If the measurement outcome is $x_i$, then the updated state will obey $(-1)^{x_i}X_i\ket{\psi}=\ket{\psi}$, and we find that $(-1)^{x_i}X_i$ is a new generator of the stabilizer state with $N_x\rightarrow N_x+1$.
Then, to ensure all $g_j$ are commuting such that the state is a valid stabilizer state, we must, as described in appendix <ref>, first change the representation of the stabilizer state so only one $g_k$ is non-commuting, and then remove it from the set of generators.
The net effect is to replace the non commuting generator $g_k$ with the measurement operator $(-1)^{x_i}X_i$.
Such an update can only be performed while at the same time maintaining the CSS gauge if one of the $\{g^z_j\}$ generators is chosen to be replaced (See Appendix <ref> for why).
This results in $N_z\rightarrow N_z-1$, and reflects the fact, that after the measurement, the qubit $i$ is in a superposition state of $Z_i$ basis states such that the coherence in the $Z$ basis, $C_z=L-N_z$, has increased by one bit.
Overall, we find that the $X_i$ measurement on a maximally entangled state increases the number of bits known about the $X$ basis states by 1 and decreases the number bits known about the $Z$ basis states by $1$: $N_x\rightarrow N_x+1$ and $N_z\rightarrow N_z-1$.
The effects of $Y_i$ and $Z_i$ measurements are determined similarly (See Appendix <ref>), and we find that the non-zero conditional probabilities for the Markov chain are given as:

\begin{eqnarray}\label{eq:pm0probs}
P(N_x+1,N_z-1|N_x, N_z)=p_x \\ \nonumber
P(N_x-1,N_z+1|N_x, N_z)=p_z \\ \nonumber
P(N_x-1,N_z-1|N_x, N_z)=p_y
\end{eqnarray}

for $N_x$ and $N_z$ away from the Markov chain's boundaries: $N_x\geq 0$, $N_z\geq 0$ and $N_x+N_z\leq L$.
The second boundary condition is because the number of generators allowed in the stabilizer state can not exceed $L$.
The transition rates at the boundary are similarly determined and are shown in Fig. <ref>.
We can therefore consider the dynamics of $(N_x,N_z)$ as a two dimensional random walk and derive the diffusion equation for the evolution of $P(N_x,N_z,m)=P(x,z,m)$ as

\begin{eqnarray}
\partial_m P = (p_y+p_z-p_x)\partial_x          &P& \\ \nonumber
+ (p_y+p_x-p_z)\partial_z          &P& \\ \nonumber
+ 1/2(\partial_x^2+\partial_z^2)      &P& \\ \nonumber
- (p_x+p_z-p_y)\partial_x\partial_y&P&
\end{eqnarray}

which has drift velocity $(p_y+p_z-p_x)\hat{x}+(p_y+p_x-p_z)\hat{z}$ leading to the rate Eq. <ref> discussed in the main text.
Notice that special care must be taken on the $N_z=0$ (and $N_x=0$ by symmetry) boundaries.
At $N_z=0$ boundary the conditional probabilities are given as $P(N_x+1,0|N_x, 0)=p_x $ and $P(N_x-1,0|N_x, 0)=p_y$ and result in the diffusion equation

\begin{eqnarray*}
\partial_m P(z=0) = \left(\left(p_y+p_z-p_x\right)\partial_x  + \frac{1}{2}\partial_x^2\right)     P(z=0).
\end{eqnarray*}

The steady state solution on this boundary gives the localization length $\lambda\sim 1/(p_y+p_z-p_x)$ in the $x$ direction as discussed in the main text.

§.§ Finite measurement rate dynamics

Steady state coherence $C_x$ as a function of $p_m$ and $\Delta_x=(p_x-p_z)$ for $p_y=p_R=p_e=0$.
The top figure shows volume-law coherence $C_x$ for $\Delta_x=0.5$ obtained via Clifford simulations. It shows no evidence of a phase with area-law coherence for finite $p_m$.
The bottom figure shows the steady state coherence $C_x$ for different values of $p_m$ and a system size $L=32$.
The colored dots are data computed via Clifford simulations, while the solid lines correspond to the predictions using the coherence rate equations derived in the text.  The three black lines correspond to the rate equation Eq. <ref> where the lengths scales $\xi=(2.7,5, 8)$ are found by best fit for the Clifford simulation data at $p_m=(0.06, 0.01, 0.0025$) respectively.

The two limits discussed above offer solutions for the steady state coherence in two extremes: 1) $p_m/p_u\rightarrow \infty$, in which the probability that a measurement of $X_i$ is uncertain depends on the number of bits known about the $X$ basis; and 2) $p_m/p_u\rightarrow 1/L^2$ in which the probability of an uncertain measurement outcome depends only on the rates $p_x$, $p_z$ and $p_y$.
When $p_x>p_y+p_z$ these two extremes are distinguished by volume-law v.s. area-law coherence in the $X$ basis.
This suggest the possibility of a coherence transition as a function of measurement rate $p_m$.
This possibility is ruled out by the Clifford simulations shown in the top panel of Fig. <ref>, which only shows volume-law coherence.
Thus, the weak measurement limit, $p_m<1/L^2$ is a finite size effect and does not exist as $L\rightarrow \infty$ for finite $p_m$.

To access the finite measurement rate limit, we first note that the two extremes discussed above correspond to two distinct structures of the generators in the late time stabilizer states: 1) when $p_m/p_u\rightarrow \infty$ and $g_i$ are single site Pauli operators, and 2) when $p_m/p_u\rightarrow 1/L^2$ and the generators $g_i$ have extent scaling with system size.
To interpolate between these two limits and access the finite $p_m$ dynamics, we make the assumption that the generators of the stabilizer state have instead a finite extent $\xi$ and are centered at sites evenly spaced throughout the chain.
We proceed as before and determine the probability that the measurement of $X_i$ changes the state and the coherence.
This occurs if one of the generators $g_i$ does not commute with $X_i$, which under the above assumption, is only possible for generators centered at most $\xi$ sites away.
If we take $\beta_x$ as the probability one of these generators commutes with $X_i$, then the probability all generators commute with support on site $i$ is $\beta_x^\xi$.
Thus the probability a measurement of $X_i$ is uncertain and obtains information about the $X$ basis is $1-\beta_x^\xi$, yielding the rate equation

\begin{eqnarray}\label{eq:ratexiapx}
\partial_m\overline{N}_x=p_x(1-\beta_x^\xi)-p_z(1-\beta_z^\xi)-p_y(1-\beta_y^\xi)
\end{eqnarray}

In the weak measurement limit, we expect $\xi\sim L \rightarrow \infty$, since the generators are assumed to have extent over the entire system.
This is consistent with the fact that in the limit $\xi\rightarrow \infty$, Eq. <ref> reproduces the weak measurement rate equation Eq. <ref> for $p_y=0$.
If instead we work in the measurement-only limit,the stabilizer state becomes a product state with $\xi=1$.
If we choose $\beta_x=\frac{\overline{N}_x}{L}$, $\beta_x=\frac{\overline{N}_z}{L}$ and $\beta_y=\frac{L-\overline{N_x}-\overline{N}_z}{L}$, then the rate equation Eq. <ref> will have the form of Eq. <ref>.
Therefore, we have a single phenomenological parameter $\xi$ to interpolate between the two extreme limits of strong and weak measurement.
The steady state coherence for a given $\xi$ is then given by the following implicit equation $\partial_m \overline{N}_x=0$, which can be solved numerically.
Numerical solutions for $C_x=L-\overline{N}_x$ are shown in the bottom panel of Fig. <ref>, and agree well with Clifford simulations of $C_x$ when $\Delta_x>0.1$ for a single choice of the length scale $\xi$.

§.§ Markov Chain Effects of $Z$ and $Y$ measurements

Above, in section <ref>, we presented the conditional probabilities, Eq. <ref> for the Markov chain in the weak measurement limit and derived the contribution from the $X_i$ measurements.
That derivation relied on the fact that the measurement of $X_i$ can only be performed on a stabilizer state while maintaining the CSS gauge if one of the $\{g^z_j\}$ generators is used to perform gaussian elimination.
This is seen as follows.
First, all generators which do not commute with $X_i$ contain a $Z_i$ Pauli operator.
Since there are generally $O(L)$ $\{g^z_j\}$ operators, with stabilizer extent $L$, at least $C_z> 1$ of them is likely to contain $Z_i$.
This is also true of the $\{g^y_j\}$ generators, but if one of them, say $g^y_k$, is used to perform the gaussian elimination of $Z_i$, the $N_z$ $\{g^z_i\}$ generators will all contain the $X$ Pauli string of the $g^y_k$ operator used for elimination.
After this procedure, $g^x_y$ will then contain $C_z-1$ linear dependent rows and the state will not be in the CSS gauge.
This does not occur if one of the $\{g^z_j\}$ generators is uses for elimination.

The derivation of the contribution from $Z$ measurements is exactly the same as the first due to the duality between $X$ and $Z$ measurements in the stabilizer gauge.
The contribution from $Y$ measurements occurs because it becomes, with high probability, a generator in $\{g^x_j\}$ and another in $\{g^z_j\}$ will not commute with the $Y_i$ measurement in the $L\rightarrow \infty $ limit.
Thus one generator (say the one in $\{g^x_j\}$), will have to eliminate the other ($\{g^z_j\}$) leading to $N_z\rightarrow N_z-1$ and $N_y \rightarrow N_y+1$.
Then the row in $g^x_x$ used for elimination will be replaced with $Y_i$ leading to $N_x\rightarrow N_x-1$ and $N_y\rightarrow N_y+1$.
Thus for a $Y$ measurement $N_x$ and $N_z$ both decrease by $1$ as in the third equation in Eq. <ref>.

Scaling collapse for tripartite mutual information $I_3$. Data for the different system sizes cross at $\Delta_x=\Delta_x^c=0.333\pm 0.005$ confirming the $\Delta_x=1/3$ critical point predicted by the dynamics of coherence. The curves collapse for $\nu=1.2\pm0.05$ where the error source is sampling error from the finite, $O(2000)$, circuit realizations performed which we estimate to be $\Delta I_3\approx 0.5$. Here we do not show lines for $L\leq 128$ owing to non-universal finite size effects causing a slight drift in the apparent critical point. For example, the curves for $L=128$ and $L=256$ cross at $\Delta_x^c=0.35$.

§ CRITICAL PROPERTIES OF THE COHERENCE CONTROLLED ENTANGLEMENT TRANSITION

In section <ref>, we presented a phase transition controlled by the relative rate of $X$, $Y$ and $Z$ measurements of a circuit composed of CNOTs and measurements at a fixed overall measurement rate $p_m=0.01$.
The phase transition was observable in the half cut entanglement entropy at late times, $S(L/2)$, the biparitite, $I_2$, and tripartite, $I_3$, mutual informations and the coherent information, $\mathcal{C}$ between the initial and final state of the system.
These quantities identified a critical point of $\Delta_x^c=(p_x-p_z)/(1-p_y)=1/3$ when $p_y=1/4$, and $p_x+p_z=1-p_y$.
In this appendix, we determine the length critical exponent, $\xi\sim (\Delta_x-\Delta_x^c)^{-\nu}$ and the exponent $\beta$ for the coherent information, $\mathcal{C}\sim(\Delta_x-\Delta_x^c)^{\beta}$.

Scaling collapse for the coherent information, $\mathcal{C}$, occurring at three different times $t=5L,15L$ and $t/L=40$ as labeled in the figure. In this figure $\Delta_x^c=0.33\pm0.02$, $\nu=1.09\pm0.05$ and $\beta=0.65\pm0.05$, where the variation in the exponents arises from both sampling error as in Fig. <ref> and from variations in the best fit critical parameters at the three different times. The length critical exponent, $\nu$ is compatible by a single standard deviation with the one obtained in Fig. <ref> for the tripartite mutual information.

To identify these critical exponents, we make the following scaling hypothesis

\begin{eqnarray}
I_3(\Delta_x) &=& f\left( \left( \Delta_x-\Delta_x^c \right)L^{-\nu}\right) \\ \nonumber
\mathcal{C}(\Delta_x)&=& L^{\beta/\nu}g\left( \left( \Delta_x-\Delta_x^c \right)L^{-\nu}\right),
\end{eqnarray}

and find that data from our numerical simulations confirms these hypothesises in Fig. <ref> and Fig. <ref>.
There, we determine that the critical parameters, $\Delta_x^c=0.33\pm0.02$, $\nu=1.09\pm0.05$ and $\beta=0.65\pm0.05$ obtain the best fits to the data.
These exponents are distinct from the critical exponents found for the transition described in Ref. <cit.>, where different from the circuit considered here, the two-qubit unitaries are not just CNOTs and are chosen from the full two-qubit Clifford group, and the transition is controlled by the rate of measurements, $p_m$.
In particular, Ref. <cit.> finds the coherent information exponent as $\beta=0$, which is inconsistent with our data.
The critical exponent $\nu=1.09$ is consistent with directed percolation which describes the classical transition discussed in section <ref> and in Refs. <cit.>.
Future work could find it interesting to better understand if the transition is in the directed percolation universality class or not.
An obstacle to this is apparent large finite size effects occurring in these circuits as discussed in the caption of Fig. <ref>.


# Chiral nematic and fluctuation-induced first-order phase transitions in AB-
stacked kagome bilayers

A. Zelenskiy  Department of Physics and Atmospheric Science, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5  M. L. Plumer  Department of
Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia,
Canada B3H 3J5  Department of Physics and Physical Oceanography, Memorial
University of Newfoundland, St. John’s, Newfoundland, A1B 3X7, Canada  B. W.
Southern  Department of Physics and Astronomy, University of Manitoba,
Winnipeg, Manitoba, Canada R3T 2N2  M. E. Zhitomirsky  Université Grenoble
Alpes, Grenoble INP, CEA, IRIG, PHELIQS, 38000 Grenoble, France  Institut
Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France
T. L. Monchesky  Department of Physics and Atmospheric Science, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5

###### Abstract

We study a Heisenberg-Dzyaloshinskii-Moriya Hamiltonian on AB-stacked kagome
bilayers at finite temperature. In a large portion of the parameter space, we
observe three transitions upon cooling the system: a crossover from Heisenberg
to the XY chiral paramagnet, Kosterlitz-Thouless transition to a chiral
nematic phase, and a fluctuation-induced first-order transition to an Ising-
like phase. We characterize the properties of phases numerically using Monte
Carlo finite-size analysis. To further explain the nature of the observed
phase transitions, we develop an analytical coarse-graining procedure that
maps the Hamiltonian onto a generalized XY model on a triangular lattice. To
leading order, this effective model includes both bilinear and biquadratic
interactions and is able to correctly predict the two phase transitions.
Lastly, we study the Ising fluctuations at low temperatures and establish that
the origin of the first-order transition stems from the quasi-degenerate ring
manifold in the momentum space.

###### pacs:

Valid PACS appear here

Introduction.– Competing interactions are at the root of complex behavior for
a broad variety of physical systems Muthukumar  _et al._ (1997); Toledano  _et
al._ (2009); Ochi  _et al._ (2018); Akça  _et al._ (2011); Winter  _et al._
(2016). In magnetic systems the competition can arise from lattice geometry
(geometric frustration) or from the spin-orbit interactions Savary and Balents
(2016); Takagi  _et al._ (2019). The resulting ordered states, often possess
important properties, such as topological stability and non-zero chirality,
and are attractive for device applications Fert  _et al._ (2013). On the other
hand, a considerable amount of research has been devoted to phases that either
remain disordered down to zero temperature (spin liquids), or exhibit partial
ordering, such as spin nematics Savary and Balents (2016); Knolle and Moessner
(2019); Penc and Läuchli (2010). The experimental discovery of these phases
remains challenging, since they lack conventional dipolar ordering.
Theoretical studies have shown that the stabilization of spin nematic states
is non-trivial Jiang  _et al._ (2023) and often requires higher-order spin
interactions, such as the biquadratic exchange Blume and Hsieh (2003); Andreev
and Grishchuk (1984); Shannon  _et al._ (2010). Furthermore, since spin
nematics break rotational, but not the time-reversal symmetry, the
orientations of the spins in these phases continue to fluctuate in the Ising-
like fashion. Geometric frustration, among other things, was shown to
accommodate for these fluctuations Shannon  _et al._ (2006); Momoi  _et al._
(2012); Iqbal  _et al._ (2016).

The Heisenberg antiferromagnet on a kagome lattice is the paradigmatic example
of geometrically frustrated spin system with macroscopic degeneracy Chalker
_et al._ (1992); Harris  _et al._ (1992); Huse and Rutenberg (1992). At low
temperatures, the classical spins form the 120 coplanar structure with
dominant octupolar correlations Zhitomirsky (2008).

Figure 1: (a) Crystal structure of the AB-stacked kagome compounds: the A and
B layers are shown in two shades of gray, dashed and solid lines represent the
($J_{1}$, $D_{1}$) and ($J_{2}$, $D_{2}$) interactions respectively. The
arrows on the top left cell indicate bond directions taken for calculating
chirality $\chi(\mathbf{r})$. (b) Part of the phase diagram (from Ref. I)
where the Ising-like phases are stable. (c) Fragment of the spin structure in
the Ising-like phase. The color corresponds to
$\mathbf{A}(\mathbf{r})\cdot\mathbf{A}(0)$ (purple and yellow for $\pm 1$
respectively). (d) Definition of the LOPs.

The macroscopic degeneracy in 2D kagome is generally unstable with respect to
anisotropic interactions, such as Dzyaloshinskii-Moriya (DM) or changes in the
geometry that introduce competing interactions Chernyshev and Zhitomirsky
(2015); McCoombs  _et al._ (2023).

Figure 2: (a) High-temperature properties of the heat-capacity, LOP magnitude
susceptibility, and average chirality alignment. (b), (c) Fragment of the spin
configuration in the chiral nematic phase. In (b), we plot the spins on AB-SKL
and color the unit cells using the overlap of the LOP vectors. In (c), we plot
the same configuration but in terms of $\mathbf{A}(\mathbf{r})$, where the
collinearity parameter (see text) is used to color the directors. The dark
spots correspond to disclinations. (d) Dipolar and (e) quadrupolar correlation
functions plotted as function of distance per kagome bond-length.

A previous study of the AB-stacked kagome lattice (AB-SKL) Zelenskiy  _et al._
(2022) (Ref. I) revealed that the symmetry of the model introduces a large
number of duality transformations, allowing for a unified description of
magnetic phases in different parts of the parameter space. Furthermore, a
minimal Heisenberg-DM (HDM) Hamiltonian was shown to stabilize various single-
and multiple-$q$ structures. Among these, the most intriguing are magnetic
phases where the spins in individual unit cells have a distorted 120 degree
structure, which alternates throughout the system, forming Ising-like
structures (see fig. 1 (c)).

In this article, we study the finite-temperature properties of the Ising-like
phase in a single AB-SKL bilayer. We show through both numerical and
analytical calculations that in a large region of parameter space thermal
fluctuations stabilize a phase that exhibits simultaneous chiral and nematic
order. The coexistence of nematicity and chirality is extremely unusual, since
these properties are typically associated with opposite types of structures
(collinear and non-collinear, respectively). Moreover, we find that at lower
temperatures, the chiral nematic phase breaks the time-reversal symmetry and
transforms into Ising-like structures via a fluctuation-induced first-order
transition.

Our study is relevant to the magnetic properties of compounds with AB-SKL
structure, such as $\mathrm{Mn}_{3}X$ ($X=\mathrm{Sn}$, $\mathrm{Ge}$,
$\mathrm{Ga}$) and $\mathrm{Fe}_{3}\mathrm{Sn}_{2}$. These systems have
received a considerable amount of attention due to their unusual transport
properties, including recent discoveries of the anomalous Hall effect
Nakatsuji  _et al._ (2015); Nayak  _et al._ (2016); Kiyohara  _et al._ (2016);
Kida  _et al._ (2011). The non-magnetic atoms in these materials induce a weak
spin-orbit coupling, which by symmetry should result in the intra- and
interlayer DM interactions Zelenskiy  _et al._ (2021). Despite the
experimental reports of helical Park  _et al._ (2018); Fenner  _et al._
(2009), skyrmion Hirschberger  _et al._ (2019), magnetic bubble, and spin
glass Hou  _et al._ (2017) phases in AB-SKL materials, a theoretical
description of the magnetism of these systems is still lacking.

Model Hamiltonian.– We consider classical $\mathrm{O}(3)$ spins on an AB-SKL.
The minimal HDM model was derived in Zelenskiy  _et al._ (2021) and can be
written as follows:

$\displaystyle\mathcal{H}_{JD}=\mathcal{H}_{J}+\mathcal{H}_{D},$ (1)
$\displaystyle\mathcal{H}_{J}=\frac{1}{2}\sum_{\mathbf{r}\mathbf{r^{\prime}}}\sum_{ij}J_{ij}(\mathbf{r}-\mathbf{r}^{\prime})\mathbf{S}_{i}(\mathbf{r})\cdot\mathbf{S}_{j}(\mathbf{r}^{\prime}),$
$\displaystyle\mathcal{H}_{D}=\frac{1}{2}\sum_{\mathbf{r}\mathbf{r^{\prime}}}\sum_{ij}D_{ij}(\mathbf{r}-\mathbf{r}^{\prime})\mathbf{\hat{z}}\cdot\left[\mathbf{S}_{i}(\mathbf{r})\times\mathbf{S}_{j}(\mathbf{r}^{\prime})\right],$

where $\mathbf{r}$ and $\mathbf{r}^{\prime}$ label the positions of the unit
cells in a triangular superlattice, and $i,j$ label the six sublattices. The
four parameters correspond to the interlayer ($J_{1}$, $D_{1}$) and intralayer
($J_{2}$, $D_{2}$) exchange and DM couplings, respectively (see fig. 1 (a)).
The interlayer interactions stabilize $Q_{z}=0$ for all relevant parameter
values, and so the properties of a single bilayer should be representative of
the properties in the bulk. In the following, we will also take advantage of
the self-duality, by defining local coordinates for the six sublattices.

Figure 3: Magnetic phases studied in this work. (Top) Finite-size data for
low-temperature heat capacity and average quadrupole moment. (Bottom) Average
dipolar structure factors from MC data (left half) and calculated from the
effective XY Hamiltonian (right half).

The dual version of the model (1) is written in terms of new local spin
variables $\tilde{\mathbf{S}}_{i}(\mathbf{r})$ as well as dual parameters
($\widetilde{J}_{1}$, $\widetilde{D}_{1}$, $\widetilde{J}_{2}$,
$\widetilde{D}_{2}$) Sup . An important property of the dual parameters,
pointed out in Ref. I, is that in the stability region of the Ising-like
phases (fig. 1 (b)) we generally have
$|\widetilde{D}_{2}|/|\widetilde{J}_{1}|<1$ and
$|\widetilde{J}_{2}|/|\widetilde{J}_{1}|\ll 1$.

Details of numerical simulations.– We perform Monte Carlo (MC) simulations
using standard heat-bath updates combined with the over-relaxation method
Creutz (1987). A single MC step consists of one heat-bath update, followed by
5 over-relaxation steps. Simulations are performed in bilayer systems with
$N=L^{2}$ unit cells with $18\leq L\leq 108$, and in the temperature range
$0.01\leq T\leq 5$. A single run typically consists of $10^{5}$ MC steps at
each temperature. Finally, the results are averaged over 10 independent
simulations to estimate the statistical errors. A list of definitions of
average quantities is provided in the Supplemental Material Sup .

For consistency with the results for the 3D systems in Ref. I, we fix
$J_{1}=2$ and $J_{2}=1$ and vary the values of $D_{1}$ and $D_{2}$. Thus, the
temperatures can be assumed to have units of $|J_{2}|$. In this article, we
present the results for two representative systems with $D_{1}=-J_{1}$ and
$D_{2}=-0.5J_{2}$ and provide data for the extended range of parameters in Sup
.

Monte Carlo results.– First, we report our numerical findings. Since the DM
term in our model (1) breaks the out-of-plane $C_{2}$ spin symmetry, the
system develops non-zero chirality in each unit cell, which we define for the
bilayer system as $\chi(\mathbf{r})=\hat{\mathbf{z}}\cdot\sum_{\langle
ij\rangle}\mathbf{S}_{i}(\mathbf{r})\times\mathbf{S}_{j}(\mathbf{r})$, where
indices $i,j$ label sites on a kagome triangle (see fig. 1). As the system is
cooled, qualitative changes in the spin structure occur at temperatures
$T^{*}$, $T_{Q}$, $T_{D}$ (fig. 2 (a)). Above $T^{*}$, we observe a broad
Schottky-like peak in the heat capacity. A closer analysis reveals that in
this temperature range the spins in each unit cell form an approximately 120
degree planar structure, with spins on the A triangle parallel to those on the
B triangle (fig. 1). The corresponding “local” order parameter (LOP)
$\mathbf{A}(\mathbf{r})$ is a two-dimensional vector, which transforms as an
irreducible representation $E_{g}^{(14)}$, as discussed in Ref. I. Below
$T^{*}$, the chirality in each unit cell becomes negative, i.e.
$\langle\mathrm{sgn}(\chi(\mathbf{r}))\rangle=-1$. At the same time, the
fluctuations in magnitude of LOPs become very small, as seen from the
temperature dependence of
$C_{|A|}=\frac{1}{N}\sum_{\mathbf{r}}\langle(|A(\mathbf{r})|^{2}-\langle|A(\mathbf{r})|^{2}\rangle)^{2}\rangle$
in fig. 2 (a).

Despite the apparent ordering of the chiralities, the in-plane spin
fluctuations remain large, which poses a question about the global ordering of
the system. To analyze the spin structure on the global scale, we define the
dipole and quadrupole correlation functions respectively as

$\displaystyle\mathcal{S}_{D}(\bm{\rho})$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{r}}\langle\hat{\mathbf{A}}(\mathbf{r})\cdot\hat{\mathbf{A}}(\mathbf{r}+\bm{\rho})\rangle,$
(2) $\displaystyle\mathcal{S}_{Q}(\bm{\rho})$
$\displaystyle=\frac{1}{N}\sum_{\mathbf{r}}\langle\bm{\mathcal{Q}}(\mathbf{r})\cdot\bm{\mathcal{Q}}(\mathbf{r}+\bm{\rho})\rangle,$
(3)

where ${\bm{\rho}=\mathbf{r}-\mathbf{r}^{\prime}}$, and the quadrupole tensor
is defined as
${\mathcal{Q}_{\alpha\beta}(\mathbf{r})=\hat{A}_{\alpha}(\mathbf{r})\hat{A}_{\beta}(\mathbf{r})-\frac{1}{2}\delta_{\alpha\beta}}$.
As seen from fig. 2 (d), (e), in the temperature range between $T_{Q}$ and
$T^{*}$, both types of correlations decay exponentially with distance. Thus,
in this region, the state of the system can be thought of as chiral paramagnet
Pappas  _et al._ (2009).

Further decreasing the temperature of the system, we observe an appearance of
spontaneous quadrupole moment at $T_{Q}$ (fig. 3). Below $T_{Q}$,
$\mathcal{S}_{Q}(\rho=|\bm{\rho}|)$ displays a clear algebraic decay, with a
correlation length that strongly depends on the temperature (fig. 2 (e)). This
is a strong indication of the emergent quasi-long-range KT ordering of the
nematic degrees of freedom (NDOF), as a result of the algebraic breaking of
the continuous $U(1)$ symmetry. Since chirality vectors remain ordered, this
nematic phase is also chiral. We confirm the KT nematic order by defining the
collinearity parameter
${\psi(\mathbf{r})=\frac{1}{3}\sum_{\bm{\rho}}\langle\bm{\mathcal{Q}}(\mathbf{r})\cdot\bm{\mathcal{Q}}(\mathbf{r}+\bm{\rho})\rangle}$,
where the sum runs over the nearest neighbors Sup . Fig. 2 (c) and Sup  shows
the formation of topological defects, identical in character to the
disclinations in conventional nematics de Gennes and Prost (1993). Below
$T_{Q}$, the defects appear in pairs, which is further consistent with the KT
theory of 2D nematics Stein (1978).

In the reciprocal space, $\mathcal{S}_{Q}(\mathbf{q})$ becomes sharply peaked
at $\mathbf{q}=0$ at $T_{Q}$ Sup . In contrast, $\mathcal{S}_{D}(\mathbf{q})$
shows a broad ring feature at incommensurate wavevectors (fig. 3). As the
system is cooled down below $T_{Q}$, the number of wavevectors contributing to
the ring decreases and the six-fold anisotropy becomes more pronounced. Since
the Ising degrees of freedom (IDOF) continue to fluctuate, we conclude that
the time-reversal $\mathbb{Z}_{2}$ symmetry must remain unbroken in the
nematic phase.

Figure 4: Energy histograms for a range of temperatures near $T=T_{D}$ for
$L=54$ system.

Finally, at $T_{D}$ the IDOF freeze, and the LOP vectors form complicated
network patterns, as reported in Ref. I. Analysis of the histograms of energy
components at $T_{D}$ (collected using $2\times 10^{6}$ MC steps) reveal
multiple peaks (fig. 4), which signals a weak first-order transition. Unlike
in the conventional first-order transitions, we observe 3-7 peaks in the
energy histograms for a range of temperatures. The heat capacity in the same
range of temperatures appears noisy and does not display a clear anomaly.

These observations lead us to believe that the free energy landscape of the
Ising-like phases consists of many near-degenerate minima. Therefore, we
suspect that the characteristic network patterns form when the system fails to
reach equilibrium as a result of the large configurational entropy. Our
speculations are supported by the fact that for most systems in our study we
can impose a single-$q$ stripe structure which is slightly lower in energy
than the random configurations ($\Delta E\sim 10^{-5}$). Nevertheless, these
ordered states almost never occur in 2D system, even for longer MC runs
($>10^{6}$ MC updates).

Coarse-graining.– To better understand the MC data, we provide analytical
analysis of the model (1). When the system is in the chiral paramagnet state,
its properties are effectively described by $N$ fluctuating LOP vectors.
Therefore, it is desirable to construct an effective Hamiltonian, written
explicitly in terms of $\mathbf{A}(\mathbf{r})$ variables. To do this, we
consider the normal modes of spin fluctuations within a single unit cell
Dasgupta and Tchernyshyov (2020). In total, there are twelve modes, half of
which ($\alpha_{\\{0-5\\}}(\mathbf{r})$) describe the in-plane fluctuations,
and the remaining ($\gamma_{\\{0-5\\}}(\mathbf{r})$) describe the out-of-plane
fluctuations. Among these, only the uniform in-plane rotations, which we
denote $\alpha_{0}(\mathbf{r})$, do not change the magnitude of the LOP
vectors. As a result, we may construct a coarse-graining procedure, whereby
the hard modes $\alpha_{\\{1-5\\}}(\mathbf{r})$,
$\gamma_{\\{0-5\\}}(\mathbf{r})$ are integrated out, leaving effective
interactions written in terms of the soft modes $\alpha_{0}(\mathbf{r})$. The
procedure follows closely the method presented in Ref. I, and is given in Sup
. In the derivation, we take advantage of the smallness of
$|\widetilde{J_{2}}|/|\widetilde{J}_{1}|\sim 10^{-2}$ and calculate the
effective Hamiltonaian up to the smallest power of
$|\widetilde{D_{2}}|/|\widetilde{J}_{1}|\sim 10^{-1}$. The result is a
generalized XY Hamiltonian on a triangular lattice:

$\displaystyle\mathcal{H}_{\text{eff}}$
$\displaystyle=E_{0}+\mathcal{H}_{D}+\mathcal{H}_{Q}+\mathcal{H}_{DQ},$ (4)
$\displaystyle\mathcal{H}_{D}$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{r}\mathbf{r}^{\prime}}\mathcal{J}_{D}(\bm{\rho})\hat{\mathbf{A}}(\mathbf{r})\cdot\hat{\mathbf{A}}(\mathbf{r}^{\prime}),$
(5) $\displaystyle\mathcal{H}_{Q}$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{r}\mathbf{r}^{\prime}}\mathcal{J}_{Q}(\bm{\rho})\bm{\mathcal{Q}}(\mathbf{r})\cdot\bm{\mathcal{Q}}(\mathbf{r}^{\prime}),$
(6) $\displaystyle\mathcal{H}_{DQ}$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{r}\mathbf{r}^{\prime}\mathbf{r}^{\prime\prime}}\mathcal{J}_{DQ}(\bm{\rho};\bm{\rho}^{\prime})\hat{\mathbf{A}}^{T}(\mathbf{r}^{\prime})\bm{\mathcal{Q}}(\mathbf{r})\hat{\mathbf{A}}(\mathbf{r}^{\prime\prime}),$
(7)

where $E_{0}$ is a constant, ${\bm{\rho}=\mathbf{r}-\mathbf{r}^{\prime}}$, and
${\bm{\rho}^{\prime}=\mathbf{r}-\mathbf{r}^{\prime\prime}}$. Importantly,
$\mathcal{H}_{Q}$ is equivalent to a biquadratic coupling of the LOP vectors
Sup . The dipolar couplings $\mathcal{J}_{D}(\bm{\rho})$ extend to the third
neighbors and lead to geometric frustration, whereas
$\mathcal{J}_{Q}(\bm{\rho})$ only couple nearest neighbors and stabilize
collinear (nematic) configurations of the LOPs. Generally, the biquadratic
couplings are larger than the dipolar
(${|\mathcal{J}_{Q}(\bm{\rho})|\sim~{}5|\mathcal{J}_{D}(\bm{\rho})|}$).

A family of similar generalized XY models has been studied numerically Granato
_et al._ (1991); Carpenter and Chalker (1989); Poderoso  _et al._ (2011); Shi
_et al._ (2011); Canova  _et al._ (2014, 2016); Serna  _et al._ (2017);
Žukovič (2018) and using renormalization techniques Lee and Grinstein (1985);
Korshunov (1985, 1986). In these works, the dipolar interactions are typically
unfrustrated and stabilize ferromagnetic order. When the coupling term is
zero, the phase diagram in the
$|\mathcal{J}_{D}(\bm{\rho})|/|\mathcal{J}_{Q}(\bm{\rho})|\ll 1$ limit has
been well established: as the system is cooled down, it first undergoes a
nematic KT transition ($T=T_{Q}$), followed by an Ising transition ($T=T_{D}$)
leading to a phase with a quasi-long-range ferromagnetic order. This model was
proposed to be relevant for a range of systems, including liquid crystals, and
superconductors Lee and Grinstein (1985); Korshunov (1985); Shi  _et al._
(2011); Serna  _et al._ (2017). In the vast majority of magnetic systems, the
biquadratic term ($\mathcal{H}_{Q}$), if present, is smaller than the exchange
interaction, meaning that the split transition cannot occur through this
mechanism. Our coarse-graining procedure uncovers that the effective
interactions impose a large quadrupolar coupling through DM interactions. We
note that the validity of the effective model in (4) extends beyond the Ising-
like phases into the $\mathbf{Q}=0$ phase. Using duality transformations in
Ref. I, we can quickly construct similar models for other $\mathbf{Q}=0$
phases. Since these phases occupy most of the parameter space, and are known
to be the ground states of the $\mathrm{Mn}_{3}X$ compounds, the properties of
(4) are extremely relevant for the future experimental studies.

In the case of the Ising-like phases, the situation is complicated by both the
geometric frustration on the dipolar interaction and the presence of the
coupling term. $\mathcal{J}_{D}(\mathbf{q})$ produces a degenerate ring,
similar to the $\mathcal{S}_{D}(\mathbf{q})$, which leads to a competition
between different incommensurate configurations. The $\mathcal{H}_{DQ}$ may
influence a variety of properties, including the universality classes of
transitions, as well as the nature of topological defects Granato  _et al._
(1991); Drouin-Touchette  _et al._ (2022); Jiang  _et al._ (1993, 1996). This
analysis is outside the scope of this work and will be reported elsewhere
Zelenskiy  _et al._ Our calculations show that $\mathcal{H}_{DQ}$ changes the
value of $T_{D}$ as well as the radius of the ring in
$\mathcal{J}_{D}(\mathbf{q})$, but does not break its degeneracy Sup .
Therefore, our results for the decoupled model ($\mathcal{H}_{DQ}=0$) still
apply to the physics of the system.

Mean-field theory.– In order to study the properties of the effective model in
eq. (4), we construct a mean-field theory using variational methods Reimers
_et al._ (1991); Berlinsky and Harris (2019). The derivation of the model is
given in the Supplemental Material Sup . We denote $Q$ and $\phi(\mathbf{q})$
as the order parameters for the NDOF and the IDOF, respectively. We obtain the
following Landau expansion

$\displaystyle f_{L}$ $\displaystyle=f_{0}+f_{D}+f_{Q},$ (8) $\displaystyle
f_{D}$
$\displaystyle=\tau_{D}\Phi+3\lambda_{D}\Phi^{2}-\frac{3\lambda_{D}}{2}\sum_{\mathbf{q}}|\phi(\mathbf{q})|^{4},$
(9) $\displaystyle f_{Q}$
$\displaystyle=\frac{\tau_{Q}}{2}Q^{2}+\frac{\lambda_{Q}}{4}Q^{4},$ (10)

where $f_{0}$ is a constant, and
$\Phi=\sum_{\mathbf{q}}|\phi(\mathbf{q})|^{2}$. Here, we restrict the
wavevectors to lie on the degenerate ring. The coefficients
$\tau_{D},\tau_{Q}$ change sign at $T_{D}$ and $T_{Q}$ respectively, and are
related to the corresponding bare susceptibilities evaluated at the critical
wavevectors, and $\lambda_{D},\lambda_{Q}$ are positive constants Sup .

The mean-field theory predicts two phase transitions, consistent with the
numerical results. We note that $f_{Q}$ has exactly the same form as the mean-
field expansion for a 2D XY model, which is unsurprising given that
$\mathcal{H}_{Q}$ can be mapped onto an XY Hamiltonian by changing
$\alpha_{0}(\mathbf{r})\longrightarrow\frac{1}{2}\alpha_{0}(\mathbf{r})$.
Therefore, in the decoupled limit, the nematic transition should belong to the
XY universality class. This is in contrast to a system with a 3D nematic order
parameter, where a first-order transition is predicted at the level of the
mean-field theory.

The model further predicts the Ising transition to be continuous. Assuming
that the order parameter is defined by $m$ magnetic wavevectors, the
corresponding free energy in the ordered state is

$f_{D}^{\text{min}}=-\frac{\tau_{D}^{2}}{6\lambda_{D}}\frac{m}{2m-1}.$ (11)

The structure of the equilibrium Ising order parameter depends on the sign of
$\lambda_{D}$. Thus, in the mean-field limit, the free energy is minimized by
a single-$q$ solution ($m~{}=~{}1$).

Effects of fluctuations.– This predicted nature of the phase transition is
inconsistent with our numerical observations, which indicates that the role of
thermal fluctuations is not negligible. Indeed, the degenerate ring of
critical wavevectors $\mathcal{H}_{D}$ signals that the phase space of
fluctuations is very large, even if their amplitudes are small. Near
$T=T_{D}$, the bare dipolar susceptibility can be parameterized according to

$\chi_{0,D}(\mathbf{q})\approx\frac{1}{\tau_{D}+c(q-q_{0})^{2}},$ (12)

where we ignored the effects of the hexagonal anisotropy. We further consider
only wavevectors with radius $q$ close to the critical ring ($q_{0}$).

This scenario was first studied in three-dimensional isotropic systems by
Brazovskii Brazovskiǐ (1975), who, for the case of a 3D system, showed that
the large volume of fluctuations stabilizes the disordered state down to
$T=0$. This prevents the system from undergoing a continuous phase transition.
Nevertheless, Brazovskii’s analysis indicated that the system may still have a
first-order transition, even in the absence of a cubic term in the Landau
theory.

Since the field theory for the IDOF in the AB-SKL is identical to that
contained in Ref Brazovskiǐ (1975), up to the dimension of the system, we
follow the same steps to obtain the renormalized values of $\tau_{D}$ and
$\lambda_{D}$ Chaikin and Lubensky (2000); Altland and Simons (2010); Amit and
Martin-Mayor (2005). The procedure is described in the Supplemental Material
Sup . For the renormalized susceptibility, we obtain

$\chi_{D}(\mathbf{q})=\frac{1}{t_{D}+c(q-q_{0})^{2}},$ (13)

where the renormalized parameter $t_{D}$ is defined through a simple self-
consistency relation:

$t_{D}=\tau_{D}+\frac{3\lambda_{D}q_{0}}{2\sqrt{ct_{D}}}.$ (14)

Since $t_{D}$ is non-negative for all values of $\tau_{D}$, the fluctuations
stabilize the nematic phase for all $T<T_{Q}$. Furthermore, the renormalized
value of the vertex $\lambda_{D}$ is calculated to be

$l_{D}=\lambda_{D}\frac{1-2\Pi}{1+\Pi},$ (15)

where $\Pi\propto t_{D}^{-\frac{3}{2}}$. Since $l_{D}$ changes sign at
$2\Pi=1$, and since mean-field theory predicts a positive sixth-order term Sup
, we conclude that thermal fluctuations induce a first-order transition. The
negative value of $l_{D}$ and the form of eq. (12) further indicate that
fluctuations will prefer multiple-$q$ solutions, which can contribute to the
formation of random Ising-like structures.

The Brazovskii transition has been mostly discussed in the context of weak
crystallization Brazovskii  _et al._ (1987); Kats  _et al._ (1993),
cholesteric liquid crystals Brazovskiǐ and Dmitriev (1975); Seul and Andelman
(1995), as well as some biological systems Lavrentovich  _et al._ (2016);
Bates  _et al._ (1988). However, the symmetry of the order parameter in these
systems implies a cubic term in the Landau free energy, and a first-order
transition is generally not surprising.

In magnetic systems, Brazovskii scenario remains largely unstudied. To our
knowledge, the only other magnetic system where this type of transition has
been clearly demonstrated is a helical magnet $\mathrm{MnSi}$ Mühlbauer  _et
al._ (2009); Janoschek  _et al._ (2013); Bauer  _et al._ (2013); Kindervater
_et al._ (2019). Nevertheless, we believe that Brazovskii scenario applies to
many other frustrated systems with large ground state degeneracy Gvozdikova
and Zhitomirsky (2005). Our results demonstrate that the same arguments still
hold for 2D systems with a 1D degenerate manifold, which is applicable to a
large number of frustrated 2D magnets.

We note that higher-order effective interactions, which we ignored in this
work, will break the degeneracy of the ring in $\chi_{0,D}(\mathbf{q})$.
However, provided that this splitting is small, thermal fluctuations will
still populate the whole ring, meaning that the analysis above should still
apply.

Concluding remarks.– Our theoretical study of the Ising-like phases in AB-SKL
bilayers uncovered rich physical phenomena. These phenomena bridge the
properties of a broad range of magnetic and non-magnetic systems, such as
liquid crystals, helical magnets, and glasses.

The description of the ordered phases in our system depends crucially on the
partial magnetic ordering of the unit cells, i.e. the transition from
Heisenberg to the XY chiral paramagnet. The coarse-graining procedure that
reflects this transition unveils the effective biquadratic interaction between
LOPs, which is responsible for the stabilization of the chiral nematic phase.
To our knowledge, this work is the first to discuss a magnetic phase that is
simultaneously nematic and chiral. Even more remarkable is the fact that this
phase is stabilized over a large range of parameters, thanks in part to the
dual properties of the HDM model. Furthermore, our analytical procedure can be
generalized to other triangular systems, which could aid the experimental
realization of chiral nematics.

We also note that our numerical results in the chiral paramagnet phase are
qualitatively similar to the experimental results in the “fluctuation
disordered” phase of $\mathrm{MnSi}$, appearing above $T_{c}$ at low fields
Pappas  _et al._ (2009); Janoschek  _et al._ (2013). Adding Brazovskii
scenario to these similarities hints at universal properties of the HDM models
that apply to crystals with and without inversion symmetry.

Finally, the glassy properties of the Ising-like phases deserve further
investigation. It is not clear if a single-$q$ stripe always provides the
ground state or if the Ising constraint may lead to additional frustration
and, as a result, large degeneracy. A combination of chirality and non-uniform
magnetic structure makes these states an interesting subject for spintronic
studies, since the itinerant electrons will couple to the emergent
electromagnetic fields Schulz  _et al._ (2012); Everschor-Sitte and Sitte
(2014).

To conclude, we hope that the richness of magnetic properties discussed in
this work will serve as a motivation for future studies of the AB-SKL.

The work of A. Z., M. L. P., and T. L. M. was supported by the Natural
Sciences and Engineering Research Council of Canada (NSERC). The work of M. E.
Z. was supported by ANR, France.

## References

  * Muthukumar  _et al._ (1997) M. Muthukumar, C. K. Ober, and E. L. Thomas, Science 277, 1225 (1997), https://www.science.org/doi/pdf/10.1126/science.277.5330.1225 .
  * Toledano  _et al._ (2009) J. C. F. Toledano, F. Sciortino, and E. Zaccarelli, Soft Matter 5, 2390 (2009).
  * Ochi  _et al._ (2018) M. Ochi, M. Koshino, and K. Kuroki, Physical Review B (2018).
  * Akça  _et al._ (2011) S. Akça, A. Foroughi, D. Frochtzwajg, and H. W. C. Postma, PLoS ONE 6 (2011).
  * Winter  _et al._ (2016) S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Physical Review B 93, 214431 (2016).
  * Savary and Balents (2016) L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2016).
  * Takagi  _et al._ (2019) H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nature Reviews Physics 1, 264 (2019).
  * Fert  _et al._ (2013) A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnology 8, 152 (2013).
  * Knolle and Moessner (2019) J. Knolle and R. Moessner, Annual Review of Condensed Matter Physics 10, 451 (2019), https://doi.org/10.1146/annurev-conmatphys-031218-013401 .
  * Penc and Läuchli (2010) K. Penc and A. M. Läuchli, in  _Introduction to frustrated magnetism: materials, experiments, theory_ (Springer, 2010) pp. 331–362.
  * Jiang  _et al._ (2023) S. Jiang, J. Romhányi, S. R. White, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. Rev. Lett. 130, 116701 (2023).
  * Blume and Hsieh (2003) M. Blume and Y. Y. Hsieh, Journal of Applied Physics 40, 1249 (2003), https://pubs.aip.org/aip/jap/article-pdf/40/3/1249/10557522/1249_2_online.pdf .
  * Andreev and Grishchuk (1984) A. Andreev and I. Grishchuk, Sov. Phys. JETP 60, 267 (1984).
  * Shannon  _et al._ (2010) N. Shannon, K. Penc, and Y. Motome, Phys. Rev. B 81, 184409 (2010).
  * Shannon  _et al._ (2006) N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev. Lett. 96, 027213 (2006).
  * Momoi  _et al._ (2012) T. Momoi, P. Sindzingre, and K. Kubo, Phys. Rev. Lett. 108, 057206 (2012).
  * Iqbal  _et al._ (2016) Y. Iqbal, P. Ghosh, R. Narayanan, B. Kumar, J. Reuther, and R. Thomale, Phys. Rev. B 94, 224403 (2016).
  * Chalker  _et al._ (1992) J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender, Phys. Rev. Lett. 68, 855 (1992).
  * Harris  _et al._ (1992) A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45, 2899 (1992).
  * Huse and Rutenberg (1992) D. A. Huse and A. D. Rutenberg, Phys. Rev. B 45, 7536 (1992).
  * Zhitomirsky (2008) M. E. Zhitomirsky, Phys. Rev. B 78, 094423 (2008).
  * Chernyshev and Zhitomirsky (2015) A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 92, 144415 (2015).
  * McCoombs  _et al._ (2023) J. S. R. McCoombs, A. Zelenskiy, M. L. Plumer, B. W. Southern, and T. L. Monchesky, Phys. Rev. B 107, 064407 (2023).
  * Zelenskiy  _et al._ (2022) A. Zelenskiy, T. L. Monchesky, M. L. Plumer, and B. W. Southern, Phys. Rev. B 106, 144433 (2022).
  * Nakatsuji  _et al._ (2015) S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
  * Nayak  _et al._ (2016) A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. Kübler, C. Felser, and S. S. P. Parkin, Science Advances 2, 10.1126/sciadv.1501870 (2016).
  * Kiyohara  _et al._ (2016) N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Applied 5, 064009 (2016).
  * Kida  _et al._ (2011) T. Kida, L. A. Fenner, A. A. Dee, I. Terasaki, M. Hagiwara, and A. S. Wills, Journal of Physics: Condensed Matter 23, 112205 (2011).
  * Zelenskiy  _et al._ (2021) A. Zelenskiy, T. L. Monchesky, M. L. Plumer, and B. W. Southern, Phys. Rev. B 103, 144401 (2021).
  * Park  _et al._ (2018) P. Park, J. Oh, K. Uhlířová, J. Jackson, A. Deák, L. Szunyogh, K. H. Lee, H. Cho, H.-L. Kim, H. C. Walker, D. Adroja, V. Sechovský, and J.-G. Park, npj Quantum Materials 3, 63 (2018).
  * Fenner  _et al._ (2009) L. A. Fenner, A. A. Dee, and A. S. Wills, Journal of Physics: Condensed Matter 21, 452202 (2009).
  * Hirschberger  _et al._ (2019) M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkawa, T. Kurumaji, M. Kriener, Y. Yamasaki, H. Sagayama, H. Nakao, K. Ohishi, K. Kakurai, Y. Taguchi, X. Yu, T.-h. Arima, and Y. Tokura, Nature Communications 10, 5831 (2019).
  * Hou  _et al._ (2017) Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang, Q. Zhang, Y. Zhang, E. Liu, F. Xu, W. Wang, G. Wu, X. Zhang, B. Shen, and Z. Zhang, Advanced Materials 29, 1701144 (2017), https://onlinelibrary.wiley.com/doi/pdf/10.1002/adma.201701144 .
  * (34) See Supplemental Material at [URL will be inserted by publisher] for the details on duality, supplementary finite-size scaling results, derivation of the effective Hamiltonian and mean-field theory, as well as the details of the fluctuation analysis.
  * Creutz (1987) M. Creutz, Phys. Rev. D 36, 515 (1987).
  * Pappas  _et al._ (2009) C. Pappas, E. Lelièvre-Berna, P. Falus, P. M. Bentley, E. Moskvin, S. Grigoriev, P. Fouquet, and B. Farago, Phys. Rev. Lett. 102, 197202 (2009).
  * de Gennes and Prost (1993) P. de Gennes and J. Prost,  _The Physics of Liquid Crystals_, International Series of Monographs on Physics (Clarendon Press, 1993).
  * Stein (1978) D. L. Stein, Phys. Rev. B 18, 2397 (1978).
  * Dasgupta and Tchernyshyov (2020) S. Dasgupta and O. Tchernyshyov, Phys. Rev. B 102, 144417 (2020).
  * Granato  _et al._ (1991) E. Granato, J. M. Kosterlitz, J. Lee, and M. P. Nightingale, Phys. Rev. Lett. 66, 1090 (1991).
  * Carpenter and Chalker (1989) D. B. Carpenter and J. T. Chalker, Journal of Physics: Condensed Matter 1, 4907 (1989).
  * Poderoso  _et al._ (2011) F. C. Poderoso, J. J. Arenzon, and Y. Levin, Phys. Rev. Lett. 106, 067202 (2011).
  * Shi  _et al._ (2011) Y. Shi, A. Lamacraft, and P. Fendley, Phys. Rev. Lett. 107, 240601 (2011).
  * Canova  _et al._ (2014) G. A. Canova, Y. Levin, and J. J. Arenzon, Phys. Rev. E 89, 012126 (2014).
  * Canova  _et al._ (2016) G. A. Canova, Y. Levin, and J. J. Arenzon, Phys. Rev. E 94, 032140 (2016).
  * Serna  _et al._ (2017) P. Serna, J. T. Chalker, and P. Fendley, Journal of Physics A: Mathematical and Theoretical 50, 424003 (2017).
  * Žukovič (2018) M. Žukovič, Physics Letters A 382, 2618 (2018).
  * Lee and Grinstein (1985) D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541 (1985).
  * Korshunov (1985) S. E. Korshunov, JETP Lett. (Engl. Transl.); (United States) 41:5,  (1985).
  * Korshunov (1986) S. E. Korshunov, Journal of Physics C: Solid State Physics 19, 4427 (1986).
  * Drouin-Touchette  _et al._ (2022) V. Drouin-Touchette, P. P. Orth, P. Coleman, P. Chandra, and T. C. Lubensky, Phys. Rev. X 12, 011043 (2022).
  * Jiang  _et al._ (1993) I. M. Jiang, S. N. Huang, J. Y. Ko, T. Stoebe, A. J. Jin, and C. C. Huang, Phys. Rev. E 48, R3240 (1993).
  * Jiang  _et al._ (1996) I. M. Jiang, T. Stoebe, and C. C. Huang, Phys. Rev. Lett. 76, 2910 (1996).
  * (54) A. Zelenskiy, M. L. Plumer, B. W. Southern, and T. L. Monchesky, (unpublished).
  * Reimers  _et al._ (1991) J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, Phys. Rev. B 43, 865 (1991).
  * Berlinsky and Harris (2019) A. J. Berlinsky and A. B. Harris, Density matrix mean-field theory and landau expansions, in  _Statistical Mechanics: An Introductory Graduate Course_ (Springer International Publishing, Cham, 2019) pp. 201–222.
  * Brazovskiǐ (1975) S. A. Brazovskiǐ, Soviet Journal of Experimental and Theoretical Physics (1975).
  * Chaikin and Lubensky (2000) P. Chaikin and T. Lubensky,  _Principles of Condensed Matter Physics_ (Cambridge University Press, 2000).
  * Altland and Simons (2010) A. Altland and B. D. Simons,  _Condensed Matter Field Theory_, 2nd ed. (Cambridge University Press, 2010).
  * Amit and Martin-Mayor (2005) D. J. Amit and V. Martin-Mayor,  _Field Theory, the Renormalization Group, and Critical Phenomena_, 3rd ed. (World Scientific, 2005) https://www.worldscientific.com/doi/pdf/10.1142/5715 .
  * Brazovskii  _et al._ (1987) S. A. Brazovskii, I. E. Dzyaloshinskii, and A. R. Muratov, Sov. Phys. JETP 66, 625 (1987).
  * Kats  _et al._ (1993) E. I. Kats, V. V. Lebedev, and A. R. Muratov, Physics Reports 228, 1 (1993).
  * Brazovskiǐ and Dmitriev (1975) S. Brazovskiǐ and S. Dmitriev, Soviet Journal of Experimental and Theoretical Physics 42, 497 (1975).
  * Seul and Andelman (1995) M. Seul and D. Andelman, Science 267, 476 (1995), https://www.science.org/doi/pdf/10.1126/science.267.5197.476 .
  * Lavrentovich  _et al._ (2016) M. O. Lavrentovich, E. M. Horsley, A. Radja, A. M. Sweeney, and R. D. Kamien, Proceedings of the National Academy of Sciences 113, 5189 (2016), https://www.pnas.org/doi/pdf/10.1073/pnas.1600296113 .
  * Bates  _et al._ (1988) F. S. Bates, J. H. Rosedale, G. H. Fredrickson, and C. J. Glinka, Phys. Rev. Lett. 61, 2229 (1988).
  * Mühlbauer  _et al._ (2009) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, R. A., N. A., G. R., and B. P., Science 323, 915 (2009), https://www.science.org/doi/pdf/10.1126/science.1166767 .
  * Janoschek  _et al._ (2013) M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii, P. Böni, and C. Pfleiderer, Phys. Rev. B 87, 134407 (2013).
  * Bauer  _et al._ (2013) A. Bauer, M. Garst, and C. Pfleiderer, Phys. Rev. Lett. 110, 177207 (2013).
  * Kindervater  _et al._ (2019) J. Kindervater, I. Stasinopoulos, A. Bauer, F. X. Haslbeck, F. Rucker, A. Chacon, S. Mühlbauer, C. Franz, M. Garst, D. Grundler, and C. Pfleiderer, Phys. Rev. X 9, 041059 (2019).
  * Gvozdikova and Zhitomirsky (2005) M. V. Gvozdikova and M. E. Zhitomirsky, Journal of Experimental and Theoretical Physics Letters 81, 236 (2005).
  * Schulz  _et al._ (2012) T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Nature Physics 8, 301 (2012).
  * Everschor-Sitte and Sitte (2014) K. Everschor-Sitte and M. Sitte, Journal of Applied Physics 115, 172602 (2014).