has been used to model some condensed matter systems and QCD at finite isospin in the light quarks limit. In particular, it may be useful to describe kaon condensation, which has been predicted to take place in neutron stars and could therefore influence their dynamics.
Similarly to what happens for standard Goldstone bosons like gapless pions, the fields of gapped Goldstone bosons are required to realized the symmetry in a nonlinear fashion, and this has consequences on their interactions at low energy. For example, we will see how the velocity of a gapped Goldstone boson controls any amplitude containing it as external state. In order to have a clear framework to work on, we will first provide a brief review of spontaneous breaking of internal symmetries, proving Goldstone’s theorem and discussing the coset construction, both powerful tools to describe systems with nonlinearly realized symmetries.
Then we will focus on the setup we are interested in, that is systems with an internal symmetry at finite density for one of the charges Q, where the ground state breaks Q as well as some other charges Q_i, together with time translations and boosts. As we proceed in exposing the theoretical aspects, we will try and provide concrete examples to clarify our statements and have some tangible applications. In particular, we will refer to the case of a fully broken SO(3) symmetry at finite density for one of the charges. This model will be approached both from a UV perspective, using a renormalizable Lagrangian for a real triplet, and in a low energy contest.
We also discuss the ambiguities that arise in power counting: when describing an effective field theory, one has to set a scale Λ, where the interactions become strong and the low energy construction is no longer consistent. When breaking only internal symmetries, we can always find a regime where the Goldstone bosons’ energies are << Λ, since they are massless. In our case, however, we have to deal with another energy scale µ determining the mass of some of the Goldstone modes. It is shown that setting µ << Λ lets us consistently include the gapped Goldstone excitations in the effective theory, as one would expect. This is the limit we will adopt through most of this work. However, cases where µ ∼ Λ are possible and physically relevant, and we will briefly discuss this possibility towards the end of this thesis.