Are there Goldstone bosons in ${d\leq z+1}$ ?

Understand

We study the viability of spontaneous breaking of continuous symmetries in theories with Lifshitz scaling, according to the number of space-time dimensions $d$ and the dynamical scaling $z$.

• Then, the answer to the question in the title is no (quantum field theoretically) and yes (holographically).
• With field theory tools, we show that symmetry breaking is indeed prevented by large quantum fluctuations when $d\leq z+1$, as expected from scaling arguments.
• With holographic tools, on the other hand, we find nothing that prevents the existence of a vacuum expectation value.