# Stacking-engineered ferroelectricity and multiferroic order in van der Waals
magnets

Daniel Bennett<EMAIL_ADDRESS>John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
02138, USA  Gabriel Martínez-Carracedo  Departamento de Física, Universidad de
Oviedo, 33007 Oviedo, Spain  Centro de Investigación en Nanomateriales y
Nanotecnología, Universidad de Oviedo-CSIC, 33940 El Entrego, Spain  Xu He
Theoretical Materials Physics, Q-MAT, University of Liège, B-4000 Sart Tilman,
Belgium  Jaime Ferrer  Departamento de Física, Universidad de Oviedo, 33007
Oviedo, Spain  Centro de Investigación en Nanomateriales y Nanotecnología,
Universidad de Oviedo-CSIC, 33940 El Entrego, Spain  Philippe Ghosez
Theoretical Materials Physics, Q-MAT, University of Liège, B-4000 Sart Tilman,
Belgium  Riccardo Comin  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA, USA  Efthimios Kaxiras  John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
02138, USA  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA

###### Abstract

Two-dimensional (2D) materials that exhibit spontaneous magnetization,
polarization or strain (referred to as ferroics) have the potential to
revolutionize nanotechnology by enhancing the multifunctionality of nanoscale
devices. However, multiferroic order is difficult to achieve, requiring
complicated coupling between electron and spin degrees of freedom. We propose
a universal method to engineer multiferroics from van der Waals magnets by
taking advantage of the fact that changing the stacking between 2D layers can
break inversion symmetry, resulting in ferroelectricity and possibly
magnetoelectric coupling. We illustrate this concept using first-principles
calculations in bilayer $\mathrm{NiI_{2}}$, which can be made ferroelectric
upon rotating two adjacent layers by $180^{\circ}$ with respect to the bulk
stacking. Furthermore, we discover a novel multiferroic order induced by
interlayer charge transfer which couples the interlayer spin order and
electronic polarization. Our approach is not only general but also systematic,
and can enable the discovery of a wide variety of 2D multiferroics.

Introduction.—The ability to synthesize stable layered van der Waals (vdW)
materials has opened up new possibilities for mechanically-assembled stacks of
heterostructures and led to significant advances in materials science [1, 2,
3]. These new artificial materials display a range of unprecedented
multifunctional properties [4, 5, 6], one notable example being ferroic orders
such as ferromagnetism [7, 8] and ferrroelectricity [9, 10, 11]. 2D magnetic
materials show great promise as a new class of magnets because their
properties can be controlled through the application of external stimuli
beyond temperature and magnetic field, such as mechanical strain or gate
voltage [12, 13, 14, 15]. Recently a new type of ferroelectricity in vdW
materials was proposed [9] and experimentally observed [10, 11, 16, 17, 18,
19, 20]. For vdW materials such as hexagonal boron nitride (hBN) or transition
metal dichalcogenides (TMDs) where centrosymmetry is broken by stacking-
engineering, an electronic out-of-plane polarization $P_{\perp}$ occurs
through interlayer charge transfer. The magnitude and orientation of
$P_{\perp}$ is determined by the relative stacking between the layers and can
be switched by a relative sliding between the layers, which results in ‘vdW
ferroelectricity’ [10].

The functional properties of vdW materials can be further tuned by introducing
a relative twist or lattice mismatch between neighboring layers, resulting in
a moiré superlattice on a scale much larger than the underlying crystal
periodicity. This has been the source of many interesting and unconventional
emergent phenomena, one notable example being superconductivity in ‘magic
angle’ twisted bilayer graphene [21, 22, 23, 24]. Creating a moiré
superlattice from a vdW ferroelectric results in the formation of moiré polar
domains (MPDs) [25, 26]. These MPDs have been experimentally shown to result
in ferroelectricity [11, 18] through the asymmetric change in domain size in
response to an applied electric field. Furthermore, the MPDs also have an in-
plane polarization component ${\bf P}_{\parallel}$ and as a result they have
nontrivial topology [27, 28, 29], analogous to the polar textures observed in
oxide perovskites [30, 31, 32, 33]. Similar topological magnetic textures have
been reported by twisting vdW magnets, such as $\mathrm{CrI_{3}}$ [34].

The interplay between ferroelectricity and magnetism can give rise to
multiferroic order [35, 36], which is promising for enhanced functionality in
nanodevices. While 2D multiferroics are less common than bulk counterparts
such as $\mathrm{BiFeO_{3}}$ [37], the vdW magnet $\mathrm{NiI_{2}}$ [38, 39,
40] has shown signals of multiferroic order down to the monolayer limit [41].
The proposed mechanism in $\mathrm{NiI_{2}}$ is the appearance of improper
ferroelectricity in response to long-range magnetic order which breaks
inversion symmetry [42, 43, 44, 45]. There are a few disadvantages to this
relatively complicated magnetoelectric coupling mechanism. Specifically, the
strength of the coupling depends on the strength of the Rashba spin-orbit
coupling (SOC) in a material and may be weak in materials that do not contain
heavy elements. Moreover, it also requires magnetization gradients such as
complex spin textures.

Figure 1:  Illustration of tuning the ferroic properties of vdW magnets with
stacking engineering. (a) A vdW bilayer with an AFM interlayer spin
configuration is sketched (spins denoted by red arrows), with color-coded
edges to help identify the relative orientation of the layers. The bilayer has
an inversion center and is nonpolar. (b) Rotating the layers by $180^{\circ}$
breaks the inversion center, resulting in a stacking dependent polarization
$P_{\perp}({\bf x})$, denoted by the green (down) and purple (up) arrows. (c)
A relative sliding ${\bf x}$ between the layers inverts the polarization. (d)
Changing the interlayer spin order to FM (spins denoted by blue arrows)
affects the magnitude of the polarization, but not the direction. The
corresponding stackings in bilayer $\mathrm{NiI_{2}}$ are illustrated below in
each case. Most of the magnetic moment is concentrated on the Ni atoms, with a
small magnetic moment transferred to the I atoms.

We propose a simple and general approach for engineering multiferroics from
vdW magnets through artificial stacking (see Fig. 1). This approach has
already been successfully been applied to fabricate vdW ferroelectrics [11,
16, 18]. Using bilayer $\mathrm{NiI_{2}}$ as an example, we show with ab
initio calculations that by rotating one of the layers by 180∘ with respect to
the natural bulk stacking, the bilayer becomes ferroelectric (see Fig. 2).
Whether or not a system can be made ferroelectric can be determined simply
from knowledge of the space groups of the different stackings. It turns out
that the symmetries of bilayers of $\mathrm{NiI_{2}}$ and of TMDs in the H1
structure (such as $\mathrm{MoS_{2}}$) are exactly the same upon rotation by
180∘ with respect to the natural bulk stacking (see Table 1). This feature
makes their ferroelectric properties identical. Furthermore, we discover a
novel multiferroic order in artifically stacked bilayer $\mathrm{NiI_{2}}$,
between the interlayer electronic polarization and spin ordering. Changing the
interlayer spin order changes the polarization by a factor of two, which leads
to additional jumps in the ferroelectric hysteresis loop and implies that the
ferroelectric polarization may be tuned using a magnetic field.

Stacking | $\mathrm{NiI_{2}}$ | $\mathrm{MoS_{2}}$
---|---|---
Aligned | Anti-aligned | Aligned | Anti-aligned
AA |  $P\bar{3}m1$ (#164) |  $P\bar{6}m2$ (#187) |  $P\bar{6}m2$ (#187) |  $P\bar{3}m1$ (#164)
AB |  $P\bar{3}m1$ (#164) |  $P3m1$ (#156) |  $P3m1$ (#156) |  $P\bar{3}m1$ (#164)
DW |  $C2/m$ (#12) |  $Abm2$ (#39) |  $Abm2$ (#39) |  $C2/m$ (#12)
BA |  $P\bar{3}m1$ (#164) |  $P3m1$ (#156) |  $P3m1$ (#156) |  $P\bar{3}m1$ (#164)
Table 1:  Space group analysis of the high-symmetry relative stackings in
bilayer $\mathrm{NiI2}$ and $\mathrm{MoS_{2}}$. The bold font indicates the
natural bulk stacking.

Results.—We performed first-principles density functional theory (DFT)
calculations to simulate the properties of bilayer $\mathrm{NiI_{2}}$ (see
Methods). We considered the aligned stacking which occurs naturally in bulk
$\mathrm{NiI_{2}}$, as well as the artificial anti-aligned stacking (see Fig.
1), with antiferromagnetic (AFM) and ferromagnetic (FM) interlayer spin
configurations. We emphasize that throughout this work, AFM and FM always
refer to the interlayer alignment of spins and not the alignment within a
single layer. Because the wavelength of the spin spiral in $\mathrm{NiI2}$ is
relatively long (around 7 unit cells [45]), we consider uniform arrangements
of spins pointing in the out-of-plane direction in each layer in order to
avoid calculations involving large supercells.

Fig. 2 shows the stacking energy $\mathcal{V}_{\rm stack}$ and out-of-plane
electronic dipole moment $p_{\perp}$ as a function of relative displacement
$x$ between the layers along the unit cell diagonal. Results for aligned and
anti-aligned stackings as well as AFM and FM interlayer spin configurations
are shown (see Supplementary Material (SM) for additional details). The unit
cell diagonal contains all of the high symmetry stacking configurations,
namely AA ($x=0,1$), AB ($x=\tfrac{1}{3}$) and BA ($x=\tfrac{2}{3}$), and the
domain wall (DW) stacking ($x=\tfrac{1}{2}$). The AFM configuration was found
to be lower in energy for all stackings in both aligned and anti-aligned
$\mathrm{NiI_{2}}$. The aligned bilayer is non-polar for all stacking
configurations as the inversion center between the layers is preserved upon
sliding. For the anti-aligned stacking, the bilayer attains a stacking-
dependent electronic dipole moment which is largest and of opposite sign for
the energetically favorable AB and BA stackings. The polarization being an odd
function of stacking and respecting the $\mathcal{C}_{3}$ rotation symmetry of
the crystal takes the form [26]:

$p_{\perp}(x,y)=p^{\text{odd}}_{1}\bigg{[}\sin{(2\pi x)}+\sin{(2\pi
y)}-\sin{(2\pi(x+y))}\bigg{]}\;,$ (1)

with ${\bf x}=(x,y)$ in fractional coordinates of the unit cell lattice
constant. $p^{\text{odd}}_{1}$ is a coefficient to be fit to DFT calculations.
Additionally, anti-aligned $\mathrm{NiI_{2}}$ also has a stacking-dependent
in-plane polarization similar to rhombohedral hBN and TMDs (see SM).

Figure 2:  Stacking energy $\mathcal{V}_{\rm stack}$ and out-of-plane dipole
moment $p_{\perp}$ for aligned and anti-aligned bilayer $\mathrm{NiI2}$, as a
function of relative stacking $x$ along the unit cell diagonal. The labels AA,
AB/BA and DW refer to the relative stackings, with AA (aligned) and AB/BA
(ani-aligned) sketched in Fig. 1. Results for an AFM (FM) interlayer spin
order are shown in red (blue). The dots show results from first-principles
calculations and the solid lines show fits of the data to
$\mathcal{C}_{3}$-symmetric basis functions, such as Eq. (1) for $p_{\perp}$.

Fig. 2 shows that the magnitude of the polarization in anti-aligned
$\mathrm{NiI_{2}}$ is sensitive to the interlayer spin configuration. Upon
switching from AFM to FM the polarization decreases in magnitude by
approximately a factor of 2. Fig. 3 illustrates the ferroelectric hysteresis
loop in anti-aligned $\mathrm{NiI_{2}}$ including the effects of this
multiferroic order. In addition to the first-order interlayer sliding jumps in
polarization there are also jumps in the magnitude of polarization, but not in
sign, driven by a change in interlayer spin order. Thus, the spin order in
anti-aligned $\mathrm{NiI_{2}}$ can be switched using an applied electric
field and the magnitude of the polarization can be tuned using a magnetic
field. This can be explained by considering the free energy:

$\mathcal{V}^{\alpha}({\bf x},\mathcal{E})=\mathcal{V}_{\rm
stack}^{\alpha}({\bf x})-\mathcal{E}\cdot p^{\alpha}({\bf x})\;,$ (2)

where $\alpha=$ AFM, FM, which is sketched in Fig. 3 at different points
around the hysteresis loop. Applying an electric field $\mathcal{E}$ which is
anti-aligned with the dipole moment (AB/BA), the FM configuration eventually
becomes lower in energy. This results in a change in the interlayer spin order
driven by a decrease in polarization. Increasing the electric field further,
the other stacking configuration will eventually become preferable (BA/AB),
resulting in polarization reversal through interlayer sliding.

Figure 3:  Illustration of the ferroelectric hysteresis loop for anti-aligned
bilayer $\mathrm{NiI2}$. The free energies of the AFM (red) and FM (blue)
interlayer spin configurations are sketched as a function of stacking, at six
different points around the hysteresis loop. The dot shows the state that
$\mathrm{NiI_{2}}$ adopts for each electric field value, which is possibly
metastable. The red (blue) background indicates that $\mathrm{NiI_{2}}$ has an
AFM (FM) interlayer spin order. The purple (green) arrows indicate that
$\mathrm{NiI_{2}}$ has polarization up (down). The jumps that decrease the
magnitude of the polarization but do not change its sign are driven by a
change in interlayer spin order (AFM $\rightarrow$ FM). The jumps which change
the sign in polarization are driven by interlayer sliding (AB $\rightarrow$
BA) and are also accompanied by a change in interlayer spin order (FM
$\rightarrow$ AFM).

The intralayer magnetic exchange parameters were obtained using the
relativistic LKAG formalism [46] within a non-orthogonal localized basis
framework, implemented in the grogu code [47] (see Methods and SM). The
exchange and intra-atomic anisotropy tensors to any desired neighbor shell can
be calculated, corresponding to the following Heisenberg Hamiltonian:

$H_{\rm intra}=\,\frac{1}{2}\,\sum_{i\neq j}\,J^{ij}_{\parallel}\,{\bf
e}_{i}\cdot{\bf e}_{j}+\frac{1}{2}\,\sum_{i\neq j}\,{\bf
e}_{i}\,J_{S,\parallel}^{ij}\,{\bf e}_{j}+\,\frac{1}{2}\,\sum_{i\neq j}\,{\bf
D}^{ij}\cdot({\bf e}_{i}\times{\bf e}_{j})\;.$ (3)

Here ${\bf e}_{i}$ is a unit vector pointing along the spin angular momentum
of the localized magnetic entity placed at site $i$. $J^{ij}_{\parallel}$,
$J_{S,\parallel}^{ij}$ and ${\bf D}^{ij}$ are the isotropic exchange
constants, symmetric traceless exchange tensor and Dzyaloshinskii-Moriya (DM)
vectors between sites $i$ and $j$, respectively. The interlayer isotropic
exchange parameter $J_{\perp}$ was obtained from total energy differences
between the FM and AFM bilayer spin alignments, mapped onto the effective
Hamiltonian:

$H_{\rm inter}=J_{\perp}{\bf e}_{1}\cdot{\bf e}_{2}\;,$ (4)

where ${\bf e}_{1}$ and ${\bf e}_{2}$ are unit vectors describing the spin
directions in the first and second layers of bilayer $\mathrm{NiI_{2}}$. The
interlayer exchange changes by as much as a factor of 2 as a function of
stacking (see SM) which suggests that the magnetic transition temperature is
also stacking-dependent. The interlayer spin order does not change upon
changing the stacking in contrast to $\mathrm{CrI_{3}}$ [48], and the AFM spin
order is always energetically preferable at zero electric field.

From our calculations the electric field required to change the interlayer
spin order is $\mathcal{E}=2.77$ V/Å for the AB/BA stackings in anti-aligned
bilayer $\mathrm{NiI_{2}}$. The coercive magnetic field $B_{\rm c}$ required
to switch from AFM to FM spin-order in anti-aligned $\mathrm{NiI_{2}}$ and
tune the electronic polarization is shown in Fig. 4 (a). $B_{\rm c}$ modulates
with stacking, ranging between 25–40 T. In order to investigate the switching
dynamics in more detail, we perform classical Monte Carlo (MC) calculations
with a Heisenberg Hamiltonian (see Methods) to simulate anti-aligned bilayer
$\mathrm{NiI_{2}}$ in the AB stacking using a $64\times 64$ supercell. The
intralayer magnetic interaction is frustrated due to the competition between
the FM nearest neighbor (NN) exchange interactions and the AFM 3rd NN exchange
interaction. The ground state from MC simulations is a spin spiral state with
the in-plane spin and AFM interlayer ordering, as was found in Ref. [49]. By
measuring the specific heat as a function of temperature, we find a Néel
temperature of $T_{N}=50$ K (Fig. 4 (b)). The average out-of-plane
magnetization $M_{\perp}$ in response to an out-of-plane magnetic field
$B_{\perp}$ is shown in Fig. 4 (c), for temperatures ranging from 0 K to 100
K. The magnetic field required to align the magnetic moment is $B_{\perp}\sim
800$ T at $T=0$ K, and is reduced to about 200 T at finite temperature. In
order to consider the magnetic field required to change the interlayer spin
order, we calculate ${\bf e}_{1}\cdot{\bf e}_{2}$, averaged over the
supercell, as a function of magnetic field and temperature (Fig. 4 (d)).
Comparing Figs. 4 (c) and (d), the magnetic field required to change the
interlayer spin order is approximately a factor of two smaller than the field
required to align all of the spins with the field (destroy the spin spiral
state).

Discussion and conclusions.—We propose that vdW magnets can be made
ferroelectric by fabricating multilayers in artificial stackings which break
inversion symmetry, and we illustrate this concept using bilayer
$\mathrm{NiI_{2}}$ as an example. This approach can be generalized to other
vdW magnets, although the resulting polar properties may differ depending on
the crystal symmetries [50]. This has the potential to uncover new types of
multiferroic order: since the polarization is electronic, changes in the
electronic structure induced by spin ordering are likely to result in
magnetoelectric coupling. As a case in point, we find a new type of
multiferroic order in bilayer $\mathrm{NiI_{2}}$ upon rotation of the layers
to the anti-aligned stacking. We predict additional jumps in the ferroelectric
hysteresis loop, with the polarization changing in magnitude by a factor of
two as a result of changes in the interlayer spin order.

Figure 4:  (a) Estimate of the coercive magnetic field $B_{\rm c}(x)$ in anti-
aligned bilayer $\mathrm{NiI_{2}}$. The points show the values of $B_{\rm c}$
obtained from DFT calculations at each stacking $x$. The solid line shows the
values obtained from the fits to the stacking energies in Fig. 2. (b) Specific
heat $C_{V}$ of the system as a function of temperature $T$ from MC
simulations. $C_{V}$ is normalized such that $C_{V}(T\to 0)=1$. The peak at 50
K indicates the Néel temperature $T_{N}$. (c) Average out-of-plane
magnetization $M_{\perp}$, in units of the spontaneous magnetization $M_{\rm
S}$ as a function of perpendicular magnetic field $B_{\perp}$ and temperature
$T$. (d) Average interlayer spin order ${\bf e}_{1}\cdot{\bf e}_{2}$ as a
function of perpendicular magnetic field $B_{\perp}$ and temperature $T$.

The coercive fields required to achieve this switching are predicted to be
unrealistically large by our calculations, with electric fields of order 1
V/Åand magnetic fields of order 10–100 T (similar values were predicted in
Ref. [49]). We note that the large coercive fields obtained in DFT/MC
calculations correspond to the physical mechanism of
polarization/magnetization reversal. Experimentally however, switching occurs
due to nano-domain nucleation and subsequent growth due to domain wall
propagation. This switching mechanism needs to overcome much lower energy
barriers and therefore requires much smaller coercive fields, typically below
$\mathcal{E}=0.2$ V/nm for vdW ferroelectrics [11, 16, 18] and $B=1$ T for vdW
magnets [51, 7, 52].

In the present work we considered the interplay between electronic
polarization and interlayer spin order assuming homogeneous spin textures
within each layer. The coupling between polarization and complex spin textures
may lead to more interesting and exotic multiferroic coupling. For example,
suppose the spin spirals in both layers are offset from one another, which can
be described by a relative phase shift. If the relative phase shift is the 0
($\pi$), the interlayer spin order is FM (AFM). By using an electric field to
switch the polarization through interlayer sliding, the relative phase between
the two spin spirals and hence the interlayer spin order may also change.

Finally, we propose that stacking vdW magnets to form moiré superlattices may
be a route to engineering multiferroic topological structures, in which polar
and spin topology may both be controlled with magnetic and electric fields.
Considering the out-of-plane and in-plane polarization in anti-aligned bilayer
$\mathrm{NiI_{2}}$, a twist by a small angle about the $180^{\circ}$
orientation will result in a regular network of polar merons and antimerons
(winding numbers $Q\pm\frac{1}{2}$ in each domain) [27]. Additionally, it is
known that magnetic skyrmions can spontaneously form in $\mathrm{NiI_{2}}$ in
the few-layer limit [49]. By precisely engineering a moiré superlattice of
period commensurate with the magnetic skyrmion wavelength, one may achieve the
coexistence of polar and magnetic topological structures in a single device.

##  I Methods

###  I.1 First-principles calculations

First-principles density functional theory (DFT) calculations were performed
using the siesta [53] code, version 5.0, with norm-conserving [54] psml
pseudopotentials [55], obtained from Pseudo-Dojo [56]. A basis of
triple-$\zeta$ double polarized (TZDP) orbitals were used for all
calculations, the cutoff radii of which were determined with an energy shift
of 20 meV. Diffuse orbitals [57] were added ($5s$ and $5p$ for Ni, $6s$ and
$6p$ for I) in order to better treat the interlayer interactions and the decay
of the charge density into the vacuum region. We employed cutoff radii of 15
bohr for all diffuse orbitals, which were found to yield electronic band
structures in good agreement with results obtained from the abinit plane-wave
code [58] using the same psml pseudopotentials. A Monkhorst-Pack
$\mathbf{k}$-point grid [59] of $18\times 18$ was used in all calculations
and, the real space grid was determined with an energy cutoff of 1000 Ry. We
use the PBE exchange-correlation functional [60] in all calculations. A DFT-D3
dispersion correction [61] was included in order to treat the long range
interactions between the layers, with a dipole correction in the vacuum region
to prevent dipole-dipole interactions between periodic images [62].

A Hubbard-$U$ correction was included on the Ni $3d$ orbitals with a value of
$U=1.8$ eV, following Ref. [49]. SOC was included in all calculations except
for geometry relaxation of the interlayer separation in the bilayers. The
initial spin configuration was taken to be $\pm 2\mu_{\rm B}$ in the out-of-
plane direction on the Ni atoms of each layer, the opposite (same) sign for
AFM (FM) interlayer spin configurations. The spins did not deviate much from
the initial configuration after convergence of the calculations, except for a
small transfer of some spin to the I atoms.

A lattice constant of 4.00 Å was obtained for monolayer $\mathrm{NiI_{2}}$, in
good agreement with the experimentally reported value of 3.98 Å, which was
used in the bilayer calculations. Geometry relaxations were then performed
(without SOC) in order to determine the interlayer separation, using a force
tolerance of 0.1 meV/Å. The top layer was translated above the bottom layer
along the unit cell diagonal, for both aligned and anti-aligned stacking
configurations, and for both AFM and FM interlayer spin configurations. After
relaxation, calculations were then repeated including SOC. The polarization
was calculated from Berry phases.

###  I.2 Magnetic exchange parameters

The grogu code [47] was used to calculate the intralayer magnetic exchange
parameters of the Heisenberg Hamiltonian in Eq. (3). As the dominant DM
interaction is given by first nearest neighbors and the radial component is
negligible, it is convenient to express DM vectors in cylindrical coordinates
${\bf D}^{ij}=(D^{r},D^{\theta},D^{z})$ where the radial component points
along the bond.

grogu calculations were performed using a dense Monkhorst-Pack grid of
$100\times 100$ which take a self-consistent Hamiltonian obtained from siesta
calculations (including SOC). The localized magnetic entities were projected
onto $3d$-orbitals of the Ni atoms.

###  I.3 Monte Carlo spin simulations

MC simulations were performed using the Heisenberg model:

$H=H_{\rm intra}+H_{\rm inter}-\sum_{i}\mathbf{B}\cdot\mathbf{S}_{i}\;,$ (5)

where $H_{\rm intra}$ and $H_{\rm inter}$ are given by Eqs. (3) and (4), and
the last term is the Zeeman interaction. The intralayer exchange parameters
for monolayer $\mathrm{NiI_{2}}$ from Ref. [49] were used, which were
calculated using the supercell method, because the exchange parameters
calculated using the LKAG method underestimate the intralayer exchanges (1st
NN in particular) due to the strong hybridization between the Ni and I atoms
in each layer. The 1st, 2nd and 3rd NN exchange parameters are
$J^{1}_{\parallel}=-7.0$ meV, $J^{2}_{\parallel}=-0.3$ meV and
$J^{3}_{\parallel}=+5.8$ meV, respectively. In addition, the anisotropic
exchange parameters for the 1st NNs between Ni atoms are taken into account.
For the interlayer interaction, we take only the NN exchange parameter, and
estimate that with the energy difference between the ferromagnetic and
antiferromagnetic interlayer alignments, which gives a $J_{\perp}=7$ meV.
Calculations were repeated using the exchange parameters obtained in this
work. The results using both sets of exchange parameters are compared in SM.

We then perform Metropolis MC simulations to estimate the magnetic field
required to align the interlayer magnetic moments. The calculations are
performed using a $64\times 64$ supercell, with periodic boundary conditions
in the in-plane directions. An additional calculation using a $1\times 1$ cell
was performed so that intralayer interaction is constrained to be FM. We note
that if the intralayer spin is constrained to be FM, the coercive magnetic
field is much smaller, approximately 25 T with an in-plane magnetic field,
which is consistent with the estimation from firs-principles calculations, or
approximately 120 T with an out-of-plane magnetic field.

##  II Acknowledgements

D.B. and E.K. acknowledge the US Army Research Office (ARO) MURI project under
grant No. W911NF-21-0147 and the Simons Foundation award No. 896626. G.M.-C.
and J.F. have been funded by MCIN/AEI/10.13039/501100011033/FEDER, UE via
project PID2022-137078NB-100 and by Asturias FICYT under grant AYUD/2021/51185
with the support of FEDER funds. G.M.-C. has been supported by Programa
“Severo Ochoa” de Ayudas para la investigación y docencia del Principado de
Asturias. X.H. and Ph.G. acknowledge financial support from F.R.S.-FNRS
Belgium through the PDR project PROMOSPAN (Grant No. T.0107.20). R.C.
acknowledges support by the Department of Energy, Office of Science, Office of
Basic Energy Sciences, under Award No. DE-SC0019126. J.F. acknowledges a
discussion with A. Hierro on the role of magnetization reversal versus domain
growth in experimental magnetic materials.

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./SM.pdf


# Techni-Composite Higgs models with (a)symmetric dark matter candidates

Giacomo Cacciapaglia<EMAIL_ADDRESS>Institut de Physique des 2
Infinis (IP2I), CNRS/IN2P3, UMR5822, 69622 Villeurbanne, France  Université de
Lyon, Université Claude Bernard Lyon 1, 69001 Lyon, France  Mads T. Frandsen
<EMAIL_ADDRESS>CP3-Origins, University of Southern Denmark, Campusvej 55,
DK-5230 Odense M, Denmark  Wei-Chih Huang<EMAIL_ADDRESS>CP3-Origins,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Martin Rosenlyst<EMAIL_ADDRESS>CP3-Origins, University of Southern
Denmark, Campusvej 55, DK-5230 Odense M, Denmark  Philip Sørensen
<EMAIL_ADDRESS>II. Institute of Theoretical Physics, Universität
Hamburg, 22761 Hamburg, Germany  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

We propose a novel class of composite models that feature both a technicolor
and a composite Higgs vacuum limit, resulting in an asymmetric dark matter
candidate. These Techni-Composite Higgs models are based on an extended left-
right electroweak symmetry with a pseudo-Nambu Goldstone boson Higgs and
stable dark matter candidates charged under a global $\mathrm{U}(1)_{X}$
symmetry, connected to the baryon asymmetry at high temperatures via the
$\mbox{SU}(2)_{\rm R}$ sphaleron. We consider, as explicit examples, four-
dimensional gauge theories with fermions charged under a new confining gauge
group $G_{\rm HC}$.

††preprint: DESY-21-196

The nature of Dark Matter (DM) and the origin of its relic density are
arguably among the most important open questions in particle physics
Bertone:2018krk . An answer could be found within existing solutions to other
problems in nature: for instance, we know that the bulk of the baryonic mass
is due to strong dynamics while the baryonic relic density is due to a
particle/anti-particle asymmetry. Furthermore, the baryonic and DM relic
densities are of the same order. Within Technicolor (TC) models of the
electroweak (EW) symmetry breaking Weinberg:1975gm ; Eichten:1979ah , this
observation has motivated DM candidates whose mass is due to new strong
interactions and whose stability is due to $\mbox{U}(1)_{X}$ techni-baryon
charge Nussinov:1985xr ; Barr:1990ca . The baryon and DM relic densities can
share a common asymmetric origin if this $\mbox{U}(1)_{X}$ global symmetry is
anomalous under the EW sphalerons, paralleling the EW anomaly of the baryon
number Kuzmin:1985mm . This realizes asymmetric dark matter (ADM) Barr:1990ca
in TC models. In view of the progress at the LHC, the main drawback of TC
models is that, in general, there is no simple parametric limit in which a
Standard Model (SM)-like Higgs is recovered.

One way out consists in engineering a vacuum misalignment of TC into a
Composite Higgs (CH) model Kaplan:1983fs ; Dugan:1984hq ; Contino:2003ve ,
which is possible for a subset of TC theories Cacciapaglia:2014uja ;
Cacciapaglia:2020kgq . The composite Higgs state now arises as a pseudo-Nambu-
Goldstone boson (pNGB) from the spontaneously broken (chiral) global symmetry.
Hence, it can be parametrically close to the SM Higgs. However, in CH models
the $\mbox{U}(1)_{X}$ techni-baryon or specie number is either broken or no
longer anomalous under the EW sphalerons, thus the link between the baryon and
DM asymmetric relic densities is severed (see Frigerio:2012uc ; Ma:2015gra ;
Ma:2017vzm ; Ballesteros:2017xeg ; Balkin:2017aep ; Cai:2018tet ;
Balkin:2018tma ; Cacciapaglia:2019ixa  for models with thermal composite DM).

In this letter, we propose a new model building avenue where the CH and TC
limits are simultaneously present, thus allowing for a successful description
of the EW symmetry breaking and ADM. This requires:

  * i)

a SM-like composite pNGB Higgs multiplet with custodial symmetry Georgi:1984af
;

  * ii)

a composite DM candidate, stable due to a $\mbox{U}(1)_{X}$ symmetry of the
strong interactions;

  * iii)

an EW anomaly of the $\mbox{U}(1)_{X}$ symmetry allowing for shared asymmetry
of baryons and DM;

  * iv)

a suppressed DM thermal relic density.

The first ingredient i) is a key feature of CH models Kaplan:1983fs ,
holographic extra dimensions Contino:2003ve ; Hosotani:2005nz , Little Higgs
ArkaniHamed:2001nc ; ArkaniHamed:2002qx , Twin Higgs Chacko:2005pe  and
elementary Goldstone Higgs models Alanne:2014kea . Extensions of the global
symmetries can accommodate ii), however these model types typically do not
satisfy ii) and iii) together.

To realize all four requirements of the Techni-Composite Higgs scenario, we
are led to consider an extended Left-Right EW sector, with gauged
$\mbox{SU}(2)_{\rm L}\otimes\mbox{SU}(2)_{\rm R}\otimes\mbox{U}(1)_{\rm
Y^{\prime}}$ symmetry, dynamically broken by the strong dynamics via a left-
right coset: $G/H\supset G_{L}/H_{L}\otimes G_{R}/H_{R}$. The left sub-coset
$G_{L}/H_{L}$ is pinned in a CH direction by appropriate interactions, while
the right sub-coset $G_{R}/H_{R}$ is in the TC vacuum with an unbroken global
$\mbox{U}(1)_{X}$. This symmetry is anomalous under $\mbox{SU}(2)_{\rm R}$,
and the lightest composite state carrying $X$-charge plays the role of ADM.
Imposing the requirements i)-iv) also constrains non-trivially the form of the
operators that generate the SM-fermion masses. Note that the interplay between
TC and CH limits was used in Ref. Cai:2019cow , where the TC vacuum is only
present at high temperatures. In our paradigm, the two limits coexist at all
temperatures.

For concreteness, we consider four-dimensional gauge theories with a single
strongly interacting gauge group $G_{\rm HC}$, with hyper-fermions that
generate the global symmetry breaking via condensation. For models with a
single fermionic representation, the symmetry breaking patterns are known
Witten:1983tx ; Kosower:1984aw : Given $N$ Weyl spinors transforming as the HC
$\mathcal{R}$ representation, the three possible classes of vacuum cosets are
$\mbox{SU}(N)/\mbox{SO}(N)$ for real $\mathcal{R}$,
$\mbox{SU}(N)/\mbox{Sp}(N)$ for pseudo-real $\mathcal{R}$ and
$\mbox{SU}(N)\otimes\mbox{SU}(N)\otimes\mbox{U}(1)/\mbox{SU}(N)\otimes\mbox{U}(1)$
for complex $R$ Peskin:1980gc . The minimal CH cosets that fulfil requirement
i), within these three classes, contain $N=5$ in the real case Dugan:1984hq ,
$N=4$ in both the pseudo-real Galloway:2010bp  and the complex cases
Ma:2015gra . In terms of pNGB spectrum, the pseudo-real case is the most
minimal, with only 5 states. Similarly, the minimal TC cosets that fulfil the
requirements ii)-iv) contain $N=4$ in the real case Sannino:2004qp ;
Dietrich:2006cm ; Foadi:2007ue ; Frandsen:2009mi , $N=4$ in the pseudo-real
Ryttov:2008xe  and $N=2$ in the complex one Kaplan:1983fs . In the first two,
the ADM candidates are pNGBs, while in the complex case it is a baryon
Kaplan:1983fs .

To realize our scenario, we need to introduce two sets $\mathbb{S}_{L,R}$ of
hyper-fermions, charged under $\mbox{SU}(2)_{\rm L,R}$, respectively. The
representations $\mathcal{R}_{L,R}$ may be different or identical, and they
determine the resulting coset structure. We will assume that the pattern of
the $L$ and $R$ cosets are the same as above even when the two representations
are different, while for $\mathcal{R}_{L}\equiv\mathcal{R}_{R}$ the coset is
enlarged. The strong $G_{\rm HC}$ interactions produce condensates of the
$\mathbb{S}_{L,R}$ fermions at scales $f_{L,R}$, which are of similar size.
Hence, the breaking of the EW gauge symmetry occurs as follows:

$\displaystyle\mbox{SU}(2)_{\rm L}\otimes\mbox{SU}(2)_{\rm
R}\otimes\mbox{U}(1)_{\rm Y^{\prime}}\xrightarrow{f_{R}}\mbox{SU}(2)_{\rm
L}\otimes\mbox{U}(1)_{\rm Y}$ (1)
$\displaystyle\xrightarrow{f_{L}\sin\theta_{L}=v_{\rm EW}}\mbox{U}(1)_{\rm
EM}\,,$

where the hierarchy between the EW scale $v_{\rm EW}$ and the compositeness
scale $f_{L}\sim f_{R}$ is generated in the $\mathbb{S}_{L}$-sector and is
parametrized by a (small) angle $\theta_{L}$.

The most minimal choice for the $\mathbb{S}_{L}$ sector consists in four Weyl
spinors $Q_{L}$, arranged in one $\mbox{SU}(2)_{\rm L}$ doublet $(U,D)$ and
two singlets $\widetilde{U}$ and $\widetilde{D}$, transforming as a pseudo-
real representation $\mathcal{R}_{L}$ of $G_{\rm HC}$ and as a fundamental of
a global $G_{L}=\mbox{SU}(4)_{L}$. The minimal $L$ coset will, therefore,
contain the longitudinal components of the $W^{\pm}$ and $Z$ bosons, a Higgs
candidate and a singlet $\eta$. We will focus on this scenario in the
following, as shown in Table 1.111The same gauge charge assignment can be used
for a complex $\mathcal{R}_{L}$, at the price of including right-handed hyper-
fermions with the same quantum numbers. For real $\mathcal{R}_{L}$ one would
need two $\mbox{SU}(2)_{\rm L}$ doublets with opposite $\mbox{U}(1)_{\rm
Y^{\prime}}$ charges and a neutral singlet, thus 5 Weyl spinors in total. The
minimal $\mathbb{S}_{R}$ also contains four Weyl spinors $Q_{R}$, arranged in
an $\mbox{SU}(2)_{\rm R}$ doublet $(C,S)$ and two singlets $\widetilde{C}$ and
$\widetilde{S}$. The quantum numbers, together with the $\mbox{U}(1)_{X}$
charges, are shown in Table 1, and are valid for all possible representations
$\mathcal{R}_{R}$: in the complex case, however, the singlets $\widetilde{C}$
and $\widetilde{S}$ transform as the conjugate $\mathcal{R}_{R}^{\ast}$, while
$\mbox{U}(1)_{X}$ is the techni-baryon number. In the pseudo-real case, the
coset is $\mbox{SU}(4)_{R}/\mbox{Sp}(4)_{R}$ and, besides the longitudinal
modes of the $W_{\rm R}^{\pm}$ and $Z_{\rm R}$ bosons, the spectrum contains a
complex neutral pNGB carrying $X$-charge. In the real case, the coset
$\mbox{SU}(4)_{R}/\mbox{SO}(4)_{R}$ contains a complex neutral pNGB and two
charged ones carrying $X$-charge. In the complex case, the coset
$\mbox{SU}(2)_{R1}\otimes\mbox{SU}(2)_{R2}\otimes\mbox{U}(1)_{X}/\mbox{SU}(2)_{VR}\otimes\mbox{U}(1)_{X}$
does not contain pNGBs carrying $X$-charge, so the DM candidate is played by a
baryon-like state. In all cases, the $\mbox{U}(1)_{X}$ symmetry has a gauge
anomaly with respect to the $\mbox{SU}(2)_{\rm R}\otimes\mbox{U}(1)_{\rm
Y^{\prime}}$ symmetry. Note that, if $\mathcal{R}_{L}\neq\mathcal{R}_{R}$,
there exists a global $\mbox{U}(1)_{\Theta}$ symmetry under which both sets of
hyper-fermions are charged, which is spontaneously broken by the condensates
and generates a light singlet pNGB, along the lines of Refs Ferretti:2016upr ;
Belyaev:2016ftv . Finally, if
$\mathcal{R}_{L}=\mathcal{R}_{R}\equiv\mathcal{R}$, the coset is enhanced: for
pseudo-real $\mathcal{R}$, the coset is $\mbox{SU}(8)/\mbox{Sp}(8)$; for real
$\mathcal{R}$, the minimal case is $\mbox{SU}(9)/\mbox{SO}(9)$; for complex
$\mathcal{R}$, we have
$\mbox{SU}(6)_{1}\otimes\mbox{SU}(6)_{2}\otimes\mbox{U}(1)_{TB}/\mbox{SU}(6)_{V}\otimes\mbox{U}(1)_{TB}$,
where the $\mbox{U}(1)_{X}$ in Table 1 is a linear combination of
$\mbox{U}(1)_{TB}$ and a $\mbox{U}(1)$ factor inside $\mbox{SU}(6)_{V}$.

| $G_{\rm HC}$ | $\mbox{SU}(3)_{\rm QCD}$ | $\mbox{SU}(2)_{\rm L}$ | $\mbox{SU}(2)_{\rm R}$ | $\mbox{U}(1)_{\rm Y^{\prime}}$ | $\mbox{U}(1)_{X}$
---|---|---|---|---|---|---
$(U,D)$ | $\mathcal{R}_{L}$ | 1 | $\Box$ | 1 | 0 | 0
$\widetilde{U}$ | $\mathcal{R}_{L}$ | 1 | 1 | 1 | $-1/2$ | 0
$\widetilde{D}$ | $\mathcal{R}_{L}$ | 1 | 1 | 1 | $+1/2$ | 0
$(C,S)$ | $\mathcal{R}_{R}$ | 1 | 1 | $\Box$ | 0 | $+1$
$\widetilde{C}$ | $\mathcal{R}_{R}$ | 1 | 1 | 1 | $-1/2$ | $-1$
$\widetilde{S}$ | $\mathcal{R}_{R}$ | 1 | 1 | 1 | $+1/2$ | $-1$
$q_{{\rm L},i}$ | 1 | $\Box$ | $\Box$ | 1 | $+1/6$ | 0
$q_{{\rm R},i}$ | 1 | $\Box$ | 1 | $\Box$ | $+1/6$ | 0
$l_{{\rm L},i}$ | 1 | 1 | $\Box$ | 1 | $-1/2$ | 0
$l_{{\rm R},i}$ | 1 | 1 | 1 | $\Box$ | $-1/2$ | 0
Table 1: Fermion field content and their charges of the Techni-Composite Higgs
template models with the full left-right gauge symmetry. All groups are gauged
except for $\mbox{U}(1)_{X}$, which is a global symmetry in the $R$ sector
responsible for dark matter stability.

The remaining important ingredient for model building is the list of operators
that generate the SM-fermion masses. The operators play an important role in
determining the vacuum alignment in both $L$ and $R$ sectors, particularly via
the top mass Panico:2015jxa . To connect the $\mbox{U}(1)_{X}$ anomaly to the
baryon number via $\mbox{SU}(2)_{\rm R}$, we require that the right-handed SM
fermions transform as doublets, as shown in the bottom rows of Table 1. The
operators that generate the fermion masses, therefore, appear as 6-fermion
operators with the generic structure $Q_{L}Q_{L}Q_{R}Q_{R}\psi_{\rm
L}\psi_{\rm R}$. For instance, for the top quark

$\frac{\xi_{t}}{\Lambda_{t}^{5}}(Q_{L}^{T}P_{L}Q_{L})(Q_{R}^{T}P_{R}Q_{R})q_{{\rm
L},3}q_{{\rm R},3}+{\rm h.c.}\,,$ (2)

where $P_{L}$ and $P_{R}$ are two-index matrices in the $G_{L}$ and $G_{R}$
space, respectively, selecting the appropriate combinations of the hyper-
fermions that ensure gauge invariance and couple the top fields with the
components that acquire a non-zero condensate. As such, $P_{L,R}$ transform as
doublets of $\mbox{SU}(2)_{\rm L,R}$, respectively. Similar operators can be
added for all SM fermions. Note also that, for complex $\mathcal{R}_{L,R}$, it
suffices to replace $Q_{L,R}^{T}$ by the conjugate hyper-fermions. We also
remark that the operator in Eq. (2), reminiscent of the mass terms in the
original TC models Eichten:1979ah , can be generated via partial compositeness
Kaplan:1991dc  in models proposed in Refs Barnard:2013zea ; Ferretti:2013kya
via operators of the form:

$\displaystyle\frac{y_{L}}{\Lambda_{t}}q_{{\rm
L},3}(Q_{L}^{T}P_{L}Q_{L}\chi_{t})+\frac{y_{R}}{\Lambda_{t}}q_{{\rm
R},3}(Q_{R}^{T}P_{R}Q_{R}\chi_{t}^{\dagger})+{\rm h.c.}\,,$ (3)

where $\chi_{t}$ is a new hyper-fermion, transforming in a suitable
representation of $G_{\rm HC}$, and carrying appropriate quantum numbers under
the SM gauge symmetry. To illustrate the new scenario, in the following we
will focus on a specific minimal model, and leave other examples to the
Supplementary Material.

The minimal scenario we illustrate here is based on $G_{\rm
HC}=\mbox{Sp}(6)_{\rm HC}$ with $\mathcal{R}_{L}=F$ (fundamental, pseudo-real)
and $\mathcal{R}_{R}=A$ (antisymmetric, real). We will also include masses for
the hyper-fermions in $\mathbb{S}_{L}$, because they help stabilize the CH
vacuum Cacciapaglia:2014uja ; Dong:2020eqy . The relevant physics can be
described below the condensation scale in terms of an effective theory,
following the CCWZ prescription Marzocca:2012zn , and based uniquely on the
coset symmetry:

$\frac{\mbox{SU}(4)_{L}\times\mbox{SU}(4)_{R}\times\mbox{U}(1)_{\Theta}}{\mbox{Sp}(4)_{L}\times\mbox{SO}(4)_{R}}\,.$
(4)

At lowest order, the effective Lagrangian has the form

$\mathcal{L}_{\rm EFT}=\mathcal{L}_{\chi pt}-V_{\rm eff}\,,$ (5)

where the first term corresponds to the usual chiral perturbation theory for
the pNGBs, and the second term contains the effective potential generated by
loops of the SM fields, namely the EW gauge bosons and the fermions (top). The
latter plays a crucial role in determining the vacuum alignment and the gauge
symmetry breaking (see the Supplementary Material).

To investigate the asymmetric relic, we analyse the dynamics of the sphalerons
associated with the left and right gauge symmetries. The $\mbox{SU}(2)_{\rm
L}\times\mbox{SU}(2)_{\rm R}$ sphaleron equations, when in equilibrium above
$f_{L,R}$, yield chemical potential equations of the form

$\displaystyle\begin{aligned} &\left(\mu_{u{\rm L},i}+2\mu_{d{\rm
L},i}\right)+\mu_{\nu{\rm
L},i}+\frac{d(\mathcal{R}_{L})}{2}(\mu_{U}+\mu_{D})=0,\\\ &\left(\mu_{u{\rm
R},i}+2\mu_{d{\rm R},i}\right)+\mu_{\nu{\rm
R},i}+\frac{d(\mathcal{R}_{R})}{2}\left(\mu_{C}+\mu_{S}\right)=0,\\\
\end{aligned}$ (6)

where sums over the generations are left understood and $d(\mathcal{R})$ is
the dimension of representation $\mathcal{R}$. The labelling of the chemical
potentials $\mu$ follows Ref. Ryttov:2008xe . The alignment of the
$\mathbb{S}_{L}$ sector vacuum also results in the separate equation
$\mu_{U}+\mu_{D}=0$. Together with equilibrium conditions from the 6-fermion
operators in Eq. (2) and conditions on the relevant charges, these sphaleron
processes yield a system that can be solved for the relic density of
$X$-charged states after condensation (see the Supplementary Material).

If all three families of SM fermions are gauged under $\mbox{SU}(2)_{\rm R}$
and all the operators, including both sphalerons, are in equilibrium, then the
total $\mbox{U}(1)_{X}$ asymmetry is zero. However, due to the high scale of
$f_{L,R}$ not all families are in equilibrium. A fermion $\psi$, receiving its
mass $m_{\psi}$ from a Yukawa interaction generated by a 6-fermion operator
such as Eq. (2), is in equilibrium at a temperature $T$ if Davidson:2008bu ;
Deppisch:2017ecm

$\displaystyle 2.3\times 10^{4}\times\sqrt{\frac{10\text{
TeV}}{f}}\frac{m_{\psi}}{v_{\rm EW}}\gtrsim\left(\frac{f}{T}\right)^{9/2}.$
(7)

The system will have non-trivial solutions if the 6-fermion operators are
inefficient for at least one charged fermion but efficient for at least one
other charged fermion. By evaluating Eq. (7) at the condensation temperature
$T=f$, until which expect the sphalerons to be active, we find this to be
realised for 20 GeV $\lesssim f\lesssim 3\times 10^{12}$ GeV.

In this window, 6-fermion operators are inefficient for at least the electron
but efficient for at least the top quark. Solving the set of equilibrium under
these conditions we find, for both first- and second-order phase transitions,
the following ratio:

$\displaystyle\begin{aligned}
\left|\frac{X}{B}\right|=&2\left(3+\frac{L}{B}\right),\end{aligned}$ (8)

where $X$, $B$ and $L$ are the $X$-charge, lepton, baryon number densities
respectively, and where we leave $L$ and $B$ as free parameters in this work.
For second-order phase transitions, Eq. (8) applies even when all 6-fermion
operators are inefficient. For first-order phase transitions, the system is
under-constrained in absence of efficient 6-fermion operators. The ADM relic
density can be expressed in terms of the charge densities as

$\displaystyle\frac{\Omega_{DM}}{\Omega_{B}}=\left|\frac{X}{B}\right|\frac{m_{DM}}{2m_{p}}\sigma\left(\frac{m_{DM}}{2T_{F}}\right),$
(9)

where $T_{F}$ is the temperature of the phase transition and $\sigma(x)$ is
the Boltzmann suppression factor Harvey:1990qw ; Ryttov:2008xe .

In absence of Boltzmann suppression ($m_{DM}\ll T_{F}$), the asymmetry sharing
naturally fixes the DM number density to the order of the baryonic number
density, so that $\Omega_{DM}/\Omega_{B}\sim\mathcal{O}(1)$ is found for
$m_{DM}\sim\mathcal{O}(m_{p})$. The correct relic density
$\Omega_{DM}/\Omega_{B}=5.36$ Planck:2018vyg , therefore, requires either a
small DM mass $m_{DM}\ll f_{R}$, a tuning in $|X/B|\ll 1$, or an exponential
suppression in the Boltzmann factor for $T_{F}\ll m_{DM}$. Another possibility
is to allow the decay of the heavy $X$-charged pNGB into another light stable
state Frandsen:2011kt .

To assure condition iv), the DM pNGB needs to efficiently annihilate to
suppress the thermal component of the relic density. For heavy DM candidates
($m_{DM}>m_{W_{\rm R}}=g_{\rm R}f_{R}/2$), the dominant annihilation channel
is in a pair of $\mbox{SU}(2)_{\rm R}$ gauge bosons, which has a cross section
of the form:

$\displaystyle\langle\sigma v\rangle_{W_{\rm R}}=$ $\displaystyle\frac{g_{\rm
R}^{4}m_{DM}^{2}}{32\pi m_{W_{\rm R}}^{4}}\sqrt{1-\frac{m_{W_{\rm
R}}^{2}}{m_{DM}^{2}}}$ (10)
$\displaystyle\quad\quad\times\left(1-\frac{m_{W_{\rm
R}}^{2}}{m_{DM}^{2}}+\frac{3}{4}\frac{m_{W_{\rm
R}}^{4}}{m_{DM}^{4}}\right)\,.$

This channel is effective in wiping out the thermal relic if $\langle\sigma
v\rangle_{W_{\rm R}}\gg 3\times 10^{-26}~{}\text{cm}^{3}\text{s}^{-1}$
Steigman:2012nb , implying

$m_{DM}\gg
0.073~{}\text{TeV}\times\left(\frac{f_{R}}{\text{TeV}}\right)^{2}\,,$ (11)

for $m_{DM}\gg m_{W_{\rm R}}$. For lighter DM masses, $m_{DM}<m_{W_{\rm R}}$,
the main annihilation mode involves a pair of pNGBs that do not carry
$X$-charges: by studying the potential, we found that the dominant channel
involves the $\mbox{U}(1)_{\Theta}$ pNGB $\Theta$, which can be parametrically
lighter than the other pNGBs Belyaev:2016ftv . The annihilation cross section
reads

$\displaystyle\langle\sigma
v\rangle_{\Theta}=\frac{\lambda_{XX\Theta\Theta}^{2}}{32\pi
m_{DM}^{2}}\sqrt{1-\frac{m_{\Theta}^{2}}{m_{DM}^{2}}}\,,$ (12)

where the quartic coupling is suppressed by the misalignment in the $L$-coset,
$\lambda_{XX\Theta\Theta}\sim\lambda_{0}(v_{\rm SM}/f_{L})^{2}$. This process
can wipe out the thermal density for

$m_{DM}\ll
0.21~{}\text{TeV}\times\left(\frac{\text{TeV}}{f_{L}}\right)^{2}\lambda_{0}\,,$
(13)

for $m_{\Theta}\ll m_{DM}$.

Figure 1: Constraints on $m_{DM}$ and the compositeness scale $f\equiv
f_{L}=f_{R}$. The red shaded region is excluded by thermal overproduction,
while an upper limit on $m_{DM}\lesssim 110$ TeV comes from unitarity bounds
Baldes:2017gzw . The low mass limit of the excluded region depends on
$\lambda_{0}$, and we show three sample values. A lower limit on $f$ comes
from direct searches of $W_{\rm R}$ at colliders, which crucially depends on
the value of $g_{\rm R}$ (we show three sample values: $g_{\rm R}=1,\ g_{\rm
L},\ g_{\rm Y}$, with the last being the minimum allowed value). The high
$m_{DM}$ edge of the red region has a mild dependence on $g_{\rm R}$,
illustrated by the thin orange band, moving to the right for larger $g_{\rm
R}$.

The strongest bound on the compositeness scale comes from direct searches at
colliders for $W_{\rm R}$, as this state can be produced via Drell-Yann if it
couples to the first generation. The most recent CMS bound from di-jet
resonant searches Sirunyan:2018mpc  reads $m_{W_{\rm R}}\gtrsim 5.2$ TeV,
which implies for $g_{\rm R}=g_{\rm L}$ a bound $f_{R}\gtrsim 16.4$ TeV. If
$f_{L}=f_{R}$, this bound also implies a bound on the misalignment in the EW
sector ($L$-coset), which we can best express in terms of the fine tuning
parameter $\xi=v_{\rm SM}^{2}/f_{L}^{2}$ Contino:2010rs : the $W_{\rm R}$
searches imply a bound $\xi\lesssim 2.25\cdot 10^{-4}$. This is much stronger
in this model than the bounds on Higgs compositeness from the Higgs couplings
to the SM particles set by the LHC data deBlas:2018tjm  ($\xi\lesssim 0.1$),
and the EW precision measurements Cacciapaglia:2020kgq  ($\xi\lesssim 0.04$).
On the contrary, as we expect $m_{DM}\sim f_{R}$, the values seem compatible
with the limit in Eq. (11). The fine-tuning of one part in $4\cdot 10^{3}$
should not discourage the study of this model, as it represents a huge
improvement over the fine-tuning in the SM, which for the Higgs mass amounts
to one part in $10^{34}$ against the Planck scale. As already discussed, due
to the large DM masses, additional tuning is needed in Eq. (9) to obtain the
correct relic density via the asymmetric production.

Another valid possibility consists in tuning the mass of the DM to be
$m_{DM}\ll f_{R}$. Assuming $|X/B|=\mathcal{O}(1)$ in Eq. (9) (and $T_{F}\gg
m_{DM}$), saturating the relic density would require $m_{DM}\approx 1$ GeV.
This low mass can be achieved by properly tuning some couplings in the
effective potential in Eq. (5). As the DM pNGB is nearly massless, this limit
is technically natural according to ’t Hooft naturalness principle
tHooft:1979rat  as it reveals the restoration of a global symmetry.

In summary, we have presented a family of Techni-Composite Higgs models which
employ new strong dynamics to produce both a pNGB Higgs and an asymmetric DM
candidate. The key novel ingredient is a Left-Right symmetry and the
contemporary presence of a Composite Higgs vacuum in the L-sector and a
Technicolor vacuum in the R-sector. An illustrative example of the allowed
parameter space is shown in Fig. 1. This shows model-independently the main
features of Techni-Composite models: a successful ADM relic density can be
obtained for masses above $10$ TeV or for masses around the GeV scale. This
family of models can thereby naturally explain the observed Higgs mass and DM
abundance with a minimum of tuning, although some tuning remains necessary to
ensure the correct vacuum alignment and DM mass.

## Acknowledgements

GC acknowledges partial support from the Labex-LIO (Lyon Institute of Origins)
under grant ANR-10-LABX-66 (Agence Nationale pour la Recherche), and FRAMA
(FR3127, Fédération de Recherche “André Marie Ampère”). MTF, WCH, and MR
acknowledge partial funding from The Council For Independent Research, grant
number DFF 6108-00623. The CP3-Origins centre is partially funded by the
Danish National Research Foundation, grant number DNRF90. PS acknowledges
support by the Deutsche Forschungsgemeinschaft under Germany’s Excellence
Strategy - EXC 2121 “Quantum Universe” - 390833306.

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Supplementary material

##  I Effective Lagrangian and vacuum alignment

We consider the scenario with $\mathcal{R}_{R}$ real and $\mathcal{R}_{L}$
pseudoreal of $G_{\rm HC}$, which applies to the model studied in the main
text. To describe the general vacuum alignment in the effective Lagrangian of
this scenario, we identify an $\mbox{SU}(2)_{\rm L}\times\mbox{U}(1)_{\rm
Y^{\prime}}$ subgroup in $\mbox{SU}(4)_{L}$ by the generators

$\displaystyle T_{L{\rm L}}^{i}$ $\displaystyle=\begin{pmatrix}\sigma_{i}&0\\\
0&0\end{pmatrix},\quad T_{L{\rm Y^{\prime}}}^{3}=\begin{pmatrix}0&0\\\
0&-\sigma_{3}^{T}\end{pmatrix},$ (E1)

and an $\mbox{SU}(2)_{\rm R}\times\mbox{U}(1)_{\rm Y^{\prime}}$ subgroup in
$\mbox{SU}(4)_{R}$ by the generators

$\displaystyle T_{R{\rm R}}^{i}$ $\displaystyle=\begin{pmatrix}\sigma_{i}&0\\\
0&0\end{pmatrix},\quad T_{R{\rm Y^{\prime}}}^{3}=\begin{pmatrix}0&0\\\
0&-\sigma_{3}^{T}\end{pmatrix},$ (E2)

where $\sigma_{i}$ are the Pauli matrices with $i=1,2,3$. Note that the $\rm
Y^{\prime}$-charge generator is identified as ${\rm Y^{\prime}}\equiv
T^{3}_{L{\rm Y^{\prime}}}+T^{3}_{R{\rm Y^{\prime}}}$, while the standard
hypercharge is given by ${\rm Y}\equiv T_{L{\rm Y^{\prime}}}^{3}+T_{R{\rm
R}}^{3}+T_{R{\rm Y^{\prime}}}^{3}$ after the breaking of $\mbox{SU}(2)_{\rm
R}$. Furthermore, $T_{L{\rm Y^{\prime}}}^{3}$ and $T_{R{\rm Y^{\prime}}}^{3}$
are part of global $\mbox{SU}(2)$ symmetries that define a custodial symmetry
in both cosets Georgi:1984af . This is required in the $L$-coset to reproduce
the correct $Z$/$W$ mass ratio.

The alignment between the extended EW subgroup $\mbox{SU}(2)_{\rm
L}\times\mbox{SU}(2)_{\rm R}\times\mbox{U}(1)_{\rm Y^{\prime}}$ and the
stability group $\mbox{Sp}(4)_{L}\times\mbox{SO}(4)_{R}$ can then be
conveniently parameterized by one misalignment angle, $\theta_{L}$. To do so,
we identify the vacua that leave the $\mbox{SU}(2)_{\rm
L}\times\mbox{U}(1)_{\rm Y}$ symmetry intact, $E_{L}^{\pm}$, and the one
breaking $\mbox{SU}(2)_{\rm L}\times\mbox{U}(1)_{\rm Y}$ to $\mbox{U}(1)_{\rm
EM}$, $E_{L}^{B}$. In the $R$-coset, there is no vacuum that preserves
$\mbox{SU}(2)_{\rm R}\times\mbox{U}(1)_{\rm Y^{\prime}}$, hence we define the
one breaking the gauge group to $\mbox{U}(1)_{\rm Y}$, $E_{R}^{B}$. They are
given in term of two-index $\mbox{SU}(4)$ matrices as:

$\displaystyle E_{L}^{\pm}$ $\displaystyle=\begin{pmatrix}i\sigma_{2}&0\\\
0&\pm i\sigma_{2}\end{pmatrix},\quad
E_{L}^{B}=\begin{pmatrix}0&\mathbb{1}_{2}\\\ -\mathbb{1}_{2}&0\end{pmatrix},$
(E3) $\displaystyle E_{R}^{B}$
$\displaystyle=\begin{pmatrix}0&\mathbb{1}_{2}\\\
\mathbb{1}_{2}&0\end{pmatrix}.$

Either choice of $E_{L}^{\pm}$ is equivalent Galloway:2010bp , and in this
letter we have chosen $E_{L}^{-}$. The true $\mbox{SU}(4)_{L}$ vacuum can be
written as a linear combination of the above vacua,
$E_{L}(\theta_{L})=c_{\theta_{L}}E_{L}^{-}+s_{\theta_{L}}E_{L}^{B}$ (a CH
vacuum), while the vacuum of the $\mathbb{S}_{R}$ sector is $E_{R}\equiv
E_{R}^{B}$ (a TC-like vacuum). We use the short-hand notations
$s_{x}\equiv\sin x$, $c_{x}\equiv\cos x$ and $t_{x}\equiv\tan x$ throughout.

The Goldstone excitations around the vacuum are then parameterized by

$\displaystyle\begin{split}\Sigma_{L}(x)&=\exp\left[2\sqrt{2}\,i\left(\frac{\Pi_{L}(x)}{f_{L}}+\frac{\Pi_{\Theta}(x)}{f_{\Theta}}\right)\right]E_{L}(\theta_{L}),\\\
\Sigma_{R}(x)&=\exp\left[2\sqrt{2}\,i\left(\frac{\Pi_{R}(x)}{f_{R}}-\frac{\Pi_{\Theta}(x)}{f_{\Theta}}\right)\right]E_{R},\end{split}$
(E4)

where the pion matrices $\Pi_{x}$ are defined as

$\displaystyle\begin{split}&\Pi_{L}(x)=\sum^{5}_{i=1}\Pi_{L}^{i}X_{L}^{i}\in
G_{L}/H_{L},\\\ &\Pi_{R}(x)=\sum^{9}_{a=1}\Pi_{R}^{a}X_{R}^{a}\in
G_{R}/H_{R},\\\ &\Pi_{\Theta}(x)=\Theta(x)\
\frac{\mathbb{1}_{4}}{4}\in\mbox{U}(1)_{\Theta}/\varnothing\,.\end{split}$
(E5)

The last one encodes the diagonal pNGB state $\Theta$ associated with the
broken global symmetry group $\mbox{U}(1)_{\Theta}$, acting on the two fermion
representations and having no gauge anomaly with the HC group. The matrices
$X_{L}^{i}$ are the $\theta_{L}$-dependent broken generators of
$\mbox{SU}(4)_{L}$, while $X_{R}^{a}$ are the broken generators of
$\mbox{SU}(4)_{R}$. While in general the decay constants are different, for
simplicity from now on we will assume $f\equiv f_{L,R}=f_{\Theta}$.

In the $\mathbb{S}_{L}$ sector, we identify the would-be Higgs boson as
$h\equiv\Pi_{L}^{4}\sim
c_{\theta_{L}}(\overline{U}U+\overline{D}D)+s_{\theta_{L}}{\rm Re}\
U^{T}\mathcal{C}D$ and the singlet pNGB as $\eta\equiv\Pi_{L}^{5}\sim{\rm Im}\
U^{T}\mathcal{C}D$, while the remaining three $\Pi_{L}^{1,2,3}$ are exact
Goldstones eaten by the massive $W^{\pm}$ and $Z$. In the $\mathbb{S}_{R}$
sector, we identify the complex isotriplet scalars,

$\displaystyle\begin{split}&\Pi_{CS}^{0}\equiv\frac{\Pi_{R}^{8}+i\Pi_{R}^{9}}{\sqrt{2}}\sim
C^{T}\mathcal{C}S,\\\
&\Pi_{CC}^{+}\equiv\frac{\Pi_{R}^{4}+i\Pi_{R}^{5}+\Pi_{R}^{6}+i\Pi_{R}^{7}}{2}\sim
C^{T}\mathcal{C}C,\\\
&\Pi_{SS}^{-}\equiv\frac{\Pi_{R}^{4}+i\Pi_{R}^{5}-\Pi_{R}^{6}-i\Pi_{R}^{7}}{2}\sim
S^{T}\mathcal{C}S,\end{split}$ (E6)

where $\Pi_{CS,\overline{CS}}^{0}$ is identified as the DM candidate. The
remaining three, $\Pi_{R}^{1,2,3}$, are exact Goldstones eaten by the massive
$W_{\rm R}^{\pm}$ and $Z_{\rm R}$. Note that, following Ref. Ryttov:2008xe ,
we have used Dirac spinors to indicate the hyperfermions, combining the
$\mbox{SU}(2)_{\rm L,R}$ doublet and singlet Weyl spinors.

Below the condensation scale $\Lambda_{\rm HC}\sim 4\pi f$, the effective
Lagrangian is given by

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=\mathcal{L}_{\rm
kin}-V_{\rm eff}.$ (E7)

The kinetic part of the Lagrangian is given by

$\displaystyle\mathcal{L}_{\mathrm{kin}}=\frac{f^{2}}{8}\text{Tr}[D_{\mu}\Sigma_{L}^{\dagger}D^{\mu}\Sigma_{L}]+\frac{f^{2}}{8}\text{Tr}[D_{\mu}\Sigma_{R}^{\dagger}D^{\mu}\Sigma_{R}].$
(E8)

with

$\displaystyle
D_{\mu}\Sigma_{L/R}=\partial_{\mu}\Sigma_{L/R}-i(G_{L/R,\mu}\Sigma_{L/R}+\Sigma_{L/R}G_{L/R,\mu}^{T}),$
$\displaystyle G_{L/R,\mu}=g_{\rm L/R}W_{{\rm L/R},\mu}^{i}T_{\rm
L/R}^{i}+g_{\rm Y^{\prime}}B_{\mu}^{\prime}T_{L{\rm Y^{\prime}}/R{\rm
Y^{\prime}}}^{3},$

where $g_{\rm L,R,Y^{\prime}}$ are the gauge couplings of $\mbox{SU}(2)_{\rm
L}$, $\mbox{SU}(2)_{\rm R}$ and $\mbox{U}(1)_{\rm Y^{\prime}}$, respectively.
Here the hypercharge coupling is given by $g_{\rm Y}^{-2}=g_{\rm
Y^{\prime}}^{-2}+g_{\rm R}^{-2}$. Henceforth, the minimal value $g_{\rm R}$
can acquire is $g_{\rm R}^{\rm min}=g_{\rm Y}$.

At leading order, each source of symmetry breaking contributes independently
to the effective potential in Eq. (E7):

$\displaystyle V_{\rm eff}\supset V_{\rm gauge}+V_{\rm top}+V_{\rm m}+V_{\rm
B}.$ (E9)

Here the EW gauge interactions in Eq. (E8) yield the gauge loop contributions
to the potential $V_{\rm eff}$:

$\displaystyle V_{\text{gauge}}=$ $\displaystyle-C_{L}f^{4}\left\\{g_{\rm
L}^{2}\sum_{i}\text{Tr}\left[T^{i}_{L{\rm L}}\Sigma_{L}(T^{i}_{L{\rm
L}}\Sigma_{L})^{*}\right]+g_{\rm Y^{\prime}}^{2}\text{Tr}\left[T_{L{\rm
Y^{\prime}}}^{3}\Sigma_{L}(T_{L{\rm
Y^{\prime}}}^{3}\Sigma_{L})^{*}\right]\right\\}$ (E10)
$\displaystyle+C_{R}f^{4}\left\\{g_{\rm
R}^{2}\sum_{i}\text{Tr}\left[T^{i}_{R{\rm R}}\Sigma_{R}(T^{i}_{R{\rm
R}}\Sigma_{R})^{*}\right]+g_{\rm Y^{\prime}}^{2}\text{Tr}\left[T_{R{\rm
Y^{\prime}}}^{3}\Sigma_{L}(T_{R{\rm
Y^{\prime}}}^{3}\Sigma_{L})^{*}\right]\right\\},$

where $C_{L,R}$ encode non-perturbative low energy constants. The top loop
contributions arising from the six-fermion operator

$\displaystyle\frac{\xi_{t}}{\Lambda_{t}^{5}}(Q_{L}^{T}P_{L}Q_{L})(Q_{R}^{T}P_{R}Q_{R})q_{{\rm
L},3}q_{{\rm R},3}+{\rm h.c.}$ (E11)

yield the effective potential contribution:

$\displaystyle V_{\rm top}$
$\displaystyle=-\frac{1}{4}C_{t}y_{t}^{2}f^{4}\Big{\\{}\left|\Tr[P_{L}^{1}\Sigma_{L}]\right|^{2}\left|\Tr[P_{R}^{2}\Sigma_{R}]\right|^{2}$
(E12)
$\displaystyle+\Tr[P_{L}^{1}\Sigma_{L}]\Tr[P_{R}^{2}\Sigma_{R}]\Tr[P_{L}^{2}\Sigma_{L}^{\dagger}]\Tr[P_{R}^{1}\Sigma_{R}^{\dagger}]$
$\displaystyle+(\text{L}\leftrightarrow\text{R})\Big{\\}},$

where $C_{t}$ is a non-perturbative coefficient for the top loop, $y_{t}$
(proportional to $(\Lambda_{\rm HC}/\Lambda_{t})^{5}\xi_{t}$) is identified by
the top Yukawa coupling and the projectors,

$\displaystyle(P_{L,R}^{1})_{ij}=\frac{1}{2}(\delta_{1i}\delta_{3j}\pm\delta_{3i}\delta_{1j}),\qquad(P_{L,R}^{2})_{ij}=\frac{1}{2}(\delta_{2i}\delta_{4j}\pm\delta_{4i}\delta_{2j}),$
(E13)

select the components of $Q_{L,R}^{T}Q_{L,R}$ that transform as doublets of
$\mbox{SU}(2)_{\rm L,R}$. Moreover, the explicit hyper-fermion bilinear mass
terms yield potential contributions:

$\displaystyle V_{\rm m}=2\pi Z_{L}\Tr[M_{L}\Sigma_{L}^{\dagger}]+\rm h.c.,$
(E14)

where $M_{L}={\rm diag}(m_{1}\epsilon,-m_{2}\epsilon)$ is the mass matrix of
the $\mathbb{S}_{L}$ hyper-fermions. Finally, it is relevant to consider
possible four-hyper-fermion operators in the $\mathbb{S}_{R}$ sector of the
form $Q_{R}Q_{R}Q_{R}Q_{R}$ which yield potential contributions Foadi:2007ue :

$\displaystyle V_{\rm B}$
$\displaystyle=C_{B}g_{B}^{2}f^{4}\Tr[B\Sigma_{R}^{\dagger}B\Sigma_{R}].$
(E15)

The coefficients $C_{L,R}$, $C_{t}$, $Z_{L}$ and $C_{B}$ in Eq. (E10)-(E15)
are $\mathcal{O}(1)$ form factors that can be computed on the lattice
Arthur:2016ozw . At leading order, the effective potential for the
misalignment angle is given by

$\displaystyle V_{\rm eff}^{0}=$ $\displaystyle-8\pi
f^{3}Z_{L}m_{UD}c_{\theta_{L}}-f^{4}C_{L}\widetilde{g}_{L}^{2}c_{\theta_{L}}^{2}-f^{4}C_{t}y_{t}^{2}s_{\theta_{L}}^{2},$
(E16)

where $m_{UD}\equiv m_{1}+m_{2}$ and $\widetilde{g}_{L}^{2}\equiv(3g_{\rm
L}^{2}+g_{\rm Y^{\prime}}^{2})/2$. By minimizing the above potential,
$\partial_{\theta_{L}}V_{\rm eff}^{0}=0$, we obtain

$\displaystyle c_{\theta_{L}}=\frac{4\pi
Z_{L}m_{UD}}{f(C_{t}y_{t}^{2}-C_{L}\widetilde{g}_{L}^{2})},$ (E17)

where the $G_{L}/H_{L}$ part of the vacuum is aligned in a composite Higgs
direction ($0<c_{\theta_{L}}<1$), while the $G_{R}/H_{R}$ part of the vacuum
remains in a TC direction. The conditions for this vacuum alignment to be a
stable minimum in the presence of these terms are
$\partial_{\theta_{L}\theta_{L}}V_{\rm eff}^{0}>0$ and that all the squared
masses of the pNGBs are positive. However, the condition
$\partial_{\theta_{L}\theta_{L}}V_{\rm eff}^{0}>0$ is fulfilled by requiring
$m_{h}^{2}>0$ due to the fact that $\partial_{\theta_{L}\theta_{L}}V_{\rm
eff}^{0}=f^{2}m_{h}^{2}$. Therefore, we need that all the pNGBs have positive
squared masses. The Higgs and $\eta$ masses are

$\displaystyle m_{h}^{2}=2(C_{t}y_{t}^{2}-C_{L}\widetilde{g}_{L}^{2})v_{\rm
EW}^{2},\quad\quad m_{\eta}^{2}=m_{h}^{2}/s_{\theta_{L}}^{2},$ (E18)

while the masses of the complex isotriplet composite scalars,
$\Pi_{CS,\overline{CS}}^{0},\Pi^{\pm}_{CC},\Pi_{SS}^{\pm}$, in the
$\mathbb{S}_{R}$ sector are

$\displaystyle
m_{\Pi_{CS}^{0}}^{2}=2(C_{B}g_{B}^{2}-C_{R}\widetilde{g}_{R}^{2}+C_{t}y_{t}^{2}s_{\theta_{L}}^{2})f^{2},$
(E19) $\displaystyle
m_{\Pi_{CC}^{\pm}}^{2}=m_{\Pi_{SS}^{\pm}}^{2}=m_{\Pi_{CS}^{0}}^{2}+2C_{R}g_{Y^{\prime}}^{2}f^{2},$

where $\widetilde{g}_{R}^{2}\equiv(-g_{\rm R}^{2}+g_{\rm Y^{\prime}}^{2})/2$.
To obtain a stable vacuum, we therefore need $m_{h}^{2}>0$ and
$m_{\Pi_{CS}^{0}}^{2}>0$. Following the common lore that the top loops tend to
break the electroweak symmetry while gauge loops preserve it, we assume that
the form factors $C_{L,R,t}>0$. Hence, the masses squared are always positive,
as long as the top loops dominate, as expected as the Yukawa coupling is
larger than the gauge ones. Furthermore, in the $R$-coset, we find that
$\widetilde{g}_{R}^{2}\geq 0$ for $g_{\rm R}\geq\sqrt{2}g_{\rm Y}$, while it
becomes positive for $g_{\rm Y}<g_{\rm R}<\sqrt{2}g_{\rm Y}$. In the latter
range, it tends to cancel the contribution of the top. As
$m_{\Pi_{CS}}^{0}=m_{DM}$, the magnitude of the Dark Matter candidate mass
crucially depends on the value of these coefficients. In general, the first
Eq. (E19) implies that the DM mass is of the order of the pNGB decays
constants, i.e. $m_{DM}\sim f$. Small masses, of the order of GeV, could be
obtained if a tuned cancellation is enacted. This could happen if the gauge
contribution is large and positive, for $g_{\rm R}\gtrsim g_{\rm Y}$, or for
$C_{B}<0$.

As discussed in the main text, for light DM mass the annihilation is
dominantly into the light pNGB associated to the $\mbox{U}(1)_{\Theta}$
symmetry. If either of the vector-like masses $m_{1}$ or $m_{2}$ are
vanishing, the pNGB $\Theta$ state mixes with the $\mathbb{S}_{L}$ pNGB $\eta$
state, resulting in that the mass eigenstate $\widetilde{\Theta}$, consisting
mostly of $\Theta$, is massless while $\widetilde{\eta}$ has a mass of order
$f$:

$\displaystyle m_{\widetilde{\Theta}}^{2}=0,\quad\quad
m_{\widetilde{\eta}}^{2}=\frac{1}{4}\frac{m_{h}^{2}}{s_{\theta_{L}}^{2}}(5+c_{2\theta_{L}})\approx\frac{3m_{h}^{2}}{2s_{\theta_{L}}^{2}}.$
(E20)

However, the $\widetilde{\Theta}$ state can achieve a small mass from its
mixing with a $\Theta^{\prime}$ state corresponding to the $\mbox{U}(1)$
symmetry which is quantum anomalous. In addition, the mass of
$\Theta^{\prime}$ is generated by instanton effects related to the
$\mbox{U}(1)$ anomaly Belyaev:2016ftv .

Thus, for DM masses below the $W_{\rm R}$ mass, the dominant annihilation
channel is $\Pi_{X}\overline{\Pi}_{X}\to\widetilde{\Theta}\widetilde{\Theta}$
with the coupling $\lambda_{XX\widetilde{\Theta}\widetilde{\Theta}}$ given by

$\displaystyle\mathcal{L}_{\rm
eff}\supset\lambda_{XX\widetilde{\Theta}\widetilde{\Theta}}\Pi_{CS}^{0}\Pi_{\overline{CS}}^{0}\widetilde{\Theta}\widetilde{\Theta}$
(E21)

with

$\displaystyle\lambda_{XX\widetilde{\Theta}\widetilde{\Theta}}=$
$\displaystyle
4C_{t}y_{t}^{2}s_{\theta_{L}}^{2}\frac{c_{\theta_{L}}}{\sqrt{2+c_{\theta_{L}}^{2}}}\approx\frac{4}{\sqrt{3}}C_{t}y_{t}^{2}\left(\frac{v_{\rm
EW}}{f}\right)^{2}\equiv\lambda_{0}\left(\frac{v_{\rm EW}}{f}\right)^{2}.$

This implies that $\lambda_{0}\sim\frac{4}{\sqrt{3}}C_{t}y_{t}^{2}$, which
could be an order 1 number.

In the scenario with $\mathcal{R}_{R}=F$ (fundamental, pseudo-real) of $G_{\rm
HC}=\mbox{Sp}(2N_{\rm HC})$, the DM candidate is identified by a complex
isosinglet, $\Pi_{CS,\overline{CS}}$, where its mass is given by

$\displaystyle
m_{\Pi_{CS}}^{2}=2(C_{B}g_{B}^{2}-C_{R}\widetilde{g}_{R}^{2}+C_{t}y_{t}^{2}s_{\theta_{L}}^{2})f^{2},$
(E22)

where $\widetilde{g}_{R}^{2}=(3g_{\rm R}^{2}+g_{\rm Y^{\prime}}^{2})/2$ which
is always positive. Due to the fact that $s_{\theta_{L}}\ll 1$,
$m_{\Pi_{CS}}^{2}$ is negative when $C_{B}g_{B}^{2}=0$ and therefore a small
DM mass, $m_{\Pi_{CS}}\ll f$, can be achieved by tuning $C_{B}g_{B}^{2}$ to a
certain value of order unity. Furthermore, in the scenarios with the top mass
arising from PC operators, the DM mass can also be tuned by the term
$C_{B}g_{B}^{2}$ to small values in both scenarios. These models are inspired
by the work in Ref. Ferretti:2013kya .

##  II Examples of theories featuring the Techni-Composite Higgs mechanism

In the main body of the article, we studied in detail one specific model based
on a gauge symmetry $\mbox{Sp}(2N)_{\rm HC}$ and with fermions in two
different representations. However, the same mechanism can be found in many
other models, with different possibilities for the $L$ and $R$-cosets, as
listed in Tables T1 and T2, respectively. Here we list the quantum numbers of
the HC fermions needed to obtain the minimal cosets, for the three classes of
HC representations: pseudo-real, real and complex.

| $G_{\rm HC}$ | $\mbox{SU}(2)_{\rm L}$ | $\mbox{SU}(2)_{\rm R}$ | $\mbox{U}(1)_{\rm Y^{\prime}}$ | $\mbox{U}(1)_{TB,L}$
---|---|---|---|---|---
pseudo-real
$(U,D)$ | $\mathcal{R}_{L}$ | $\Box$ | 1 | 0 | 0
$\widetilde{U}$ | $\mathcal{R}_{L}$ | 1 | 1 | $-1/2$ | 0
$\widetilde{D}$ | $\mathcal{R}_{L}$ | 1 | 1 | $+1/2$ | 0
complex
$(U,D)$ | $\mathcal{R}_{L}$ | $\Box$ | 1 | 0 | 1
$\widetilde{U}$ | $\mathcal{R}_{L}$ | 1 | 1 | $-1/2$ | 1
$\widetilde{D}$ | $\mathcal{R}_{L}$ | 1 | 1 | $+1/2$ | 1
$(U^{c},D^{c})$ | $\mathcal{R}_{L}^{\ast}$ | $\Box$ | 1 | 0 | -1
$\widetilde{U}^{c}$ | $\mathcal{R}_{L}^{\ast}$ | 1 | 1 | $+1/2$ | -1
$\widetilde{D}^{c}$ | $\mathcal{R}_{L}^{\ast}$ | 1 | 1 | $-1/2$ | -1
real
$(U,D)$ | $\mathcal{R}_{L}$ | $\Box$ | 1 | 1/2 | 0
$(\widetilde{U},\widetilde{D})$ | $\mathcal{R}_{L}$ | $\Box$ | 1 | -1/2 | 0
$X$ | $\mathcal{R}_{L}$ | 1 | 1 | 0 | 0
Table T1: Fermion field content and their charges for the minimal L-cosets. | $G_{\rm HC}$ | $\mbox{SU}(2)_{\rm L}$ | $\mbox{SU}(2)_{\rm R}$ | $\mbox{U}(1)_{\rm Y^{\prime}}$ | $\mbox{U}(1)_{X}$ | $\mbox{U}(1)_{TB,R}$
---|---|---|---|---|---|---
pseudo-real
$(C,S)$ | $\mathcal{R}_{R}$ | 1 | $\Box$ | 0 | $+1$ | 0
$\widetilde{C}$ | $\mathcal{R}_{R}$ | 1 | 1 | $-1/2$ | $-1$ | 0
$\widetilde{S}$ | $\mathcal{R}_{R}$ | 1 | 1 | $+1/2$ | $-1$ | 0
complex
$(C,S)$ | $\mathcal{R}_{R}$ | 1 | $\Box$ | 0 | $+1$ | 1
$\widetilde{C}$ | $\mathcal{R}_{R}^{\ast}$ | 1 | 1 | $-1/2$ | $-1$ | $-1$
$\widetilde{S}$ | $\mathcal{R}_{R}^{\ast}$ | 1 | 1 | $+1/2$ | $-1$ | $-1$
real
$(C,S)$ | $\mathcal{R}_{R}$ | 1 | $\Box$ | 0 | $+1$ | 0
$\widetilde{C}$ | $\mathcal{R}_{R}$ | 1 | 1 | $-1/2$ | $-1$ | 0
$\widetilde{S}$ | $\mathcal{R}_{R}$ | 1 | 1 | $+1/2$ | $-1$ | 0
Table T2: Fermion field content and their charges for the minimal R-cosets.

Finally, in Table T3 we provided some examples of gauge-fermion theories
generating various combinations for the $L$ and $R$-cosets. In the cases with
$\mathcal{R}_{L}=\mathcal{R}_{R}$, the global symmetry is extended to a single
simple group that contains the $L$ and $R$ sub-cosets

$G_{\rm HC}$ | $\mathcal{R}_{L}$ | dim($\mathcal{R}_{L}$) | $\mathcal{R}_{R}$ | dim($\mathcal{R}_{R}$) | Annotations
---|---|---|---|---|---
(
$\mbox{SU}(4)/\mbox{Sp}(4)_{L}\otimes\mbox{SU}(4)/\mbox{SO}(4)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing$
$\mbox{Sp}(2N)_{\rm HC}$ | $F$ | $2N$ | $A$ | $N(2N-1)-1$ |  $N$ odd to avoid Witten anomalies
$\mbox{Sp}(2N)_{\rm HC}$ | $F$ | $2N$ | $Adj$ | $N(2N+1)$ |  $N$ even to avoid Witten anomalies
(
$\mbox{SU}(4)^{2}/\mbox{SU}(4)_{L}\otimes\mbox{SU}(4)/\mbox{SO}(4)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing\otimes\mbox{U}(1)_{TB,L}$
$\mbox{SU}(4)_{\rm HC}$ | $F$ | $4$ | $A_{2}$ | $6$ | -
$\mbox{SO}(10)_{\rm HC}$ | $Spin$ | $16$ | $F$ | $10$ | -
(
$\mbox{SU}(4)^{2}/\mbox{SU}(4)_{L}\otimes\mbox{SU}(2)^{2}/\mbox{SU}(2)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing\otimes\mbox{U}(1)_{TB,L}\otimes\mbox{U}(1)_{TB,R}$
$\mbox{SU}(5)_{\rm HC}$ | $F$ | $5$ | $A_{2}$ | $10$ | -
(
$\mbox{SU}(5)/\mbox{SO}(5)_{L}\otimes\mbox{SU}(4)/\mbox{SO}(4)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing$
$\mbox{SO}(7)_{\rm HC}$ | $F$ | $7$ | $Spin$ | $8$ | -
(
$\mbox{SU}(5)/\mbox{SO}(5)_{L}\otimes\mbox{SU}(4)/\mbox{Sp}(4)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing$
$\mbox{Sp}(2N)_{\rm HC}$ | $A$ | $N(2N-1)-1$ | $F$ | $2N$ | $N\geq 2$
(
$\mbox{SU}(5)/\mbox{SO}(5)_{L}\otimes\mbox{SU}(2)^{2}/\mbox{SU}(2)_{R}\otimes\mbox{U}(1)_{\Theta}/\varnothing\otimes\mbox{U}(1)_{TB,R}$
$\mbox{SU}(4)_{\rm HC}$ | $A$ | $6$ | $F$ | $4$ | -
$\mbox{SO}(10)_{\rm HC}$ | $F$ | $10$ | $Spin$ | $16$ | -
( $\mbox{SU}(8)/\mbox{Sp}(8)$
$\mbox{Sp}(2N)_{\rm HC}$ | $F$ | $2N$ | $F$ | $2N$ | -
( $\mbox{SU}(9)/\mbox{SO}(9)$
$\mbox{SU}(4)_{\rm HC}$ | $A$ | $6$ | $A$ | $6$ | -
$\mbox{SO}(10)_{\rm HC}$ | $F$ | $10$ | $F$ | $10$ | -
( $\mbox{SU}(6)^{2}/\mbox{SU}(6)\otimes\mbox{U}(1)_{TB}$
$\mbox{SU}(N)_{\rm HC}$ | $F$ | $N$ | $F$ | $N$ | -
$\mbox{SU}(5)_{\rm HC}$ | $A$ | $10$ | $A$ | $10$ | -
Table T3:  Examples of gauge fermion theories leading to the Techni-Composite
Higgs models.

##  III Asymmetry sharing

We here in detail show how the asymmetric dark matter relic can be calculated
Harvey:1990qw ; Ryttov:2008xe . First, note that at high temperatures the
particle number density $n_{+}$ and the anti-particle number density $n_{-}$
of a given species are given by

$\displaystyle
n_{\pm}=g\int\frac{d^{3}k}{(2\pi)^{3}}\frac{g}{e^{(E\mp\mu)\beta}-\eta}\mbox{\quad
with\quad}\begin{aligned} \eta&=+1\text{ for bosons}\\\\[-4.30554pt]
\eta&=-1\text{ for fermions}\end{aligned}$ (E23)

where $g$ is the number of internal degrees of freedom, $\mu$ is the chemical
potential of the particle species and $\beta=1/T$ (with $k_{B}=1$). At the
freeze-out temperature of sphalerons, $T_{F}$, we have $\mu/T_{F}\ll 1$ such
that the difference in particle numbers of a given species is given by

$\displaystyle
n=n_{+}-n_{-}=gT_{F}^{3}\frac{\mu}{T_{F}}\frac{\sigma(m/T_{F})}{6},$ (E24)

which reveals that the chemical potentials are the relevant quantities. Here
the statistical suppression factor $\sigma$ is

$\displaystyle\sigma(m/T_{F})=\begin{cases}\frac{6}{4\pi^{2}}\int_{0}^{\infty}dxx^{2}\cosh[-2](\frac{1}{2}\sqrt{x^{2}+(m/T_{F})^{2}})&\text{for
fermions}\\\
\frac{6}{4\pi^{2}}\int_{0}^{\infty}dxx^{2}\sinh[-2](\frac{1}{2}\sqrt{x^{2}+(m/T_{F})^{2}})&\text{for
bosons.}\end{cases}$ (E25)

The statistical factor $\sigma(m/T_{F})$ is normalized such that

$\displaystyle\lim_{m/T_{F}\to 0}\sigma(m/T_{F})=\begin{cases}1&\text{for
fermions,}\\\ 2&\text{for bosons,}\end{cases}$ (E26)

while $\sigma(m/T_{F})\simeq 2(m/2\pi T_{F})^{3/2}e^{-m/T_{F}}$ for large
$m/T_{F}\gg 1$. In therms of chemical potentials, the ratio of DM and baryon
energy densities can be expressed as

$\displaystyle\frac{\Omega_{DM}}{\Omega_{B}}=\frac{\frac{1}{2}m_{DM}}{m_{P}}\left|\frac{\mu_{X}}{\mu_{B}}\right|\sigma\left(\frac{m_{DM}}{2T_{F}}\right),$
(E27)

where we neglected the $\sigma(m/2T_{F})$ for the SM fermions. The task is
then to calculate the total chemical potential $\mu_{X}$ of all fermions
charged under $\mbox{U}(1)_{X}$ and the total chemical potential of all
baryons $\mu_{B}$

To study the chemical potentials we adopt the we adopt the notation of Ryttov-
Sannino Ryttov:2008xe  and Harvey-Turner Harvey:1990qw . The equilibrium
condition from sphalerons, as written in this convention, was already given in
the main text. Additionally, the 6-fermion operators, given by Eq. (E11), lead
to equilibrium conditions of the form

$\displaystyle\begin{aligned} \mu_{UL}-\mu_{UR}+\mu_{SR}-\mu_{SL}-\mu_{u{\rm
L},i}+\mu_{u{\rm R},i}&=0,\\\ \mu_{UL}-\mu_{UR}+\mu_{SR}-\mu_{SL}+\mu_{d{\rm
L},i}-\mu_{d{\rm R},i}&=0,\\\ \mu_{UL}-\mu_{UR}+\mu_{SR}-\mu_{SL}-\mu_{\nu{\rm
L},i}+\mu_{\nu{\rm R},i}&=0,\\\ \mu_{UL}-\mu_{UR}+\mu_{SR}-\mu_{SL}+\mu_{e{\rm
L},i}-\mu_{e{\rm R},i}&=0,\end{aligned}$ (E28)

and

$\displaystyle\begin{aligned} \mu_{DL}-\mu_{DR}+\mu_{CR}-\mu_{CL}-\mu_{u{\rm
R},i}+\mu_{u{\rm L},i}&=0,\\\ \mu_{DL}-\mu_{DR}+\mu_{CR}-\mu_{CL}+\mu_{d{\rm
R},i}-\mu_{d{\rm L},i}&=0,\\\ \mu_{DL}-\mu_{DR}+\mu_{CR}-\mu_{CL}-\mu_{\nu{\rm
R},i}+\mu_{\nu{\rm L},i}&=0,\\\ \mu_{DL}-\mu_{DR}+\mu_{CR}-\mu_{CL}+\mu_{e{\rm
R},i}-\mu_{e{\rm L},i}&=0,\\\ \end{aligned}$ (E29)

where the index i refers to the three generations of SM fermions, each of
which has equilibrium conditions of this form. However, depending on the
temperature, only some generations of quarks and leptons have efficient
6-fermion operators. As was pointed out in the letter, a fermion $\psi$,
receiving its mass $m_{\psi}$ from such an interaction is in equilibrium at a
temperature $T$ if Davidson:2008bu ; Deppisch:2017ecm

$\displaystyle 2.3\times 10^{4}\times\sqrt{\frac{10\text{
TeV}}{f}}\frac{m_{\psi}}{v_{\rm EW}}\gtrsim\left(\frac{f}{T}\right)^{9/2}.$
(E30)

We assume that the conditions corresponding to (E28) and (E29) apply for any
fermions for which the condition (E30) is satisfied for all participating
species at the condensation temperature $T=f$.

Furthermore, thermal equilibrium in the electroweak interactions imply the
following conditions:

$\displaystyle\begin{aligned} \mu_{u{\rm L},i}&=\mu_{d{\rm L},i}+\mu_{W\rm
L},&\mu_{u{\rm R},i}&=\mu_{d{\rm R},i}+\mu_{W\rm R},\\\ \mu_{e{\rm
L},i}&=\mu_{\nu{\rm L},i}+\mu_{W\rm L},&\mu_{e{\rm R},i}&=\mu_{\nu{\rm
R},i}+\mu_{W\rm R},\\\ \mu_{DL}&=\mu_{UL}+\mu_{W\rm
L},&\mu_{SL}&=\mu_{CL}+\mu_{W\rm R}.\\\ \end{aligned}$ (E31)

The equilibrium is constrained by conditions on the net charge of plasma. To
quantify this, consider the eigenvalues of third the components of the
isospins of $\mbox{SU}(2)_{\rm L,R}$, which are

$\displaystyle T_{L\rm L}^{3}=$ $\displaystyle
3\times\sum_{i}\left(\frac{1}{2}\mu_{u{\rm L},i}-\frac{1}{2}\mu_{d{\rm
L},i}\right)+\sum_{i}\left(\frac{1}{2}\mu_{\nu{\rm L},i}-\frac{1}{2}\mu_{e{\rm
L},i}\right)-4\mu_{W\rm
L}+2\left(\frac{1}{2}\mu_{UL}-\frac{1}{2}\mu_{DL}\right),$ (E32)
$\displaystyle T_{R\rm R}^{3}=$ $\displaystyle
3\times\sum_{i}\left(\frac{1}{2}\mu_{u{\rm R},i}-\frac{1}{2}\mu_{d{\rm
R},i}\right)+\sum_{i}\left(\frac{1}{2}\mu_{\nu{\rm R},i}-\frac{1}{2}\mu_{e{\rm
R},i}\right)-4\mu_{W\rm
R}+2\left(\frac{1}{2}\mu_{CR}-\frac{1}{2}\mu_{SR}\right).$ (E33)

The overall electric charge, $Q=T_{L\rm L}^{3}+T_{R\rm R}^{3}+Y^{\prime}$, is
then

$\displaystyle Q=$ $\displaystyle 3\times\sum_{i}\left(\frac{2}{3}\mu_{u{\rm
L},i}+\frac{2}{3}\mu_{u{\rm R},i}-\frac{1}{3}\mu_{d{\rm
L},i}-\frac{1}{3}\mu_{d{\rm R},i}\right)+\sum_{i}\left(-1\mu_{e{\rm
L},i}-1\mu_{e{\rm R},i}+0\mu_{\nu{\rm L},i}+0\mu_{\nu{\rm R},i}\right)$
$\displaystyle+d(\mathcal{R}_{L})\left(\frac{1}{2}\mu_{UL}+\frac{1}{2}\mu_{UR}-\frac{1}{2}\mu_{DL}-\frac{1}{2}\mu_{DR}\right)+d(\mathcal{R}_{R})\left(\frac{1}{2}\mu_{CL}+\frac{1}{2}\mu_{CR}-\frac{1}{2}\mu_{SL}-\frac{1}{2}\mu_{SR}\right)$
(E34) $\displaystyle-4\mu_{W\rm L}-4\mu_{W\rm R}.$

The conditions imposed depend on the type of phase transition. Above the
transition, where $\mbox{SU}(2)_{\rm L,R}$ are good symmetries, $T_{{L\rm
L},{R\rm R}}^{3}$ must vanish. If the phase transition is sudden, i.e. first
order, then this property is assumed to be inherited by the relic such that
relevant condition is

$\displaystyle Q=0,\quad T_{L\rm L}^{3}=0,\quad T_{R\rm R}^{3}=0.$ (E35)

Below the transition, $\mbox{SU}(2)_{\rm L,R}$ are broken, such that $T_{{L\rm
L},{R\rm R}}^{3}$ need no longer vanish. If the phase transition is gradual,
i.e. second order, then this leads to violation of $T_{{L\rm L},{R\rm R}}^{3}$
conditions. Instead, the VEV drives the chemical potentials of neutral
condensates to zero, such that the relevant condition is

$\displaystyle\begin{gathered}Q=0,\quad\mu_{UL}-\mu_{UR}=0,\quad\mu_{CR}-\mu_{CL}=0.\end{gathered}$
(E37)

We are ultimately interested in comparing the total baryon and DM densities,
which depend on the total chemical potentials of the baryons and technibaryons
(charged under $\mbox{U}(1)_{X}$), which are

$\displaystyle\mu_{B}$ $\displaystyle=\sum_{i}\left(\mu_{u{\rm
L},i}+\mu_{u{\rm R},i}+\mu_{d{\rm L},i},+\mu_{d{\rm R},i}\right),$ (E38)
$\displaystyle\mu_{X}$
$\displaystyle=d(\mathcal{R}_{R})\left(\mu_{CR}+\mu_{CL}+\mu_{SL}+\mu_{SR}\right).$
(E39)

Solving all the above conditions for $\mu_{X}$ and $\mu_{B}$ we find

$\displaystyle\left|\frac{\mu_{X}}{\mu_{B}}\right|=2\left(3+\frac{L}{B}\right),$
(E40)

if 6-fermion operations are inefficient for at least one generation but
efficient for at least one other generation. Since the occupation numbers are
proportional to the chemical potentials, we can identify
$\left|\mu_{X}/\mu_{B}\right|=\left|X/B\right|$. If all generations are in
equilibrium, then the lepton number is constrained to $L=-3B$ so that
$\mu_{X}=0$. Furthermore, if 6-fermion operators are inefficient for all
generations, then $\mu_{X}$ is unconstrained in the case of first-order
transitions while Eq. (E40) applies for the case of second-order transitions.