aainstitutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. INDIA.
bbinstitutetext: Centre de Physique Theorique, Ecole Polytechnique de Paris,
91128 Palaiseau Cedex, France.
ccinstitutetext: Okinawa Institute of Science and Technology (OIST), 1919-1
Tancha, Onna-son, Okinawa 904-0495, JAPAN
ddinstitutetext: Department of Physics, BITS-Pilani, K K Birla Goa Campus,
Zuarinagar, Goa-403726. INDIA.

# Magic Fermions: Carroll and Flat Bands

a,b Arjun Bagchi c Aritra Banerjee d Rudranil Basu a Minhajul Islam a Saikat
Mondal
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Carroll algebra is constructed as the $c\to 0$ limit of the Poincare
algebra and is associated to symmetries on generic null surfaces. In this
paper, we begin investigations of Carrollian fermions or fermions defined on
generic null surfaces. Due to the availability of two different (degenerate)
metrics on Carroll spacetimes, there is the possibility of two different
versions of Carroll Clifford algebras. We consider both possibilities and
construct explicit representations of Carrollian gamma matrices and show how
to build higher spacetime dimensional representations out of lower ones.
Actions for Carroll fermions are constructed with these gamma matrices and the
properties of these actions are investigated.

We show that in condensed matter systems where the dispersion relation becomes
trivial i.e. the energy is not dependent on momentum and bands flatten out,
Carroll symmetry generically appears. We give explicit examples of this
including that of twisted bi-layer graphene, where superconductivity appears
at so called magic angles and connect this to Carroll fermions.

##  1 Introduction

“Everyone wants a magical solution to their problem, and everyone refuses to
believe in magic.”

The physics of flat bands in condensed matter has attracted a lot of recent
interest with wide ranging applications in exotic systems like fractional
quantum Hall systems, spin liquids and twisted bi-layer graphene where “magic”
superconductivity appears at certain specific angles of twist. In this paper,
we will propose a generic solution to flat bands that has a touch of magic, —
fermions with Carrollian symmetry.

### The rise and rise of Carroll

Non-Lorentzian physics or the study of physics beyond relativistic invariance
and without Lorentz or Poincare symmetry and the geometry underlying these
Non-Lorentzian structures have become very fashionable of late. This is
primarily due to the discovery of uses of Non-Lorentzian structures in various
branches of physics including condensed matter physics, classical and quantum
gravity. In this work, we will be interested in Carrollian theories, where the
Carroll group replaces the Poincare group as the symmetry group of interest.

Starting from Poincare group, one can obtain Carroll group as a contraction by
taking the speed of light going to zero limit. This was first introduced by
Levy-Leblond LevyLeblond  and Sen Gupta NDS  as a “degenerate cousin of the
Poincare group”. Numerous counter-intuitive things occur in this limit. The
light-cone closes completely, the space-time metric becomes degenerate and new
non-Lorentzian structures emerge on the spacetime manifold Henneaux:1979vn .
These Carrollian manifolds can be understood mathematically as fibre-bundles
where there are two distinct metrics on the base and the fibre instead of a
single non-degenerate metric for the entire spacetime. Interestingly, any null
hypersurface is a Carroll manifold. Two of the most important examples of null
hypersurfaces are null boundary of asymptotically flat spacetimes and event
horizons of generic black holes. Thus Carrollian and conformal Carrollian
structures are inexorably linked with scatterings in flat spacetimes and black
hole physics.

Conformal Carrollian symmetry has been discovered to be isomorphic to Bondi-
Metzner-Sachs (BMS) symmetry in generic dimensions Duval:2014uva , following
closely related observations in two dimensions Bagchi:2010zz . BMS groups
emerge as the asymptotic symmetry groups for asymptotically flat spacetimes at
null infinity Bondi:1962px ; Sachs:1962zza . These groups are infinite-
dimensional in dimensions three Barnich:2006av  and four 111In higher
dimensions, depending on boundary conditions the BMS can be truncated to the
Poincare algebra or can still remain infinite dimensional Kapec:2015vwa .. A
considerable body of research has been built up relating asymptotic
symmetries, soft theorems and memory effect following the lead of Strominger
Strominger:2013jfa . The formulation of holography for asymptotically flat
spacetimes has followed two routes, Celestial and Carrollian holography. In
the Celestial formulation, it has been proposed that a 2d conformal field
theory living on the celestial sphere can compute scattering amplitudes in 4d
asymptotically flat spacetimes. This has led to novel results about asymptotic
symmetries and soft theorems. See Strominger:2017zoo ; Pasterski:2021rjz ;
Raclariu:2021zjz  for recent reviews on Celestial holography and references
therein. Carroll holography on the other hand postulates codimension one
holography between flat spacetimes and a Carrollian CFT living on the null
boundary of flatspace Bagchi:2010zz ; Bagchi:2012cy ; Bagchi:2016bcd . This
has seen quite a bit of success, particularly in lower dimensions. See
Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2013qva ; Bagchi:2015wna ;
Bagchi:2014iea ; Jiang:2017ecm ; Barnich:2012aw ; Duval:2014lpa ;
Hartong:2015usd  for a non-exhaustive list of important contributions in this
regard. The connection between 4d scattering and 3d Carrollian CFTs has been
recently proposed in Bagchi:2022emh  drawing inspiration from the Celestial
approach 222See also Donnay:2022aba  for further connections between the
Celestial and Carrollian formulations of flat holography..

We have mentioned above that Carrollian structures emerge naturally on all
null surfaces. The event horizons of generic black holes form an important
class of these null surfaces and defining theories on these null surfaces
naturally gives rise of Carrollian (conformal) field theories. The emergence
of Carrollian and BMS symmetries (as well as “BMS-like” symmetries) on black
hole even horizons have been addressed in Donnay:2019jiz ; Hawking:2016msc ;
Donnay:2015abr ; Afshar:2016wfy ; Penna:2017bdn ; Carlip:2017xne . If one
wishes to define quantum field theories living on black hole event horizons,
it is but natural that these theories would imbibe the Carroll structures that
appear geometrically on these degenerate backgrounds. The investigation of
Carrollian field theories become important in this context as well.

Carroll symmetry also appears in the tensionless limit of string theory. In
this limit the worldsheet metric becomes degenerate and BMS3 replaces the
worldsheet conformal symmetry Bagchi:2013bga ; Bagchi:2015nca ; Bagchi:2019cay
; Bagchi:2020fpr ; Bagchi:2020ats . It has been shown in Bagchi:2021ban  that
strings probing black hole geometries become effectively tensionless near the
horizon of the black hole. The background Carroll structure of the blackhole
event horizon induces a Carroll structure on the worldsheet of the string.
Rather intriguingly, in a recent paper Bagchi:2022iqb , it has been shown that
null strings wrapping the horizon of a BTZ black hole in asymptotically AdS
spacetimes can give an explicit counting of its microstates, reproducing not
only the leading Bekenstein-Hawking entropy but amazingly even its logarithmic
corrections correctly.

A brief word on the other applications. A substantial amount of research has
been done regarding Carroll aspects of gravity Bergshoeff:2017btm ;
Duval:2017els ; Hartong:2015xda ; Morand:2018tke ; Ciambelli:2018ojf ;
Henneaux:2021yzg . Carroll fluids have been addressed in deBoer:2017ing ;
Ciambelli:2018wre ; Ciambelli:2018xat ; Ciambelli:2020eba ; Petkou:2022bmz ;
Freidel:2022bai ; Freidel:2022vjq . New links have emerged between Carroll
symmetry and cosmology and dark energy deBoer:2021jej , as well as connections
to condensed matter physics through the theory of fractons Bidussi:2021nmp .

### In this paper: Carroll Fermions and Flat Bands

There has been many recent attempts at understanding various aspects of field
theories having Carrollian and Conformal Carrollian symmetry. Initially the
effort was to understand these as the systematic $c\to 0$ limit of
relativistic field theories Bagchi:2019xfx  and the Carrollian version of
scalars, electrodynamics and Yang-Mills theories were constructed and studied.
Later, there has been the effort in constructing actions for these theories
Bergshoeff:2017btm ; Bagchi:2019clu  and studying theories from these actions.
Scalars Gupta:2020dtl ; deBoer:2021jej ; Bagchi:2022eav ; Baiguera:2022lsw ;
Rivera-Betancour:2022lkc , gauge fields, and gravity Bergshoeff:2017btm ;
Duval:2017els ; Hartong:2015xda ; Morand:2018tke ; Ciambelli:2018ojf ;
Henneaux:2021yzg  have received much attention. Rather surprisingly, fermions
have been left out of the fun 333See however Bergshoeff:2017btm  for an
earlier attempt (we will comment briefly about this in Section (4)) and
Yu:2022bcp ; Hao:2022xhq  for very recent explorations in $d=2$.. In this
paper we aim to rectify this and build towards an understanding of Carroll
fermions in general dimension $d$. A companion paper 2dcarroll  contains
details of Carroll fermions specifically for $d=2$.

In condensed matter physics, one of the very important notions is that of the
dispersion relation which captures the dependence of energy with spatial
momentum in a system. The very simple non-relativistic dispersion relation
$E=\frac{p^{2}}{2m}$ captures complex phenomena like Bose-Einstein
condensation. Of late, there has been a great deal of interest in theories
having trivial dispersion relations, where energies don’t depend on momentum
at all. These so called flat band structures are at the heart of many exotic
phenomena including fractional quantum Hall effect, and magic
superconductivity in twisted bi-layer graphene. We will show in this paper
that the appearance of flat bands is directly connected to the appearance of
Carrollian symmetry in the systems under consideration. As these systems are
usually fermionic systems, the Carrollian theories that are connected to these
are the Carrollian fermions we build in the first part of this paper.

The rest of the paper is organised as follows. We give a short review of
Carroll symmetry from an algebraic and a geometric perspective in Section (2).
Section (3) is our first encounter with Carroll fermions. Here we define the
Carroll Clifford algebra and find two distinct types of algebras and discuss
their properties and representations. In Section (4), we build actions for the
two different fermions and discuss structures and symmetries related to the
actions. In Section (5), we move on to show how Carroll symmetry emerges in
flat band materials in condensed matter systems and follow that with explicit
examples in Section (6). We conclude with some remarks in Section (7). An
appendix has details of Carrollian and conformal Carrollian isometries.

##  2 Carroll Symmetry

In this section we will give a quick recap of Carroll symmetry. The Carroll
group is be obtained by taking the speed of light going to zero limit of the
relativistic Poincare symmetry. This is the algebraic approach we go on to
describe below. There is also a geometric way of describing the Carroll group
in terms of degenerate metrics and Carroll manifolds. We will elaborate on
this as well.

###  2.1 Contraction, transformations and algebra

Carroll group arises from an Inonu-Wigner contraction from the Poincare group.
This is performed by taking the limit $c\rightarrow 0$ of the Poincare group.
This is equivalent to the scaling

$x^{i}\rightarrow x^{i},\quad t\rightarrow\epsilon t,\quad\epsilon\rightarrow
0$ (1)

where $i=1,...,(d-1)$. Under this limit the Carroll generators are defined by
re-scaling the parent Poincare generators in the following way

$H\equiv\epsilon P_{0}^{\text{P}},\,P_{i}\equiv
P_{i}^{\text{P}},\,C_{i}\equiv\epsilon J_{0i}^{\text{L}},\,J_{ij}\equiv
J_{ij}^{\text{P}},$ (2)

where the superscript P refers to the Poincare generators. The explicit form
of the Carroll generators are:

$H=\partial_{t},\quad P_{i}=\partial_{i},\quad C_{i}=x_{i}\partial_{t},\quad
J_{ij}=x_{i}\partial_{j}-x_{j}\partial_{i}.$ (3)

Here $H,P_{i},C_{i},J_{ij}$ are time translation, spatial translations,
Carroll boosts and $(d-1)$ spatial rotations respectively. These generators
generate the Carroll transformation on space-time coordinate

$t^{{}^{\prime}}=t+a-\vec{v}.\vec{x},\quad\vec{x}^{{}^{\prime}}=\boldsymbol{R}\vec{x}+\vec{b}$
(4)

where the parameters of the group $(a,\vec{b},\vec{v},\boldsymbol{R})$
describes time-translation, space-translation, Carroll boosts and $SO(d-1)$
rotation respectively. The Lie algebra of the Carroll group is given by
following non-zero commutation relations

$\displaystyle[J_{ij},J_{kl}]=4\delta_{[i[k}J_{l]j]},\,[J_{ij},P_{k}]=2\delta_{k[j}P_{i]},\,[J_{ij},C_{k}]=2\delta_{k[j}C_{i]},\,[C_{i},P_{j}]=-\delta_{ij}H.$
(5)

Crucially, the commutation relation between the Carroll boosts becomes
$[C_{i},C_{j}]=0$, reflecting the non-Lorentzian nature of the algebra. When
we generalise to conformal Carroll symmetry we have additional generators,
Dilatation $(D)$, temporal $(K_{0})$ and spatial $(K_{i})$ special Conformal
Carroll transformation:

$D=t\partial_{t}+x_{i}\partial_{i},\quad K_{0}=x_{i}x_{i}\partial_{t},\quad
K_{i}=2x_{i}(t\partial_{t}+x_{j}\partial_{j})-x_{j}x_{j}\partial_{i}.$ (6)

The additional non-vanishing commutation relations are

$\displaystyle[D,P_{i}]=-P_{i},\,[D,H]=-H\,[D,K_{i}]=K_{i},\,[D,K_{0}]=K_{0},$
$\displaystyle[K_{0},P_{i}]=-2C_{i}\,[K_{i},H]=-2C_{i},\,[K_{i},P_{j}]=-2\delta_{ij}D-2J_{ij}.$
(7)

This is the finite Conformal Carrollian Algebra (f-CCA). The algebra can be
obtained by the $c\to 0$ limit of the relativistic Conformal algebra. It is
important to note that this finite algebra can be rewritten in a suggestive
form and be given an infinite dimensional lift for all spacetime dimensions
Bagchi:2016bcd . This infinite dimensional lift makes connection with the
important fact that CCA is isomorphic to the BMS algebra in one higher
dimension

$\mathfrak{CCarr}_{d}=\mathfrak{bms}_{d+1}.$ (8)

The BMS algebra is the symmetry of the null boundary of asymptotically flat
spacetime in $(d+1)$ dimensions. The topology of null boundary is
$R_{u}\otimes S^{d}$, where $R_{u}$ is a null line.

The details of the infinite lift of the CCA are dimension dependent. The
infinite extension in $d=2$ gives the BMS3 algebra:

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c_{L}}{12}(n^{3}-n)\delta_{n+m,0},$
(9)
$\displaystyle[L_{n},M_{m}]=(n-m)M_{n+m}+\frac{c_{M}}{12}(n^{3}-n)\delta_{n+m,0},$
(10) $\displaystyle[M_{n},M_{m}]=0$ (11)

Here the supertranslations $M_{n}$ are the generators of angle-dependent
translations of the null direction $u$ and $L_{n}$, the superrotations,
generate the diffeomorphism of the circle at infinity. There is a similar
enhancement possible for $d=3$ which gives rise to the BMS4 algebra. Even for
$d\geq 4$ it is possible to give an infinite extension for the
supertranslations Bagchi:2016bcd . For $d=4$ this is

$M_{f}=f(x,y,z)\partial_{t},$ (12)

for any arbitrary tensor field $f$, transforming under $SO(3)$. These close to
form an infinite Abelian ideal along with the (finite) conformal generators on
the sphere. It is possible to have an extension to the full diffeomorphism
group on the sphere with other boundary conditions, but we will not be
interested in this and other generalisations of the BMS group in this paper.

###  2.2 Carroll Geometry

Let us now give a more geometric picture of Carroll symmetry and review the
non-Lorentzian structures that arise in this context.

We begin by considering the Carroll limit of ordinary Minkowski spacetime to
get a first idea of the degenrate structures we would encounter. Consider the
$d$-dimensional Minkowski space with line element:

$ds^{2}=-c^{2}dt^{2}+(dx^{i})^{2}$ (13)

Then the covariant metric and its contravariant inverse are:

$\displaystyle\eta_{\mu\nu}=\begin{pmatrix}-c^{2}&0\\\
0&I_{d-1}\end{pmatrix}\qquad\eta^{\mu\nu}=\begin{pmatrix}-1/c^{2}&0\\\
0&I_{d-1}\end{pmatrix}$ (14)

Now taking the Carroll limit $(c\rightarrow 0)$ we get a degenerate covariant
spatial metric $\tilde{h}_{\mu\nu}$ with one zero eigenvalue and a degenerate
contravariant temporal metric $\Theta^{\mu\nu}$ with one non-zero eigenvalue:

$\eta_{\mu\nu}\rightarrow\tilde{h}_{\mu\nu}=\begin{pmatrix}0&0\\\ 0&\quad
I_{d-1}\end{pmatrix},\qquad-c^{2}\eta^{\mu\nu}\rightarrow\Theta^{\mu\nu}=\begin{pmatrix}1&0\\\
0&\quad 0_{d-1}\end{pmatrix}$ (15)

These two are the invariant tensors for the Carroll group. As
$\Theta^{\mu\nu}$ is basically 1$\times$1 matrix, we can define vector
$\theta^{\mu}$ such that

$\Theta^{\mu\nu}=\theta^{\mu}\theta^{\nu}$ (16)

and degeneracy implies

$\tilde{h}_{\mu\nu}\theta^{\nu}=0.$ (17)

Having had a brief encounter with the degeneracy we will encounter in the
limit, let us now see how we can formalise this structure from an
instrinsically Carrollian point of view Henneaux:1979vn . To this end, we now
review Carroll geometry following Duval:2014uoa . A Carroll structure is a
quadruplet $(\mathcal{C},\tilde{h},\theta,\nabla)$, where

  * •

$\mathcal{C}$ is a $d$ dimensional manifold, on which one can choose a
coordinate chart $(t,x^{i})$.

  * •

$\tilde{h}$ is a covariant, symmetric, positive, tensor field of rank $d-1$
and of signature $(0,\underbrace{+1,\ldots,+1}_{d-1})$.

  * •

$\theta$ is a non-vanishing vector field which generates the kernel of
$\tilde{h}$ .

  * •

$\nabla$ is a symmetric affine connection that parallel transports both
$\tilde{h}_{\mu\nu}$ and $\theta^{\nu}$.

This $d$ dimensional Carrollian manifold can be described by a fibre bundle
with the $(d-1)$ dimensional spatial directions forming the base space and and
the temporal direction forming a $1d$ fibre is on top of this. Of this class
of manifolds, we will be most interested in the flat Carrollian manifold:

$\mathcal{C}=\underbrace{\mathcal{R}}_{fibre}\times\underbrace{\mathcal{R}^{d-1}}_{base},\quad\tilde{h}=\delta_{ij}dx^{i}\otimes
dx^{j},\quad\theta=\partial_{t}$ (18)

Carroll Lie algebra is identified with those vector fields
$\xi=\xi^{a}\partial_{a}$ of $\mathcal{C}$, which satisfy the isometry
conditions:

$\displaystyle\mathcal{L}_{\xi}\tilde{h}_{\mu\nu}=0,\quad\mathcal{L}_{\xi}\theta=0.$
(19)

Solving these one obtains (for details the reader is pointed to Appendix A)

$\displaystyle\xi^{i}=\omega^{i}_{j}x^{j}+b^{i},\qquad\xi^{0}=a+f(x^{k})$ (20)

So the isometry group of Carroll structure is infinite-dimensional. Further
requiring metric compatibility with the connection defined by $\nabla$, the
function $f(x^{k})$ is restricted to be linear and the Carroll algebra becomes
finite dimensional and is given by (5). To include Conformal Carroll
isometries, one has to modify the isometry eq. $(\ref{CLa})$ to the condition
for conformal isometries in the Carroll background:

$\displaystyle\mathcal{L}_{\xi}\tilde{h}=\lambda_{1}\tilde{h},\quad\mathcal{L}_{\xi}\theta={\lambda_{2}}\theta.$
(21)

for some function $\lambda$ on $\mathcal{C}$. It can be shown that these
conformal Carroll generators in $d$ dimensions close to form exactly the
BMSd+1 algebra Duval:2014uoa  when

$N=-\frac{\lambda_{1}}{\lambda_{2}}=2.$ (22)

This specific value is the one for which space and time scale in the same way
under the Dilatation generator. For other values of $N$, one gets more generic
Carroll conformal algebras.

##  3 Carroll Clifford Algebra

We wish to build towards an understanding of fermions in Carrollian
spacetimes. To this end, we first postulate a Carrollian version of the
relativistic Clifford algebra and then we will discuss some aspects of its
representation theory. Before we embark on this journey, let us quickly remind
ourselves of relativistic fermions and the properties which we would need to
construct Carroll analogues of.

###  3.1 Relativistic Fermions

Fermions make up all the matter around us. The fundamental particles leptons,
quarks are all fermions. Not only are fermions of central physical
significance, their mathematical structure is also very rich. They live not on
the group manifold that defines the spacetime but on the universal cover of
it. One of the principal objects that define fermions is the Clifford algebra
associated with it.

For relativistic fermions, the fundamental equation is of course the Dirac
equation

$(i\gamma^{\mu}\partial_{\mu}-m)\Psi=0,$ (23)

which can be derived from the corresponding Dirac action

$S=\int d^{d}x\sqrt{-g}\,\bar{\Psi}(i\gamma^{\mu}\partial_{\mu}-m)\Psi.$ (24)

$\Psi$ is a four-component Dirac spinor and $\bar{\Psi}$ is defined as
$\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$. Here $\gamma$ matrices obey the
Clifford algebra:

$\\{\gamma^{\mu},\gamma^{\nu}\\}=-2g^{\mu\nu}.$ (25)

where $g^{\mu\nu}$ is the spacetime metric whose determinant is given by
$\sqrt{-g}$ and appears in the Dirac action (24). Let us focus on $d=4$ and
flat spacetimes $g^{\mu\nu}=\eta^{\mu\nu}=\text{diag}(-1,1,1,1)$. A
particularly useful representation of the Clifford algebra here is given by:

$\displaystyle\gamma^{0}=\begin{pmatrix}0&I\\\
I&0\end{pmatrix}\qquad\gamma^{i}=\begin{pmatrix}\sigma^{i}&0\\\
0&-\sigma^{i}\end{pmatrix},\\{i=1,2,3\\}$ (26)

with $\sigma^{i}$ being the Pauli matrices. The gamma matrices obviously carry
the information of the spacetime as is clear from the Clifford algebra. This
becomes even more transparent when we link $\sigma$ with the spacetime
isometry algebra. Using the $\gamma$ matrices we can construct a set of
matrices

$\Sigma^{\mu\nu}=\frac{1}{4}[\gamma^{\mu},\gamma^{\nu}].$ (27)

These matrices form a representation of Lorentz algebra

$[\Sigma^{\mu\nu},\Sigma^{\rho\sigma}]=-\eta^{\mu\rho}\Sigma^{\nu\sigma}+\eta^{\nu\rho}\Sigma^{\mu\sigma}-\eta^{\nu\sigma}\Sigma^{\mu\rho}+\eta^{\mu\sigma}\Sigma^{\nu\rho}$
(28)

Here $\Sigma^{0i}$ is representation of Lorentz boost, whereas $\Sigma^{ij}$
is of $SO(d-1)$ rotation.

###  3.2 The Carrollian Algebra

For Carrollian spacetimes, we have two metrics given by $\tilde{h}_{\mu\nu}$
and $\Theta^{\mu\nu}$. We need both to generate the isometry of Carroll
spacetimes. In keeping with this we propose that the Carroll Clifford algebra
is given by:

${\boxed{\big{\\{}\tilde{\gamma}_{\mu},\tilde{\gamma}_{\nu}\big{\\}}=2\tilde{h}_{\mu\nu},\quad\big{\\{}\hat{\gamma}^{\mu},\hat{\gamma}^{\nu}\big{\\}}=2\Theta^{\mu\nu}}}$
(29)

For flat Carroll spacetimes, which is the case we will be interested in
throughout this work:

$\tilde{h}_{\mu\nu}=\text{diag}(0,1,1,\ldots)\quad\Theta^{\mu\nu}=\text{diag}(1,0,0,\ldots).$
(30)

Contrary to the relativistic case, here we see that there are two sets of
$\gamma$ matrices which obey different versions of the Carroll Clifford
algebras. In principle, these could lead us to different types of fermions on
a Carrollian manifold. In order to investigate if there is a preferred type of
Clifford algebra or a preferred type of fermion, let us push on.

As a further test of our proposal, we will need to check that we can construct
the set of matrices $\tilde{\Sigma}$ and $\hat{\Sigma}$ in analogy with (28),
which should now close to the equivalent of the Lorentz (and not Poincare)
algebra. This is the sub-algebra of Carroll algebra which contains the Carroll
boosts and rotations but excludes translation generators. For the lower gammas
$\tilde{\gamma}$, we construct

$\tilde{\Sigma}_{ab}\equiv\frac{1}{4}[\tilde{\gamma}_{a},\tilde{\gamma}_{b}]$
(31)

and find

$\displaystyle[\tilde{\Sigma}_{0i},\tilde{\Sigma}_{0j}]=0,\quad[\tilde{\Sigma}_{0i},\tilde{\Sigma}_{jk}]=-\delta_{ij}\tilde{\Sigma}_{0k}+\delta_{ik}\tilde{\Sigma}_{0j}$
(32a)
$\displaystyle[\tilde{\Sigma}_{ij},\tilde{\Sigma}_{kl}]=-\delta_{il}\tilde{\Sigma}_{jk}+\delta_{ik}\tilde{\Sigma}_{jl}-\delta_{jk}\tilde{\Sigma}_{il}+\delta_{jl}\tilde{\Sigma}_{ik}.$
(32b)

Here $\tilde{\Sigma}_{0i}$ are the Carroll boosts and $\tilde{\Sigma}_{ij}$
are spatial rotations. As seen from the above, the algebra does close to form
the homogeneous part of the Carroll algebra. So, the lower gammas pass this
test.

However, the same is not apparently true for the upper gammas. We again define

$\hat{\Sigma}^{ab}\equiv\frac{1}{4}[\hat{\gamma}^{a},\hat{\gamma}^{b}].$ (33)

Although we get

$[\hat{\Sigma}^{0i},\hat{\Sigma}^{0j}]=0$ (34)

which means that the Carroll boosts are recovered, the rotation matrices are
identically zero as the product of $\hat{\gamma}^{i}$ in Carroll case is zero:

$\hat{\Sigma}^{ij}=0.$ (35)

The rotation-boost and the rotation-rotation commutators are hence identically
zero and the $\hat{\Sigma}^{ab}$ do not form a faithful representation of the
Carroll algebra in the background. We will present a way around this potential
hurdle in Section (4). Below we construct representations for both sets of
gamma matrices, and later in the paper we construct actions for both sets of
gamma matrices.

###  3.3 Representation of the Carroll Clifford algebra

In this subsection, we will construct an explicit representation of our
Carroll Clifford algebra in $d=4$ and show how to get this from a lower
dimensional representation. This will help us build representations for
arbitrary dimensions.

####  3.3.1 Representation in $d=4$

A representation of our Carroll Clifford algebra in $d=4$ flat Carroll
spacetimes that would prove particularly important is the following:

$\displaystyle\begin{split}\tilde{\gamma}_{0}=\begin{pmatrix}0&0&0\,\,\,&0\\\
0&0&0&0\\\ i&0&0&0\\\
0&\,\,i&0&0\end{pmatrix},\tilde{\gamma}_{1}=\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix},\tilde{\gamma}_{2}=\begin{pmatrix}0&-i&0&0\\\ i&0&0&0\\\
0&0&0&i\\\
0&\,\,0&-i&0\end{pmatrix},\tilde{\gamma}_{3}=\begin{pmatrix}1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0\,\,\,&1\end{pmatrix}\end{split}$ (36)

These matrices obey the following algebra

$\big{\\{}\tilde{\gamma}_{\mu},\tilde{\gamma}_{\nu}\big{\\}}=2\tilde{h}_{\mu\nu}.$
(37)

Again following our arguments above, we can construct the $\tilde{\Sigma}$
matrices (31) and show that the Carrollian spacetime algebra (32) is satisfied
by these $\tilde{\Sigma}$ matrices. This is thus a proper representation of
the Carroll Clifford algebra.

Now we construct a representation for the upper gammas. Here we will have
another set of matrices given by

$\displaystyle\begin{split}\hat{\gamma}^{0}=\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{pmatrix}\quad\hat{\gamma}^{1}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&i&0&0\\\
i&0&0&0\end{pmatrix}\quad\hat{\gamma}^{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&1&0&0\\\
-1&0&0&0\end{pmatrix}\quad\hat{\gamma}^{3}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ i&0&0&0\\\ 0&-i&0&0\end{pmatrix}\end{split}$ (38)

It can be checked that these matrices obey the algebra

$\\{\hat{\gamma}^{\mu},\hat{\gamma}^{\nu}\\}=2\Theta^{\mu\nu}.$ (39)

As discussed before, the ${\hat{\Sigma}}^{ab}$ matrices constructed out of
these $\hat{\gamma}^{a}$’s have the peculiar property that the rotation
matrices ${\hat{\Sigma}}^{ij}$ vanish identically. The boosts
${\hat{\Sigma}}^{0i}$ are non-trivial and form an abelian algebra, as expected
from Carrollian boosts.

One must note again that the absence of rotation generators in the upper gamma
representation can be traced back to their degenerate structure. However, we
have not yet discussed the structure of discrete symmetries associated to
these representations, and whether or not the above generators are compliant
with those symmetries. In a later section we will ponder further over these
discrete symmetries in $d=4$, and come back to a modified proper definition of
Carroll rotation generators which will form a faithful representation of the
algebra.

####  3.3.2 Representation in general even dimensions

Consider $d=2$. The degenerate Clifford algebra and its solutions were
previously discussed in the context of tensionless superstrings Bagchi:2017cte
. We will borrow some of the nomenclature here. Let us begin by considering
the lower Carroll Clifford algebra in $d=2$ (37). This reduces to the matrix
equations

$(\tilde{\gamma}_{0})^{2}=0,\quad(\tilde{\gamma}_{1})^{2}=I,\quad\tilde{\gamma}_{0}\tilde{\gamma}_{1}+\tilde{\gamma}_{1}\tilde{\gamma}_{0}=0.$
(40)

A trivial solution is $\tilde{\gamma}_{0}=0$ and $\tilde{\gamma}_{1}=I$. This
is what we will call the homogeneous case, keeping with the nomenclature
borrowed from Bagchi:2017cte . We will briefly comment on this later in the
paper. The non-trivial solution has the form

$\displaystyle\tilde{\gamma}_{0}=\begin{pmatrix}0&0\\\
a&0\end{pmatrix}\qquad\tilde{\gamma}_{1}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ (41)

with $a\in\mathcal{R}$. Now we will use these 2d matrices as seed matrices to
generate generic higher dimensional matrices. The prescription we use to get
the lower $\tilde{\gamma}$ matrices for higher even dimensions is the
following

$\displaystyle\tilde{\gamma}_{{}^{\prime}0}=\tilde{\gamma}_{0}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\,\tilde{\gamma}_{{}^{\prime}i}=\tilde{\gamma}_{i}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},\,\tilde{\gamma}_{{}^{\prime}d-2}=I\otimes\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,\tilde{\gamma}_{{}^{\prime}d-1}=I\otimes\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}.$ (42)

Here $\tilde{\gamma}_{\mu}$ are the $2^{k}\times 2^{k}$ matrices in $d-2$
dimensions and $I$ is the $2^{k}\times 2^{k}$ identity matrices. It can be
checked that with the construction outlined above, starting with the 2d gamma
matrices (41) one reproduces the 4d gamma matrices (36).

Now, we come to the upper gammas. The upper Clifford algebra in $d=2$ (39)
reduces to the matrix equations

$(\hat{\gamma}^{0})^{2}=I,\quad(\hat{\gamma}^{1})^{2}=0,\quad\hat{\gamma}^{0}\hat{\gamma}^{1}+\hat{\gamma}^{1}\hat{\gamma}^{0}=0.$
(43)

Again we have a homogeneous (or trivial) solution is $\hat{\gamma}^{0}=I$ and
$\hat{\gamma}^{1}=0$. The non-trivial or inhomogeneous solutions are of the
form

$\displaystyle\hat{\gamma}^{0}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\qquad\tilde{\gamma}^{1}=\begin{pmatrix}0&0\\\
a&0\end{pmatrix}$ (44)

with $a\in\mathbb{R}$. We can get higher dimensional upper $\hat{\gamma}$
matrices starting from $d=2$. In $d=2k+2$, $(k=1,2,...)$

$\displaystyle\hat{\gamma}^{{}^{\prime}0}=\hat{\gamma}^{0}\otimes
I,\,\hat{\gamma}^{{}^{\prime}i}=\hat{\gamma}^{i}\otimes
I,\,\hat{\gamma}^{{}^{\prime}d-2}=\begin{pmatrix}0&0\\\
I&0\end{pmatrix}\otimes\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,\hat{\gamma}^{{}^{\prime}d-1}=\begin{pmatrix}0&0\\\
I&0\end{pmatrix}\otimes\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}.$ (45)

Note that the construction for moving upwards in dimension for the lower and
the upper gamma matrices are not identical. It can again be checked that with
the prescription outlined the upper $2d$ gammas (44) become the upper $4d$
gammas we presented earlier in (38).

####  3.3.3 Representation in general odd dimensions

Having understood how to construct higher even dimensional representations
from the 2d representation, we now address how to construct general odd
dimensional representations for our deformed Clifford algebra (29). For this,
we will explicitly construct a three dimensional representation and use our
previous method of generating higher dimensional representations (42)-(45). In
$d=3$, one of the representations which obey lower Carroll Clifford algebra
(37) is given by

$\displaystyle\begin{split}\tilde{\gamma}_{0}=\begin{pmatrix}0&0&0\,\,\,&0\\\
0&0&0&0\\\ i&0&0&0\\\
0&\,\,i&0&0\end{pmatrix}\qquad\tilde{\gamma}_{1}=\begin{pmatrix}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix}\qquad\tilde{\gamma}_{2}=\begin{pmatrix}0&-i&0&0\\\
i&0&0&0\\\ 0&0&0&i\\\ 0&\,\,0&-i&0\end{pmatrix}.\end{split}$ (46)

Notice that unlike the relativistic case, we use a higher $4\times 4$
dimensional representation for these matrices. In order to generate higher odd
dimensional matrices, we use these 3d matrices as seed matrices and then
follow the method outlined earlier in (42). For example, going from 3d to 5d,
the representation matrices will be constructed such as

$\displaystyle\tilde{\gamma}^{{}^{\prime}}_{0}=\tilde{\gamma}_{0}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\quad\tilde{\gamma}^{{}^{\prime}}_{i}=\tilde{\gamma}_{i}\otimes\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\quad\tilde{\gamma}^{{}^{\prime}}_{3}=I\otimes\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\quad\tilde{\gamma}^{{}^{\prime}}_{4}=I\otimes\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}.$ (47)

Here $\tilde{\gamma}_{0},\tilde{\gamma}_{i}$ are the matrices in 3d
representation $(\ref{3dg})$, and $I$ is $4\times 4$ matrix.

For the upper Clifford algebra (39), in $d=3$, we get the following matrices:

$\displaystyle\hat{\gamma}^{0}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&-1&0\\\
0&0&0&-1\end{pmatrix}\qquad\tilde{\gamma}^{1}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&i&0&0\\\
i&0&0&0\end{pmatrix}\qquad\hat{\gamma}^{2}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&1&0&0\\\ -1&0&0&0\end{pmatrix}.$ (48)

These can be used along with eq $(\ref{2t4})$ to construct representations for
higher odd dimensional upper Carroll gamma matrices.

###  3.4 Structure of Carroll spinors in $d=4$

There are further representation dependent structures in a theory of spinors,
which form the basis of discrete symmetries. Since we will be moving on to
discuss the structure of Carroll invariant fermion actions, it will be
instructive to clear up the nature of some of these symmetries for the Carroll
case. It would turn out that the structures discussed here provide a solution
to the apparent problem with the upper gammas not providing a faithful
representation of the background Carroll algebra.

We first start with spinor $\Psi$ and wish to define its adjoint, like we do
for the Lorentz invariant case. The reader is reminded that relativistic Dirac
adjoints are usually defined as $\overline{\Psi}=\Psi^{\dagger}\gamma^{0}$,
and these lead to an array of bilinears that change covariantly under Lorentz
transformations that map spinors according to $\Psi\rightarrow
S(\hat{\Sigma})\Psi$. Now, since Carroll Clifford algebra is manifestly
degenerate, defining an adjoint with $\gamma^{0}$ would be problematic. We
thus introduce an analogue of the Dirac adjoint:

$\bar{\Psi}=\Psi^{\dagger}\Lambda,\quad\text{where}\quad\Lambda=\begin{pmatrix}0&iI\\\
-iI&0\end{pmatrix}$ (49)

such that $\Lambda^{2}=I$. The matrix $\Lambda$ is defined so that the term
$\bar{\Psi}\Psi$ behaves like a scalar under Carroll transformations. To
justify this choice, note that hermitian conjugate of the upper $\gamma$
matrices now satisfy

$\hat{\gamma}^{\mu^{\dagger}}=-\Lambda\hat{\gamma}^{\mu}\Lambda^{-1}$ (50)

which implies the Carroll invariant condition for the generators,

$\hat{\Sigma}^{\mu\nu\dagger}=-\Lambda\hat{\Sigma}^{\mu\nu}\Lambda^{-1}\implies
S(\hat{\Sigma})^{\dagger}=\Lambda S(\hat{\Sigma})^{-1}\Lambda,$ (51)

which in turn keeps the mass terms invariant under Carroll transformations.
Note that, for representation with lower $\gamma$ matrices one can argue the
existence of a similar adjoint structure in the same vein. In fact, a short
calculation leads us to the adjoint for lower gamma matrices as:

$\Lambda=\begin{pmatrix}0&I\\\ I&0\end{pmatrix},$ (52)

which is still idempotent. We will be using these definitions to write down
our spinor actions throughout the paper.

We now focus on the discussion about the structure of Charge Conjugation
symmetries in the Carroll case. In general for the Lorentz case, one would
like to work with real fermions that stay real under Lorentz boosts, and hence
a Majorana condition $\Psi^{*}=\Psi$ is imposed on the spinors at every
spacetime point. In a generic Clifford basis, one would define the charge
conjugate as:

$\Psi\mapsto\Psi^{c}=\,\mathcal{C}{\bar{\Psi}}^{{T}},$ (53)

where $\mathcal{C}$ is a $4\times 4$ matrix satisfying

$\displaystyle\quad\,\hat{\gamma}^{\mu^{T}}=-\mathcal{C}\hat{\gamma}^{\mu}\mathcal{C}^{-1},~{}~{}\mathcal{C}^{\dagger}\mathcal{C}=\mathbf{I}.$
(54)

One can easily check this is a good definition, i.e. the conjugate spinors
transforms accordingly under Carroll transformations. A viable solution to the
above conditions can be written here as:

$\displaystyle\mathcal{C}=\begin{pmatrix}0&i\sigma^{2}\\\
i\sigma^{2}&0\end{pmatrix}$ (55)

where it can be further checked that

$\displaystyle\hat{\Sigma}^{\mu\nu^{T}}=-\mathcal{C}\hat{\Sigma}^{\mu\nu}\mathcal{C}^{-1},\quad\hat{\gamma}^{\mu^{*}}={(\Lambda\mathcal{C})}\hat{\gamma}^{\mu}(\Lambda\mathcal{C})^{-1},\quad\hat{\Sigma}^{\mu\nu^{*}}={(\Lambda\mathcal{C})}\hat{\Sigma}^{\mu\nu}(\Lambda\mathcal{C})^{-1}.$
(56)

Once we put this structure in place, we can go ahead and define a Majorana-
like condition for Carroll spinors which simply reads $\Psi^{c}=\Psi$. One can
similarly find an equivalent representation for charge conjugation
corresponding to the lower gamma matrices, however they would not be very
important for our purpose here.

##  4 Action for Carroll fermions and its symmetries

Equipped with the basic structures and required tools, we now want to
construct Carroll invariant actions for fermions specialising to the case for
$d=4$. In this section, we will start with a two component Carroll bispinor,
and write down actions in the Dirac form involving both lower and upper gamma
matrices.

As we have seen before, although lower gamma matrices can be used to construct
perfectly faithful representations of the Carroll algebra, the upper gammas
fail in this regard due to vanishing rotation generators. We will remedy this
via demanding the rotation generators to be compliant with charge conjugation
symmetries, and this leads to a general faithful representation even for the
upper gamma theory. We will discuss the explicit invariance of these actions
under Carroll symmetries.

Once that is done, we would present a general Carroll invariant action in 4d
by just demanding all individual terms to be Carroll boost and rotation
invariant, and show these invariant terms encompass both lower gamma action
and the modified upper gamma action.

Finally, we will give a physical meaning to the two different varieties of
Carroll fermions arising out of the lower and upper gammas.

###  4.1 Action with lower gammas

We will begin by looking at an action constructed out of the lower gamma
matrices in (36). Since the $\tilde{\Sigma}$ matrices constructed out of these
gammas give a proper representation of the spacetime Carroll algebra (minus
translations), we expect that the action constructed out of these matrices
would have invariance under the full Carroll group and when the mass terms are
turned off, we expect to see conformal Carroll invariance appearing.

Evidently, due the faithful representation furnished by the the $\gamma_{\mu}$
matrices, we have $\bar{\Psi}\tilde{\gamma}_{0}\partial_{t}\Psi$ and
$\bar{\Psi}\Psi$ as two Carroll invariant terms, corresponding to the kinetic
and the mass term for a theory of a 4-component fermion $\Psi$. Here adjoint
spinors are defined as (52). Hence the following action is a good point to
start from:

$S_{\text{lower}}=\int
dtd^{3}x\big{(}\bar{\Psi}\tilde{\gamma}_{0}\partial_{t}\Psi-m\bar{\Psi}\Psi\big{)}.$
(57)

One can also write a second type of mass term here given by

$\mathcal{L}_{\text{mass}}=m\bar{\Psi}\tilde{\gamma}_{0}\Psi.$ (58)

It is easily checked that this mass term is indeed Carroll invariant. We will
use this second mass term in the discussion in terms of the two-component
spinors below, although our main interest lies in the massless or conformal
theory.

Due to the degeneracy in the $\gamma$ matrices displayed above, only two of
the four components of the Carroll-Dirac fermion $(\Psi)$:

$\displaystyle\Psi=\begin{pmatrix}\phi\\\ \chi\end{pmatrix}$ (59)

take part in the dynamics and the action, now with second type of mass term,
boils down to:

$S_{\text{lower}}=\int
dtd^{3}x\left(i\phi^{\dagger}\partial_{t}\phi-m\phi^{\dagger}\phi\right),$
(60)

where $\phi$ is a 2-component spinor. For the massive case, this enjoys a
$SU(2)$ symmetry at each space point, which enlarges to $SO(4)$ for the
massless case. Even for the global parts of these transformations, these are
richer than the relativistic case, where one encounters $U(1)\times U(1)$
symmetry only for Weyl (massless fermions) and just a $U(1)$ for the massive
one.

We recall that due to the degeneracy in the representation of the Carrollian
Clifford algebra, we have a theory of two component spinors, resulting into
the dynamics of a single degree of freedom. Moreover the components of the
spinor $\phi$ are decoupled in (60), indicating the said representation to be
reducible. Hence the minimalistic Carrollian fermion theory should have half
degree of freedom 444See, eg. chapter seven of Henneaux:1992ig ..

### Continuous symmetries

For $m\neq 0$, the free theory (60) enjoys space-time translation, spatial
rotation and the infinite number of supertranslation symmetries. Rotation
acting on the two component spinor $\phi$ can be easily read off-from the 4
dimensional representation of the $\Sigma_{ij}$ matrices as:

$\displaystyle\delta_{n}\phi=\epsilon^{ijk}n_{i}x_{j}\partial_{k}\phi-\frac{i}{2}n^{l}\sigma_{l}\phi.$
(61)

for a constant $3$ vector $n^{i}$. On the other hand the supertranslations act
on the 2-component spinor $\phi$ respectively as:

$\displaystyle\delta_{f}\phi=f(x)\partial_{t}\phi.$ (62)

Here by $f$ we denote any tensor field on $\mathbb{R}^{3}$. The special cases
$f=1,f=x^{i}$ and $x^{i}x_{i}$ respectively correspond to time translation,
Carrollian boost and the temporal part of the special conformal
transformation. It is to be noted that although the mass term breaks special
conformal symmetry, the temporal part of it still holds to be a symmetry. Also
note that, since the components of $\phi$ are decoupled, the spin matrix
doesn’t appear in the spatial rotation generator.

For the massless case, as expected, the dilatation and the spatial part of the
special conformal transformation are symmetries as well:

$\displaystyle\delta_{\Delta}\phi=(t\partial_{t}+x_{k}\partial_{k}+\Delta)\phi,$
$\displaystyle\delta_{k}\phi=k^{j}\left(2\Delta
x_{j}+2x_{j}t\partial_{t}+2x_{i}x_{j}\partial_{i}\phi-
x_{i}x_{i}\partial_{j}-i\epsilon_{ijk}x^{i}\sigma_{k}\right)\phi.$ (63)

For this case invariance under dilatation imposes that $\Delta=\frac{d-1}{2}$.
For $d=4$, $\Delta=\frac{3}{2}$, as in the relativistic case. This matches
with the previous exposition with Carrollian fermions Bagchi:2019xfx . It is
interesting to note here that even though the whole finite CCA does not turn
out to be symmetries of the massive system, due to the obvious breaking of
scaling invariance by the mass term, the infinite dimensional
supertranslations remain a symmetry of (57), even with this mass term.

It’s time we discuss the internal symmetries of the 2-component theory of
Carrollian fermions (60). Rewriting the 2-component Grassman field, stripped
to the real parts as:

$\phi=\begin{pmatrix}\phi_{1}+i\phi_{2}\\\ \phi_{3}+i\phi_{4}\end{pmatrix},$

we see the that the action (60) becomes:

$\displaystyle S_{\text{lower}}=i\int
dtd^{3}x\,\left(\sum_{l=1}^{4}\phi_{l}\partial_{t}\phi_{l}-2m\left(\phi_{1}\phi_{2}+\phi_{3}\phi_{4}\right)\right).$
(64)

The kinetic part is manifestly invariant under $SO(4)$ symmetry
transformations, with arbitrary spatial dependence, which breaks to $SU(2)$
for the mass term. The corresponding Noether charges respectively for the
massless and the massive case are:

$\displaystyle Q_{(m=0)}=\int
d^{3}x\,\lambda^{ij}(x)\phi_{i}\phi_{j},~{}~{}~{}Q_{(m\neq 0)}=\int
d^{3}x\,\phi^{\dagger}\mathbf{n}(x)\cdot\boldsymbol{\sigma}\phi.$ (65)

Here $\lambda$ are $\mathbf{R}^{3}$ valued lie algebra elements of
$\mathfrak{so}(4)$ and $\mathbf{n}(x)$ are vector fields on $\mathbf{R}^{3}$.

We discuss a rather intriguing feature of these lower Carroll fermion before
ending this subsection regarding the space-time rotation symmetry of the
action (60). It is evident that the spin-matrix part appearing in the rotation
transformation rule (61) is derived from the Clifford algebra representation
($\Sigma_{ij}$), substantiating $\phi$ as a spin-1/2 field. However, the
action (60), having now spin mixing involved is invariant, even if we treat
$\phi$ as a scalar (spin-0) under space-time rotation. This indicates that
Carroll symmetry doesn’t necessitate fermions (anti-commuting fields) to have
half-integral spin representations. We will give a more intuitive picture of
why this peculiar feature arises here at the end of this section.

### The Hamiltonian

Before we go on to actions for upper gammas, let us point out another
interesting feature of this action that will be of interest later in the
paper. The momentum conjugate to the spinor is given by

$\displaystyle\Pi=\partial\mathcal{L}/\partial\dot{\Psi}=\bar{\Psi}\tilde{\gamma}_{0}.$
(66)

The Hamiltonian for the Carroll fermions is given by

$\mathcal{H}_{\text{lower}}=\Pi\dot{\Psi}-\mathcal{L}=m\phi^{\dagger}\phi,$
(67)

where we have used the Weyl mass term (58). We note that the Hamiltonian
vanishes for the massless case.

The vanishing of the Hamiltonian in the massless case is also borne out by the
computation of the energy-momentum tensor for the action. Starting from the
Lagrangian (57), one can use Noether’s procedure to arrive at the energy-
momentum tensor for these lower Carroll fermions:

$T^{a}_{\,\,\,\ b}=\frac{\partial
L}{\partial(\partial_{a}\Psi)}\partial_{b}\Psi+\frac{\partial
L}{\partial(\partial_{a}\bar{\Psi})}\partial_{b}\bar{\Psi}-\delta^{a}_{\,\,b}\mathcal{L}$
(68)

Component-wise the above turns out to be:

$\displaystyle T^{t}_{\,\,\,t}=m\phi^{\dagger}\phi,\quad
T^{i}_{\,\,\,t}=0,\quad T^{t}_{\,\,\,i}=i\phi^{\dagger}\partial_{i}\phi,\quad
T^{i}_{\,\,\,j}=-\delta^{i}_{j}\mathcal{L}.$ (69)

The vanishing of $T^{i}_{\,\,\,t}$ is a tell-tale sign of a Carroll invariant
system, arising from Carroll boost invariance. The further vanishing of
$T^{t}_{\,\,\,t}$, in the massless case, indicates the consequent vanishing of
the Hamiltonian of the system. Using equations of motion, it is also obvious
that the stress tensor defined is traceless for the massless case indicating
the emergence of conformal Carroll symmetry in the system.

###  4.2 Action with upper gammas

We now consider upper gamma matrices to construct our action. Since the sigma
matrices arising out of these have the peculiar feature that the ones
corresponding to rotations identically vanish, we can see actions written from
here seemingly will not be invariant under the whole Carroll group.

We start with an action in the Dirac form as in the following, again
concentrating on the $4d$ case:

$S_{\text{upper}}=\int
dtd^{3}x\,\bar{\Psi}(\hat{\gamma}^{0}\partial_{t}+\hat{\gamma}^{i}\partial_{i})\Psi.$
(70)

Note that there is no $i$ in this Lagrangian to make it Hermitian with the
choice of $\Lambda$ in (49). Here as we have described before,
$(\hat{\gamma}^{0})^{2}=1$ and $(\hat{\gamma}^{i})^{2}=0$. For the moment, we
will only consider the massless case. Under boost transformation the above
action is invariant but we can clearly see that the vanishing of the
$\hat{\Sigma}^{ij}$ in this representation of upper gammas leads to a
variation of the action which is not seemingly rotation invariant, as all
brackets containing $\hat{\Sigma}^{ij}$ simply drop out. For the special case
of $d=2$, however, since there are no rotation generators, this action becomes
invariant under the whole Conformal Carroll group. Interestingly, the
symmetries get infinitely enhanced to the full BMS3 algebra (125). The details
of this and other important implication of Carroll fermions in $d=2$ are
addressed in a companion paper 2dcarroll .

However, all is not lost for the upper gammas. As we commented earlier, this
apparent non-invariance stems from the peculiar degenerate structure of the
spatial $\hat{\gamma}^{i}$ matrices. Remember, those do not have a clear
hermitian structure. Hence, we need to find a generator for the $SO(3)$
subalgebra which is manifestly compliant with the charge conjugation
symmetries. In this regard, let us define a modified rotation generator that
makes the spin current symmetric under charge conjugation:

$\displaystyle\hat{\Sigma}^{ij}_{c}=\frac{1}{4}[\hat{\gamma}^{i},\hat{\gamma}^{j}_{c}]+\frac{1}{4}[\hat{\gamma}^{i}_{c},\hat{\gamma}^{j}]$
(71)

where we have put in a subscript $c$ to remind the reader of the fact that the
charge conjugation plays a crucial role in making up these modified rotation
generators. Here

$\hat{\gamma}^{i}_{c}=-\mathcal{C}\hat{\gamma}^{i}\mathcal{C}^{-1},$ (72)

and $\mathcal{C}$ is as given in (55). What this effectively does is to impose
proper Hermiticity onto the rotation subalgebra, which was erstwhile missing.
Note here that spatial lower gammas $\tilde{\gamma}_{i}$ are not degenerate
and hence do not need this modified definition. As a matter of fact, in any
representation where the Hermitian conjugate of spatial gamma matrices are
related to itself upto a sign (relativistic Dirac matrices for example), the
above definition boils down to the usual definition of rotation generators.
Using these above representation the generators of Carroll algebra become
matrices of the form:

$\displaystyle\hat{\Sigma}^{0i}_{c}=-\frac{1}{2}\begin{pmatrix}0&0\\\
\sigma^{i}&0\end{pmatrix},\quad\hat{\Sigma}^{ij}_{c}=\frac{i}{2}\epsilon^{ijk}\begin{pmatrix}\sigma^{k}&0\\\
0&\sigma^{k}\end{pmatrix}.$ (73)

They can be checked to explicitly close the Carroll boost-rotation algebra,

$\displaystyle[\hat{\Sigma}^{0i}_{c},\hat{\Sigma}^{0j}_{c}]=0$ (74a)
$\displaystyle[\hat{\Sigma}^{0i}_{c},\hat{\Sigma}^{jk}_{c}]=-\delta^{ik}\hat{\Sigma}^{0j}_{c}+\delta^{ij}\hat{\Sigma}^{0k}_{c}$
(74b)
$\displaystyle[\hat{\Sigma}^{ij}_{c},\hat{\Sigma}^{kl}_{c}]=\delta^{il}\hat{\Sigma}^{jk}_{c}-\delta^{ik}\hat{\Sigma}^{jl}_{c}+\delta^{jk}\hat{\Sigma}^{il}_{c}-\delta^{jl}\hat{\Sigma}^{ik}_{c}.$
(74c)

Also this furthermore implies

$\displaystyle(\hat{\Sigma}^{0i}_{c})^{\dagger}=-\Lambda\hat{\Sigma}^{0i}_{c}\Lambda,\quad(\hat{\Sigma}^{ij}_{c})^{\dagger}=-\Lambda\hat{\Sigma}^{ij}_{c}\Lambda.$
(75)

Equipped with a now proper faithful representation of the Carroll generators,
we can now go ahead to investigate the symmetries of our massless Dirac upper
gamma action in (70).

### Continuous symmetries

Of course the action is trivially invariant under translations. In what
follows let us mention the non-trivial symmetries of the total bispinor
$\Psi$. Under the action of Carroll boosts, the spinors and their adjoints
change as:

$\displaystyle\delta_{C}\Psi=(x_{i}\partial_{t}-\hat{\Sigma}^{0i}_{c})\Psi,\quad\delta_{C}\bar{\Psi}=(x_{i}\partial_{t}\bar{\Psi}+\bar{\Psi}\hat{\Sigma}^{0i}_{c})$
(76)

then the action is invariant under these changes as the Lagrangian variation
$\delta_{C}\mathcal{L}=\partial_{t}(x^{i}\mathcal{L})$ is a total
derivative555The reader may be a bit perturbed by the structure of the indices
in the above equations, but we reminder her that we are not thinking of a
Carroll covariant formulation in this work. The $\hat{\Sigma}_{c}$ are just
the rotation matrices constructed out of the upper gammas, taking into account
the charge conjugation.. Under rotations, the fields change as

$\displaystyle\delta_{J}\Psi=(x_{i}\partial_{j}-x_{j}\partial_{i}-\hat{\Sigma}^{ij}_{c})\Psi,\quad\delta_{J}\bar{\Psi}=(x_{i}\partial_{j}-x_{j}\partial_{i})\bar{\Psi}+\bar{\Psi}\hat{\Sigma}^{ij}_{c}.$
(77)

Remember this was the non trivial one in the case of rotation generators
beforehand. Since the modified generator and commutators involving it do not
vanish, one can see that the change in the Lagrangian would be

$\displaystyle\delta_{J}\mathcal{L}=\partial_{j}\big{(}x_{i}\mathcal{L}\big{)}-\partial_{i}\big{(}x_{j}\mathcal{L}\big{)},$
(78)

and hence, invariance will be restored. One can go further ahead and check for
all global generators of the Carroll Conformal Algebra in $4d$, including:

Dilatation:

$\delta_{D}\Psi=(t\partial_{t}+x_{k}\partial_{k}+\Delta)\Psi,~{}~{}~{}\delta_{D}\bar{\Psi}=(t\partial_{t}+x_{k}\partial_{k}+\Delta)\bar{\Psi}$
(79)

Temporal Special Conformal Transformation:

$\delta_{K}\Psi=-x_{i}x_{i}\partial_{t}\Psi+2x_{i}\hat{\Sigma}^{0i}_{c}\Psi~{}~{}~{}\delta_{K}\bar{\Psi}=-x_{i}x_{i}\partial_{t}\bar{\Psi}-2x_{i}\bar{\Psi}\hat{\Sigma}^{0i}_{c}.$
(80)

Spatial Special Conformal Transformation:

$\displaystyle\delta_{K_{j}}\Psi=2\Delta
x_{j}\Psi+2x_{j}t\partial_{t}\Psi+2x_{i}x_{j}\partial_{i}\Psi-
x_{i}x_{i}\partial_{j}\Psi-2x_{i}\hat{\Sigma}^{ij}_{c}\Psi-2t\hat{\Sigma}^{0j}_{c}\Psi$
(81a) $\displaystyle\delta_{K_{j}}\bar{\Psi}=2\Delta
x_{j}\Psi+2x_{j}t\partial_{t}\bar{\Psi}+2x_{i}x_{j}\partial_{i}\bar{\Psi}-x_{i}x_{i}\partial_{j}\bar{\Psi}+2x_{i}\bar{\Psi}\hat{\Sigma}^{ij}_{c}+2t\bar{\Psi}\hat{\Sigma}^{0j}_{c}.$
(81b)

One can show clear invariance of the Lagrangian under all these finite CCA
transformations. Specifically, under Dilatations, the spinors change again
with $\Delta=\frac{3}{2}$ as in the case before.

### Hamiltonian and Stress Tensor

In the case of Lower gamma representations, we were concerned with only one
two component block of the bispinor. Since in the current case all four
components are involved in the dynamics, let us clearly define our spinor
structure:

$\displaystyle\Psi=\begin{pmatrix}\phi\\\ \chi\end{pmatrix}$ (82)

where both $\phi,\chi$ are two component spinors. With this structure, the
lagrangian (70) can be written in terms of components:

$\mathcal{L}_{\text{upper}}=-i\big{(}\chi^{\dagger}\dot{\phi}+\phi^{\dagger}\dot{\chi}-\phi^{\dagger}\sigma^{i}\partial_{i}\phi\big{)},$
(83)

with a clear presence of space derivatives unlike the lower gamma case. One
can compare this action with the $2d$ worldsheet actions found in relation to
Inhomogeneous super-BMS symmetries for tensionless strings in Bagchi:2017cte ;
Bagchi:2018wsn  and see this is a natural higher dimensional analogue of the
one written there. Now we can easily find out the equations of motion:

$\displaystyle\dot{\phi}=0,\quad\dot{\chi}=\sigma^{i}\partial_{i}\phi.$ (84)

Noting the conjugate momenta $\Pi_{\Psi}=\bar{\Psi}\tilde{\gamma}^{0}$, it is
then also easy to construct the Hamiltonian density:

$\displaystyle\mathcal{H}_{\text{upper}}=\Pi\dot{\Psi}-\mathcal{L}_{\text{upper}}=-\bar{\Psi}\hat{\gamma}^{i}\partial_{i}\Psi=i\phi^{\dagger}\sigma^{i}\partial_{i}\phi.$
(85)

Again, note that contrary to the lower gamma case, the massless Hamiltonian
does not vanish.

Starting from the Lagrangian using the Noether’s procedure we can calculate
the energy-momentum tensor for upper Carroll fermions:

$T^{\mu}_{\,\,\,\nu}=\frac{\partial
L}{\partial(\partial_{\mu}\Psi)}\partial_{\nu}\Psi+\frac{\partial
L}{\partial(\partial_{\mu}\bar{\Psi})}\partial_{\nu}\bar{\Psi}-\delta^{\mu}_{\,\,\,\nu}\mathcal{L}.$
(86)

Here we can write component wise,

$\displaystyle
T^{t}_{\,\,\,t}=-\bar{\Psi}\hat{\gamma}^{i}\partial_{i}\Psi,\quad
T^{i}_{\,\,\,t}=\bar{\Psi}\hat{\gamma}^{i}\partial_{t}\Psi,\quad
T^{t}_{\,\,\,i}=\bar{\Psi}\hat{\gamma}^{0}\partial_{i}\Psi,\quad
T^{i}_{\,\,\,j}=\bar{\Psi}\hat{\gamma}^{i}\partial_{j}\Psi-\delta^{i}_{j}\mathcal{L}.$
(87)

Trace of the EM tensor $T^{\mu}_{\,\,\,\mu}=-3\mathcal{L}$ which vanishes
using equation of motion. Futhermore, we can see $T^{i}_{\,\,\,t}$ vanishes
on-shell for this case, signifying the onset of Carroll physics.

An indication of emergent Carrollian symmetry in a system is the vanishing of
the commutator of Hamiltonians at different spacetime points Henneaux:2021yzg
. Although in the case of the lower fermions, this was trivial, due to the
presence of spatial derivatives, this is no longer obvious for the upper
fermions. We check this explicitly below. One can impose the canonical
commutation relations, which read

$\displaystyle\\{\Pi(x),\Psi(x^{\prime})\\}=\delta(x-x^{\prime})\rightarrow\\{\bar{\Psi}(x),\Psi(x^{\prime})\\}=\hat{\gamma}^{0}\delta(x-x^{\prime})$
$\displaystyle\\{\Psi(x),\Psi(x^{\prime})\\}=\\{\bar{\Psi}(x),\bar{\Psi}(x^{\prime})\\}=0$
(88)

As a further consistency check of Carroll invariance, one can try to evaluate
the commutator between Hamiltonian densities,

$\displaystyle[\mathcal{H}(x),\mathcal{H}(x^{\prime})]$
$\displaystyle=[\bar{\Psi}(x)\hat{\gamma}^{i}\partial_{i}\Psi(x),\bar{\Psi}(x^{\prime})\hat{\gamma}^{j}\partial_{j}\Psi(x^{\prime})]$
$\displaystyle=\bar{\Psi}(x)\hat{\gamma}^{i}\hat{\gamma}^{0}\hat{\gamma}^{j}\partial_{i}\delta(x-x^{\prime})\partial_{j}\Psi(x^{\prime})-\bar{\Psi}(x^{\prime})\hat{\gamma}^{j}\hat{\gamma}^{0}\hat{\gamma}^{i}\partial_{j}\delta(x-x^{\prime})\partial_{i}\Psi(x)$
$\displaystyle=-\bar{\Psi}(x)\hat{\gamma}^{0}\hat{\gamma}^{i}\hat{\gamma}^{j}\partial_{i}\delta(x-x^{\prime})\partial_{j}\Psi(x^{\prime})+\bar{\Psi}(x^{\prime})\hat{\gamma}^{0}\hat{\gamma}^{j}\hat{\gamma}^{i}\partial_{j}\delta(x-x^{\prime})\partial_{i}\Psi(x)$

Now using Clifford algebra , the product of two $\hat{\gamma}^{i}$ matrices is
identically zero. So the above commutator vanishes:

$[\mathcal{H}(x),\mathcal{H}(x^{\prime})]=0.$ (89)

###  4.3 Generic Carroll boost invariant action

As the reader noticed, in the last section we modified the definition of
rotation generators to retrace the invariance of our upper gamma action under
$SO(3)$ subalgebra. Since this may have come in as a novelty, and the seeming
non-invariance of the case earlier was puzzling enough, let us give some more
arguments in favour of our upper gamma action.

We set out to find the most general $4d$ spinor lagrangian invariant under
Carroll boosts are rotations. In terms of two 2-component spinors $\phi$ and
$\chi$, one can write the most generic Lagrangian involving all possible
temporal and spatial derivative terms as

$\mathcal{L}=a\phi^{\dagger}\dot{\phi}+b\chi^{\dagger}\dot{\chi}+c\phi^{\dagger}\dot{\chi}+d\chi^{\dagger}\dot{\phi}+e\phi^{\dagger}\sigma\cdot\partial\phi+f\chi^{\dagger}\sigma\cdot\partial\chi+g\phi^{\dagger}\sigma\cdot\partial\chi+h\chi^{\dagger}\sigma\cdot\partial\phi$
(90)

Here all the coefficients are in general complex. The two component spinors
transform under Carroll boost and rotation as

$\displaystyle\delta_{b}\phi=b^{i}x_{i}\partial_{t}\phi,~{}~{}~{}~{}\delta_{b}\chi=b^{i}(x_{i}\partial_{t}\chi+\alpha\sigma_{i}\phi)$
(91)
$\displaystyle\delta_{n}\phi=\epsilon^{ijk}n_{k}x_{i}\partial_{j}\phi+\frac{i}{2}n^{i}\sigma_{i}\phi,~{}~{}~{}~{}\delta_{n}\chi=\epsilon^{ijk}n_{k}x_{i}\partial_{j}\chi+\frac{i}{2}n^{i}\sigma_{i}\chi.$
(92)

Note that $\phi$ and $\chi$ mix under boosts for the transformation of $\chi$.
The parameter $\alpha$ comes from a ambiguity in the Clifford Algebra, since a
rescaling of $\hat{\gamma}^{i}$ matrices do not change the algebra. This
constant rescaling parameter appears in the spin-matrix part of the boost
transformation rule. The effect of this (the ‘$a$’ in (44)) is very
interesting in the $2d$ Conformal Carroll case, where this scaling gives rise
to an automorphism of the superrotation algebra 2dcarroll . Invariance of the
test lagrangian under these symmetries imply certain constraints on the
constant coefficients:

$\displaystyle c\alpha+d\alpha^{*}+e=0,\quad b=f=g=h=0.$ (93)

Hence the form of boost-rotation invariant Lagrangian gets modified as

$\mathcal{L}=a\phi^{\dagger}\dot{\phi}+c\phi^{\dagger}\dot{\chi}+d\chi^{\dagger}\dot{\phi}+e\phi^{\dagger}\sigma\cdot\partial\phi.$
(94)

We have to also ensure that the Lagrangian is Hermitian. This requires further
constraints on the parameters:

$\displaystyle a^{*}=-a,\,c=-d^{*},\,e^{*}=-e.$ (95)

Then without any loss of generality, we could choose $c=d=-i,~{}a=e=i$, and
the boost ambiguity can be chosen as $\alpha=\frac{1}{2}$ consistently. Now
with this choice, we can clearly see our most general invariant Lagrangian
(94) splits into a chiral part depending on $\phi$, which turns out to be the
massless lower gamma action (60), and the rest is simply identified with the
upper gamma cousin of it (83). So, in general, any Carroll invariant fermionic
Lagrangian can be written as

$\mathcal{L}=a_{L}\mathcal{L}_{\text{lower}}+a_{U}\mathcal{L}_{\text{upper}},$
(96)

where we have introduced two constants $a_{L}$ and $a_{U}$. We have not
written any term that introduces an interaction between these two types of
fermions. It would be interesting to see if such terms are allowed from
geometric considerations of generic Carroll manifolds.

###  4.4 Physics of two types of Carroll fermions

Let us pause to summarise our findings so far. The geometry of Carrollian
manifolds suggested that there may be different types of fermions residing
here. This was due to the two different Clifford algebras one could write down
(125). Following an analysis of the representation theory, in this section, we
wrote down two distinct actions for the two different types of fermions (125)
and (125), which we will keep calling lower and upper. These led to theories,
both Carroll invariant, but with different distinguishing features, e.g. a
zero and a non-zero Hamiltonian. So what are these two different types of
fermions, and can we say something more fundamental about them? Below we try
to paint a more physical picture of what has so far been a mostly mathematical
exercise.

Figure 1: Carroll fermions viewing $\mathscr{I}^{\pm}$: (A) Lower fermions,
(B) Upper fermions

##### Lower fermions:

The lower fermions come from the “time-like” part of the Clifford algebra.
This Clifford algebra does not know about what is happening in the spatial
part of the Carroll spacetime. The action (125) also suggests that the spatial
parts just go for a ride here. This is most apparent in the feature discussed
below (65). These lower fermions, although anti-commuting, need not have half-
integral spins. This just means that the spin part, which resides in the
$(d-1)$ dimensional spatial fibre of the $d$ dimensional Carrollian spacetime,
does not have a defining role for the lower fermions. In the closed up
lightcone, they reside purely along the null-timelike direction. In a sense,
they behave like a one-dimensional theory living only on the null line,
oblivious to the spatial fibre. E.g. when we consider lower fermions on the
null boundary of 4d Minkowski spacetimes $\mathscr{I}^{\pm}$, which is given
by R${}_{u}\times$S2, these fermions would only see the null line parametrized
by the retarded time $u$ and fail to see the celestial spheres attached to
each point on the null line. This is depicted by part (A) in Figure (1).

##### Upper fermions:

The upper fermions, on the other hand, come from the “spatial” part of the
Carroll Clifford algebra. In the action (125) there are now spatial
derivatives along with the temporal ones. The upper fermions clearly see the
base of the fibre bundle, i.e. the null time direction, as well as the spatial
fibres. These fermions are more usual, more like their relativistic
counterparts. Their anticommuting nature comes hand in hand with their half-
integral spins. Going back to the example of the null boundary of 4d flat
spacetimes, the upper fermions see the whole R${}_{u}\times$S2 structure, as
shown in part (B) in Figure (1).

The fact that the massless version of lower fermions have a vanishing
Hamiltonian and the upper massless fermions have a non-vanishing one and they
behave in very different ways can also be traced back to the representation
theory of the Carroll algebra, where the Hamiltonian is a central element and
the two cases $H=0$ and $H\neq 0$ leads to different classes of
representations. See deBoer:2021jej  for more details on these classes of
representations.

Carroll theories, generically, are of two types, Electric and Magnetic. These
names have been derived from the Galilean counterparts and specifically the
theory of electromagnetism in the non-relativistic world where one can have
two distinct limits called the electric and magnetic limits where the electric
effects dominate over magnetic ones and vice versa. Similar phenomena have
been seen in generic field theories in the Carrollian regime, and the electric
theories are defined as ones where temporal features dominate over spatial
ones and similarly for magnetic theories. Even Carroll scalars come in these
forms 666There is even a third type of Carroll scalar in two spacetime
dimensions Bagchi:2022eav .. From our above discussion, it seems that lower
fermions are Electric and the upper ones are Magnetic.

In generic field theories, the electric theory is obtained as the leading term
in a $c\to 0$ limit of the corresponding relativistic theory, while the
magnetic theory comes from the sub-leading term. A naive analysis of the $c\to
0$ limit of a relativistic fermionic field theory runs into some hurdles which
we have not immediately been able to sort out. We hope to return to this in
the near future.

##  5 Flat Bands and the Emergence of Carroll

The relation between energy and momentum, the so-called dispersion relation,
is at the heart of many complicated phenomena in real life condensed matter
systems. For non-relativistic free particles, energy depends on the square of
the spatial momentum and this simple fact coupled with the requirement of a
minimum energy state leads to bosons bunching together and forming a Bose-
Einstein condensate at zero momentum. Fermions on the other hand avoid each
other and fill up the dispersion curve up to the Fermi energy. This simple
quadratic dispersion explains a lot of the physics of metals, semi-conductors
and simple insulators. Low temperature properties of magnetic materials are
understood in terms of linear or quadratic dispersion relations.

Of late, there has been a surge of interest in understanding flat band
physics, where the energy don’t depend on momentum at all. Usual intuitions
don’t hold in this regime, — bosons don’t condensate, fermions don’t form
usual Fermi surfaces. A lot of the known physics breaks down. Flat band
physics seems appear in some very interesting systems including spin liquids,
fractional quantum hall systems, and twisted bi-layer graphene, where
superconductivity appears at unexpected temperatures.

###  5.1 A quick argument

Below we will go on to show that Carroll invariance is a generic feature of
such flat band physics and then give a flavour of this in the twisted bi-layer
graphene system. But before we proceed with the calculations and explicit
demonstrations, here is a quick physical argument of why one should expect
Carroll invariance to appear in these contexts.

We are attempting to understand flat bands. This is a limit where the cone in
position space with the energy and momentum collapses onto the momentum plane.
A priori this looks like a non-relativistic limit where the speed of light
goes to infinity and the lightcone opens up completely. But we must remember
that we are in momentum space. Opening up of the cone in momentum space is
equivalent to the actual position space lightcone closing down. This is
obviously the Carrollian limit and hence whenever one encounters flat bands in
systems, one will generically end up with Carrollian structures 777This
understanding came about in a quick discussion with Blagoje Oblak. AB thanks
him for asking the question which led to this picture..

Figure 2: (a) Flattening out of the momentum space dispersion relation and (b)
closing down of space-time light cone $c\rightarrow 0$.

Our discussion above and explicit calculations below also tie up very nicely
with the claims in the condensed matter literature that flat bands generically
lead to localisation in position space. From a Carroll perspective, this
localisation in space is a manifest consequence of the closing up of the
lightcone as the speed of light goes to zero. Having understood this very
quick intuitive and physical picture, we make the emergence of Carroll
symmetry very explicit below.

###  5.2 Emergence of Carroll

Let us consider the following free theory Hamiltonian, expressed in terms of
infinite number of Heisenberg picture oscillators (assume the system to be
autonomous):

$\displaystyle H=\sum_{n}E_{n}a^{\dagger}_{n}a_{n},$ (97)

where we will take the liberty of considering the oscillators to be
uncountably infinite as well, thus being able to convert the sum to integral
when focusing on specific models. Also the index $n$ should be taken as a
mnemonic for a number of indices, denoting other charges present in the
system.

Let us start with the case of fermions, $i.e.$ equal-time anti-commuting
oscillators

$\\{a^{\dagger}_{n},a_{m}\\}=\delta_{mn}$ (98)

where it is understood that the Kronecker delta is to be replaced by Dirac
delta as and when needed. Now we make an assumption that there exists an
operator $\mathfrak{h}$, which is self adjoint on the Hilbert space of square
integrable (generalizable to multi-component/ vector-values, as in the case of
Weyl or Dirac spinors) functions on $\mathbb{R}^{d}$, with suitable boundary
conditions. Also we demand that the eigenvalues of $\mathfrak{h}$ are $E_{n}$
as appearing in (97). $u_{n}(\mathbf{r})$ are orthonormal eigenfunctions:

$\mathfrak{h}u_{n}(\mathbf{r})=E_{n}u_{n}(\mathbf{r}).$ (99)

Hence the Hamiltonian can be repackaged as:

$\displaystyle H=\sum_{m,n}\int
d^{d}\mathbf{r}\,a^{\dagger}_{m}(t)u^{\star}_{m}(\mathbf{r})\mathfrak{h}\left(a_{n}(t)u_{n}(\mathbf{r})\right)=\int
d^{d}\mathbf{r}\,\Psi^{\dagger}(\mathbf{r},t)\mathfrak{h}\Psi(\mathbf{r},t)$
(100) $\displaystyle\mbox{where
}\Psi(\mathbf{r},t):=\sum_{n}a_{n}(t)u_{n}(\mathbf{r}).$

The definition of the field operator $\Psi$ above, in terms of the oscillators
directly imply the following non-zero anti-commutator:

$\displaystyle\\{\Psi(\mathbf{r},t),\Psi^{\dagger}(\mathbf{r}^{\prime},t)\\}=\delta^{d}(\mathbf{r}-\mathbf{r}^{\prime}).$
(101)

At the classical level, identifying $\Psi^{\star}$ as the canonically
conjugate momentum of $\Psi$, the action can be written via Legendre
transformation 888This involves a second class constraint, which is easily
solved. Also, for $\mathfrak{h}$ being a matrix-valued differential operator,
ie, $u$ being multi-component functions, the inner product needs to be
adjusted with insertion of another matrix for the sake of self-adjointness.:

$\displaystyle S=\int
dt\,d^{d}\mathbf{r}\Psi^{\star}\left(i\partial_{t}-\mathfrak{h}\right)\Psi.$
(102)

As is well known, for $d=3$ Weyl fermions, the dispersion in (97) is
$E_{\mathbf{k}}=c|\mathbf{k}|$, while for massive Dirac fermions,
$E_{\mathbf{k}}=c\left(\mathbf{k}^{2}+m^{2}c^{2}\right)^{1/2}$ and
$\mathfrak{h}$ is 2 and 4 dimensional matrix valued differential operator
respectively for these cases. The large $c$ limit of the later gives us:
$E_{\mathbf{k}}\approx\frac{\mathbf{k}^{2}}{2m}$, which describes the
Schrödinger field theory and is described by
$\mathfrak{h}=-\frac{1}{2m}\nabla^{2}$, on single component fermions.

In line of the above discussion, let’s for simplicity take a single component
continuum field $\Psi$ to describe the real space degrees of freedom. We now
impose the supertranslation transformations on it as:

$\displaystyle\delta_{\mathbf{f}}\Psi(\mathbf{r},t)=f(\mathbf{r})\partial_{t}\Psi,$
(103)

for any real function $f$ on $\mathbb{R}^{d}$. As described earlier in this
paper, $f=1$ is the time translation and $f=\mathbf{b}\cdot\mathbf{r}$, for
any $d$-vector $\mathbf{b}$ is the Carrollian boost. We don’t have an
inhomogeneous term because this is a single component case. On the space of
Heisenberg picture oscillators

$a_{m}(t)=e^{-iE_{m}\,t}a_{m}(0)$ (104)

this transformation acts as:

$\displaystyle\delta_{f}a_{m}(t)=-i\sum_{n}E_{n}\int
d^{d}\mathbf{r}\,f(\mathbf{r})\,u_{n}(\mathbf{r})u^{\star}_{m}(\mathbf{r})a_{n}(t)$
(105)

and similarly on $a^{\dagger}_{m}$. The transformation of the fundamental
anti-commutators is

$\displaystyle\delta_{f}\\{a_{m},a^{\dagger}_{n}\\}=-i\int
d^{d}\mathbf{r}\,f(\mathbf{r})u_{n}(\mathbf{r})u^{\star}_{m}(\mathbf{r})(E_{n}-E_{m})$
(106)

Note that the above expression vanishes identically for $f=1$ due to
orthonormality of the eigenfunctions $u_{n}$. This is expected, since by
definition the system is time translation invariant. A sufficient condition
for the above to vanish for an arbitrary $f$ is

$E_{n}=E_{m}\quad\forall\,m,n,$

ie, the dispersion is trivial. Or in other words, the Hamiltonian has to be a
flat-band one. The invariance of the brackets $\\{a_{m},a_{n}\\}$ and
$\\{a^{\dagger}_{m},a^{\dagger}_{n}\\}$ does not put any more conditions on
the dispersion. Furthermore, with the flat-dispersion Hamiltonian
$H=E\sum_{n}a^{\dagger}_{n}a_{n}$ also is invariant under the super-
translation transformations:

$\displaystyle\delta_{f}H=0.$ (107)

The canonical structure and the Hamiltonian for the flat-band system remaining
invariant implies that the system enjoys infinite number of Carrollian
supertranslation symmetries.

Since $E_{n}$ are the eigenvalues of the differential operator $\mathfrak{h}$,
one comes across a flat-dispersion situation whenever $\mathfrak{h}$ doesn’t
contain any derivative at all and only is a constant. This is certainly true
if the spatial manifold is $\mathbb{R}^{d}$ and one has open boundary
conditions. However there exists non-trivial $\mathfrak{h}$ containing
derivatives operating on Hilbert space of functions, whose domains are compact
submanifolds of $\mathbb{R}^{d}$ with periodic boundary conditions. We will
illustrate a couple of such cases in what follows.

A further caveat is that, for a multi-component $\Psi$, in general the
$\mathfrak{h}$ dispersion set comes in the form of a number of disconnected
branches/ bands, some of which may cross at a finite number of points. Since
the distinct bands describe dynamics of decoupled degrees of freedom, for any
band that is pristine and flat (i.e. not crossing any other bands), the
corresponding degrees of freedom will have Carrollian supertranslations as
symmetry. Moreover, such a pristine flat band, if it is the lowest energy one,
would capture effective quantum dynamics from the Wilsonian perspective.

Before delving into examples emerging from low energy physics, let’s try to
understand connection between flat-bands and Carroll symmetry starting from
Lorentz covariant systems having trivial open boundary conditions. The
Hamiltonian of a free massive Lorentz covariant theory has the structure, in
terms of fourier basis oscillator modes:

$\displaystyle H=\int
d^{d}\mathbf{k}\sqrt{c^{2}\mathbf{k}^{2}+m^{2}c^{4}}\,a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}.$
(108)

We have just one component and suppressed the spin and polarization
information for convenience. The dispersion relation
$E_{k}=\sqrt{c^{2}\mathbf{k}^{2}+m^{2}c^{4}}$ bears the fact that the theory
is Lorentz covariant. Written in real space, the integrand in the Hamiltonian
will involve spatial derivatives in fields, resulting in to non-commuting
Hamiltonian densities. But, a direct consequence of Carroll invariance is
commuting Hamiltonian densities , as per Henneaux:2021yzg . Therefore it
follows that theories with no spatial derivative in real space Hamiltonian and
hence having a non-dispersive Hamiltonian in terms of Fourier modes should
have Carrollian symmetry. We have also seen that there are Carrollian theories
having spatial derivatives, like in the case of our upper fermions. But still
the Hamiltonian densities commute and there is emergent Carrollian symmetry.

One straightforward way to construct such a theory is starting with
relativistic dispersion $E_{\mathbf{k}}=\sqrt{c^{2}\mathbf{k}^{2}+m^{2}c^{4}}$
and take $c\rightarrow 0$ with $mc^{2}=E_{0}$ remaining finite. $E_{0}$
defines the UV cut-off at which the theory is defined. Hence $E_{k}=E_{0}$
becomes dispersion-less (or flat-band, as commonly referred to in condensed
matter physics literature). For the special case of $m=0$, the dispersionless
Hamiltonian is identically zero.

##  6 Carroll Fermions in Condensed Matter Systems

In this section, we look at two examples of Carrollian fermions in real life
condensed matter systems. Our first example will be that of a $1d$ chain,
while the second example be in $2+1$ dimensions and will involve twisted bi-
layered graphene, which shows emergent superconductivity. We will give some
details of usual graphene and its bi-layer cousin for the benefit of people
like ourselves who may be unfamiliar with these systems.

###  6.1 Carroll fermions in one dimensional chain

We begin in $1+1$ dimensions, with a 1 dimensional lattice model of fermions
known as the diamond network ladder leykam2018artificial  or the Creutz ladder
creutz2001aspects . Consider, as per the figure below, a ladder consisting of
a couple of parallel 1 dimensional crystals (arbitrarily long), each having
same translational lattice symmetry with lattice parameter $A$. In the end of
the analysis, we will take $A\rightarrow 0$. As per our model, for each chain,
tightly bound electrons can experience nearest neighbour hopping with energy
$t$. Also there are inter-chain hoppings, which are both nearest ($\gamma$) as
well as next-to-nearest neighbour ($\gamma^{\prime}$).

Figure 3: The coupled chain system with all the intra-chain and inter-chain
hopping strengths shown.

The Hamiltonian in real space oscillator basis is:

$\displaystyle
H=\sum_{p}t(a^{\dagger}_{p+1}a_{p}+b^{\dagger}_{p+1}b_{p})+\gamma
a^{\dagger}_{p}b_{p}+\gamma^{\prime}(a^{\dagger}_{p+1}b_{p}+a^{\dagger}_{p}b_{p+1})+\mathrm{h.c.}$
(109)

For periodic boundary conditions, one may go to the Fourier basis defined by

$c_{k}=\sum_{p}a_{p}e^{ikAp},d_{k}=\sum_{p}b_{p}e^{ikAp}\quad\text{for}\quad
k\in(-\pi/A,\pi/A).$ (110)

As a consequence, the Hamiltonian becomes:

$\displaystyle H=\sum_{k}\begin{pmatrix}c_{k}^{\dagger}\\\
d_{k}^{\dagger}\end{pmatrix}\begin{pmatrix}2t\,\cos(kA)&\gamma+2\gamma^{\prime}\cos(kA)\\\
\gamma+2\gamma^{\prime}\cos(kA)&2t\,\cos(kA)\end{pmatrix}\begin{pmatrix}c_{k}&d_{k}\end{pmatrix}.$
(111)

Diagonalizing, we get two bands:

$E^{\pm}(k)=2(t\pm\gamma^{\prime})\cos(k)\pm\gamma.$ (112)

We readily observe that for $t=\gamma^{\prime}$, $E^{-}$ becomes independent
of $k$ and hence a flat-band appears. Hence for $t=\gamma^{\prime}$, in
another basis of modes, in which the matrix in (111) is diagonal, we have the
Hamiltonian (109) $H=H_{\mathrm{flat}}+H^{\prime}$, with

$H_{\mathrm{flat}}=-\gamma\sum_{k}\tilde{c}^{\dagger}_{k}\tilde{c}_{k}~{}~{}~{}\mbox{
and
}~{}~{}~{}~{}H^{\prime}=\sum_{k}\left(4t\cos(k)+\gamma\right)\tilde{d}^{\dagger}_{k}\tilde{d}_{k}$
(113)

Now with the choice of $\gamma>2t$, the bands don’t cross and the flat-band
Hamiltonian contains the lowest energy modes.

Figure 4: The two dispersion bands, with the lower one being perfectly flat,
for $\gamma^{\prime}=\gamma>2t$

.

Since the modes of $H_{\mathrm{flat}}$ contains the lowest lying modes, the
effective dynamics is driven by it. In the continuum limit, as $A\rightarrow
0$, the Brillouin zone ($k\in[-\pi/A,\pi/A]$) covers the whole of real line
and $\tilde{c}_{k}\rightarrow\tilde{c}(k)$. Defining the fourier transform of
it as the single component fermionic field $\psi(x)$, we have the Hamiltonian
describing the low energy dynamics as:

$\displaystyle H_{\mathrm{flat}}=-\gamma\int dx\,\psi^{\dagger}\psi,$ (114)

$\gamma$ playing the role of the mass term.

This above Hamiltonian is very clearly the Hamiltonian for our lower fermions
(67). We have thus identified the physics of this condensed matter system to
the formal Carrollian structures we had previously discussed.

###  6.2 Magic angle bilayer graphene

Our next example would be a more involved one, that of bi-layer graphene
twisted at particular angles called magic angles, where superconductivity
appears.

Before delving in to the topic of bilayer graphene, let us remind ourselves
that the effective low energy description of electrons in monolayer graphene
is both scale (massless) and Lorentz invariant, with the Hamiltonian
describing the free dynamics of two component spinor $\psi$:

$\displaystyle H=iv_{F}\int
d^{2}\mathbf{r}\,\Psi^{\dagger}(\mathbf{r},t)\,\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\,\Psi(\mathbf{r},t),$
(115)

giving rise to linear dispersion $E(\mathbf{k})=v_{F}|\mathbf{k}|$. Here we
keep in mind that the Fermi velocity $v_{F}\approx 3\times 10^{6}m/s$ plays
the role of speed of light, but is 3 order in magnitude smaller than $c$ and
is determined by the product of the hopping energy $\tau$ and the lattice
parameter $a$. Also, the effective description (115) and the space-time
symmetries enjoyed by it are valid for energy scales $\ll 1/a$.

When one stacks a couple of graphene layers on top of each other with a twist
by an angle $\theta$, one needs to take into account interlayer couplings,
which extends modifies the above monoloayer Hamiltonian (115). The low energy
continuum model capturing these features and robustly accounting for
dispersionless states is the so-called ‘chiral’ one tarnopolsky2019origin :

$\displaystyle\tilde{H}=v_{F}\int
d^{2}\mathbf{r}\,\Phi^{\dagger}(\mathbf{r},t)D\Phi(\mathbf{r},t).$ (116)

Here,
$\Phi^{\dagger}=\begin{pmatrix}\psi_{1}^{\dagger}&\psi_{2}^{\dagger}&\chi_{1}^{\dagger}&\chi_{2}^{\dagger}\end{pmatrix}$
is a 4-component spinor, containing 2 for each $(\psi,\chi)$ graphene layer
and $D$ is the matrix differential operator:

$\displaystyle D=\begin{pmatrix}0&\mathcal{D}^{\ast}(-\mathbf{r})\\\
\mathcal{D}(\mathbf{r})&0\end{pmatrix},~{}~{}\mbox{ where
}~{}~{}\mathcal{D}(\mathbf{r})=\begin{pmatrix}-2i\bar{\partial}&\alpha
U(\mathbf{r})\\\ \alpha U(-\mathbf{r})&-2i\bar{\partial}\end{pmatrix}\mbox{
and }$ (117) $\displaystyle U(\mathbf{r})=\sum_{j=0}^{2}e^{i\left(\frac{2\pi
j}{3}-\mathbf{q}_{j}\cdot\mathbf{r}\right)}\mbox{ for
}\mathbf{q}_{0}=\frac{8\pi}{3a}\sin(\theta/2)(0,-1),\mathbf{q}_{1,2}=\frac{8\pi}{3a}\sin(\theta/2)(\pm\sqrt{3}/2,1/2)$

The dimensionless parameter $2\pi\alpha=\frac{w_{1}}{\tau}\csc(\theta/2)$
controls the coupling between the layers through the interlayer hopping
strength $w_{1}$. Any two of the $\mathbf{q}$ appearing in the equation above
serve as the basis vectors in the ‘mini’-Brillouin zone. Note that, in the
commensurate Moire pattern formed by the two mono-layer graphene sheets, the
real space periodicity scales as $\csc(\theta/2)$ and is quite large for small
twist angle $\theta$. In effect, the $\mathbf{k}$ space periodicity scales as
$\theta/2$ and that is reflected in the form of the $\mathbf{q}$ vectors
above. We can compare (116) with the generic Hamiltonian (100), identifying
the operator $D$ of (117) with $\mathfrak{h}$. In continuation of our previous
discussion, the operator $D$ is a matrix-valued one, and contains differential
operator. Hence if it has to have at least one flat band, it should be
supplemented with appropriate periodic boundary condition.

Note that by setting $w_{1}=0$ in (116) one recovers two decoupled Dirac
fermions each corresponding to a monolayer graphene sheet. For an angle
$\theta$, in the Fourier space one looks for Bloch waves in smaller ‘scale’,
mini-Brillouin zone (mBZ), depicted in the figure 5. In the left panel, the
blue and the red hexagons stand for Brillouin zones corresponding to
individual mono-layer graphanes, whereas the construction of the mBZ is shown
in purple outline. In the absence of inter-layer coupling, the low energy
dispersion is given as a pair of filled (sea) and vacant Dirac cones at each
vertex (Dirac points) of the mBZ.

Figure 5: Couple of Dirac points depicting the dispersion relation in Moire
Brillouin zone (momentum space), in absence of inter-layer coupling. The
separation between the Dirac points $k_{\theta}=\frac{8\pi}{3a}\sin(\theta/2)$
is twist angle dependent. Here $a$ is the lattice parameter. The fermi
velocity is, as expected, same as that of monolayer.

We need to keep in mind that introduction of the Moire crystal potential
$U(\mathbf{r})$, for arbitrary $\theta$, breaks the space-time symmetries
enjoyed previously by the pristine low energy effective theory of monolayer
graphene. In the following we will see how those symmetries (and many more)
are restored at the magic angle.

For a fixed $\theta$, as one turns on $w_{1}$ the Dirac cones start to deform.
However the gapless nature of the Dirac points (at the points $K,K^{\prime}$
as per figure 5) is protected by symmetries. This means that one still gets a
couple of zero modes corresponding to each of the the points $K$ and
$K^{\prime}$ in Brillouin zone even for nonzero $\alpha$ perturbation in the
Hamiltonian (116):

$D\Phi_{(K,K^{\prime})}=0.$

In the following, we keep track of the mode $K$, whereas all the properties
hold same for the point $K^{\prime}$ as well. For nonzero $\alpha$ the fermi
velocity at the $K$ point is given in terms of the zero-mode:

$\displaystyle
v_{F}(\alpha)=\Big{|}\partial_{\mathbf{k}}\frac{\Phi^{\dagger}_{K}G_{k}\Phi_{K}}{\Phi^{\dagger}_{K}\Phi_{K}}\Big{|}_{\mathbf{k}=0}$
(118)

where $G_{k}$ is the momentum space Green’s function of the operator $D$. One
can perturbatively calculate $v_{F}(\alpha)$ for $\alpha<1$ and up to first
order in perturbation, one finds that $v_{F}(\alpha=1/\sqrt{3})=0$. This
ensures flattening of the erstwhile Dirac cone at the point $\mathbf{k}=0$, ie
the $K$ point. For a value of $w_{1}$ phenomenologically collected, this
happens for the angle $\theta\approx 1.05^{\circ}$ tarnopolsky2019origin ,
which is famously known as the ‘magic angle’.

Now we have found 4 zero modes of the operator $D$, to each for the points
$K,K^{\prime}$. If there exists at least one flat-band containing one of those
modes, the whole dispersion relation for that band must be identically zero.
Hence one next looks for zero modes away from the $K$ point. But the
Hamiltonian (116) has charge-conjugation (particle-hole) symmetry, which tells
us that there will be two such pristine flat bands containing the zero modes
at $K$. By virtue of $C_{3}$ symmetry, the same argument applies to the two
other bands containing zero modes of $K^{\prime}$. Hence if one band flattens,
all the 4 of them should follow suit, resulting in vanishing of the
Hamiltonian.

The essential way to capture this is by looking for zero mode solutions for
$D$ throughout the entire mBZ, $i.e.$ finding a basis
$\\{\Psi_{\mathbf{k}}|\mathbf{k}\in\mathrm{mBZ}\\}$, such that
$D\Psi_{\mathbf{k}}(\mathbf{r})=0$. The main caveat in finding the kernel of
the operator $D$ on the Hilbert space of square integrable 4 component
fermions is in boundary conditions. The Moire pattern generated by the twist
implies the periodicity:

$\mathbf{r}\rightarrow\mathbf{r}+m\mathbf{a}_{+}+n\mathbf{a}_{-}\quad\text{for}\,(m,n)\in\mathbb{Z}\times\mathbb{Z}\quad\text{and}\quad\mathbf{a}_{\pm}=\dfrac{a}{4\sin(\theta/2)}\left(\pm\sqrt{3},1\right).$
(119)

This implies that, when brought in the fundamental domain by a
$SL(2,\mathbb{Z})$ transformation, the base manifold is a torus of fixed
modular parameter $\tau=-1+\sqrt{3}i$. Also note that the spatial periodicity
imposes a discretization in the mBZ points:
$\mathbf{k}\cdot\mathbf{a}_{\pm}\in 2\pi\mathbb{Z}$.

Now, the Fermi velocity (118) vanishing at some nonzero value of $\alpha$
implies that the kernel of $D$ is spanned by 4 component spinors whose
components are given by theta functions with rational characteristic

$\left(\mathbf{k}\cdot\mathbf{a}_{+}/(2\pi)-1/6,1/6-\mathbf{k}\cdot\mathbf{a}_{-}/(2\pi)\right).$

Hence one concludes that for each point on the mBZ, there are non-trivial zero
mode of $D$. This implies that the Hamiltonian (116) itself vanishes, with all
the 4 bands being pristine zero energy ones. In a full fledged numerical
calculation, there are other non-flat higher energy bands, not captured by the
present model though. However they are substantially gapped from the pristine
flat-bands and don’t play significant role in determining low energy effective
dynamics. Therefore according to our arguments presented in section 5.2, the
low energy theory of ‘magic’ angle bilayer graphene is Carrollian invariant.

Although it is tempting to conclude that since the Hamiltonian of the system
is identically zero, the fermions in question are the massless versions of
lower Carrollian fermions, we refrain from making this identification right
away as the physics involved in the above is complicated and is crucially
dependent on boundary conditions. We have not addressed boundary conditions in
our discussions of Carroll fermions and a full blown comparison with magic
angle graphene would require us to do so. But the emergent physics is clearly
Carrollian and the actors on the stage are fermions, so there is no doubt that
a more detailed understanding of Carroll fermions would lead us to a better
understanding of magic angle bi-layer graphene.

### Unconventional superconductivity

Observation of emergent superconductivity in magic angle bilayer graphene led
experimentalists to test 3,4 and 5 multilayer stacked graphene sheets twisted
alternatively at $\pm$ magic angle. Tri-layer carr2020ultraheavy  onwards
khalaf2019magic , these systems exhibit flatbands corresponding to Carrollian
modes, as well as a set of decoupled Dirac fermions 999Signifying both the
signatures of Lorentzian as well as Carrollian physics, decoupled at different
energy scales. and show robust superconductivity park2021tunable ;
park2022robust . This gives rise to an ever stronger conviction regarding the
positive role of flatbands behind robust superconductivity. It is important to
note that, although rare in mesoscopic scale physics, flatbands have been
observed even in a few systems apart from graphene. What sets graphene apart
is the absence of robust emergent superconductivity in these systems, where
notably the pristine phenomena of flattening of Dirac cones is absent too.

In this era of when a full-blown microscopic theory describing the formation
of cooper pairs resulting in this superconductivity is unavailable, we propose
from the perspective of global symmetry arguments is that the Carrollian
contraction of the Lorentz covariant theory describing low energy of graphene
plays a significant role.

### Looking ahead

This is of course just the start and we have identified the symmetry structure
associated with generic flat bands to be that of Carroll and in cases of
gapless excitations, Conformal Carroll. We remind the reader that these
symmetries are infinite dimensional and just as $2d$ relativistic conformal
field theories have played a starring role in understanding e.g. the theory of
fixed points and phase transitions, it is likely that Carroll and its
conformal cousin would be vital to a deeper understanding of the exotic and
exciting new physics arising out of flat band structures.

##  7 Conclusions and discussions

###  7.1 Summary of the paper

In our paper, we have begun an investigation into fermions on Carrollian
manifolds and hence, among other things, tried to address what happens if you
put fermions on generic null surfaces.

We found that the degenerate nature of Carroll manifolds leads to not one, but
two Carrollian Clifford algebras. We investigated the representations of both
these versions of Carrollian gamma matrices, constructed explicit matrix
representations in $d=4$ and showed how to build up representations of these
gamma matrices in arbitrary even and odd dimensions. While the lower gamma
matrices furnished a faithful representation of the underlying Carroll
algebra, we found that the upper gamma matrices did not.

We called upon charge conjugation to fix this apparent problem and went on to
construct actions from both types of gamma matrices. These led us to two
distinct theories of Carroll fermions, the lower or time-like fermions and the
upper space-like fermions.

The lower fermions were exhibited curious properties: even in higher
dimensions, they were effectively one dimensional, they were anti-commuting
but did not necessary lead to half integral spins. For massless lower
fermions, the Hamiltonian vanished.

The upper fermions were more like usual relativistic fermions. Their action
contained both spatial and temporal derivatives. This led to non-vanishing
Hamiltonians even in the massless limit. The fermions did not behave lower
dimensional and had half integral spins.

In perhaps the most interesting part of our paper, we argued that Carrollian
physics naturally emerges in condensed matter systems where one encounters the
physics of flat bands, that are in vogue. We argued that the flattening of
energy bands essentially means lightcones closing up in position space and the
emergence of Carroll. We then explicitly showed Carroll invariance in these
theories in general.

We then went on to give explicit examples of Carrollian fermions in condensed
matter systems. We showed that a particular fermionic lattice model in 1+1
dimensions exactly reduces to our lower fermions. We also showed how
Carrollian physics would show up in magic bi-layer graphene.

###  7.2 Future directions

There are numerous directions that this work opens up. Let us list a few of
them below.

  * •

Covariantisation: We have looked at fermionic theories on flat Carroll
backgrounds. One of the immediate things to do is to find a covariant
formulation in terms of the underlying geometric data and see how fermions
couple of non-flat Carroll backgrounds.

  * •

Understanding the limit: As we briefly discussed earlier, it is of importance
to understand the Carrollian limit of relativistic fermions and figure out how
the structures we discussed in this paper arises naturally from their
relativistic parent theories. This would give more insight to the origin of
the two different species of fermions we have found in the Carrollian world.
Clarification as to whether the lower and upper fermions are electric and
magnetic theories in the current nomenclature would also help clarify some
properties of the theories we have discovered from the intrinsic analysis.

  * •

Dimensional reduction: We discussed that the lower fermions behaved like one-
dimensional objects. It would be useful to quantify this further.
Investigation of the spin-statistics theorem for Carrollian theories may hold
important clues in this regard. It is possible that the lower fermions are
like anyons in the relativistic set up. Relatedly, it would be of interest to
investigate whether there is a reduction of our upper fermions. Recently it
has been shown that the spacelike mass Carroll scalar theory can be reduced to
a relativistic Euclidean CFT in one lower dimension Baiguera:2022lsw . Whether
a similar thing holds for the spacelike or upper fermions would be rather
intriguing to work out.

  * •

More condensed matter applications: We have just begun to uncover Carroll
symmetry in condensed matter systems. Even the case of twisted bi-layer
graphene which we discussed in this paper needs to be understood better by
looking at non-trivial boundary conditions. It would be exciting to look at
the physics of e.g. quantum Hall systems. Again the possible connection to
lower fermions which behave somewhat like anyons is tantalizing.

  * •

Supersymmetry: Supersymmetry in Carrollian systems, especially in conformal
Carrollian ones, have been studied from the point of view of the algebra and
different contractions of the parent relativistic algebras. See e.g.
Bagchi:2022owq  for a recent account of this in generic dimensions and
references therein. But now, armed with the knowledge of the Carroll Clifford
algebra, and the properties of Carroll fermions, it will be instructive to
revisit this and construct supersymmetry in the Carroll case from scratch.
This is work in progress.

Clearly there are numerous very interesting things to follow up on. Of these,
the emerging connections to real life condensed matter examples are most
intriguing.

### Acknowledgements

It is a pleasure to thank Kinjal Banerjee, Sudipta Dutta, Daniel Grumiller,
Marc Henneaux, Blagoje Oblak, Marios Petropoulos for helpful conversations.
Some preliminary results were presented at the 2nd Carroll Workshop at the
University of Mons, Belgium and AB thanks the participants for discussions.

The work of AB was partially supported by a Swarnajayanti fellowship
(SB/SJF/2019-20/08) from the Science and Engineering Research Board (SERB)
India, the SERB grant (CRG/2020/002035), a Research-in-Groups grant at the
Erwin Schrödinger Institute, Vienna and a visiting professorship at Ecole
Polytechnique Paris. AB also acknowledges the warm hospitality of Vienna
University of Technology and the Niels Bohr Institute, Copenhagen during
various stages of this work. The work of ArB is supported by the Quantum
Gravity Unit of the Okinawa Institute of Science and Technology Graduate
University (OIST). ArB would like to thank TU Wien for hospitality during the
early stages of this project. The work of RB is supported by the following
grants from the SERB, India: CRG/2020/002035, SRG/2020/001037. SM is supported
by grant number 09/092(1039)/2019-EMR-I from Council of Scientific and
Industrial Research (CSIR).

## APPENDICES

##  Appendix A Carroll Isometries

For Carroll case the vector fields $X$ obey

$\displaystyle\mathcal{L}_{\xi}\tilde{h}_{ab}=0,\quad\mathcal{L}_{\xi}\theta_{a}=0$
(120)

Now from the definition of lie derivative

$\mathcal{L}_{\xi}\tilde{h}_{ab}=\xi^{c}\partial_{c}\tilde{h}_{ab}+\tilde{h}_{cb}\partial_{a}\xi^{c}+\tilde{h}_{ac}\partial_{b}\xi^{c}$

Using the expression of $\tilde{h}_{ab}$ we get,

$\displaystyle\partial_{A}\xi_{B}+\partial_{B}\xi_{A}$ $\displaystyle=$
$\displaystyle 0\qquad{A=1,2,...,d}$ (121)
$\displaystyle\partial_{A}\xi^{B}+\partial_{B}\xi^{A}$ $\displaystyle=$
$\displaystyle 0$ (122)

In the last line we have used $\xi_{A}=\delta_{AB}\xi^{B}$. Also from

$\mathcal{L}_{\xi}\theta^{a}=\xi^{b}\partial_{b}\theta^{a}-\theta^{b}\partial_{b}\xi^{a}$
(123)

using $\theta^{a}=-\delta_{a}^{0}$, we get

$\displaystyle\partial_{0}\xi^{0}=0,\quad\partial_{0}\xi^{A}=0.$ (124)

Solving these equations give

$\xi^{0}=f(x^{k})+a,\quad\xi^{A}=\omega^{A}_{B}x^{B}+b^{A}.$ (125)

So the isometry group of the degenerate Carrollian metric is infinite-
dimensional. If one restricts the transformation to be linear, one gets the
finite-dimensional Carroll group. So the vector field for the finite
dimensional Carroll group

$\displaystyle\xi=\xi^{0}\partial_{0}+\xi^{A}\partial_{A}=a\partial_{t}+\omega^{A}_{B}x^{B}\partial_{A}+v_{A}x^{A}\partial_{t}+b^{A}\partial_{A}$
(126)

For Conformal case the equations are

$\displaystyle\mathcal{L}_{\xi}\tilde{h}=\lambda\tilde{h},\quad\mathcal{L}_{\xi}\theta=-\frac{\lambda}{2}\theta$
(127)

By solving in the similar manner we get

$\displaystyle\partial_{A}\xi_{B}+\partial_{B}\xi_{A}$ $\displaystyle=$
$\displaystyle\lambda\delta_{AB}\qquad{A=1,2,...,d}$ (128)
$\displaystyle\partial_{0}\xi^{0}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{2},\quad\partial_{0}\xi^{A}=0$ (129)

One can find the solution as

$\displaystyle\xi=f(x)\partial_{t}+\omega^{A}_{B}x^{B}\partial_{A}+b^{A}\partial_{A}+\Delta(t\partial_{t}+x^{A}\partial_{A})+$
$\displaystyle\kappa_{A}\big{(}2x^{A}(t\partial_{t}+x^{A}\partial_{A})-x^{C}x_{C}\delta^{AB}\partial_{B}\big{)}$
(130)

Here $b$,$\Delta$,$\kappa$ are arbitrary coefficients , $\omega^{A}_{B}$ is
antisymmetric.

## References

  * (1) L. Leblond, _Une nouvelle limite non-relativiste du group de Poincaré_ , _Annales Poincare Phys.Theor. 3 (1965) 1_ .
  * (2) N. Sen Gupta, _On an Analogue of the Galileo Group_ , _Nuovo Cim. 54 (1966) 512 • DOI: 10.1007/BF02740871_ .
  * (3) M. Henneaux, _Geometry of Zero Signature Space-times_ , _Bull. Soc. Math. Belg._ 31 (1979) 47.
  * (4) C. Duval, G. W. Gibbons and P. A. Horvathy, _Conformal Carroll groups and BMS symmetry_ , _Class. Quant. Grav._ 31 (2014) 092001 [1402.5894].
  * (5) A. Bagchi, _Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories_ , _Phys. Rev. Lett._ 105 (2010) 171601 [1006.3354].
  * (6) M. Bondi, H. van der Burg and A. Metzner, _Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems_ , _Proc. Roy. Soc. Lond._ 21 .
  * (7) R. Sachs, _Asymptotic symmetries in gravitational theory_ , _Phys. Rev._ (1962) .
  * (8) G. Barnich and G. Compere, _Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions_ , _Class. Quant. Grav._ 24 (2007) F15 [gr-qc/0610130].
  * (9) D. Kapec, V. Lysov, S. Pasterski and A. Strominger, _Higher-Dimensional Supertranslations and Weinberg’s Soft Graviton Theorem_ , 1502.07644.
  * (10) A. Strominger, _On BMS Invariance of Gravitational Scattering_ , _JHEP_ 07 (2014) 152 [1312.2229].
  * (11) A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , 1703.05448.
  * (12) S. Pasterski, _Lectures on celestial amplitudes_ , _Eur. Phys. J. C_ 81 (2021) 1062 [2108.04801].
  * (13) A.-M. Raclariu, _Lectures on Celestial Holography_ , 2107.02075.
  * (14) A. Bagchi and R. Fareghbal, _BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries_ , _JHEP_ 10 (2012) 092 [1203.5795].
  * (15) A. Bagchi, R. Basu, A. Kakkar and A. Mehra, _Flat Holography: Aspects of the dual field theory_ , _JHEP_ 12 (2016) 147 [1609.06203].
  * (16) A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, _Holography of 3D Flat Cosmological Horizons_ , _Phys. Rev. Lett._ 110 (2013) 141302 [1208.4372].
  * (17) G. Barnich, _Entropy of three-dimensional asymptotically flat cosmological solutions_ , _JHEP_ 10 (2012) 095 [1208.4371].
  * (18) A. Bagchi and R. Basu, _3D Flat Holography: Entropy and Logarithmic Corrections_ , _JHEP_ 03 (2014) 020 [1312.5748].
  * (19) A. Bagchi, D. Grumiller and W. Merbis, _Stress tensor correlators in three-dimensional gravity_ , _Phys. Rev. D_ 93 (2016) 061502 [1507.05620].
  * (20) A. Bagchi, R. Basu, D. Grumiller and M. Riegler, _Entanglement entropy in Galilean conformal field theories and flat holography_ , _Phys. Rev. Lett._ 114 (2015) 111602 [1410.4089].
  * (21) H. Jiang, W. Song and Q. Wen, _Entanglement Entropy in Flat Holography_ , _JHEP_ 07 (2017) 142 [1706.07552].
  * (22) G. Barnich, A. Gomberoff and H. A. Gonzalez, _The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes_ , _Phys. Rev. D_ 86 (2012) 024020 [1204.3288].
  * (23) C. Duval, G. W. Gibbons and P. A. Horvathy, _Conformal Carroll groups_ , _J. Phys._ A47 (2014) 335204 [1403.4213].
  * (24) J. Hartong, _Holographic Reconstruction of 3D Flat Space-Time_ , _JHEP_ 10 (2016) 104 [1511.01387].
  * (25) A. Bagchi, S. Banerjee, R. Basu and S. Dutta, _Scattering Amplitudes: Celestial and Carrollian_ , _Phys. Rev. Lett._ 128 (2022) 241601 [2202.08438].
  * (26) L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, _Carrollian Perspective on Celestial Holography_ , _Phys. Rev. Lett._ 129 (2022) 071602 [2202.04702].
  * (27) L. Donnay and C. Marteau, _Carrollian Physics at the Black Hole Horizon_ , _Class. Quant. Grav._ 36 (2019) 165002 [1903.09654].
  * (28) S. W. Hawking, M. J. Perry and A. Strominger, _Soft Hair on Black Holes_ , _Phys. Rev. Lett._ 116 (2016) 231301 [1601.00921].
  * (29) L. Donnay, G. Giribet, H. A. Gonzalez and M. Pino, _Supertranslations and Superrotations at the Black Hole Horizon_ , _Phys. Rev. Lett._ 116 (2016) 091101 [1511.08687].
  * (30) H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo et al., _Soft Heisenberg hair on black holes in three dimensions_ , _Phys. Rev. D_ 93 (2016) 101503 [1603.04824].
  * (31) R. F. Penna, _Near-horizon BMS symmetries as fluid symmetries_ , _JHEP_ 10 (2017) 049 [1703.07382].
  * (32) S. Carlip, _Black Hole Entropy from Bondi-Metzner-Sachs Symmetry at the Horizon_ , _Phys. Rev. Lett._ 120 (2018) 101301 [1702.04439].
  * (33) A. Bagchi, _Tensionless Strings and Galilean Conformal Algebra_ , _JHEP_ 05 (2013) 141 [1303.0291].
  * (34) A. Bagchi, S. Chakrabortty and P. Parekh, _Tensionless Strings from Worldsheet Symmetries_ , _JHEP_ 01 (2016) 158 [1507.04361].
  * (35) A. Bagchi, A. Banerjee and P. Parekh, _Tensionless Path from Closed to Open Strings_ , _Phys. Rev. Lett._ 123 (2019) 111601 [1905.11732].
  * (36) A. Bagchi, A. Banerjee, S. Chakrabortty, S. Dutta and P. Parekh, _A tale of three — tensionless strings and vacuum structure_ , _JHEP_ 04 (2020) 061 [2001.00354].
  * (37) A. Bagchi, A. Banerjee and S. Chakrabortty, _Rindler Physics on the String Worldsheet_ , _Phys. Rev. Lett._ 126 (2021) 031601 [2009.01408].
  * (38) A. Bagchi, A. Banerjee, S. Chakrabortty and R. Chatterjee, _A Rindler Road to Carrollian Worldsheets_ , 2111.01172.
  * (39) A. Bagchi, D. Grumiller and M. M. Sheikh-Jabbari, _Horizon Strings as 3d Black Hole Microstates_ , 2210.10794.
  * (40) E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, _Carroll versus Galilei Gravity_ , _JHEP_ 03 (2017) 165 [1701.06156].
  * (41) C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, _Carroll symmetry of plane gravitational waves_ , _Class. Quant. Grav._ 34 (2017) 175003 [1702.08284].
  * (42) J. Hartong, _Gauging the Carroll Algebra and Ultra-Relativistic Gravity_ , _JHEP_ 08 (2015) 069 [1505.05011].
  * (43) K. Morand, _Embedding Galilean and Carrollian geometries I. Gravitational waves_ , _J. Math. Phys._ 61 (2020) 082502 [1811.12681].
  * (44) L. Ciambelli and C. Marteau, _Carrollian conservation laws and Ricci-flat gravity_ , _Class. Quant. Grav._ 36 (2019) 085004 [1810.11037].
  * (45) M. Henneaux and P. Salgado-Rebolledo, _Carroll contractions of Lorentz-invariant theories_ , _JHEP_ 11 (2021) 180 [2109.06708].
  * (46) J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, _Perfect Fluids_ , _SciPost Phys._ 5 (2018) 003 [1710.04708].
  * (47) L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos and K. Siampos, _Flat holography and Carrollian fluids_ , _JHEP_ 07 (2018) 165 [1802.06809].
  * (48) L. Ciambelli, C. Marteau, A. C. Petkou, P. M. Petropoulos and K. Siampos, _Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids_ , _Class. Quant. Grav._ 35 (2018) 165001 [1802.05286].
  * (49) L. Ciambelli, C. Marteau, P. M. Petropoulos and R. Ruzziconi, _Gauges in Three-Dimensional Gravity and Holographic Fluids_ , _JHEP_ 11 (2020) 092 [2006.10082].
  * (50) A. C. Petkou, P. M. Petropoulos, D. R. Betancour and K. Siampos, _Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars_ , _JHEP_ 09 (2022) 162 [2205.09142].
  * (51) L. Freidel and P. Jai-akson, _Carrollian hydrodynamics from symmetries_ , 2209.03328.
  * (52) L. Freidel and P. Jai-akson, _Carrollian hydrodynamics and symplectic structure on stretched horizons_ , 2211.06415.
  * (53) J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, _Carroll symmetry, dark energy and inflation_ , 2110.02319.
  * (54) L. Bidussi, J. Hartong, E. Have, J. Musaeus and S. Prohazka, _Fractons, dipole symmetries and curved spacetime_ , 2111.03668.
  * (55) A. Bagchi, A. Mehra and P. Nandi, _Field Theories with Conformal Carrollian Symmetry_ , _JHEP_ 05 (2019) 108 [1901.10147].
  * (56) A. Bagchi, R. Basu, A. Mehra and P. Nandi, _Field Theories on Null Manifolds_ , _JHEP_ 02 (2020) 141 [1912.09388].
  * (57) N. Gupta and N. V. Suryanarayana, _Constructing Carrollian CFTs_ , _JHEP_ 03 (2021) 194 [2001.03056].
  * (58) A. Bagchi, A. Banerjee, S. Dutta, K. S. Kolekar and P. Sharma, _Carroll covariant scalar fields in two dimensions_ , 2203.13197.
  * (59) S. Baiguera, G. Oling, W. Sybesma and B. T. Søgaard, _Conformal Carroll Scalars with Boosts_ , 2207.03468.
  * (60) D. Rivera-Betancour and M. Vilatte, _Revisiting the Carrollian scalar field_ , _Phys. Rev. D_ 106 (2022) 085004 [2207.01647].
  * (61) Z.-f. Yu and B. Chen, _Free field realization of the BMS Ising model_ , 2211.06926.
  * (62) P.-X. Hao, W. Song, Z. Xiao and X. Xie, _A BMS-invariant free fermion model_ , 2211.06927.
  * (63) A. Banerjee, S. Dutta and S. Mondal, _Carroll fermions in two dimensions_ , .
  * (64) C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, _Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time_ , _Class. Quant. Grav._ 31 (2014) 085016 [1402.0657].
  * (65) A. Bagchi, A. Banerjee, S. Chakrabortty and P. Parekh, _Inhomogeneous Tensionless Superstrings_ , _JHEP_ 02 (2018) 065 [1710.03482].
  * (66) M. Henneaux and C. Teitelboim, _Quantization of gauge systems_. 1992.
  * (67) A. Bagchi, A. Banerjee, S. Chakrabortty and P. Parekh, _Exotic Origins of Tensionless Superstrings_ , _Phys. Lett. B_ 801 (2020) 135139 [1811.10877].
  * (68) D. Leykam, A. Andreanov and S. Flach, _Artificial flat band systems: from lattice models to experiments_ , _Advances in Physics: X_ 3 (2018) 1473052.
  * (69) M. Creutz, _Aspects of chiral symmetry and the lattice_ , _Reviews of Modern Physics_ 73 (2001) 119.
  * (70) G. Tarnopolsky, A. J. Kruchkov and A. Vishwanath, _Origin of magic angles in twisted bilayer graphene_ , _Physical review letters_ 122 (2019) 106405.
  * (71) S. Carr, C. Li, Z. Zhu, E. Kaxiras, S. Sachdev and A. Kruchkov, _Ultraheavy and ultrarelativistic dirac quasiparticles in sandwiched graphenes_ , _Nano letters_ 20 (2020) 3030.
  * (72) E. Khalaf, A. J. Kruchkov, G. Tarnopolsky and A. Vishwanath, _Magic angle hierarchy in twisted graphene multilayers_ , _Physical Review B_ 100 (2019) 085109.
  * (73) J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi and P. Jarillo-Herrero, _Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene_ , _Nature_ 590 (2021) 249.
  * (74) J. M. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T. Taniguchi et al., _Robust superconductivity in magic-angle multilayer graphene family_ , _Nature Materials_ (2022) 1.
  * (75) A. Bagchi, D. Grumiller and P. Nandi, _Carrollian superconformal theories and super BMS_ , _JHEP_ 05 (2022) 044 [2202.01172].