Symmetry Realization via a Dynamical Inverse Higgs Mechanism

TL;DR

This paper introduces the concept of a Dynamical Inverse Higgs Mechanism (DIHM), showing how certain broken space-time symmetries can be realized without Goldstone bosons through operator constraints, with applications to Fermi liquids.

Contribution

It proposes the DIHM as a new way to realize broken symmetries without Goldstones and analyzes its role in Fermi liquids, extending the understanding of symmetry realization mechanisms.

Findings

DIHM provides an alternative to Goldstone bosons for symmetry realization.

In 2D Fermi liquids, DIHM is the only consistent realization for broken boosts and dilatations.

The paper presents the action for the rotational Goldstone (angulon) and discusses constraints for DIHM.

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. This phenomena is called an Inverse Higgs Mechanism (IHM). 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 have 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), generalizaing the results for Vishwanath and Wantanabe.