Schwartz IV - 4 : Spontaneous symmetry breaking

Spontaneous symmetry breaking is one of the most important concepts in quantum field theory. The distinction between spontaneous and explicit symmetry breaking is that, with spontaneous symmetry breaking the Lagrangian is invariant under the symmetry, but the ground state of the theory is not. With explicit symmetry breaking, there was never an exact symmetry to begin with. One usually associates spontaneous symmetry breaking with phase transitions. The amazing thing about spontaneous symmetry breaking is that one can say a tremendous amount about the broken phase with a very general effective field theory whose only input is the symmetry that was broken – no detailed microscopic description is needed. We will see a number of examples in this lecture. You are undoubtedly already familiar with spontaneous symmetry breaking in the context of ferromagnetic materials, like iron. The magnetic moment of such a material can be represented by a field M (x) related to the local direction the spins are pointing. At high temperature, the entropic term in the free energy F = E − TS dominates the energetic one and M (x) points in random directions at each point x. When a ferromagnetic material is cooled below its Curie temperature, TC (TC = 1032K for iron), the energetic contribution to the free energy, which is lower when neighboring atoms are aligned, starts to dominate. As the temperature is lowered, domains with aligned spins start to grow, and long-range order emerges. The typical size of these domains is known as the coherence length, ξ. For T <TC it is helpful to write M (x) = μ Q + σQ (x), where μQ is the expectation value of M in the vacuum (T = 0), and σ Q are the excitations around this minimum. At high temperature, the theory has a rotational symmetry (no direction is distinguished) but at low temperature, this symmetry is spontaneously broken, since μQ points in some particular direction. The field σ Q (x) which encodes excitations around the vacuum encodes spin waves whose quanta are called Goldstone bosons. In this lecture, we will see that spontaneous symmetry breaking has different implications depending on the nature of the symmetry. The simplest symmetries are discrete, like a Z2 symmetry φ(x) → −φ(x). For discrete symmetries, spontaneous symmetry breaking looks a lot like explicit symmetry breaking. On the other hand, if the symmetry is a continuous global symmetry, like φ(x)→ eφ(x) for any constant α ∈R, the breaking of the symmetry automatically implies the existence of long-range correlations and associated massless particles. This is Goldstone’s theorem, and the massless particles are known as Goldstone bosons. If the symmetry is gauged, like φ(x) → eφ(x) with an associated massless gauge field Aμ(x), then in the broken phase the gauge boson will acquire a mass. This is known as the Higgs mechanism. In this lecture, we will consider all of these cases, derive some important results about spontaneously broken theories, and show how to consistently quantize the theories in the broken phase. A number of examples, such as QCD, the electroweak theory, ferromagnets, superconductors and superfluids will be discussed.