# Time-loops to spot torsion on bidimensional Dirac materials with
dislocations

Alfredo Iorio<EMAIL_ADDRESS>IPNP - Faculty of Mathematics and
Physics, Charles University, V Holešovičkách 2, 18000 Prague 8, Czech Republic
Pablo Pais<EMAIL_ADDRESS>Instituto de Ciencias Físicas y
Matemáticas, Universidad Austral de Chile, Casilla 567, 5090000 Valdivia,
Chile  IPNP - Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 18000 Prague 8, Czech Republic

###### Abstract

Assuming that, with some care, dislocations could be meaningfully described by
torsion, we propose here a scenario based on a previously unexplored role of
time in the low-energy Dirac field theory description of two-space-dimensional
Dirac materials. Our approach is based on the realization of an exotic time-
loop that could be realized by oscillating particle-hole pairs, overcoming the
well-known geometrical obstructions due to the lack of a third spatial
dimension. General symmetry considerations allow concluding that the effects
we are looking for can only be seen if we move to the nonlinear response
regime.

##  1 Introduction

Einstein’s theory of General Relativity (GR) is the accepted theory of
gravitation, surely for its beauty, but perhaps more solidly for its many
observational successes. Recent remarkable examples of those are the discovery
by the Event Horizon Telescope (EHT) of astrophysical black holes, first in
the centre of the galaxy M87 [1], then in the centre of our Milky Way [2], and
the observation of the gravitational waves by the Laser Interferometer
Gravitational-wave Observatory (LIGO) [3], opening the way to a new era in the
exploration of the signals coming from the Universe surrounding us.

However, astronomical observations pointing to dark matter and dark energy, on
the one hand, and theoretical challenges that mostly have to do with a quantum
theory of gravity, on the other hand, call for a theory that extends GR. Since
matter is characterized, at the fundamental level, by the _mass_ and the
_spin_ of its elementary constituents111As fundamental particles locally obey
quantum mechanics and special relativity theory, they are classified by
reducible unitary representations of the Poincaré group [4], labelled by mass
$m$ and spin $s$., and since GR is a geometric theory entirely based on
curvature, whose source is the mass, a natural extension of GR is obtained by
including torsion in the geometrical properties of the spacetime. Indeed, spin
is related to torsion, as mass is related to curvature. On this, the
literature is vast, so let us point to the seminal work of Tom Kibble [5].

Therefore, one direction to explore for extensions of GR is to include torsion
as an extra feature of the spacetime geometry. This is the case of the
Einstein-Cartan (EC) theory of gravitation, where through a contorsion in the
spin-connection, torsion appears in the physical phenomena, allowing the
possibility of a direct extension of GR. On this, see the two excellent
reviews [4] and [6], which have complementary points of view.

At this time, though, there is no direct experimental evidence of the effects
of spacetime torsion, even though some argue that the very existence of spin
itself is the only manifestation of spatiotemporal torsion [7] and that
serious work is currently undertaken within certain cosmological scenarios
[8]. Therefore, it is handy that certain analog systems might furnish tabletop
experimental grounds to test fundamental theories of gravity based on torsion.

Indeed, to include torsion in the geometric description of spacetime, Cartan
himself was inspired by Cosserat’s theory of crystal dislocations, see, e.g.,
the story told in [7]. On the other hand, three-dimensional gravity models
have been employed to describe the (un)elastic properties of crystalline
materials, where disclinations and dislocations correspond to curvature and
torsion in the continuum limit, respectively, see, e.g., [9].

Given the natural occurrence of topological defects in graphene [10, 11] and
other Dirac materials [12], and given that such materials have been proposed
as versatile analogs of classical and quantum gravitational phenomena, as told
in [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and in the recent review [24],
then it is very natural to seek for the role of torsion in the analog
gravitational physics realized there. However, early attempts in that
direction (that we shall recall later in Section 2) had negative outcomes due
to the geometrical obstruction existing in two-space-dimensional materials
[11].

Here we present a possible way out from that obstruction, based on the results
of [25], where time plays an unexpected role in the $(2+1)$-dimensional Dirac-
like description of the low-energy quasiparticles of graphene [10].

##  2 The Riemann-Cartan spaces

Einstein himself [26] points out that, for a theory of gravitation as GR, more
important than the metric is the parallel transport, taken into account by the
affine connection. An $n$-dimensional (differential) manifold $M$, equipped
with a connection $\Gamma$, is said to be _linearly connected_ , and it is
denoted by $L_{n}$. Given a point $p$ on $M$, we say that a vector
$v=v^{\mu}\partial_{\mu}$ of the tangent space at that point, $T_{p}M$, is
parallel-transported along the direction $dx^{\nu}$ if its variation is given
by

$\delta_{\|}v^{\mu}=v^{\mu}_{\|}(x+dx)-v^{\mu}(x)=-\Gamma{{}^{\mu}}{{}_{\rho\nu}}\,v^{\nu}\,dx^{\rho}\;,$

where $\Gamma_{\rho\nu}^{\mu}$ is a _linear connection_. The covariant
derivative $\nabla\,v$ is defined by the difference

$v^{\mu}(x+dx)-v^{\mu}_{\|}(x+dx)=v^{\mu}(x+dx)-v^{\mu}(x)-\delta_{\|}\,v^{\mu}=\left(\partial_{\nu}v^{\mu}+\Gamma{{}^{\mu}}{{}_{\nu\rho}}v^{\rho}\right)\,dx^{\nu}\equiv(\nabla_{\nu}v^{\mu})\,dx^{\nu}\;,$
(1)

where in the second equality we expanded $v^{\mu}(x+dx)$ just up linear order
in $dx$ and, in the third equality, the order of the covariant indices of
$\Gamma$ must be noted. We stress here that the connection transforms under
coordinate changes in an appropriate way in order for the quantity
$\nabla_{\nu}v^{\mu}$ to transform as a covariant vector, and this can be
generalized to an arbitrary rank tensor. Henceforth, the difference

$\tensor{T}{{}^{\mu}}{{}_{\nu\rho}}\equiv\left(\Gamma{{}^{\mu}}{{}_{\nu\rho}}-\Gamma{{}^{\mu}}{{}_{\rho\nu}}\right)\;,$
(2)

does transform as a third-rank tensor, called the _torsion tensor_.

Parallel transport is a path-dependent process, and if we parallel transport a
vector around an infinitesimal loop, the components change in a way reflected
by the _curvature tensor_

$\tensor{R}{{}^{\mu}}{{}_{\sigma\nu\rho}}\equiv\partial_{\nu}\Gamma{{}^{\mu}}{{}_{\rho\sigma}}-\partial_{\rho}\Gamma{{}^{\mu}}{{}_{\nu\sigma}}+\Gamma{{}^{\mu}}{{}_{\nu\tau}}\,\Gamma{{}^{\tau}}{{}_{\rho\sigma}}-\Gamma{{}^{\mu}}{{}_{\rho\tau}}\,\Gamma{{}^{\tau}}{{}_{\nu\sigma}}\;,$
(3)

that is a fourth-rank tensor. Importantly, both curvature and torsion can be
defined in a manifold without the notion of a metric.

From definitions (2) and (3), it can be shown [27] that curvature and torsion
are responsible for the noncommutativity of covariant derivatives of a vector
in $L_{n}$,

$\nabla_{\nu}\nabla_{\rho}v^{\mu}-\nabla_{\rho}\nabla_{\nu}v^{\mu}=R{{}^{\mu}}{{}_{\sigma\nu\rho}}\,v^{\sigma}-\tensor{T}{{}^{\sigma}_{\nu\rho}}\nabla_{\sigma}v^{\mu}\;,$

while for a scalar, $\varphi$, the noncommutativity is entirely due to torsion
alone

$\nabla_{\nu}\nabla_{\rho}\varphi-\nabla_{\rho}\nabla_{\nu}\varphi=-\tensor{T}{{}^{\sigma}}{{}_{\nu\rho}}\partial_{\sigma}\varphi\;.$

It is when we want to make contact with the experiments, and measure angles
and distances between events in a spacetime manifold, that we introduce the
_metric tensor_ as a second-rank tensor defining the line element, i.e. the
infinitesimal distance between two points as

$ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\;,$ (4)

whereas, by integration, we can define the longitude of any curve on $L_{n}$.

All the above also makes contact with the equivalence principle, as we can
measure in inertial spaces _locally_ diffeomorphic to Minkowski flat space.
This principle was initially formulated by taking into account just point-like
test particles and does not consider internal structures as spin. Therefore,
in more general settings, the equivalence principle can be formulated in a
$U_{n}$ space. On this, see the discussion in [28].

A reasonable assumption is that the local distances do not change under
parallel transportation. This can be assured if

$\nabla_{\rho}g_{\mu\nu}=0\;.$ (5)

The condition (5) for a linear connection $\Gamma$ is called _metric
compatibility_ 222As $Q_{\rho\mu\nu}\equiv\nabla_{\rho}g_{\mu\nu}$ is a third-
rank tensor, called the _nonmetricity tensor_ , we can wonder how far we can
go by not assuming this to be zero. In fact, there are some theories where
this tensor has a physical interpretation. However, in this work we always
take $Q_{\rho\mu\nu}=0$. For more details about the jargon of different
spaces, see [7].. An $n$-dimensional manifold $M$ with a linear connection
preserving local distances, i.e. fulfilling condition (5), is called a
_Riemann-Cartan (RC) space_ , denoted by $U_{n}$. Fig. 1 gives a geometric
interpretation of torsion, with details in the caption. This is the first link
between the linear connection and the metric.

Figure 1: One geometric interpretation of torsion in Riemann-Cartan spaces.
Consider two vector fields, $X$ and $Y$, at a point $P$. First, parallel-
transport $X$ along $Y$ to the infinitesimally close point $R$. Then, again
from $P$, parallel-transport $Y$ along $X$ to reach a point $Q$, which
generally does not coincide with $R$. In fact, it is precisely the failure to
close the parallelogram the geometrical signal of a nonzero torsion. Its value
is the difference between the two resulting vectors $T(X,Y)$. In Riemannian
spaces $V_{n}$, this tensor is assumed to be zero. The picture was inspired by
[7] but with the notation of [29].

GR further assumes that the torsion tensor vanishes, i.e., that the linear
connection is symmetric. In such a case, the manifold is called
(pseudo-)Riemannian, denoted by $V_{n}$. The unique linear metric-compatible
connection without torsion, called the _Levi-Civita connection_ , can then be
deduced directly from the metric [29]

$\left\\{\begin{array}[]{c}\mu\\\
\nu\,\rho\end{array}\right\\}=\frac{1}{2}\,g^{\mu\sigma}\,(\partial_{\nu}g_{\rho\sigma}+\partial_{\rho}g_{\nu\sigma}-\partial_{\sigma}g_{\nu\rho})\;.$
(6)

The quantities (6) are called _Christoffel symbols_ , and the curvature
associated with the Levi-Civita connection is the _Riemannian curvature
tensor_ , denoted by $\tensor{\mathring{R}}{{}^{\mu}_{\nu\rho\sigma}}$. In
this way, the linear connection in a $U_{n}$ space can be written as

$\Gamma{{}^{\mu}}{{}_{\nu\rho}}=\left\\{\begin{array}[]{c}\mu\\\
\nu\,\rho\end{array}\right\\}+\tensor{K}{{}^{\mu}_{\nu\rho}}\;,$

where
$\tensor{K}{{}^{\mu}_{\nu\rho}}\equiv\frac{1}{2}\,(\tensor{T}{{}^{\mu}_{\nu\rho}}+\tensor{T}{{}_{\nu}^{\mu}{}_{\rho}}+\tensor{T}{{}_{\rho}^{\mu}{}_{\nu}})$
is called the _contorsion tensor_ 333Some references called it _contortion
tensor_ [4]. However, as we are following closer the terminology of [29], we
keep the name _contorsion_. As far as we know, there is no consensus yet about
the name.. Notice that, contrary to the torsion tensor,
$\tensor{K}{{}^{\mu}_{\nu\rho}}$ is not necessarily antisymmetric in the last
two indices, unless torsion is totally antisymmetric (see below).

Apart from $V_{n}$ spaces, there are other less standard “degenerated” cases
of $U_{n}$ spaces. One very known example is the Weitzenböck space [30], where
the tensor (3) is assumed to be zero. Still, the Riemannian curvature is not
necessarily zero, leading to the _teleparallel gravity_ theory [31]. However,
we shall not take this direction here.

It is helpful to consider a non-coordinate basis, $e^{a}_{\mu}$, not only
because this simplifies the notation, but also because we shall be dealing
with spinors. The $e^{a}_{\mu}$, also called _vielbeins_ , are defined through
the metric as

$g_{\mu\nu}=e^{a}_{\mu}\,e^{b}_{\nu}\eta_{ab}\;,$

where we choose the signature $\eta_{ab}=\mbox{diag}(1,-1,\cdots)$ for the
Minkowski metric. The _spin-connection_ ,
$\omega^{ab}_{\mu}=e^{a}_{\lambda}(\delta^{\lambda}_{\nu}\partial_{\mu}+\Gamma^{\lambda}_{\mu\nu})e^{b\nu}$,
can be decomposed into torsion-free and contorsion contributions [29, 32],
$\omega_{\mu}^{ab}=\mathring{\omega}_{\mu}^{ab}+\kappa_{\mu}^{ab}$, where
$T^{\lambda}_{\mu\nu}=E_{a}^{\lambda}\tensor{\kappa}{{}_{\nu}^{a}{}_{b}}e^{b}_{\mu}-E_{a}^{\lambda}\tensor{\kappa}{{}_{\mu}^{a}{}_{b}}e^{b}_{\nu}$,
being $E_{a}^{\mu}$ the vielbein inverse, i.e.,
$e^{a}_{\mu}\,E^{\mu}_{b}=\delta^{a}_{b}$ and
$e^{a}_{\mu}\,E^{\nu}_{a}=\delta^{\nu}_{\mu}$.

We are interested in $(2+1)$-dimensional systems so we will focus on $U_{3}$
spaces. A dynamical study of the RC spaces that shows how torsion is related
to spin angular momentum can be found in [4] and references therein. Here we
assume that the geometry is fixed and there are no propagating gravitational
degrees of freedom. Nonetheless, we shall keep in mind the natural role of
torsion when spinorial fields are at stake [5]. With this, besides possible
boundary terms, the graphene-Dirac action with the minimal GR extension in a
$U_{3}$ space with a fixed spacetime in a RC geometry is then

$S=\mathrm{i}\,\hbar\,v_{F}\int
d^{3}x\;|e|\;\overline{\psi}\left(E_{a}^{\mu}\,\gamma^{a}\mathring{\nabla}_{\mu}-\frac{\mathrm{i}}{4}\gamma^{5}\frac{\epsilon^{\mu\nu\rho}}{|e|}T_{\mu\nu\rho}\right)\psi\;.$
(7)

The details of the deduction of each term of (7) are in [25]. Let us just
mention that, being this the effective action for graphene, here we trade the
speed of light, $c$, for the Fermi velocity, $v_{F}$. Furthermore, as for
notation, $|e|=\sqrt{|g|}$, the covariant derivative,
$\mathring{\nabla}_{\mu}$, is based on the torsion-free connection,
$\mathring{\omega}_{\mu}^{ab}$, only, and $\gamma^{5}\equiv
i\gamma^{0}\gamma^{1}\gamma^{2}=\left(\begin{array}[]{cc}I_{2\times 2}&0\\\
0&-I_{2\times 2}\\\ \end{array}\right)$ (we used the conventions of [19] for
$\gamma^{0},\gamma^{1},\gamma^{2}$ giving a $\gamma^{5}$ that commutes with
the other three gamma matrices444This is due to the reducible, rather than
irreducible, representation of the Lorentz group we use.).

From the action (7), the well-known fact that torsion couples to spinors
through its totally antisymmetric component [6], can be read-off. Therefore,
as remarked above, in this case, the contorsion
$\tensor{K}{{}^{\mu}_{\nu\rho}}$ is also antisymmetric, as can be seen from
its definition. This mathematical fact is behind the obstruction pointed out
some time ago leading to the conclusion that, in two-dimensional Dirac
materials, torsion can have no physical role. On this see, e.g., [11, 33, 34].

##  3 Topological defects and their continuum limit in two-dimensional
materials

This work focuses primarily on graphene-like materials, but many of the
following considerations also apply to other two-dimensional crystals. The two
most important topological defects in these materials are _disclinations_ and
_dislocations_ 555Another important defect which could have a geometrical
meaning, and has possible impacts in using graphene-like materials as analogs,
is the _grain boundary_ [19].. These two defects are related to curvature and
torsion, respectively, in a continuum-limit description of graphene-like (and
many other) materials [35, 36].

In a honeycomb lattice, i.e., with a hexagonal structure, the dislocation
defect is an $n$-side polygon, with $n=3,\,4,\,5$, or $n=7,\,8\,9\,\ldots$.
The defects belonging to the first set ($n\leq 5$) carry an intrinsic positive
curvature, while the defects belonging to the other set ($n\geq 7$) hold
intrinsic negative curvature. Specific arrangements for such defects are then
obeyed when the overall curvature is positive, see, e.g., [37], and other
specific arrangements are obeyed when the overall curvature is negative, see,
e.g., [38].

One way to produce dislocation defects is to have a dipole of disclinations
with zero total curvature, as the pentagon-heptagon dipole of Fig. 2. This
corresponds to introducing a strip in the lower half-plane of such a figure.
The _Burgers vector_ $\vec{b}$ characterizes this type of defect, and its
modulus is the width of the introduced strip. The process of cutting and
sewing is called a _Volterra process_ [36], and it gives us the intuition of
why these defects are topological: one cannot undo them by a continuous
transformation.

Figure 2: Edge dislocation from two disclinations. Two disclinations, a
heptagon and a pentagon, by adding up to zero total intrinsic curvature. This
makes a dislocation with Burgers vector $\vec{b}$, as can be seen in the lower
half-plane. Burgers vector in the continuum limit caries torsion. This figure
is taken from [19].

The torsion tensor in crystals is related to the Burgers vector through the
formula [35, 9]

$b^{a}=\int\int_{\Sigma}e^{a}_{\lambda}T^{\lambda}_{\mu\nu}dx^{\mu}\wedge
dx^{\nu}\;,$ (8)

where $\Sigma$ is a surface containing the dislocation, but otherwise
arbitrary, $a=0,1,2$. However, the apparent simplicity of (8) should not
shadow the subtleties that it involves. In fact, given a distribution of
Burgers vector, it is not obvious how to assign a specific corresponding
torsion tensor. See, e.g., [39] and [40] to learn how the dislocation
distributions have potential electronic device applications.

From (8), we see that to have a nonzero Burgers vector giving rise to
$\epsilon^{\mu\nu\rho}T_{\mu\nu\rho}\neq 0$ (which is necessary for a nonzero
coupling in (7)), we can follow two roads: (i) a time-directed screw
dislocation (only possible if the crystal has a time direction)

$b_{t}\propto\int\int T_{012}dx\wedge dy\,,$ (9)

or (ii) an edge dislocation “felt” by an integration along a space-time
circuit (only possible if we can actually go around a loop in time), e.g,

$b_{x}\propto\int\int T_{102}dt\wedge dy\,.$ (10)

##  4 Torsion through time crystals or through time-loops

The point we want to make here is that scenarios (i) and (ii) are, in fact,
not impossible to realize in a laboratory.

Scenario (i) could be explored in the context of the fascinating time crystals
introduced by Wilczek [41, 42], being nowadays the focus of intense
experimental studies [43, 44]. Such lattices, discrete in all dimensions,
including time, would be an interesting playground to probe quantum gravity
ideas [45]. In particular, it would be precious to explore defect-based models
of classical gravity/geometry, see [46] and [35]. Despite the beauty of
scenario (i), we shall focus only on the more manageable but still quite
challenging scenario (ii).

Let us assume the Riemann curvature to be zero,
$\tensor{\mathring{R}}{{}^{\mu}_{\nu\rho\sigma}}=0$, but torsion (or
contorsion $\tensor{K}{{}^{\mu}_{\nu\rho}}\neq 0$) to be nonzero, and let us
choose a frame where $\mathring{\omega}_{\mu}^{ab}=0$. The action (7) is then

$S=\mathrm{i}\,\hbar\,v_{F}\int
d^{3}x|e|\,\left(\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{\mathrm{i}}{4}\overline{\psi}_{+}\phi\psi_{+}+\frac{\mathrm{i}}{4}\overline{\psi}_{-}\phi\psi_{-}\right)\;,$
(11)

where $\psi=(\psi_{+},\psi_{-})$ and
$\phi\equiv\epsilon^{\mu\nu\rho}T_{\mu\nu\rho}/|e|$ is what we call _torsion
field_. Even in the presence of torsion, the two irreducible spinors,
$\psi_{+}$ and $\psi_{-}$, are decoupled (however, they couple to $\phi$ with
opposite signs).

To overcome the three-dimensional geometric obstruction through the “time-
loop” in the $(y,t)$-plane of scenario (ii), see (10), our proposal is to make
use of the particle-antiparticle description of the dynamics encoded in the
action (11). The picture goes as follows.

First, one realizes that the regime of the materials we are describing is the
“half-filling” [10], for which the energy states of the valence band ($E<0$)
have the vacancies completely filled (being the analog of the Dirac sea),
while the vacancies of the conduction band ($E>0$) stay empty.

Think now of exciting a pair particle-hole out of this vacuum and making them
oscillate, say, along the $y$-axis, as described in Fig. 3. This amounts to a
circuit of the particle-antiparticle pair in the $(y,t)$-plane. What is left
to do is to fully exploit the emergent relativistic-like structure of the
model and see the portion of the circuit described by the antiparticle moving
forwards in time, as corresponding to the same particle moving backwards in
time. This realizes what we may call a time-loop.

Figure 3: Idealized _time-loop_. At $t=0$, the hole (yellow) and the particle
(black) start their journey from $y=0$, in opposite directions. Evolving
forward in time, at $t=t^{*}>0$, the hole reaches $-y^{*}$, while the particle
reaches $+y^{*}$, (blue portion of the circuit). Then they come back to the
original position, $y=0$, at $t=2t^{*}$ (red portion of the circuit). This can
be repeated indefinitely. On the far right, the equivalent time-loop, where
the hole moving forward in time is replaced by a particle moving backward in
time. Figure taken from [25].

The pictures in Fig. 3 refer to a defect-free honeycomb graphene-like sheet.
The presence of a dislocation, with Burgers vector $\vec{b}$ directed along
$x$, would result in a failure to close the loop proportional to $\vec{b}$
[25].

Therefore, going from a first to a second quantization approach, paying due
attention to the subtleties involved in describing dislocation distributions
by a suitable torsion tensor, the Dirac field theory emerging here can indeed
include a nonzero coupling with torsion. This accounts for a field theoretical
description of the effects of dislocations only when the third dimension is
taken to be time, overcoming the geometrical obstruction discussed earlier
[11, 33, 34].

##  5 Linear and non-linear response regimes

The above is a nice idea, but the real challenge is to bring this idealized
picture close to experiments. We present below the first steps in that
direction; see [25] for more details.

The simplest settings we can envisage to realize the picture above-presented
need: i) an _external electromagnetic field_ to excite the particle-hole pair
necessary for the time-loop, and ii) that a suitable disclination/torsion
provides the non-closure of the loop in the appropriate direction, something
we shall refer to as _holonomy_. In other words, we are looking for the
measurable effects of a disclination/torsion-induced holonomy in a time-loop.
It is only a suitable combination of those interactions that can produce the
effect we are looking for. Notice this is more related to the _Cartan circuit_
for torsion and curvature visualization [7] than the non-closing parallelogram
torsion interpretation given in Fig. 1.

Therefore, the action governing the relevant microscopic dynamics is

$\displaystyle S$ $\displaystyle=$ $\displaystyle\mathrm{i}\int
d^{3}x\,|e|\,\left(\overline{\Psi}\gamma^{\mu}(\partial_{\mu}-\mathrm{i}g_{\mbox{em}}\,A_{\mu})\Psi-\mathrm{i}\,g_{\mbox{tor}}\,\overline{\psi}_{+}\phi\psi_{+}+\mathrm{i}\,g_{\mbox{tor}}\,\overline{\psi}_{-}\,\phi\,\psi_{-}\right)$
(12) $\displaystyle\rightarrow$ $\displaystyle\mathrm{i}\int
d^{3}x\,\left(\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-\mathrm{i}\,g_{\mbox{em}}\,\,\hat{j}^{\mu}_{\mbox{em}}\,A_{\mu}-\mathrm{i}\,g_{\mbox{tor}}\,\hat{j}_{\mbox{tor}}\,\phi\right)\equiv
S_{0}[\overline{\psi},\psi]+S_{I}[A,\phi]\;.$ (13)

where we have set constants to one, $g_{\mbox{em}}$ and $g_{\mbox{tor}}$ are
the electromagnetic and torsion coupling constants, respectively. In the last
line, to avoid unnecessary complications, we considered only one Dirac point,
say $\psi\equiv\psi_{+}$, and the metric is taken to be flat, $|e|=1$. Hence
the indices are the flat ones, $\mu,\nu,...\to a,b,...$, but to ease the
notation, we shall use Greek letters, anyway. Finally,
$\hat{j}^{\mu}_{\mbox{em}}\equiv\overline{\psi}\gamma^{\mu}\psi$, while
$\hat{j}_{\mbox{tor}}\equiv\overline{\psi}\psi$.

The electromagnetic field is external, hence a four-vector666A different, if
not more natural $(2+1)-$dimensional setting would be to obtain $A_{\mu}$ by
suitably straining the material, see, e.g., [33, 34], and [17]. In that case,
a typical setting is $A_{t}\equiv 0$, $A_{x}\sim u_{xx}-u_{yy}$, $A_{x}\sim
2u_{xy}$, where $u_{ij}$ is the strain tensor.
$A_{\mu}\equiv(V,A_{x},A_{y},A_{z})$. Nonetheless, the dynamics it induces on
the electrons living on the membrane is two-dimensional. Therefore, the
effective vector potential may be taken to be $A_{\mu}\equiv(V,A_{x},A_{y})$,
see, e.g., [47, 48]. Alternatively, the so-called _reduced QED_ approach can
be taken [49, 50]. In such an approach, the gauge field propagates in a three-
dimensional space and one direction is integrated out to obtain an effective
interaction with the electrons constrained to move in a two-dimensional
plane.777This approach could shed some light on the appearance of a photon
Chern-Simons term. On this, see [51, 52].

As mentioned above, we do not consider the dynamics of defects here. Hence the
torsion field $\phi$ as well enters into the action as an external field. A
different view, when $\phi$ is constant, is to include it into the unperturbed
action, where it plays the role of a mass $S_{0}\to S_{m}$, see, e.g., [53],
where $S_{m}=i\int d^{3}x\;\overline{\psi}(\not{\partial}-m(\phi))\psi$.

We are in the situation described by the microscopic perturbation

$S_{I}[F_{i}]=\int d^{3}x\,\hat{X}_{i}(\vec{x},t)F_{i}(\vec{x},t)\;,$ (14)

with the system responding through $\hat{X}_{i}(\vec{x},t)$ to the external
probes $F_{i}(\vec{x},t)$. The general goal is then to find

$\hat{X}_{i}[F_{i}]\;,$ (15)

to the extent of predicting a measurable effect of the combined action of the
two perturbations $F_{i}(\vec{x},t)$: $F^{\mbox{em}}_{1}(\vec{x},t)\propto
A_{\mu}(\vec{x},t)$ that induces the response $\hat{j}^{\mu}_{\mbox{em}}$, and
$F^{\mbox{tor}}_{2}(\vec{x},t)\propto\phi(\vec{x},t)$ that induces the
response $\hat{j}_{\mbox{tor}}$:

$S_{I}[A,\phi]=\int
d^{3}x\,\left(\hat{j}^{\mu}_{\mbox{em}}A_{\mu}+\hat{j}_{\mbox{tor}}\phi\right)\,,$
(16)

where we have included the couplings, $g_{\mbox{em}}$ and $g_{\mbox{tor}}$, in
the respective currents.

In fact, in our model, described microscopically by the action (13), we can
indeed produce a prediction based on the charge conjugation invariance of that
emergent relativistic theory. Such prediction is that

$\chi^{\mbox{torem}}_{\mu}(x,x^{\prime})\sim\langle\hat{j}^{\mbox{em}}_{\mu}(x)\hat{j}^{\mbox{tor}}(x^{\prime})\rangle\equiv
0\;.$ (17)

This is nothing more than an instance of the Furry’s theorem of quantum field
theory [54], that in QED reads

$\chi^{\mbox{em}}_{\mu_{1}...\mu_{2n+1}}(x_{1},...,x_{2n+1})\sim\langle\hat{j}^{\mbox{em}}_{\mu_{1}}(x_{1})\cdots\hat{j}^{\mbox{em}}_{\mu_{2n+1}}(x_{2n+1})\rangle=0\;,$
(18)

and for us implies

$\chi^{\mbox{torem}}_{\mu_{1}...\mu_{2n+1}}(x_{1},...,x_{2n+1},y_{1},...,y_{m})\sim\langle\hat{j}^{\mbox{em}}_{\mu_{1}}(x_{1})\cdots\hat{j}^{\mbox{em}}_{\mu_{2n+1}}(x_{2n+1})\hat{j}^{\mbox{tor}}(y_{1})\cdots\hat{j}^{\mbox{tor}}(y_{m})\rangle=0\;.$
(19)

This result tells us that the effects we are looking for can only be seen if
we move to the nonlinear response regime. We can resort to a well-developed
technique, the high-order harmonic generation (HHG), which can characterize
structural changes both in atoms and molecules and, more recently, bulk
materials (for a recent review, see, e.g. [55]). Therefore, in our scheme, the
intra-band harmonics, governed by the intra-band (electron-hole) current, will
be strongly modified, depending on whether dislocations are there or not.

##  6 Conclusions

When time is duly included in the emergent relativistic-like picture of Dirac
materials, the geometric obstruction to describe the effects of dislocations
in terms of a suitable coupling with torsion within the $(2+1)$-dimensional
field theoretical description of the $\pi$-electrons dynamics can be overcome.
However, surely problems remain to be addressed, like a unique assignment of
torsion to a given distribution of Burgers vectors. Nonetheless, when this is
possible, our suggestion here opens the doors to using these materials as
analogs of many important theoretical scenarios where torsion plays a role.

We then show the first steps toward testing the realization of these
scenarios. We have focussed on one of the Gedankenexperiments of [25] on time-
loop. This could spot the presence of edge dislocations, routinely produced in
Dirac materials. The effect must be based on the interplay between an external
electromagnetic field (necessary to excite the pair particle-hole that
realizes the time-loops) and a suitable distribution of dislocations described
as torsion (that will be responsible for a measurable holonomy in the time-
loop).

We also can prove, on very general grounds, that a nonlinear response regime
is necessary. In particular, in an HHG technique, the structure of such a
response would include manifestation of the torsion-induced holonomy of the
time-loop through specific suppression patterns and generation of higher
harmonics. This sounds promising, for an experimental finding, as the laser-
graphene interaction, controlling electron dynamics on an unprecedented
precision scale, is the focus of intense studies, both theoretical and
experimental, see, e.g., [47, 48].

## Acknowledgements

We sincerely thank Thomas Elze and the other organizers for putting together
an amazing conference. We are indebted to Marcelo Ciappina and Adamantia
Zampeli for their collaboration on these exciting matters and Jorge Zanelli
for many fruitful inspirational discussions. We gladly acknowledge support
from Charles University Research Center (UNCE/SCI/013). P. P. is also
supported by Fondo Nacional de Desarrollo Científico y Tecnológico–Chile
(Fondecyt Grant No. 3200725).

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