Covariant effective action for a Galilean invariant quantum Hall system

Abstract

We construct effective field theories for gapped quantum Hall systems coupled to background geometries with local Galilean invariance i.e. Bargmann spacetimes. Along with an electromagnetic field, these backgrounds include the effects of curved Galilean spacetimes, including torsion and a gravitational field, allowing us to study charge, energy, stress and mass currents within a unified framework. A shift symmetry specific to single constituent theories constraints the effective action to couple to an effective background gauge field and spin connection that is solved for by a self-consistent equation, providing a manifestly covariant extension of Hoyos and Son's improvement terms to arbitrary order in m.

Physical stress, mass, and energy for non-relativistic matter

“…While we have worked out some illustrative examples, it would be of interest to use this formalism to extend the analysis of [6,37,49] to spinful fluids, and that of [22] to spinful electrons. One could also investigate non-relativistic scale anomalies for spinful nonrelativistic fields following [50][51][52][53][54].…”

“…Newton-Cartan geometry was originally developed by Cartan to describe Newtonian gravity within a geometric framework similar to that of General Relativity [10,11] (see also [12,13]). Recently, it has been used in the condensed matter literature as the natural setting for Galilean invariant physics, with applications that include cold atoms [14], non-relativistic fluids [6,[15][16][17], the quantum Hall effect [18][19][20][21][22], as well as non-relativistic holographic systems [23][24][25][26][27]. It is well recognized in the literature that it is necessary to couple these systems to torsionful geometries to define the full suite of currents available in a non-relativistic system and to study their linear response [9,23,24,26,28].…”

“…This method of imposing Galilean invariance cannot be used in a curved spacetime as there do not exist global inertal coordinates. As detailed in section 1.2 of [41] and 2.2 of [22], one may view the phase factor as an attempt to absorb the boost transformation of a µ into a U (1) M transformation for ψ. As such, the phase factor does not appear in our treatment and a Schrödinger field is trivial under local Galilean boosts.…”

“…Examination of quantum Hall effective actions with constraints from non-relativistic symmetries was initiated by Hoyos and Son in [18]. In this work the authors also impose additional symmetries owing to the single component nature of the quantum Hall fluid (see [22] for a manifestly covariant way of implementing these symmetries). In this section we will reproduce known results for the stress and energy current induced by the Wen-Zee term as a simple example in the use of the formalism given above (also see [47] for a similar computation).…”

Non-relativistic supersymmetry on curved three-manifolds

“…coupled condensed matter systems [14][15][16][17][18][19][20][21][22][23] have however led to a renewed interest in nonrelativistic QFTs on curved backgrounds as well. There exist various notions of nonrelativistic differential geometry among which Newton-Cartan geometry is the prime example [24].…”