# Symmetry protected 1D chains in mixed-valence iron oxides

Denis M. Vasiukov<EMAIL_ADDRESS>Division of Synchrotron Radiation
Research, Department of Physics, Lund University, Box 118, Lund 221 00, Sweden
Ghanashyam Khanal  Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA  Ilya Kupenko  Institut für Mineralogie,
University of Münster, Corrensstr. 24, 48149 Münster, Germany  Georgios
Aprilis  ESRF, The European Synchrotron, 71 Avenue des Martyrs, CS40220, 38043
Grenoble Cedex 9, France  Sergey V. Ovsyannikov  Bayerisches Geoinstitut,
Universität Bayreuth, Universitätsstr. 30, D-95447 Bayreuth, Germany  Stella
Chariton  Bayerisches Geoinstitut, Universität Bayreuth, Universitätsstr. 30,
D-95447 Bayreuth, Germany  Valerio Cerantola  ESRF, The European Synchrotron,
71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France  Vasily
Potapkin  Bayerisches Geoinstitut, Universität Bayreuth, Universitätsstr. 30,
D-95447 Bayreuth, Germany  Aleksandr I. Chumakov  ESRF, The European
Synchrotron, 71 Avenue des Martyrs, CS40220, 38043 Grenoble Cedex 9, France
Leonid Dubrovinsky  Bayerisches Geoinstitut, Universität Bayreuth,
Universitätsstr. 30, D-95447 Bayreuth, Germany  Kristjan Haule  Department of
Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
Elizabeth Blackburn  Division of Synchrotron Radiation Research, Department of
Physics, Lund University, Box 118, Lund 221 00, Sweden

###### Abstract

During the last decade of high-pressure research a whole new series of iron
oxides was discovered, like Fe4O5, Fe5O6, Fe7O9 etc. [lavina2011discovery, ;
lavina2015unraveling, ; sinmyo2016discovery, ; bykova2016structural, ;
ishii2018synthesis, ; koemets2021chemical, ], featuring closely related
structures with arrays of one-dimensional (1D) chains of trigonal prisms
embedded between slabs of octahedra. Here, we develop a unified approach to
the series based on a specific crystallographic generation mechanism which
predicts the structures of these oxides and naturally classifies them in terms
of the slab cycle. When including magnetic interactions, we show that the 1D
chains have a symmetry protection against magnetic perturbations from the iron
ions in the slabs, and that the slab size determines the type of magnetic
order, which is either ferromagnetic or antiferromagnetic. Dynamical mean-
field theory calculations reveal the orbitally selective Mott state of the Fe
ions and tendency of conductivity to low-dimensional behavior with particular
enhancement along the 1D chains. Across the series, the decoupling of the
chains increases, and so with the inherent charge ordering of the slabs, these
structures have the potential to allow experimental realization of the model
system of coupled 1D wires [vishwanath2001two, ; kane2002fractional, ;
teo2014luttinger, ; iadecola2016wire, ; meng2015coupled, ]. We point out the
possibility to stabilize these compounds in the thin-film form that, together
with a wide range of possible ionic substitutions and fact that these
compounds are recoverable at ambient pressure, makes them a very promising
platform to engineer physical systems with interesting magnetotransport
phenomena, as corroborated by the recent discovery of quantum Hall effect in
ZrTe5 [tang2019three, ].

Of the many new high-pressure iron oxides discovered during the last decade,
the set consisting of Fe4O5, Fe5O6, HP-Fe3O4 etc. [lavina2011discovery, ;
lavina2015unraveling, ; sinmyo2016discovery, ; bykova2016structural, ;
ishii2018synthesis, ; koemets2021chemical, ] is particularly special. The
oxides in this set share a common packing motif and can be united into a
$n$FeO$\cdot m$Fe2O3 homologous seriesbykova2016structural . Many members of
this series are recoverable at ambient conditions after high-pressure
synthesis and feature iron ions in mixed-valent statesovsyannikov2018pressure
; sinmyo2016discovery ; maitani2022electrical , resembling magnetite (Fe3O4),
where this electronic state results in the famous metal-insulator Verwey
transition at low temperaturesverwey1939electronic , with formation of a
peculiar charge-ordered ground state with trimer units of iron
ionssenn2012charge ; baldini2020discovery . Indeed, similar Verwey-type
transitions were found in Fe5O6 [ovsyannikov2020room, ] and Fe4O5
[ovsyannikov2016charge, ; ovsyannikov2018pressure, ].

In this paper we focus on an overlooked feature of these compounds, namely
that part of the iron ions is arranged in arrays of symmetry protected 1D
chains. We use both experimental and computational methods to explore these
arrays and argue that, under certain plausible conditions, they could host
coupled one-dimensional (1D) wires. Such systems have been extensively studied
in theory, but have proven difficult to realize
experimentallyvishwanath2001two ; kane2002fractional ; teo2014luttinger ;
iadecola2016wire ; meng2015coupled . These models yield a broad spectrum of
fascinating states, from non-Fermi liquidvishwanath2001two  to a zoo of
topological phaseskane2002fractional ; teo2014luttinger ; iadecola2016wire ,
including fractional quantum Hall states, in the case of conductive wires
(_viz_ Luttinger liquids), and spin liquids in the case of insulating
wiresmeng2015coupled .

Fig. 1: The $\\{311\\}$ tropochemical cell-twinning in the homologous series
of iron oxides as exemplified by the Fe4O5 structure. On the left, the FeO
structure is projected along $[0\bar{1}1]$. Regularly applying a mirror plane
parallel to (311) at the oxygen sheet at $5d_{311}^{\text{FeO}}$ intervals
(slab cycle 3), we derive a structure with a new unit cell. Upon this
operation, pairs of iron atoms, as outlined by the blue triangle, are brought
into proximity across twinning planes. The last step is the coalescence of
these pairs into a single iron atom that leads to a new crystallographic site
at the twinning plane with a trigonal prismatic oxygen environment, as marked
by the blue colour in the Fe4O5 structure on the right. These trigonal prisms
form infinite 1D chains along the $a$ axis in the resultant structure. The
slab width $N$ can be readily determined using either of two equivalent
octahedral chains highlighted by the violet and green cages in the Fe4O5
structure. For this particular slab cycle the twinning plane is preserved as a
symmetry element in the generated structure. The periodicity $p$ in the plane
of projection of FeO is equivalent to the $b$ lattice constant of a new unit
cell. Hereafter we use orange and blue colors to designate the octahedral and
trigonal prismatic sites respectively.

We begin with the new insight that stoichiometric variation in the
$n$FeO$\cdot m$Fe2O3 series resembles the well-known Magnéli
phasesschwingenschlogl2004vanadium . This motivated us to investigate the
underlying crystallographic mechanism responsible for generating the variety
of structures seen in this homologous series. There is a close link between
the $n$FeO$\cdot m$Fe2O3 series and the ambient pressure $n$PbS$\cdot m$Bi2S3
homologous series, with isostructural correspondence between Fe5O6 and the
mineral lillianite Pb3Bi2S6, and between Fe5O7 and PbBi4S7, the so-called V1
phase takeuchi1997tropochemical . This resemblance reveals the common
crystallographic generative mechanism behind both series. In the case of the
$n$PbS$\cdot m$Bi2S3 series, it was studied in detail and named tropochemical
cell-twinningtakeuchi1997tropochemical . We illustrate this mechanism in
Figure 1 by way of deriving the Fe4O5 structure from wüstite (FeO).

The starting point is the parent rocksalt structure of wüstite. If a mirror
plane is applied parallel to the {311} plane at a sheet of oxygen atoms, we
generate a twin. Consecutively applying this operation with some periodic
spacing (multiple of $d_{311}^{\text{FeO}}$), we get a new unit cell with
identical slabs of the FeO6 octahedra between twinning planes. As seen from
Figure 1, some iron atoms are brought together at the twinning plane upon this
procedure. Merging these atoms gives a new iron site with a trigonal prismatic
oxygen environment. The trigonal prisms share common triangular facets and
form an array of 1D chains along the $a$ axis separating the octahedral slabs
(Fig. 1). This also eliminates iron atoms from the structure, modifying the
stoichiometry.

Depending on the spacing between twinning planes, one obtains structures with
different slab widths. There are two equivalent ways to define this slab size:
(_i_) the number, $n$, of $d_{311}^{\text{FeO}}$ spacings – in other words the
number of the oxygen sheets per slab – and (_ii_) the number, $N$, of
octahedra in either of the two octahedral chains along
$\langle\bar{2}11\rangle$ and $\langle 0\bar{1}\bar{1}\rangle$ directions in
the initial wüstite structure. Hereafter we will use the latter notation, but
there is a simple conversion between them, namely $n=N+2$.

The $\\{311\\}$ cell-twinning mechanism considered here can eliminate up to
one third of the cations from the initial rocksalt structure given the
smallest slab width of one octahedron, which corresponds to the $\eta$-Fe2O3
phase bykova2016structural . Variation in stoichiometry leads to changes in
the average cation valence state, starting from pure +2 in FeO to pure +3 in
$\eta$-Fe2O3, with the formation of numerous mixed-valence compounds at
intermediate stoichiometries.

In Extended Data Table I, the known members of the $n$FeO$\cdot m$Fe2O3 series
are collected. It is natural to classify these oxides according to the slab
cycle, or sequence of slabs, which entirely determines the symmetry and
crystallographic positions in the generated structures. Most of the compounds
in Extended Data Table I can be assigned to two groups, the $N$-family and
$N$,$N$+1-family. The $N$-family has the simplest cycle, with only one type of
slab, and crystallizes in an orthorhombic space group $Cmcm$ (#63). The mirror
plane perpendicular to the $c$ axis in this family is the initial twinning
plane as illustrated in Figure 1. There is a subtle crystallographic
difference between compounds with even and odd $N$ that has important
consequences for the magnetic structure (see Supplementary Materials for
details).

The $N$,$N$+1-family contains slabs of two different widths differing by one
octahedron. This slab cycle does not allow preservation of the twinning plane
as a symmetry element in the resulting structures. Instead, only symmetry
elements relevant to the $a$ axis of $Cmcm$ space group are preserved, and the
unit cell becomes monoclinic with space group $C2/m$ (#12).

Generally speaking the slab cycle can be arbitrarily complex, as seen in the
phase-V subseries of the lillianite structuretakeuchi1997tropochemical . The
first sign of this complexity in the iron oxides is the recently discovered
Fe7O10 [koemets2021chemical, ], with a trinomial slab cycle 1,1,2 which
contains two different twinning planes between the 1,1 and 1,2 slabs, where
only the first one is preserved as a mirror plane in the resulting structure.
New iron oxides with more complex slab cycle are likely to be discovered in
the future.

Fig. 2:  Mössbauer spectra of several members of the $n$FeO$\cdot m$Fe2O3
homologous series at room temperature. The blue and orange components
correspond to iron ions in the trigonal prismatic and octahedral environment,
respectively. The pink line is the fit residual. In the paramagnetic region
all of these oxides show similar spectra consisting of two distinct doublets
corresponding to the iron ions in different oxygen environment, here
exemplified by HP-Fe3O4 and Fe7O9. The spectra of the Fe4O5 and Fe4O5 oxides
show an unexpected coexistence of the magnetic sextet from the octahedral slab
with the paramagnetic doublet of the trigonal prismatic site. This means that
magnetic ordering in the octahedral slab does not influence iron ions in the
chains of trigonal prisms. The spectrum of HP-Fe3O4 was collected at 20 GPa
whereas other spectra were acquired at ambient pressure; the spectrum of Fe7O9
was already published in [sinmyo2016discovery, ].

The representative Mössbauer spectra of the studied oxides are shown in Fig.
2. Their basic feature is the presence of two distinct components. One belongs
to the octahedral slab while the other corresponds to the chains of trigonal
prisms. The data show that in all recoverable oxides, namely Fe4O5, Fe5O6 and
Fe7O9, the chains are exclusively populated by Fe2+ ions at ambient
conditions. This observation allows us to estimate the average oxidation state
of the Fe ions in the slabs and the isomer shifts of the slab components are
indeed in a good agreement with the predicted values (Extended Data Table II).

An intriguing feature of the spectra in Fig. 2 is the apparent coexistence of
the magnetically ordered iron ions in the slabs with the disordered iron ions
in the chains of Fe4O5 and Fe5O6. Accordingly, we deduce that in both
compounds slabs and chains have two different ordering temperatures and that
the magnetic ordering of the 1D chains is independent from that of the slabs.
This peculiar magnetic behavior was also observed in the isostructural
compounds MnFe3O5 and CoFe3O5 [hong2018MnFe3O5, ; hong2018CoFe3O5, ] and it is
confirmed to be stable by our theoretical calculations. Thus our data
invalidates the previously published model of magnetic structure for Fe4O5
[ovsyannikov2016charge, ], where the simultaneous magnetic ordering of all
iron ions was assumed.

Among the oxides presented in Fig. 2 the HP-Fe3O4 is the only non-recoverable
phase and so its spectrum was acquired at 20 GPa, after in situ synthesis in
diamond anvil cell. Although HP-Fe3O4, as an $N=2$ member of the $N$-family,
could, in principle, possess iron ions in an integer oxidation state, our data
unexpectedly revealed the presence of mixed-valent states in both
crystallographic positions, i.e. in the chains and slabs.

Fig. 3:  The expected magnetic configurations for the charge-averaged state in
the $N$-family. The transition from paramagnetic to completely magnetically
ordered state proceeds via the intermediate partially ordered state (a), with
the magnetically ordered slabs but the disordered 1D chains. The complete
magnetic order differs in odd and even $N$ compounds due to the ferromagnetic
and antiferromagnetic _interchain_ ordering, respectively. We illustrate this
here with (b) $N=3$, i.e. Fe4O5, and (c) $N=2$, i.e. HP-Fe3O4, structures.

To understand the origin of the independent magnetic behavior of spins in
slabs and chains it is instructive to establish a generic scheme of exchange
interactions in these structures. Let us consider them for Fe4O5, the $N=3$
member of the $N$-family, since in this case there are neutron diffraction
data hong2018CaFe3O5 ; hong2018MnFe3O5 ; hong2018CoFe3O5  for several
isostructural compounds and solid solutions differing only by cations in the
chains (namely Ca, Mn or Co), which revealed an identical collinear
antiferromagnetic order of the slabs at room temperature with a (0 0 0)
propagation vector presented in Fig. 3a.

In this magnetic structure, the magnetic moments in the slabs align along the
$c$-axis, with ferromagnetic exchange along the $a$-axis and antiferromagnetic
exchange along the $b$\- and $c$-axis (Fig. 3a). This pattern of exchange
interactions in the octahedral slab should be relevant not only in Fe4O5 but
can be generalized to all members of the $N$-family. This conjecture is
supported by our DMFT calculations for HP-Fe3O4 and the data from CaFe5O7,
which is equivalent to the $N=5$ case, where the same magnetic configuration
is adopted in the ground statedelacotte2017morin . As for the chains,
application of the well-known Goodenough-Kanamori-Anderson (GKA) rules
suggests that they should have relatively weak _intrachain_ ferromagnetic
coupling in case of Fe2+ ions (see Suppl. Materials).

Fig. 4:  Theoretical eDMFT spectral functions of HP-Fe3O4 (top row) and Fe4O5
(bottom row) in three different states: paramagnetic (b and f), partially
ordered state with ordered magnetic slabs and disordered chains as in Fig. 3a
(c and g) and complete long range ordering established on all Fe atoms as in
Fig. 3c and 3b (d and h) in magnetic space groups #63.465 and #63.464,
respectively. In a and e the first primitive Brillouin zones are shown with
respective momentum pathway used in the plots, and between them the
corresponding crystallographic axes are displayed to show the relative
orientation of reciprocal space to the real space, namely $\Gamma-R$ is
parallel to the $a$ axis, $\Gamma-Y$ is parallel to the $b$ axis and
$\Gamma-Z$ is parallel to the $c$ axis.

The fact that ordering in the 1D chains is independent of the ordering in the
slabs is due to symmetry protection provided by the preserved twinning plane
in the $N$-family. For the case of the antiferromagnetic structure in Fig. 3a,
the induced Weiss fields from magnetically ordered adjacent octahedral slabs
exactly cancel out each other for iron atoms located at the mirror plane.
Therefore magnetic interactions in the 1D chains are _symmetry-protected_
against perturbations from the octahedral slabs and also from the interchain
interactions along the $c$-axis. Consequently, the chains have lower ordering
temperatures, as determined by the intrachain interactions.

At intermediate temperatures this leads to the peculiar magnetic structure
with coexistence of ordered slabs and disordered chains (Fig. 3a) that is
observed in our Mössbauer spectra of Fe4O5 and Fe5O6 at room temperature in
Fig. 2. From these observations it immediately follows that such independent
magnetic ordering cannot be observed in the monoclinic $N,N$+1-family. Indeed,
in our DMFT simulations for Fe7O9 we observe non-zero Weiss field in the
chains for all types of magnetic order in the slabs, contrary to Fe4O5 and HP-
Fe3O4 where coexisting magnetic state (Fig. 3a) is found to be stable, with no
induced Weiss field in the chains.

Therefore upon complete magnetic ordering for the $N$-family we expect to see
ferromagnetic chains embedded in the antiferromagnetic background of the slabs
(Fig. 3b and 3c). The subtle crystallographic difference between odd and even
$N$ compounds mentioned above results in ferromagnetic _interchain_ ordering
in the odd $N$ case (Fig. 3b) and antiferromagnetic _interchain_ ordering in
the even $N$ case (Fig. 3c). Thus the odd and even $N$ compounds should have a
ferromagnetic and an antiferromagnetic ground state, respectively (see Suppl.
Materials for more details).

Ab-initio simulations have been performed by the combination of Density
Functional Theory and embedded Dynamical Mean Field Theory (DFT+eDMFT) for two
members of the $N$-family, HP-Fe3O4 and Fe4O5, i.e. even and odd $N$ case,
respectively, in three different states at 120 K: (_i_) the paramagnetic state
with all local moments disordered, (_ii_) the intermediate partially ordered
state with magnetically ordered slabs but disordered 1D chains (Fig. 3a), and
(_iii_) the state with complete magnetic order on all Fe atoms (Fig. 3b and
3c).

In accordance with the expected mixed-valent state, both compounds are
metallic in all three magnetic configurations. Overall our results are
consistent with previous paramagnetic (DFT+eDMFT) calculations of Fe4O5 and
Fe5O6 [yang2021metallic, ; qin2021site, ] and the computed oxidation states of
Fe atoms are in good agreement with our Mössbauer data (see occupation number
$n_{d}$ in Extended Data Table III).

The obtained single-particle spectral functions are displayed in Fig. 4. Fe
atoms in the slabs are coordinated by the oxygen octahedra, hence the $t_{2g}$
orbitals are lower in energy and are partially occupied with finite density of
states at the Fermi level, whereas the $e_{g}$ orbitals are almost exactly
half-filled, and are fully gapped with the large Mott gap exceeding 1 eV, see
Extended Data Fig. 5 and 6. In the 1D chains, the Fe atoms are sitting in
trigonal prisms (see Suppl. Materials for the orbital splitting in the
trigonal prism), and here the $z^{2}$ and $x^{2}-y^{2}$ orbitals appear
metallic, while the other three orbitals are gapped.

The divergent self-energy, seen as sharp poles in Extended Data Fig. 5 and 6,
attests to the Mott insulating nature of the gapped orbitals, and shows
enormous scattering rate values for them. This implies that the Fe atoms in
the paramagnetic state appear in a very unusual configuration, namely the
orbitally selective Mott state anisimov2002orbital ; koga2004orbital ;
koga2005spin ; biermann2005non ; liebsch2005novel ; werner2007high ;
de2009orbital ; vojta2010orbital ; huang2012pressure ; herbrych2018spin ;
herbrych2020block ; kim2022orbital , which is well known for its bad metallic
behaviour. The Mott-insulating localized electrons provide a strong source of
spin scattering for the electrons in the metallic orbitals, and so the latter
cannot properly conduct electrons biermann2005non . Perhaps the most famous
example of such selective Mott states are the colossal magnetoresistive
manganites tokura1999colossal ; pascut2020role , but several other systems
were recently studied as potential candidates to harbor such unusal electronic
configurations anisimov2002orbital ; de2009orbital ; vojta2010orbital ;
kim2022orbital . Due to this large spin scattering, the spectral functions of
these paramagnetic states (Fig. 4b and 4f) are very diffuse and show no
evidence for developed conducting bands around the Fermi level. The predicted
electrical conductivity at 100 K is of the order of $10^{5}$ S/m and is rather
isotropic (see the Extended Data Table III).

Upon magnetic ordering of the slabs, the spin-scattering on octahedrally
coordinated Fe atoms is arrested, and so the electrons on these atoms
experience very small scattering rates. The metallic orbitals now start to
form bands which cross the Fermi level (see Fig. 4c and 4g), but these bands
are still relatively fuzzy, because the spin scattering in the chains remains
active. The plot of the density of states and the scattering rate in Extended
data Figs. 5 and 6 confirm that not much has changed for the Fe atoms in the
chains, even though the slabs are now magnetically ordered. Consequently, the
conductivity has increased by approximately a factor of two (see Extended Data
Table III), but it is still quite isotropic and small for a metallic system.

Finally the magnetic moments in the chains also order, as depicted in Fig. 3b
and 3c, which removes all spin scattering at the Fermi level, making the
system a normal magnetic metal with sharp bands near the Fermi level, as shown
in Fig. 4d and 4h. This state shows the peculiar tendency to low-dimensional
behavior, namely the strong dispersion of the bands along the $\Gamma-R$ and
$F(C)-Y$ direction of the first Brillouin zone, as shown in Fig. 4a and 4e,
while along the orthogonal directions the bands are mostly flat. The first
direction is along the 1D chains of trigonal prisms, i.e. the real space $a$
axis, therefore we expect enhanced conductivity along the 1D chains.

Indeed, the calculation results in Extended Data Table III show increased
anisotropic conductivity with largest values along the chains in this ordered
state. In HP-Fe3O4 conductivity increases by an order of magnitude and the
anisotropy ($2\sigma^{a}/[\sigma^{b}+\sigma^{c}]$) is about two, although we
find that it is very sensitive to doping and can become as large as five upon
a small shift of the Fermi level (see Extended Data Table IV).

It is interesting to note that in the completely ordered magnetic state of HP-
Fe3O4 the flat parts of the bands at the Fermi level are mostly formed by the
$z^{2}$ and $x^{2}-y^{2}$ orbitals of the Fe atoms in the 1D chains, as
evident by the narrow peak in the corresponding densities of states at the
Fermi level (see Extended Data Fig. 5). Therefore, the chains show a clear
tendency to act as an 1D conductors.

Up to now we have treated these compounds only in the charge-averaged state.
However, at low temperatures the inherent mixed-valence state of the slabs can
act as a natural switch between metallic and insulating behavior by means of
the charge-ordering transition ovsyannikov2016charge ; ovsyannikov2020room .
This means that the anisotropy could be enhanced even further in the true
charge-ordered ground state, leaving the arrays of chains as the only
conductive channel at low temperature.

The $N$-family is therefore a 3D system with an emergent dimensionality
reduction. This is brought about because the tropochemical cell-twinning
mechanism magnetically isolates the iron ions in the trigonal prisms from the
surrounding slabs, allowing the 1D nature of the chains to manifest in
physical properties like the conductivity. This conclusion is rather general
and this symmetry protection of magnetic interactions in the chains also
applies under ionic substitutionhong2018MnFe3O5 ; hong2018CoFe3O5  and in
other examples even with the (1/2 0 0) magnetic propagation vector, as in
stoichiometric CaFe3O5 [cassidy2019single, ].

Crucially, these materials are transferable to the ambient pressure realm, as
distinct to the superconducting high-pressure
hydridespickard2020superconducting . Half of the oxides listed in Extended
Data Table III are recoverable and, in addition, these structures show great
compositional flexibility that considerably affects their stability
fieldwoodland2013fe ; boffa2015complete ; hong2019thesis . For instance,
compounds with $N\geq 3$ can be synthesized _even at ambient pressure_ if the
Fe atoms in the chains are substituted by Ca [evrard1980mise, ;
malaman1981preparation, ] while another interesting alternative is to
substitute oxygen with other chalcogens, as exemplified by the lillianite
casetakeuchi1997tropochemical .

Furthermore, the crystallographic derivation mechanism described here suggests
that it may be possible to stabilize these compounds in the thin-film form.
The inherent connection with the rock salt structure via tropochemical cell-
twinning (Fig. 1) means that (011)-oriented substrates with rock salt
structure could provide a coherent interface for the (100)-oriented films of
the $N$-family members.

To conclude, the distinct magnetic and electronic properties of the structures
in this series make them candidates for the realization of a large variety of
interesting physical systems, particularly considering magnetoresistance. One
possible implementation should be highlighted, namely the coupled quantum wire
modelsvishwanath2001two ; kane2002fractional ; teo2014luttinger ;
iadecola2016wire ; meng2015coupled , because the arrays of conductive 1D
chains described here would be an excellent testbed for them. However the
range of possible phenomena is much wider and the recent observation of the 3D
quantum Hall effect in ZrTe5 [tang2019three, ] confirm this, as it is
equivalent to the $N=3$ member of $N$-family with empty cationic sites in the
slabs.

#### Acknowledgements

We are grateful to Daniel I. Khomskii and Pavel A. Volkov for inspiring
discussions. D.V. and E.B. thank the Swedish Research Council for support
under grant number 2018-04704. G.K. and K.H. acknowledge the support of
National Science Foundation by the grant number DMR-1709229. We acknowledge
the European Synchrotron Radiation Facility for provision of synchrotron
radiation resources at the beamline ID18.

#### Author contributions

S.O., L.D. and D.V. prepared the samples. D.V., I.K., G.A., S.C., V.C., V.P.,
L.D. and A.C. performed the Mössbauer experiments. G.K. and K.H. performed the
DFT+eDMFT calculations. D.V. developed the crystallographic approach. D.V.,
K.H. and E.B. interpreted the results and wrote the manuscript with input from
all co-authors.

## Methods

### Sample preparation

The single crystals of Fe4O5 and Fe5O6 were synthesized in a 1200-tonne multi-
anvil press at the Bavarian Research Institute of Experimental Geochemistry
and Geophysics (BGI) at 14-16 GPa and 1200-1400 ∘C from the stoichiometric
mixtures of of Fe3O4 (Aldrich, 99.99% purity) and Fe (99.99%). The details of
the synthesis were described in previous worksovsyannikov2018pressure ;
ovsyannikov2020room . For the Mössbauer experiments we selected single
crystals using a three-circle Bruker diffractometer equipped with a SMART APEX
CCD detector and a high-brilliance Rigaku rotating anode (Rotor Flex FR-D, Mo-
Kα radiation) with Osmic focusing X-ray optics.

To obtain well-crystallized pure HP-Fe3O4 phase we performed an _in situ_
synthesis in a BX90 diamond anvil cellkantor2012bx90  (DAC) using laser
heating at 20 GPa from a magnetite single crystal enriched with 57Fe. We used
a double-sided laser-heating system with infrared laserskupenko2012portable ;
aprilis2017portable . The phase transition from spinel structure to the high-
pressure phase was clearly seen from Mössbauer spectra as at this pressure the
spinel phase has two well-developed magnetic sextets while the spectrum of HP-
Fe3O4 is a superposition of two paramagnetic doublets.

### Mössbauer spectroscopy

The Mössbauer spectra were collected at the Nuclear Resonance beamline
ruffer1996nuclear  (ID18) at the European Synchrotron Radiation Facility
(ESRF) using the synchrotron Mössbauer source (SMS) potapkin201257fe . The SMS
is based on a high-quality single crystal of 57FeBO3 which acts as a nuclear
monochromator. Energy modulation is achieved by means of a standard Mössbauer
velocity transducer on which the borate crystal is mounted. The advantages of
SMS are the small beam spot size, about 10–15 $\mu$m, and the absence of non-
resonant radiation that greatly improves the data quality and decreases the
acquisition time for experiments with DACs.

The folded Mössbauer spectra contain 512 channels and were fitted using the
MossA software prescher2012mossa , version 1.01a. We used a transmission
integral fit assuming a Lorentzian-squared line shape of the SMS and a
Lorentzian or pseudo-Voigt line shape for the absorber. The line width and the
center shift of the SMS was controlled before and after collection of each
spectrum using the reference single line absorber (K2Mg57Fe(CN)6). All center
shifts are reported relative to $\alpha$-Fe at ambient conditions.

The assignment of the oxidation state of iron ions was done based on the
comparison of the determined isomer shift values with literature data of high-
spin iron ions in oxygen environmentmenil1985systematic . As a reference for
the 6-fold oxygen coordinated iron at ambient pressure we took values of
octahedral Fe2.5+ in magnetite (0.67 mm/s) and Fe3+ in hematite (0.36 mm/s),
to minimize the influence of the inductive effectmenil1985systematic . The
isomer shift of the iron ion in the +2.5 mixed-valent state is just an average
of the isomer shifts of pure divalent and trivalent iron. Therefore from the
magnetite value it follows that pure Fe2+O6 would have 0.98 mm/s isomer shift
and this allows us to calculate the isomer shift for iron ions in any
intermediate oxidation state between +2 and +3. As can been seen from Extended
Data Table II the iron ions in the trigonal prisms of Fe4O5, Fe5O6 and Fe7O9
oxides have isomer shifts very close to our estimate, confirming occupation of
the trigonal prismatic sites by pure Fe2+. Under this assumption we can
calculate the average oxidation state of iron ions in the octahedral slabs and
in the same manner predict isomer shift values for them. In Extended data
Table II one can see a remarkable agreement between experimental and predicted
values.

The spectrum of HP-Fe3O4 was collected after synthesis at 20 GPa so we need to
make a correction for the pressure dependence of isomer shift which we
thoroughly studied in our previous workVasiukov_spin_transition . Although
from the stoichiometry of HP-Fe3O4 one would expect only pure Fe2+ and Fe3+ in
the chains and slabs, respectively, the determined isomer shift values are
quite far away from the expected numbers (Extended Data Table II). Note that
the deviation from the calculated number is opposite for the chains and slabs,
which corresponds to an admixture of trivalent iron to the chains and divalent
iron to the slabs. Therefore the mixed-valent state is adapted by both
crystallographic positions in this phase. The results of our DMFT calculations
in Extended Data Table III also show a similar trend.

### DFT+eDMFT calculations

The theoretical calculations were carried out by the combination of DFT and
eDMFT kotliar2006electronic , as implemented in WIEN2k and the Rutgers eDMFT
code blaha2020wien2k ; haule2010dynamical ; haule2018structural . These
calculations are charge self-consistent and use exact double-counting between
DFT and DMFT haule2015exact . For the self-consistent calculation we set
$RK_{max}$ (which determines the size of basis) to be 7.0 and used 1000
$k$-points for the $k$-point sampling in the irreducible Brillouin zone. To
compute the resistivity, we used up to 10000 $k$-points. We adopted the local-
density approximation (LDA) perdew1992accurate , which gives the best results
for lattice properties when combined with eDMFT haule2015free . The self-
energy on the real axis was obtained using the analytical continuation by
maximum entropy method. The experimental crystal structures used in
calculations were taken from works [lavina2011discovery, ;
bykova2016structural, ], namely Fe4O5 at 10 GPa and HP-Fe3O4 at 44 GPa. We
used $U$=10.0 eV and $J_{H}=1.0$ eV for the Coulomb repulsion and Hund’s
coupling respectively, as obtained by the constrained DMFT calculation. The
auxiliary impurity problem was solved using a continuous-time quantum Monte
Carlo impurity solver haule2007quantum .

The magnetic structures were first guessed by finding the Weiss field on each
Fe atom in the presence of the exchange field of the neighboring Fe atoms,
which can provide several candidate orderings which can be further simulated
for their stability. Among several trial configurations, we found that the
theoretical ordering of Fe magnetic moments, depicted in Fig. 3, is indeed
energetically favourable to other configurations.

Note that the spin configurations used in calculations (Extended Data Figs. 7
and 8) are non-collinear, and canted along the high-symmetry axis of the
octahedra, which minimizes the sign problem in quantum Monte Carlo. Namely, in
the DMFT calculation the local coordinate axis is chosen such that it
coincides with the high symmetry axis of the polyhedron, which minimizes the
non-diagonal hybridization, and hence reduces the sign problem. The spin-
orientation along the $z$-axis then does not introduce additional off-diagonal
hybridization, which makes the sign problem in magnetic calculation
approximately equal to the paramagnetic case.

Extended data Table III shows the Fe occupation numbers $n_{d}$. In this work,
the DMFT calculations were carried out in the complete Linear Augmented Plane
Wave basis, and the occupation numbers were computed by integrating the
electronic charge within the touching Muffin-tin spheres around iron atoms and
projecting them to the relevant $d$ orbitals.

The theoretical conductivity in the same table was calculated as the zero
frequency limit of the optical conductivity, which is computed as the
convolution of the single-particle Green’s functions with the proper velocity
matrix elements. Notice that the vertex corrections to conductivity vanish
within the single site eDMFT framework because considered here structure have
the inversion symmetry. All visualizations of crystal and magnetic structures
in this work were prepared using the VESTA softwaremomma2011vesta .

## References

  * (1) Lavina, B. _et al._ Discovery of the recoverable high-pressure iron oxide Fe4O5.  _Proc. Natl. Acad. Sci._ 108, 17281–17285 (2011).
  * (2) Lavina, B. & Meng, Y.  Unraveling the complexity of iron oxides at high pressure and temperature: Synthesis of Fe5O6.  _Sci. Adv._ 1, e1400260 (2015).
  * (3) Sinmyo, R. _et al._ Discovery of Fe7O9: a new iron oxide with a complex monoclinic structure.  _Sci. Rep._ 6, 32852 (2016).
  * (4) Bykova, E. _et al._ Structural complexity of simple Fe2O3 at high pressures and temperatures.  _Nat. Commun._ 7 (2016).
  * (5) Ishii, T., Uenver-Thiele, L., Woodland, A. B., Alig, E. & Ballaran, T. B.  Synthesis and crystal structure of Mg-bearing Fe9O11: New insight in the complexity of Fe-Mg oxides at conditions of the deep upper mantle.  _Am. Mineral._ 103, 1873–1876 (2018).
  * (6) Koemets, E. _et al._ Chemical stability of FeOOH at high pressure and temperature, and oxygen recycling in early Earth history.  _Eur. J. Inorg. Chem._ 2021, 3048–3053 (2021).
  * (7) Vishwanath, A. & Carpentier, D.  Two-dimensional anisotropic non-Fermi-liquid phase of coupled Luttinger liquids.  _Phys. Rev. Lett._ 86, 676 (2001).
  * (8) Kane, C., Mukhopadhyay, R. & Lubensky, T.  Fractional quantum Hall effect in an array of quantum wires.  _Phys. Rev. Lett._ 88, 036401 (2002).
  * (9) Teo, J. C. & Kane, C.  From Luttinger liquid to non-Abelian quantum Hall states.  _Phys. Rev. B_ 89, 085101 (2014).
  * (10) Iadecola, T., Neupert, T., Chamon, C. & Mudry, C.  Wire constructions of Abelian topological phases in three or more dimensions.  _Phys. Rev. B_ 93, 195136 (2016).
  * (11) Meng, T., Neupert, T., Greiter, M. & Thomale, R.  Coupled-wire construction of chiral spin liquids.  _Phys. Rev. B_ 91, 241106 (2015).
  * (12) Tang, F. _et al._ Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5.  _Nature_ 569, 537–541 (2019).
  * (13) Ovsyannikov, S. V. _et al._ Pressure tuning of charge ordering in iron oxide.  _Nat. Commun._ 9, 4142 (2018).
  * (14) Maitani, S., Sinmyo, R., Ishii, T., Kawaguchi, S. I. & Hirao, N.  The electrical conductivity of Fe4O5, Fe5O6, and Fe7O9 up to 60 GPa.  _Phys. Chem. Miner._ 49, 1–8 (2022).
  * (15) Verwey, E.  Electronic conduction of magnetite (Fe3O4) and its transition point at low temperatures.  _Nature_ 144, 327–328 (1939).
  * (16) Senn, M. S., Wright, J. P. & Attfield, J. P.  Charge order and three-site distortions in the Verwey structure of magnetite.  _Nature_ 481, 173–176 (2012).
  * (17) Baldini, E. _et al._ Discovery of the soft electronic modes of the trimeron order in magnetite.  _Nat. Phys._ (2020).
  * (18) Ovsyannikov, S. V. _et al._ Room-temperature Verwey-type transition in iron oxide, Fe5O6.  _Angew. Chem., Int. Ed._ 59, 5632–5636 (2020).
  * (19) Ovsyannikov, S. V. _et al._ Charge-ordering transition in iron oxide Fe4O5 involving competing dimer and trimer formation.  _Nat. Chem._ 8, 501 (2016).
  * (20) Schwingenschlögl, U. & Eyert, V.  The vanadium Magnéli phases VnO2n-1.  _Ann. Phys._ 13, 475–510 (2004).
  * (21) Takèuchi, Y.  _Tropochemical Cell-twinning: A Structure-building Mechanism in Crystalline Solids_.  Material science of minerals and rocks (Terra Scientific Pub., 1997).
  * (22) Hong, K. H., Arevalo-Lopez, A., Coduri, M., McNally, G. & Attfield, J. P.  Cation, magnetic, and charge ordering in MnFe3O5.  _J. Mater. Chem. C_ 6, 3271–3275 (2018).
  * (23) Hong, K. H., Solana-Madruga, E., Coduri, M. & Attfield, J. P.  Complex cation and spin orders in the high-pressure ferrite CoFe3O5.  _Inorg. Chem._ 57, 14347–14352 (2018).
  * (24) Hong, K. H., Arevalo-Lopez, A. M., Cumby, J., Ritter, C. & Attfield, J. P.  Long range electronic phase separation in CaFe3O5.  _Nat. Commun._ 9, 2975 (2018).
  * (25) Delacotte, C. _et al._ Morin-like spin canting in the magnetic CaFe5O7 ferrite: a combined neutron and Mössbauer study.  _J. Solid State Chem._ 247, 13–19 (2017).
  * (26) Yang, A. _et al._ Metallic nature and site-selective magnetic moment collapse in iron oxide Fe4O5 at the extreme conditions of Earth’s deep interior.  _Phys. Lett. A_ 414, 127607 (2021).
  * (27) Qin, Q.-Y. _et al._ Site-selective magnetic moment collapse in compressed Fe5O6.  _Chin. Phys. Lett._ 38, 089101 (2021).
  * (28) Anisimov, V., Nekrasov, I., Kondakov, D., Rice, T. & Sigrist, M.  Orbital-selective Mott-insulator transition in Ca2-xSrxRuO4.  _Eur. Phys. J. B_ 25, 191–201 (2002).
  * (29) Koga, A., Kawakami, N., Rice, T. & Sigrist, M.  Orbital-selective Mott transitions in the degenerate Hubbard model.  _Phys. Rev. Lett._ 92, 216402 (2004).
  * (30) Koga, A., Kawakami, N., Rice, T. & Sigrist, M.  Spin, charge, and orbital fluctuations in a multiorbital Mott insulator.  _Phys. Rev. B_ 72, 045128 (2005).
  * (31) Biermann, S., de’ Medici, L. & Georges, A.  Non-Fermi-liquid behavior and double-exchange physics in orbital-selective Mott systems.  _Phys. Rev. Lett._ 95, 206401 (2005).
  * (32) Liebsch, A.  Novel Mott transitions in a nonisotropic two-band Hubbard model.  _Phys. Rev. Lett._ 95, 116402 (2005).
  * (33) Werner, P. & Millis, A. J.  High-spin to low-spin and orbital polarization transitions in multiorbital Mott systems.  _Phys. Rev. Lett._ 99, 126405 (2007).
  * (34) de’ Medici, L., Hassan, S. R., Capone, M. & Dai, X.  Orbital-selective Mott transition out of band degeneracy lifting.  _Phys. Rev. Lett._ 102, 126401 (2009).
  * (35) Vojta, M.  Orbital-selective Mott transitions: Heavy fermions and beyond.  _J. Low Temp. Phys._ 161, 203–232 (2010).
  * (36) Huang, L., Wang, Y. & Dai, X.  Pressure-driven orbital selective insulator-to-metal transition and spin-state crossover in cubic CoO.  _Phys. Rev. B_ 85, 245110 (2012).
  * (37) Herbrych, J. _et al._ Spin dynamics of the block orbital-selective Mott phase.  _Nat. Commun._ 9, 1–10 (2018).
  * (38) Herbrych, J., Alvarez, G., Moreo, A. & Dagotto, E.  Block orbital-selective Mott insulators: a spin excitation analysis.  _Phys. Rev. B_ 102, 115134 (2020).
  * (39) Kim, M., Kim, H.-S., Haule, K. & Vanderbilt, D.  Orbital-selective Mott phase and non-Fermi liquid in FePS3.  _Phys. Rev. B_ 105, L041108 (2022).
  * (40) Tokura, Y. & Tomioka, Y.  Colossal magnetoresistive manganites.  _J. Magn. Magn. Mater._ 200, 1–23 (1999).
  * (41) Pascut, G. L. & Haule, K.  Role of orbital selectivity on crystal structures and electronic states in BiMnO3 and LaMnO3 perovskites.  _arXiv preprint arXiv:2005.12179_ (2020).
  * (42) Cassidy, S. J., Orlandi, F., Manuel, P. & Clarke, S. J.  Single phase charge ordered stoichiometric CaFe3O5 with commensurate and incommensurate trimeron ordering.  _Nat. Commun._ 10 (2019).
  * (43) Pickard, C. J., Errea, I. & Eremets, M. I.  Superconducting hydrides under pressure.  _Annu. Rev. Condens. Matter Phys._ 11, 57–76 (2020).
  * (44) Woodland, A. _et al._ Fe4O5 and its solid solutions in several simple systems.  _Contrib. Mineral. Petrol._ 166, 1677–1686 (2013).
  * (45) Boffa Ballaran, T., Uenver-Thiele, L. & Woodland, A. B.  Complete substitution of Fe2+ by Mg in Fe4O5: The crystal structure of the Mg2Fe2O5 end-member.  _Am. Mineral._ 100, 628–632 (2015).
  * (46) Hong, K. H.  _Synthesis and properties of CaFe 3O5 and related materials_.  Ph.D. thesis, The University of Edinburgh (2019).
  * (47) Evrard, O. _et al._ Mise en évidence de CaFe4O6 et détermination des structures cristallines des ferrites de calcium CaFe2+nO4+n ($n=1,2,3$): nouvel exemple d’intercroissance.  _J. Solid State Chem._ 35, 112–119 (1980).
  * (48) Malaman, B. _et al._ Preparation et caracterisation des ferrites de calcium CaFe2+nO4+n a valeurs fractionnaires de $n$ (3/2, 5/2) et leur incidence sur le diagramme Fe-Ca-O a 1120 ∘C.  _Mater. Res. Bull._ 16, 1139–1148 (1981).
  * (49) Kantor, I. _et al._ BX90: A new diamond anvil cell design for X-ray diffraction and optical measurements.  _Rev. Sci. Instrum._ 83, 125102 (2012).
  * (50) Kupenko, I. _et al._ Portable double-sided laser-heating system for Mössbauer spectroscopy and X-ray diffraction experiments at synchrotron facilities with diamond anvil cells.  _Rev. Sci. Instrum._ 83, 124501 (2012).
  * (51) Aprilis, G. _et al._ Portable double-sided pulsed laser heating system for time-resolved geoscience and materials science applications.  _Rev. Sci. Instrum._ 88, 084501 (2017).
  * (52) Rüffer, R. & Chumakov, A. I.  Nuclear resonance beamline at ESRF.  _Hyperfine Interact._ 97, 589–604 (1996).
  * (53) Potapkin, V. _et al._ The 57Fe synchrotron Mössbauer source at the ESRF.  _J. Synchrotron Radiat._ 19, 559–569 (2012).
  * (54) Prescher, C., McCammon, C. & Dubrovinsky, L.  MossA: a program for analyzing energy-domain Mössbauer spectra from conventional and synchrotron sources.  _J. Appl. Crystallogr._ 45, 329–331 (2012).
  * (55) Menil, F.  Systematic trends of the 57Fe Mössbauer isomer shifts in (FeOn) and (FeFn) polyhedra. Evidence of a new correlation between the isomer shift and the inductive effect of the competing bond ${T}-{X}$ ($\rightarrow$Fe) (where ${X}$ is O or F and ${T}$ any element with a formal positive charge).  _J. Phys. Chem. Solids_ 46, 763–789 (1985).
  * (56) Vasiukov, D. M. _et al._ Pressure-induced spin pairing transition of Fe3+ in oxygen octahedra.  _arXiv preprint arXiv:1710.03192v1_ (2017).
  * (57) Kotliar, G. _et al._ Electronic structure calculations with dynamical mean-field theory.  _Rev. Mod. Phys._ 78, 865 (2006).
  * (58) Blaha, P. _et al._ WIEN2k: An APW+lo program for calculating the properties of solids.  _J. Chem. Phys._ 152, 074101 (2020).
  * (59) Haule, K., Yee, C.-H. & Kim, K.  Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn5, CeCoIn5, and CeRhIn5.  _Phys. Rev. B_ 81, 195107 (2010).
  * (60) Haule, K.  Structural predictions for correlated electron materials using the functional dynamical mean field theory approach.  _J. Phys. Soc. Jpn._ 87, 041005 (2018).
  * (61) Haule, K.  Exact double counting in combining the dynamical mean field theory and the density functional theory.  _Phys. Rev. Lett._ 115, 196403 (2015).
  * (62) Perdew, J. P. & Wang, Y.  Accurate and simple analytic representation of the electron-gas correlation energy.  _Phys. Rev. B_ 45, 13244 (1992).
  * (63) Haule, K. & Birol, T.  Free energy from stationary implementation of the DFT+DMFT functional.  _Phys. Rev. Lett._ 115, 256402 (2015).
  * (64) Haule, K.  Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base.  _Phys. Rev. B_ 75, 155113 (2007).
  * (65) Momma, K. & Izumi, F.  VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data.  _J. Appl. Crystallogr._ 44, 1272–1276 (2011).
  * (66) Foster, P. & Welch, A.  Metal-oxide solid solutions. Part 1.–lattice-constant and phase relationships in ferrous oxide (wustite) and in solid solutions of ferrous oxide and manganous oxide.  _Trans. Faraday Soc._ 52, 1626–1635 (1956).

Extended data Fig. 5:  Theoretical eDMFT electronic density of states (DOS)
and self-energy in HP-Fe3O4 for three different states: the paramagnetic state
(top row), the intermediate state with ordered slabs and disordered chains
(middle row) and the completely ordered state (bottom row). For each state we
show the projected electronic DOS on the Fe $3d$ orbitals in the upper panel,
while the lower panel displays the imaginary part of the self-energy, which
gives the electronic scattering rate. The plots for the slabs ($8f$ site) and
the chains ($4c$ site) are contoured with orange and blue frames,
respectively. The up and down arrows mark the spin-majority and spin-minority
states, respectively.

Extended data Fig. 6:  Theoretical eDMFT electronic density of states (DOS)
and self-energy in Fe4O5 for three different states: the paramagnetic state
(top row), the intermediate state with ordered slabs and disordered chains
(middle row) and the completely ordered state (bottom row). For each state we
show the projected electronic DOS on the Fe $3d$ orbitals in the upper panel,
while the lower panel displays the imaginary part of the self-energy, which
gives the electronic scattering rate. The plots for the slabs ($8f$ and $4a$
sites) and the chains ($4c$ site) are contoured with orange and blue frames,
respectively. The up and down arrows mark the spin-majority and spin-minority
states, respectively.

Extended data Fig. 7:  The canted magnetic structure and local coordinate systems used for the eDMFT calculations of HP-Fe3O4. The red, green and dark blue axes correspond to the local $x$, $y$ and $z$ axes, respectively. Extended data Fig. 8:  The canted magnetic structure and local coordinate systems used for the eDMFT calculations of Fe4O5. The red, green and dark blue axes correspond to to the local $x$, $y$ and $z$ axes, respectively, in accord with the previous figure. The light green octahedra correspond to Fe atoms at the $4a$ Wyckoff site. Extended data Table 1:  Crystallographic data of the $n$FeO$\cdot m$Fe2O3 homologous series. Among the currently known oxides of this series, two subseries can be distinguished with different type of the slab cycle: an orthorhombic $N$-family with space group $Cmcm$ and a monoclinic $N$,$N$+1-family with space group $C2/m$. The recently discovered Fe7O10 has a more complicated slab cycle. The relevant data of the parent wüstite structure (FeO) are also provided for comparison. The $a$ and $b$ lattice constants of the $N$-family are equivalent to the $b$ and $a$ lattice constants of the $N$,$N$+1-family, respectively. Compound | SG | Slab cycle | Iron sites | P, GPa |  _a_ , Å |  _b_ , Å |  _c_ , Å | $\beta$ | Ref.
---|---|---|---|---|---|---|---|---|---
$N$-family |  |  | Tr. prism | Octahedron |  |  |  | exp. | calc. |  |
$\eta$-Fe2O3 | $Cmcm$ | 1 | 4c(m2m) | 4a(2/m..) | 64 | 2.640 | 8.639 | 6.414 |  |  | [bykova2016structural, ]
HP-Fe3O4 | $Cmcm$ | 2 | 4c(m2m) | 8f(m..) | 44 | 2.694 | 9.282 | 9.309 |  |  | [bykova2016structural, ]
Fe4O5 | $Cmcm$ | 3 | 4c(m2m) | 4a(2/m..) $|$ 8f(m..) | ambient | 2.891 | 9.802 | 12.580 | 12.965 |  | [ovsyannikov2016charge, ]
Fe5O6 | $Cmcm$ | 4 | 4c(m2m) | 8f(m..) $|$ 8f(m..) | ambient | 2.877 | 9.917 | 15.340 | 15.558 |  | [ovsyannikov2020room, ]
$N$,$N$+1-family |  |  |  |  |  |  _b_ , Å |  _a_ , Å |  |  |  |
Fe5O7 | $C2/m$ | 1,2 | 4i(m) | 2a(2/m) $|$ 4i(m) | 41 | 2.733 | 9.208 | 8.270 |  | $105.50^{\circ}$ | [bykova2016structural, ]
Fe7O9 | $C2/m$ | 2,3 | 4i(m) | 2d(2/m) $|$ 4i(m) $|$ 4i(m) | ambient | 2.895 | 9.696 | 11.428 | 11.916 | $101.69^{\circ}$ | [sinmyo2016discovery, ]
111Fe9O11 was stabilized as a solid solution with Mg and its actual composition is Mg0.9Fe8.1O11 [ishii2018synthesis, ].Fe9O11 | $C2/m$ | 3,4 | 4i(m) | 2a(2/m) $|$ 4i(m) $|$ 4i(m) $|$ 4i(m) | ambient | 2.892 | 9.844 | 14.176 | 14.480 | $99.956^{\circ}$ | [ishii2018synthesis, ]
|  |  |  |  |  |  _a_ , Å |  _b_ , Å |  |  |  |
Fe7O10 | $Cmcm$ | 1,1,2 | 4c(m2m) $|$ 8f(m..) | 8f(m..) $|$ 8f(m..) | 64 | 2.652 | 8.767 | 21.96 |  |  | [koemets2021chemical, ]
Fe0.94O | $Fm\bar{3}m$ | $\infty$ |  |  | ambient |  222The periodicity along <110>.3.041 |  333The periodicity along <332>.10.084 |  444The actual cubic lattice constant.4.3 |  |  | [foster1956metal, ]
Extended data Table 2: Parameters of Mössbauer spectra of the studied high-pressure iron oxides at room temperature. $\delta_{CS}$ is the center shift, $\Delta E_{Q}$ is the quadrupole splitting, $\varepsilon$ is the quadrupole shift and $B_{hf}$ is the hyperfine magnetic field. The expected octahedral Fe2+:Fe3+ ratio and calculated center shift values (see Methods section for details) are evaluated assuming that the trigonal prismatic site is occupied only by Fe2+. Compound | HP-Fe3O4 | Fe4O5 | Fe5O6 | Fe7O9
---|---|---|---|---
Tr. prism/oct. |  1:2 |  1:3 |  1:4 |  2:5
Expected oct. Fe2+/Fe3+ ratio | 0:1 | 1:2 | 1:1 | 1:4
Pressure | 20(1) GPa | ambient | ambient | ambient
| Tr. prism | Oct. | Tr. prism | Oct. | Tr. prism | Oct. | Tr. prism | Oct.
expt. $\delta_{CS}$, mm/s | 0.73(2) | 0.45(2) | 0.98(2) | 0.59(1) | 0.97(3) | 0.70(1) | 0.954(3) | 0.561(3)
calc. $\delta_{CS}$, mm/s | $0.90\pm 0.04$ | $0.30\pm 0.02$ | 0.98 | 0.57 | 0.98 | 0.67 | 0.98 | 0.48
$\Delta E_{Q}$/$2\varepsilon$, mm/s | 2.32(4) | $\sim 0$ | 2.34(4) | 0.44(1) | 1.08(4) | 0.67(1) | 2.736(6) | 0.348(4)
$B_{hf}$, T | 0 | 0 | 0 | 24.78(5) | 0 | 32.87(4) | 0 | 0
Area, % | 36(5) | 64(5) | 21(4) | 79(4) | 19(2) | 81(2) | 26.9(6) | 73.1(6)

Extended data Table 3:  Theoretical eDMFT electric conductivity $\sigma$ at $100\,$K, occupation $n_{d}$ and magnetic moment $M$ for each Fe-site in HP-Fe3O4 and Fe4O5. Here $\sigma^{b}$ and $\sigma^{c}$ are the conductivities along the $b$ and $c$ axes, respectively, while $\sigma^{a}$ corresponds to the $a$ axis, i.e. along the 1D chains. The slabs of Fe4O5 have another Fe-site at the $4a$ Wyckoff position, in addition to the $8f$ position. Compound | Magnetic order |  $\sigma$, $10^{5}\cdot$S/m | Fe occupation $n_{d}$ | Magn. moment, $\mu_{B}$
---|---|---|---|---
|  | $\sigma^{a}$ | $\sigma^{b}$ | $\sigma^{c}$ | (4c, 8f, [4a]) | (4c, 8f, [4a])
| _Complete ordering_ | 55 | 17 | 25 | (5.80, 5.58) | (4.12, 4.28)
HP-Fe3O4 | _Ordered slabs_ | 2.4 | 2.0 | 2.3 | (5.80, 5.57) | (0, 4.28)
| _Paramagnetic_ | 1.2 | 1.1 | 0.9 | (5.80, 5.57) | (0, 0)
| _Complete ordering_ | 5.0 | 3.0 | 3.0 | (5.83, 5.80, 5.51) | (3.44, 4.08, 4.4)
Fe4O5 | _Ordered slabs_ | 2.2 | 2.3 | 2.4 | (5.90, 5.76, 5.58) | (0, 4.06, 4.34)
| _Paramagnetic_ | 1.9 | 1.5 | 1.6 | (5.92, 5.72, 5.61) | (0, 0, 0)

Extended data Table 4:  Variations of the conductivity and its anisotropy in HP-Fe3O4 on electronic doping in the completely ordered magnetic state. The discrepancy in conductivity values with Extended Data Table III is due to the use of a less dense mesh in these calculations. Doping, electron per formula unit | Conductivity, S/m
---|---
| $\sigma^{a}$ | $\sigma^{b}$ | $\sigma^{c}$
$+0.09$ | 30.6 | 16.1 | 16.4
$+0.06$ | 115.0 | 37.1 | 34.8
0 | 43.5 | 15.0 | 28.5
$-0.05$ | 15.0 | 5.2 | 6.3
$-0.07$ | 25.6 | 5.9 | 5.4


# Colonel Blotto Game: An Analysis and Extension to Networks111I thank Kevin
Ren for introducing me to the Colonel Blotto game, and Prof. Daniel Kane for
reading and offering valuable suggestions on an earlier draft. This project
was originally conceived and executed while I was in eighth grade for
submission to the 2021 Greater San Diego Science and Engineering fair.

Sidarth Erat222La Jolla High School, La Jolla, email<EMAIL_ADDRESS>
###### Abstract

The Colonel Blotto game, introduced by Borel in the 1920s, is often used for
modeling various real-life settings, such as elections, lobbying, etc. The
game is based on the allocation of limited resources by players to a set of
fields. Each field is “won” and a corresponding field-specific value is
obtained by the player who sends the most resources. In this paper, we
formulate a discrete Blotto game played on a general accessibility network
(i.e., the bipartite graph made of players and the fields they can allocate
resources to). The primary goal is to find how the topology of the
accessibility network controls the existence and uniqueness of equilibrium
allocations, and how it affects the fraction of fields that are entered and
the average payoff of players at equilibrium.

We establish that, in a 2-regular topology, when the values of fields are
close enough and the number of players is not a multiple of 4, then there is a
unique equilbrium. We also prove that players are better off and fields are
more likely to be entered in a regular topology than a random topology. We
find numerically that dispersion of field weights negatively affects average
player payoff. The main contribution is a framework for analyzing contests
where players are permitted access to some (but not necessarily all) venues of
competition.

##  1 Introduction

Multiplayer resource allocation contests are ubiquitous in daily life. The
Colonel Blotto game, introduced by Borel almost one hundred years ago, is one
of the most basic models of resource allocation contests.

As an example, consider a population of cheetahs roaming over a large domain.
In the classical Blotto model, cheetahs allocate their energy to hunting
gazelles in various locations, and the cheetah that allocates the most energy
in a given location gets the gazelle in that location. However, the classical
model discounts the possibility that not every location is accessible to every
cheetah. Specifically, a cheetah might be able to hunt only in their
“neighborhood” hunting grounds.

There is a natural way to correspond these restrictions on players’ movement
to an “accessibility network,” which is a bipartite graph with the set of
players as one part and the fields as the another part. This paper aims to
find how these network restrictions on which venues can be accessed by each
player affect the equilibria of the game and “field survival rate” (in the
phrasing of our example, the number of locations where no cheetahs chooses to
hunt).

In order to focus our attention to the network structure, we consider a
discrete Boolean Blotto game (i.e., each player has a single resource that
cannot be split between fields). We first focus on 2-regular networks and give
conditions for the existence of unique Nash equilibria. In this case, it is
also found that at equilibrium, greater heterogeneity in field weights leads
to lower average payoffs and higher survival rates. We then direct our
attention to comparing and contrasting k-regular networks with randomly
generated networks, with the additional assumption that fields have the same
weight. In this case we find that in a random topology, survival rate is
higher and average payoffs are lower.

The rest of the article is organized as follows: In §2 we offer a review of
the literature. We introduce and formalize the model in §3. §4 solves the case
of a 2-regular network, and §5 offers the comparison between random and
regular network topologies. In §6, we discuss the results.

##  2  Literature Review

The Colonel Blotto game has been extensively researched for a century, and the
literature is vast. So, we offer a selective review, focusing on a succession
of key studies that illustrate the gradual development and refinement of the
Colonel Blotto model.

The Colonel Blotto game was originally introduced by Emile Borel in the early
20th century. The game he analyzed involved 2 players, each with a fixed
amount of resources. The players allocate these resources to battlefields of
different weights, each won by the player that sends the most resources to it.
Each player tries to maximize the total weight of all fields won. The game,
which, on the surface, appears quite simple, has still not been fully solved
for arbitrary weights and resource levels. Below are several variants of the
game analyzed over the last 100 years:

Before the development of formal modern game theory, in 1921, Borel [1953]
established that the discrete 2-player Colonel Blotto (where both players have
$B$ discrete resources) has a pure strategy solution for all $B<7$. In the
1920’s, Borel and Ville further analyzed the game, presenting mixed strategy
equilibria (i.e., where strategies are probability distributions over the set
of possible plays) for the two-player, equal weight, equal-resource case of
the continuous Colonel Blotto game. Borel is responsible for the “hex
solution,” an equilibrium based on the random picking of a strategy
$(x_{1},x_{2},x_{3})$ from a recursively-defined hexagonal lattice.

In a memorandum of Gross and Wagner [1950], the 2 field case and the 3 field,
equal resource case was fully solved. This was the first step in finding
equilibria of two player Blotto games.

Roberson [2006] fully solves the equal weight case of two person Blotto,
assuming each person can access every venue of competition. Its major
methodological contribution is the use of copulas to couple marginal
equilibrium distributions, which can be done by applying Sklar’s theorem.

Boix-Adserà et al. [2021] analyzes multiplayer Blotto games in general. The
paper focuses on a Boolean Blotto game, somewhat similar in structure to the
variant used in this paper. The resources sent to any given field in Boolean
Blotto are always in the set $\\{0,1\\}$. This is the first significant paper
that fully solves a Boolean variant of multiplayer Blotto. However, the paper
uses a symmetric variant, rather than one with variable field valuations
and/or restrictions on troop movement, the latter of which is the primary
focus of the current study.

To summarize, in Borel and Ville’s original analysis of the game, solutions
were introduced for very specific cases, including the equal weight, equal
resource case. Gross and Wagner [1950] introduced full solutions for the 2
field, 2 player case, along with the 2 player, 3 field, equal resource case.
Later, Roberson [2006] utilized copulas to combine marginal equilibrium
distributions for the 2 player, equal-weight case into a viable multivariate
distribution. Recent research has found fairly elementary equilibria for
certain subcases of the game, including the multiplayer variant (both
continuous and Boolean). A solution for equilibria of the Boolean variant is
given in Boix-Adserà et al. [2021]. Through extensive analysis over the
decades, significant progress has been made both in the full solution of the
2-player case of Colonel Blotto’s game and the solution of various discrete
and/or multiplayer Blotto variants. The current study adds to this literature
by considering an orthogonal dimension, namely possible constraints on the
players ability to access different fields, and takes a first cut at
understanding how the “accessibility network” plays a role in determining the
equilibrium outcomes.

##  3 The model

Consider a Blotto game on $n$ players (which are henceforth called “cheetahs”)
and $m$ fields (which are henceforth called “gazelles”). Each cheetah has a
finite ”hunting budget” (the total amount of effort it can exert hunting). Let
these cheetahs have hunting budgets $b_{1},b_{2},\cdots b_{n}$, and let the
gazelles have weights $w_{1},w_{2},\cdots w_{m}$.

Define the accessibility matrix to be an $n\times m$ matrix with entries
$a_{ij}$ such that $a_{ij}\in\\{0,1\\}$, and $a_{ij}=1$ iff the $i$th cheetah
is connected to the $j$th gazelle (i.e., $j$th gazelle is accessible to
cheetah $i$).

Furthermore, let the cheetah $i$’s budget allocation be represented by
$\vec{x_{i}}\in\mathbb{R}^{m}$ with entries $x_{i1},x_{i2},\dots x_{im}$ such
that:

  1. (1)

$\sum_{j=1}^{m}x_{ij}\leq b_{i}$

  2. (2)

$0\leq x_{ij}\leq b_{i}a_{ij}$

with the interpretation that $x_{ij}$ is the budget allocated by cheetah $i$
to gazelle $j$.

We assume that the cheetah that exerts the most effort hunting a specific
gazelle wins that gazelle. That is, gazelle $j$ is won by cheetah $i$ if
$x_{ij}>0$ and for all $p\neq i$, $x_{ij}>x_{pj}$. If there is a tie for
highest effort among $k$ cheetahs, each cheetah gets value $w_{j}r^{k}$, for a
fixed constant $r$ (where $0\leq r<1$).

In other words, the $i$th cheetah has total winnings:

$\pi(i)=\sum_{j=1}^{m}w_{j}\prod_{p\neq i}g(i,j,p)$

where

$g(i,j,p)=\begin{cases}0\text{ if }x_{ij}<x_{pj}\\\ 1\text{ if
}x_{ij}>x_{pj}\\\ r\text{ if }x_{ij}=x_{pj},\text{ for some fixed }0<r<1\\\
\end{cases}$

The cheetahs play mixed strategies over the set of all possible $\bar{x_{i}}$.
In other words, let

$X_{i}=\\{\vec{x_{i}}|0\leq x_{ij}\leq b_{i}a_{ij};\sum_{j=1}^{m}x_{ij}\leq
b_{i}\\}$

Hence, a strategy for cheetah $i$ is a distribution over the set $X_{i}$.

We shall consider only the case where $b_{i}=1$ for all $i$, and additionally
assume that $x_{ij}\in\\{0,1\\}$. More specifically, this paper focuses only
on a discrete Boolean Colonel Blotto variant (i.e., each player has only one
unit of resources and splitting of resources between fields is disallowed).

##  4 Case of 2-regular topology

Firstly consider the case of a 2-regular topology, specifically a network
topology where $m=n$ and, for each $i$, cheetah $i$ can hunt only gazelles $i$
and $i+1$ (indices are taken mod $n$). In this case, every mixed strategy of
cheetah $i$ can be represented as some number $p_{i}$ such that $0<p_{i}<1$,
so that cheetah $i$ goes to gazelle $i$ with probability $p_{i}$ and goes to
gazelles $i+1$ with probability $1-p_{i}$. For any mixed strategy equilibrium,
the expected payoff from playing either pure strategy must be the same, or
cheetah $i$ would lower its expected payoff by mixing.

We first characterize how interior point mixed strategy equilibria should look
if they exist.

######  Lemma 1.

If the tuple of strategies $(p_{1},p_{2},\dots p_{n})$ is an interior point
equilibrium (i.e., equilibrium where $0<p_{i}<1$), then for all $i$,

$w_{i}p_{i-1}+w_{i+1}p_{i+1}=2w_{i+1}-w_{i}$

Note: Non-interior point equilibria need not satisfy the set of equations.

###### Proof.

Note that if some tuple $(p_{1},p_{2},\dots p_{n})$ 333indices are taken mod
$n$ is an equilibrium, then for every $i$, the expected payoff of exerting all
its effort hunting gazelle $i$ is:

$w_{i}\left(1*p_{i-1}+\frac{1-p_{i}}{2}\right)=\frac{w_{i}}{2}(1+p_{i})$

The expected payoff of spending its entire budget on gazelle $i+1$ is:

$w_{i+1}\left(1-\frac{p_{i+1}}{2}+1*(1-p_{i+1})\right)$

Equating these two expressions, for all $i$,

$w_{i}p_{i-1}+w_{i+1}p_{i+1}=2w_{i+1}-w_{i}$

∎

The above lemma says nothing about the existence of interior point equilibria.
So, we now analyze conditions for the existence and uniqueness of equilibria.
To do this, we first prove the following auxiliary lemma.

######  Lemma 2.

Let $M$ be an $n\times n$ matrix such that $a_{ij}=1$ if $i=j-1$ or $i=j+1$,
and $a_{ij}=0$ otherwise. Then, $M$ is invertible if and only if $n$ does not
divide 4

###### Proof.

If $4|n$, color column $\vec{x_{i}}$ red if $i\equiv 1,2\pmod{4}$, and blue if
$i\equiv 0,3\pmod{4}$.

In every row, with adequate rotation of columns to preserve color, the nonzero
entries form a block: $[1\text{ }0\text{ }1]$. Clearly, however this is
placed, each 1 must be colored differently. Hence, the sum of the blue colored
entries is equal to the sum of the red colored entries. Therefore, the columns
are linearly dependent and A is not invertible.

Now consider the case where $n\not\equiv 0\pmod{4}$. Now let
$t_{1},t_{2},\dots$ be some set of real numbers such that
$\sum_{i=0}t_{i}\vec{x_{i}}=\vec{0}$. Note that by Lemma 1, for all $i$:

$t_{i-1}+t_{i+1}=0,t_{i+1}+t_{i+3}=0\implies t_{i-1}=t_{i+3}$

where all values $i$ are taken $\pmod{n}$

For convenience, split this case into 3 subcases: (1) $n=4a+1$ for some
natural $a$, (2) $n=4a+3$ for some natural $a$, and (3) $n=4a+2$ for some
natural $a$.

If $n=4a+1$, note that from basic modular arithmetic, for all $i$,

$t_{i}=t_{i+4}=\dots=t_{n+i-1}=t_{i-1}=t_{i+3}$

Plugging in $i+3$, $t_{i+3}=t_{i}=t_{i+6}$. Hence,
$t_{i}=t_{i+6}=t_{i+2}=-t_{i}$, so $t_{i}=0$ for all $i$. Therefore, the
columns of $M$ are linearly independent and $M$ is invertible.

If $n=4a+3$, note again that:

$t_{i}=t_{i+4}=\dots=t_{n+i-3}=t_{i+1}$

Plugging in $i+1$, $t_{i+1}=t_{i+2}$ Hence, $t_{i}=t_{i+1}=t_{i+2}=-t_{i}$.
Therefore, $t_{i}=0$ for all $i$. Therefore, the columns of $M$ are linearly
independent and $M$ is invertible.

Finally, if $n=4a+2$, note that:

$t_{i}=t_{i+4}=\dots=t_{n+i-2}=t_{i-2}=t_{i+2}$

Hence, $t_{i}=t_{i+2}=-t_{i}$, so $t_{i}=0$ for all $i$. Then, the columns of
$M$ must be linearly independent, and $M$ is again invertible.

Therefore, for all cases where $4\not|n,$ $M$ is invertible. ∎

######  Theorem 1.

When $n$ does not divide 4, there exists a unique interior point equilibrium
for sufficiently small
$\dfrac{(\max_{j}{w_{j}}-\min_{j}{w_{j}})}{\max_{j}w_{j}}$.

###### Proof.

Without loss of generality, assume that $\max_{j}w_{j}=1.$ We first claim that
an equilibrium exists if $w_{i}=w_{j}$ for all $i,j$. Moreover, this
equilibrium is $p_{i}=1/2$ for all $i$.

This is true since if every cheetah $i$, for all $i\neq 1$ goes to gazelle $i$
with probablity $1/2$ and gazelle $i+1$ with probability $1/2$, then for all
strategies $p_{1}$ that cheetah $1$ has, the expected winnings $W(p_{1})$ by
playing $p_{1}$ is:

$(1/2)p_{i}+1/2(1-p_{i})+1/4(p_{i})+1/4(1-p_{i})=3/4$

regardless of $p_{i}$. Hence, this strategy is indeed optimal and the
equilibrium $(1/2,1/2,\cdots)$ is the solution of the set of equations in
Lemma 1.

Moreover, the determinant of a matrix is a continuous polynomial in the
entries, so the set of invertible matrices is open. Therefore, since the
matrix $M$ (defined as in Lemma 2) is invertible when weights are equal and
$4\not|n$, the new coefficient matrix should still be invertible when
$\max_{j}w_{j}-\min_{j}w_{j}$ is small enough. Therefore, the solution we
found to this system must be the unique set of strategies that satisfies this
constraint. Hence, this is the only equilibrium. ∎

This result immediately gives an algorithm to obtain the Nash equilibrium,
simply by solving the system given in Lemma 1. To better analyze the effect of
dispersion in weights on each players strategy, let the weights of the
gazelles be in arithmetic sequence with common difference $\epsilon$, and let
gazelle $\lceil n/2\rceil$ WLOG have weight 1. By solving our model for
various cases of $\epsilon$, we find that the average average cheetah payoff
decreases as $\epsilon$ increases, as shown below when $n=11$ (left). However,
the expected payoffs for specific cheetahs might increase or decrease, as
shown when $n=11$:

##  5 Random vs. Regular topology

We analyze two network topologies in this section, the $k-$ regular topology
and $k-$ random topology, defined as follows: (i) In the $k-$regular topology,
each cheetah $i$ is connected to $k$ gazelles, with these $k$ gazelles being
$i,i+1,\cdots,i+k-1$ (mod $m$) in the case of regular topology, and (ii) In
the $k$-random topology, while each cheetah $i$ is still connected to $k$
gazelles, these $k$ gazelles are selected randomly from the total $m$
gazelles. It is important to note that each cheetah is only assumed to be
aware of the $k$ gazelles that she is connected to, and not of the gazelles
any other cheetah may access.

We shall assume that the gazelle weights are identical. Moreover, attach some
cost $v$ to hunting a particular gazelle. All other assumptions of the
original model are preserved.

Let each cheetah not hunt at all with probability $1-p$, and let the
probability that each cheetah hunts a given gazelle accessible to it be
$\frac{p}{k}$. We seek to find the conditions where this is an equilibrium.

######  Lemma 3.

In a regular small-world topology as described, the equilibrium probability
$p^{\ast}$ of hunting is:

$p^{\ast}=\begin{cases}1&\text{if
}v<\left(1-\frac{1}{k}+\frac{r}{k}\right)^{k-1}\\\
\frac{k(1-v^{1/(k-1)})}{1-r}&\text{otherwise}\end{cases}$

Moreover, the chance that a given gazelle survives is:

$\left(1-\frac{p^{\ast}}{k}\right)^{k}$

###### Proof.

Assume that each cheetah hunts some gazelle with probability $p$ when in
equilibrium. Evidently, cheetah 1’s expected winnings $W$ by sending all
resources to $1$ is:

$W=(1-p+\frac{k-1}{k}p)^{k-1}+\binom{k}{1}(1-p+\frac{k-1}{k}p)^{k-2}\frac{1}{k}r^{2}\dots$

By the binomial theorem,

$W(p)=(1-p+\frac{k-1}{k}p+\frac{rp}{k})^{k-1}=(1-\frac{1-r}{k}p)^{k-1}$

Now, if $w\geq W(1)$, any given cheetah mixes on the strategies of not hunting
and hunting, each with positive, probability. Hence, in equilibrium:

$0=W(p)-v$ $\implies v=(1-\frac{1-r}{k}p)^{k-1}$ $\implies
p=k(\frac{1-v^{1/{k-1}}}{1-r})$

Hence the probability that no cheetahs hunt a given gazelle (and hence that
the gazelle survives) is:

$(1-\frac{p}{k})^{k}$

On the other hand, if $v<W(1)$, then it is optimal to leave with probability
$p=1$. Therefore, the probability that no cheetahs hunt a given gazelle is:
$(1-1/k)^{k}$, which is the survival probability. ∎

We also claim:

######  Lemma 4.

In a random topology, regardless of $k$, the equilibrium probability
$p^{\ast}$ of hunting is:

$p^{\ast}=\begin{cases}1&\text{if
}v<\left(1-\frac{1}{m}+\frac{r}{m}\right)^{n-1}\\\
\frac{m(1-v^{1/(n-1)})}{1-r}&\text{otherwise}\end{cases}$

Moreover, when $m=n,$ as $n\rightarrow\infty$, $p^{\ast}\rightarrow\frac{-\ln
v}{1-r}$.

The chance that a given gazelle survives is:

$\left(1-\frac{p^{\ast}}{m}\right)^{n}=\begin{cases}\left(1-\frac{1}{m}\right)^{n}&\text{if
}v<\left(1-\frac{1}{m}+\frac{r}{m}\right)^{n-1}\\\
(1-\frac{1-v^{1/(n-1)}}{1-r})^{n}&\text{otherwise}\end{cases}$

###### Proof.

Assume that any given cheetah hunts with probability $p$, in equilibrium. Note
that the probability that any given cheetah hunts gazelle $1$ is:

$\left(\frac{p}{k}\right)*(\text{Chance that at least one of the $k$ gazelles
picked is
1})=\frac{p}{k}\cdot\frac{\dbinom{m-1}{k-1}}{\dbinom{m}{k}}=\frac{p}{k}\cdot\frac{k}{m}=\frac{p}{m}$

The chance that $t$ cheetahs send resources to gazelle $1$ is clearly

$\frac{p}{m}^{t}\left(1-\frac{p}{m}\right)^{n-1-t}$

Hence, if cheetah 1 sends all resources to $1$, the expected payoff $W$
satisfies:

$W(p)=\sum_{t=0}^{n}\frac{p}{m}^{t}\left(1-\frac{p}{m}\right)^{n-t}(r^{t})=\sum_{t=0}^{n}\frac{pr}{m}^{t}\left(1-\frac{p}{m}\right)^{n-t}=\left(1-\frac{p}{m}+\frac{pr}{m}\right)^{n-1}$

Hence,

$W=\left(1-\frac{1-r}{m}p\right)^{n-1}$

This demonstrates that the expected payoff is not dependent on $k$.

Now, there are 2 cases for $v$:

1) $v<W(1)$. In this case, the payoff will be higher if the cheetah hunts with
probability $1$, so $p=1$.

2) On the other hand, if $v\geq W(1)$, for equilibrium to be achieved,
$v=W(p)$, so:

$p=\frac{m}{1-r}(1-v^{1/{(n-1)}})$

As $n\to\infty$ and $m\to n$, this approaches:

$\frac{1}{1-r}\lim_{x\to 0}\frac{1-v^{x}}{x}$

By L’Hospital’s Rule, this is:

$p=\frac{1}{1-r}(\lim_{x\to 0}\frac{1-v^{x}}{x})=\frac{-1}{1-r}(\lim_{x\to
0}\frac{v^{x}\ln v}{1})=\frac{-\ln v}{1-r}$

So, the probability that gazelle $j$ survives (or the probability that no
cheetah hunts gazelle $j$) is:

$(1-\frac{p}{m})^{n}=(1-\frac{1-v^{1/(n-1)}}{1-r})^{n}$

∎

From now on, we assume $m\to n$ so as to compare the random and regular
topology survival rates. Let the survival rate be $S_{rnd}$ in a random
topology and $S_{reg}$ in a regular topology. Similarly, let $A_{rnd}$ and
$A_{reg}$ represent the expected payoff gained by hunting a particular gazelle
in the random and regular topologies, respectively.

Consider the following cases:

Case 1) $v<A_{rnd},A_{reg}$

In this case, $S_{reg}=(1-\frac{1}{k})^{k}$, while
$S_{rnd}=(1-\frac{1}{n})^{n}$. Hence, $S_{reg}<S_{rnd}$

Case 2)$v\in(A_{rnd},A_{reg})$

Consider the graph of survival probability against $v$, with all other
variables fixed. In the random case, the graph is constant until some critical
value $v_{1}$, while in the regular case, the graph is constant until some
value $v_{2}$. If, at some point, $S_{rnd}<S_{reg}$, even though
$S_{reg}<S_{rnd}$ holds at $v_{1},v_{2}$, $S_{reg}$ must cross a constant
value twice. This is not possible, as it is increasing. Hence,
$S_{reg}<S_{rnd}$

Case 3) $v\geq A_{reg},A_{rnd}$

Clearly, in the case of a regular topology, the probability of a gazelle
surviving is: $f(k)$, while in the case of a random topology, the probability
of a gazelle surviving is $f(n)$, where
$f(x)=(1-\frac{1-v^{1/(x-1)}}{1-r})^{x}$.

Since $f(x)$ is increasing, we have that: the survival rate is greater in a
random topology than in a “small-world” regular topology

The graph below shows $f(x)=(1-\frac{1-v^{1/(x-1)}}{1-r})^{x}$ for given
values of $v$ and $r$. As can be seen, in the last case, survival probability
increases as the number of accessible gazelles decreases.

Moreover, by a simple casework argument, this implies that the average payoff
of cheetahs is higher in regular, “small-world” networks. This is proved in
the supplementary appendix, as Lemma 5.

##  6  Discussion & Conclusions

This study of the network Colonel Blotto game has yielded three main results:
First, the existence and uniqueness of interior point equilibrium solutions
for a certain class of regular network topologies is determined by the residue
(mod 4) of the number of vertices of the network. This result is technically
intriguing, since it seems to make a connection between what would, at first
sight, appear to be a purely number theoretical property of a network and
existence/uniqueness of equilibrium of a resource allocation contest.

Secondly, we proved that if the accessibility network is randomly generated,
the survival rate is higher compared to the case where the accessibility
network is regular. In very rough terms, if the game is made more ”local”
through restrictions on movement, where each field can only be accessed by a
small set of players, then, many more fields will be “invaded” by the players
on expectation. Thus, while each individual cheetah may choose to have a
greater ”range” (i.e., be able to hunt more gazelles), cheetahs are better off
when their ranges are collectively constrained.

Finally, we discovered that heterogeneity in field weights leads to lower
average payoffs and higher (average) survival rates (i.e., more fields that
are ”untouched”). While the second part of this is somewhat more intuitive (of
course fields with extremely low valuation are avoided), the first part is
quite unexpected.

The current study focused on an “all-or-nothing” variant (where each player
had a single indivisible unit resource). In the future, it would be useful to
extend this to cases where strengths of each player vary. This will allow the
model to be applied to various real-life strategy games where resources must
be allocated in a competition against an unknown competitor.

For example, in online phishing scams (which have been studied using the
general network-free Colonel Blotto game in the past by Chia and Chuang
[2011]), scammers are competing to send the maximum number of phishing emails
to various email addresses, most of which are picked at random. The study of
such scams would be facilitated by the network Blotto model introduced in this
paper, since scammers have access to only a certain subset of email records,
and as a result can only send emails to a certain fraction of accounts.
Additionally, future research should change the tie-breaking function to allow
analysis of variants where ties between players are broken randomly (i.e., one
of the players that sends the most resources is designated, uniformly at
random, a winner). This would allow more accurate analysis of real-life
discrete scenarios, such as various situations in evolutionary biology, where
species compete over a subset of the total number of niches, and no two
species can occupy the same niche in the long term.

## References

  * Boix-Adserà et al. [2021] Enric Boix-Adserà, Benjamin L. Edelman, and Siddhartha Jayanti.  The multiplayer colonel blotto game, 2021.
  * Borel [1953] Emile Borel.  The theory of play and integral equations with skew symmetric kernels.  _Econometrica: journal of the Econometric Society_ , pages 97–100, 1953.
  * Chia and Chuang [2011] Pern Hui Chia and John Chuang.  Colonel blotto in the phishing war.  In _Decision and Game Theory for Security: Second International Conference, GameSec 2011, College Park, MD, Maryland, USA, November 14-15, 2011\. Proceedings 2_ , pages 201–218. Springer, 2011.
  * Gross and Wagner [1950] Oliver Gross and Robert Wagner.  _A continuous Colonel Blotto game_.  Rand Corporation, 1950.
  * Roberson [2006] Brian Roberson.  The colonel blotto game.  _Economic Theory_ , 29(1):1–24, 2006.

## Supplementary Appendix

We prove the following claim to confirm the main prediction of Hypothesis 1:

######  Lemma 5.

When $m\to n$, the equilibrium average payoff in the random case is
independent of $k$, and is not more than the equilibrium average payoff in the
regular case, for any regular accessibility network.

###### Proof.

Let $p$ be the equilibrium probability of hunting, for every cheetah. Let
$f(k)=(1-1/k+r/k)^{k-1}$.

Firstly, note that the number of cheetahs $N$ that a given cheetah ties with
if she visits a given gazelle satisfies $N\sim\text{Binom}{(n,p/k)}$

Hence, the expected net utility by hunting a given gazelle is:

$-v+\mathbb{E}(r^{N})=-v+\mathbb{E}(e^{N\ln
r})=-v+(1-\frac{p}{k}+\frac{rp}{k})^{k-1}$ (1)

The last line derives from the fact that $\mathbb{E}(e^{Nt})$ is the moment
generating function for a binomially-distributed random variable $N$

Hence, in a regular topology:

$p=\begin{cases}1;\hskip 8.0ptv<f(k)\\\
p^{*}=\dfrac{k(1-v^{1/(k-1)})}{1-r};\hskip 8.0ptv\geq f(k)\\\ \end{cases}$

On the other hand, in a random toplogy, when $m=n$,
$N\sim\text{Binom}(n-1,p/n)$, so the expected net utility by hunting a given
gazelle is:

$-v+\mathbb{E}(r^{N})=-v+\mathbb{E}(e^{N\ln
r})=-v+(1-\frac{p}{n}+\frac{rp}{n})^{n-1}$

Therefore, in a random topology, from Lemma 4:

$p=\begin{cases}1;\hskip 8.0ptv<f(n)\\\
p^{*}=\dfrac{n(1-v^{1/(n-1)})}{1-r};\hskip 8.0ptv\geq f(n)\\\ \end{cases}$

Hence, we have 3 cases for the relative position of $v$:

1) $v<f(n)$: In this case, since $f(k)<f(n)$, $p_{reg}=1$, and $p_{rnd}=1$.

So, the expected net winnings per cheetah in the random topology is:

$W_{rnd}=\mathbb{E}(r^{N})-v=(1-\frac{1-r}{n})^{n-1}-v$

The expected net winnings in a regular topology is:

$W_{reg}=\mathbb{E}(r^{N})-v=(1-\frac{1-r}{k})^{k-1}-v$

As $f(x)=(1-\frac{1-r}{x})^{x-1}$ is decreasing, $W_{reg}>W_{rnd}$

2) $v\geq f(k)$ In this case, a cheetah in the random topology has expected
net winnings:

$W_{rnd}=p_{rnd}(\mathbb{E}(r^{N}))-p_{rnd}v=p_{rnd}(1-\frac{p_{rnd}}{n}+\frac{rp_{rnd}}{n})^{n-1}-p_{rnd}v=0$

In the regular topology, the expected net payoff for a given cheetah is:

$W_{reg}=p_{reg}(\mathbb{E}(r^{N}))-p_{reg}v=p_{reg}(1-\frac{p_{reg}}{k}+\frac{rp_{reg}}{k})^{k-1}-p_{reg}v=0$

So, the expected payoff per cheetah is equal in the random and regular
topologies.

3) $f(n)<v<f(k)$

From the previous two cases,
$W_{reg}=\mathbb{E}(r^{N})-v=(1-\frac{1-r}{k})^{k-1}-v\geq 0$, while
$W_{rnd}=0$.

Putting all of these cases together, $W_{reg}\geq W_{rnd}$. The lemma follows.
∎