#  On the validity of SLAC fermions for the 1+1D helical Luttinger liquid

Zhenjiu Wang<EMAIL_ADDRESS>Max-Planck-Institut für Physik komplexer
Systeme, Dresden 01187, Germany  Fakher Assaad<EMAIL_ADDRESS>wuerzburg.de Institut für Theoretische Physik und Astrophysik, Universität
Würzburg, 97074 Würzburg, Germany  Würzburg-Dresden Cluster of Excellence
ct.qmat, Am Hubland, 97074 Würzburg, Germany  Maksim Ulybyshev
<EMAIL_ADDRESS>Institut für Theoretische Physik und
Astrophysik, Universität Würzburg, 97074 Würzburg, Germany

###### Abstract

The Nielson-Ninomiya theorem states that a chirally invariant free fermion
lattice action, which is local, translation invariant, and real necessarily
has fermion doubling. The SLAC approach gives up on locality and long range
hopping leads to a linear dispersion with singularity at the zone boundary. We
introduce a SLAC Hamiltonian formulation of the U(1) helical Luttinger liquid
and compare our results to bosonization. We argue that non-locality and
concomitant singularity at the zone edge has important implications. Large
momentum transfers yield spurious features already in the non-interacting
case. Upon switching interactions non-locality invalidates the Mermin-Wagner
theorem and allows for long ranged magnetic ordering such that a single
particle gap opens at the Dirac point. Nonetheless, since non-locality allows
for charge fluctuations at any length scale, the ground state remains
metallic. Hence, SLAC and bosonization results show marked differences already
at intermediate coupling strengths.

SLAC fermions, Quantum Monte Carlo, Helical Luttinger liquid

##  I Introduction

The Nielson-Ninomiya theorem states that a chirally invariant free fermion
lattice action, which is local, translation invariant, and real necessarily
has fermion doubling Nielsen and Ninomiya (1981). How should one then carry
out simulations of a single Dirac cone? A possible route is to consider higher
dimensions. A single Dirac cone in say 1+1-dimensions can be realized as a
surface state of a 2+1-dimensional topological insulator. The other Dirac cone
lies on the other surface and as the system size grows mixing between the
cones will vanish such that the physics of a single cone can be studied. In
the realm of high-energy physics, this construction is referred to as domain
wall fermions Kaplan (1992). In the domain of the solid state, this idea has
been used to study correlation effects in helical Luttinger liquids Hohenadler
_et al._ (2011); Hohenadler and Assaad (2012).

Alternatively one can violate one of the assumptions of the Nielsen-Ninomiya
theorem. SLAC fermions are subject to long range hopping and thereby violate
the locality condition. They have been used in a number of solid state Lang
and Läuchli (2019); Li  _et al._ (2018); Tabatabaei  _et al._ (2022); Da Liao
_et al._ (2022) and high energy physics Wellegehausen  _et al._ (2017); Lenz
_et al._ (2019, 2020) setups, and seem to provide a simple route to simulate a
single Dirac cone in a lattice model with finite lattice constant $a$. In
particular, it avoids the potentially expensive step of dealing with higher
dimensional systems. SLAC fermions come with a singularity at the Brillouin
zone boundary at $k=\pm\pi/a$ in one dimension. The question we will ask in
this article is how the non-locality and concomitant singularity at the zone
edge effects the physical results, in comparison to a domain wall fermion
approach.

To do so, we will consider the simplest possible model, the helical Luttinger
liquid emerging at the boundary of a 2D quantum spin Hall insulator as
realized by the Kane-Mele model Kane and Mele (2005). In particular we will
consider a setup with U(1) symmetry, corresponding to conservation of z-total
spin. This choice is challenging for SLAC fermions. For short ranged
interactions the Mermin-Wagner theorem states that continuous symmetries
cannot be spontaneously broken in 1+1-dimension even in the zero temperature
limit. In fact, in conjunction with the intrinsic nesting instabilities of
1+1-dimensional systems this impossibility of ordering leads to the
fluctuation dominated physics of the Luttinger liquid Haldane (1981);
Giamarchi (2004). The non-locality of SLAC fermions violate the assumptions of
the Mermin-Wagner theorem and can hence lead to artifacts especially in the
strong coupling limit. We note that this has recently been pointed out in Ref.
Da Liao  _et al._ (2022).

Another reason for the choice of this model is that it can be solved exactly
since only forward scattering is allowed. The results of the bosonization
approach have been favorably compared to calculations based on domain wall
fermions Hohenadler  _et al._ (2011); Hohenadler and Assaad (2012). Hence the
bosonization results will be our gold standard. In this article we formulate a
SLAC lattice regularization of the helical Luttinger liquid that allows for
negative sign free auxiliary field quantum Monte Carlo (QMC) simulations. Our
aim is to assess if the SLAC fermion approach has the ability to reproduce the
continuum bosonization results.

The article is organized as follows. In the next section, II, we will discuss
the SLAC formulation of the helical Luttinger liquid. Before discussing our
results for the non-interacting and interacting cases in Sec. V, we will
summarize the bosonization results in Sec. III and the technicalities of the
Monte Carlo simulations in Sec. IV. In Sec. VI we discuss a simple model to
understand our strong coupling results. In Sec. VII we summarize the
implications of our results.

##  II  SLAC formulation of the helical liquid

We consider the following one-dimensional model of length $L$ and lattice
constant $a$:

$\displaystyle\hat{H}=$ $\displaystyle-
v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}t(r)\left(\hat{a}_{i}^{\dagger}\hat{b}_{i+r}+\hat{b}^{\dagger}_{i+r}\hat{a}_{i}\right)$
(1)
$\displaystyle+U\sum_{i}\left(\hat{n}^{a}_{i}-\frac{1}{2}\right)\left(\hat{n}^{b}_{i}-\frac{1}{2}\right)$

with

$t(r)=(-1)^{r}\frac{\pi}{L\sin\left(r\pi/L\right)}\text{ for }r\neq 0\text{
and }t(0)=0.$ (2)

Each unit cell harbors two orbitals, and $\hat{a}_{i}^{\dagger}$,
$\hat{b}_{i}^{\dagger}$, are spinless fermion creation operators.

Using periodic boundary conditions and Fourier transformation,

$\begin{pmatrix}\hat{a}_{k}\\\
\hat{b}_{k}\end{pmatrix}=\frac{1}{\sqrt{L}}\sum_{j=1}^{L}e^{ikj}\begin{pmatrix}\hat{a}_{j}\\\
\hat{b}_{j}\end{pmatrix}$ (3)

gives, up to a constant,

$\displaystyle\hat{H}=$ $\displaystyle-
v_{F}\sum_{k}t(k)\left(\hat{a}^{\dagger}_{k},\hat{b}^{\dagger}_{k}\right)\bm{\sigma}^{y}\begin{pmatrix}\hat{a}_{k}\\\
\hat{b}_{k}\end{pmatrix}$ (4)
$\displaystyle-\frac{U}{2}\sum_{i}\left[\left(\hat{a}^{\dagger}_{i},\hat{b}^{\dagger}_{i}\right)\bm{\sigma}^{x}\begin{pmatrix}\hat{a}_{i}\\\
\hat{b}_{i}\end{pmatrix}\right]^{2}$

with

$t(k)=i\sum_{r=-L/2}^{L/2}e^{-ikr}t(r).$ (5)

For any lattice size, $t(k)$ is a real and odd function. It is plotted in Fig.
1 and as apparent scales to $t(k)=k$ in the thermodynamic limit. One will also
notice the Gibbs phenomenon (on even lattices) at the zone boundary associated
to the discontinuity of $t(k)$.

Figure 1:  $t(k)$ for even and odd lattices. The Gibbs phenomenon is apparent
on even lattices. Here we set $a=1$

The rotation:

$\begin{pmatrix}\hat{a}_{i}\\\
\hat{b}_{i}\end{pmatrix}=\frac{1}{\sqrt{2}}\left(1+\bm{\sigma}^{x}\right)\begin{pmatrix}\hat{c}_{i,\uparrow}\\\
\hat{c}_{i,\downarrow}\end{pmatrix}$ (6)

gives

$\hat{H}=-v_{F}\sum_{k,\sigma}\sigma
k\hat{c}^{\dagger}_{k,\sigma}\hat{c}_{k,\sigma}+U\sum_{i}\left(\hat{n}_{i,\uparrow}-1/2\right)\left(\hat{n}_{i,\downarrow}-1/2\right)$
(7)

corresponding to a helical liquid with Hubbard interaction. Our goal is to
investigate if the SLAC approach indeed reproduces the expected results
obtained from bosonization of the helical liquid.

##  III  Results from bosonization

Let us start by stating the bosonization Giamarchi (2004); Hohenadler and
Assaad (2012) results valid in the continuum limit, $a\rightarrow 0$. In this
limit, charge fluctuations obey boson commutation rules. Owing to time
reversal symmetry as well as U(1) spin symmetry only forward scattering
processes are allowed. As a consequence the model, within bosonization can be
solved exactly and the correlation functions follow the universal form:

$\displaystyle
C_{n}(r)\equiv\langle\hat{n}(r)\hat{n}(0)\rangle\propto\frac{1}{r^{2}}$
$\displaystyle
C_{S^{z}}(r)\equiv\langle\hat{S}^{z}(r)\hat{S}^{z}(0)\rangle\propto\frac{1}{r^{2}}$
$\displaystyle
C_{S^{x}}(r)\equiv\langle\hat{S}^{x}(r)\hat{S}^{x}(0)\rangle\propto\frac{\cos(2k_{f}r)}{r^{2K_{\rho}}}$
(8) $\displaystyle
C_{\Delta}(r)\equiv\langle\text{Re}\hat{\Delta}^{\dagger}_{r}\text{Re}\hat{\Delta}^{\phantom{\dagger}}_{0}\rangle\propto\frac{1}{r^{2/K_{\rho}}}$

In the above $K_{\rho}$ is a interaction strength dependent Luttinger liquid
exponent,
$\hat{n}(r)=\sum_{\sigma}\hat{c}^{\dagger}_{r,\sigma}\hat{c}_{r,\sigma}$,
$\bm{S}(r)=\frac{1}{2}\bm{c}^{\dagger}_{r}\bm{\sigma}\bm{c}^{\phantom{\dagger}}_{r}$
with $\bm{\sigma}$ a vector of Pauli spin matrices and
$\hat{\Delta}^{\dagger}_{r}=\hat{c}^{\dagger}_{r,\uparrow}\hat{c}^{\dagger}_{r,\downarrow}$.
Here, $2k_{f}$ denotes the momentum difference of the left spin up and right
spin down movers. This wave vector is naturally picked up in $C_{S^{x}}(r)$
since it involves scattering between the two branches. In our construction,
$k_{f}=0$. Before proceeding and as mentioned earlier the bosonization results
are consistent with the domain wall fermion approach of Hohenadler  _et al._
(2011); Hohenadler and Assaad (2012) even in the rather strong coupling limit.
The above will be our reference result and we will ask the question under
which conditions we can reproduce it with the SLAC lattice regularization.

##  IV Quantum Monte Carlo simulations

The Hamiltonian of Eq. 1 does not suffer from a negative sign problem in QMC
simulations Li  _et al._ (2016); Wei  _et al._ (2016). To see this, one will
rewrite the model as,

$\displaystyle\hat{H}=$ $\displaystyle-
v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}t(r)\left(\hat{a}_{i}^{\dagger}\hat{b}_{i+r}+\hat{b}^{\dagger}_{i+r}\hat{a}_{i}\right)$
(9)
$\displaystyle-\frac{U}{2}\sum_{i}\left(\hat{a}^{\dagger}_{i}\hat{b}^{\phantom{\dagger}}_{i}+\hat{b}^{\dagger}_{i}\hat{a}^{\phantom{\dagger}}_{i}\right)^{2}$

where we have omitted a constant and adopt a Majorana representation,

$\hat{a}_{i}=\frac{1}{2}\left(\hat{\gamma}_{i,1,1}+i\hat{\gamma}_{i,2,1}\right),\,\,\,\hat{b}_{i}=-\frac{i}{2}\left(\hat{\gamma}_{i,1,2}+i\hat{\gamma}_{i,2,2}\right).$
(10)

This yields:

$\displaystyle\hat{H}=$ $\displaystyle
v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}t(r)\frac{i}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{x}\hat{\bm{\gamma}}_{i+r}$
(11)
$\displaystyle-\frac{U}{2}\sum_{i}\left(\frac{1}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{y}\hat{\bm{\gamma}}_{i}\right)^{2},$

where
$\hat{\bm{\gamma}}_{i}^{T}=\left(\hat{\gamma}_{i,1,1},\hat{\gamma}_{i,2,1},\hat{\gamma}_{i,1,2},\hat{\gamma}_{i,2,2}\right)$.
Here we have used the fact that $t(r)$ is an odd function of $r$ and adopted
the notation $\hat{\bm{\gamma}}_{i,\sigma,\tau}$ where the Pauli $\tau$
($\sigma$)-matrices act of the $\tau$ ($\sigma$) indices. A global O(2)
symmetry in the $\sigma$-indices now becomes apparent. After a real Hubbard-
Stratonovich transformation of the perfect square, the Fermion determinant
will be given by the square of a Pfaffian. Since one will show that the
Pfaffian is real, we will conclude in the absence of the negative sign
problem. Hence the absence of sign problem for this SLAC model of the helical
liquid follows the same logic as for the so called spin-less t-V model Li  _et
al._ (2015); Huffman and Chandrasekharan (2014). In the appendix we show the
absence of sign problem for the generic model:

$\displaystyle\hat{H}=$ $\displaystyle-
v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}t(r)\left(\hat{a}_{i}^{\dagger}\hat{b}_{i+r}+\hat{b}^{\dagger}_{i+r}\hat{a}_{i}\right)$
(12)
$\displaystyle-\frac{U}{2}\sum_{i}\left[\left(\hat{a}^{\dagger}_{i},\hat{b}^{\dagger}_{i}\right)\sigma_{\alpha}\left(\hat{b}^{\phantom{\dagger}}_{i},\hat{a}^{\phantom{\dagger}}_{i}\right)^{T}\right]^{2}$

where $\sigma_{\alpha}$ is a Pauli spin matrix. Note that after computing the
square one will explicitly see that the Hamiltonian is $\alpha$-independent.
For any value of $\alpha$, we can use the ALF Assaad  _et al._ (2022)
implementation of the finite temperature auxiliary field QMC algorithm
Blankenbecler  _et al._ (1981); Hirsch (1985); White  _et al._ (1989); Assaad
and Evertz (2008). In fact, Eq. 12 that formulates the interaction in terms of
a perfect square has the required form for usage of the ALF-library, and
concomitant Hubbard-Stratonovich transformation.

As mentioned above, the results are $\alpha$-independent. However the Monte
Carlo Markov chain will have a strong $\alpha$-dependence. We have seen that
we obtain the best results when considering the $\sigma_{y}$ formulation. This
stems form the fact that after the rotation of Eq. 6 the U(1) symmetry of the
helical liquid is satisfied for each Hubbard-Stratonovich field configuration.

We also would like to stress that since we are working in the Hamltonian
formulation, the resulting Lagrangian has SLAC hoppings only in the spatial
direction and is local along the Euclidean time direction.

We used the interaction strength (band width) as the energy unit for
simulations at large (small) values of $U/v_{F}$. For $U/v_{F}\leq 4$ we
choose $v_{F}\beta=L$ and $v_{F}\Delta\tau=0.1$; whereas for $U/v_{F}>4$ we
considered $U\beta/4=L$ and $U\Delta\tau/4=0.1$.

##  V Results

We will show that the SLAC approach suffers from two basic issues.

The first one can be seen already in the non interacting limit and originates
from processes with large momentum transfer. This deficiency can be
illustrated as the violation of the anomaly relation in lattice Schwinger
model with SLAC fermions Nason (1985). If we consider the continuum theory and
turn on a constant electrical field pointing to the right, the right movers
will acquire momentum and energy and will fill their branch of the dispersion
relation up to some positive level. At the same time the left movers will
loose energy and hence their branch of the cone will be filled only up to the
same but negative level. Thus the axial charge will appear as an imbalance
between the right and left movers. However, this is not true for the SLAC
fermions due to the finite size of the Brillouin zone and finite depth of the
Dirac sea. E.g. the right movers at the bottom of the Dirac sea will also
acquire energy thus the very bottom of this branch of the dispersion relation
will not be filled any more. These effects will compensate the difference
between right- and left-movers in the low momentum modes leading to the axial
charge being always zero. Though this is only qualitative illustration, it
shows the presence of the non trivial dynamics near the edge of the Brillouin
zone.

Another issue can be seen upon switching on intermediate to strong
correlations as measured in unit of the band-width. In this case we observe a
long ranged order again contradicting the results from continuum theory.

Both points will be carefully studied in the subsequent sub-sections.

###  V.1 The non-interacting case

SLAC fermions become very transparent when introducing a length scale $\xi$ in
the hopping:

$t_{\xi}(r)=t_{0}(\xi)\frac{(-1)^{r/a}\pi}{\frac{L}{a}\sin(r\pi/L)}e^{-\sin(r\pi/L)/\xi}.$
(13)

In the above, we adjust $t_{0}(\xi)$ so as to fix the bandwidth to $2\pi$.
Clearly, the Fourier transform of a short range function has to be smooth and
one will see that for any finite value of $\xi$ we observe two crossings of
the Fermi surface albeit with very different values of the velocity. In fact
the velocity at the zone boundary diverges with growing values of $\xi$.

Figure 2:  $t_{\xi}(k)/g(\xi)$ as a function of the length scale $\xi$, for
$L=257$. $g(\xi)$ is the $k$ independent factor which normalizes the band
width to be $\pi v_{F}$.

In principle, for any finite value of $\xi$ we expect Umklapp processes to be
relevant such that any finite value of $U$ should lead to an insulating state.
However, since the velocity at the zone boundary diverges as $\xi$, the phase
space available to these Umklapp process will vanish in the
$\xi\rightarrow\infty$ limit. Another consequence of the singularity at the
zone boundary is that large momentum transfer will always provide a
discrepancy with the bosonization even in the non-interacting case. One can
illustrate this by considering the charge-charge correlation functions for
SLAC Hamiltonian for the half-filled case, $\mu=0$, at zero temperature:

$\langle\hat{n}(r)\hat{n}(0)\rangle=\frac{1}{2\pi^{2}}\frac{\cos(\pi
r)-1}{r^{2}}.$ (14)

In the above
$\hat{n}(r)=\sum_{\sigma}\hat{c}^{\dagger}_{r,\sigma}\hat{c}_{r,\sigma}$. This
result is independent on the value of $\xi$ and merely relies on the fact that
the dispersion relation intersects the Fermi energy at wave vectors $k=0$ and
$k=\frac{\pi}{a}$. The above expression already deviates from the bosonization
result III and shows that already at this level one will obtain the same
result as for the continuum model, where the zone edge diverges, only if one
blocks large momentum transfers. This can be done introducing point-splitting
operators on the lattice, as was already suggested in Weinstein (1982); Lenz
_et al._ (2020).

###  V.2 Monte Carlo results

Figure 3:  Real space correlation functions (a)-(e) and corresponding
structure factors (f)-(j). Here we consider the charge, $C_{n}$, x
(z)-component of spin $C_{S^{x}}$ /$C_{S^{z}}$ ), pairing $C_{\Delta}$ and
single particle $G_{\uparrow}$ correlation functions. All the subfigures share
the same legend color as the one in (a). For $C_{n}$, $C_{S^{x}}$,
$C_{\Delta}$ and $G_{\uparrow}$, we took $L=243$; whereas for $C_{S^{z}}$, we
took $L=203$. The reason of this mismatch is the large QMC fluctuation of
$C_{S^{z}}$ for large sizes.  Figure 4:  $B(k=0,L)L$ as function of $1/L$ for
charge (a), $\hat{S}^{x}$ (b), $\hat{S}^{z}$ (c), pairing (d). (b) and (c)
share the same legend color as the one in (a).

We have computed structure factors:

$S_{\bullet}(k)=\sum_{r}e^{ikr}C_{\bullet}(r)$ (15)

where the bullet refers to charge, spin along the z- or x-spin quantization
axis or paring correlations (see Eq. III). To obtain an estimate of the power
law decay at a given wave vector, one can consider:

$\displaystyle b_{\bullet}(k,L/2)\equiv
2C_{\bullet}(k)-C_{\bullet}(k+2\pi/L)-C_{\bullet}(k-2\pi/L).$ (16)

$\frac{L}{2\pi}b_{\bullet}(k,L/2)$ corresponds to the left minus the right
derivative at a given $k$-vector. Hence for a smooth function this quantity
scales to zero as a function of system size. However for $k$-vectors where one
observes a cusp, it will scale to a finite value. One will show that:

$C_{\bullet}(L/2)=\frac{1}{4L}\sum_{k}e^{ikL/2}b_{\bullet}(k,L/2)$ (17)

such that the scaling of $b_{\bullet}(k,L/2)$ at wave vectors k where one
observes a cusp will reflect the decay of the correlation function Assaad and
Würtz (1991) at this wave vector.

Fig. 3 plots the real and k-space correlation functions for the above
mentioned quantities. To better understand the results, we consider the
behavior of the cusps in the corresponding structure factors by plotting
$b_{\bullet}(k=0,L/2)$ as a function of system size and coupling strength.

Let us start with the charge. From Fig. 3(a) we see that irrespective of the
coupling constant in the range $U\in\left[0,10\right]$ the real space charge
correlation decays as $1/r^{2}$. In the weak coupling limit we observe a
$(-1)^{r}$ modulation alongside the uniform decay. This weak coupling behavior
gives way to a uniform decay at strong coupling. In k-space, Fig. 3(f) we see
that the cusp at $k=\pi$ rounds off as a function of growing interaction
strength but that the cusp at $q=0$ remains robust. We also notice that as a
function of growing interaction strength the charge response is suppressed. We
can pin down the charge exponent by analyzing $Lb_{n}(k=0,L/2)$ in Fig. 4(a).
Irrespective of the interaction strength, it is to an accurate degree
$L$-independent thus reflecting a $1/r^{2}$ decay of the charge correlations.

At weak coupling the z-component of spin is very similar to the charge (at
$U=0$ they are identical), see Figs. 3(c,h). In contrast however, the cusp at
$k=0$ becomes more pronounced at strong coupling. Fig. 4(c), shows that the
z-spin correlations acquire a non-trivial exponent in the strong coupling
limit. This stands at odds with the bosonization result of Eq. III.

The spin-correlations in the x-spin quantization, Figs. 3(b,g), direction are
most intriguing results. At weak coupling and on our system sizes, this
correlation function follows roughly a $1/r^{2}$ form consistent with
$Lb_{S^{x}}(k=0,L)$ constant, Fig. 4(b). $Lb_{S^{x}}(k=0,L/2)$ has a marked
U-dependence. It is remarkable to see that at strong coupling
$Lb_{S^{x}}(k=0,L/2)\propto L^{2}$ thus suggesting long range magnetic order
as is confirmed by the very strong peak in the structure factor a $k=0$ and
the lack of decay in real space. We note that $Lb_{S^{x}}(k=0,L/2)\propto
L^{2}$ is consistent with $C_{S^{x}}(k=0)\propto L$. This result seems at odd
with the Mermin-Wagner theorem, that states that a continuous symmetry cannot
be broken at $T=0$ in the ground state. However, the assumptions for the
theorem to be valid requires short ranged interactions. The non-locality of
the SLAC fermions may very well invalidate this assumption. We note that long
range order can be stabilized by coupling spin-chains locally to an ohmic bath
thus introducing long ranged interactions along the imaginary time Weber  _et
al._ (2022).

Long ranged order along the x-spin quantization axis breaks time reversal
symmetry and allows for elastic scattering between the right-moving spin down
and left moving spin up electrons. At the single particle, mean-field level we
expect:

$\hat{H}_{MF}=\sum_{k}\hat{\bm{c}}^{\dagger}_{k}\left[-v_{f}k\sigma_{z}+m_{x}\sigma_{x}\right]\hat{\bm{c}}^{\phantom{\dagger}}_{k}$
(18)

where $m_{x}$ denotes the ordered moment. This symmetry breaking generates a
mass gap at the Fermi momentum $k_{f}=0$:
$E_{k}=\pm\sqrt{\left(v_{f}k\right)^{2}+m_{x}^{2}}$. Fig. 3(j), plots the
single particle equal time Green function,
$G_{\sigma}(k)=\langle\hat{c}^{\dagger}_{k,\sigma}\hat{c}^{\phantom{\dagger}}_{k,\sigma}\rangle$.
As $U$ grows, the singularity at $k=0$ evolves to a smooth feature. Fig. 3(e)
confirms this: at weak coupling, $G_{\sigma}(r)\propto 1/r$ as expected for
Dirac electrons in 1+1D, and in the strong coupling limit the $L$-independent
from of $LB(k=0,L)$ is consistent with a mass gap. $G_{\sigma}(k)$ has another
singularity at $k=\pi$ that dominates the long-ranged real space behavior.
Putting all together, the data in the strong coupling limit is consistent with
the form: $G_{\sigma}(r)\propto\left(ae^{-r/\xi}+(-1)^{r}\right)/r$ where
$\xi$ is set by the inverse ordered moment $m_{x}$. We also notice that the
overall amplitude of $G_{\sigma}(r)$ diminishes as a function of U in the
strong coupling.

Finally, we consider the pairing correlations in Figs. 3(d,i) as well as in
Fig. 4(d). We again observe non-analyticities at $k=0$ and $k=\pi$ in the
structure factor. The non-analytical behavior at $k=0$ survives the strong
coupling limit, whereas $C_{\Delta}(k)$ evolves towards a smooth function in
the vicinity of $k=\pi$. The singularity at $k=0$ leads to a $1/r^{2}$ decay
of the pair correlation, and again the overall amplitude of the correlation
function decreases as as function of increasing $U$.

##  VI Interpretation of the strong coupling limit

In this section, we provide consistent interpretation of the strong coupling
limit. In this limit the QMC data shows long ranged magnetic ordering along
the x-spin quantization axis. The single particle Green function decays as
$(-1)^{r}/r$, and the density as well as the pairing correlations follow a
$1/r^{2}$ law. On the other hand the z-component of spin correlations have a
power-law decay with exponent depending on the interaction strength. Clearly
this behavior lies at odds with the bosonization results. A simple model can
account for most of the above. Let us start with a mean-field representation
of the strong coupling ground state:

$|\psi_{0}\rangle=\prod_{r}\frac{1}{\sqrt{2}}\left(\hat{c}^{\dagger}_{r,\uparrow}+\hat{c}^{\dagger}_{r,\downarrow}\right)|0\rangle$
(19)

Since the above wave function has no charge fluctuations and precisely one
electron per site it is the ground state 111Here we omit the spin degeneracy
and just choose spin-symmetry broken state as suggested by the Monte Carlo
result. of the Hubbard interaction term, $\hat{H}_{U}$ with energy $E_{0}$.
Consider the small hopping limit such that the ground state wave function can
be estimated perturbatively in the hopping $\hat{H}_{t}$ Messiah (1999):

$|\psi\rangle=|\Psi_{0}\rangle+\hat{Q}_{0}\frac{1}{\hat{H}_{U}-E_{0}}\hat{Q}_{0}\hat{H}_{t}|\Psi_{0}\rangle.$
(20)

In the above $\hat{Q}_{0}=1-|\Psi_{0}\rangle\langle\Psi_{0}|$. Let us now
compute the charge fluctuations,
$C_{n}(r)=\langle\Psi|\left(\hat{n}_{r}-1\right)\left(\hat{n}_{0}-1\right)|\Psi\rangle$
for $r\neq 0$. The sole contribution reads:

$\displaystyle C_{n}(r)=\langle$
$\displaystyle\Psi_{0}|\hat{H}_{t}\hat{Q}_{0}\frac{1}{\hat{H}_{U}-E_{0}}\hat{Q}_{0}\left(\hat{n}_{r}-1\right)\left(\hat{n}_{0}-1\right)\times$
(21)
$\displaystyle\hat{Q}_{0}\frac{1}{\hat{H}_{U}-E_{0}}\hat{Q}_{0}\hat{H}_{t}|\Psi_{0}\rangle.$

Since $\hat{H}_{t}$ has hopping processes on all scales it contains the
operator
$t(r)\sum_{\sigma}\sigma\hat{c}^{\dagger}_{r,\sigma}\hat{c}^{\phantom{\dagger}}_{0,\sigma}$.
Applied on $|\Psi_{0}\rangle$ it will generate a doublon and holon at distance
$r$ with an energy cost with respect to $E_{0}$ set by $U$. This charge
fluctuation will be picked up by
$\left(\hat{n}_{r}-1\right)\left(\hat{n}_{0}-1\right)$. Finally the doublon
holon pair will be destroyed by the operator
$t(r)\sum_{\sigma}\sigma\hat{c}^{\dagger}_{0,\sigma}\hat{c}^{\phantom{\dagger}}_{r,\sigma}$
again contained in $\hat{H}_{t}$. As a result, we estimate:

$C_{n}(r)\simeq\frac{t^{2}(r)}{U^{2}}\propto\frac{1}{U^{2}r^{2}}.$ (22)

The power-law is confirmed by the QMC data of Fig. 3(a). It is also
interesting to note that the magnitude of the charge-charge correlations are
predicted to scale as $1/U^{2}$. Comparison between the $U=6$ and $U=10$ data
in Fig. 3(f) supports this scaling.

We note that the very same argument can be carried out for the pairing
correlations. Let us pick up the above argument at the point where a doublon
is created on site $r$ and a holon on site $0$. Applying the pairing operator
$\hat{\Delta}_{r}\hat{\Delta}^{\dagger}_{0}$ on this state, will yield a non-
zero result and transfer the doublon (holon) to the origin (site r). The
operator
$t(r)\sum_{\sigma}\sigma\hat{c}^{\dagger}_{r,\sigma}\hat{c}^{\phantom{\dagger}}_{0,\sigma}$
will then annihilate the doublon-holon pair, and we will obtain a finite
overlap with the mean-field ground state. Hence we also expect:

$C_{\Delta}(r)\simeq\frac{t^{2}(r)}{U^{2}}\propto\frac{1}{U^{2}r^{2}}$ (23)

in the strong coupling limit, which is consistent with our QMC data, but
inconsistent with the bosonization results Eq. III.

Doublons and holons are spin singlets. In the above approximation
$C_{S^{z}}(r)$ will hence vanish identically. The data for the z-component of
spin have to be understood in terms of the transverse fluctuations of the
ordered moment along the spin-x quantization axis.

We now consider the single particle Green function. Here, the relevant terms
in
$\langle\Psi|\hat{c}^{\dagger}_{r,\sigma}\hat{c}^{\phantom{\dagger}}_{0,\sigma}|\Psi\rangle$
are the mixed terms of the form:

$\langle\Psi_{0}|\hat{c}^{\dagger}_{r,\sigma}\hat{c}^{\phantom{\dagger}}_{0,\sigma}\hat{Q}_{0}\frac{1}{\hat{H}_{U}-E_{0}}\hat{Q}_{0}\hat{H}_{t}|\Psi_{0}\rangle.$
(24)

The doublon-holon pair created by $\hat{H}_{t}$ will be annihilated by the
single particle transfer
$\hat{c}^{\dagger}_{r,\sigma}\hat{c}^{\phantom{\dagger}}_{0,\sigma}$. In
accordance with the QMC results this approximation gives:

$G_{\sigma}(r)\propto\frac{t(r)}{U}\propto\frac{(-1)^{r}}{U}.$ (25)

Finally we comment on the nature, metallic or insulating, of the strong
coupling wave function. The very fact that the charge correlations follow a
power-law, suggest a metallic ground state. An accepted definition of an
insulating or metallic state is the Drude weight Kohn (1964), that probes the
localization of the wave function. Here, one considers a ring geometry and
threads a magnetic flux $\Phi$ through the the center of the ring. Such a flux
will have an effect if the charge carriers are delocalized and can circle
around it and, owing to the Aharonov-Bohm effect, acquire a phase factor
$e^{2\pi i\Phi/\Phi_{0}}$ where $\Phi_{0}$ is the flux quanta. Here we assume
that the charge carriers have the electron charge. The Drude weight in
d-spatial dimensions is defined as:

$D(L)=\frac{1}{L^{d-2}}\left.\frac{\partial^{2}E_{0}(\Phi)}{\partial\Phi^{2}}\right|_{\Phi=0}.$
(26)

For the insulating state $D(L)$ vanishes exponentially with $L$ reflecting the
localization length of the wave function. For a metallic state the Drude
weight is finite. Let us now use this accepted criterion to the SLAC fermions,
in the strong coupling limit. A glimpse at the wave function in second order
perturbation theory (see Eq. 20) shows that it contains holon-doublon
excitations, at all length scales. The fact that they are costly in energy,
means that they are short lived, but during this short time, they can
propagate over large distances due to the non-locality of the hopping. Hence
we expect the Drude weight to be finite. To substantiate this statement we
carry out the following estimations. The flux leads to a twist in the boundary
condition:

$\hat{c}_{r+L,\sigma}=e^{2\pi i\frac{\Phi}{\Phi_{0}}}\hat{c}_{r,\sigma}$ (27)

that we can rid of with the canonical transformation:

$\hat{d}_{r,\sigma}=e^{-2\pi
i\frac{\Phi}{\Phi_{0}}\frac{r}{L}}\hat{c}_{r,\sigma}.$ (28)

Under this canonical transformation, the Hubbard term remains invariant, the
hopping reads,

$\hat{H}_{t}(\Phi)=v_{F}i\sum_{i,\sigma}\sum_{r}\sigma
t(r)\hat{d}^{\dagger}_{i,\sigma}\hat{d}^{\phantom{\dagger}}_{i+r,\sigma}e^{2\pi
i\frac{\Phi}{\Phi_{0}}\frac{r}{L}},$ (29)

and $\hat{d}_{r,\sigma}$ satisfies periodic boundary conditions:
$\hat{d}_{r,\sigma}=\hat{d}_{r+L,\sigma}$. Let us now compute the second order
contribution to the energy that will pick up the dependence on the flux:

$E_{2}(\Phi)=\langle\Psi_{0}|\hat{H}_{t}(\Phi)\hat{Q}_{0}\frac{1}{\hat{H}_{U}-E_{0}}\hat{Q}_{0}\hat{H}_{t}(\Phi)|\Psi_{0}\rangle$
(30)

Starting from $|\Psi_{0}\rangle$ one will create for example a holon at
position $i$ and a doublon at position $i+r$ by applying the hopping. This
process has matrix element $iv_{F}t(r)e^{-2\pi
i\frac{\Phi}{\Phi_{0}}\frac{r}{L}}$ and energy cost set by $U$. The only way
to perceive the flux is for the charge excitation to encircle it. Hence the
second hopping process should destroy the doublon in favor of single occupancy
at site $i+r$ and create an electron at site $i+L\equiv i$ thereby restoring
single occupancy on this site such that the overlap with $|\Psi_{0}\rangle$
does not vanish. This second process comes with matrix element:
$iv_{F}t(L-r)e^{-2\pi i\frac{\Phi}{\Phi_{0}}\frac{L-r}{L}}$. Putting
everything together one obtains:

$E_{2}(\Phi)\propto-\frac{v_{F}^{2}}{U}\sum_{i}\sum_{r}t(r)t(L-r)\cos\left(2\pi\frac{\Phi}{\Phi_{0}}\right).$
(31)

We hence see that in this approximation, the Drude weight reads:

$D\propto
L^{2}\left(\frac{2\pi}{\Phi_{0}}\right)^{2}\frac{v_{F}^{2}}{U}\sum_{r}t(r)t(L-r).$
(32)

One will check that $\sum_{r}t(r)t(L-r)$ takes a finite value. Hence we obtain
the result that the Drude weight actually diverges as $L^{2}$, and only at
$U=\infty$ will we have an insulating state on any finite lattice.

##  VII  Summary and Conclusions

The U(1) helical Luttinger liquid provides a unique playground to test
interaction effects within the SLAC fermion approximation of the model. Our
key result is that giving up on locality leads to marked differences between
the SLAC and bosonization results at least in the strong coupling limit.

One dimensional systems are generically nested. For the helical Luttinger
liquid this leas to $\chi_{x}(k=0,\omega=0)\propto\log\frac{v_{f}}{k_{B}T}$.
As a consequence, a mean-field approach to correlation effects will generate
long-ranged magnetic order along the spin-x quantization axis and a charge
gap. Both the charge gap and the ordered moment will follow an essentially
singularity in the weak coupling limit.

For generic local one-dimensional models we know that the above Stoner
arguments cannot be made due to the Mermin-Wagner theorem that tells us that
fluctuations will destroy the ordering even in the ground state. For our
specific case, continuous U(1) spin-symmetry breaking is not allowed. This
competition between the Stoner instability and the Mermin-Wagner theorem is at
the very origin of Luttinger liquid behavior generic to 1+1D interacting
systems. This is exemplified by the helical Luttinger liquid: a metallic state
with no single particle gap and an interaction strength dependent power-law
decay of the spin-spin correlations in the transverse direction. We note that
due to U(1) spin-symmetry Umklapp processes are symmetry forbidden such that
the system will remain metallic for arbitrary large interactions. This
understanding of the helical Luttinger liquid, has been confirmed numerically
within a domain-wall fermion approach Hohenadler  _et al._ (2011); Hohenadler
and Assaad (2012) in which interaction effects are included only on one set of
domain-wall fermions.

The non-locality of the SLAC fermion approach brings major differences to the
above picture. The key-point is that it violates the assumptions of the
Mermin-Wagner theorem and allows for Stoner type physics. This aspect of SLAC
fermions was recently pointed out in Ref. Da Liao  _et al._ (2022). Our
numerical results explicitly confirm this in the strong coupling limit where
long ranged magnetic order along the x-spin quantization axis and global U(1)
spin symmetry breaking is observed. This allows for a mass term and in fact we
observe a single particle gap opening at the Fermi wave vector. Generically,
spin-orbit coupling will reduce the U(1) continuous symmetry to a $Z_{2}$
discrete one. In this case, the aforementioned issues of SLAC action with the
Mermin-Wagner theorem will be waived. However, some artefacts will likely
survive even for the discrete $Z_{2}$ symmetry. In particularly, the deviation
of the behaviour of the correlation functions in Eq. 23 from the strong
coupling limit does not involve a continuous symmetry for its derivation.
Hence these discrepancies will remain even for models with discrete
symmetries. Another important point is the nature of the ordered state
observed in our QMC simulations. In contrast to Dirac systems where magnetic
mass terms are generated spontaneously Assaad and Herbut (2013), this ordered
state remains metallic. This is again a consequence of non-locality inherent
to SLAC approach that produces doublon-holon pairs at any length scale.
Equivalently, the current operator becomes long-ranged. Strictly speaking
Gross-Neveu transitions that have beed studied in the realm of SLAC fermions
Lang and Läuchli (2019); Tabatabaei  _et al._ (2022); Da Liao  _et al._ (2022)
are not metal to insulator transitions but metal-to-metal ones.

At vanishing coupling strength, the results of the Helical-Luttinger liquid
with $K_{\rho}=1$ become exact provided that we block large momentum
transfers. As mentioned above, this non-interacting point is unstable.
However, since the correlation induced gap is exponentially small, one can
conjecture that exponentially large lattices will be required to perceive the
effects of non-locality.

###### Acknowledgements.

We would like to thank L. Janssen, Z.Y. Meng , and A. Wipf for comments.
Computational resources were provided by the Gauss Centre for Supercomputing
e.V. (www.gauss-centre.eu) through the John von Neumann Institute for
Computing (NIC) on the GCS Supercomputer JUWELS Jülich Supercomputing Centre
(2019) at Jülich Supercomputing Centre (JSC). MU thanks the DFG for financial
support under the projects UL444/2-1. FFA acknowledges financial support from
the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter - ct.qmat (EXC 2147, Project No. 390858490) as well
as the SFB 1170 on Topological and Correlated Electronics at Surfaces and
Interfaces (Project No. 258499086).

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##  Appendix A Absence of negative sign problem

There are various ways of formulating the QMC. Although in principle
equivalent, the various formulations will lead to different Markov time
series. Thereby the autocorrelation time for a given observable may be
formulation dependent. The various formulations stem from different ways
writing the interaction term:

$\hat{H}_{U}=-\frac{U}{2}\sum_{i}\left[\left(\hat{a}^{\dagger}_{i},\hat{b}^{\dagger}_{i}\right)\sigma_{\alpha}\left(\hat{b}^{\phantom{\dagger}}_{i},\hat{a}^{\phantom{\dagger}}_{i}\right)^{T}\right]^{2}.$
(33)

In the above, $\hat{H}_{U}$ is independent on the choice of the Pauli spin
matrix $\sigma_{\alpha}$. For the repulsive values of $U>0$, we can carry out
a HS decomposition of the perfect square term to obtain:

$Z\propto\int D\left\\{\phi(i,\tau)\right\\}e^{-S_{\alpha}(\phi(i,\tau))}$
(34)

with

$S_{\alpha}(\phi(i,\tau))=\int_{0}^{\beta}d\tau\sum_{i}\frac{\phi^{2}(i,\tau)}{2U}-\log\text{Tr}{\cal
T}e^{-\int_{0}^{\beta}d\tau\hat{h},_{\alpha}(\tau)}$ (35)
$\hat{h}_{\alpha}(\tau)=\hat{H}_{0}+\sum_{i}\phi(i,\tau)\left(\hat{a}^{\dagger}_{i},\hat{b}^{\dagger}_{i}\right)\sigma_{\alpha}\left(\hat{b}^{\phantom{\dagger}}_{i},\hat{a}^{\phantom{\dagger}}_{i}\right)^{T},$
(36)

and

$\hat{H}_{0}=-v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}t(r)\left(\hat{a}_{i}^{\dagger}\hat{b}_{i+r}+\hat{b}^{\dagger}_{i+r}\hat{a}_{i}\right).$
(37)

We again stress that the partition function is $\alpha$-independent but that
$\hat{h}_{\alpha}(\tau)$ has an explicit $\alpha$-dependency.

Using the Majorana representation of Eq. 10 we obtain different expressions
for various choices of $\alpha$. Below we go through them one by one.

$\bullet$ $\sigma_{x}$
In this case,

$\hat{h}_{x}(\tau)=v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}\frac{it(r)}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{x}\hat{\bm{\gamma}}_{i+r}+\sum_{i}\frac{\phi(i,\tau)}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{y}\hat{\bm{\gamma}}_{i}.$
(38)

We see that the operators $\hat{T}^{+}$, and $\hat{T}^{-}$ with

$\hat{T}^{+}\gamma_{i}\hat{T}^{+,-1}=T^{+}\gamma_{i}\,\,\text{ with
}T^{+}=i\tau_{y}\tau_{x}\sigma_{x}K$ (39)

and

$\hat{T}^{-}\gamma_{i}\hat{T}^{-,-1}=T^{-}\gamma_{i}\,\,\text{ with
}T^{-}=\tau_{z}i\sigma_{y}K$ (40)

where $K$ denotes complex conjugation leaves $\hat{h}(\tau)$ invariant.
Furthermore $\left(T^{\pm}\right)^{2}=\pm 1$ and
$\left\\{T^{+},T^{-}\right\\}=0$. Hence according to Ref. Li  _et al._ (2016)
the Hamiltonian falls into the so called Majorana class and does not suffer
for the negative sign problem.

$\bullet$ $\sigma_{y}$
For this choice,

$\hat{h}_{y}(\tau)=v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}\frac{it(r)}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{x}\hat{\bm{\gamma}}_{i+r}+\sum_{i}\frac{\phi(i,\tau)}{4}\hat{\bm{\gamma}}_{i}^{T}\sigma_{y}\tau_{x}\hat{\bm{\gamma}}_{i}.$
(41)

such that we have to choose:

$T^{-}=i\tau_{y}K\,\,\text{ and }\,\,T^{+}=\tau_{z}K$ (42)

to show that the model is in the Majorana class.

$\bullet$ $\sigma_{z}$
For this choice,

$\hat{h}_{z}(\tau)=v_{F}\sum_{i=1}^{L}\sum_{r=-L/2}^{L/2}\frac{it(r)}{4}\hat{\bm{\gamma}}_{i}^{T}\tau_{x}\hat{\bm{\gamma}}_{i+r}-\sum_{i}\frac{\phi(i,\tau)}{4}\hat{\bm{\gamma}}_{i}^{T}\sigma_{y}\tau_{z}\hat{\bm{\gamma}}_{i}.$
(43)

such that we have to choose:

$T^{-}=i\tau_{y}K\,\,\text{ and }\,\,T^{+}=\tau_{z}\sigma_{x}K$ (44)

to show that the model is in the Majorana class.


11institutetext:  Universidade Rio Grande do Norte, Brazil
11email: aderson@geofisica.ufrn.br,mam<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: CNRS, Univ Lyon,
INSA Lyon, UCBL, LIRIS, UMR5205,69622 Villeurbanne, France
22email<EMAIL_ADDRESS>
# Data Centred Intelligent Geosciences:
Research Agenda and Opportunities
Position Paper††thanks: This work is funded by the project ADAGEO, IEA CNRS
collaboration with Federal University of Rio Grande do Norte
https://adageo.github.io.

Aderson Farias do Nascimento 11 Martin A. Musicante 11 Umberto Souza da Costa
11 Bruno M. Carvalho 11 Marcus Alexandre Nunes 11 Genoveva Vargas-Solar 22

###### Abstract

This paper describes and discusses our vision to develop and reason about best
practices and novel ways of curating data-centric geosciences knowledge (data,
experiments, models, methods, conclusions, and interpretations). This
knowledge is produced from applying statistical modelling, Machine Learning,
and modern data analytics methods on geo-data collections. The problems
address open methodological questions in model building, models’ assessment,
prediction, and forecasting workflows.

##  1 Introduction

Massive data production is a critical aspect of experimental sciences. It has
not been different for geoscience. Examples of geoscientific data include any
physical observable related to the energy industry, mining, monitoring
hazardous areas (e.g. effects of salt mining in populated areas), etc.
Nowadays, with the relative facility and lowering the cost to collect data,
the data processing to exploit their value is a challenge. It requires
expertise in data maintenance and processing, data analysis, and the design of
experiments of target domains for which data will provide insight and
knowledge.

This paper describes and discusses our vision to develop and reason about best
practices and novel ways of curating [11] data-centric geosciences knowledge
(data, experiments, models, methods, conclusions, and interpretations). This
knowledge is produced from applying statistical modelling, Machine Learning,
and modern data analytics methods on geo-data collections. The problems
address open methodological questions in model building, models’ assessment,
prediction, and forecasting workflows.

This paper is organised as follows. Section 2 discusses related work regarding
existing disaggregated data centres approaches and data science workflow
execution. Section 3 describes our vision and research challenges and
opportunities of data centred smart geosciences. Section 4 describes examples
of use cases addressed through data centred strategies using mathematical and
Machine Learning or artificial intelligence algorithms. Section 5 concludes
the paper and discusses future work.

##  2 Related Work

In France, portals like SISMER111https://data.ifremer.fr/SISMER/Missions and
Form@Ter222https://www.poleterresolide.fr are initiatives willing to share
data about target observation in geosciences and then share analytics
experiments results. Data Terra333https://www.data-terra.org is a research
infrastructure dedicated to Earth System observation data. In general, the
objective of these platforms and portals is to facilitate access to satellite,
airborne and in-situ data collected and managed by research laboratories or
federative structures, by national infrastructures, the oceanographic fleet,
aircraft, balloons, and by space missions (e.g., Data Terra). They manage,
archive, and share TB of data. For example, Data Terra represented 50,000 TB
in 2017 and is estimated to reach 100,000 TB by 2022. Beyond multi-source
data, they also share products and services through a unified portal. Data is
curated with metadata, included under accepted standards like the European
standard INSPIRE. The challenge is to define common bases for all data
producers and make the data sets interoperable so that their resources are
consistent, shareable, exploitable, and, in a multidisciplinary approach,
required to study the Earth. In this sense, the ODATIS Ocean Cluster offers
several services for data producers similar to data labs for referencing,
hosting, dissemination and interoperability. They also provide access to
computing services for running experiments (models) that require important
computing resources.

At the European level, actions adopting a data science perspective, for
example, the project EPOS444https://www.epos-ip.org and the Alan Turing
Institute extend these initiatives to European partners willing to take full
advantage of the possibilities provided by analytics and data science to run
experiments and contribute to solving leading problems addressed by the
discipline. Indeed, with the advent of digital technologies, libraries
proposing analytics models have been run on mainframes and high-performance
computing centres (HPC) to produce visualisation, modelling and simulation
systems to accelerate interpretation and planning.

Brazilian scientific agenda has widely installed and developed data centres
like https://www.eveo.com.br/en/ and https://baxtel.com/data-center/brazil-
brasil. These data centres aim to provide mainly large-scale computing
resources to run experiments, for example, those regarding geosciences,
particularly those key for the national economies in France and Brazil in oil
and hydrocarbon exploitation, extraction of minerals, and its interaction with
populated areas.

##  3 Towards Smart Data Centred Geosciences

Lately, geoscience researchers have been discovering the power of Machine
Learning in solving problems in their field. Bergen et al. [1], for example,
show that random forests were used on continuous acoustic emission in a
laboratory shear experiment to model instantaneous friction and to predict
time-to-failure [9, 6] surveyed the applications of Machine Learning in
seismology and presented five research areas in in which Machine Learning
classification, regression, clustering algorithms show promise: earthquake
detection and phase picking, earthquake early warning (EEW), ground-motion
prediction, seismic tomography, and earthquake geodesy. In exploration
geophysics, Machine Learning has been used in seismic data processing and
reservoir characterization [5, 8]. Clustering methods were used to identify
key geophysical signatures and determine their relationship to rock types for
geological mapping in the Brazilian Amazon [3]. However, many researchers in
the area are still not prepared to take advantage of data-driven approaches to
their analyses at scale. In this context existing projects and actions are
emerging to provide specialized portals and systems that can encourage the
sharing of collected data (observations), experiments, and analytics results
that should even promote reproducibility.

In this context, we can see the emergence of multidisciplinary teams to
collaborate in the search for computational solutions. These teams are formed
by experts in geology/geosciences, computer science, statistics and physics,
among others. The work of these teams usually relies on the use of
mathematical/computational tools to process large amounts of data. Big data
analytical techniques and Machine Learning has been used with success.

Many scientists and companies believe that they can generate fresh insight,
reduce decision cycle times and steal a march on their competition by
automating the search for patterns and relationships in their data. Therefore,
geophysics and data science, including algorithms, mathematical models and
computing, must converge for developing experiments for obtaining insight and
foresight about the observations contained in data collections. Experiments
represent best practices for addressing problems and questions on geophysics
that must be treated as data and knowledge to be shared and reused by
scientists and practitioners.

Data collections issued in situ observations shared in pivot formats are vital
for developing experiments that can lead to relevant governmental, economic,
and social decision making. Information about how these data have been
collected, used, curated, and maintained, including the conditions in which
analyses are run and associated results and their use to lead to specific
policies, should be managed and shared.

Merging data-centric techniques with modelling and simulation to answer
questions in geoscience and make timely, clever, and disruptive decisions can
lead to a new geoscience perspective that will benefit from data curation and
analytics. In our vision, three important directions can be considered
described in the following lines.

### Collected data, models and knowledge integration.

A wide variety of geophysical data (potential fields, electromagnetic data,
seismic data, weather data, etc.) has been acquired with extensive wavelength
ranges from surface sensor arrays, drilled wells, satellites and many other
sources. These data sets are among the most significant science data sets in
use, comparable in size and complexity only to those from astronomy and
particle physics. Integrating access to data collections and their curated
versions under a global knowledge graph can promote its maintenance, analysis,
and experimentation. It can also show the knowledge of the discipline with its
vocabulary, concepts, and relations in a synthetic manner. Inspired by
existing public data labs like Kaggle or CoLab of Google, it can be essential
to work to extend existing portals. These portals can be revisited towards
specialized data science labs on geosciences. Through these labs, scientists
and practitioners can share raw data, models, and experiments’ return of
experience and run and reproduce other experiments with almost no requirement
of interacting with specialized engineering support for accessing CPU and GPU
clusters.

### Curation, maintenance, exploration of data collections

for bringing value to petabytes of data produced from in situ observations and
also from experiments. Given that data act as a backbone in modelling
phenomena for understanding their behaviour, it is critical to developing good
collection and maintenance: which are available data collections? Are they
complete? Which is their provenance? In which conditions were they collected?
have they been processed? In which cases have they been used, and what are the
associated results?

Data curation is a set of techniques to process (raw, distributed,
heterogeneous) data to extract their value. It proposes methods to explore
data collections using well-adapted data structures like graphs that can be
explored and enriched while new data and analytics results are produced. Data
curation means also keeping track of the type of experiments run on data,
their results, and the conditions in which they were performed.

Maintaining a catalogue of questions and experiments related to data can help
provide a new vision of data and the scientific community’s knowledge. This
catalogue can extend existing meta-data and associated data collections
information provided by actions like ODATIS and Data Terra.

### Modelling and simulating experiments to answer questions in geoscience and
make timely decisions.

Both data sources and models come with recognized issues that existing
methodologies have difficulties coping with but which novel data science-based
approaches can address. For example, features for which exact physical models
are unknown (e.g., subsurface geology, earthquakes) or models that are
difficult to reconcile (e.g., seismic measurements vs social media alerts).
This will imply:

  * •

Designing ad hoc experiment programming languages for enabling friendly,
context-aware, and declarative construction of complex experiments in
geosciences.

  * •

Enabling the execution of experiments fusing different data collocations at
different scales to maintain data, prepare experiments, and manage associated
results.

  * •

Programming experiments

    * –

Applying statistical methods to investigate and unveil new patterns in
geophysical data, answering open problems, or leading to further research
questions.

    * –

Building predictive models to describe better or approximate geophysical
phenomena, increasing the knowledge about our planet.

    * –

Parallelizing algorithms for processing geophysical data, thus, allowing for
the processing of very large data sets in reasonable times.

### Discussion

From the Geophysics point of view, proposing best practices and ad-hoc
strategies for developing data centred experiments to solve geosciences
problems will impact different vital areas of the economy. For example, oil
companies that ride this wave will significantly increase the current
productivity of their knowledge workers, optimize business processes, and
reduce operational costs in a way that is not possible through incremental
change. Some companies now use algorithms to define optimal drilling
locations, using automated or semi-automated systems that deliver results on
much shorter cycle times than traditional methods.

From the Data Science/Data Processing perspective, this kind of
multidisciplinary research can provide the ground to devise new data curation
techniques, to propose a domain-specific query language, or to define new
methods for processing heterogeneous data [10]. In addition, statistical
knowledge is essential for extracting information from the massive amount of
data we will process. New methods and models will be crucial to model data and
make conclusions in a timely fashion [12].

##  4 Use Cases

To illustrate the type of data analytics challenges introduced by geosciences
problems, we describe in this section three examples. These use cases can be
solved with different techniques and can call for data science strategies for
specifying solutions and deploying them in target architectures.

### Estimating the approximate earthquake epicentres.

The understanding of earthquake occurrence in intraplate areas has been one of
the most challenging tasks in Seismology [4]. Compared to border plate
regions, interplate areas suffer less attenuation of seismic waves. As a
consequence, a significant hazard may rise from moderate magnitude
earthquakes. Understanding the earthquake generating mechanisms depends on
assessing the stress field in the intraplate areas.

Seismic stations collect signals that can represent earthquakes produced in a
specific area. The challenge is to determine whether signals represent
earthquakes in such a case compute the epicentres. For addressing the
challenge, it is possible to use mathematical, Machine Learning or artificial
intelligence methods [7]. The first task to address this question is to
compute the earthquake epicentre’s direction using the sensor’s initial
movement polarity when the waves P and S are discovered. Then, compute the
distance considering how the sensor moves from North-South (it should be the
same as the East-West), as shown in Figure 1.

Figure 1: Sensor movement. Martin: Referencia?

### Estimation of stacking velocity using CDP semblance.

Semblance analysis is a technique used in the study and refinement of seismic
data. Along with other methods, this technique enables the improvement of the
resolution of data, even in the presence of background noise. The data yielded
by semblance analysis tends to be easier to interpret when discovering the
underground structure of an area (see Figure 2).

Figure 2: NMO correction after velocity picking on the semblance.Martin:
Referencia?

Estimating the stacking velocities is one of the essential steps in the CMP
(Common Mid Point) seismic processing. This is because the better the
estimation of the stacking velocities, the better the quality of the zero-
offset section obtained. Currently, the most convenient velocity analysis
method consists of manually picking the stacking velocities in the velocity
spectrum, using the semblance as a coherence measure. The semblance gives us a
measure of multichannel coherence. It is necessary to define an analytics
workflow with the following phases to perform this task: (i) transform the CDP
or CMP gathered traces from the offset and time coordinates into the coherence
semblances in coordinates of time and stacking velocities. (ii) Pick local
maxima of these coherence semblances and assign zero offset time and
corresponding stacking velocities. (iii) Correct the CDP or CMP gathers for
normal moveout (NMO).

### Denoising data from sensors.

The Brazilian Seismographic Network (RSBR) operates since 2011. Station
installation began in 2011 in southeast (SE) Brazil and finished in 2014 in
the Amazon forest. The network integrates 84 stations (as of December 2017)
operated by four institutions in different regions of Brazil. Seismic stations
collect signals that can represent earthquakes produced in a specific area.
This data usually contains noise produced by the context where the sensor is
placed and by the technology of the sensor itself. The challenge is to filter
this data to make it ready to be analyzed. This consortium is responsible for
the Brazilian Seismic Bulletin [2].

##  5 Conclusions and Future Work

This paper proposes our vision about the multidisciplinary agenda for
developing data centred solutions for geosciences problems. The amount of data
collected through observing the Earth and its geophysical phenomena call for
agile data and knowledge curation techniques to manage both data, experiments,
and results. The research agenda includes (i) integrating and describing data
collected with different technology, (ii) estimating its quality, and
preparing it to be used as input of different methods. Research on smart data
centred geoscience also calls for curation tasks, including data tracking the
way data is cleaned, the experiments that use it and the obtained results.
Exploration methods and systems must be associated with curated data and
knowledge to facilitate an agile understanding of this content. Finally,
execution environments providing computing resources necessary for setting and
deploying experiments are vital for promoting multidisciplinary global
experimental sciences. The research performed within the project ADAGEO
555https://adageo.github.io funded by the IEA program of the French CNRS. is
willing to address these problems through a Brazilian and French collaborative
community.

## References

  * [1] Bergen, K.J., Johnson, P.A., Maarten, V., Beroza, G.C.: Machine learning for data-driven discovery in solid earth geoscience. Science 363(6433) (2019)
  * [2] Bianchi, M.B., Assumpção, M., Rocha, M.P., Carvalho, J.M., Azevedo, P.A., Fontes, S.L., Dias, F.L., Ferreira, J.M., Nascimento, A.F., Ferreira, M.V., et al.: The brazilian seismographic network (rsbr): improving seismic monitoring in brazil. Seismological Research Letters 89(2A), 452–457 (2018)
  * [3] Carneiro, C.d.C., Fraser, S.J., Crósta, A.P., Silva, A.M., Barros, C.E.d.M.: Semiautomated geologic mapping using self-organizing maps and airborne geophysics in the brazilian amazon. Geophysics 77(4), K17–K24 (2012)
  * [4] Fonsêca, J., Ferreira, J., do Nascimento, A., Bezerra, F., Neto, H.L., de Menezes, E.: Intraplate earthquakes in the potiguar basin, brazil: Evidence for superposition of local and regional stresses and implications for moderate-size earthquake occurrence. Journal of South American Earth Sciences 110, 103370 (2021)
  * [5] Jia, Y., Ma, J.: What can machine learning do for seismic data processing? an interpolation application. Geophysics 82(3), V163–V177 (2017)
  * [6] Kong, Q., Trugman, D.T., Ross, Z.E., Bianco, M.J., Meade, B.J., Gerstoft, P.: Machine learning in seismology: Turning data into insights. Seismological Research Letters 90(1), 3–14 (2019)
  * [7] Leandro, W.P., Santana, F.L., Carvalho, B.M., do Nascimento, A.F.: Parallel source scanning algorithm using gpus. Computers & Geosciences 140, 104497 (2020)
  * [8] Li, S., Huang, X., Cao*, H.: Seismic data prediction lithology sequence model based on machine learning. In: SEG 2018 Workshop: Reservoir Geophysics, Daqing, China, 5-7 August 2018. pp. 249–251. Society of Exploration Geophysicists and the Chinese Geophysical Society (2020)
  * [9] Rouet-Leduc, B., Hulbert, C., Lubbers, N., Barros, K., Humphreys, C.J., Johnson, P.A.: Machine learning predicts laboratory earthquakes. Geophysical Research Letters 44(18), 9276–9282 (2017)
  * [10] Vargas-Solar, G., Farokhnejad, M., Espinosa-Oviedo, J.: Towards human-in-the-loop based query rewriting for exploring datasets. In: Proceedings of the Workshops of the EDBT/ICDT 2021 Joint Conference (2021)
  * [11] Vargas-Solar, G., Kemp, G., Hernández-Gallegos, I., Espinosa-Oviedo, J., Da Silva, C.F., Ghodous, P.: Demonstrating data collections curation and exploration with curare. In: EDBT/ICDT Conference 2019. p. 4 (2019)
  * [12] Vargas-Solar, G., Zechinelli-Martini, J.L., Espinosa-Oviedo, J.A.: Enacting data science pipelines for exploring graphs: from libraries to studios. In: ADBIS, TPDL and EDA 2020 Common Workshops and Doctoral Consortium. pp. 271–280. Springer (2020)


branes), acting with (B.24) on a supercharge $\epsilon$ we get
${\mathcal{E}}_{{{NP}}}\Gamma^{{P}}\epsilon=0$. For a Majorana spinor
$\epsilon$ in $d=11$, the bilinear
$K_{{M}}=\bar{\epsilon}\Gamma_{{M}}\epsilon$ is either time-like or null. If
it is timelike, $\epsilon$ has no one-form that annihilates it; so all
components of ${\mathcal{E}}_{{{NP}}}$ are zero. If $K_{{M}}$ is null, it is
the only one-form that annihilates $\epsilon$. In an adapted vielbein such
that $K=e^{+}$, all components of ${\mathcal{E}}_{{{NP}}}$ except
${\mathcal{E}}_{{{N-}}}$ are zero; since ${\mathcal{E}}$ is symmetric, in fact
all components except ${\mathcal{E}}_{--}$ are zero.171717If there is at least
another supercharge with a different $K$ (which is the case for vacuum
compactifications with a $d\geq 3$ external spacetime) then all components
${\mathcal{E}}_{{{NP}}}=0$.

##  Appendix C Inclusion of M2-branes

In this appendix we provide details of the derivation of (4.29) and of the
bound (4.30).

###  C.1 M2 BPS-energy

We first derive the second term appearing on the r.h.s. of (4.29), which comes
from the contribution of the last term of (4.26) inside (4.7):

$\displaystyle\frac{1}{4}\int_{S}*{\rm d}x_{{N}}\,\big{(}*\big{[}{\rm
d}x^{{N}}\wedge\delta^{(8)}({\mathcal{C}})\big{]}\big{)}\cdot\Omega^{\text{\tiny(M2)}}=-\frac{1}{4}\int_{S}*e^{{{\underline{0}}}}\,\big{(}*\big{[}e^{{{\underline{0}}}}\wedge\delta^{(8)}({\mathcal{C}})\big{]}\big{)}\cdot\Omega^{\text{\tiny(M2)}}$
(C.1) $\displaystyle=-\frac{1}{4}\int_{S}{\rm
dvol}_{S}\,\big{(}*_{S}\delta^{(8)}({\mathcal{C}})|_{\Sigma}\big{)}\cdot\Omega^{\text{\tiny(M2)}}=-\frac{1}{4}\int_{S}\Omega^{\text{\tiny(M2)}}\wedge\delta^{(8)}({\mathcal{C}})=-\frac{1}{4}\int_{{\mathcal{C}}\cap
S}\Omega^{\text{\tiny(M2)}}\,,$

where in the second line we have used the adapted vielbein
$e^{{A}}=(e^{{{\underline{0}}}},e^{a})$ introduced in (4.18). Notice that
instead the final result does not depend on such a choice.

In order to obtain the first term on the r.h.s. of (4.29), we first derive the
formula (4.25) of $T^{{{MN}}}_{\text{\tiny(M2)}}$. This can be extracted from
the variation of the M2 action (4.21) under a metric deformation

$\displaystyle\frac{1}{2}\int\delta
g_{{{MN}}}T^{{{MN}}}_{\text{\tiny(M2)}}*1\equiv\frac{1}{2\pi}\delta
S_{\text{\tiny(M2)}}=-\frac{1}{2}\int_{\cal C}{\rm
d}^{3}\sigma\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X^{{M}}\partial_{\beta}X^{{N}}\delta
g_{{{MN}}}$ (C.2) $\displaystyle=-\frac{1}{2}\int_{\cal
C}e^{\underline{012}}\,\eta^{\underline{\alpha\beta}}e_{\underline{\alpha}}(X^{{M}})e_{\underline{\beta}}(X^{{N}})\delta
g_{{{MN}}}=-\frac{1}{2}\int\delta
g_{{{MN}}}e_{\underline{\alpha}}^{{M}}e_{\underline{\beta}}^{{N}}\eta^{\underline{\alpha\beta}}\,e^{\underline{012}}\wedge\delta^{(8)}({\cal
C})$ $\displaystyle=\frac{1}{2}\int\delta
g_{{{MN}}}e_{\underline{\alpha}}^{{M}}e_{\underline{\beta}}^{{N}}\eta^{\underline{\alpha\beta}}\,*[e^{\underline{012}}\wedge\delta^{(8)}({\cal
C})]*1\,\,,$

where we have used the adapted vielbein (4.24).

By using (4.25) in (4.26), we see that it contributes to (4.7) by the term

$\displaystyle\frac{1}{4}\int
K_{{M}}e_{\underline{\alpha}}^{{M}}e_{\underline{\beta}}^{{N}}\eta^{\underline{\alpha\beta}}\,*[e^{\underline{012}}\wedge\delta^{(8)}({\cal
C})]*{\rm d}x_{{N}}=\frac{1}{4}\int
K_{\underline{\alpha}}*e^{\underline{\alpha}}\,*[e^{\underline{012}}\wedge\delta^{(8)}({\cal
C})]$ (C.3) $\displaystyle=\frac{1}{4}\int
K^{\underline{\alpha}}\,\iota_{\underline{\alpha}}\,[e^{\underline{012}}\wedge\delta^{(8)}({\cal
C})]=\frac{1}{4}\int
K^{\underline{0}}\,e^{\underline{12}}\wedge\delta^{(8)}({\cal
C})=\frac{1}{4}\int_{{\mathcal{C}}\cap S}K^{\underline{0}}\,{\rm
vol}_{{\mathcal{C}}\cap S}\,,$

where we have used a further adapted vielbein combining the properties of
(4.18) and (4.24). This completes our derivation of the M2 BPS energy (4.29).

###  C.2 M2 bound

Consider a spacetime foliated by space-like leaves $S$ parametrized by a
‘time’ $t$. By using adapted coordinates $x^{{M}}=(t,x^{m})$, the line element
can be written as

$\displaystyle{\rm d}s^{2}$ $\displaystyle=\mathrm{e}^{2D}{\rm
d}t^{2}+\hat{g}_{mn}({\rm d}x^{m}+\hat{V}^{m}{\rm d}t)({\rm
d}x^{n}+\hat{V}^{n}{\rm d}t)\,,$ (C.4)

where $\hat{g}_{mn}$ is the metric induced on $S$. The adapted vielbein (4.18)
is then

$e^{\underline{0}}=\mathrm{e}^{D}{\rm d}t\,,\qquad e^{a}=\hat{e}^{a}_{m}({\rm
d}x^{m}+\hat{V}^{m}{\rm d}t)\,,$ (C.5)

where $\hat{e}^{a}=\hat{e}^{a}_{m}{\rm d}x^{m}$ is a vielbein on $S$. Note
also that the dual frame $e_{{A}}=(e_{\underline{0}},e_{a})$ is such that

$e_{\underline{0}}=\mathrm{e}^{-D}(\partial_{t}-\hat{V}^{m}\partial_{m})\,,\qquad
e_{a}=\hat{e}_{a}\,.$ (C.6)

Now consider an M2-brane world-volume ${\mathcal{C}}$. One can impose a
partial static gauge and use $t$ as time coordinate along ${\mathcal{C}}$. The
world-volume coordinates $\sigma^{\alpha}=(t,\sigma^{i})$ induce a
decomposition of the world-volume line-element

$\displaystyle{\rm d}s^{2}_{\mathcal{C}}$ $\displaystyle=\mathrm{e}^{2D}{\rm
d}t^{2}+\hat{h}_{ij}({\rm d}\sigma^{i}+\hat{v}^{i}{\rm d}t)({\rm
d}\sigma^{j}+\hat{v}^{j}{\rm d}t)\,,$ (C.7)

where $\hat{h}_{ij}$ is the induced metric on ${\mathcal{C}}\cap S$.
Furthermore, we can pick an adapted bulk vielbein
$e^{{A}}=(e^{\underline{0}},e^{a})=(e^{\underline{\alpha}},e^{\tilde{a}})$
satisfying the properties of (4.18) and (4.24). Hence, the pull-back of
$e^{\underline{\alpha}}$ to ${\cal C}$ gives a vielbein
${\mathfrak{e}}^{\underline{\alpha}}$ of ${\rm d}s^{2}_{\mathcal{C}}$. Note
also that on ${\mathcal{C}}$ we can identify the push-forward of the dual
frame ${\mathfrak{e}}_{\underline{\alpha}}$ with the bulk
$e_{\underline{\alpha}}$.

By following [31], we can introduce the world-volume momentum density

${\cal P}^{{M}}=-\sqrt{-h}\,h^{t\alpha}\partial_{\alpha}X^{{M}}\,.$ (C.8)

Notice that ${\cal P}^{{M}}$ and $\bar{\chi}\Gamma^{{M}}\chi$ are both causal
and future-pointing, for any spinor $\chi$. Hence

$-{\cal P}^{{M}}\bar{\chi}\Gamma_{M}\chi\geq 0\,,$ (C.9)

where the inequality is saturated if and only if $\chi=0$. By choosing
$\chi=(\mathbf{1}-\Gamma_{\text{\tiny M2}})\epsilon$ with

$\Gamma_{\text{\tiny
M2}}\equiv\frac{\epsilon^{\alpha\beta\gamma}}{3!\sqrt{-h}}\partial_{\alpha}X^{{M}}\partial_{\beta}X^{{N}}\partial_{\gamma}X^{P}\Gamma_{{MNP}}\,,$
(C.10)

and recalling that $K^{{M}}=\bar{\epsilon}\Gamma^{{M}}\epsilon$, we get the
inequality

$-{\cal P}^{{M}}K_{{M}}\geq-{\cal
P}^{{M}}\bar{\epsilon}\Gamma_{{M}}\Gamma_{\text{\tiny M2}}\epsilon\,.$ (C.11)

This is saturated iff $\Gamma_{\text{\tiny M2}}\epsilon=\epsilon$.

By using the adapted vielbein above, we can write

$\displaystyle-{\cal P}^{{M}}\bar{\epsilon}\Gamma_{{M}}\Gamma_{\text{\tiny
M2}}\epsilon$ $\displaystyle=-\frac{1}{3!}{\cal
P}^{{M}}e_{{M}}^{\underline{\alpha}}\,\epsilon^{\underline{\beta\gamma\delta}}\,\bar{\epsilon}\Gamma_{\underline{\alpha}}\Gamma_{\underline{\beta\gamma\delta}}\epsilon=-\frac{1}{2}{\cal
P}^{{M}}e_{{M}}^{\underline{\alpha}}\eta_{\underline{\alpha\beta}}\,\epsilon^{\underline{\beta\gamma\delta}}\,\bar{\epsilon}\Gamma_{\underline{\gamma\delta}}\epsilon$
(C.12) $\displaystyle=-\frac{1}{2}{\cal
P}^{{M}}e_{{M}}^{\underline{\alpha}}\eta_{\underline{\alpha\beta}}\,\epsilon^{\underline{\beta\gamma\delta}}\,\Omega^{\text{\tiny(M2)}}_{\underline{\gamma\delta}}=-\frac{1}{2\sqrt{-h}}{\cal
P}^{M}e^{\underline{\alpha}}_{M}\eta_{\underline{\alpha\beta}}{\mathfrak{e}}^{\underline{\beta}}_{\beta}\epsilon^{\beta\gamma\delta}\Omega^{\text{\tiny(M2)}}_{\gamma\delta}$
$\displaystyle=\frac{1}{2}h^{t\alpha}h_{\alpha\beta}\epsilon^{\beta\gamma\delta}\Omega^{\text{\tiny(M2)}}_{\gamma\delta}=\frac{1}{2}\epsilon^{t\alpha\beta}\Omega^{\text{\tiny(M2)}}_{\beta\gamma}\,,$

and then

$-{\cal P}^{{M}}\bar{\epsilon}\Gamma_{{M}}\Gamma_{\text{\tiny
M2}}\epsilon\,{\rm d}\sigma^{1}\wedge{\rm
d}\sigma^{2}=\Omega^{\text{\tiny(M2)}}|_{{\mathcal{C}}\cap S}\,.$ (C.13)

Hence, with respect to the oriented two-form ${\rm d}^{2}\sigma\equiv{\rm
d}\sigma^{1}\wedge{\rm d}\sigma^{2}>0$ along ${\mathcal{C}}\cap S$, we can
write (C.11) in the form

$-{\cal P}^{{M}}K_{{M}}{\rm
d}^{2}\sigma\geq\Omega^{\text{\tiny(M2)}}|_{{\mathcal{C}}\cap S}\,.$ (C.14)

This is the M2 counterpart of [31, Eq. (3.25)] for F1 strings. On the other
hand

$\displaystyle-{\cal P}^{{M}}K_{{M}}{\rm d}^{2}\sigma$
$\displaystyle=\sqrt{-h}\,h^{t\alpha}\partial_{\alpha}X^{{M}}e^{{A}}_{{M}}K_{{A}}{\rm
d}^{2}\sigma=\sqrt{-h}\,h^{t\alpha}{\mathfrak{e}}^{\underline{\alpha}}_{\alpha}K_{\underline{\alpha}}{\rm
d}^{2}\sigma$ (C.15)
$\displaystyle=\sqrt{-h}\,{\mathfrak{e}}^{t}_{\underline{\alpha}}K^{\underline{\alpha}}\,{\rm
d}^{2}\sigma=K^{\underline{0}}\,{\rm d}{\rm vol}_{{\mathcal{C}}\cap S}\,,$

where in the last step we have used
${\mathfrak{e}}^{t}_{\underline{\alpha}}=\delta^{\underline{0}}_{\underline{\alpha}}\mathrm{e}^{-D}$
and

$\sqrt{-h}\,{\rm d}t\wedge{\rm
d}^{2}\sigma={\mathfrak{e}}^{\underline{0}}\wedge{\mathfrak{e}}^{\underline{1}}\wedge{\mathfrak{e}}^{\underline{2}}=\mathrm{e}^{D}\sqrt{\hat{h}}\,{\rm
d}t\wedge{\rm d}^{2}\sigma=\mathrm{e}^{D}{\rm d}t\wedge{\rm d}{\rm
vol}_{{\mathcal{C}}\cap S}\,.$ (C.16)

Hence (C.14) is equivalent to

$K^{\underline{0}}\,{\rm d}{\rm vol}_{{\mathcal{C}}\cap
S}\geq\Omega^{\text{\tiny(M2)}}|_{{\mathcal{C}}\cap S}\,,$ (C.17)

as stated in (4.30). Furthermore, supersymmetry is preserved if and only if
the bound is saturated.

##  Appendix D Details on type II

In this appendix we describe the computation leading from (5.10) to (5.13). We
will be less detailed than in section B.3; we advise the reader to read that
first.

The first step is subtracting the terms on the first line of (5.13); this is
lengthy and tedious, but in principle straightforward. Useful identities for
this include:

$\Gamma_{{M}}H\Gamma^{H}=-4H\,,\qquad\Gamma_{{M}}H\Gamma^{{MN}}=-4H\Gamma^{{N}}-\Gamma^{{N}}H\,,$
(D.1)

which follow from (A.4) and from
$\Gamma^{{M}}\Gamma^{{N}}=\Gamma^{{MN}}+g^{{MN}}1$; the usual definition
$\\{\Gamma^{{M}},\Gamma^{{N}}\\}=2g^{{MN}}1$ of Clifford algebra; and (A.15).
The result is

$\begin{split}\mathrm{e}^{2\phi}\nabla_{{M}}E^{{MN}}&=\overline{\left({\mathcal{D}}_{{M}}-\frac{1}{8}\Gamma_{{M}}{\mathcal{O}}\right)\epsilon}\,\Gamma^{{MPN}}\left({\mathcal{D}}_{{P}}-\frac{1}{8}\Gamma_{{P}}{\mathcal{O}}\right)\epsilon-\frac{1}{8}\overline{{\mathcal{O}}\epsilon}\,\Gamma^{{N}}{\mathcal{O}}\epsilon\\\
&-\bar{\epsilon}\,\Gamma^{{MNP}}(D_{{M}}D_{{P}}+[D_{{M}},\Gamma^{{MNP}}{\mathcal{A}}_{{P}}])\epsilon-\bar{\epsilon}\,\left[\sum_{\alpha=1}^{6}Q^{{N}}_{\alpha}\right]\epsilon\end{split}$
(D.2)

where the $Q_{\alpha}^{N}$ are terms quadratic in the fields, which we will
write below; they will need further, less trivial processing.

###  D.1 Linear terms

We start from the linear terms in (D.2).

The $\bar{\epsilon}\Gamma^{{MNP}}D_{{M}}D_{{P}}\epsilon$ is evaluated with
(4.13). For the remaining terms, we recall (5.7) and (A.16):

$[D_{{M}},\Gamma^{{MNP}}{\mathcal{A}}_{{P}}]=+\frac{1}{4}\Gamma^{{[M}}\nabla_{{M}}H\Gamma^{{N]}}\otimes\sigma_{3}-\Gamma^{{MN}}\nabla_{{M}}\mathrm{d}\phi-2\Gamma^{{[M}}\nabla_{{M}}{\mathcal{F}}\,\Gamma^{{N]}}\,.$
(D.3)

The dilaton term gives

$-\bar{\epsilon}\,\Gamma^{{MN}}\nabla_{{M}}\mathrm{d}\phi\,\epsilon=-\bar{\epsilon}(\Gamma^{{MNP}}+2\Gamma^{{[M}}g^{{N]P}})\nabla_{{M}}\partial_{{P}}\phi\epsilon=-(\nabla^{{P}}\nabla^{{N}}-g^{{NP}}\nabla^{2})\phi\,\bar{\epsilon}\Gamma_{{P}}\epsilon\,.$
(D.4)

The $H$ term simplifies similar to (B.11):

$\begin{split}&\frac{1}{4}\bar{\epsilon}\,\Gamma^{{[M}}\nabla_{{M}}H\Gamma^{{N]}}\otimes\sigma_{3}\epsilon=-\frac{1}{4}\bar{\epsilon}[(\mathrm{d}x\wedge+\iota)(\mathrm{d}x\wedge-\iota)]^{{[MN]}}\nabla_{{M}}H\otimes\sigma_{3}\epsilon\\\
&=\frac{1}{4}\bar{\epsilon}(\mathrm{d}x^{{N}}\wedge\mathrm{d}H-\iota^{{N}}*\mathrm{d}*H)\otimes\sigma_{3}\epsilon\,.\end{split}$
(D.5)

Following the same steps, recalling the definition of ${\mathcal{F}}$ in (5.3)
and the self-duality property (5.1), the RR term becomes

$-2\bar{\epsilon}\Gamma^{{[M}}\nabla_{{M}}{\mathcal{F}}\,\Gamma^{{N]}}\epsilon=\pm\frac{1}{4}\bar{\epsilon}_{1}(\mathrm{d}x^{{N}}\wedge\mathrm{d}(\mathrm{e}^{\phi}F))\epsilon_{2}+\frac{1}{4}\bar{\epsilon}_{2}(\mathrm{d}x^{{N}}\wedge\mathrm{d}(\mathrm{e}^{\phi}\lambda
F))\epsilon_{1}\,.$ (D.6)

By (A.15), the two terms on the right-hand side are in fact equal to each
other.

###  D.2 Quadratic tems

We now turn to the $Q_{\alpha}^{N}$ in (D.2). We organized them depending on
what fields they contain.

We first give some useful identities. One is obtained by applying the Fierz
identities to the bispinors $\epsilon_{a}\otimes\bar{\epsilon}_{a}$. As in
(B.16), the only non-zero terms in the sum are those for $k=1$, 5 and 9.
Moreover the last is dual to the first. This results in

$\begin{split}&32\epsilon_{1}\otimes\bar{\epsilon}_{1}=(\bar{\epsilon}_{1}\,\Gamma_{{M}}\epsilon_{1})(1+\Gamma)\Gamma^{{M}}+\frac{1}{5!}(\bar{\epsilon}_{1}\,\Gamma_{{M_{5}\cdots
M_{1}}}\epsilon_{1})\Gamma^{{M_{1}\cdots M_{5}}}\,,\\\
&32\epsilon_{2}\otimes\bar{\epsilon}_{2}=(\bar{\epsilon}_{2}\,\Gamma_{{M}}\epsilon_{2})(1\mp\Gamma)\Gamma^{{M}}+\frac{1}{5!}(\bar{\epsilon}_{2}\,\Gamma_{{M_{5}\cdots
M_{1}}}\epsilon_{2})\Gamma^{{M_{1}\cdots M_{5}}}\,.\end{split}$ (D.7)

The first is purely quadratic in $H$:

$\displaystyle 64\bar{\epsilon}\,Q^{{N}}_{1}\epsilon$
$\displaystyle\equiv\bar{\epsilon}\,\Gamma_{{M}}H\Gamma^{{N}}H\Gamma^{{M}}\epsilon=\mathrm{Tr}(\Gamma_{{M}}H\Gamma^{{N}}H\Gamma^{{M}}(\epsilon_{1}\otimes\bar{\epsilon}_{1}+\epsilon_{2}\otimes\bar{\epsilon}_{2})))$
(D.8) $\displaystyle\buildrel(\ref{eq:ee-
fierz}),(\ref{eq:gag})\over{=}-\frac{1}{4}\bar{\epsilon}_{1}\Gamma_{{M}}\epsilon_{1}\mathrm{Tr}(H\Gamma^{{N}}H(1-\Gamma)\Gamma^{{M}})-\frac{1}{4}\bar{\epsilon}_{2}\Gamma_{{M}}\epsilon_{2}\mathrm{Tr}(H\Gamma^{{N}}H(1\pm\Gamma)\Gamma^{{M}})$
$\displaystyle\buildrel(\ref{eq:ii-
tr})\over{=}16T^{{MN}}_{{(H)}}\bar{\epsilon}\,\Gamma_{{M}}\epsilon\,.$

With similar steps,

$\displaystyle\bar{\epsilon}\,Q^{{N}}_{2}\epsilon$
$\displaystyle\equiv\bar{\epsilon}\,(\not{\partial}\phi\Gamma^{{N}}\not{\partial}\phi-2\nabla^{{N}}\phi\not{\partial}\phi)\epsilon=-g^{{MN}}|\mathrm{d}\phi|^{2}\bar{\epsilon}\,\Gamma_{{M}}\epsilon\,;$
(D.9) $\displaystyle\bar{\epsilon}\,Q^{{N}}_{3}\epsilon$
$\displaystyle\equiv\bar{\epsilon}\,\Gamma_{{M}}{\mathcal{F}}\Gamma^{{N}}{\mathcal{F}}\Gamma^{{M}}\epsilon$
(D.10)
$\displaystyle=\frac{\mathrm{e}^{2\phi}}{8}\left((\bar{\epsilon}_{1}\,\Gamma_{{M}}\epsilon_{1})(T^{{MN}}_{{F}}-*(F\wedge\lambda
F)^{{MN}})+(\bar{\epsilon}_{2}\,\Gamma_{{M}}\epsilon_{2})(T^{{MN}}_{{F}}+*(F\wedge\lambda
F))^{{MN}}\right)\,.$

In (D.10) we have also used (A.12b), and

$(\mathrm{d}x^{{M}}\wedge\mathrm{d}x^{{N}}\wedge F\wedge\lambda
F)_{10}=-(\iota^{{M}}F)\cdot(\mathrm{d}x^{{N}}\wedge F)\,,$ (D.11)

which follows from repeated use of (A.8) and self-duality (5.1).

The remaining three $Q^{N}_{\alpha}$ require use of (A.3). We have

$\begin{split}8Q^{{N}}_{4}&=[\\{H,\Gamma^{{N}}\\},\not{\partial}\phi]=\partial_{{M}}\phi(\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{M}}}}\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{N}}}}-\stackrel{{\scriptstyle\to}}{{\Gamma^{{M}}}}\stackrel{{\scriptstyle\to}}{{\Gamma^{{N}}}}-2\stackrel{{\scriptstyle\to}}{{\Gamma^{{[M}}}}\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{N]}}}})H\\\
&=\partial_{{M}}\phi\left(-(\mathrm{d}x\wedge-\iota)^{2}-(\mathrm{d}x\wedge+\iota)^{2}+2(\mathrm{d}x\wedge-\iota)(\mathrm{d}x\wedge+\iota)\right)^{{[MN]}}H\\\
&=-4\partial_{{M}}\phi\iota^{{M}}\iota^{{N}}H=-4\iota_{\mathrm{d}\phi}\iota^{{N}}H=4\iota^{{N}}*\mathrm{d}\phi\wedge*H\,,\end{split}$
(D.12)

recalling (A.8) in the last step. A similar computation, skipping a few steps,
gives

$\begin{split}Q^{{N}}_{5}&=\not{\partial}\phi{\mathcal{F}}\Gamma^{{N}}-\Gamma^{{N}}{\mathcal{F}}\not{\partial}\phi=2\partial_{{M}}\phi\stackrel{{\scriptstyle\to}}{{\Gamma^{{[M}}}}\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{N]}}}}{\mathcal{F}}\\\
&=\frac{\mathrm{e}^{\phi}}{8}\left(\mp(1-*\lambda)\mathrm{d}x^{{N}}\wedge\mathrm{d}\phi\wedge
F\otimes
b^{\dagger}-(1-\lambda*)\mathrm{d}x^{{N}}\wedge\mathrm{d}\phi\wedge\lambda
F\otimes b\right)\,.\end{split}$ (D.13)

Finally, the most complicated term is

$\displaystyle 8Q^{{N}}_{6}$
$\displaystyle=[\Gamma_{{M}},H\Gamma^{{N}}]{\mathcal{F}}\Gamma^{{M}}+\Gamma^{{M}}{\mathcal{F}}[\Gamma^{{N}}H,\Gamma_{{M}}]$
(D.14) $\displaystyle=\mp
8((H\wedge+\iota_{H})(\mathrm{d}x-\iota)^{{N}}+(H\wedge-\iota_{H})(\mathrm{d}x+\iota)^{{N}}){\mathcal{F}}=\mp
16(H\wedge\mathrm{d}x^{{N}}-\iota_{H}\iota^{{N}}){\mathcal{F}}$
$\displaystyle=\mp\mathrm{e}^{\phi}(1-*\lambda)H\wedge\mathrm{d}x^{{N}}\wedge
F\otimes
b^{\dagger}-\mathrm{e}^{\phi}(1-\lambda*)H\wedge\mathrm{d}x^{{N}}\wedge\lambda
F\otimes b\,;$

we defined
$\iota_{H}\equiv\frac{1}{3!}H_{{MNP}}\iota^{{M}}\iota^{{N}}\iota^{{P}}$, and
we used

$\begin{split}\stackrel{{\scriptstyle\to}}{{H}}+\stackrel{{\scriptstyle\to}}{{\Gamma^{{M}}}}\stackrel{{\scriptstyle\leftarrow}}{{H_{{M}}}}&=4(H\wedge+\iota_{H})\,,\\\
\stackrel{{\scriptstyle\leftarrow}}{{H}}+\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{M}}}}\stackrel{{\scriptstyle\to}}{{H_{{M}}}}&=4(H\wedge-\iota_{H})(-1)^{\rm
deg}\,,\end{split}$ (D.15)

which can in turn be derived from (A.3).

Collecting all the results in this subsection finally takes (D.2) to (5.13).

##  Appendix E From M-theory to IIA

Consider the compactification of M-theory on a circle with periodic variable
$y\simeq y+1$ with $\ell_{\rm s}=\ell_{\textrm{\tiny P}}=1$. Hatted quantities
will refer to M-theory. The 11d coordinates are
$x^{\,\hat{\\!{{M}}}}=(x^{{M}},y)$ and the $d=11$ vielbein
$\hat{e}^{\,\hat{\\!{{A}}}}=(\hat{e}^{{A}},\hat{e}^{\underline{0}})$ is
related to the 10d one $e^{{{A}}}$ by:

$\hat{e}^{{A}}=\mathrm{e}^{-\phi/3}e^{{A}}\,,\qquad\hat{e}^{\underline{10}}\,\mathrm{e}^{2\phi/3}=\left(\mathrm{d}y-C_{1}\right)\,.$
(E.1)

Correspondingly,
$\hat{\epsilon}\,=\mathrm{e}^{-\phi/6}\epsilon\equiv\mathrm{e}^{-\phi/6}(\epsilon_{1}+\epsilon_{2})$
while the 11d gravitino splits as follows into string frame gravitino and
dilatino:

$\hat{\psi}=\hat{\psi}_{\,\hat{\\!{{M}}}}\mathrm{d}x^{\,\hat{\\!{{M}}}}=\mathrm{e}^{-\phi/6}\left(\psi-\frac{1}{6}\Gamma_{(1)}\lambda\right)+\frac{1}{3}e^{5\phi/6}\Gamma^{\underline{10}}\lambda\left(\mathrm{d}y-C_{1}\right)\,.$
(E.2)

Consider now a foliation into space-like slices $\hat{S}=S\times S^{1}$ and
the corresponding asymptotic boundaries $\partial\hat{S}=\partial S\times
S^{1}$. By identifying $\Gamma^{\underline{10}}\equiv\Gamma$, the
corresponding conserved supercharge reduces as follows

$\displaystyle\hat{Q}(\hat{\epsilon})=$
$\displaystyle\,-\int_{\partial\hat{S}}\,\overline{\hat{\epsilon}}\hat{\Gamma}_{(8)}\wedge\hat{\psi}$
(E.3) $\displaystyle=$
$\displaystyle\,-\int_{\partial\hat{S}}\mathrm{e}^{-2\phi}\bar{\epsilon}[e^{-\phi}\Gamma_{(8)}+\Gamma_{(7)}\Gamma\wedge(\mathrm{d}y-C_{1})]\wedge[\psi-\frac{1}{6}\Gamma_{(1)}\lambda+\frac{1}{3}e^{\phi}\Gamma\lambda(\mathrm{d}y-C_{1})]$
$\displaystyle=$
$\displaystyle\,-\int_{\partial\hat{S}}\mathrm{e}^{-2\phi}\bar{\epsilon}\Gamma\Gamma_{(7)}\wedge(\psi-\frac{1}{6}\Gamma_{(1)}\lambda)\wedge(\mathrm{d}y-C_{1})+\frac{1}{3}\int_{\hat{\mathcal{S}}}\mathrm{e}^{-2\phi}\bar{\epsilon}\Gamma\Gamma_{(8)}\lambda\wedge(\mathrm{d}y-C_{1})$
$\displaystyle=$
$\displaystyle\,-\int_{\partial\hat{S}}\mathrm{e}^{-2\phi}\bar{\epsilon}\Gamma(\Gamma_{(7)}\wedge\psi-\Gamma_{(8)}\lambda)\wedge(\mathrm{d}y-C_{1})$
$\displaystyle=$ $\displaystyle\,-\int_{\partial
S}\mathrm{e}^{-2\phi}\bar{\epsilon}\Gamma\left(\Gamma_{(7)}\wedge\psi-\Gamma_{(8)}\lambda\right)\equiv
Q(\epsilon)\,.$

So we indeed get the type IIA supercharge (5.4).

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