Are there Goldstone bosons in d≤z+1 ?

Abstract

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 ď 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. This difference is made possible by the large N limit of holography. An important subtlety in this last framework is that in order to get a proper description of a conserved current, renormalization of the temporal mode of the bulk vector requires an alternative quantization. We also comment on the implications of turning on temperature.

Topics Not Covered in This Book

“…Due to the discrete nature of the Matsubara sum, the contributions of all terms with .k / = 0 are infrared-finite. An infrared divergence can only arise from the zero Matsubara mode, and will be absent provided .d ≥ 2n+1 or .D ≥ 2n+2 [12] (the last column of Table 15.1). While the critical dimension is the same for type-.…”

Independent Goldstone modes for translations and shift symmetry from a real modulated scalar

“…Apart from exploring the working of higher dimensional gradient Mexican hats and the stability of the corresponding vacua, the increase of the dimensionality of the system has an obvious interest in order to circumvent the no-go theorems for symmetry breaking in low dimensionality [55][56][57]. It is possible that the present models serve as an effective description for some holographic systems, in this case the destabilizing fluctuations which in low-dimensions prevent the ordering are suppressed by large-N effects [58][59][60][61].…”