Symmetry realization via a dynamical inverse Higgs mechanism

Abstract

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.

Spontaneously broken boosts and the Goldstone continuum

“…The usual ideas of non-linearly realizing the broken symmetries through local Goldstone fields simply do not apply. It is not obvious what replaces them though: Rothstein and Shrivastava have studied what it means for the momentum-space Fermi liquid effective action to be consistent with spontaneously broken boosts [17,18], but it is not clear to us what the relationship is between their results and ours. For instance, they find that spontaneously broken boost invariance requires interactions in the IR to take the standard Landau form (back-to-back and forward scattering.)…”

“…Finally, let us mention that here has been a lot of theoretical work on Fermi liquids and many of its properties, including some of the reasons behind the remarkable success of Landau's Fermi liquid theory [15,16] have been fully or partly understood [11,[17][18][19]. Also, even more exotic states of matter that are not contained in the classification of [10] have been found in the past decades, like bad metals [20,21], non-Fermi liquids [19] and anisotropic Fermi liquids [22], to name a few.…”

“…More precisely, we can take O(x) = J j (x), so that 16) where n ≡ Ω|J 0 (x)|Ω is the charge density of |Ω . In that case, our theorem only involves matrix elements and expectation values of conserved currents-T µν and J µ : 17) or, in the language of spectral densities,…”

Counting Rules of Nambu–Goldstone Modes

“…(61, 79,80,81)). More generally, the reduction of independent degrees of freedom for space-time symmetries through this mechanism is called the inverse-Higgs effect (78,82,83,84,85). This, however, fails to explain the situations which there is no unbroken continuous translation symmetries.…”

The spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals

“…This EFT has been worked out by [52] and may be adaptable to the Schrödinger invariant case in a trap. Especially interesting would be systems with a Fermi surface, however such a critical state must necessarily be a non-fermi liquid following the results of [53].…”