Abstract
Abstract The maximal algebra of symmetries of the free single-particle Schrödinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schrödinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N ) model, a putative bulk dual of the unitary and the ideal Fermi gases is briefly discussed.
Highlights
The quantum many-body problem of a non-relativistic two-component Fermi gas with short-range attractive interactions is a longstanding problem in condensed matter physics
The maximal algebra of symmetries of the free single-particle Schrodinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schrodinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra
We present the general arguments of [34] and demonstrate that, in the large-N limit, the generating functionals of the unitary Fermi gas and of the ideal Fermi gas are related by a Legendre transformation
Summary
The quantum many-body problem of a non-relativistic two-component Fermi gas with short-range attractive interactions is a longstanding problem in condensed matter physics. In this limit the results obtained from the free theory are of direct interest for the theoretically more challenging critical regime at the unitarity point This important observation motivated us to focus in this paper on a collection of free non-relativistic massive fermions in the fundamental representation of Sp (2N ) and to study its symmetries and currents.
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