118 research outputs found
Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
The Eisenhart-Duval lift allows embedding non-relativistic theories into a
Lorentzian geometrical setting. In this paper we study the lift from the point
of view of the Dirac equation and its hidden symmetries. We show that
dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in
general gives rise to the non-relativistic Levy-Leblond equation in lower
dimension. We study in detail in which specific cases the lower dimensional
limit is given by the Dirac equation, with scalar and vector flux, and the
relation between lift, reduction and the hidden symmetries of the Dirac
equation. While there is a precise correspondence in the case of the lower
dimensional massive Dirac equation with no flux, we find that for generic
fluxes it is not possible to lift or reduce all solutions and hidden
symmetries. As a by-product of this analysis we construct new Lorentzian
metrics with special tensors by lifting Killing-Yano and Closed Conformal
Killing-Yano tensors and describe the general Conformal Killing-Yano tensor of
the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Lastly,
we show how dimensionally reducing the higher dimensional operators of the
massless Dirac equation that are associated to shared hidden symmetries it is
possible to recover hidden symmetry operators for the Dirac equation with flux.Comment: 18 pages, no figures. Version 3: some typos corrected, some
discussions clarified, part of the abstract change
Curvatronics with bilayer graphene in an effective spacetime
We show that in AB stacked bilayer graphene low energy excitations around the
semimetallic points are described by massless, four dimensional Dirac fermions.
There is an effective reconstruction of the 4 dimensional spacetime, including
in particular the dimension perpendicular to the sheet, that arises dynamically
from the physical graphene sheet and the interactions experienced by the
carriers. The effective spacetime is the Eisenhart-Duval lift of the dynamics
experienced by Galilei invariant L\'evy-Leblond spin particles
near the Dirac points. We find that changing the intrinsic curvature of the
bilayer sheet induces a change in the energy level of the electronic bands,
switching from a conducting regime for negative curvature to an insulating one
when curvature is positive. In particular, curving graphene bilayers allows
opening or closing the energy gap between conduction and valence bands, a key
effect for electronic devices. Thus using curvature as a tunable parameter
opens the way for the beginning of curvatronics in bilayer graphene.Comment: 8 pages, 3 figures. Revised version with additional materia
Is the pseudogap a topological state?
We conjecture that the pseudogap is an inhomogeneous condensate above the
homogeneous state whose existence is granted by topological stability. We
consider the simplest possible order parameter theory that provides this
interpretation of the pseudogap and study its angular momentum states. The
normal state gap density, the breaking of the time reversal symmetry and the
checkerboard pattern are naturally explained under this view. The pseudogap is
a lattice of skyrmions and the inner weak local magnetic field falls below the
experimental threshold of observation given by NMR/NQR and SR experiments.Comment: 12 pages, six figures, one tabl
Topologically stable gapped state in a layered superconductor
We show that a layered superconductor, described by a two-component order
parameter, has a gapped state above the ground state, topologically protected
from decay, containing flow and counter flow in the absence of an applied
magnetic field. This state is made of skyrmions, breaks time reversal symmetry
and produces a weak local magnetic field below the present threshold of
detection by SR and NMR/NQR. We estimate the density of carriers that
condense into the pseudogap.Comment: 6 pages, 4 figure
Eisenhart lifts and symmetries of time-dependent systems
Certain dissipative systems, such as Caldirola and Kannai's damped simple
harmonic oscillator, may be modelled by time-dependent Lagrangian and hence
time dependent Hamiltonian systems with degrees of freedom. In this paper
we treat these systems, their projective and conformal symmetries as well as
their quantisation from the point of view of the Eisenhart lift to a Bargmann
spacetime in dimensions, equipped with its covariantly constant null
Killing vector field. Reparametrization of the time variable corresponds to
conformal rescalings of the Bargmann metric. We show how the Arnold map lifts
to Bargmann spacetime. We contrast the greater generality of the
Caldirola-Kannai approach with that of Arnold and Bateman. At the level of
quantum mechanics, we are able to show how the relevant Schr\"odinger equation
emerges naturally using the techniques of quantum field theory in curved
spacetimes, since a covariantly constant null Killing vector field gives rise
to well defined one particle Hilbert space. Time-dependent Lagrangians arise
naturally also in cosmology and give rise to the phenomenon of Hubble friction.
We provide an account of this for Friedmann-Lemaitre and Bianchi cosmologies
and how it fits in with our previous discussion in the non-relativistic limit.Comment: 34 pages, no figures. Minor corrections, some references adde
Electronic properties of curved few-layers graphene: a geometrical approach
We show the presence of non-relativistic L\'evy-Leblond fermions in flat
three- and four-layers graphene with AB stacking, extending the results
obtained in [Curvatronics2017] for bilayer graphene. When the layer is curved
we obtain a set of equations for Galilean fermions that are a variation of
those of L\'evy-Leblond with a well defined combination of pseudospin, and that
admit L\'evy-Leblond spinors as solutions in an approriate limit. The local
energy of such Galilean fermions is sensitive to the intrinsic curvature of the
surface. We discuss the relationship between two-dimensional pseudospin,
labelling layer degrees of freedom, and the different energy bands. For
L\'evy-Leblond fermions an interpretation is given in terms of massless
fermions in an effective 4D spacetime, and in this case the pseudospin is
related to four dimensional chirality. A non-zero energy band gap between
conduction and valence electronic bands is obtained for surfaces with positive
curvature.Comment: 16 pages, 4 figures. Matches the published version. Refined theory
that describes the unique combination of isospin states ocurring in curved
bilayer graphene sheet
Godel-type Metrics in Various Dimensions II: Inclusion of a Dilaton Field
This is the continuation of an earlier work where Godel-type metrics were
defined and used for producing new solutions in various dimensions. Here a
simplifying technical assumption is relaxed which, among other things,
basically amounts to introducing a dilaton field to the models considered. It
is explicitly shown that the conformally transformed Godel-type metrics can be
used in solving a rather general class of Einstein-Maxwell-dilaton-3-form field
theories in D >= 6 dimensions. All field equations can be reduced to a simple
"Maxwell equation" in the relevant (D-1)-dimensional Riemannian background due
to a neat construction that relates the matter fields. These tools are then
used in obtaining exact solutions to the bosonic parts of various supergravity
theories. It is shown that there is a wide range of suitable backgrounds that
can be used in producing solutions. For the specific case of (D-1)-dimensional
trivially flat Riemannian backgrounds, the D-dimensional generalizations of the
well known Majumdar-Papapetrou metrics of general relativity arise naturally.Comment: REVTeX4, 17 pp., no figures, a few clarifying remarks added and
grammatical errors correcte
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