# Giant anomalous Hall effect in quasi-two-dimensional layered antiferromagnet
Co1/3NbS2

Giulia Tenasini  Department of Quantum Matter Physics, University of Geneva,
24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland  Group of Applied
Physics, University of Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva,
Switzerland  Edoardo Martino  Laboratory of Physics of Complex Matter, École
polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland  Nicolas Ubrig
Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest
Ansermet, CH-1211 Geneva, Switzerland  Group of Applied Physics, University of
Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland  Nirmal J.
Ghimire  Department of Physics and Astronomy, George Mason University,
Fairfax, Virginia 22030, USA  Quantum Materials Center, George Mason
University, Fairfax, Virginia 22030, USA.  Materials Science Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  Helmuth Berger  Laboratory
of Physics of Complex Matter, École polytechnique Fédérale de Lausanne, 1015
Lausanne, Switzerland  Oksana Zaharko  Laboratory for Neutron Scattering and
Imaging, Paul Scherrer Institut, Villigen PSI, Switzerland  Fengcheng Wu
Materials Science Division, Argonne National Laboratory, Lemont, Illinois
60439, USA  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA  J. F. Mitchell  Materials Science Division, Argonne National Laboratory,
Lemont, Illinois 60439, USA  Ivar Martin  Materials Science Division, Argonne
National Laboratory, Lemont, Illinois 60439, USA  László Forró  Laboratory of
Physics of Complex Matter, École polytechnique Fédérale de Lausanne, 1015
Lausanne, Switzerland  Alberto F. Morpurgo<EMAIL_ADDRESS>Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest
Ansermet, CH-1211 Geneva, Switzerland  Group of Applied Physics, University of
Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland

###### Abstract

The discovery of the anomalous Hall effect (AHE) in bulk metallic
antiferromagnets (AFMs) motivates the search of the same phenomenon in two-
dimensional (2D) systems, where a quantized anomalous Hall conductance can in
principle be observed. Here, we present experiments on micro-fabricated
devices based on Co1/3NbS2, a layered AFM that was recently found to exhibit
AHE in bulk crystals below the Néel temperature $T_{N}$ = 29 K. Transport
measurements reveal a pronounced resistivity anisotropy, indicating that upon
lowering temperature the electronic coupling between individual atomic layers
is increasingly suppressed. The experiments also show an extremely large
anomalous Hall conductivity of approximately 400 S/cm, more than one order of
magnitude larger than in the bulk, which demonstrates the importance of
studying the AHE in small exfoliated crystals, less affected by crystalline
defects. Interestingly, the corresponding anomalous Hall conductance, when
normalized to the number of contributing atomic planes, is
$\sim\,0.6\;e^{2}/h$ per layer, approaching the value expected for the
quantized anomalous Hall effect. The observed strong anisotropy of transport
and the very large anomalous Hall conductance per layer make the properties of
Co1/3NbS2 compatible with the presence of partially filled topologically non-
trivial 2D bands originating from the magnetic superstructure of the
antiferromagnetic state. Isolating atomically thin layers of this material and
controlling their charge density may therefore provide a viable route to
reveal the occurrence of the quantized AHE in a 2D AFM.

The anomalous Hall Effect (AHE) –i.e., the presence of a finite Hall
resistance in the absence of an applied magnetic field, $H$– has been long
considered to be a phenomenon characteristic of metallic ferromagnets,
originating from the interplay of a finite magnetization and spin-orbit
interaction Nagaosa  _et al._ (2010). Only recently, it has been appreciated
that the AHE can arise more generally in systems with broken time-reversal
symmetry (TRS). Based on this notion, the occurrence of AHE in
antiferromagnets (AFMs) with vanishing magnetization has been predicted
theoretically Shindou and Nagaosa (2001); Y. Taguchi, Y. Oohara, H. Yoshizawa,
N. Nagaosa and Tokura (2001); Martin and Batista (2008); Chen  _et al._ (2014)
and confirmed experimentally on materials such as Mn3Sn Nakatsuji  _et al._
(2015); Li  _et al._ (2019a) Sung  _et al._ (2018), Mn3Ge Kiyohara  _et al._
(2016); Nayak  _et al._ (2016), Mn3Ga Liu  _et al._ (2017) and Mn5Si3 Sürgers
_et al._ (2014, 2016, 2017). In all these systems, the AHE originates from the
non-vanishing integral of the Berry curvature $\Omega_{n}(\textbf{q})$ over
the occupied states, with the transverse conductivity given by
$\sigma_{xy}=\frac{e^{2}}{\hbar}\int_{BZ}\frac{d^{3}\textbf{q}}{(2\pi)^{3}}f[E_{n}(\textbf{q})]\Omega_{n}(\textbf{q})$,
where $E_{n}(\textbf{q})$ is the energy of the $n$-th band in the
antiferromagnetic Brillouin zone Xiao  _et al._ (2010); Chen  _et al._ (2014);
Nayak  _et al._ (2016); Suzuki  _et al._ (2017); Manna  _et al._ (2018a, b)
($f(E)$ is the Fermi function and the subscripts $xy$ denote the plane in
which transport is measured).

Figure 1: Properties of Co1/3NbS2. a, Crystallographic unit cell of Co1/3NbS2.
b, Top-view of the lattice displaying the regular triangular arrangement of
the Co atoms which gives rise to a $\sqrt{3}a_{0}\times\sqrt{3}a_{0}$
superstructure –where $a_{0}$ is the lattice constant of the parent compound
(2H-NbS2)– typical of 1/3 fractional intercalation Nakayama  _et al._ (2006).
The crystallographic unit cell of Co1/3NbS2 is indicated by the yellow shaded
area. c, Temperature dependence of the in-plane resistivity, $\rho_{xx}$,
(blue) and of the magnetic susceptibility, $\chi$, (green) –acquired in an
applied field of 0.1 T along the $c$-axis– both measured in a bulk crystal.
The two curves show a clear kink in correspondence of the antiferromagnetic
transition temperature at $T_{N}$ $\simeq$ 29 K. d, Magnetization, $M$, as a
function of magnetic field ($H\parallel c$) for different temperatures above
and below $T_{N}$ measured in a bulk crystal; $M$ is negligible at $H=0$. e,
Optical micrographs of two devices investigated in the present work: a device
sculpted using a focused ion beam (top left; the scale bar is 20 $\mu$m) and a
device based on an exfoliated crystal approximately 50 nm thick (top right;
the scale bar is 5 $\mu$m). Temperature dependence of the ratio between the
resistivity out-of-plane ($\rho_{z}$) and the resistivity in-plane
($\rho_{xx}$), showing a pronounced anisotropy below $T_{N}$. The inset shows
the resistivity along the $c$-axis, $\rho_{z}$, as a function of temperature.

The possibility to express the Hall conductivity in terms of the Berry
curvature underscores the topological origin of the phenomenon Nagaosa  _et
al._ (2010). Accordingly, dimensionality is expected to play a key role in
determining how the AHE manifests itself, because –as it is well-established–
three-dimensional (3D) systems with broken TRS are all topologically
equivalent unless additional crystalline symmetries are imposed, whereas in
two dimensions topologically distinct states can exist Yang and Nagaosa
(2014). Indeed, in two dimensions the integral of the Berry curvature of an
individual band $n$ is a topological invariant (the Chern number $C_{n}$) that
can assume any possible integer value. In two-dimensional (2D) magnetic
systems, therefore, topologically non-trivial bands can occur that –whenever
completely filled– give a quantized contribution $C_{n}\frac{e^{2}}{h}$ to the
measured Hall conductivity Haldane (1988); Ohgushi  _et al._ (2000); Wang  _et
al._ (2013); Chang and Li (2016); Liu  _et al._ (2016); Tokura  _et al._
(2019). Such a quantum AHE has been reported in thin films of a ferromagnetic
topological insulator Chang  _et al._ (2013) (for related interesting new
developments see also Serlin  _et al._ (2019); Deng  _et al._ (2019)), but in
2D AFMs the conditions to observe the same phenomenon have never been realized
experimentally, since all known antiferromagnetic materials hosting a large
AHE display rather isotropic transport properties Nakatsuji  _et al._ (2015);
Kiyohara  _et al._ (2016); Nayak  _et al._ (2016); Sürgers  _et al._ (2017).
Here, we perform transport measurements on exfoliated crystals of the layered
AFM Co1/3NbS2 and show that this system exhibits an anomalous Hall conductance
as high as 0.6 $e^{2}/h$ per atomic layer as well as a strongly anisotropic
transport, compatible with the existence of topologically non-trivial 2D
bands.
Co1/3NbS2 is a well-known AFM, consisting of magnetic Co ions intercalated
between NbS2 planes, as represented in Fig. 1a. Past transport and
magnetization measurements on bulk crystals Barišić  _et al._ (2011)
(reproduced on the crystals used here, see Fig. 1c) established that the
transition to the antiferromagnetic state occurs at $T_{N}\simeq$ 29 K (the
value of $T_{N}$ slightly varies depending on the exact stoichiometric
composition in intercalated compounds), and that for $T<T_{N}$ the system
remains metallic, with vanishing magnetization at zero applied magnetic field
(Fig. 1d). Neutron diffraction experiments were performed Parkin  _et al._
(1983), and were fit to a collinear AFM structure with the magnetic unit cell
double the size of the structural unit cell. In those experiments all six
symmetry-related in-plane $q$ vectors were observed with nearly equal weights,
and attributed to a multi-domain structure with each domain having single-$q$
ordering. However, subsequent experiments performed on this material [note1]In
the course of this work in order to determine the full magnetic structure of
Co1/3NbS2 we have performed neutron diffraction experiments on the single
crystal diffractometer Zebra at Swiss Spallation Source SINQ. We found that
the data are compatible with the collinear multi-domain structure proposed by
Parkin  _et al._ (1983) but also with a single domain multi-q structure. could
not distinguish between the multi-domain single-$q$ AFM and single domain
multi-$q$ AFM. As we will argue, this distinction is critical: based on
symmetry arguments, the AHE is incompatible with a simple collinear AFM with
single-$q$ lying in the $a$-$b$ plane that was proposed in Parkin  _et al._
(1983). On the other hand, it is naturally expected in a non-coplanar AFM with
three-$q$, all lying in the $a$-$b$ plane. Despite being more complex than the
single-$q$ state, such a multi-$q$ AFM has no uniform spin magnetization, with
AHE arising due to the large uncompensated Berry curvature in the band
structure Shindou and Nagaosa (2001); Y. Taguchi, Y. Oohara, H. Yoshizawa, N.
Nagaosa and Tokura (2001); Martin and Batista (2008); Kübler and Felser
(2014); Nakatsuji  _et al._ (2015); Machida  _et al._ (2010); Ndiaye  _et al._
(2019). Indeed, AHE was observed recently at $T<T_{N}$ in bulk crystals
Ghimire  _et al._ (2018), despite the absence of any sizable magnetization.
This observation, as well as the layered nature of Co1/3NbS2, motivates us to
study the AHE in detail using exfoliated crystals.

Figure 2: AHE in Co1/3NbS2. (a) Hall resistivity, $\rho_{xy}$ and (b)
anomalous Hall resistivity, $\rho_{xy}^{A}$, measured in a representative
device upon cycling the magnetic field back and forth from -14 T to +14 T
($H\parallel c$), for selected temperatures above and below $T_{N}$. The
anomalous Hall resistivity in b is obtained by subtracting from the curves
shown in a the linear contribution due to the ordinary Hall effect.
$\rho_{xy}^{A}$ exhibits sharp jumps at values of coercive fields, $H_{c}$,
that rapidly grow upon lowering $T$. c. Longitudinal resistivity, $\rho_{xx}$,
as function of magnetic field, acquired simultaneously to $\rho_{xy}$; a small
magnetoresistance (MR $\sim$ 2%) is observed below $T_{N}$. The small jumps
visible in the curves in correspondence of $H_{c}$ (less than 10% of the
absolute change in $\rho_{xy}^{A}$) can be attributed to small material
inhomogeneities mixing the longitudinal and transverse resistivity.

For our transport studies we realized micro-fabricated devices based on
Co1/3NbS2 crystals, either exfoliated with an adhesive tape (Fig. 1e, top
right), or sculpted using a focused ion beam (FIB) Moll (2018) (Fig. 1e, top
left; for more details see Supplemental Material Sec. S1). Being much smaller
than bulk crystals, exfoliated layers are considerably less affected by
structural defects, facilitating the observation of the intrinsic properties
of Co1/3NbS2. For instance, as we show below and of key importance for our
results, the in-plane resistivity, $\rho_{xx}$, measured on exfoliated layers
is found to be significantly lower than in the bulk Ghimire  _et al._ (2018),
a clear indication of the reduction of scattering from structural defects.
Micro-fabricated devices also allow us to compare transport along specific
crystallographic directions, which is essential to reveal a pronounced
anisotropy. We find that the out-of-plane resistivity, $\rho_{z}$, (see inset
in Fig. 1e), is more than one order of magnitude larger than the in-plane
resistivity, with the ratio $\rho_{z}/\rho_{xx}$ increasing upon cooling,
without saturating at the lowest temperature reached in the measurements
($T\simeq 4$ K; see Fig. 1e).
The AHE is readily detected by measuring the Hall resistivity, $\rho_{xy}$,
upon sweeping the magnetic field applied parallel to the $c$-axis from
$\mu_{0}H=-14$ T to $\mu_{0}H=+14$ T and back. The data in Fig. 2 are
representative of the behavior observed in five devices based on exfoliated
crystals having different thickness, $t$, (varying between 40 and 90 nm) and
lateral dimensions (ranging from approximately $4\times 8$ $\mu$m2 to
$15\times 26$ $\mu$m2). For temperatures $T$ above $T_{N}$ ($T=30$ K in Fig.
2a), $\rho_{xy}$ exhibits a $H$-linear ordinary Hall effect (grey curve), but
as $T$ is lowered below $T_{N}$ (blue curve measured at $T=28$ K), a
pronounced hysteresis becomes visible, such that $\rho_{xy}\neq 0$ at $H=0$.
Upon lowering $T$, the coercive magnetic field, $H_{c}$, rapidly increases
(pink curve), and below 24 K becomes larger than the highest field accessible
in our experimental set-up (14 T), so that the hysteresis cycle can no longer
be observed.
In this temperature range, the anomalous contribution of the Hall resistivity,
$\rho_{xy}^{A}$, can be straightforwardly isolated by subtracting the linear
contribution due to the ordinary Hall effect, as shown in Fig. 2b.
Interestingly, despite the presence of large, sharp jumps in $\rho_{xy}^{A}$
at $H_{c}$, the longitudinal resistivity, $\rho_{xx}$, does not exhibit
significant variations (see Fig. 2c): only a small magnetoresistance (MR
$\sim$ 2%) originating from the presence of the AFM state is observed (there
is no MR at 30 K, see dark grey curve), with no significant change at $H_{c}$.
The comparison of the $H$-dependence of the anomalous Hall resistivity (Fig.
2b) and of the bulk magnetization (Fig. 1d) shows that $\rho_{xy}^{A}$ is not
simply proportional to $M$, i.e., the AHE in Co1/3NbS2 is very different from
that commonly observed in ferromagnets.

Figure 3: Temperature evolution of AHE in an exfoliated Co1/3NbS2 device. (a)
Anomalous Hall resistivity, $\rho_{xy}^{A}$, and (b) longitudinal resistivity,
$\rho_{xx}$, as a function of temperature measured in a representative device,
after field-cooling (FC) in $\mu_{0}H_{FC}=\pm$8 T (red), $\pm$14 T (blue) and
0 T (ZFC, grey curve), with $H\parallel c$. The yellow diamonds in a represent
the values of $\rho_{xy}^{A}$ extracted at $H=0$ from measurements of the
hysteresis loops done at fixed $T$ (see Fig. 2b). $\rho_{xx}$ shows no
dependence on $H_{FC}$: the curves in b fall exactly on top of each others and
an identical $T$-dependence is observed at ZFC. c, Remnant magnetization, $M$,
as a function of temperature for $H_{FC}$ of 8 T (red) and 14 T (blue) with
$H\parallel c$. d, Temperature dependence of the corresponding anomalous Hall
conductivity, $\sigma_{xy}^{A}$, for FC in 8 T (red) and 14 T (blue). The blue
diamond represents the maximum value of the anomalous Hall conductivity
reported previously in bulk crystals (27 S/cm at 23 K, see Ghimire  _et al._
(2018)). The right axis indicates the anomalous Hall conductance per atomic
layer, $G_{xy}^{A}/N$, obtained normalizing the measured $\sigma_{xy}^{A}$
(left axis) to the total number of contributing crystalline planes [note2]The
anomalous Hall conductivity normalized to the number of crystalline unit cells
(each composed by two NbS2 planes with intercalated Co atoms) would give an
anomalous Hall conductance larger than $e^{2}/h$ per layer. Here, we normalize
to the number of NbS2 planes since each atomic layer –if considered as a fully
isolated entity– can give rise to 2D topologically non-trivial bands in the
antiferromagnetic state.. e, Anomalous Hall conductance per atomic layer,
$G_{xy}^{A}/N$, measured at 5 K for different samples as function of their
thickness. The pink dashed line and the shaded area represent the mean value
and the standard deviation of all measured values.

The rapid increase in coercive field upon lowering $T$ prevents the hysteresis
cycle in $\rho_{xy}$ to be measured at low temperature. Therefore, to
investigate systematically the evolution of the AHE, we follow the strategy
used earlier in Mn3Sn Nakatsuji  _et al._ (2015) and Mn3Ge Kiyohara  _et al._
(2016), and perform measurements of $\rho_{xy}$ as a function of increasing
$T$ at zero magnetic field, after cooling down the devices from $T>T_{N}$ to
$T=5$ K in the presence of an applied field. Irrespective of the specific
antiferromagnetic state responsible for the occurrence of the AHE, which is,
in general, different in different materials, the idea is as follows. If the
device is cooled in the absence of an applied field, chiral microscopic
domains (that we refer to hereafter as _microdomains_) of the
antiferromagnetic state contributing to the AHE with different sign form with
equal probability, and coexist in the material as $T$ is lowered below
$T_{N}$. As a result, the total anomalous Hall conductivity vanishes. On the
other hand, field-cooling (FC) in a sufficiently large applied magnetic field
from $T>T_{N}$ down to low temperature tends to align the microdomains
–favoring one chirality– making $\rho_{xy}^{A}$ finite. The microscopic
mechanism of coupling between the antiferromagnetic microdomains and the
applied magnetic field may vary. The minimal coupling that is expected to be
always present is to the electronic orbital magnetization Shi  _et al._ (2007)
(along the $c$-axis) which generically accompanies the AHE, both being caused
by the band Berry curvature. Another possibility is the coupling to a weak
out-of-plane spin canting due to spin-orbit interaction.
The temperature dependence of the anomalous Hall resistivity measured after
field-cooling at $\mu_{0}H_{FC}=\pm 8$ T (red lines) and $\mu_{0}H_{FC}=\pm
14$ T (blue lines) is shown in Fig. 3a, for one of our Co1/3NbS2 devices,
which exhibits a representative behavior. Below $T_{N}$ a finite
$\rho_{xy}^{A}$ emerges, which follows exactly the same temperature dependence
for both field values. As expected for field induced alignment of
antiferromagnetic microdomains, cooling in fields of opposite polarity selects
microdomains of opposite chirality, resulting in equal and opposite values of
$\rho_{xy}^{A}$. The soundness of the FC procedure is illustrated by the
perfect agreement between the $T$-dependent $\rho_{xy}^{A}$ curve measured
after field-cooling and the values of $\rho_{xy}^{A}$ extracted at $H=0$ in
measurements of the hysteresis loops done at fixed $T$ (see the yellow
diamonds in Fig. 3a). As expected, no AHE is observed upon cooling down the
sample in the absence of an applied magnetic field (i.e., upon zero-field-
cooling, ZFC, see grey curve in Fig. 3a).
We measured the magnetization of the Co1/3NbS2 bulk crystals used for
exfoliation at $H=0$ after FC (Fig. 3c), and found an extremely low residual
magnetic moment smaller than 10-3 $\mu_{B}/Co$ for $T<T_{N}$, which may
originate either from a small spin polarization in the antiferromagnetic state
Ghimire  _et al._ (2018) or from the orbital magnetization of the conducting
electrons Chen  _et al._ (2018). Irrespective of the origin of the residual
moment, finding a very large anomalous Hall resistivity in the presence of a
negligibly small magnetization confirms that the mechanism responsible for the
AHE observed in Co1/3NbS2 is different from that at work in ferromagnets. Note
that the longitudinal resistivity measured simultaneously to $\rho_{xy}^{A}$
on the same device, is entirely insensitive to FC: the exact same
$T$-dependence is observed irrespective of the value and the polarity of $H$
applied during FC (see Fig. 3b; an identical $T$-dependence for $\rho_{xx}$ is
observed at ZFC). Note also that the low temperature longitudinal resistivity
measured on the exfoliated layers is approximately four times smaller than in
the bulk crystals used for exfoliation (compare Fig. 3b and Fig. 1c).

Figure 4: Detecting structural domain switching in the AHE. Temperature
dependence of (a) anomalous Hall resistivity, $\rho_{xy}^{A}$, and (b)
longitudinal resistivity, $\rho_{xx}$, after FC in $\pm$8 T (red) and $\pm$14
T (blue) with $H_{FC}\parallel c$, measured in one of our devices. In a, the
red curves display a trend similar to the one observed in Fig. 3a, whereas the
blue curves measured at $\mu_{0}H_{FC}$ = 14 T exhibit a larger value and a
jump at 17 K with a drop to the value measured at $\mu_{0}H_{FC}$ = 8 T. In
analogy with Fig. 3a, the yellow (light-blue) diamonds correspond to the
values of $\rho_{xy}^{A}$ extracted at $H=0$ (14 T) in measurements of the
hysteresis loops done at fixed $T$ (see Supplemental Material Fig. S2b). The
light-blue dashed line is a guide to the eye. In b, $\rho_{xx}$ shows no
sizable variation in correspondence of the jump observed in $\rho_{xy}^{A}$,
indicating that at the microscopic level the magnetic state of the system
remains the same across the jump. c, Temperature dependence of the anomalous
Hall conductance per atomic layer, $G_{xy}^{A}/N$, for FC in 8 T (red) and 14
T (blue). The inset shows a sketch of the envisioned magnetic configuration of
the exfoliated crystal, with a structural defect that _splits_ the crystal in
two regions (colored in red and blue) with thickness $t_{1}$ and $t_{2}$ (the
arrow indicates the direction of the $c$-axis). The two regions have opposite
chirality upon FC in 8 T, whereas they have the same chirality upon FC in 14
T. d, $G_{xy}^{A}/N$ as a function of $H_{FC}$ measured at 5 K with
$H\parallel c$; the inset shows a zoom-in at small $H_{FC}$. e, $G_{xy}^{A}/N$
as a function of temperature for different $H_{FC}$ applied in the NbS2 planes
($H\perp c$). Note that the value of $G_{xy}^{A}/N$ upon $H_{FC}$ in 8 T
corresponds approximately to the value obtained for FC in 0.05 T when
$H\parallel c$.

To discuss the AHE data quantitatively we invert the resistivity tensor to
calculate the anomalous Hall conductivity
$\sigma_{xy}^{A}=\frac{\rho^{A}_{xy}}{{\rho_{xx}}^{2}+{\rho^{A}_{xy}}^{2}}$.
The temperature dependence of the anomalous Hall conductivity measured (at
$H=0$) after FC at 8 T and 14 T is shown in Fig. 3d. The onset of a non-
vanishing $\sigma_{xy}^{A}$ corresponds to the Néel temperature of Co1/3NbS2,
and the increase in $\sigma_{xy}^{A}$ upon lowering $T$ exhibits a behavior
analogous to that expected for the order parameter of a second order phase
transition. Notably, due mainly to the small longitudinal resistivity of
exfoliated crystals, the values of $\sigma_{xy}^{A}$ measured in our devices
reach up to approximately 400 S/cm at low temperature, more than one order of
magnitude larger than the maximum value measured in Co1/3NbS2 bulk crystals
(27 S/cm; see Ghimire  _et al._ (2018)). Such a value is comparable to –or
possibly even larger than– the highest ever reported so far for the anomalous
Hall conductivity of an AFM. The corresponding anomalous Hall conductance per
atomic layer is $G_{xy}^{A}\sim 0.6$ $e^{2}/h$ at $T=5$ K note (2). Similar
values are found in all the different exfoliated devices that we measured (see
Fig. 3e; device-to-device fluctuations are consistent with the uncertainty in
the detailed device geometry, originating from the small size of the
exfoliated crystals).
Together with the strongly anisotropic resistivity indicative of transport
occurring in the quasi-2D regime (see Fig. 1e), the very large observed
anomalous Hall conductance per layer suggests that topologically non-trivial
2D bands with a non-zero Chern number $C_{n}$ are present in the
antiferromagnetic state of Co1/3NbS2. The observed metallic dependence of the
in-plane resistivity on temperature implies that the Fermi level is not
located in an energy gap and that, accordingly, no quantization of the
anomalous Hall conductance should be expected. Nevertheless, even if the Fermi
level is located within a topologically non-trivial band, the occupied states
experience the effects of Berry curvature, explaining the presence of a large
AHE. In this metallic regime, the longitudinal conductivity is typically much
larger than the transverse one (as it is the case here) and the relation
$\rho_{xy}=\rho_{xx}^{2}\sigma_{xy}$ can be used to extract the transverse
conductivity from the value of $\rho_{xy}$ and $\rho_{xx}$ (which are the
actually measured quantities). With $\sigma_{xy}=\rho_{xy}/\rho_{xx}^{2}$, it
is clear that any extrinsic disorder increasing $\rho_{xx}$, while leaving
$\rho_{xy}$ unaffected, causes the value of $\sigma_{xy}$ extracted from the
measurements to be reduced. This is indeed what appears to happen in
Co1/3NbS2: owing to the larger resistivity, measurements performed on bulk
materials lead to a Hall conductance per layer of approximately 0.01 $e^{2}/h$
–masking the possible topologically non-trivial nature of the bands– rather
than to values of the order of $e^{2}/h$ that we report here [note3]In view of
the envisioned topological nature of the electronic bands in the magnetic
state of Co1/3NbS2, one should consider the possibility that the lower
resistivity measured in exfoliated layers as compared to bulk crystals
originates from the presence of a surface conduction contribution.
Experimentally this possibility can be ruled out, because the value of the
longitudinal resistivity measured in different devices, with crystal thickness
ranging between approximately 40 and 100 nm, is the same within the
experimental precision, irrespective of thickness. In contrast, a significant
contribution of surface conduction would manifest itself in a smaller
resistivity value for thinner exfoliated crystals..
An additional feature is observed in one of the devices investigated. Namely,
the device shows very different temperature dependencies of the anomalous Hall
resistivity upon FC for field values smaller and larger than approximately
$\mu_{0}H\simeq$ 8 T (see Fig. 4a). Upon cooling down in a field smaller than
(or equal to) 8 T, the anomalous Hall resistivity displays the usual
qualitative behavior, whereas, when the device is field-cooled in a larger
field, $\rho_{xy}^{A}$ has a higher low-temperature value and a downward jump
at $T\simeq 17$ K is observed. In contrast, the longitudinal resistivity
measured concomitantly with $\rho_{xy}^{A}$ remains entirely insensitive to
the conditions of FC (see Fig. 4b). The corresponding anomalous Hall
conductivity obtained by inverting the resistivity tensor is shown in Fig. 4c,
for FC done in a low and in a high magnetic field. Consistently with the
apparent presence of two states in this device, for $T<T_{N}$ the evolution of
$\rho_{xy}^{A}$ upon sweeping the magnetic field up and down also exhibits an
anomalous hysteresis comprising two loops (see Supplemental Material Sec. S2).
The absence of any sizable change in $\rho_{xx}$ in correspondence of the jump
in $\sigma_{xy}^{A}$ indicates that –at a microscopic level– the magnetic
state of the system remains the same across the jump, i.e., the jump observed
in $\sigma_{xy}^{A}$ is not due to a change in the magnetic ordering on the
atomic scale. As the jump in the anomalous Hall conductivity is observed only
in one out of five devices, the phenomenon can be attributed to the presence
in that device of two macroscopic domains (that we refer to hereafter as
_macrodomains_) which tend to contribute to the AHE with opposite sign. A
possible origin is a structural stacking fault in the exfoliated layer which
_splits_ the device into two separate regions, as illustrated in the inset of
Fig. 4c. The magnetic coupling across the fault energetically favors states
with opposite chirality. Thus, if FC is performed with $\mu_{0}H\leq 8$ T, the
interfacial interaction dominates and the macrodomains remain anti-aligned,
leading to a reduced value of AHE. For a stronger magnetic field, the energy
benefit of the alignment (which goes as the product of the volume of the
smaller macro-domain and the magnetic field) dominates and the chiralities of
the macrodomains align, leading to a larger combined AHE. At low temperatures,
even after the strong aligning field is removed, macrodomains can remain
aligned, pinned by the effective Ising anisotropy of chirality. Upon heating,
the barrier to realignment can be overcome by the thermal fluctuations,
causing an abrupt drop in $\sigma_{xy}^{A}$.
The different nature of such macrodomains, as compared to the chiral
microdomains discussed earlier, present at zero field in all the devices that
we investigated, can be fully appreciated by looking at the evolution of the
low-temperature anomalous Hall conductance per atomic layer as a function of
field applied during FC (Fig. 4d). A very small magnetic field is sufficient
to align microdomains commonly present in all devices, since FC with
$\mu_{0}H=0.05$ T already stabilizes a low-$T$ value of $G_{xy}^{A}/N$
corresponding to a considerable fraction of $e^{2}/h$ per atomic layer, as
visible in the inset of Fig. 4d. On the other hand, as shown in Fig. 4d, the
macrodomains due to the stacking fault require a much larger field to be
switched. More studies are necessary to fully understand the origin of the
structural macrodomains and what determines their stability. However, in
contrast to most common antiferromagnetic conductors, in AFMs that (like
Co1/3NbS2) exhibit AHE, magnetic field and simple transport measurements allow
different types of magnetic domains and their switching to be controlled and
studied experimentally.
So far we focused on the effects of the out-of-plane magnetic field (i.e.,
$H\parallel c$). We now address the effect of an in-plane magnetic field
(i.e., parallel to the NbS2 layers). The temperature dependence of the AHE is
shown in Fig. 4e for different values of field applied in-plane during FC,
from 2 to 14 T. The value of the anomalous Hall conductance per atomic layer,
$G_{xy}^{A}/N$, obtained by FC in a 8 T in-plane field, is slightly less than
0.2 $e^{2}/h$, comparable to the anomalous Hall conductance per atomic layer
induced by FC in a much smaller out-of-plane field of 0.05 T. This very large
disparity in field values indicates that the effect of the in-plane field can
be most likely attributed to a very small misalignment of the applied field
with respect to the NbS2 planes, causing a finite out-of-plane component (an
explanation certainly compatible with the precision with which our devices are
mounted in the experimental set-up). Similar behavior is observed irrespective
of the orientation of the in-plane field relative to the crystallographic axis
of the material, i.e., an in-plane magnetic field has no significant
influence, irrespective of its direction. This is an important conclusion
because it further reinforces that the alignment of chiral antiferromagnetic
microdomains during the FC process is due to the coupling of the applied field
to the out-of-plane magnetization Shi  _et al._ (2007). The latter could be
due to the orbital magnetization of itinerant electrons, whose orbital
electronic motion takes place predominantly in the NbS2 planes, as evidenced
by the quasi-2D nature of transport in Co1/3NbS2 (see Fig. 1e). Such a motion
gives origin to an orbital magnetization pinned to the direction perpendicular
to the NbS2 planes that naturally couples to an out-of-plane magnetic field
and not to an in-plane one. It can also be due to a small, spin-orbit induced
canting of the Co moments.
The precise origin of the AHE in Co1/3NbS2 cannot be identified at this stage,
because existing neutron diffraction experiments have not yet allowed the full
determination of the magnetic structure (in part due to the small size of the
available single crystals). This is preventing reliable calculations of the
electronic band structure in the antiferromagnetic state of the material. We
nevertheless note that the collinear antiferromagnetic structure proposed in
Parkin  _et al._ (1983) cannot lead to AHE, since it has a symmetry that
corresponds to spatial translation in the direction of the ordering vector $q$
combined with time reversal. Since spatial translation cannot affect the Hall
response, and time reversal flips its sign, such symmetry immediately implies
zero AHE. We also note that the scenario suggested in Šmejkal  _et al._
(2019), based on a collinear AFM with an in-plane ferromagnetic order and an
out-of-plane antiferromagnetic order is incompatible with the neutron data
that show antiferromagnetic order in the plane and hence cannot be used to
explain the large observed AHE.
The pronounced anisotropy of transport, the coupling of the magnetic field to
the orbital magnetization, and the anomalous Hall conductance per atomic layer
reaching values close to $e^{2}/h$ all strongly suggest that the AHE
originates from the presence of 2D topologically non-trivial bands induced by
the antiferromagnetic state. In the truly 2D case, models predicting the
occurrence of the quantum Hall effect in antiferromagnetic states with
vanishing magnetization have been known for over 10 years Shindou and Nagaosa
(2001); Martin and Batista (2008). An exact quantization of the Hall
conductance occurs when the chemical potential lies within a band gap, and the
magnetic structure causes some of the fully occupied bands to become
topologically non-trivial. This has been demonstrated theoretically to happen
for a variety of non-collinear and non-coplanar spin structures Shindou and
Nagaosa (2001); Martin and Batista (2008); Ito and Nomura (2017); Zhang  _et
al._ (2017); Ndiaye  _et al._ (2019); Li  _et al._ (2019b). For instance, the
three-$q$ non-coplanar _tetrahedral_ antiferromagnetic state Martin and
Batista (2008) in each triangular Co layer, would be consistent with the
magnitude and the direction of the antiferromagnetic ordering vectors observed
experimentally, as well as with the fact that all six symmetry-related $q$
(three $\pm q$ pairs) are observed with equal weight. Our preliminary
calculations based on a realistic tight-binding model [note4]F. Wu, and I.
Martin, Unpublished show that this state can indeed yield large AHE with very
weak orbital magnetization, consistently with our experimental observations.
As we already mentioned earlier, in the case of our Co1/3NbS2 crystals, the
system is metallic and the Hall conductance (per atomic layer) cannot be
precisely quantized. Even so, it can remain comparable to $e^{2}/h$, as it is
expected for nearly fully filled or almost empty topological bands. In this
regime, small electron density variations can cause large variations in the
observed anomalous Hall conductance, particularly if the Fermi level is
located close to a band edge, where the Berry curvature is typically the
highest. Experimentally, it would therefore be interesting to change the
carrier density in Co1/3NbS2 to see whether large changes in Hall conductance
can actually be detected. It may even be possible to induce full quantization,
if the density variation allows to completely fill/empty all the partially
filled bands. Viable strategies to change the electron density by a
significant amount include the use of ionic liquid gated devices, if the
thickness of Co1/3NbS2 can be reduced to a few layers, or intercalation of
atoms (either alkali or halogen) in between the NbS2 planes (i.e., where the
Co atoms also reside). Yet another route consists in investigating other
NbS2-based compounds intercalated with different transition metals: Fe1/3NbS2
for instance is also an antiferromagnetic conductor Gorochov  _et al._ (1981);
Nair  _et al._ (2019), with a different expected band filling as compared to
Co1/3NbS2, owing to the difference valence of Fe as compared to Co atoms.
Although challenging experimentally, all these different strategies are worth
considering because, if successful, they would allow unveiling a quantum
anomalous Hall antiferromagnet, i.e., an exotic state of matter that has never
been observed before.
In summary, we performed transport experiments on exfoliated and micro-
fabricated Co1/3NbS2 devices that provide a new perspective on the nature of
the anomalous Hall effect in the antiferromagnetic state of this material.
These devices enable the observation of a much larger AHE as compared to bulk
crystals –with an anomalous Hall conductance reaching values close to
$e^{2}/h$ per atomic layer– as well as of a large anisotropy of the
longitudinal resistivity, that becomes increasingly pronounced upon lowering
temperature. These findings suggest that Co1/3NbS2 represents a very promising
platform to search for a topological 2D quantum anomalous Hall antiferromagnet
exhibiting quantized Hall conductance at zero applied magnetic field. They
also show the importance of performing experiments on small exfoliated devices
to reduce extrinsic effects originating from disorder. Indeed, disorder and
all sources of inhomogeneity can be drastically reduced in exfoliated
crystals, simply because their volume can be easily made smaller by a factor
of approximately $10^{7}$-$10^{8}$ as compared to bulk crystals commonly used
to perform transport experiments . In the case of Co1/3NbS2 the reduced
disorder results in a significantly smaller value of longitudinal resistivity
that translates into a value of Hall conductance per layer reaching up to a
large fraction of $e^{2}/h$. One may expect similar effects to be equally
important in the analysis and interpretation of the measured anomalous Hall
effect in other antiferromagnetic materials that have been discussed earlier.

We sincerely acknowledge Alexandre Ferreira for technical assistance, Enrico
Giannini and Bhushan Hegde for helpful support in magnetization measurements,
Ignacio Gutiérrez-Lezama and Marco Gibertini for fruitful discussion and
Christian Rüegg for support. E.M. and L.F. gratefully acknowledge financial
support from the Swiss National Science Foundation Grant No. 200021-175836.
E.M. acknowledges Philip Moll for help and training in the FIB samples
preparation, supported from the Max-Planck Society. Work in the Materials
Science Division at Argonne National Laboratory (crystal synthesis,
characterization, theoretical studies) was supported by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences, Materials Science and
Engineering Division. F.W. is also supported by the Laboratory for Physical
Sciences. A.F.M. gratefully acknowledges financial support from the Swiss
National Science Foundation (Division II) and from the EU Graphene Flagship
project.

The authors declare no competing financial interests.

## References

  * Nagaosa  _et al._ (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Shindou and Nagaosa (2001) R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001).
  * Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa and Tokura (2001) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa and Y. Tokura, Science 291, 2573 (2001).
  * Martin and Batista (2008) I. Martin and C. D. Batista, Phys. Rev. Lett. 101, 156402 (2008).
  * Chen  _et al._ (2014) H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * Nakatsuji  _et al._ (2015) S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
  * Li  _et al._ (2019a) X. Li, C. Collignon, L. Xu, H. Zuo, A. Cavanna, U. Gennser, D. Mailly, B. Fauqué, L. Balents, Z. Zhu, and K. Behnia, Nat. Commun. 10, 3021 (2019a).
  * Sung  _et al._ (2018) N. H. Sung, F. Ronning, J. D. Thompson, and E. D. Bauer, Appl. Phys. Lett. 112, 132406 (2018).
  * Kiyohara  _et al._ (2016) N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl. 5, 064009 (2016).
  * 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, Sci. Adv. 2, e1501870 (2016).
  * Liu  _et al._ (2017) Z. H. Liu, Y. J. Zhang, G. D. Liu, B. DIng, E. K. Liu, H. M. Jafri, Z. P. Hou, W. H. Wang, X. Q. Ma, and G. H. Wu, Sci. Rep. 7, 515 (2017).
  * Sürgers  _et al._ (2014) C. Sürgers, G. Fischer, P. Winkel, and H. V. Löhneysen, Nat. Commun. 5, 3400 (2014).
  * Sürgers  _et al._ (2016) C. Sürgers, W. Kittler, T. Wolf, and H. V. Löhneysen, AIP Adv. 6, 055604 (2016).
  * Sürgers  _et al._ (2017) C. Sürgers, T. Wolf, P. Adelmann, W. Kittler, G. Fischer, and H. v. Löhneysen, Sci. Rep. 7, 42982 (2017).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * Suzuki  _et al._ (2017) M.-T. Suzuki, T. Koretsune, M. Ochi, and R. Arita, Phys. Rev. B 95, 094406 (2017).
  * Manna  _et al._ (2018a) K. Manna, L. Muechler, T.-H. Kao, R. Stinshoff, Y. Zhang, J. Gooth, N. Kumar, G. Kreiner, K. Koepernik, R. Car, J. Kübler, G. H. Fecher, C. Shekhar, Y. Sun, and C. Felser, Phys. Rev. X 8, 041045 (2018a).
  * Manna  _et al._ (2018b) K. Manna, Y. Sun, L. Muechler, J. Kübler, and C. Felser, Nat. Rev. Mater. 3, 244 (2018b).
  * Yang and Nagaosa (2014) B.-J. Yang and N. Nagaosa, Phys. Rev. Lett. 112, 246402 (2014).
  * Haldane (1988) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
  * Ohgushi  _et al._ (2000) K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, R6065 (2000).
  * Wang  _et al._ (2013) J. Wang, B. Lian, H. Zhang, and S.-C. Zhang, Phys. Rev. Lett. 111, 086803 (2013).
  * Chang and Li (2016) C.-Z. Chang and M. Li, J. Phys.: Condens. Matter 28, 123002 (2016).
  * Liu  _et al._ (2016) C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Condens. Matter Phys. 7, 301 (2016).
  * Tokura  _et al._ (2019) Y. Tokura, K. Yasuda, and A. Tsukazaki, Nat. Rev. Phys. 1, 126 (2019).
  * Chang  _et al._ (2013) C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
  * Serlin  _et al._ (2019) M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Science 367, 900 (2020).
  * Deng  _et al._ (2019) Y. Deng, Y. Yu, M. Z. Shi, J. Wang, X. H. Chen, and Y. Zhang, Science 367, 895 (2020).
  * Nakayama  _et al._ (2006) M. Nakayama, K. Miwa, H. Ikuta, H. Hinode, and M. Wakihara, Chem. Mater. 18, 4996 (2006).
  * Barišić  _et al._ (2011) N. Barišić, I. Smiljanić, P. Popčević, A. Bilušić, E. Tutiš, A. Smontara, H. Berger, J. Jaćimović, O. Yuli, and L. Forró, Phys. Rev. B 84, 75157 (2011).
  * Parkin  _et al._ (1983) S. S. P. Parkin, E. A. Marseglia, and P. J. Brown, J. Phys. C: Solid State Phys. 16, 2765 (1983).
  * note (1) In the course of this work to determine the full magnetic structure of Co1/3NbS2 we performed neutron diffraction experiments on the single crystal diffractometer Zebra at Swiss Spallation Source SINQ. We found that the data are compatible with the collinear multi-domain structure proposed by Parkin  _et al._ (1983) but also with a single domain multi-$q$ structure.
  * Kübler and Felser (2014) J. Kübler and C. Felser, Europhys. Lett. 108, 67001 (2014).
  * Machida  _et al._ (2010) Y. Machida, S. Nakatsuji, S. Onoda, T. Tayama, and T. Sakakibara, Nature 463, 210 (2010).
  * Ndiaye  _et al._ (2019) P. B. Ndiaye, A. Abbout, V. M. L. D. P. Goli, and A. Manchon, Phys. Rev. B 100, 144440 (2019).
  * Ghimire  _et al._ (2018) N. J. Ghimire, A. S. Botana, J. S. Jiang, J. Zhang, Y. S. Chen, and J. F. Mitchell, Nat. Commun. 9, 3280 (2018).
  * Moll (2018) P. J. Moll, Annu. Rev. Condens. Matter Phys. 9, 147 (2018).
  * note (2) The anomalous Hall conductivity normalized to the number of crystalline unit cells (each composed of two NbS2 planes with intercalated Co atoms) would give an anomalous Hall conductance larger than $e^{2}/h$ per layer. Here, we normalize to the number of NbS2 planes since each atomic layer –if considered as a fully isolated entity– can give rise to 2D topologically non-trivial bands in the antiferromagnetic state.
  * Shi  _et al._ (2007) J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99, 197202 (2007).
  * Chen  _et al._ (2018) H. Chen, T.-C. Wang, D. Xiao, G.-Y. Guo, Q. Niu, and A. H. MacDonald, Phys. Rev. B 101, 104418 (2020).
  * note (3) In view of the envisioned topological nature of the electronic bands in the magnetic state of Co1/3NbS2, one should consider the possibility that the lower resistivity measured in exfoliated layers as compared to bulk crystals originates from the presence of a surface conduction contribution. Experimentally this possibility can be ruled out, because the value of the longitudinal resistivity measured in different devices, with crystal thickness ranging between approximately 40 and 100 nm, is the same within the experimental precision, irrespective of thickness. In contrast, a significant contribution of surface conduction would manifest itself in a smaller resistivity value for thinner exfoliated crystals.
  * Šmejkal  _et al._ (2019) L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova,  arXiv:1901.00445 .
  * Ito and Nomura (2017) N. Ito and K. Nomura, J. Phys. Soc. Jpn. 86, 063703 (2017).
  * Zhang  _et al._ (2017) Y. Zhang, Y. Sun, H. Yang, J. Železný, S. P. P. Parkin, C. Felser, and B. Yan, Phys. Rev. B 95, 075128 (2017).
  * Li  _et al._ (2019b) X. Li, A. H. MacDonald, and H. Chen,  arXiv:1902.10650 .
  * note (4) F. Wu and I. Martin, (unpublished).
  * Gorochov  _et al._ (1981) O. Gorochov, A. L. Blanc-soreau, J. Rouxel, P. Imbert, and G. Jehanno, Philos. Mag. B 43, 621 (1981).
  * Nair  _et al._ (2019) N. L. Nair, E. Maniv, C. John, S. Doyle, J. Orenstein, and J. G. Analytis, Nat. Mater. 19, 153 (2020).


# Better bounds for the union-closed sets conjecture using the entropy
approach

Stijn Cambie Extremal Combinatorics and Probability Group (ECOPRO), Institute
for Basic Science (IBS), Daejeon, South Korea, supported by the Institute for
Basic Science (IBS-R029-C4), E-mail<EMAIL_ADDRESS>
###### Abstract

We improve the best known constant $\frac{3-\sqrt{5}}{2}$ for which the union-
closed conjecture is known to be true, by using dependent samples as suggested
by Sawin and the entropy approach on this problem initiated by Gilmer.
Meanwhile, we focus on the intuition behind this entropy approach and its
boundaries.

##  1 Introduction

The union-closed conjecture is a challenging conjecture in extremal set
theory, see e.g. [10, sec. 32], which became famous due to its elegance. A
union-closed family $\mathcal{F}$ is a collection of sets such that the union
of any two sets belongs to $\mathcal{F}$ as well. The union-closed conjecture
states that if $\mathcal{F}$ contains at least one nonempty set, then there is
an element that belongs to at least half of the sets in $\mathcal{F}.$ This
can be formally stated as follows, where we recommend the reader who is not
familiar with some of the terminology, notations or definitions to first have
a look at Subsection 1.1.

######  Conjecture 1 (Union-closed conjecture).

If $\mathcal{F}\not=\\{\emptyset\\}$ is a union-closed family with ground set
$[n],$ then there exists an element $i\in[n]$ such that at least half of the
sets in $\mathcal{F}$ contain $i$, i.e.,
$\lvert\mathcal{F}(i)\rvert\geq\frac{\lvert\mathcal{F}\rvert}{2}.$

This would be tight by taking all subsets of a fixed ground set. Indeed, if
$\mathcal{F}=2^{[m]},$ then every integer $i\in[m]$ appears in exactly
$2^{m-1}$ sets of $\mathcal{F}.$ Possibly, these are essentially the only
tight examples (duplicating elements does not change the structure), as also
suggested in the blog on the Polymath project;
https://gowers.wordpress.com/2016/01/21/.

According to [2, 3, 10], the union-closed conjecture was already a folklore
conjecture since the late 1960s or beginning 1970s, and was made well-known by
Frankl, who rediscovered it in the late 1970s (1979 according to [9]), and Ron
Graham. Nonetheless, the first formal publication containing it might be [18].
In the $90$s, it was proven in [15, 20] that there is an element that appears
in at least a
$\Omega\left(\frac{\log\lvert\mathcal{F}\rvert}{\lvert\mathcal{F}\rvert}\right)$
fraction of all sets. The latter result is also implied by the union-closed
size problem by Reimer [17], which was fully resolved in [3]. In contrast to
the union-closed size problem, which was solved in $10$ years, the union-
closed conjecture is still open. It has been proven in various specific cases,
e.g. for certain random generated union-closed families it has been proven to
be true with high probability [2]. More on the history till $2015$, with other
equivalent formulations of the conjecture, can be found in the survey [5]. One
equivalent formulation by [4] states that every bipartite graph with at least
one edge has at least one vertex in each bipartition class that belongs to at
most half of the maximal independent sets. At the beginning of $2017$, Karpas
[14] proved the union-closed conjecture for families that contain roughly at
least half of all sets.

Very recently, Gilmer [11] proved the first linear bound using an elegant
entropy-based method. As such he resolved the $\varepsilon$-Union-Closed Sets
Conjecture, as stated in [12]. Gilmer claimed that a tight version of his
method could prove a fraction equal to $\frac{3-\sqrt{5}}{2},$ which was soon
verified by [1, 6, 19]. The tight version heavily depends on determining the
minimum of a function $\frac{h(x^{2})}{xh(x)}$, where $h$ is the binary
entropy function. Chase and Lovett [6] gave a clear, short proof using the
minimum determined by [1].

A sketch for a fourth variant of the proof with an additional claim, that can
be verified as well, was given in [16]. A question and conjecture of Gilmer
were soon answered in the negative, by [19, 8]. By working with approximate
union-closed families in [6], $\frac{3-\sqrt{5}}{2}$ seemed possibly the best
constant one could aim for with the idea of Gilmer [11]. Nevertheless, as
suggested by Sawin [19], this is not the case. For this, we address the
following question, from which the improved constant can be concluded later.
Note the inequality cannot be strict, since for any $\\{0,1\\}$-valued random
variable equality does hold.

######  Question 2 (Sawin).

What is the maximum value $c$ for which the following is true? For every
$p,q,r$ identically distributed $[0,1]$-valued random variables with
expectation less than $c$, where $p$ and $q$ are independent, but $p$ and $r$
are not necessarily independent, we have

$(1-\alpha)\,\mathbb{E}[H(p+q-pq)]+\alpha\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]\geq\,\mathbb{E}[H(p)].$
(1)

Yu [21] considered the approach and question of Sawin in larger generality and
derived bounds expressed in general optimisation forms.

Our contribution consists in solving Question 2 completely (partially
computational) and as such improving the constant for which the union-closed
conjecture is true. The core content is written in Section 2. Here, we start
with explaining the entropy approach in general. After that, we give some
intuition why the bound can be improved despite the sharpness of the
approximate union-closed conjecture by [6] and why the direct use of Sawin’s
idea cannot improve the constant too much. For this, we provide the upper
bound for $c$ in Question 2, which later will turn out to be sharp. As a last
part of this section, in Subsection 2.4, we summarize the additional steps of
the proof for the improved constant.

In the next section, Section 3, we prove the best bound of $c$ in Question 2.
In Subsections 3.1 and 3.2, we reduce the possible probability distributions
one has to consider and prove that the critical probability distributions have
a support containing at most $3$ values. This is as such the technical core to
work out Sawin’s idea and to answer Question 2. After this, we can express the
problem as a minimisation problem of a function in $4$ variables and the
optimisation problem can be verified with a computer-verification. In contrast
to the detailed work in e.g. [1] for the constant $\frac{3-\sqrt{5}}{2}$, this
is done slightly less rigorous. We need to take into account that the
minimisation problem finds a local minimum and there is a finite computer
precision involved. By considering some plots, we note that there are two
regimes to verify. By observing that the extremal probability distribution is
atomic (support has $2$ elements) in one regime, we prove it more precisely
for that regime and conclude. Since the constant is not $\frac{1}{2}$ and
another core method will be needed for the full resolution, an even more
rigorous analysis than has been done seems unnecessary, especially as it does
not contribute to further understanding the underlying principles. Finally, in
Section 4 we summarize the proof for the improved constant on the union-closed
conjecture, based on the answer for Question 2. That is, for $c\sim 0.3823455$
we prove the following theorem.

######  Theorem 3.

Let $\mathcal{F}\subset 2^{[n]}$ be a nonempty union-closed family. Then there
is some element $i\in[n]$ that appears in at least $c\lvert\mathcal{F}\rvert$
many sets of $\mathcal{F}.$

So to recap, in this paper, we prove that by taking a linear combination of
probability distributions, not all of them being independent, the bound on the
union-closed conjecture derived with the entropy approach of Gilmer [11] can
be improved slightly.

###  1.1 Terminology and notation

In this subsection, we collect some basic notation and definitions used in
extremal set theory and the entropy method used in the papers related to the
work of Gilmer. For more of this, we refer to [10] for extremal set theory and
[7] for entropy.

The standard finite set of cardinality $n$ is denoted with
$[n]=\\{1,2,\ldots,n\\}.$ A subset of $[n]$ is a set containing some elements
from $[n]$ and can possibly be the empty set $\emptyset.$ A collection of
subsets of $[n]$ is called a set system or family $\mathcal{F}$. The family
$2^{[n]}$ (the power set of $[n]$) contains all $2^{n}$ possible subsets of
$[n]$ and in general a family $\mathcal{F}\subset 2^{[n]}$ is a subset of this
power set. A uniform family is a family whose sets all have the same
cardinality $k.$ The largest $k$-uniform family on the ground set $[n]$ is
$\binom{[n]}{k}=\\{A\subset[n]\colon\lvert A\rvert=k\\}.$ Similarly we use
$\binom{[n]}{\geq k}$ to denote $\\{A\subset[n]\colon\lvert A\rvert\geq k\\}.$

A family $\mathcal{F}$ is called union-closed if for every $A,B\in\mathcal{F}$
also the union $A\cup B\in\mathcal{F}.$ Using
$\mathcal{F}\cup\mathcal{F}=\\{A\cup B\colon A,B\in\mathcal{F}\\},$ a family
$\mathcal{F}$ is union-closed if and only if
$\mathcal{F}\cup\mathcal{F}=\mathcal{F}.$ It is approximate union-closed if
the latter is true for almost every choice of $A,B\in\mathcal{F}.$ The sets
containing a fixed element $i$ are essentially presented by the family
$\mathcal{F}(i)=\\{A\backslash\\{i\\}\colon i\in A\in\mathcal{F}\\}.$
Similarly, $\mathcal{F}(\overline{i})=\\{A\colon i\not\in A\in\mathcal{F}\\}.$

We will use some Landau-notation. Given two functions
$f,g\colon\mathbb{R}\to\mathbb{R}$, we write

  * •

$f=o(g)$ if $\lim_{x\to\infty}f(x)/g(x)=0,$

  * •

$f\sim g$ if $\lim_{x\to\infty}f(x)/g(x)=1.,$

  * •

$f=O(g)$ if there is a constant $C$ such that $\lim_{x\to\infty}\lvert
f(x)/g(x)\rvert<C,$

  * •

$f=\Omega(g)$ if $g=O(f),$

  * •

$f=\Theta(g)$ if $g=O(f)$ and $f=O(g).$

A random variable $X$ can be related with the probability distribution of the
outcomes. The entropy $H(X)$ of a discrete random variable $X$ equals the
Shannon entropy of its probability sequence and is denoted by $H(X).$ The
support of a probability distribution or random variable, is the subset of the
possible outcomes which have positive probability (density function). If the
support of $X$ is a finite set $A$, and each outcome $x\in A$ has a
probability $p_{x},$ then

$H(X)=-\sum_{x\in A}p_{x}\log_{2}p_{x}.$

It is a fundamental result, a corollary of Jensen’s inequality, that this is
bounded by $\log_{2}\lvert A\rvert.$ For a random variable with only two
outcomes, occurring with probabilities $p$ and $1-p$, we denote the entropy
with the binary entropy function
$h(p)=-(p\log_{2}p+(1-p)\log_{2}(1-p)).$111For clarity, the binary entropy
function is denoted with $h$ and the entropy function of a variable with $H$.
A conditional entropy of a random variable $Y$ given $X$ can be computed as

$H(Y|X)\ =-\sum_{x\in X,y\in
Y}\,\mathbb{P}(x,y)\log_{2}{\frac{\,\mathbb{P}(x,y)}{\,\mathbb{P}(x)}}.$

Here $\,\mathbb{P}(x,y)$ is the probability on the outcome $\\{X=x,Y=y\\}.$
The expectation of a random variable $X$ is $\,\mathbb{E}[X]=\sum_{x\in
A}p_{x}x.$ We use $\vee$ for the logical or, that is $x\vee y$ implies that
$x$ or $y$ has to be satisfied.

##  2 Preliminaries

In this section, we start with giving the main idea behind the entropy
approach introduced by Gilmer [11]. Next, we give some intuition why the
examples from [11, 6], that at first sight show that the constant
$\frac{3-\sqrt{5}}{2}$ could not be improved with the method, actually can be
improved. We give examples why the linear combination in Question 2 is
necessary and why the constant could not be improved much by solving this
question. Finally, we summarize the essence/ intuition behind the proof.

###  2.1 Why the entropy approach works for the union-closed conjecture

The contraposition of the union-closed conjecture states that if every element
$i\in[n]$ appears in strictly less than half of the sets of a family
$\mathcal{F}\subset 2^{[n]}$, then $\mathcal{F}$ is not union-closed and thus
$\lvert\mathcal{F}\rvert<\lvert\mathcal{F}\cup\mathcal{F}\rvert.$ The idea
behind the entropy approach initiated by Gilmer [11] is that if one can find a
way to sample sets from $\mathcal{F}\cup\mathcal{F}$ such that the entropy is
strictly larger than $\log_{2}\lvert\mathcal{F}\rvert,$ one can conclude that
$\lvert\mathcal{F}\rvert<\lvert\mathcal{F}\cup\mathcal{F}\rvert.$ The latter
since the entropy of a random variable with $N$ possible outcomes is bounded
by $\log_{2}N.$ Proving that $H(A\cup B)>\log_{2}\lvert\mathcal{F}\rvert$
where $A,B$ are sampled from $\mathcal{F}$, is hard when lacking information
on $\mathcal{F}.$ It is easier to compare $H(A\cup B)$ with $H(A)$. For the
conclusion to hold, $A$ needs to be sampled uniform random from $\mathcal{F}$.
Gilmer [11] did this by taking two uniform independent (iid) random samples
$A,B$ from $\mathcal{F}$ and considering the entropy of the union $A\cup B.$

For alternatives, one could sample $B$ non-uniform from $\mathcal{F}$, or have
two sampled random variables which are not independent. In that case one tries
to do this in such a way that $A\cup B$ is as uniformly distributed over
$\mathcal{F}\cup\mathcal{F}$ as possible. Exactly uniformly distributed seems
impossible. If $\mathcal{F}$ would be union-closed, $A\cup B$ needs to be
equal to $A$, but for a non-union-closed family it should give all of
$\mathcal{F}\cup\mathcal{F}$ as possible outcomes. Once one is sampling
dependent samples, one would need additional ideas to know more about the
conditional probability distributions. Sawin [19] gave the most natural choice
when sampling the sets element-wise.

###  2.2 Observations on approximate union-closed set system

Let $\psi=\frac{3-\sqrt{5}}{2}$ be the smallest root of
$2x-x^{2}=1-x\Leftrightarrow x^{2}-3x+1=0$ and
$g\colon\mathbb{N}\to\mathbb{N}\colon n\mapsto g(n)$ be a function which is
both $o(n)$ and $\omega(n^{0.5}).$ In [6], the authors take $g(n)\sim
n^{2/3}.$ The family

$\mathcal{F}=\\{A\subset[n]\colon\lvert A\rvert=\psi n+g(n)\vee\lvert
A\rvert\geq(1-\psi)n\\}\\\ =\binom{[n]}{\psi
n+g(n)}\cup\binom{[n]}{\geq(1-\psi)n}$

is an approximate union-closed family. That is, the union of two sets (iid,
independent identically distributed, uniform random chosen) in $\mathcal{F}$
belongs with high probability to $\mathcal{F}$ as well. Nevertheless, since
$\psi>\frac{1}{3}$, one can also observe that
$\mathcal{F}\cup\mathcal{F}=\binom{n}{\geq\psi n+g(n)}.$ That is, only a small
proportion of $\mathcal{F}\cup\mathcal{F}$ belongs to $\mathcal{F}$. As such,
this construction that would indicate that improving the constant $\psi$ might
be impossible with the entropy approach, does not necessarily imply the latter
conclusion. If unions of two sets in $\mathcal{F}$ are taken with a non-
uniform measure or in a dependent way, such that it is more likely that sets
with small union are taken, the initial idea about sharpness by this
construction is not valid anymore. The entropy of the union of two random
variables taking values in $\mathcal{F}$ in a dependent way can have larger
entropy. With the dependency proposed by Sawin, the union of $2$ sets of the
above approximate union-closed family will almost surely be of size
$(0.5+o(1))n$ for example.

###  2.3 Upper bound for Question 2 and limits on the approach of Sawin

Let $h(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x).$ Let the roots of the function
$h(x)(2-h(x))-h(2x-x^{2})$ be $0<b_{1}<b_{2}<1.$ Then $b_{1}\sim
0.139499451909862$ and $b_{2}\sim 0.329454738503037.$ Choose $b=b_{2}$ and
$a=\frac{1-h(b)}{2-h(b)}\sim 0.0788772927059232.$

Let $p,q,r$ be three identically distributed $[0,1]$-valued random variables,
where $\,\mathbb{P}(p=1)=a$ and $\,\mathbb{P}(p=b)=1-a,$ such that $p$ and $q$
are independent and $p$ and $r$ are as negatively correlated as possible in
the sense that $\,\mathbb{P}(p=r=1)=0.$ For these choices, we have
$\,\mathbb{E}[p]=0.382345533366703$222The computation can be found in
https://github.com/StijnCambie/UCconjecture/blob/main/Sharpness.sagews and
$(1-a)^{2}h(2b-b^{2})=(1-2a)=(1-a)h(b)$ which is equivalent with

$\,\mathbb{E}[H(p+q-pq)]=\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]=\,\mathbb{E}[H(p)].$

Hence no linear combination satisfies Equation 2 with a strict inequality and
local perturbations (increasing $a$) will result in counterexamples when $c$
is allowed to be slightly larger.

On a different note, we observe that taking a linear combination, as done in
Equation 2, will be necessary when aiming to improve the constant
$\frac{3-\sqrt{5}}{2}$ in the progress on the union-closed conjecture. For
this, observe that
$\,\mathbb{E}\left[H(max(p,r,\min(p+r,1/2)))\right]<\,\mathbb{E}[H(p)]$ for
$p,r$ identically distributed with $\,\mathbb{P}(p=r=1)=0$,
$\,\mathbb{P}(p=b)=1-a$ and $\,\mathbb{P}(p=1)=a$, where $b=\frac{1}{4}$ and
$a=\frac{1-h(b)}{2-h(b)}+\varepsilon$ for some small $\varepsilon>0.$ Since
$\,\mathbb{E}[p]<0.37<\frac{3-\sqrt{5}}{2}$, looking to the single term could
not lead to the improvement.

From these observations, one can conclude that one cannot aim to prove the
result with a constant better than $0.382345533366703$ with the exact
suggested approach of Sawin.

###  2.4 Summary of the proof

The idea from Sawin [19] is to sample sets twice element-wise. Here
iteratively, one samples $A\cap[k]$ and $B\cap[k]$ for $0\leq k\leq n$ based
on the probability that a set in $\mathcal{F}$, given the intersection with
$[k]$, would contain the additional element $k+1$. Now one can do the sampling
of the element $k+1$ in a dependent way for $A$ and $B$, such that at the end,
both $A$ and $B$ are uniform samples over $\mathcal{F}.$ Once some elements
are sampled, one looses control over the conditional probabilities (the
distribution) and so we assume the worst-case scenario. Since the worst-case
scenario for the two different strategies is different, a better bound is
obtained by taking a linear combination of these two different ways of
sampling in $\mathcal{F}\cup\mathcal{F}.$ Since the entropy of the whole
sample can be obtained by summing (conditional) entropies for every element
$k\in[n]$, it is sufficient to prove the inequality for these entropies for
one element. At that point, one gets a question purely depending on the
probability distribution of random values and their expectation.

To attack that question, we do local optimisation to find properties of an
optimal distribution; redistributing the probability mass function and
verifying convexity and concavity. As such, we reduce the problem to a simpler
inequality, depending on only $4$ unknown parameters, associated with a
probability distribution whose support contains at most $3$ elements. Finally,
this finite problem is verified numerically with a computer program in a few
ways, and solved more precisely for the main region corresponding to atomic
probability distributions (the support contains only two elements).

##  3 Proof for the optimal constant of Question 2

In this section, we prove that the constant $c\sim 0.382345533366,$ i.e., the
upper bound derived in Subsection 2.3, is the answer to Question 2. Here, we
first prove that it is sufficient to consider the case where $p,q,r$ are
probability distributions which are not supported on $(0.5,1).$ Next, by doing
some analytical computations on the behaviour of the functions in two
intervals, similar to what Sawin [19] did, we show that the problem boils down
to knowing the minima of a continuous function in three variables on a bounded
region. As such, the remaining problem is a minimisation problem for which the
statement can be checked by a computer. This implies that an exact rigorous
calculus proof is missing. Since this would not add intuition to the core
problem (Conjecture 1) and the entropy approach, and the improvement of the
constant is small, the verification by computer seems sufficient to us as
evidence for the exact bounds of the idea proposed by Sawin [19].

###  3.1 Reduction of support of the probability distribution

First we prove that it is sufficient to consider $[0,1]$-valued random
variables which do not attain values in $(0.5,1).$

For readibility and since it is sufficient to consider finite supported
measures for the application on Conjecture 1, we prove the following lemmas
only in the case of discrete probability distributions. The proof of the
following lemma can be modified for general probability distributions by
replacing sums by integrals and probability $\,\mathbb{P}$ by probability
distribution $\mu$, and additional care for some subtleties that arise.

######  Lemma 4.

Assume $p,q$ are independent identically distributed (iid) $[0,1]$-valued
random variables with expectation $\,\mathbb{E}[p]=c\leq 0.39$ such that
$\,\mathbb{P}(p=y)>0$ for some $1/2<y<1$. Then the modified common probability
distribution $p^{\prime},q^{\prime}$ of $p,q$ for which
$\,\mathbb{P}(p^{\prime}=1)=\,\mathbb{P}(p=1)+(2y-1)\,\mathbb{P}(p=y),\,\mathbb{P}(p^{\prime}=y)=0,\,\mathbb{P}(p^{\prime}=0.5)=\,\mathbb{P}(p=0.5)+(2-2y)\,\mathbb{P}(p=y)$
and $\,\mathbb{P}(p^{\prime}=y^{\prime})=\,\mathbb{P}(p=y^{\prime})$ for every
$y^{\prime}\in[0,1]\backslash\\{0.5,y,1\\}$ satisfies

  * •

$\,\mathbb{E}[p^{\prime}]=\,\mathbb{E}[p]$ and

  * •

$w\,\mathbb{E}[H(p^{\prime})]-\,\mathbb{E}[H(p^{\prime}+q^{\prime}-p^{\prime}q^{\prime})]>w\,\mathbb{E}[H(p)]-\,\mathbb{E}[H(p+q-pq)]$
for every $w\leq 1.044$, where $p^{\prime},q^{\prime}$ are iid.

###### Proof.

The first part is immediate since the choice of redistribution is chosen in
such a way that the following two linear combinations are true;
$(2y-1)+(2-2y)0.5=y$ and $(2y-1)+(2-2y)=1$. The latter to ensure that we still
have a probability distribution. Hence it remains to prove the second part.
For this, we first take a very small value
$\varepsilon=\frac{\,\mathbb{P}(p=y)}{N}$ by choosing a large positive integer
$N$. Let $I$ be the support (set with all values $x$ for which
$\,\mathbb{P}(p=x)>0$) with $1/2$ and $1$ included as well.

First, we do the redistribution of only an $\varepsilon$-fraction in the
probability distribution, that is
$\,\mathbb{P}(p^{\prime}=1)=\,\mathbb{P}(p=1)+(2y-1)\varepsilon,\,\mathbb{P}(p^{\prime}=y)=\,\mathbb{P}(p=y)-\varepsilon,\,\mathbb{P}(p^{\prime}=0.5)=\,\mathbb{P}(p=0.5)+(2-2y)\varepsilon.$
Now
$\,\mathbb{E}[H(p^{\prime}+q^{\prime}-p^{\prime}q^{\prime})]-\,\mathbb{E}[H(p+q-pq)]$
is equal to $\varepsilon\lambda+O(\varepsilon^{2})$, where $\lambda$ equals

$\displaystyle 2\sum_{x\in
I}\left((2-2y)h(0.5(1-x))-h((1-y)(1-x))\right)\,\mathbb{P}(p=x).$

Here we have used that $h(x+y-xy)=h((1-x)(1-y))$ (by symmetry of $h$ and
$1-(x+y-yx)=(1-x)(1-y)$) and $h(0)=0$. Let
$g(x)=(2-2y)h(0.5(1-x))-h((1-y)(1-x)).$ Note that

$\displaystyle\ln 2\frac{d}{dx}g(x)$
$\displaystyle=(y-1)\ln\left(\frac{(1-y)(1+x)}{x+y-xy}\right)>0\mbox{ and}$
$\displaystyle\ln 2\frac{d^{2}}{dx^{2}}g(x)$
$\displaystyle=-\frac{(1-y)(2y-1)}{(x+1)(x+y-xy)}<0$

since $0<(1-y)(1+x)<x+y-xy$ for $1>y>0.5$ and every $1\geq x\geq 0.$ Due to
Jensen’s inequality for the concave function $g$,
$\lambda=2\,\mathbb{E}[g(p)]$ is upper bounded by $2g(c).$ This upper bound is
independent of $\,\mathbb{P}(p=y).$ Hence we can do this $N$ times and
conclude that for $p^{\prime},q^{\prime}$ distributed as in the lemma, we have
$\,\mathbb{E}[H(p^{\prime}+q^{\prime}-p^{\prime}q^{\prime})]-\,\mathbb{E}[H(p+q-pq)]\leq
2g(c)\,\mathbb{P}(p=y)+O(\varepsilon).$ It is also straightforward to compute
that
$\,\mathbb{E}[H(p)]-\,\mathbb{E}[H(p^{\prime})]=\,\mathbb{P}(p=y)\left(h(y)-(2-2y)h(0.5)\right)=-g(0)\,\mathbb{P}(p=y).$
Finally, it suffices to prove that

$2g(c)-g(0)<0$

since then $\varepsilon$ can be chosen sufficiently small such that after
adding the $O(\varepsilon)$ term, it is still negative. Since $g$ is an
increasing function and $g(0)<0$ (due to $h$ being concave), it suffices to
prove that $2g(0.39)-1.044g(0)<0.$ This is the case for every
$\frac{1}{2}<y<1.$333Verification at
https://github.com/StijnCambie/UCconjecture/blob/main/reduction_UC.sagews ∎

######  Lemma 5.

Let $p,r$ be identically distributed $[0,1]$-valued random variables, not
necessarily independent. Then one can modify the underlying common probability
distribution by distributing the probability mass function on $(0.5,1)$ over
$0.5$ and $1$ such that $\,\mathbb{E}[p]$ is the same and

$\,\mathbb{E}\left[H\left(max\left(p^{\prime},r^{\prime},\min\left(p^{\prime}+r^{\prime},1/2\right)\right)\right)\right]\leq\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right].$

###### Proof.

We do the following procedure as long as there is some value in $(0,1)$ with
positive probability. Let $y=\max\\{y\mid 0.5<y<1\wedge\,\mathbb{P}(p=y)>0\\}$
be the largest value in $(0,1)$ with positive probability and $y^{\prime}$ the
second largest such value, or $0.5$ if no other value in $(0,1)$ has positive
probability. We distribute the probability mass function of $y$ over
$y^{\prime}$ and $1$ (such that we still end with a probability measure). We
let $p^{\prime}$ and $r^{\prime}$ be dependent as before, with the
corresponding distribution taken into account (made clear below). We claim
that the considered quantity
$\,\mathbb{E}\left[H\left(\max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]$
did not increase by doing so. If $\,\mathbb{P}(p=y,r=y)>0$, we increase
$\,\mathbb{P}(p=y^{\prime},r=y^{\prime})$ and $\,\mathbb{P}(p=1,r=1)$
accordingly and conclude by concavity of $h$. If $\,\mathbb{P}(p=y,r=1)>0$,
then $\max\\{p,r,\min(p+r,1/2)\\}=1$ both before and after the local
adaptation of $p$ (similarly when $p$ and $r$ are switched) and so there is no
change by this term. If $\,\mathbb{P}(p=y,r=z)>0$, for some $z\leq
y^{\prime}$, then we conclude again by concavity of $h.$ By iterating this
process, the probability measure on $(0.5,1)$ is distributed over $0.5$ and
$1$ and the condition in the lemma is satisfied. ∎

Now, assume there are random variables $p,q,r$ satisfying the conditions of
Question 2 for which $\,\mathbb{E}[H(p)]\leq c$ and
$(1-\alpha)\,\mathbb{E}[H(p+q-pq)]+\alpha\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]\leq\,\mathbb{E}[H(p)]$
for some $\alpha\in[0,1].$ Next, we consider the modified random variables
$p^{\prime},q^{\prime},r^{\prime}$, where the probability distribution is
iteratively adapted by distributing the probability $\,\mathbb{P}(p=y)$ for
some $0.5<y<1$ over $\,\mathbb{P}(p=0.5)$ and $\,\mathbb{P}(p=1)$. Since
$1.044>\frac{1}{1-\alpha},$ Lemma 4 implies that
$\,\mathbb{E}[H(p^{\prime})]-(1-\alpha)\,\mathbb{E}[H(p^{\prime}+q^{\prime}-p^{\prime}q^{\prime})]>\,\mathbb{E}[H(p)]-(1-\alpha)\,\mathbb{E}[H(p+q-pq)]$.
Also
$\,\mathbb{E}\left[H\left(max\left(p^{\prime},r^{\prime},\min\left(p^{\prime}+r^{\prime},1/2\right)\right)\right)\right]\leq\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]$
for the natural choice of the adapted dependency of $p$ and $r$ by Lemma 5. So
if for some value of $c$, there are probability distributions $p,q$ and $r$
for which Equation 2 is not satisfied, there are such probability
distributions for which the support is disjoint from $(0.5,1).$

###  3.2 Reduction to small support of the probability distribution

In the previous subsection, we proved that for Question 2 one can assume that
$(0.5,1)$ is not in the support of the probability distribution. In this
subsection, we narrow the support down to a finite set with at most $3$
elements.

First, we observe that the quantity
$\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]$
is minimized when $\,\mathbb{P}(p=r=1)=0$. If
$\,\mathbb{P}(p=r=1),\,\mathbb{P}(p=x,r=y)>\varepsilon>0$ for some choice
where $0<\max\\{x,y\\}<1,$ increasing the probabilities for
$\,\mathbb{P}(p=x,r=1)$ and $\,\mathbb{P}(p=1,r=y)$ with $\varepsilon$ and
decreasing $\,\mathbb{P}(p=r=1),\,\mathbb{P}(p=x,r=y)$ with $\varepsilon,$ the
quantity
$\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]$
decreases. Note that in the remaining case $\,\mathbb{P}(p=r=0)$ would be the
only other positive probability and so the whole quantity is zero. Also now
one can do the same modifications, since we always assume that
$\,\mathbb{P}(p=1)\leq\frac{1}{2}.$ Hence we can assume that
$\,\mathbb{P}(p=r=1)=0$.

Now, by the result of the previous subsection, whenever $p,r<1,$ we have
$p,r\leq\frac{1}{2}$ and hence
$max\left(p,r,\min\left(p+r,1/2\right)\right)=\min\left(p+r,1/2\right).$

Let $\,\mathbb{P}(p=1)=a$ in the remaining of this subsection. Since $h$ is an
increasing function on $[0,1/2]$, we can assume that the values $x,y$ for
which $\,\mathbb{P}(p=x,r=y)>0$ satisfy $x,y\leq x_{0}$ where $x_{0}$ is the
minimum value for which $\,\mathbb{P}(p\leq x_{0})\geq 1-2a.$ If
$\,\mathbb{P}(p=x,r=y),\,\mathbb{P}(p=x^{\prime},r=y^{\prime})\geq\varepsilon>0,$
where $x<x^{\prime}$ and $y>y^{\prime}$, we can increase them with
$\varepsilon$ and increase
$\,\mathbb{P}(p=x,r=y^{\prime}),\,\mathbb{P}(p=x^{\prime},r=y)$ with
$\varepsilon.$ The studied quantity does not increase, by the follow claim.

######  Claim 6.

The function $g\colon[0,1]\to[0,1]\colon x\mapsto h(min(x,1/2))$ is a concave
function. For every choice of reals $x<x^{\prime}$ and $y>y^{\prime}$ in
$[0,1/2]$, $g(x+y)+g(x^{\prime}+y^{\prime})\geq
g(x+y^{\prime})+g(x^{\prime}+y).$

###### Proof.

The second derivative of $g$ equals that one of $h$ on $[0,1/2)$ and is as
such strictly negative on this interval. The second derivative of $g$ is zero
for $x\geq 1/2.$

Since $(x+y)+(x^{\prime}+y^{\prime})=(x+y^{\prime})+(x^{\prime}+y)$ and
$x+y^{\prime}<x+y,x^{\prime}+y^{\prime}<x^{\prime}+y,$ the pair
$\\{x^{\prime}+y,x+y^{\prime}\\}$ majorizes $\\{x+y,x^{\prime}+y^{\prime}\\}.$
The claim now follows from Karamata’s inequality [13]. ∎

We conclude that we can assume that $\,\mathbb{P}(p=r)=1-2a$ and this happens
exactly for the $(1-2a)$-quantile, that is, for every $x<x_{0},$ we have
$\,\mathbb{P}(p=r=x)=\,\mathbb{P}(p=x)$ and
$\,\mathbb{P}(p=r=x_{0})=1-2a-\,\mathbb{P}(p<x_{0}).$

We now use the same approach as Sawin. Let $\alpha\in(0,1)$ and
$c<\frac{1}{2}$ be fixed constants. Let $\nu$ be a probability distribution
which minimizes

$H_{\nu}=(1-\alpha)\,\mathbb{E}[H(p+q-pq)]+\alpha\,\mathbb{E}^{\prime}\left[H\left(\min\left(2p,1/2\right)\right)\right]-\,\mathbb{E}[H(p)]$
(2)

among all probability distributions with expectation bounded by $c$. Such a
distribution does exist, as explained in the proof of Sawin’s Lemma 3. Here
$\,\mathbb{E}^{\prime}$ has to be interpreted as the expectation over the
$(1-2a)$-quantile. Let $\mu$ be a (possibly different from $\nu$) optimal
probability measure, for which the value of $x_{0}$ and $a=\,\mathbb{P}(p=1)$
correspond with the one of $\nu.$ Consider the combination
$\mu^{\prime}=(1-\varepsilon)\mu+\varepsilon\nu.$ Then
$\frac{H_{\mu^{\prime}}-H_{\mu}}{\varepsilon}$ equals, up to a
$O(\varepsilon)$ function,

$\displaystyle
2(1-\alpha)\left(\left(\,\mathbb{E}_{(p,q)\sim\mu\times\nu}-\,\mathbb{E}_{(p,q)\sim\mu\times\mu}\right)[H(p+q-pq)]\right)-(\,\mathbb{E}_{p\sim\nu}-\,\mathbb{E}_{p\sim\mu})[H(p)]$
$\displaystyle+\alpha\left((\,\mathbb{E}^{\prime}_{p\sim\nu}-\,\mathbb{E}^{\prime}_{p\sim\mu})\left[H\left(\min\left(2p,1/2\right)\right)\right]\right).$

Now the function
$F_{\mu}(q)=\,\mathbb{E}_{p\sim\mu}[2(1-\alpha)H(p+q-pq)-H(q)]$ satisfies
$\frac{d}{dq}\left(q(1-q)\frac{d^{2}}{dq^{2}}F_{\mu}(q)\right)<0$, as verified
in the proof of [19, Lem. 3].444Here $q$ is a single variable instead of
random variable. Similarly, we verify that $\ln
2\frac{d^{2}}{dq^{2}}h\left(2q\right)=\frac{-2}{(1-2q)q}$ and
$\frac{d}{dq}\left(q(1-q)\frac{d^{2}}{dq^{2}}h\left(2q\right)\right)=\frac{-2}{(1-2q)^{2}}<0.$

This implies that if we consider the function $F_{\mu}(q)+H\left(2q\right)$ on
the interval $[0,min\\{x_{0},1/4\\}]$ and $F_{\mu}(q)$ on the interval
$[min\\{x_{0},1/4\\},1/2]$ separately, in both cases the function is either
strictly convex on one part and strictly concave at the other part, or convex/
concave on the whole interval. On each interval, the minimum is attained by a
probability distribution that either has only one value with positive
probability, or two, one of them being the maximum of the interval. When the
latter occurs on the second interval, one can extend the interval to
$[\min\\{x_{0},1/4\\},1]$ and distribute additional mass over $1.$ This
results into at most $4$ different values. If $min\\{x_{0},1/4\\}=x_{0}$ has
positive probability on the first interval and this is different from the
(smallest) value $y_{0}$ on the second interval that got positive probability,
one can repeat the argument by replacing $x_{0}$ by $\min\\{y_{0},1/4\\}.$
This implies that in case there are $4$ values with positive probability, the
values $1$ and $\frac{1}{4}$ are among them. But when we would have
$x_{0}\geq\frac{1}{4},$ we know that
$\,\mathbb{E}^{\prime}[H(\min\\{2p,1/2\\})$ does not depend on the
distribution on the interval $[1/4,1]$ and as such, we can repeat the argument
about the extremum for $F_{\mu}(q)$. At the end, we conclude that the support
has no more than $3$ elements with positive probability. Furthermore, if the
support contains exactly $3$ elements with positive probability, at least one
of them is at most $1/4.$

###  3.3 Verification for distributions with support of size at most $3$

Once the support is reduced to $3$ elements, $\\{a_{1},a_{2},1\\}$, by knowing
the associated probabilities $p_{1},p_{2},1-p_{1}-p_{2}$ of each element, the
inequality in Question 2 can be checked. As such, we find an optimisation
problem in $4$ variables. Using Maple, it has been checked in multiple ways;
by solving a minimisation problem in multiple regimes, and by plotting an
implicit plot, as well as plots with a fixed choice for $a_{1}$ assuming
$\,\mathbb{E}[p]=c$. From these, we note that there are two local regions
where the minima occur; around $a_{1}=0$ and around $p_{1}=0$555See
https://github.com/StijnCambie/UCconjecture, documents FinalComputation23,
corresponding with the cases where $p$ is $\\{0,1\\}$-valued and the atomic
one used to show sharpness in Subsection 2.3.

Finally, we also give a more rigorous proof for the case where the
distribution is atomic, i.e., $\,\mathbb{P}(p=b)=1-a$ and
$\,\mathbb{P}(p=1)=a$ and $\,\mathbb{E}[p]=a+(1-a)b\leq c$, where $c\sim
0.3823455$ is the claimed optimum. Then
$\,\mathbb{E}[H(p+q-pq)]=(1-a)^{2}h(2b-b^{2}),\,\mathbb{E}[H(p)]=(1-a)h(b)$
and
$\,\mathbb{E}\left[H\left(max\left(p,r,\min\left(p+r,1/2\right)\right)\right)\right]\geq(1-2a)h(\min(2b,1/2)).$
Since the case where $\,\mathbb{P}(p=r=1)=0$ is the worst case, we need to
show that

$(1-\alpha)(1-a)^{2}h(2b-b^{2})+\alpha(1-2a)h(\min(2b,1/2))-(1-a)h(b)\geq
0,\mbox{ or equivalently}$
$(1-\alpha)h(2b-b^{2})a^{2}+\left(-2(1-\alpha)h(2b-b^{2})-2\alpha
h(\min(2b,1/2))+h(b)\right)a+O_{b,\alpha}(1)\geq 0$

For fixed (non-zero) $b$, this is a quadratic function in $a$ with positive
leading coefficient, which attains its minimum at $a=1+\frac{2\alpha
h(\min(2b,1/2))-h(b)}{2(1-\alpha)h(2b-b^{2})}>\frac{c-b}{1-b}$ and thus it is
sufficient to prove this in the case where $a=\frac{c-b}{1-b}.$666See
https://github.com/StijnCambie/UCconjecture/blob/main/Sharpness.sagews

##  4 Proof of bound for sharper union-closed conjecture

In this section, for completeness of the exposition, we give the formal proof
of Theorem 3, sketched in [19]. Let $\alpha\sim 0.0356069$ and $c\sim
0.3823455$ be the previously determined optimal constants for Question 2.

###### Proof of Theorem 3.

Assume there is a (nonempty) union-closed family $\mathcal{F}\subset 2^{[n]}$
for which every element $i\in[n]$ appears in at most a $c$-fraction of the
sets in $\mathcal{F}$. Without loss of generality, we can assume that $1$
appears in at least one set in $\mathcal{F}.$ We consider random variables
$A,B,C$, which are three uniform samples from $\mathcal{F}$, defined as
follows. The uniform sampling of $B$ happens independent of the sampling of
$A$ and $C.$ The latter two are sampled element-wise and in a dependent way.
We denote $A_{i}=1$ if $i\in A$ and otherwise $A_{i}=0$, i.e., it is the
indicator function $1_{i\in A}$, and $A_{<i}=(A_{1},\ldots,A_{i-1})$ is the
sequence of the first $i-1$ indicator random variables. Analogously $C_{i}$
and $C_{<i}$ are defined. For every $i\in[n],$ we consider

$p_{i}=\frac{\lvert\\{S\in\mathcal{F}\colon i\in
S,S_{<i}=A_{<i}\\}\rvert}{\lvert\\{S\in\mathcal{F}\colon
S_{<i}=A_{<i}\\}\rvert}\mbox{ and }r_{i}=\frac{\lvert\\{S\in\mathcal{F}\colon
i\in S,S_{<i}=C_{<i}\\}\rvert}{\lvert\\{S\in\mathcal{F}\colon
S_{<i}=C_{<i}\\}\rvert}.$

If $\max\\{p_{i},r_{i}\\}>0.5,$ we take $x\in U([0,1])$, a uniformly random
element from $[0,1]$, and take $A_{i}=[x<p_{i}]$ and $C_{i}=[x<q_{i}]$. That
is, conditionally on $A_{<i},$ $A_{i}=1$ if $p_{i}<x$ and analogous. Note that
there are possibly $2^{i-1}$ different values for $p_{i}$, depending on the
realisation of $A_{<i},$ i.e., $p_{i}$ is a conditional probability
distribution. Similarly, if $p_{i},r_{i}\leq 1/2,$ we take $A_{i}=[x<p_{i}]$
and $C_{i}=[0.5-r_{i}<x<0.5]$ for the uniform random generated $x\in[0,1].$
This implies that for the union $A\cup C,$ we have

$\,\mathbb{P}[(A\cup C)_{i}\mid
A_{<i},C_{<i}]=\max\\{p_{i},r_{i},\min(p_{i}+r_{i},1/2)\\}.$

By the product rule applied to the conditional probabilities $p_{i}$, we have
that $A$ (and similarly $C$) will be uniformly distributed over $\mathcal{F}$,
that is for a particular set $T\in\mathcal{F},$ we have

$\,\mathbb{P}(A=T)=\prod_{i\in T}\frac{\lvert\\{S\in\mathcal{F}\colon i\in
S,S_{<i}=T_{<i}\\}\rvert}{\lvert\\{S\in\mathcal{F}\colon
S_{<i}=T_{<i}\\}\rvert}\cdot\prod_{i\not\in
T}\frac{\lvert\\{S\in\mathcal{F}\colon i\not\in
S,S_{<i}=T_{<i}\\}\rvert}{\lvert\\{S\in\mathcal{F}\colon
S_{<i}=T_{<i}\\}\rvert}=\frac{1}{\lvert\mathcal{F}\rvert}.$

Hence $A,B,C$ all have the (same) uniform probability distribution. Let
$q_{i}=\,\mathbb{P}(i\in B\mid B_{<i}).$ The chain rule and data processing
inequality respectively yield

$H((A\cup B)_{<i+1})=H((A\cup B)_{<i})+H((A\cup B)_{<i+1}\mid(A\cup
B)_{<i})\geq H((A\cup B)_{<i})+H((A\cup B)_{<i+1}\mid A_{<i},B_{<i}),$

while $H(A_{<i+1})=H(A_{<i})+H(A_{<i+1}\mid A_{<i}).$ As such to prove that

$(1-\alpha)H(A\cup B)+\alpha H(A\cup C)>H(A),$

it is sufficient to prove that $(1-\alpha)H((A\cup B)_{1})+\alpha H((A\cup
C)_{1})>H(A_{1})$ and $(1-\alpha)H((A\cup B)_{<i+1}\mid A_{<i},B_{<i})+\alpha
H((A\cup C)_{<i+1}\mid A_{<i},C_{<i})\geq H(A_{<i+1}\mid A_{<i})$ for every
$i\geq 2.$ But with the conditional probabilities of $(A\cup B)_{i}$ being
$p_{i}+q_{i}-p_{i}q_{i}$ (by the principle of inclusion-exclusion) and of
$(A\cup C)_{i}$ being $\max\\{p_{i},r_{i},\min(p_{i}+r_{i},0.5)\\}$ (by the
choice of the samples), where $p_{i},q_{i},r_{i}$ all have expectation less
than $c$, this follows from the answer to Question 2. Since equality cannot
appear for $i=1$, the inequality is strict. We conclude that $\max\\{H(A\cup
C),H(A\cup B)\\}>H(A)=\log_{2}\lvert\mathcal{F}\rvert$, which is a
contradiction. So no such family $\mathcal{F}$ as initially assumed does
exist. ∎

#### Open access statement.

For the purpose of open access, a CC BY public copyright license is applied to
any Author Accepted Manuscript (AAM) arising from this submission.

## References

  * [1] R. Alweiss, B. Huang, and M. Sellke.  Improved Lower Bound for the Union-Closed Sets Conjecture.  arXiv e-prints, page arXiv:2211.11731, Nov. 2022.
  * [2] P. Balister and B. Bollobás.  Random union-closed families.  In Number theory, analysis, and combinatorics, De Gruyter Proc. Math., pages 1–9. De Gruyter, Berlin, 2014.
  * [3] I. Balla, B. Bollobás, and T. Eccles.  Union-closed families of sets.  J. Combin. Theory Ser. A, 120(3):531–544, 2013.
  * [4] H. Bruhn, P. Charbit, O. Schaudt, and J. A. Telle.  The graph formulation of the union-closed sets conjecture.  European J. Combin., 43:210–219, 2015.
  * [5] H. Bruhn and O. Schaudt.  The journey of the union-closed sets conjecture.  Graphs Combin., 31(6):2043–2074, 2015.
  * [6] Z. Chase and S. Lovett.  Approximate union closed conjecture.  arXiv e-prints, page arXiv:2211.11689, Nov. 2022.
  * [7] T. M. Cover and J. A. Thomas.  Elements of information theory.  Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, second edition, 2006\.
  * [8] D. Ellis.  Note: a counterexample to a conjecture of Gilmer which would imply the union-closed conjecture.  arXiv e-prints, page arXiv:2211.12401, Nov. 2022.
  * [9] P. Frankl.  Extremal set systems.  In Handbook of combinatorics, Vol. 1, 2, pages 1293–1329. Elsevier Sci. B. V., Amsterdam, 1995.
  * [10] P. Frankl and N. Tokushige.  Extremal problems for finite sets, volume 86.  American Mathematical Soc., 2018.
  * [11] J. Gilmer.  A constant lower bound for the union-closed sets conjecture.  arXiv e-prints, page arXiv:2211.09055, Nov. 2022.
  * [12] Y. Hu.  On the Union-Closed Sets Conjecture.  arXiv e-prints, page arXiv:1706.06167, June 2017.
  * [13] J. Karamata.  Sur une inégalité rélative aux fonctions convexes.  Publ. Math. Univ. Belgrade, 1:145–148, 1932.
  * [14] I. Karpas.  Two Results on Union-Closed Families.  arXiv e-prints, page arXiv:1708.01434, Aug. 2017.
  * [15] E. Knill.  Graph generated union-closed families of sets.  arXiv preprint math/9409215, 1994.
  * [16] L. Pebody.  Extension of a Method of Gilmer.  arXiv e-prints, page arXiv:2211.13139, Nov. 2022.
  * [17] D. Reimer.  An average set size theorem.  Combin. Probab. Comput., 12(1):89–93, 2003.
  * [18] I. Rival.  Graphs and order, nato asi series, vol. 147, 1985.
  * [19] W. Sawin.  An improved lower bound for the union-closed set conjecture.  arXiv e-prints, page arXiv:2211.11504, Nov. 2022.
  * [20] P. Wójcik.  Union-closed families of sets.  Discrete Math., 199(1-3):173–182, 1999.
  * [21] L. Yu.  Dimension-Free Bounds for the Union-Closed Sets Conjecture.  arXiv e-prints, page arXiv:2212.00658, Dec. 2022.