# Magnon squeezing in conical spin spirals

D. Wuhrer<EMAIL_ADDRESS>Fachbereich Physik, Universität
Konstanz, D-78457 Konstanz, Germany  L. Rózsa<EMAIL_ADDRESS>Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany
Department of Theoretical Solid State Physics, Institute of Solid State
Physics and Optics, Wigner Research Centre for Physics, H-1525 Budapest,
Hungary  Department of Theoretical Physics, Budapest University of Technology
and Economics, H-1111 Budapest, Hungary  U. Nowak  Fachbereich Physik,
Universität Konstanz, D-78457 Konstanz, Germany  W. Belzig  Fachbereich
Physik, Universität Konstanz, D-78457 Konstanz, Germany

###### Abstract

We investigate squeezing of magnons in a conical spin spiral configuration. We
find that while the energy of magnons propagating along the $\boldsymbol{k}$
and the $-\boldsymbol{k}$ directions can be different due to the non-
reciprocal dispersion, these two modes are connected by the squeezing, hence
can be described by the same squeezing parameter. The squeezing parameter
diverges at the center of the Brillouin zone due to the translational
Goldstone mode of the system, but the squeezing also vanishes for certain wave
vectors. We discuss possible ways of detecting the squeezing.

##  I Introduction

Magnons are collective excitations of magnetically ordered systems, which may
be interpreted as a wave propagating through the material carrying spin
angular momentum and magnetic moment Dyson (1956); Kittel (2004); Nolting and
Ramakanth (2009). Due to their relatively low dissipation, magnons have risen
as possible candidates to process and transport information in computing
architectures Chumak  _et al._ (2015); Yuan  _et al._ (2022); Kamra  _et al._
(2020); et al. (2021).

Recently there is a growing interest in taking advantage of the quantum-
mechanical nature of magnons Yuan  _et al._ (2022). One of these properties is
squeezing, where the uncertainty in one observable is reduced at the cost of
an increased variance in the conjugated observable Kamra  _et al._ (2020). The
squeezing implies entanglement both in the ground state of the system Kamra
and Belzig (2016); Kamra  _et al._ (2019); Wuhrer  _et al._ (2022); Zou  _et
al._ (2020) as well as in the excited states carrying a non-integer multiple
of $\hbar$ as spin Kamra and Belzig (2016). This entanglement may be utilized
by coupling the magnetic system to quantum dots Skogvoll  _et al._ (2021); Zou
_et al._ (2022).

In contrast to squeezing in photonic systems which may be achieved under non-
equilibrium conditions Walls (1983); Wu  _et al._ (1986); Gerry and Knight
(2004), the squeezing of magnons is an intrinsic property. The degree of
squeezing in ferromagnets can already achieve relatively large values compared
to photonic systems Kamra and Belzig (2016), where it is caused by the
relatively weak magnetocrystalline and dipolar anisotropy terms. Squeezing is
further enhanced in antiferromagnets Kamra  _et al._ (2019); Wuhrer  _et al._
(2022), where the Heisenberg exchange interaction is responsible for the
squeezing, which is typically the largest magnetic energy scale. Magnon
squeezing has also been studied in two-sublattice ferrimagnets Kamra and
Belzig (2017), which interpolate between the ferromagnetic and
antiferromagnetic limits by tuning the magnitude of the magnetic moment on one
of the sublattices.

An alternative approach to transforming the parallel spin alignment in
ferromagnets to the antiparallel alignment in antiferromagnets is by
continuously increasing the angle between neighboring spins, leading to the
formation of a spin spiral. Spin spirals may be stabilized by the competition
between ferromagnetic and antiferromagnetic exchange interactions with
different neighbors, as is common in, e.g., the rare-earth metals Ho, Tb or Dy
Izyumov (1984). The Dzyaloshinsky–Moriya interaction (DMI) Dzyaloshinsky
(1958); Moriya (1960) is present in materials with broken inversion symmetry,
where it creates spin spirals with a preferred rotational sense. These have
been studied extensively over the last decades both in bulk samples, such as
the B20 class including FeCoSi, MnSi or Cu2OSeO3 Uchida  _et al._ (2006); Seki
_et al._ (2012); Bauer and Pfleiderer (2012), as well as in atomically thin
magnetic layers including Mn mono- and double-layers on W(110) Bode  _et al._
(2007); Yoshida  _et al._ (2012); von Bergmann  _et al._ (2014); Hasselberg
_et al._ (2015). A planar spin spiral state often transforms into a conical
spin spiral under the application of an external magnetic field, possessing a
finite net magnetic moment along the cone axis parallel to the field. The
external field may also be utilized to create magnetic domain walls or
skyrmions Bogdanov and Yablonskii (1989); Nagaosa and Tokura (2013), which
themselves have been suggested as suitable candidates for unconventional
information processing Hayashi  _et al._ (2008); Parkin  _et al._ (2008); Fert
_et al._ (2013). Magnon excitations of spin spirals and skyrmions have been
analyzed in the classical limit Garst  _et al._ (2017), for example from the
standpoint of the topology of the magnon band structure Weber  _et al._
(2022). Quantum effects and in particular squeezing in such magnetic
configurations seem to have eluded attention so far.

In this work, we investigate magnon squeezing in conical spin spirals
stabilized by frustrated Heisenberg interactions and DMI in the presence of an
external field. The system enables the analytical description of the magnon
dispersion Michael and Trimper (2010), providing a clear insight into the
entanglement between the modes. The considered system displays non-reciprocal
spin-wave propagation common in non-collinear spin structures Garst  _et al._
(2017), meaning that the frequency of magnons with opposite wave vectors
differs from each other. We find a high degree of squeezing which typically
increases with the angle between the spins when going from the ferromagnetic
toward the antiferromagnetic configuration. The squeezing parameter typically
decreases when moving away from the center of the Brillouin zone, but we find
certain curves along which it exactly vanishes. We establish that the non-
reciprocity of the magnon dispersion is not observed in the squeezing
parameter due to the entanglement between the $\boldsymbol{k}$ and
$-\boldsymbol{k}$ modes.

This work is structured as follows. In Sec. II we discuss the theory of
squeezing in a general spin model forming a conical spin spiral ground state
in the classical limit. We discuss in Sec. II.1 the properties of the ground
state. Sec. II.2 copes with the determination of the magnon dispersion.
Subsequently, in Sec. II.3, we calculate the squeezing parameter
$r_{\boldsymbol{k}}$ to quantify the degree of squeezing. Finally, in Sec.
III, as a specific example we discuss a two-dimensional square lattice magnet
on a substrate with nearest-neighbor (NN) and next-nearest-neighbor (NNN)
Heisenberg interaction and NN DMI.

##  II General spin spirals

We consider the Hamiltonian

$\displaystyle\hat{\mathcal{H}}=\hat{\mathcal{H}}_{\text{H}}+\hat{\mathcal{H}}_{\text{DMI}}+\hat{\mathcal{H}}_{\text{Z}}\,,$
(1)

where

$\displaystyle\hat{\mathcal{H}}_{\text{ex}}=\frac{1}{2}\sum_{\boldsymbol{R}_{i},\boldsymbol{R}_{j}}J_{ij}\boldsymbol{\hat{S}}_{i}\cdot\boldsymbol{\hat{S}}_{j}\,,$
(2)

with $J_{ij}$ being the strength of the symmetric Heisenberg exchange
interaction between spins at sites $\boldsymbol{R}_{i}$ and
$\boldsymbol{R}_{j}$. Note that with this sign convention, negative and
positive values of $J_{ij}$ denote ferromagnetic and antiferromagnetic
coupling, respectively. $\boldsymbol{\hat{S}}_{i}$ stands for the
dimensionless spin operator at site $\boldsymbol{R}_{i}$. The DMI is given by

$\displaystyle\begin{split}\hat{\mathcal{H}}_{\text{DMI}}&=\frac{1}{2}\sum_{\boldsymbol{R}_{i},\boldsymbol{R}_{j}}\boldsymbol{D}_{ij}\cdot\left(\boldsymbol{\hat{S}}_{i}\times\boldsymbol{\hat{S}}_{j}\right)\,.\end{split}$
(3)

$\boldsymbol{D}_{ij}$ expresses the strength of the antisymmetric exchange
between spins at sites $\boldsymbol{R}_{i}$ and $\boldsymbol{R}_{j}$. It is a
vector quantity whose direction strongly depends on the lattice symmetry as
shown by Moriya Moriya (1960). As an antisymmetric interaction, the DMI
changes sign under exchange of the positions
$\boldsymbol{D}_{ij}=-\boldsymbol{D}_{ji}$.

The last term in Eq. (1) reads

$\displaystyle\hat{\mathcal{H}}_{\text{Z}}$
$\displaystyle=-\mu_{\textrm{B}}gB_{\boldsymbol{n}}\sum_{\boldsymbol{R}_{i}}\boldsymbol{n}\boldsymbol{\hat{S}}_{i}\,,$
(4)

with $B_{\boldsymbol{n}}$ being the external magnetic field oriented along the
unit vector $\boldsymbol{n}$ in this model.

###  II.1 Classical ground state

As a starting point for the quantisation procedure, we determine the classical
spin configuration minimizing the energy. For this we substitute
$\boldsymbol{\hat{S}}_{i}$ by
$\boldsymbol{S}_{i}=\left<\boldsymbol{\hat{S}}_{i}\right>$ in the Hamiltonian,
with the ansatz

$\displaystyle\boldsymbol{S}_{i}=S$
$\displaystyle\left[\sin\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\sin(\vartheta)\boldsymbol{e}_{1}+\cos\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\sin(\vartheta)\boldsymbol{e}_{2}\right.$
$\displaystyle\left.+\cos(\vartheta)\boldsymbol{n}\right],$ (5)

where the unit vectors $\boldsymbol{e}_{1},\boldsymbol{e}_{2},$ and
$\boldsymbol{n}$ form a right-handed system. This expression describes a
harmonic conical spin spiral state with the axis of the cone along the
$\boldsymbol{n}$ direction, as illustrated in Fig. 1. Here $S$ is the spin
quantum number, $\vartheta$ is the opening angle of the cone and
$\boldsymbol{q}_{0}$ is the wave vector of the spin spiral. In Eq. (4) it was
assumed that the magnetic field is pointing along the cone axis
$\boldsymbol{n}$. Deviations from this would induce distortions in the spin
spiral compared to the harmonic form, making a general analytical treatment
impossible.

Figure 1: Illustration of the classical conical spin spiral state. The
configuration is the ground state of the two-dimensional square lattice model
discussed in Sec. III, with $J_{2}=D=0.2|J_{1}|$,
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0.025|J_{1}|$ and $\vartheta\approx
0.4\pi$. The color indicates the phase of the rotation
$\boldsymbol{r}_{i}\cdot\boldsymbol{q}_{0}$ around the cone opening direction
$\boldsymbol{n}$.

Substituting $\boldsymbol{S}_{i}$ into Eq. (1) we end up with the classical
energy

$\displaystyle\begin{split}E_{\textrm{cl}}=&\frac{SN}{2}\left[S\left(\cos^{2}(\vartheta)J_{0}+\sin^{2}(\vartheta)\left(J_{\boldsymbol{q}_{0}}+iD_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\right)\right.-\\\
&\left.\phantom{\frac{NS}{2}}-2\mu_{\textrm{B}}gB_{\boldsymbol{n}}\cos(\vartheta)\right]\,.\end{split}$
(6)

$J_{\boldsymbol{q}_{0}}$ and $D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$ are the
spatial Fourier transforms of the symmetric and antisymmetric exchange
interactions for wave vector $\boldsymbol{q}_{0}$,

$\displaystyle J_{\boldsymbol{q}_{0}}$
$\displaystyle=\sum_{\boldsymbol{\delta}}J_{\boldsymbol{\delta}}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{q}_{0}}\,,$
$\displaystyle\boldsymbol{D}_{\boldsymbol{q}_{0}}=\sum_{\boldsymbol{\delta}}\boldsymbol{D}_{\boldsymbol{\delta}}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{q}_{0}}\,,$
(7)

where $\boldsymbol{\delta}=\boldsymbol{R}_{i}-\boldsymbol{R}_{j}$ runs over
all possible lattice vectors.
$D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}=\boldsymbol{n}\cdot\boldsymbol{D}_{\boldsymbol{q}_{0}}$
is given by the projection of the Fourier transform of the DMI onto the cone
opening direction.

The exact form of Fourier transforms of the DMI and the Heisenberg interaction
depend on the symmetry of the considered lattice chosen. One should note that
since $J_{ij}$ is symmetric, $J_{-\boldsymbol{q}_{0}}=J_{\boldsymbol{q}_{0}}$
is always a real symmetric quantity, while the antisymmetry of
$\boldsymbol{D}_{ij}$ implies
$\boldsymbol{D}_{-\boldsymbol{q}_{0}}^{\boldsymbol{n}}=-\boldsymbol{D}_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$
to be purely imaginary, i.e.,
$i\boldsymbol{D}_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$ is real and
antisymmetric.

We need to minimize Eq. (6) with respect to the parameters $\vartheta$ and
$\boldsymbol{q}_{0}$. Minimizing the energy with respect to the opening angle
$\vartheta$ of the spin cone yields two possible solutions,

$\displaystyle\sin\vartheta=0\quad\text{and}\quad\cos{\vartheta}=\frac{\mu_{\textrm{B}}gB_{\boldsymbol{n}}}{A_{\boldsymbol{q}_{0}}}\,,$
(8)

where the latter depends on
$A_{\boldsymbol{q}_{0}}=S\left[J_{\boldsymbol{0}}-\left(J_{\boldsymbol{q}_{0}}+iD^{\boldsymbol{n}}_{\boldsymbol{q}_{0}}\right)\right]$
and therefore on $\boldsymbol{q}_{0}$. The minimum is given by
$\sin\vartheta=0$ for $A_{\boldsymbol{q}_{0}}<0$ or
$|\mu_{\textrm{B}}gB_{\boldsymbol{n}}|>|A_{\boldsymbol{q}_{0}}|$. In this
case, the spins align ferromagnetically along the magnetic field direction.
The second solution describes a spin spiral ground state, which is planar
($\vartheta=\pi/2$) for $B_{{\boldsymbol{n}}}=0$ and conical otherwise.
Increasing the magnetic field eventually closes the cone and forces the system
into the collinear configuration.

Minimizing the energy with respect to $\boldsymbol{q}_{0}$ is equivalent to
solving

$\displaystyle\partial_{\boldsymbol{q}_{0}}\left(J_{\boldsymbol{q}_{0}}+iD_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)$
$\displaystyle=0\,,$ (9)

and proving that the solution is a minimum. This condition implicitly depends
on the choice of the ${\boldsymbol{n}}$ direction through
$D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$. We use the following procedure for
the minimization: First, we determine $\boldsymbol{D}_{\boldsymbol{q}_{0}}$
based on the symmetries of the system Moriya (1960). Second, for each
$\boldsymbol{q}_{0}$ we set the ${\boldsymbol{n}}$ direction to be
antiparallel to $i\boldsymbol{D}_{\boldsymbol{q}_{0}}$, since this minimizes
$iD_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$ when $\boldsymbol{q}_{0}$ is fixed.
Third, we solve Eq. (9) to find the wave vector. Note that this establishes a
connection between the direction of $\boldsymbol{q}_{0}$ and the
${\boldsymbol{n}}$ direction. Since Eq. (9) is independent of the magnetic
field, this only means that the field must be oriented along the
${\boldsymbol{n}}$ direction determined from the minimization procedure.
Fourth, we calculate $\vartheta$ from Eq. (8). Note that the wave vector
specified by Eq. (9) does not depend on the opening angle $\vartheta$ assuming
that the configuration is a real spin spiral, i.e., $\sin\vartheta\neq 0$.

An example for determining the classical ground state will be discussed for a
specific system in Sec. III.

###  II.2 Magnon spectrum

We determine the single-particle excitations of the quantum-mechanical system
Eq. (1) by an expansion around the classical ground state. With the parameters
given by Eqs. (8) and (9) we define

$\displaystyle\boldsymbol{e}_{i,1}=\begin{pmatrix}\sin\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\cos(\vartheta)\\\
\cos\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\cos(\vartheta)\\\
-\sin(\vartheta)\end{pmatrix}\,,\boldsymbol{e}_{i,2}=\begin{pmatrix}-\cos\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\\\
\sin\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\\\
0\end{pmatrix}\,,$
$\displaystyle\boldsymbol{e}_{i,3}=\frac{1}{S}\boldsymbol{S}_{i}=\begin{pmatrix}\sin\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\sin(\vartheta)\\\
\cos\left(\boldsymbol{R}_{i}\cdot\boldsymbol{q}_{0}\right)\sin(\vartheta)\\\
\cos(\vartheta)\end{pmatrix}\,,$ (10)

expressed in the global basis
$\left\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{n}\right\\}$. The
vectors $\boldsymbol{e}_{i,1},\boldsymbol{e}_{i,2},\boldsymbol{e}_{i,3}$ form
the basis of the right-handed coordinate system for the spin at site
$\boldsymbol{R}_{i}$, with the spin along the $\boldsymbol{e}_{i,3}$ direction
in the classical ground state. To calculate the magnon spectrum, we apply the
Holstein–Primakoff transformation Holstein and Primakoff (1940) in its
linearised version,

$\displaystyle\hat{S}_{i,1}=\sqrt{\frac{S}{2}}\left(\hat{a}_{i}+\hat{a}^{\dagger}_{i}\right)\,,\qquad\hat{S}_{i,2}=-i\sqrt{\frac{S}{2}}\left(\hat{a}_{i}-\hat{a}^{\dagger}_{i}\right)\,,$
$\displaystyle\hat{S}_{i,3}=S-\hat{a}^{\dagger}_{i}\hat{a}^{\phantom{\dagger}}_{i}\,,$
(11)

with
$\hat{S}_{i,\alpha}=\boldsymbol{\hat{S}}_{i}\cdot\boldsymbol{e}_{i,\alpha}$
using the coordinate system defined in Eq. (10), and choosing the operators
corresponding to the right-handed ordering of the eigenvectors.
$\left[\hat{a}_{i},\hat{a}^{\dagger}_{j}\right]=\delta_{ij}$ are bosonic
creation and annihilation operators. This so-called linear spin-wave
approximation is an expansion of $\sqrt{1-\hat{a}^{\dagger}\hat{a}/(2S)}$ for
small magnon occupation numbers $\left<\hat{a}^{\dagger}\hat{a}\right>$
compared to the spin length $2S$. We want to emphasize that since the spin
waves are delocalized as will be seen below, this only means that the total
number of magnons must be smaller than the total spin
$\sum_{\boldsymbol{k}}\left<\hat{a}^{\dagger}_{\boldsymbol{k}}\hat{a}^{\phantom{\dagger}}_{\boldsymbol{k}}\right>\ll
2NS$ with $N$ being the number of lattice sites. Due to this, larger magnon
numbers are possible without violating the linear spin-wave condition.

After performing Fourier transformation in real space, the spin-wave
Hamiltonian takes the form

$\displaystyle\hat{\mathcal{H}}_{\textrm{SW}}=\sum_{\boldsymbol{k}\in\text{BZ}}\left[\omega_{\boldsymbol{k}}\hat{a}^{\dagger}_{\boldsymbol{k}}\hat{a}^{\phantom{\dagger}}_{\boldsymbol{k}}+\mu_{\boldsymbol{k}}\hat{a}_{\boldsymbol{k}}\hat{a}_{-\boldsymbol{k}}+\mu_{\boldsymbol{k}}^{\ast}\hat{a}^{\dagger}_{\boldsymbol{k}}\hat{a}^{\dagger}_{-\boldsymbol{k}}\right]\,$
(12)

where $\boldsymbol{k}$ runs over the Brillouin zone (BZ) of the system. Note
that the Fourier transformation can only be used to diagonalize the
Hamiltonian since the spin spiral is harmonic and can be described by a single
wave vector $\boldsymbol{q}_{0}$. For anharmonic spirals, the Brillouin zone
would have to be reduced based on the period of the spiral, and multiple bands
would appear.

The parameters in Eq. (12) are given by

$\displaystyle\begin{split}\mu_{\boldsymbol{k}}=&\frac{S}{4}s_{\vartheta}^{2}\left[J_{\boldsymbol{k}}-\frac{1}{2}\left(J_{\boldsymbol{k}-\boldsymbol{q}_{0}}+J_{\boldsymbol{k}+\boldsymbol{q}_{0}}\right)+\right.\\\
&\phantom{\frac{S^{2}}{2}s_{\vartheta}^{2}[J_{\boldsymbol{k}}}+\left.\frac{i}{2}\left(D_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}-D_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\right]\,,\end{split}$
(13)

and

$\displaystyle\omega_{\boldsymbol{k}}=$ $\displaystyle
2\mu_{\boldsymbol{k}}-S\left(J_{\boldsymbol{q}_{0}}+iD_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)+$
$\displaystyle+\frac{S}{2}\left(1-c_{\vartheta}\right)\left(J_{\boldsymbol{k}+\boldsymbol{q}_{0}}+iD_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)+$
(14)
$\displaystyle+\frac{S}{2}\left(1+c_{\vartheta}\right)\left(J_{\boldsymbol{k}-\boldsymbol{q}_{0}}-iD_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\,,$

where we introduced $\sin\vartheta=s_{\vartheta}$ and
$\cos\vartheta=c_{\vartheta}$ to shorten the expressions. In the determination
of $\mu_{\boldsymbol{k}}$ and $\omega_{\boldsymbol{k}}$ we used the second
solution in Eq. (8) and substituted the parts containing the magnetic field
through $A_{\boldsymbol{q}_{0}}$ and $\cos{(\vartheta)}$. In case of
$\sin(\vartheta)=0$ being the appropriate ground state, one would end up with
$\mu_{\boldsymbol{k}}=0$. This corresponds to the field-polarized
ferromagnetic case and would already diagonalise the Hamiltonian in the Fock
space for magnons with wave vector $\boldsymbol{k}$, with energies equal to

$\displaystyle\omega_{\boldsymbol{k}}=S(J_{\boldsymbol{k}}-iD_{\boldsymbol{k}}^{\boldsymbol{n}})-SJ_{\boldsymbol{0}}+\mu_{\textrm{B}}gB_{\boldsymbol{n}}S\,.$
(15)

It can be shown that $\mu_{-\boldsymbol{k}}=\mu_{\boldsymbol{k}}$ and
$\omega_{-\boldsymbol{k}}\neq\omega_{\boldsymbol{k}}$ by using

$\displaystyle\begin{split}J_{\boldsymbol{k}+\boldsymbol{q}}&\quad\overset{\boldsymbol{k}\to-\boldsymbol{k}}{\longrightarrow}\quad
J_{-\boldsymbol{k}+\boldsymbol{q}}=\phantom{-}J_{\boldsymbol{k}-\boldsymbol{q}}\,,\\\
D^{\boldsymbol{n}}_{\boldsymbol{k}+\boldsymbol{q}}&\quad\overset{\boldsymbol{k}\to-\boldsymbol{k}}{\longrightarrow}\quad
D^{\boldsymbol{n}}_{-\boldsymbol{k}+\boldsymbol{q}}=-D^{\boldsymbol{n}}_{\boldsymbol{k}-\boldsymbol{q}}\,.\end{split}$
(16)

The final step in the diagonalization procedure is to perform a Bogoliubov
transformation by introducing new bosonic operators
$\hat{\alpha}_{\boldsymbol{k}}$ which are connected to the original operators
$\hat{a}_{\boldsymbol{k}}$ by the Bogoliubov matrix

$\displaystyle\begin{pmatrix}\hat{a}_{\boldsymbol{k}}\\\
\hat{a}^{\dagger}_{-\boldsymbol{k}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}u_{\boldsymbol{k}}&v_{\boldsymbol{k}}\\\
v_{\boldsymbol{k}}^{\ast}&u_{\boldsymbol{k}}^{\ast}\end{pmatrix}\begin{pmatrix}\hat{\alpha}_{\boldsymbol{k}}\\\
\hat{\alpha}^{\dagger}_{-\boldsymbol{k}}\end{pmatrix}\,,$
$\displaystyle|u_{\boldsymbol{k}}|^{2}-|v_{\boldsymbol{k}}|^{2}$
$\displaystyle=1\,.$ (17)

After performing the Bogoliubov transformation, the Hamiltonian takes the form

$\displaystyle\hat{\mathcal{H}}=\sum_{\boldsymbol{k}\in\text{BZ}}\Omega_{\boldsymbol{k}}\hat{\alpha}^{\dagger}_{\boldsymbol{k}}\hat{\alpha}^{\phantom{\dagger}}_{\boldsymbol{k}}\,,$
(18)

which is diagonal in the Fock space of the new magnons
$\hat{\alpha}_{\boldsymbol{k}}$. The Bogoliubov parameters take the values

$\displaystyle u_{\boldsymbol{k}}$
$\displaystyle=\sqrt{\frac{\omega_{\boldsymbol{k}}+\Omega_{-\boldsymbol{k}}}{\Omega_{\boldsymbol{k}}+\Omega_{-\boldsymbol{k}}}}\,,$
$\displaystyle v_{\boldsymbol{k}}$
$\displaystyle=-e^{-i\varphi_{\mu_{\boldsymbol{k}}}}\sqrt{\frac{\omega_{\boldsymbol{k}}-\Omega_{\boldsymbol{k}}}{\Omega_{\boldsymbol{k}}+\Omega_{-\boldsymbol{k}}}}\,,$
(19)

with $\varphi_{\mu_{\boldsymbol{k}}}$ being the complex phase of
$\mu_{\boldsymbol{k}}$. The frequency of a magnon with wave vector
$\boldsymbol{k}$ is given by

$\displaystyle\Omega_{\boldsymbol{k}}$
$\displaystyle=\frac{1}{2}\left[\left(\omega_{\boldsymbol{k}}-\omega_{-\boldsymbol{k}}\right)+\sqrt{\left(\omega_{\boldsymbol{k}}+\omega_{-\boldsymbol{k}}\right)^{2}+4|\mu_{\boldsymbol{k}}|^{2}}\right]=Sc_{\vartheta}\left[J_{\boldsymbol{k}+\boldsymbol{q}_{0}}-J_{\boldsymbol{k}-\boldsymbol{q}_{0}}+i\left(D^{\boldsymbol{n}}_{\boldsymbol{k}+\boldsymbol{q}_{0}}+D^{\boldsymbol{n}}_{\boldsymbol{k}-\boldsymbol{q}_{0}}\right)\right]+$
$\displaystyle+S\sqrt{-\left(J_{\boldsymbol{q}_{0}}+iD^{\boldsymbol{n}}_{\boldsymbol{q}_{0}}\right)+\frac{1}{2}\left(J_{\boldsymbol{k}+\boldsymbol{q}_{0}}+J_{\boldsymbol{k}-\boldsymbol{q}_{0}}\right)+\frac{i}{2}\left(D_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}-D_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)}\times$
(20)
$\displaystyle\times\sqrt{s_{\vartheta}^{2}J_{\boldsymbol{k}}+\frac{c_{\vartheta}^{2}}{2}\left[J_{\boldsymbol{k}+\boldsymbol{q}_{0}}+J_{\boldsymbol{k}-\boldsymbol{q}_{0}}+i\left(D_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}-D_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\right]-\left(J_{\boldsymbol{q}_{0}}+iD^{\boldsymbol{n}}_{\boldsymbol{q}_{0}}\right)}\,.$

Using $\omega_{\boldsymbol{k}}\neq\omega_{-\boldsymbol{k}}$ it becomes clear
that $\Omega_{\boldsymbol{k}}\neq\Omega_{-\boldsymbol{k}}$ when the spin
spiral is not planar, i.e., for $\cos\vartheta\neq 0$. This is known as non-
reciprocal magnon propagation. While the non-reciprocal propagation is often
connected to the presence of the DMI, which is indeed required for it in the
ferromagnetic limit of Eq. (15), in this case the symmetry between
$\boldsymbol{k}$ and $\boldsymbol{-k}$ magnons is broken by the rotational
sense of the spin spiral characterized by $\boldsymbol{q}_{0}$, and the
presence of a finite net magnetic moment that breaks time-reversal symmetry.

The spin-wave dispersion in Eq. (20) is equivalent to the zero-temperature
limit of the dispersion relation derived in Ref. Michael and Trimper (2010)
for $S=1/2$.

###  II.3 Squeezing parameters

Figure 2: Variances of the spin components orthogonal to the ground state
direction. (a) Ferromagnetic ordering with no squeezing in the variances of
the $x$ and $y$ component of the spins. Variances at different sites lie in
the same plane. (b) Spin spiral ordering with squeezed variances in orthogonal
directions. Variances lie in different planes depending on the site and the
corresponding local frame Eq. (10).

The Bogoliubov transformation in Eq. (17) connects the magnon operators with
wave vectors $\boldsymbol{k}$ and $-\boldsymbol{k}$. This is analogous to the
situation in ferromagnets Kamra and Belzig (2016) and in antiferromagnets
Kamra  _et al._ (2019), which are included in the present formalism for spin-
spiral wave vectors $\boldsymbol{q}_{0}=\boldsymbol{0}$ and
$\boldsymbol{q}_{0}=\boldsymbol{b}/2$, where $\boldsymbol{b}$ is a basis
vector of the atomic reciprocal lattice, respectively. Squeezing between
$\boldsymbol{k}$ and $-\boldsymbol{k}$ is described by the two-mode-squeezing
operator

$\displaystyle\hat{S}_{2}(r_{\boldsymbol{k}})=\exp\left(r_{\boldsymbol{k}}^{\ast}\hat{a}_{\boldsymbol{k}}\hat{a}_{-\boldsymbol{k}}-r_{\boldsymbol{k}}\hat{a}^{\dagger}_{\boldsymbol{k}}\hat{a}^{\dagger}_{-\boldsymbol{k}}\right)\,.$
(21)

The degree of squeezing in a system can be given in terms of the absolute
value of the complex squeezing parameter $r_{\boldsymbol{k}}$. In a biaxial
ferromagnet, squeezing can be imagined as a reduction of the standard
deviation of the spin component along the hard axis and a simultaneous
increase of the standard deviation of the spin component along the
intermediate axis. The product of the standard deviations, as limited by the
Heisenberg uncertainty principle, remains conserved. In a conical spin spiral
the squeezing also describes the different values of the standard deviations
of the spin components perpendicular to the equilibrium direction, but the
situation is more complicated since these directions are defined in the local
coordinate system of Eq. (10) which changes from site to site, as can be seen
in Fig. 2. The phase $\varphi_{r_{\boldsymbol{k}}}$ of $r_{\boldsymbol{k}}$
determines along which direction the standard deviation is reduced or
enhanced. However, the value of the phase is not gauge-invariant, i.e., it
depends on the choice of $\boldsymbol{e}_{i,1}$ and $\boldsymbol{e}_{i,2}$ in
Eq. (10) which may be freely chosen as long as the right-handed orientation of
the frame is conserved.

The product of the squeezing operators over the $\boldsymbol{k}$ vectors in
half of the Brillouin zone
$\prod_{\boldsymbol{k}}\hat{S}_{2}(r_{\boldsymbol{k}})$ transforms the
classical vacuum state which is destroyed by each original magnon operator
$\hat{a}_{\boldsymbol{k}}$ to the approximate quantum or squeezed vacuum of
linear spin-wave theory which is destroyed by the magnon eigenstates
$\hat{\alpha}_{\boldsymbol{k}}$. Note that the squeezing operator creates
pairs of the original magnons with opposite wave vectors, similarly to Cooper
pairs in Bardeen–Cooper–Schrieffer theory; the difference is that higher
magnon occupations are also present in the squeezed vacuum due to the bosonic
commutation relations. Due to the construction of the bosonic Fock space, the
original magnon states are highly entangled in the squeezed vacuum Kamra  _et
al._ (2019).

The connection between the vacua implies that the transformation of the
operators has to follow

$\displaystyle\begin{split}\hat{\alpha}_{\boldsymbol{k}}&=\hat{S}_{2}(r_{\boldsymbol{k}})\hat{a}_{\boldsymbol{k}}\hat{S}^{-1}_{2}(r_{\boldsymbol{k}})=\\\
&=\cosh(|r_{\boldsymbol{k}}|)\hat{a}_{\boldsymbol{k}}+e^{i\varphi_{r_{\boldsymbol{k}}}}\sinh(|r_{\boldsymbol{k}}|)\hat{a}^{\dagger}_{-\boldsymbol{k}}\,.\end{split}$
(22)

The connection between the squeezing parameter and the matrix elements of the
Bogoliubov matrix in Eq. (17) is given by

$\displaystyle u_{\boldsymbol{k}}$
$\displaystyle=\cosh(|r_{\boldsymbol{k}}|)\,,$ $\displaystyle
v_{\boldsymbol{k}}$
$\displaystyle=-e^{i\varphi_{r_{\boldsymbol{k}}}}\sinh(|r_{\boldsymbol{k}}|)\,.$
(23)

Using this equation and Eq. (19), one can determine the squeezing parameter
for a mode with wave vector $\boldsymbol{k}$ in a conical spin spiral,

$\displaystyle\tanh(|r_{\boldsymbol{k}}|)$
$\displaystyle=\sqrt{\frac{\omega_{\boldsymbol{k}}-\Omega_{\boldsymbol{k}}}{\omega_{\boldsymbol{k}}+\Omega_{-\boldsymbol{k}}}}\,,$
$\displaystyle\varphi_{r_{\boldsymbol{k}}}$
$\displaystyle=-\varphi_{\mu_{\boldsymbol{k}}}\,,$ (24)

constituting the central result of this work.

Using Eq. (20) one can show that

$\displaystyle\omega_{\boldsymbol{k}}\mp\Omega_{\pm\boldsymbol{k}}$
$\displaystyle=\omega_{-\boldsymbol{k}}\mp\Omega_{\mp\boldsymbol{k}}\,.$ (25)

Together with Eq. (24) this leads to

$\displaystyle\tanh(|r_{-\boldsymbol{k}}|)$
$\displaystyle=\sqrt{\frac{\omega_{-\boldsymbol{k}}-\Omega_{-\boldsymbol{k}}}{\omega_{-\boldsymbol{k}}+\Omega_{\boldsymbol{k}}}}$
$\displaystyle\overset{\text{Eq.\eqref{eq:Omega_Condition}}}{=}\sqrt{\frac{\omega_{\boldsymbol{k}}-\Omega_{\boldsymbol{k}}}{\omega_{\boldsymbol{k}}+\Omega_{-\boldsymbol{k}}}}=$
$\displaystyle=\tanh(|r_{\boldsymbol{k}}|)\,,$ (26)

meaning that the squeezing parameter is invariant under wave-vector inversion
despite the non-reciprocal magnon propagation. In the present choice of gauge,
the phase $\varphi_{r_{\boldsymbol{k}}}=-\varphi_{\mu_{\boldsymbol{k}}}$ is
also invariant since $\mu_{\boldsymbol{k}}$ is even under inverting the wave
vector. This is explained by the fact that the two-mode-squeezing operator in
Eq. (21) assigns a single squeezing parameter to the pair of magnon operators
at wave vectors $\boldsymbol{k}$ and $-\boldsymbol{k}$. It is also worth
noting that the squeezing vanishes for $\mu_{\boldsymbol{k}}=0$, since this
leads to $\Omega_{\boldsymbol{k}}=\omega_{\boldsymbol{k}}$, for example in the
collinear polarized state with the spectrum given in Eq. (15).

It is also important to note that the squeezing parameter diverges as
$\boldsymbol{k}\to\boldsymbol{0}$, and is not defined for the uniform mode
$\boldsymbol{k}=\boldsymbol{0}$. For this mode
$\omega_{\boldsymbol{0}}=2\mu_{\boldsymbol{0}}$ and
$\Omega_{\boldsymbol{0}}=0$ which makes it impossible to diagonalize the
Hamiltonian using a Bogoliubov transformation fulfilling Eq. (17). In the
formalism of non-Hermitian eigenvalue equations, this is known as an
exceptional point; note that while the spin-wave Hamiltonian is Hermitian, the
equation of motion is enforced to be non-Hermitian by the bosonic commutation
relations Flynn  _et al._ (2020).

The divergence of the squeezing parameter at $\boldsymbol{k}=\boldsymbol{0}$
was also found in isotropic antiferromagnets in Ref. Kamra  _et al._ (2019),
and is connected to the Goldstone mode of the system. In a commensurate spin
structure such as a collinear antiferromagnet, the Goldstone mode related to
the global spin rotation may be lifted by an arbitrarily weak anisotropy term.
However, for the incommensurate spin spirals discussed here, the Goldstone
mode is related to the translation of the spiral along the wave vector
direction $\boldsymbol{q}_{0}$, and including an anisotropy term in the plane
of the spiral would only distort its shape but would not obstruct its
translation, unless the anisotropy term is strong enough to lock the spiral
into a commensurate state. This implies that the magnon modes with low wave
vectors in conical spin spirals are always very strongly squeezed.

Figure 3: Phase diagram of the classical ground state of a two-dimensional
square lattice magnet as a function of NNN Heisenberg coupling $J_{2}$ and DMI
$D$, displaying field-polarized (FP), conical spin spiral (SS) and
intermediate spin spiral (ISS) configurations with different orientations of
the wave vector $\boldsymbol{q}_{0}$. We fixed the field energy to be
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0.025|J_{1}|$. The red dot and the blue
triangle correspond to $J_{2}=0.2|J_{1}|$, $D=0.2|J_{1}|$, and
$J_{2}=0.4|J_{1}|$, $D=0.2|J_{1}|$, respectively. The black dashed line is the
analytic boundary of the FP state given by Eq. (30). Figure 4: Energy
dispersion, in units of $|J_{1}|$, for magnons with wave vector
$\boldsymbol{k}$ when the system is in the classical SS configuration along
$(1,0,0)$ with $\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0.025|J_{1}|$,
$J_{2}=0.4|J_{1}|$ and $D=0.2|J_{1}|$.

Since the squeezing parameter is related to, although not simply proportional
to, the parameter $\mu_{\boldsymbol{k}}$ in Eq. (13), analyzing this equation
will give a qualitative understanding of the squeezing parameter. The value of
$\mu_{\boldsymbol{k}}$ decreases as $\sin\vartheta$ is decreased, i.e., as the
external magnetic field closes the cone angle and drives the system into the
collinear state. For $\boldsymbol{q}_{0}=0$ describing a ferromagnetic
alignment, the squeezing vanishes as no anisotropy is present, as already
discussed in Ref. Kamra and Belzig (2016). With increasing values of
$\boldsymbol{q}_{0}$, $\mu_{\boldsymbol{k}}$ increases as the Heisenberg
interactions and the DMI start contributing to the squeezing. This is relevant
in particular for magnon wave vectors
$\lvert\boldsymbol{k}\rvert\ll\lvert\boldsymbol{q}_{0}\rvert$, and the
squeezing is expected to decrease for larger $\boldsymbol{k}$. However, apart
from this qualitative decrease, the condition $\mu_{\boldsymbol{k}}=0$
implying a vanishing squeezing parameter defines a subspace of codimension one
in reciprocal space, i.e., a curve in two dimensions and a surface in three
dimensions. Magnons with opposite wave vectors located on this subspace are
not present in the squeezed vacuum, and thus are exempt from the entanglement.

##  III Application to an example system

(a)

(b)

Figure 5: Magnon non-reciprocity
$\Delta\Omega_{\boldsymbol{k}}=\Omega_{\boldsymbol{k}}-\Omega_{-\boldsymbol{k}}$
for different directions of the spin spiral wave vector $\boldsymbol{q}_{0}$.
(a) SS along $(1,1,0)$ with $J_{2}=0.2|J_{1}|$ and $D=0.2|J_{1}|$. The
colorbar is capped at $|\Delta\Omega_{\boldsymbol{k}}|=0.3|J_{1}|$ with
maximal energy difference
$\Delta\Omega_{\boldsymbol{k}}^{\text{max}}=0.7|J_{1}|$. (b) SS along
$(1,0,0)$ with $J_{2}=0.4|J_{1}|$ and $D=0.2|J_{1}|$. The colorbar is capped
at $|\Delta\Omega_{\boldsymbol{k}}|=0.5|J_{1}|$ with maximal energy difference
$\Delta\Omega_{\boldsymbol{k}}^{\text{max}}=1.4|J_{1}|$. The arrows show the
direction of $\boldsymbol{q}_{0}$ for each spin spiral but not its proper
length. Both figures use $\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0.025|J_{1}|$
and $S=1$.

As an example system, we investigate a 2D square lattice lying in the $xy$
plane. We consider a NN ferromagnetic Heisenberg interaction $J_{1}<0$, and
express all other parameters in units of $\lvert J_{1}\rvert$. Besides the
field energy $\mu_{\textrm{B}}gB_{\boldsymbol{n}}$, we take into account NNN
Heisenberg interaction $J_{2}$ and NN DMI of strength $D$. We assume the
square lattice to be on a non-magnetic substrate with $C_{4\textrm{v}}$
symmetry. The substrate is necessary for breaking the inversion symmetry
required for a finite DMI. The interactions are sketched in Fig. 7 in Appendix
B.

Generally, both an antiferromagnetic NNN Heisenberg term $J_{2}>0$ and the DMI
$D$ could stabilize a spin spiral when competing with $J_{1}$. As will be
discussed below, in the present system only ferromagnetic and row-wise
antiferromagnetic states are stabilized in the absence of DMI. The $J_{2}$
term enables to tune the wave vector of the spin spiral in the whole range
between the ferromagnetic and the antiferromagnetic limits, while the DMI
would only be able to open a maximum of $\pi/2$ angle between neighbouring
spins. Another motivation for taking into account $J_{2}$ is that for only NN
Heisenberg and DMI terms, the magnon dispersion would be reciprocal in the
conical spin spiral state, $\omega_{\boldsymbol{k}}=\omega_{-\boldsymbol{k}}$
in Eq. (14). This situation has to be avoided since it would be related to the
choice of parameters, not to the symmetry of the system.

The Fourier transform of the interactions are given by

$\displaystyle
J_{\boldsymbol{k}}=2J_{1}\left[\cos(ak_{x})+\cos(ak_{y})\right]+4J_{2}\cos(ak_{x})\cos(ak_{y})\,,$
(27)

and

$\displaystyle\boldsymbol{D}_{\boldsymbol{k}}$
$\displaystyle=-2iD\begin{pmatrix}-\sin(ak_{y})\\\ \sin(ak_{x})\\\
0\end{pmatrix}\,,$ (28)

where $a$ is the lattice constant.

(a)

(b)

Figure 6: Absolute value of the squeezing parameter $r_{\boldsymbol{k}}$ for
different classical ground states. (a) SS along $(1,1,0)$ with
$J_{2}=0.2|J_{1}|$ and $D=0.2|J_{1}|$. The color scale is capped at
$|r_{\boldsymbol{k}}|=0.01$, such that finer details get revealed. (b) SS
along $(1,0,0)$ with $J_{2}=0.4|J_{1}|$ and $D=0.2|J_{1}|$. The color scale is
capped at $|r_{\boldsymbol{k}}|=0.1$. In both cases the squeezing diverges
when approaching the uniform $\boldsymbol{k}=0$ mode. Both figures use
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0.025|J_{1}|$.

As discussed in Sec. II.2, for each spiral wave vector $\boldsymbol{q}_{0}$ we
select the cone axis $\boldsymbol{n}$ to be antiparallel to
$\boldsymbol{D}_{\boldsymbol{q}_{0}}$, leading to

$\displaystyle D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$
$\displaystyle=+2i|D|\sqrt{\sin^{2}(aq_{0}^{y})+\sin^{2}(aq_{0}^{x})}\,.$ (29)

Note that the $\boldsymbol{n}$ direction always lies in the $xy$ plane.

With Eqs. (27) and (29) we can now determine the minima of the classical
energy using Eq. (9). An analytical calculation of the extrema detailed in
Appendix B yields four kinds of possible configurations, which are displayed
in the phase diagram in Fig. 3.

The first one is the field-polarized (FP) configuration, corresponding to a
collinear alignment of the spins along the magnetic field direction with
$\vartheta\in\\{0,\pi\\}$. This is the ground state if the magnetic field
becomes stronger than the interactions between the spins, represented by
$A_{\boldsymbol{q}_{0}}$ in Eq. (8), thereby closing the cone angle. The
boundary of the FP configuration in the phase diagram can be estimated for
small field strengths by

$\displaystyle\lvert
D\rvert\leq\sqrt{-\mu_{\textrm{B}}gB_{\boldsymbol{n}}(J_{1}+2J_{2})}\,$ (30)

for $J_{1}+2J_{2}<0$. This is denoted by the black dashed line in Fig. 3. As
expected, the field-polarized region becomes more extended as the magnetic
field is increased.

The other three phases correspond to conical spin spiral (SS) configurations
differing in the direction of the wave vector $\boldsymbol{q}_{0}$, on which
the NNN Heisenberg coupling $J_{2}$ has the strongest effect. We get
$\boldsymbol{q}_{0}\parallel\boldsymbol{e}_{x/y}$ for large values of
$J_{2}/\lvert J_{1}\rvert$. This corresponds to ferromagnetic rows along the
$\boldsymbol{e}_{y/x}$ direction while along the perpendicular direction the
spins are rotating. For small $J_{2}/\lvert J_{1}\rvert$ the wave vector lies
along the diagonal of the square lattice with $q_{0}^{x}=\pm q_{0}^{y}$. The
two regions are connected by the intermediate spin spiral (ISS) configuration
where $\boldsymbol{q}_{0}$ continuously rotates from the $(1,0,0)$ direction
along the axis towards the $(1,1,0)$ along the diagonal, given by the
expression

$\displaystyle\sqrt{\sin^{2}(aq_{0}^{x})+\sin^{2}(aq_{0}^{y})}=\frac{|D|}{2J_{2}}\,.$

This region becomes wider for higher $D$ values. Note that in all spin spiral
configurations, energetically equivalent states are found if
$\boldsymbol{q}_{0}$ and the cone axis direction are transformed by the
symmetries of the square lattice.

We also discuss the configurations along the line $D=0$. For
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}=0$, a ferromagnetic state
($\boldsymbol{q}_{0}=\boldsymbol{0}$) is formed for $J_{2}<\lvert
J_{1}\rvert/2$, and the classical ground state is row-wise antiferromagnetic
($\boldsymbol{q}_{0}=\left(\pi/a,0,0\right)$) for $J_{2}>\lvert
J_{1}\rvert/2$. At the point $J_{2}=\lvert J_{1}\rvert/2$, all spin spirals
with wave vectors along the $\left(1,0,0\right)$ direction are degenerate.
This demonstrates that the DMI is necessary for finding a unique spin spiral
ground state. Under applying a magnetic field, the ferromagnetic and the
antiferromagnetic configurations transform into a tilted collinear and a spin-
flop state, respectively. The magnetic field also selects the
$\boldsymbol{q}_{0}=\boldsymbol{0}$ wave vector as the energetically preferred
one at $J_{2}=\lvert J_{1}\rvert/2$. These configurations are difficult to
denote in the two-dimensional phase diagram since they are restricted to a
line. However, they can be included in the SS configuration with
$\boldsymbol{q}_{0}\parallel\left(1,0,0\right)$, for specific values of the
wave vector.

We investigate the magnon spectrum and the squeezing in two points of the
phase diagram, $J_{2}=0.2|J_{1}|$, $D=0.2|J_{1}|$ with $\vartheta\approx
0.4\pi$ and $a|\boldsymbol{q}_{0}|\approx 0.1\pi$ along $(1,1,0)$, and
$J_{2}=0.4|J_{1}|$, $D=0.2|J_{1}|$ with $\vartheta\approx 0.45\pi$ and
$a|\boldsymbol{q}_{0}|=\pi/4$ along $(1,0,0)$, denoted by a red dot and a blue
triangle in Fig. 3, respectively.

The magnon dispersion for the blue triangle in Fig. 3 is shown in Fig. 4,
while the magnon non-reciprocity given by

$\displaystyle\Delta\Omega_{\boldsymbol{k}}$
$\displaystyle=\left(\Omega_{\boldsymbol{k}}-\Omega_{-\boldsymbol{k}}\right)=$
$\displaystyle=2Sc_{\vartheta}\left[J_{\boldsymbol{k}+\boldsymbol{q}_{0}}-J_{\boldsymbol{k}-\boldsymbol{q}_{0}}+i\left(D_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}+D_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\right]\,$
(31)

is illustrated in Fig. 5(a) and (b) for the two states mentioned above.

The spectrum in Fig. 4 displays a Goldstone mode with zero frequency
$\boldsymbol{k}=\boldsymbol{0}$, which arises because translating the spin
spiral along $\boldsymbol{q}_{0}$ costs no energy, as discussed in Sec. II.3.
Overall, the magnon frequencies increase when moving away from the center of
the Brillouin zone, but there is an asymmetry between wave vectors
$\boldsymbol{k}$ and $-\boldsymbol{k}$ which is easier to see in Fig. 5. This
asymmetry reaches high values of the order of $\lvert J_{1}\rvert$ away from
the center of the Brillouin zone. Comparing Figs. 5(a) and (b), it can be
concluded that the energy difference completely vanishes orthogonal to
$\boldsymbol{q}_{0}$ as well as along the lattice vectors
$\boldsymbol{e}_{x/y}$. The disappearance of the non-reciprocity perpendicular
to $\boldsymbol{q}_{0}$ is enforced by the $C_{4\textrm{v}}$ symmetry of the
system when the spin spiral wave vector lies in a mirror plane. The vanishing
of $\Delta\Omega_{\boldsymbol{k}}$ along the main axes appears to be specific
to the choice of interaction parameters, and may be lifted if interactions
with further neighbors are taken into account. Even for a generic direction of
$\boldsymbol{q}_{0}$ (e.g., in the ISS) and for a spin model containing more
interaction terms, there must be a line crossing the whole Brillouin zone
along which the non-reciprocity vanishes, since
$\Delta\Omega_{\boldsymbol{k}}$ is continuous and by definition has to change
sign when reversing the wave vector direction. Overall, magnons travelling
along $\boldsymbol{q}_{0}$ opposed to along $-\boldsymbol{q}_{0}$ seem to have
a lower energy, therefore they are easier to excite.

The squeezing parameters from Eq. (24) are displayed in Fig. 6 for the two
considered spin spiral states. This displays the patterns discussed in Sec.
II.3. The squeezing, in contrast to the dispersion, is always symmetric
between $\boldsymbol{k}$ and $-\boldsymbol{k}$. In particular, the squeezing
parameter respects a $C_{2\textrm{v}}$ symmetry, as expected by reducing the
$C_{4\textrm{v}}$ symmetry by the direction of the spin spiral wave vector
which is located in a mirror plane. The squeezing parameter diverges as the
Goldstone mode $\boldsymbol{k}=\boldsymbol{0}$ is approached. It generally
decreases towards the boundary of the Brillouin zone, similarly to what was
observed for the antiferromagnetic configuration in Ref. Wuhrer  _et al._
(2022). However, at the brightest yellow curves inside the atomic Brillouin
zone the parameter $\mu_{\boldsymbol{k}}$ in Eq. (13), and consequently the
squeezing $r_{\boldsymbol{k}}$, vanish. For the wave vector along the
$\left(1,1,0\right)$ direction in Fig. 6(a), these curves are almost
perpendicular to $\boldsymbol{q}_{0}$. For
$\boldsymbol{q}_{0}\parallel\left(1,0,0\right)$ in Fig. 6(b), the curves with
vanishing squeezing are almost parallel to the $x$ and $y$ directions, which
are parallel and perpendicular to $\boldsymbol{q}_{0}$, respectively. Here,
the squeezing does vanish for certain wave vectors perpendicular to
$\boldsymbol{q}_{0}$. The distance of the curves with vanishing
$r_{\boldsymbol{k}}$ from the center of the Brillouin zone depends on the
magnitude of $\boldsymbol{q}_{0}$, but the quantitative relationship between
these quantities appears to be dependent on the model parameters.

##  IV Conclusion

We calculated the squeezing of magnons in a conical spin spiral state. Within
linear spin-wave theory, the ground state of the system may be described by a
vacuum where pairs of magnons with wave vectors $\boldsymbol{k}$ and
$-\boldsymbol{k}$ undergo squeezing. Although the spin spiral structure
together with a finite net magnetization leads to a non-reciprocal propagation
of magnons, the squeezing parameter is symmetric under reversing the direction
of the wave vector since it describes a pair of magnons with opposite wave
vectors.

The degree of squeezing in the spin spiral state interpolates between the
ferromagnetic limit, where it vanishes due to the absence of an anisotropy
term, to the antiferromagnetic limit, where it is exchange-dominated, by
changing the wave vector $\boldsymbol{q}_{0}$ of the spiral or by closing the
cone angle. The squeezing is found to diverge when approaching the
$\boldsymbol{k}=\boldsymbol{0}$ Goldstone mode, and this divergence is not
possible to be removed by magnetic anisotropy in incommensurate spin spiral
states, in contrast to the commensurate ferromagnetic or antiferromagnetic
states. The squeezing parameter is qualitatively found to decrease away from
the center of the Brillouin zone, but it exactly vanishes for wave vectors
located on certain curves in two-dimensional and on surfaces in three-
dimensional systems.

The results were illustrated on a two-dimensional square lattice taking
nearest-neighbor and next-nearest-neighbor Heisenberg and Dzyaloshinsky–Moriya
interactions into account. We identified the regime in the parameter space
where the conical spin spiral state is stable, and discussed how the preferred
orientation of the spin spiral wave vector is changing while varying the
interactions. The direction of the wave vector is found to be reflected in the
non-reciprocity of the magnon dispersion and the squeezing parameter in the
Brillouin zone.

The squeezing of magnons describes the decrease in the standard deviation of
one spin component at the cost of increasing the standard deviation in the
conjugate spin component in the plane perpendicular to the magnetization
direction in the classical ground state. As described in Ref. Udvardi  _et
al._ (2003), this effect is intrinsically related to the classical concept of
elliptic spin-wave polarization, where the spins precess on an elliptic path
around their equilibrium direction. The analogy relies on the equivalence of
the calculation of magnon frequencies and eigenvectors in quantum and
classical linear spin-wave theory. This makes it possible to assess certain
signatures of squeezing in classical observables Kamra  _et al._ (2019): for
example, the elliptic polarization of spin waves is reflected in their
linewidth in resonance experiments Rózsa  _et al._ (2018), while the different
components of transversal spin correlations are accessible in spin-polarized
electron and neutron scattering experiments. Importantly, the squeezing also
leads to a decrease in the longitudinal spin component in the quantum limit,
which is not observed in the classical case. This enables its detection
through longitudinal spin oscillations, for example via the measurement of the
light polarization rotation after the optical excitation of magnon pairs with
opposite wave vectors in Ref. Bossini  _et al._ (2019). The creation of such
magnon pairs in the conical spin spiral states discussed here would be
particularly intriguing, since the magnons travelling along opposite
directions possess different frequencies due to the non-reciprocity, which
might be used to spatially separate them to take advantage of the entanglement
encoded in their common squeezing parameter.

## Acknowledgment

This work was financially supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) via the Collaborative Research Center SFB
1432 (project no. 425217212), by the National Research, Development, and
Innovation Office (NRDI) of Hungary under Project Nos. K131938 and FK142601,
by the Ministry of Culture and Innovation and the National Research,
Development and Innovation Office within the Quantum Information National
Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004), and by the Young
Scholar Fund at the University of Konstanz.

##  Appendix A Calculation of the magnon dispersion

Here, we give a short derivation of the energy dispersion given in Eq. (20).
We start from the Hamiltonian in Eq. (1), assume that the classical ground
state was already determined, and at each site we apply the local eigensystem
$(\boldsymbol{e}_{i,1},\boldsymbol{e}_{i,2},\boldsymbol{e}_{i,3})$ given in
Eq. (10). We define the scalar product of the local eigenvectors with the
global eigenvectors of the cone system
$(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{n})$ as
$e_{i,m}^{\alpha}=\boldsymbol{e}_{\alpha}\cdot\boldsymbol{e}_{i,m}$ with
$\alpha\in\\{1,2,\boldsymbol{n}\\}$, $m\in\\{1,2,3\\}$ and
$\boldsymbol{e}_{\boldsymbol{n}}=\boldsymbol{n}$.

Using this, we get the following equations for the different combinations of
spin products appearing in Eqs. (2)-(4):

$\displaystyle\boldsymbol{\hat{S}}_{i}\cdot\boldsymbol{\hat{S}}_{j}$
$\displaystyle=\sum_{m,l\in\\{1,2,3\\}}\left(\boldsymbol{e}_{i,m}\cdot\boldsymbol{e}_{j,l}\right)\hat{S}_{i,m}\hat{S}_{j,l}$
(32) $\displaystyle\hat{S}_{i}^{\alpha}\hat{S}_{j}^{\beta}$
$\displaystyle=\sum_{m,l\in\\{1,2,3\\}}e_{i,m}^{\alpha}e_{j,l}^{\beta}\hat{S}_{i,m}\hat{S}_{j,l}$
(33) $\displaystyle\hat{S}_{i}^{\alpha}$
$\displaystyle=\sum_{m\in\\{1,2,3\\}}e_{i,m}^{\alpha}\hat{S}_{i,m}\,.$ (34)

This enables rewriting the Hamiltonian in the following manner:

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\sum_{\boldsymbol{R}_{i},\boldsymbol{R}_{j}}\boldsymbol{\hat{S}}_{i}^{\top}\Xi_{ij}\boldsymbol{\hat{S}}_{j}+\sum_{\boldsymbol{R}_{i}}\boldsymbol{\xi}_{i}\boldsymbol{\hat{S}}_{i}\,,$
(35)

with

$\displaystyle\begin{split}\Xi_{ij}^{ml}=&\frac{1}{2}J_{ij}\left(\boldsymbol{e}_{i,m}\cdot\boldsymbol{e}_{j,l}\right)+\\\
+&\frac{1}{2}D^{\boldsymbol{n}}_{ij}\left(e_{i,m}^{1}e_{j,l}^{2}-e_{i,m}^{2}e_{j,l}^{1}\right)\,,\end{split}$
$\displaystyle i$ $\displaystyle\neq j$ (36) $\displaystyle\Xi_{ii}^{ml}=$
$\displaystyle\,0\,,$ (37) $\displaystyle\xi_{i}^{m\phantom{l}}=$
$\displaystyle-\mu_{\textrm{B}}gB_{\boldsymbol{n}}e_{i,m}^{\boldsymbol{n}}\,.$
(38)

We perform the Holstein–Primakoff transformation given by Eq. (11), resulting
in the linearized spin-wave Hamiltonian

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\sum_{\boldsymbol{R}_{i},\boldsymbol{R}_{j}}\hat{\boldsymbol{a}}_{i}^{\dagger}\chi_{ij}\hat{\boldsymbol{a}}_{j}\,,$
(39)

with $\hat{\boldsymbol{a}}_{i}=(\hat{a}_{i},\hat{a}_{i}^{\dagger})^{\top}$ and
where we dropped constant terms.

The components of the different $\chi_{ij}$ for $i\neq j$ are connected to the
$\Xi_{ij}$ and $\xi_{i}$ parameters as follows:

$\displaystyle\chi_{ij}^{a^{\dagger}a}$
$\displaystyle=\frac{S}{2}\left[\Xi_{ij}^{11}+\Xi_{ij}^{22}-i\left(\Xi_{ij}^{12}-\Xi_{ij}^{21}\right)\right]=$
(40) $\displaystyle=\left(\chi_{ij}^{aa^{\dagger}}\right)^{\ast}\,,$
$\displaystyle\chi_{ij}^{aa}$
$\displaystyle=\frac{S}{2}\left[\Xi_{ij}^{11}-\Xi_{ij}^{22}-i\left(\Xi_{ij}^{12}+\Xi_{ij}^{21}\right)\right]=$
(41) $\displaystyle=\left(\chi_{ij}^{a^{\dagger}a^{\dagger}}\right)^{\ast}\,,$

and for $i=j$ one obtains

$\displaystyle\chi_{ii}^{a^{\dagger}a}$
$\displaystyle=\frac{S}{2}\left[\Xi_{ii}^{11}+\Xi_{ii}^{22}-i\left(\Xi_{ii}^{12}-\Xi_{ii}^{21}\right)\right]-$
$\displaystyle\phantom{=}-S\sum_{j}\left(\Xi_{ij}^{33}+\Xi_{ji}^{33}\right)-2S\Xi_{ii}^{33}-\xi_{i}^{3}=$
$\displaystyle=-S\sum_{j}\left(\Xi_{ij}^{33}+\Xi_{ji}^{33}\right)-\xi_{i}^{3}\,,$
(42) $\displaystyle\chi_{ii}^{aa^{\dagger}}$
$\displaystyle=\frac{S}{2}\left[\Xi_{ii}^{11}+\Xi_{ii}^{22}+i\left(\Xi_{ii}^{12}-\Xi_{ii}^{21}\right)\right]=0\,,$
(43) $\displaystyle\chi_{ii}^{aa}$
$\displaystyle=\frac{S}{2}\left[\Xi_{ii}^{11}-\Xi_{ii}^{22}-i\left(\Xi_{ii}^{12}+\Xi_{ii}^{21}\right)\right]=0$
(44)

Since the system is expanded around the classical ground state, the terms
linear in the creation and annihilation operators must vanish.

In the next step, we perform Fourier transformation on the creation and
annihilation operators,

$\displaystyle\hat{a}_{\boldsymbol{k}}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{i}e^{+i\boldsymbol{k}\cdot\boldsymbol{R}_{i}}\hat{a}_{i}\,,$
$\displaystyle\hat{a}_{i}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{\boldsymbol{k}}e^{-i\boldsymbol{k}\cdot\boldsymbol{R}_{i}}\hat{a}_{\boldsymbol{k}}\,,$
(45)

which leads to the Hamiltonian in Eq. (12). The coefficients are then given by

$\displaystyle\mu_{\boldsymbol{k}}$
$\displaystyle=\chi_{ii}^{aa}+\sum_{\boldsymbol{\delta}}\chi_{i,i+\delta}^{aa}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{k}}\,,$
(46) $\displaystyle\omega_{\boldsymbol{k}}$
$\displaystyle=\chi_{ii}^{a^{\dagger}a}+\chi_{ii}^{aa^{\dagger}}+\sum_{\boldsymbol{\delta}}\left[\chi_{i,i+\delta}^{a^{\dagger}a}e^{i\boldsymbol{\delta}\cdot\boldsymbol{k}}+\chi_{i,i+\delta}^{aa^{\dagger}}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{k}}\right]\,,$
(47)

where $\boldsymbol{\delta}$ is a lattice vector. Using trigonometric
identities to express the products of the components of the eigenvectors
$\boldsymbol{e}_{i,m}$, Fourier transforms of the interaction coefficients
with wave vectors shifted by $\boldsymbol{q}_{0}$ appear in the Hamiltonian,

$\displaystyle J_{\boldsymbol{k}\mp\boldsymbol{q}_{0}}$
$\displaystyle=\sum_{\boldsymbol{\delta}}J_{\boldsymbol{\delta}}e^{\pm
i\boldsymbol{\delta}\cdot\boldsymbol{q}_{0}}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{k}}\,$
(48) $\displaystyle D^{\boldsymbol{n}}_{\boldsymbol{k}\mp\boldsymbol{q}_{0}}$
$\displaystyle=\sum_{\boldsymbol{\delta}}D^{\boldsymbol{n}}_{\boldsymbol{\delta}}e^{\pm
i\boldsymbol{\delta}\cdot\boldsymbol{q}_{0}}e^{-i\boldsymbol{\delta}\cdot\boldsymbol{k}}\,.$
(49)

Collecting the terms yields

$\displaystyle\omega_{\boldsymbol{k}}=S$
$\displaystyle\left[-c_{\vartheta}^{2}J_{0}-\frac{s_{\vartheta}^{2}}{2}\left(2J_{\boldsymbol{q}_{0}}-J_{\boldsymbol{k}}\right)-is_{\vartheta}^{2}D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}+\right.$
$\displaystyle+\frac{1}{4}\left(c_{\vartheta}-1\right)^{2}\left(J_{\boldsymbol{k}-\boldsymbol{q}_{0}}-iD_{\boldsymbol{k}-\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)+$
$\displaystyle\left.+\frac{1}{4}\left(c_{\vartheta}+1\right)^{2}\left(J_{\boldsymbol{k}+\boldsymbol{q}_{0}}+iD_{\boldsymbol{k}+\boldsymbol{q}_{0}}^{\boldsymbol{n}}\right)\right]+$
$\displaystyle\phantom{[}+\mu_{\textrm{B}}gB_{\boldsymbol{n}}Sc_{\vartheta}\,,$
(50)

and Eq. (13) for $\mu_{\boldsymbol{k}}$. Inserting the second equation of Eq.
(8) into Eq. (50) yields Eq. (14).

The magnon frequencies $\Omega_{\boldsymbol{k}}$ and the coefficients of the
Bogoliubov transformation in Eq. (19) are determined by solving the eigenvalue
problem of

$\displaystyle\begin{pmatrix}\omega_{\boldsymbol{k}}&\mu_{\boldsymbol{k}}^{\ast}\\\
-\mu_{\boldsymbol{k}}&-\omega_{-\boldsymbol{k}}\end{pmatrix}\begin{pmatrix}u_{\boldsymbol{k}}\\\
v_{\boldsymbol{k}}^{\ast}\end{pmatrix}=\Omega_{\boldsymbol{k}}\begin{pmatrix}u_{\boldsymbol{k}}\\\
v_{\boldsymbol{k}}^{\ast}\end{pmatrix}\,.$ (51)

##  Appendix B Calculation of the ground state

###  B.1 General interactions

Here, we derive Eqs. (8) and (9) as well as the direction of the cone axis
direction $\boldsymbol{n}$, which will be represented by the angle variable
$\boldsymbol{n}\cdot
i\boldsymbol{D}_{\boldsymbol{q}_{0}}=\lvert\boldsymbol{D}_{\boldsymbol{q}_{0}}\rvert\cos\gamma$.
We start from the classical energy given by Eq. (6) and define
$\tilde{E}_{\text{cl}}=2E_{\text{cl}}/(SN)$. This yields for the derivatives
with respect to $(\vartheta,\boldsymbol{q}_{0},\gamma)$

$\displaystyle\frac{\partial\tilde{E}_{\text{cl}}}{\partial\vartheta}$
$\displaystyle=-2\sin(\vartheta)\left[\cos(\vartheta)A_{\boldsymbol{q}_{0}}-\mu_{\textrm{B}}gB_{\boldsymbol{n}}\right]=0\,,$
(52) $\displaystyle\frac{\partial\tilde{E}_{\text{cl}}}{\partial
q_{0}^{\alpha}}$
$\displaystyle=\sin^{2}(\vartheta)\partial_{q_{0}^{\alpha}}\left(J_{\boldsymbol{q}_{0}}+iD^{\boldsymbol{n}}_{\boldsymbol{q}_{0}}\right)=0\,,$
(53)
$\displaystyle\begin{split}\frac{\partial\tilde{E}_{\text{cl}}}{\partial\gamma}&=i\sin^{2}(\vartheta)\partial_{\gamma}D^{\boldsymbol{n}}_{\boldsymbol{q}_{0}}\\\
&=-\sin^{2}(\vartheta)|\boldsymbol{D}_{\boldsymbol{q}_{0}}|\sin(\gamma)=0\,.\end{split}$
(54)

Equation (52) is fulfilled for the solutions given in Eq. (8). As already
mentioned in the main text, solving Eq. (53) requires knowing the specific
form of the Fourier transforms, which will be performed for the square lattice
in Appendix B.2. For the opening direction of the cone one obtains from Eq.
(54) that $\boldsymbol{n}$ should be either parallel or antiparallel to
$\boldsymbol{D}_{\boldsymbol{q}_{0}}$.

Finding the minima from the stationary points satisfying Eqs. (52)-(54) may be
performed by substituting them back in the energy expression, or by evaluating
the Hessian

$\displaystyle H$ $\displaystyle=\begin{pmatrix}H_{\vartheta}&0&0\\\
0&H_{\gamma}&0\\\ 0&0&H_{\boldsymbol{q}_{0}}\end{pmatrix}$ $\displaystyle
H_{\alpha}$
$\displaystyle=\left.\frac{\partial^{2}E_{\text{cl}}}{\partial^{2}\alpha}\right|_{\text{ex}}\,,$
(55)

where $H_{\boldsymbol{q}_{0}}$ is the $d\times d$ matrix containing the second
derivatives with respect to the components of $\boldsymbol{q}_{0}$, where $d$
is the dimension of the system. Note that the off-diagonal components of $H$
are generally non-zero, but they vanish in the stationary points. The second
derivatives with respect to $\vartheta$ and $\gamma$ decouple and therefore
need to be positive in the minimum.

For $\vartheta$ this yields

$\displaystyle\frac{\partial^{2}\tilde{E}_{\text{cl}}}{\partial\vartheta^{2}}$
$\displaystyle=2\left[\cos(\vartheta)\mu_{\textrm{B}}gB_{\boldsymbol{n}}-\left(\cos^{2}(\vartheta)-\sin^{2}(\vartheta)\right)A_{\boldsymbol{q}_{0}}\right]>0\,.$
(56)

This implies that $\vartheta\in\\{0,\pi\\}$ will only be a minimum if
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}>A_{\boldsymbol{q}_{0}}$ or
$\mu_{\textrm{B}}gB_{\boldsymbol{n}}<-A_{\boldsymbol{q}_{0}}$, and
$\cos(\vartheta)=\mu_{\textrm{B}}gB_{\boldsymbol{n}}/A_{\boldsymbol{q}_{0}}$
only for $A_{\boldsymbol{q}_{0}}>0$ and
$\mu_{B}g|B_{\mathrm{n}}|<|A_{\boldsymbol{q}_{0}}|$.

The direction of $\boldsymbol{n}$ is determined by

$\displaystyle\frac{\partial^{2}\tilde{E}_{\text{cl}}}{\partial\gamma^{2}}$
$\displaystyle=-\sin^{2}(\vartheta)|\boldsymbol{D}_{\boldsymbol{q}_{0}}|\cos(\gamma)>0\,,$
(57)

which together with Eq. (54) yields $\gamma=\pi$, and therefore

$\displaystyle\boldsymbol{n}=-\frac{i\boldsymbol{D}_{\boldsymbol{q}_{0}}}{|\boldsymbol{D}_{\boldsymbol{q}_{0}}|}\,.$
(58)

As mentioned in the main text, this condition holds for all wave vectors
$\boldsymbol{q}_{0}$, and the direction of the external field will be rotated
to agree with the $\boldsymbol{n}$ direction determined from this condition.
To determine $\boldsymbol{q}_{0}$ via solving Eq. (53) and calculating the
eigenvalues of $H_{\boldsymbol{q}_{0}}$, we need to specify the system and
determine $J_{\boldsymbol{q}_{0}}$ and
$\boldsymbol{D}_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$, which will be
performed for the square lattice next.

###  B.2 Square lattice

Figure 7: Interactions in the square-lattice model. The interaction $J_{1}$
acts between the middle site and the neighboring sites (green lines, parallel
to the $x$ ($y$) direction). The interaction $J_{2}$ acts between the middle
site and the next-neighboring sites (blue, diagonal lines). The
Dzyaloshinsky–Moriya vectors $\boldsymbol{D}$ (red arrows) between neighboring
atoms are perpendicular to the connecting lines in the $C_{4\textrm{v}}$
symmetry class. The magnetic field $\boldsymbol{B}$ (purple arrow) is
indicated in the upper left corner. Arrows inside the spheres illustrate the
spin spiral.

We regard the system described in Sec. III and shown in Fig. 7. The Fourier
transform of the Heisenberg exchange reads

$\displaystyle\begin{split}J_{\boldsymbol{k}}&=\phantom{+}2J_{1}\left[\cos{(ak_{x})}+\cos{(ak_{y})}\right]+\\\
&\phantom{=}+2J_{2}\left[\cos{(a(k_{x}+k_{y}))}+\cos{(a(k_{x}-k_{y}))}\right]\end{split}$
(59)
$\displaystyle\begin{split}&=\phantom{+}2J_{1}\left[\cos{(ak_{x})}+\cos{(ak_{y})}\right]+\\\
&\phantom{=}+4J_{2}\cos{(ak_{x})}\cos{(ak_{y})}\,.\end{split}$ (60)

For determining the directions of the Dzyaloshinsky–Moriya vectors, we assume
a square-lattice magnet on a substrate with $C_{4\textrm{v}}$ symmetry. The
substrate is necessary to break inversion symmetry between two spins,
otherwise no DMI would be present. Following the rules for the DMI direction
as listed by Moriya Moriya (1960), we conclude that $\boldsymbol{D}_{ij}$
points along the $x(y)$-direction if the vector connecting the two spins point
along the $y(x)$-direction, as can be seen in Fig. 7. Furthermore, using
$\boldsymbol{D}_{ij}=-\boldsymbol{D}_{ji}$ we get

$\displaystyle\boldsymbol{D}_{\boldsymbol{k}}$
$\displaystyle=2iD\left(\begin{pmatrix}-\sin{(ak_{y})}\\\ 0\\\
0\end{pmatrix}+\begin{pmatrix}0\\\ \sin{(ak_{x})}\\\ 0\end{pmatrix}\right)\,,$
(61) $\displaystyle D_{\boldsymbol{q}_{0}}^{\boldsymbol{n}}$
$\displaystyle=2i|D|\sqrt{\sin^{2}{(aq_{0}^{x})}+\sin^{2}{(aq_{0}^{y})}}\,.$
(62)

With the expressions for $J_{\boldsymbol{k}}$ and
$\boldsymbol{D}_{\boldsymbol{k}}$, we can calculate the derivative of the
classical energy Eq. (53) with respect to $q_{0}^{x}$,

$\displaystyle\frac{\partial\tilde{E}_{\text{cl}}}{\partial q_{0}^{x}}=-$
$\displaystyle
2a\sin^{2}(\vartheta)\sin(aq_{0}^{x})\Biggl{[}J_{1}+2J_{2}\cos(aq_{0}^{y})+$
$\displaystyle+\frac{|D|\cos(aq_{0}^{x})}{\sqrt{\sin^{2}(aq_{0}^{y})+\sin^{2}(aq_{0}^{x})}}\Biggr{]}=0\,.$
(63)

One gets a similar equation for the derivative with respect to $q_{0}^{y}$
with the $x$ and $y$ components of $\boldsymbol{q}_{0}$ exchanged.

From these equations we are able to derive the SS configuration with different
wave vectors discussed in Sec. III, while the FP configuration stems from the
first solution in Eq. (8). In case of vanishing DMI, the stationary points are
$q_{0}^{x}a\in\left\\{0,\pi\right\\}$ and
$q_{0}^{y}a\in\left\\{0,\pi\right\\}$, and the minimum is given by the
ferromagnetic state for $J_{2}/\lvert J_{1}\rvert<1/2$ and the row-wise
antiferromagnetic state for $J_{2}/\lvert J_{1}\rvert>1/2$. For $J_{2}/\lvert
J_{1}\rvert=1/2$, states with all values of $q_{0}^{x}$ are stationary for
$q_{0}^{y}=0$, and they are also energetically degenerate.

For a finite DMI, the SS along $(1,0,0)$ is a stationary state if
$\sin(aq_{0}^{i})=0$ with $i\in\\{x,y\\}$. For the other component of
$\boldsymbol{q}_{0}$, which we will call $q_{0}^{j}$, this would lead to

$\displaystyle
J_{1}+2r_{i}J_{2}+\frac{|D|\cos(aq_{0}^{j})}{|\sin(aq_{0}^{j})|}$
$\displaystyle=0\,,$ (64)

where $r_{i}=\pm 1$ if $q_{0}^{i}=0$ or $\pi$. Solving this equation yields
$q_{0}^{j}$.

If we rewrite the system of equations (63) by subtracting the derivatives with
respect to $q_{0}^{x}$ and $q_{0}^{y}$, we end up with

$\displaystyle
2J_{2}\left[\cos(aq_{0}^{x})-\cos(aq_{0}^{y})\right]-\frac{|D|\left[\cos(aq_{0}^{x})-\cos(aq_{0}^{y})\right]}{\sqrt{\sin^{2}(aq_{0}^{y})+\sin^{2}(aq_{0}^{x})}}$
$\displaystyle=0\,.$ (65)

This equation already yields the two other directions for the wave vector. For
$\cos(aq_{0}^{x})\neq\cos(aq_{0}^{y})$, $\boldsymbol{q_{0}}$ is between the
$(1,0,0)$ and $(1,1,0)$ directions, found as the solution of

$\displaystyle\sqrt{\sin^{2}(aq_{0}^{y})+\sin^{2}(aq_{0}^{x})}$
$\displaystyle=\frac{|D|}{2J_{2}}\,,$ (66)

together with

$\displaystyle J_{1}+2J_{2}\left[\cos(aq_{0}^{x})+\cos(aq_{0}^{y})\right]$
$\displaystyle=0\,,$ (67)

which is obtained when substituting the first equation into the sum over the
derivatives along $q_{0}^{x}$ and $q_{0}^{y}$. The two equations yield that
$0<-J_{1}/\left(2\sqrt{2}\right)<J_{2}$ and $0<\lvert
D\rvert/\left(2\sqrt{2}\right)<J_{2}$ are required for this solution to be
found.

Again regarding Eq. (65) and the second possible solution of $q_{0}^{x}=\pm
q_{0}^{y}=q_{0}$, which corresponds to a SS with wave vector along $(1,1,0)$,
we find the sum of the derivatives with respect to $q_{0}^{x}$ and $q_{0}^{y}$
to yield

$\displaystyle
J_{1}+2J_{2}\cos{(aq_{0})}+|D|\frac{\cos(aq_{0})}{|\sin{(aq_{0})}|}$
$\displaystyle=0\,.$ (68)

The case of $\sin(aq_{0})=0$ is excluded in all the above discussed cases as
it would correspond to the case of the vanishing DMI. Determining $q_{0}$ from
the above equation requires solving a quartic equation in $|\sin{(aq_{0})}|$,
which can be done analytically only in certain limits.

As mentioned in Appendix B.1, finding the global minimum can be performed by
comparing the energies of the different stationary points. Calculating the
eigenvalues of the Hessian can also be used to decide whether a certain
stationary point is a minimum, but this is omitted here since the expressions
are rather convoluted, it cannot be used to determine which of the local
minima is the global minimum, and it is known in advance that one of the
stationary points has to be the global minimum since the configuration space
is compact.

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