Abstract
The Ward identities associated with spontaneously broken symmetries can be saturated by Goldstone bosons. However, when space-time symmetries are broken, the number of Goldstone bosons necessary to non-linearly realize the symmetry can be less than the number of broken generators. The loss of Goldstones may be due to a redundancy or the generation of a gap. In either case the associated Goldstone may be removed from the spectrum. This phenomena is called an Inverse Higgs Mechanism (IHM) and its appearance has a well defined mathematical condition. However, there are cases when a Goldstone boson associated with a broken generator does not appear in the low energy theory despite the lack of the existence of an associated IHM. In this paper we will show that in such cases the relevant broken symmetry can be realized, without the aid of an associated Goldstone, if there exists a proper set of operator constraints, which we call a Dynamical Inverse Higgs Mechanism (DIHM). We consider the spontaneous breaking of boosts, rotations and conformal transformations in the context of Fermi liquids, finding three possible paths to symmetry realization: pure Goldstones, no Goldstones and DIHM, or some mixture thereof. We show that in the two dimensional degenerate electron system the DIHM route is the only consistent way to realize spontaneously broken boosts and dilatations, while in three dimensions these symmetries could just as well be realized via the inclusion of non-derivatively coupled Goldstone bosons. We present the action, including the leading order non-linearities, for the rotational Goldstone (angulon), and discuss the constraint associated with the possible DIHM that would need to be imposed to remove it from the spectrum. Finally we discuss the conditions under which Goldstone bosons are non-derivatively coupled, a necessary condition for the existence of a Dynamical Inverse Higgs Constraint (DIHC), generalizing the results for Vishwanath and Wantanabe.
Highlights
Do not play a role at leading order.1 if there are no other gapless modes, or if the Goldstones couple to sources, they are of primary importance
We show that in the two dimensional degenerate electron system the Dynamical Inverse Higgs Mechanism (DIHM) route is the only consistent way to realize spontaneously broken boosts and dilatations, while in three dimensions these symmetries could just as well be realized via the inclusion of non-derivatively coupled Goldstone bosons
We discuss the conditions under which Goldstone bosons are non-derivatively coupled, a necessary condition for the existence of a Dynamical Inverse Higgs Constraint (DIHC), generalizing the results for Vishwanath and Wantanabe
Summary
As was pointed out in [10] there are cases for which there is no inverse Higgs constraints and yet the Goldstones still do not appear. As will be discussed below a resolution of the framon puzzle is closely related to the fact that when space-time symmetries are broken, Goldstone bosons need not be derivatively coupled. To explore this possibility we utilize the coset construction which will. We will go a step further and discuss a more general symmetry breaking pattern where both boost invariance and Schrodinger invariance are spontaneously broken, which we call a “type-III framid” In this case one might expect both a framon and a non-relativistic dilaton to arise. This analysis was recently used to prove that degenerate electrons interacting in the unitary limit can not behave like a Fermi liquid [6] in the unbroken phase
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