JHEP10(2017)063
algebra. It is also worth mentioning that, while such a Galilean symmetry often appears
as the small velocity limit of a full Lorentzian relativistic symmetry, it may also appear as
an emergent symmetry at low energy.
In physical systems, the particle number M generally has a non-negative spectrum,
where the only state with M = 0 is the empty state that contains no particles. The case of
hMi = 0 (with zero particle number density ρ) is therefore not physically interesting for the
study of SSB, as we don’t expect any symmetry to be spontaneously broken in such cases.
However, in the interest of mathematical completeness (and since, as we point out later,
the study of the dilaton effective action is useful for analyzing other QFT properties outside
of SSB), we will briefly consider this case. The more physically interesting case, however,
is that for which hMi > 0. In this case, the Galilean boost symmetries are necessarily
spontaneously broken by the particle number current J
i
(x) since h[K
j
, J
i
(x)]i ∝ δ
ij
hρ(x)i 6=
0. In both cases, for the purposes of this work, we will assume that space translations and
space rotation symmetries (as well as the modified time translation generated by
˜
H) are
not broken.
We will consider systems for which the non-relativistic scale symmetry D is sponta-
neously broken, that is, there is some dimensionful operator O such that h[D, O]i ∝ hOi 6=
0. There are then two possible cases depending on whether the operator O is invariant
under the particle number U(1) symmetry.
If the operator O is U(1) invariant (and assuming all other operators with non van-
ishing expectation values are as well), the U(1) symmetry is not spontaneously broken.
Assuming non-vanishing particle number hMi 6= 0,
4
this is a case where only the scale and
Galilean boost symmetries are broken. Such a system is similar to the “framids” discussed
in [15], in the non-relativistic limit. One might expect the appearance of d + 1 NG modes,
corresponding to the broken scale and Galilean boosts. Since it is not clear whether such
a scenario indeed appears in any physical system, we relegate this case to appendix A.
If the operator O is not U(1) invariant, and h[M, O]i 6= 0, the U(1) symmetry is
spontaneously broken as well. Examples of such systems may be found in superfluids,
superconductors and other types of non-relativistic condensates. In these systems, since
both the scale, the boosts and the U(1) symmetries are spontaneously broken, one might
initially expect to find separate NG modes for each of them. However, as explained in
e.g. [12], this is not the case: due to the well-known “Inverse Higgs” mechanism, the com-
mutation relation between K
i
and P
j
in (2.1) imposes a relation between the modes v
i
that
would otherwise correspond to the broken Galilean boosts and the mode θ corresponding
to the broken U(1) symmetry, of the form: v
i
∝ ∂
i
θ. Therefore, in this case, the broken
Galilean boosts play no role when building the possible effective action for the NG modes
— one need only consider the modes corresponding to the broken scale and U(1) symme-
tries (while the effective action is, of course, required to be invariant under all of these
symmetries, including Galilean boosts).
4
Note that the condition hM i 6= 0 does not necessarily imply that the U(1) symmetry is spontaneously
broken (which requires the existence of an operator O such that h[M, O]i 6= 0). If the system is in any
eigenstate of M , for example, then M cannot be spontaneously broken, as pointed out in [14].
– 4 –
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