# Quantum spin state stabilized by coupling with classical spins

Hironori Yamaguchi1, Tsuyoshi Okubo2, Akira Matsuo3, Takashi Kawakami4,
Yoshiki Iwasaki5, Taiki Takahashi1, Yuko Hosokoshi1, and Koichi Kindo3
1Department of Physics, Osaka Metropolitan University, Osaka 599-8531, Japan
2Department of Physics, the University of Tokyo, Tokyo 113-0033, Japan
3Institute for Solid State Physics, the University of Tokyo, Chiba 277-8581,
Japan
4Department of Chemistry, Osaka University, Toyonaka, Osaka 560-0043, Japan
5Department of Physics, College of Humanities and Sciences, Nihon University,
Tokyo 156-8550, Japan

###### Abstract

We introduce a model compound featuring a spin-1/2 frustrated square lattice
partially coupled by spin-5/2. A significant magnetization plateau exceeding
60 T could be observed, indicating a quantum state formed by $S$ = 1/2 spins
in the square lattice. The remaining $S$ = 5/2 spins exhibited paramagnetic
behavior in the low-field regions. The numerical analysis confirmed that the
observed quantum state is a many-body entangled state based on the dominant AF
interactions and is strongly stabilized by coupling with spin-5/2. The
stabilization of this quantum state can be attributed to a compensation effect
similar to magnetic field-induced superconductivity, which serves as a
strategy to control the stability of quantum spin states in magnetic fields.

††preprint: APS/123-QED

Quantum spins in condensed-matter physics give rise to various entangled
states, thus facilitating the studies on quantum computation achieved through
single-qubit measurements on quantum spin states MBQC ; MBQC_rev . The main
objective is to stabilize various quantum states in spin systems to serve as
resources. The gain of the magnetic energy through the formation of quantum
states could be essential to quantize magnetic states in condensed-matter
systems. As a representative example, frustrated spin systems with competing
exchange interactions induce quantum states to lower the ground state energy
by suppressing the development of the magnetic moment. Extraordinary quantum
states, such as the quantum spin liquid and the spin nematic, are proposed in
low-dimensional frustrated systems SL0 ; kita0 ; ne1 . If there is no
frustration, slight perturbations, such as lattice distortion dimer1 ; dimer2
; dimer3  and bond randomness uematsu , in 2D systems can easily stabilize
quantum states. Even in one-dimensional (1D) systems with strong quantum
fluctuations, highly entangled nonmagnetic quantum states are preferentially
formed. The singlet state in the antiferromagnetic (AF) spin-1/2 chain with
the lattice alternation, which corresponds to the dominant correlation in the
present study, has lower energy than the ground state in the uniform AF chain.
In some 1D chain compounds, such singlet states can be formed even by
distorting the lattice at a certain temperature, leading to a spin-Peierls
transition SP_exp1 ; SP_exp2 ; SP_exp3 ; SP_exp4 ; mySP .

The flexibility of molecular orbitals in organic radical systems can generate
advanced spin-lattice designs, leading to realization of a variety of quantum
spin states. Our previous attempts in spin-lattice design using verdazyl-based
quantum organic materials (V-QOM) have resulted in the formation of diverse
quantum spin systems, including ferromagnetic-leg ladders 3Cl4FV ; 3Br4FV ;
3IV , quantum pentagon pentagon , honeycomb with randomness random , and
frustrated square lattices square1 ; square2 ; square3 ; square4 , which were
not achievabe with conventional organic and inorganic materials. These spin
models formed through V-QOM lead to unconventional quantum states that
originate from highly entangled spins. The next stage in spin-lattice design
using V-QOM involves the synthesis of verdazyl-based salts by combining
cationized verdazyl radicals with magnetic anions. This approach is aimed at
achieving more diverse quantum spin states through the coupling between $\pi$
electrons in the radicals and 3$d$ electrons in the anion molecules ($\pi$-$d$
interaction). This coupling results in magnetic field-induced quantum behavior
FeCl4 ; FeCl4_Br  and Lieb-Mattice ferrimagentic state LM1 ; LM2 ; LM3 . In
the realm of $\pi$-$d$ systems, organic conductors with conducting $\pi$
electrons and localized $d$ spins were extensively investigated, particularly
magnetic field-induced superconductivity super_FeCl4 ; super_FeBr4 , wherein
the polarized $d$ spins stabilized the superconductivity by inducing an
effective zero-field condition in the $\pi$ site JP . Similarly, in the
present work, the quantum state formed in the radical salt is also expected to
be stabilized through a similar mechanism through $\pi$-$d$ interactions.

In this letter, we present a model compound featuring a spin-1/2 spatially
anisotropic frustrated square lattice partially coupled with spin-5/2. Single
crystals of the verdazyl-based salt ($m$-MePy-V)2MnCl4 were successfully
synthesized. Molecular orbital (MO) calculations revealed four types of
exchange interactions, forming the spin-1/2 square lattice and partially
coupling with spin-5/2. High-field measurements indicate a significant
magnetic plateau exceeding 60 T, indicating the formation of a nonmagnetic
quantum state by $S$ = 1/2 spins in the square lattice. Numerical analysis
demonstrates that this observed quantum state exhibits singlet-like
correlation and is strongly stabilized by the coupling with spin-5/2.

To prepare $m$-MePy-V, we used a conventional procedure gosei , and
($m$-MePy-V)2MnCl4 was synthesized following a reported method for salts with
similar chemical structures square3 ; 3Dhoneycomb . The dark-brown
($m$-MePy-V)2MnCl4 crystals were obtained through recrystallization using
methanol. The crystal structure was determined from intensity data collected
using a Rigaku AFC-8R Mercury CCD RA-Micro7 diffractometer. High-field
magnetization measurements were carried out using a nondestructive pulse
magnet, subjecting the samples to pulsed magnetic fields of up to
approximately 60 T. The magnetic susceptibility was measured using a
commercial SQUID magnetometer (MPMS-XL, Quantum Design). The experimental
results were corrected considering the diamagnetic contributions calculated
via Pascal’s method. Specific heat measurements were performed using a
commercial calorimeter (PPMS, Quantum Design). All experiments were conducted
on small randomly oriented single crystals, considering the isotropic nature
of organic radical systems.

The verdazyl radical $m$-MePy-V and MnCl4 anion have spin-1/2 and 5/2,
respectively supple1 . The crystallographic parameters at room-temperature are
as follows: monoclinic, space group $P$212121, $a$ = 10.6892(3) $\rm{\AA}$,
$b$ = 15.2412(4) $\rm{\AA}$, $c$ = 24.4578(7) $\rm{\AA}$, $V$ = 3984.57(19)
$\rm{\AA}^{3}$, $Z$ = 4, $R$ = 0.0389, and $R_{\rm{w}}$ = 0.1003. We performed
MO calculations MOcal  to evaluate the exchange interactions. Subsequently,
four types of dominant interactions were found between the $S$ = 1/2 spins on
the radicals supple1 , as shown in Fig. 1(a). They are identified as
$J_{1}/k_{\rm{B}}$ = $72$ K, $J_{2}/k_{\rm{B}}$ = 45 K, $J_{3}/k_{\rm{B}}$ =
12 K, and $J_{4}/k_{\rm{B}}$ = $-8$ K, which are defined in the Heisenberg
spin Hamiltonian given by $\mathcal{H}=J_{n}{\sum_{\langle
i,j\rangle}}\textbf{{{S}}}_{i}{\cdot}\textbf{{{S}}}_{j}$, where $\sum_{\langle
i,j\rangle}$ denotes the sum over the neighboring spin pairs. Moreover, one
dominant AF interaction, $J_{\rm{Mn}}$, is expected between spin-1/2 and 5/2
supple1 , as shown in Fig. 1(b). Considering that the evaluation of absolute
values of interactions between radical and MnCl4 anion is difficult, we
roughly evaluated as $J_{\rm{Mn}}$/$J_{1}$ $\simeq$ 0.2 using the following
analysis of magnetic susceptibility. Consequently, $S$ = 1/2 spins form a
spatially anisotropic square lattice through $J_{\rm{1}}$–$J_{\rm{4}}$ in the
$ab$ plane, to which $S$ =5/2 spins are partially connected, as shown in Fig.
1(c). Each square unit has frustration caused by three AF interactions and one
ferromagnetic interaction, which is divided into the following two patterns:
$J_{\rm{1}}$-$J_{\rm{2}}$-$J_{\rm{3}}$-$J_{\rm{4}}$ and
$J_{\rm{1}}$-$J_{\rm{3}}$-$J_{\rm{4}}$-$J_{\rm{2}}$.

Figure 1: (color online) (a) Crystal structure of ($m$-MePy-V)2MnCl4 in the
(a) $ab$ and (b) $bc$ planes. Hydrogen atoms are omitted for clarity. The
broken lines represent the bonds forming the spin lattice. (c) Two-dimensional
spin model in the $ab$ plane. $J_{1}$, $J_{2}$, $J_{3}$, and $J_{4}$ form the
frustrated square lattice of $S$ = 1/2 spins, and one of two $S$ = 1/2 sites
is coupled by $S$ = 5/2 spins through $J_{\rm{Mn}}$.

Figure 2(a) shows the temperature dependence of the magnetic susceptibility (
$\chi$ = $M/H$) at 0.1 T, revealing a monotonic paramagnetic increase with
decreasing temperatures. In the high-temperature region, $\chi$$T$ exhibits a
rounded minimum at approximately 20 K, as shown in the inset of Fig. 2(a).
This behavior is typical of mixed spin systems LM1 ; mix1 ; mix2 ; mix3 ,
where interactions between spins of different sizes lead to excited states
composed of smaller effective spins than those in the ground state, resulting
in a rounded minimum in $\chi$$T$. In our system, the $\chi$$T$ behavior
strongly depends on the value of $J_{\rm{Mn}}$ between spin-1/2 and -5/2.
Based on MO calculations that indicate $J_{\rm{1}}$ to be the most dominant
interaction between $S$ =1/2 spins, we assumed a spin-(1/2,1/2,5/2) trimer
coupled by $J_{\rm{1}}$ and $J_{\rm{Mn}}$ with the spin Hamiltonian given by
$\mathcal{H}=J_{\rm{1}}\textbf{{{S}}}_{\rm{V1}}{\cdot}\textbf{{{S}}}_{\rm{V2}}+J_{\rm{Mn}}\textbf{{{S}}}_{\rm{V2}}{\cdot}\textbf{{{S}}}_{\rm{Mn}}$,
where $\textbf{{{S}}}_{\rm{V}i}$ and $\textbf{{{S}}}_{\rm{Mn}}$ are the
spin-1/2 and spin-5/2 operators, respectively. The calculated results, as
shown in the inset of Fig. 2(a), well reproduce the rounded minimum, allowing
us to evaluate $J_{\rm{Mn}}$/$J_{\rm{1}}$ $\simeq$ 0.2.

Figure 2(b) displays the magnetization curve at 4.2 K measured in pulsed
magnetic fields, with its inset showing the low-field region measured in
static magnetic fields. The observed hysteresis in pulsed magnetic fields
originates from the slow relaxation characteristic of organic radical systems.
The magnetization gradually increases up to a value of 5
${\mu}_{B}/\rm{f.u.}$, which persists up to at least 60 T. Considering the
isotropic $g$ values ($\sim$2.0) of the verdazyl radical and Mn2+ ion, the
saturation value is expected to be 7 ${\mu}_{B}/\rm{f.u.}$. Therefore, the
magnetic moment of 5 ${\mu}_{B}/\rm{f.u.}$ corresponds to the full
polarization of spin-5/2 along the field direction, representing 5/7 of the
full saturation value. The gradual increases in the low-field region can be
effectively explained by the Brillouin function for spin-5/2 monomer, as shown
in Fig. 2(b). These behaviors suggest that the $S$ = 1/2 spins in the square
lattice form a nonmagnetic quantum state.

The temperature dependence of the specific heat $C_{\rm{p}}$ is shown in Fig.
3(a). No peak indicative of a phase transition to a magnetic order is
observed, but a significant change occurs below approximately 15 K when
magnetic fields are applied. As the low-temperature magnetization exhibits
spin-5/2 paramagnetic behavior, the strong magnetic field dependence of
$C_{\rm{p}}$ is considered to originate from the Zeeman splitting of spin-5/2,
leading to Schottky contributions when magnetic fields are applied. To
evaluate the Schottky-type specific heat, we assumed Zeeman energy given by
-$g{\mu}_{B}S_{z}$, where $S_{z}$ = $\pm$5/2, $\pm$3/2, $\pm$1/2. Furthermore,
the magnetic specific heat $C_{\rm{m}}$ was determined by subtracting the
lattice contribution $C_{\rm{l}}$ from the experimental results. Below 15 K,
$C_{\rm{l}}$ was approximated as
$C_{\rm{l}}=a_{1}T^{3}+a_{2}T^{5}+a_{3}T^{7}$, a form that has been confirmed
to be effective for verdazyl-based compounds a235Cl3V ; Zn_ferro . The
constants $a_{1}-a_{3}$ were determined to reproduce the calculated Schottky-
type specific heat. As shown in Fig. 3(b), the magnetic specific heats at 3 T
and 6 T are well explained by the calculated results assuming $C_{\rm{l}}$
with $a_{1}=0.022$, $a_{2}=$-$5.6\times 10^{-5}$, and $a_{3}=5.1\times
10^{-8}$. The value of $a_{1}$ corresponds to a Debye temperature of 44 K,
consistent with those used for other verdazyl-based compounds 3Cl4FV ; 3Br4FV
; square3 ; a235Cl3V . The evaluated $C_{\rm{m}}$ at zero-field is quite
small, further supporting that the observed specific heats mainly originate
from the Zeeman splitting of spin-5/2. Figure 3(c) shows the magnetic entropy
$S_{\rm{m}}$ obtained via the integration of $C_{\rm{m}}/T$, where the
experimental data for 3 and 6 T have been shifted up by 4.7 and 1.2 J/ mol K,
respectively. The calculated results exhibit an asymptotic behavior toward the
total magnetic entropy of spin-5/2 ($R$ln6 $\simeq$ 14.9). Although the
experimental temperature is not sufficient to observe the entire entropy
shift, the temperature dependence of the experimental results is well
reproduced by the entropy change from the higher temperature region.

Figure 2: (color online) (a) Temperature dependence of the magnetic
susceptibility ($\chi$ = $M/H$) of ($m$-MePy-V)2MnCl4 at 0.1 T. The inset
shows the temperature dependence of $\chi T$. The lines represent the
calculated results for the spin-(1/2,1/2,5/2) trimer coupled by $J_{\rm{1}}$
and $J_{\rm{Mn}}$. (b) Magnetization curves of ($m$-MePy-V)2MnCl4 at 4.2 K in
pulsed magnetic fields. The inset shows low-field region measured in static
magnetic fields at 1.8 and 4.2 K. The solid lines represent the Brillouin
function for spin-5/2. Figure 3: (color online) (a) Temperature dependence of
the specific heat $C_{\rm{p}}/T$ of ($m$-MePy-V)2MnCl4 at 0, 3 and 6 T. (b)
Low-temperature region of $C_{\rm{m}}/T$. The solid lines represent the
Schottky-type specific heat originating from the Zeeman splitting of spin-5/2.
(c) Temperature dependence of the magnetic entropy $S_{\rm{m}}$. The
experimental values for 3 and 6 T have been shifted up by 4.7 and 1.2 J/ mol
K, respectively, to overlap with the calculated results.

Here, we considered the ground state of the present system. Our previous
studies have demonstrated that MO calculations for verdazyl-based compounds
reliably provide values of exchange interactions between radical spins, thus
enabling the examination of their intrinsic behavior 3Cl4FV ; 3Br4FV ; 3IV ;
pentagon ; square1 ; square2 . Accordingly, considering the MO evaluations, we
assumed the exchange interactions as $J_{2}/J_{1}$ = 0.63 and $J_{3}/J_{1}$ =
0.17 K. Additionally, for the quantum Monte Carlo calculation QMC , we fixed
$J_{4}$ = 0 to avoid the negative sign problem, leading to a non-frustrated
case. Regarding $J_{\rm{Mn}}$, we assumed $J_{\rm{Mn}}/J_{1}$ = 0.2, as
evaluated from the magnetization analysis above. We confirm that the 5/7
magnetization plateau appears up to approximately $H/J_{1}$ = 1.15, as shown
in Fig. 4(a). Consequently, the observed plateau up to 60 T indicates that the
actual value of $J_{1}$ is more than 70 K. Given that the magnetic properties
below the plateau phase can be described by the spin-5/2 monomer, the plateau
is considered to form an AF quantum state with fully polarized spin-5/2 along
the field direction. As the dominant AF interaction $J_{1}$ plays a crucial
role and contributes more significantly than the other interactions, we
anticipate the formation of a singlet-like quantum state in the 2D plane, as
schematically illustrated in Fig. 4(a).

Finally, we examine the stability of the quantum state in association with
frustration and coupling with spin-5/2. Frustrated systems tend to lower their
ground-state energies through lattice decouplings dimen1 ; dimen2 ; dimen3 ;
dimen4 ; dimen5 . In the present lattice, the weakest ferromagnetic
interaction $J_{4}$ is expected to be decoupled to minimize the energy loss
due to frustration, thus stabilizing the singlet-like quantum state. Moreover,
ferromagnetic interactions are known to enhance the decoupling of the AF
quantum state square4 ; hagiwara1 , which aligns well with the $J_{4}$
decoupling scenario. Accordingly, although it is difficult to quantitatively
verify the precise effects of frustration, we can expect that the frustration
acts to stabilize the quantum state in the present spin lattice. In terms of
the effects of the coupling with spin-5/2, we calculated $J_{\rm{Mn}}/J_{1}$
dependence of the magnetization curves with fixed $J_{2}/J_{1}$ = 0.63 and
$J_{3}/J_{1}$ = 0.17 K, as shown in Fig. 4(b). We confirm that a case with no
spin-5/2 coupling leads to a significant reduction in the plateau region.
These findings indicate that the quantum state in the square lattice is highly
stabilized by coupling with spin-5/2. This stabilization of the quantum state
can be understood as a compensation effect, similar to the Jaccarino-Peterin
mechanism for the magnetic field-induced superconductivity super_FeCl4 ;
super_FeBr4 ; JP , where the external magnetic field compensates the internal
magnetic field caused by the $\pi$-$d$ interaction. In our case, the internal
magnetic field induced by the fully-polarized spin-5/2 through $J_{\rm{Mn}}$
compensates the external magnetic field in the plateau phase, resulting in the
expansion of the plateau region with the singlet-like quantum state as
$J_{\rm{Mn}}$ increases.

Figure 4: (color online) (a) Calculated magnetization curve at $T/J_{1}$ =
0.01 for the expceted spin model assuming non-frustraed case ($J_{4}$ = 0)
with $J_{2}/J_{1}$ = 0.63, $J_{3}/J_{1}$ = 0.17 K, and $J_{\rm{Mn}}/J_{1}$ =
0.2. The illustrations describe the singlet-like quantum state protected by
the fully-polarized spin-5/2 in the plateau phase. (b) $J_{\rm{Mn}}/J_{1}$
dependence of the normalized magnetization curve at $T/J_{1}$ = 0.01.

In summary, we succeeded in synthesizing single crystals of the verdazyl-based
salt ($m$-MePy-V)2MnCl4. MO calculations indicated the formation of a spin-1/2
spatially anisotropic frustrated square lattice partially coupled with
spin-5/2. In the low-field region, the magnetizations and specific heats were
well explained by spin-5/2 monomer that become fully polarized at
approximately 10 T. In the high-field measurement, we observed a large
magnetization plateau exceeding 60 T, indicating a nonmagnetic quantum state
formed by $S$ = 1/2 spins in the square lattice. The numerical analysis
demonstrated that the observed quantum state exhibits singlet-like
correlations due to the dominant AF interaction. Furthermore, we confirmed
that the quantum state is strongly stabilized by coupling with spin-5/2,
attributed to the compensation mechanism with the internal magnetic field
induced by the fully-polarized spin-5/2. Our study reveals a mechanism to
stabilize quantum spin states in condensed materials and suggests a strategy
to control the stability of quantum spin states in magnetic fields, which may
inspire future technologies for quantum computation based on materials
science.

###### Acknowledgements.

We thank Y. Kono for valuable discussions. This research was partly supported
by the Murata Science Foundation and KAKENHI (Grants No. 23K13065 and No.
23H01127). A part of this work was performed as the joint-research program of
ISSP, the University of Tokyo and the Institute for Molecular Science.

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# Root distributions in Moebius–Kantor complexes

Sylvain Barré  and  Mikaël Pichot  Sylvain Barré, UMR 6205, LMBA, Université
de Bretagne-Sud,BP 573, 56017, Vannes, France<EMAIL_ADDRESS>Mikaël Pichot, McGill University, 805 Sherbrooke St W., Montréal, QC H3A 0B9,
Canada<EMAIL_ADDRESS>
###### Abstract.

We study the distribution of roots of rank 2 in nonpositively curved
2-complexes with Moebius–Kantor links. For every face in such a complex, the
parity of the number of roots of rank 2 in a neighbourhood of the face is a
well-defined geometric invariant determined by the root distribution. We study
the relation between the root distribution and the parity distribution. We
prove that there exist parity distributions in flats which are disallowed in
Moebius–Kantor complexes. This contrasts with the fact that every root
distribution can be realized. We classify the root distributions associated
with an even parity distribution (i.e., such that every face is even) on a
flat plane. We prove that there exists up to isomorphism a unique even simply
connected Moebius–Kantor complex—namely, the Pauli complex.

##  1\. Introduction

A Moebius–Kantor complex is a 2-complex with triangle faces whose links are
isomorphic to the Moebius–Kantor graph (i.e., the unique cubic symmetric graph
with 16 vertices). Every Moebius–Kantor complex can be viewed as a
nonpositively curved 2-complex, in which every face is isometric to an
equilateral triangle with sides of length 1 [6, II.5.1].

We shall begin by reviewing a few definitions.

###  1.1. Roots

Let $\Delta$ be a nonpositively curved 2-complex, $x\in\Delta$ be a vertex,
and $L_{x}$ the link at $x$, endowed with the angular metric. We call _root_
at $x$ a isometric embedding
$\alpha\colon[0,\pi]\operatorname{\hookrightarrow}L_{x}$ such that $\alpha(0)$
is a link vertex (of order $>2$). Every root has a _rank_
$\mathrm{rk}(\alpha)$, which is a rational number in $[1,2]$ defined by

$\mathrm{rk}(\alpha):=1+{N(\alpha)\over q_{\alpha}}$

where

$N(\alpha):=|\\{\beta\in\Phi_{x}\mid\alpha\neq\beta,\alpha(0)=\beta(0),\alpha(\pi)=\beta(\pi)\\}|,$

$\Phi_{x}$ denotes the set of roots at $x$ and, for a root $\alpha$,
$q_{\alpha}$ denotes the order of $\alpha(0)$ minus 1:

$q_{\alpha}:=\mathrm{val}(\alpha(0))-1.$

We refer to [1, §4] for more details on these definitions.

###  1.2. Parity in Moebius–Kantor complexes

Let $\Delta$ be a Moebius–Kantor complex. Let $f$ be a face in $\Delta$. The
parity of $f$ is defined as follows (cf. [3, §2]). Let $\tilde{f}$ denote the
equilateral triangle formed by the union of $f$ and three faces in $\Delta$
corresponding to a choice of a face $\neq f$ adjacent to every side of $f$. We
call $\tilde{f}$ a _large triangle_ containing $f$. For every vertex $x$ of
$f$, the triangle $\tilde{f}$ determine a root $\alpha_{x}$ at $x$ in
$\Delta$.

######  Definition 1.1.

We call _parity_ of $f$ the parity of the number of roots $\alpha_{x}$ which
are of rank 2, when $x$ runs over the three vertices of the face $f$.

This is a well-defined invariant, i.e., the parity of $f$ does not depend on
the choice of the large triangle $\tilde{f}$ containing $f$, by [3, Lemma
2.1].

###  1.3. Root distributions

Let $\Pi$ be a flat plane in a Moebius–Kantor complex $\Delta$, i.e., the
image of an isometric embedding $\mathbb{R}^{2}\to\Delta$, where
$\mathbb{R}^{2}$ is endowed with the Euclidean metric. Every flat is
tessellated by equilateral triangles with sides of length 1.

Let $x\in\Pi$ be a vertex. For every simplicial line $\ell\subset\Pi$ through
$x$, the rank of every root $\alpha\in\Phi_{x}$ whose image is included in
$\Pi$ and whose extremities belong to $\ell$ is constant. We call $\ell$ a
simplicial direction at $x$, and the common value $\mathrm{rk}(\alpha)$ the
rank of $\ell$. Observe that precisely two of the three simplicial directions
at $x$ in $\Pi$ are of rank 2.

######  Definition 1.2.

The _root distribution of $\Pi$_ is the map $\delta$ which associates to every
vertex $x\in\Pi$ the unique simplicial direction at $x$ which is not of rank
2.

Graphically, one may represent $\delta$ as a field of segments of length
$2\varepsilon$ which takes a vertex $x$ to the segment
$(x-\varepsilon,x+\varepsilon)$ containing $x$ in $\delta(x)$, for some small
fixed $\varepsilon>0$.

We call _abstract root distribution_ on $\mathbb{R}^{2}$ the choice, for every
vertex $x$ of the tessellation by equilateral triangle, of a simplicial
direction $\delta(x)$ at $x$.

######  Definition 1.3.

An abstract root distribution is _realized_ in a Moebius–Kantor complex
$\Delta$ if it is the root distribution of some flat $\Pi$ in $\Delta$.

We shall first prove that every abstract root distribution can be realized in
some Moebius–Kantor complex:

######  Proposition 1.4.

Every abstract root distribution can be realized in a Moebius–Kantor complex.

We refer to §2 for the proof. This result does not hold for abstract parity
distributions (see Cor. 1.7).

###  1.4. Parity prescriptions in a flat

Every abstract root distribution $\delta$ on $\mathbb{R}^{2}$ induces a parity
distribution $p$ on the triangle faces of the tessellation. If $f$ is a face,
then the parity $p(f)\in\\{0,1\\}=\\{\text{even, odd}\\}$ is the parity of the
image of $f$ under an isometric embedding $\mathbb{R}^{2}\to\Delta$ which
realizes $\delta$ in some Moebius–Kantor complex $\Delta$ (which exists by
Prop. 1.4). Since the value of $p(f)$ does not depend on the choice of the
large triangle $\tilde{f}$, one may always assume that $\tilde{f}$ is included
in the image of $\mathbb{R}^{2}\to\Delta$. Thus, $p(f)$ depends only on the
abstract root distribution $\delta$. It computes the parity of the number of
sides of $\tilde{f}$ which are not in the simplicial directions selected by
$\delta$.

######  Definition 1.5.

We call $p$ the _parity distribution of $\delta$_.

An _abstract parity distribution_ on $\mathbb{R}^{2}$ is the choice, for every
face $f$ of the tessellation by equilateral triangle, of value a
$p(f)\in\\{0,1\\}$.

We aim to show (in contrast with Prop. 1.4) that there exist abstract parity
distributions which are not the parity distribution of any flat in any
Moebius–Kantor complex. This will be an immediate corollary of the following
result.

######  Theorem 1.6.

There exists an abstract parity distribution which is not the parity
distribution of an abstract root distribution.

We refer to §3 for the proof.

An abstract parity distribution is said to be _realized_ in a Moebius–Kantor
complex $\Delta$ if it is the induced parity distribution of some flat in
$\Delta$.

######  Corollary 1.7.

There exists an abstract parity distribution on $\mathbb{R}^{2}$ which is not
realized in a Moebius–Kantor complex.

###### Proof.

The parity distribution given by Theorem 3.1 is not realized in a Moebius-
Kantor complex $\Delta$, for it would be the parity distribution of the root
distribution induced by $\Delta$. ∎

On the other hand, we can prove the following result.

######  Proposition 1.8.

Every parity distribution on a ball of radius 1 (an hexagon) in
$\mathbb{R}^{2}$ can be realized by an abstract root distribution, and
therefore can be realized in a Moebius–Kantor complex.

This is established in §4.

###  1.5. Parity prescriptions in a Moebius-Kantor complex

By Theorem 3.1, there exists an abstract parity distribution on
$\mathbb{R}^{2}$ which cannot realized in a Moebius–Kantor complex. This
should be compared with the results in §6 of [3], according to which the
parity can be freely prescribed in Moebius–Kantor complexes, and therefore
that any abstract parity distribution can always “be realized” in some
Moebius–Kantor complex.

The point here is that in the case of a parity prescription in a
Moebius–Kantor complex, the parity distribution can only be pre-assigned and
then realized inductively in successive spheres, as opposed to being
prescribed in advance as a 0-1 valued function (since there does not exist a
predefined space that would serve as domain for this function).

The free prescription theorem (from [3, §5]) for Moebius–Kantor complexes can
be stated as follows. Let $B_{n}$ denote a ball of radius $n$ centered at a
vertex in a Moebius–Kantor complex, and
$S_{n}:=B_{n}\setminus(B_{n-1})^{\circ}$ denote the closed simplicial sphere
of radius $n$, where $(B_{n-1})^{\circ}$ denotes the interior of $B_{n-1}$.
One can realize an arbitrary parity function on $S^{n}$ in a Moebius–Kantor
complex by iterating the following construction.

######  Theorem 1.9.

Given an arbitrary function $p\colon S_{n}\to\\{0,1\\}$ defined on the
2-skeleton of $S_{n}$, there exists a ball $B_{n+1}$ in a Moebius–Kantor
complex, containing $B_{n}$, such that the parity of the faces of $S_{n}$ in
$B_{n+1}$ is given by $p$.

Free construction theorems of this sort originated in the theory of buildings,
in particular in works of Ronan and Tits (see for instance [8, 9, 10, 11] and
references therein).

It was mentioned in [3] that the proof of Theorem 1.9 seems to provide
additional free parameters in the constructions, i.e., that the parity
function $p$ should not determine the isomorphism type of $B_{n+1}$ in
general. That these additional degrees of freedom ultimately exist depends
again on the balance between the root distribution at the vertices in the
boundary of $B_{n}$, and the parity distribution on the faces of $S_{n}$. A
direct computation can be made which shows, very roughly, what the degree of
freedom for the parity is with respect to the degree of freedom for the roots.

Namely, let $f_{n}$ denote the number of faces in $S_{n}$. There can be two
types of vertices in the boundary of $B_{n}$, those of order 3 and those of
order 4. Let $a_{n}$ and $b_{n}$ denote their respective number. Then the
relative degree of freedom is computed as follows: there are $2^{f_{n}}$
choices for the parity in $S_{n}$, and at most $8^{a_{n}}16^{b_{n}}$ choices
for the roots at the boundary of $B_{n}$; the quantities $a_{n},b_{n},f_{n}$
are related by the equality:

$f_{n+1}=18a_{n}+15b_{n};$

the quantities $a_{n}$ and $b_{n}$ can be computed explicitly by the
recurrence relation:

$\begin{pmatrix}a_{n+1}\\\ b_{n+1}\end{pmatrix}=\begin{pmatrix}6&2\\\
3&4\end{pmatrix}\begin{pmatrix}a_{n}\\\ b_{n}\end{pmatrix}.$

Therefore,

$2^{f_{n}}<8^{a_{n}}16^{b_{n}}.$

Since we do not need them except for a general estimate of the degrees of
freedom, we leave to the reader the exercise of verifying these relations.

A Moebius–Kantor complex is said to be _even_ if every face is even. In [3,
Remark 5.3(1)], it was suggested to study the case of even complexes, and
that—in view of the above estimates—there “ought to exist” uncountably many
non isomorphic constructions of such complexes. It turns out that there exists
a unique construction up to isomorphism. In the remainder of this
introduction, we shall state this uniqueness result more precisely, together
with some applications which resolve some questions that we left open in
earlier investigations.

###  1.6. Even root distributions

We shall first describe the relation between root distributions and parity
distributions in the even case.

The even parity distribution on $\mathbb{R}^{2}$ is the contant distribution
$p\equiv 0$ in which every face is even. A root distribution is said to be
even if its parity distribution is the even parity distribution.

We say that two abstract root distribution $\delta$ and $\delta^{\prime}$ on
$\mathbb{R}^{2}$ are isomorphic if there exists a simplicial isometry $\alpha$
of $\mathbb{R}^{2}$ such that $\delta\circ\alpha=\delta^{\prime}$.

######  Theorem 1.10.

The even root distributions can be classified up to isomorphism.

This result can be proved directly, or it can be deduced from [5]. We provide
both arguments in the present paper. The second approach is shorter since it
only requires to provide a few additions to the results of [5]. The direct
approach is given in §7.

In [5], a Moebius–Kantor complex $\Delta_{P}$ called the _Pauli complex_ was
introduced. It is defined using a triangle of groups construction associated
with the group generated by the Pauli matrices in $\mathrm{SU}(2)$. The
fundamental group of this triangle of group is a group $G_{P}$ which is
developable and therefore acts properly isometrically on a CAT(0) complex, the
complex $\Delta_{P}$. (More details on this construction are given in §5
below.)

######  Proposition 1.11.

The Pauli complex $\Delta_{P}$ is even.

This proposition is established in §5 below; it relies on basic relations
verified by the Pauli matrices.

In order to deduce Theorem 1.10 from [5], it is sufficient to prove the
following result (see §5).

######  Proposition 1.12.

Every even root distribution can be realized in the Pauli complex.

One can reformulate this proposition by saying that the map
$\Pi\mapsto\delta_{\Pi}$ which to a flat $\Pi$ of the Pauli complex associates
its unique root distribution is surjective onto the set of even root
distributions. In [5], the flats of the Pauli complex were classified by means
of ring puzzles. The map $\Pi\mapsto\delta_{\Pi}$ can be reinterpreted as a
map from ring puzzles to root distributions. It follows by Theorem 5.7 in [5]
that the Pauli ring puzzles can be classified (i.e., listed explicitly);
therefore, so can the even root distributions.

Therefore, one can deduce Theorem 1.10, and indeed, an explicit classification
of the even root distributions, from the classification of Pauli puzzles given
in Theorem 5.7 in [5]. In the terminology of [5], the classification
comprises:

  1. (1)

a unique even root distribution associated with the Pauli $M$-puzzles; and,

  2. (2)

an infinite family pairwise non-isomorphic root distributions associated with
the Pauli $T$-puzzles;

every even root distribution is isomorphic to a root distribution in case (1)
or (2). This statement is the more precise version of Theorem 1.10. As
mentioned above, a more direct approach to this result, which classifies the
even root distributions, is given in §7.

###  1.7. Even Moebius–Kantor complexes

Several examples of even Moebius–Kantor complexes were given in [4]. In fact,
it was shown there that there exist precisely four pairwise non isomorphic
even Moebius–Kantor complexes having a single vertex. In the notation of [4],
these complexes are $V_{0}^{1}$, $V_{0}^{2}$, $\check{V}_{0}^{2}$, and
$V_{4}^{1}$; we refer to [4] for their description (see also [3, Prop. 3.2]).

######  Theorem 1.13.

Suppose that $\Delta$ and $\Delta^{\prime}$ are even Moebius–Kantor complex,
and let $x\in\Delta$ and $x^{\prime}\in\Delta^{\prime}$ be two vertices. Let
$n\geq 1$, and let $\varphi_{n}\colon B_{n}(x)\to B_{n}(x^{\prime})$ be an
isomorphism between the balls of radius $n$ around $x$ and $x^{\prime}$ in
$\Delta$ and $\Delta^{\prime}$, respectively. Then there exists a unique
isomorphism $\varphi\colon\Delta\to\Delta^{\prime}$ such that
$\varphi_{|B_{n}(x)}=\varphi_{n}$.

The following are immediate corollaries from Theorem 1.13.

######  Corollary 1.14.

Every even simply connected Moebius–Kantor complex is isometrically isomorphic
to the Pauli complex.

In fact, we will prove Theorem 1.13 by cross referencing the additional
structure found in the Pauli complex with that of an arbitrary even simply
connected Moebius–Kantor complex.

The second corollary follows by choosing an appropriate isomorphism
$\varphi_{1}\colon B_{1}(x)\to B_{1}(y)$ where $x$ and $y$ are two points in
the Pauli complex:

######  Corollary 1.15.

The symmetry group of the unique even simply connected Moebius–Kantor is flag
transitive.

(We recall that a flag is a triple $(x,e,f)$ where $x$ is a vertex, $e$ an
edge, and $f$ a face, such that $x\in e\in f$.)

The following two corollaries were left open in [4] and [5], respectively.

######  Corollary 1.16.

The universal covers of $V_{0}^{1}$, $V_{0}^{2}$, $\check{V}_{0}^{2}$, and
$V_{4}^{1}$, are pairwise isometrically isomorphic.

We recall that $G_{P}$ denotes the fundamental group of the complex of groups
defining the Pauli complex (see §1.6 and §5).

######  Corollary 1.17.

The group $G_{P}$ contains a torsion free subgroup of finite index.

###### Contents

  1. 1 Introduction
    1. 1.1 Roots
    2. 1.2 Parity in Moebius–Kantor complexes
    3. 1.3 Root distributions
    4. 1.4 Parity prescriptions in a flat
    5. 1.5 Parity prescriptions in a Moebius-Kantor complex
    6. 1.6 Even root distributions
    7. 1.7 Even Moebius–Kantor complexes
  2. 2 Abstract root distributions
  3. 3 Abstract parity distributions
  4. 4 Parity prescription in the neighbourhood of a vertex
  5. 5 The Pauli complex $\Delta_{P}$
  6. 6 Every even complex is isomorphic to $\Delta_{P}$
  7. 7 Classifying the even root distributions

##  2\. Abstract root distributions

######  Proposition 2.1.

Every abstract root distribution can be realized in a Moebius–Kantor complex.

###### Proof.

Let $\Pi$ be a flat plane endowed with a root distribution. Fix a vertex
$x\in\Pi$. Let $H_{n}$ denote the hexagon of center $x$ and radius $n$ in
$\Pi$. The proof proceed by induction to construct a ball $B_{n}$ of radius
$n$ in a Moebius–Kantor complex, containing $H_{n}$, and realizing the root
distribution.

For $H_{1}\subset B_{1}$, it suffices to embed a 6-cycle with marked roots in
a Moebius–Kantor graph respecting the rank of every root. Suppose
$H_{n}\subset B_{n}$ has been constructed. Let $\tilde{B}_{n}$ be obtained by
adding a pair of faces for every edge in the boundary $S_{n}$ of $B_{n}$,
together with $H_{n+1}$, to $B_{n}$. For every vertex in $S_{n}$, the link is
either a tree (of diameter at most 5) or the union of such a tree and the
1-neighbourhood of a 6-cycle (adding at most two edges to the tree). In the
first case, the tree can be completed arbitrarily into a Moebius–Kantor graph.
In the second case, the roots are pre-positioned in the 6-cycle, and there
exists a completion which respects the roots.

Finally, the links can be completed to form $B_{n+1}$ from $\tilde{B}_{n}$,
and the direct limit of the $B_{n}$’s constructed in a such a way provides one
(among uncountably many non pairwise isomorphic) Moebius–Kantor complex
realizing the root distribution. ∎

Every root distribution defines uniquely the parity of its faces.

######  Example 2.2.

A root distribution with parallel roots defines an even flat; however, the
same flat can also be obtained with different root distributions.

We shall prove in the forthcoming section that there exists a parity
distribution on a flat plane which is not realized by a root distribution. The
notion of abstract root (and parity) distribution extends to simplicial
subsets of $\mathbb{R}^{2}$ (for example, strips or balls of radius $n$
centered at a vertex $x$) in the obvious way. Every parity (resp. root)
distributions on a simplicial subset of $\mathbb{R}^{2}$ is the restriction of
a parity (resp. root) distribution on $\mathbb{R}^{2}$.

Observe that every parity distribution on a bi-infinite strip of hight 1 can
be realized in a flat in a Moebius–Kantor complex. Indeed, it suffices to
choose appropriately a root distribution on its boundary to realize the
parity, extend the partial distribution in a flat, and apply Prop. 2.1. The
same holds true for balls of radius 1 as we shall see in §4.

##  3\. Abstract parity distributions

######  Theorem 3.1.

There exists an abstract parity distribution which is not the parity
distribution of an abstract root distribution.

###### Proof.

We shall start with an even face, say $x$, and define a parity distribution
stepwise building outward from the face $x$ in such a way that the root
distribution is analytically determined, until a contradiction arises.

Consider the following root distribution:

$x$

In the figure above, we represent a root distribution graphically as mentioned
in the introduction (see §1.3).

In the figure below, an extension of this root distrubution is shown. The
triangles labelled with label “1” indicate the odd faces, every other face is
even.

$x$11$y$

It is obvious that the face with label $y$, which we define to be even, is odd
according to the root distribution. This provides a contradiction.

We may now extend the parity distribution by an order 3 symmetry as follows:

$x$111111

If follows that for every parity distribution on $\mathbb{R}^{2}$ containing
the above distribution, the twelve ($=3\times 4$) root distributions on $x$ in
which the number of roots of rank 2 is 2, are disallowed.

Finally, the remaining even root distribution on $x$:

$x$

is also disallowed, for the same reason (i.e., it extends as the initial root
distribution). This concludes the proof of the proposition. ∎

##  4\. Parity prescription in the neighbourhood of a vertex

######  Proposition 4.1.

Every parity distribution on a ball of radius 1 (hexagon) can be realized by
an abstract root distribution.

###### Proof.

Let $B$ be a ball of radius 1 around a vertex. We distinguish three cases:
either $B$ contains at most two odd faces, at most two even faces, or
precisely three odd faces and three even faces.

The first two cases consists of five subcases each, which correspond to the
number of ways to assign a parity to $B$ with at most two even (or odd) faces.
In these cases, the root distributions can be chosen as follows:

Figure 1. Root distributions with at most 2 odd faces

It is easily seen that the above root distributions exhaust the parity
distributions with at most two odd faces. Similarly:

Figure 2. Root distributions with at most 2 even faces

The last case, with 3 even faces and 3 odd faces, consists of 3 subcases, and
for each of these cases, a root distribution exists:

Figure 3. Root distributions with exactly three odd faces

This concludes the proof of the proposition. ∎

##  5\. The Pauli complex $\Delta_{P}$

The Pauli complex is a simply connected Moebius–Kantor complex $\Delta_{P}$
obtained by a standard construction [6, Chap. 3] using a triangle of groups
associated with the group $P$ generated by the three Pauli matrices in
$\mathrm{SU}(2)$:

$X:=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\ \ Y:=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\ \ \text{ and }\ \ Z:=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

The direct limit $G_{P}$ of this triangle of groups acts on $X_{P}$ in a flag
transitive way. The complex $\Delta_{P}$ is obtained by developing around a
fundamental face $f$ using the group $P$.

The complex $\Delta_{P}$ has an important property, which was used in [5] to
classify the flat planes that it contains: every face in $\Delta_{P}$ can be
assigned with a Pauli matrix $X$, $Y$ or $Z$ in an equivariant way [5, §3].
Under this assignment, the link edges are naturally labeled, and the link is a
Moebius–Kantor graph isomorphic to the Cayley graph of $P$ with respect to
$\\{X,Y,Z\\}$. The 6-cycles in the link (which correspond to rings in the ring
puzzles) are given by the following relations between the Pauli matrices

  1. $XYXZYZ=\mathrm{Id}$

  2. $YZYXZX=\mathrm{Id}$

  3. $ZXZYXY=\mathrm{Id}$

in $P$. A root in $\Delta_{P}$ corresponds to a path of length 3 in the link
with an edge labelling in $\\{X,Y,Z\\}$.

######  Lemma 5.1.

A root is of rank 2 if and only if it contains the three labels $X$, $Y$, $Z$.

###### Proof.

Since the Moebius–Kantor graph is transitive on path of length 2, it is enough
to prove the lemma for the roots having labels $XYZ$ and $XYX$. It is clear
that a root with labels $XYZ$ can be extended in two ways using the 6-cycles
$XYZXZY$ and $XYZYXZ$, and therefore is of rank 2. A root with labels $XYX$
can be extended in a single way using a 6-cycle, namely, $XYXZYZ$. The word
$XYXY$ is not a subword of a relation of length 6 for the group $P$. ∎

This implies:

######  Proposition 5.2.

The Pauli complex $\Delta_{P}$ is an even Moebius–Kantor complex.

###### Proof.

Since $\Delta_{P}$ is flag transitive, it is enough to prove that the
fundamental face is even. We assume that the face $f$ is labelled with the
matrix $Y$, and extend $f$ into an equilateral triangle $\tilde{f}$ using
three faces with the same labels, say $X$. Then the three roots associated
with $f$ have labels $XYX$. By the previous lemma, these roots are not of rank
2. Therefore, $\Delta_{P}$ is even. ∎

As shown in the introduction, in order to deduce Theorem 1.10 from [5], it is
sufficient to prove the following result.

######  Proposition 5.3.

Every even root distribution can be realized in the Pauli complex.

One argument would be to modify the proof of Prop. 2.1, to ensure that the
Moebius–Kantor complex is even, and apply Theorem 1.13.

We shall proceed as follows.

######  Proposition 5.4.

Let $\Pi$ be a flat plane endowed with an even root distribution, and consider
a labelling by $X$ and $Y$, of two consecutive faces in $\Pi$. There exists a
unique labelling of the faces of $\Pi$ with labels in $\\{X,Y,Z\\}$ such that
the 6-cycles in the links correspond to the relations $XYXZYZ=\mathrm{Id}$,
$YZYXZX=\mathrm{Id}$, and $ZXZYXY=\mathrm{Id}$. In other words, there exists a
unique Pauli puzzle associated with $\Pi$ and the labelling of two consecutive
faces.

Prop. 5.3 follows from Prop. 5.4 and the fact that every Pauli puzzle is
realized in the Pauli complex (Theorem 3.3 in [5]).

###### Proof.

Uniqueness is clear. We prove existence by induction on the radius $n$ of the
ball $B_{n}$ centered at one of the two common vertices of the labelled
consecutive triangles.

Suppose that there exists a labelling of $B_{n}$. For every vertex $x$ in the
boundary of $B_{n}$, at least two faces of the ball $B_{1}(x)$ of radius 1 at
$x$ are labelled, and therefore there exists a unique labelling of $B_{1}(x)$
which is consistent with the parity at $x$.

Let $[x,y]$ be an edge in the boundary of $B_{n}$, and let $t$ and
$t^{\prime}$ denotes the two triangles adjacent to $[x,y]$, where $t$ is the
triangle included in $B_{n}$.

We have to show that the labelling of $t^{\prime}$ from $B_{1}(x)$ coincides
with the labelling from $B_{1}(y)$. However, since two of the three faces
adjacent to $t$ are included in $B_{n}$ and therefore labelled, only one of
the two possible labels of $t^{\prime}$ ensure, by Lemma 5.1, that $t$ is an
even face (compare Lemma 6.2). ∎

##  6\. Every even complex is isomorphic to $\Delta_{P}$

Let $\Delta$ be a simply connected even complex, $x_{0}\in\Delta$ be a vertex,
$B_{n}$ be the ball of center $x_{0}$ and radius $n$ in $\Delta$, and
$S_{n}:=\operatorname{\partial}B_{n}$ be the sphere of center $x_{0}$ and
radius $n$. We fix an isometric embedding

$\varphi_{1}\colon B_{1}\to\Delta_{P}.$

Using $\varphi_{1}$, we may label every face in $B_{1}$ with one of the Pauli
matrices $X,Y,Z$.

######  Theorem 6.1.

There exists a unique isometric embedding $\varphi\colon\Delta\to\Delta_{P}$
such that ${\varphi}_{|B_{1}}=\varphi_{1}$.

###### Proof.

We prove by induction that there exists a unique isometric embedding
$\varphi_{n}\colon B_{n}\to\Delta_{P}$ such that
${\varphi_{n}}_{|B_{1}}=\varphi_{1}$. The theorem follows immediately.

Suppose that $\varphi_{n}\colon B_{n}\to\Delta_{P}$ is an isometric
isomorphism onto its image. Using $\varphi_{n}$, we may label every face in
$B_{n}$ with one of the Pauli matrices $X,Y,Z$. Let us construct
$\varphi_{n+1}$. This construction will also establish Theorem 1.13.

For every vertex $x\in S_{n}$, there exists a unique isometric embedding
$\varphi_{x}\colon L_{x}\to\Delta$ whose restriction to $B_{n}$ coincides with
$\varphi_{n}$.

For every edge $[x,y]$ in $S_{n}$, let $h_{x,y}^{k}$, $k=0,1$, denote the
height, perpendicular to $[x,y]$, in the two faces containing $[x,y]$ which
are not contained in $B_{n}$. We write $H_{x,y}^{k}$ and $H_{y,x}^{k}$ for the
two half faces, separated by $h_{x,y}^{k}$, respectively containing $x$ and
$y$. Thus, $H_{x,y}^{k}\cup H_{y,x}^{k}$, $k=0,1$, are the two faces on
$[x,y]$ not contained in $B_{n}$.

For every $x\in S_{n}$ we may label, using $\varphi_{x}$, every half face
$H_{x,y}^{k}$, $k=0,1$, with one of the Pauli matrices $X,Y,Z$.

We claim (Lemma 6.2):

> for every $[x,y]$ and $k=0,1$, the labels of the half faces $H_{x,y}^{k}$
> and $H_{y,x}^{k}$ coincide.

This shows that there exists a well-defined isometric embedding
$\varphi_{n+1}\colon B_{n+1}\to\Delta$ extending $\varphi_{n}$, which
coincides with $\varphi_{x}$ on $L_{x}$ for every $x\in S_{n}$. It takes
$H_{x,y}^{k}\cup H_{y,x}^{k}$ to the unique face in $\Delta_{P}$ which
contains $\varphi_{n}([x,y])$, and whose label is the common label of
$H_{x,y}^{k}$ and $H_{y,x}^{k}$. The uniqueness of such an embedding is clear.
∎

We now prove the claim.

######  Lemma 6.2.

For every $[x,y]$ and $k=0,1$, the labels of the half faces $H_{x,y}^{k}$ and
$H_{y,x}^{k}$ coincide.

###### Proof.

Consider the face $[x,y,z]$ in $B_{n}$ containing $[x,y]$, and two faces
$F_{x}$ and $F_{y}$ in $B_{n}$, adjacent to $[z,x]$ and $[z,y]$ respectively,
and having identical labels, say $X$. Let $k,l\in\\{0,1\\}$ denote the two
indices such that $H_{x,y}^{k}$ and $H_{y,x}^{l}$ have label $X$. We must show
that $k=l$.

Since the labels of the faces $F_{x}$, $F_{y}$, and half faces $H_{x,y}^{k}$
and $H_{y,x}^{l}$, coincide, the rank of the roots corresponding three roots
at $x$, $y$ and $z$ is $\frac{3}{2}$ (Lemma 5.1).

Since $F$ is an even face, the number of roots of rank $\frac{3}{2}$ in a
large triangle containing $F$ is odd. Since the root at $z$ is of rank
$\frac{3}{2}$, the roots at $x$ and $y$ must have the same rank. Therefore,
$k=l$. For the corresponding face in $\Delta$, the large triangle adjoins two
roots of rank $\frac{3}{2}$ at $x$ and $y$ for a total of 3 such roots. ∎

##  7\. Classifying the even root distributions

In this section, we provide a direct approach to the classification of even
root distributions in a flat, which rely only on the root distribution rather
than the Pauli complex.

We will in fact start with an arbitrary even Moebius–Kantor complex, not a
priori known to be isomorphic to $\Delta_{P}$, and show how its flats can be
classified by studying the root distributions. The same proof applies to
classify the abstract even root distributions.

In §2 of [2], a notion of $w$-block in an $\mathrm{Aut}(F_{2})$-puzzle was
introduced. It is a geometric configuration of shapes having the property of
being forward analytic in $\mathrm{Aut}(F_{2})$ puzzles. We show here that a
configuration leading to a similar behaviour can be found in every even simply
connected Moebius–Kantor complex.

Let $\Delta$ be an even simply connected Moebius–Kantor complex.

######  Lemma 7.1.

Consider a trapezoid $t$ of height 1 with a base of length 3, such that the
roots on the base have rank 2, and the root on the top, rank 3/2. If $t$
belongs to a flat $\Pi$, then $t$ admits a unique extension into a sector of
$\Pi$ of base $t$ and rank 2 boundary.

###### Proof.

The rank 3/2 root in $t$ extends in a unique way. Since $\Delta$ is even, the
slanted sides of length 2 are of rank 3/2. This follows from the fact that the
inner vertices are both of rank 2. Thus, $t$ extends uniquely into a 2-strip
$S_{2}$ of length 3 in $\Pi$. Since $\Delta$ is even, the two upper roots in
$\Pi$ are of rank 3/2, and therefore extend uniquely into $\Pi$. Again, the
slanted sides of length 2 are of rank 3/2, and $S_{2}$ extends uniquely into
$\Pi$ into a partial cone $S_{3}$ of height 3. By induction, the upper roots
in $S_{n}$ are always of rank $3/2$, which provides a uniquely defined sector
$S:=\bigcup S_{i}$ of base $t$ in $\Pi$. ∎

Figure 4. The figure shows the sector $S$ and its rank 2 boundary

######  Lemma 7.2.

The sector $S$ (in the notation of Lemma 7.1) belongs to a unique flat.

###### Proof.

Let $S\subset\Pi$ be a copy of $S$ in a flat, and let $\overline{t}$ denote
the reflection of $t$ with respect to its rank 2 boundary. If the lower
boundary of $\overline{t}$ is of rank 3/2, we have an opposite copy of $t$ in
$\Pi$, and Lemma 7.1 shows that $\overline{t}$ extends to a copy
$\overline{S}$ of $S$ in $\Pi$. By convexity, there exists at most one flat of
$\Delta$ containing $S\cup\overline{S}$. It is easy to check that such a flat
indeed exists. (It is represented below.)

Let us show that the lower boundary of $\overline{t}$ must be of rank 3/2.
Otherwise, it is of rank 2. Since the central triangle in $t$ is even, the
2-triangle $T$ (which is formed of 4 equilateral triangles) of base the lower
base of $\overline{t}$ admits two sides of rank 3/2 or two sides of rank 2.
Since the center triangle in $\overline{t}$ is even, this proves that the
lower side of $T$ is of rank 3/2. ∎

Figure 5. The $t$-flat $\Pi_{t}$; three intersecting rank 2 (in dark)
geodesics are drawn, and one of the trapezoid $t$ is shown; the symmetries
$S_{3}$ of the center triangle extend to the $t$-flat.

######  Remark 7.3.

The flat $\Pi_{t}$ corresponds to one of the Pauli $M$-puzzles in [5], where
$M\in\\{X,Y,Z\\}$.

By strip of rank 2, we mean a simplicial flat strip whose boundary roots are
of rank 2.

######  Lemma 7.4.

If $\Delta$ is even, the minimal strips of rank 2 are of height 1.

###### Proof.

Suppose that there exists a minimal strip $\Sigma$ with rank 2 boundaries. If
it is not of height 1, then there exists a root of rank 3/2 at distance 1 from
the boundary. Since the boundary is of rank 2, $\Sigma$ contains a trapezoid
$t$. By Lemma 7.1 and Lemma 7.2, $\Sigma$ must be included in the $t$-flat.
However, the latter does not contain such a strip. ∎

By geodesic of rank 2 in a flat, we mean a simplicial straight line containing
exclusively roots of rank 2.

######  Lemma 7.5.

If a flat $\Pi$ contains a geodesic $l$ of rank 2, then either $\Pi$ coincides
with the $t$-flat $\Pi_{t}$, or it is a union of minimal strips of rank 2
parallel to $l$.

###### Proof.

Let $l$ be a rank 2 geodesic. If $\Pi$ is not a union of minimal strips of
rank 2 parallel to $l$, then there exists a parallel geodesic
$l^{\prime}\parallel l$ of rank 2, and a parallel geodesic at distance 1 from
$l^{\prime}$ which contains a root of rank 3/2. Thus, $\Pi$ contains the
trapezoid $t$ which implies $\Pi=\Pi_{t}$. ∎

We shall now consider another ‘glider’ (i.e., a configuration which is forward
analytic in flats) which behaves similarly to the $w$-block in
$\mathrm{Aut}(F_{2})$ puzzles. Furthermore, if $\Pi\neq\Pi_{t}$ then it will
also be backward analytic.

######  Lemma 7.6.

Let $\Delta$ be an even simply connected Moebius–Kantor complex. Consider a
trapezoid $t^{\prime}$ of height 1 with a base of length 3, such that
precisely one of the two roots on the base have rank 2, and such that the top
root has rank 3/2. Every flat containing $t^{\prime}$ contains a geodesic of
rank 2.

###### Proof.

Suppose that $t^{\prime}$ belongs to a flat $\Pi$. We first show that
$t^{\prime}$ admits a unique upward extension into a half-strip
$S^{\mathrm{\uparrow}}$ transverse to $t^{\prime}$ of height 3 with rank 2
boundaries. By symmetry, we may assume that the bottom rank 2 root is on the
right. Consider the unique upward extension of $t^{\prime}$ (whose boundary is
of rank 3/2). Since the left top face of the trapezoid is even, the righthand
side of the extension is of rank 3/2, and therefore admits a unique extension
into $\Pi$. This implies that the top root of the resulting extension is of
rank 3/2. Therefore, a copy of $t^{\prime}$ sits on top of it; by induction,
we find a half strip $S^{\mathrm{\uparrow}}$ piling up infinitely many copies
of $t^{\prime}$.

Next we prove that if $\Pi\neq\Pi_{t}$ then $S^{\mathrm{\uparrow}}$ extends
uniquely into a strip $S$ of height 3 with rank 2 boundary. Consider a
trapezoid $t^{\prime\prime}$ extending $S^{\mathrm{\uparrow}}$ downwards.
Since the faces of $t^{\prime}$ are even, the right bottom root of
$t^{\prime\prime}$ is of rank 2. If the left bottom root of $t^{\prime\prime}$
is even, then $t^{\prime\prime}$ is isomorphic to $t$, so $\Pi=\Pi_{t}$ by
Lemmas 7.1 and 7.2. Therefore $t^{\prime\prime}$ is isomorphic to
$t^{\prime}$; by induction, we find a strip $S$ containing
$S^{\mathrm{\uparrow}}$ and piling up infinitely many copies of $t^{\prime}$.

To prove the lemma, we may assume that $\Pi\neq\Pi_{t}$; note then that the
strip $S\subset\Pi$ is a union of strips with rank 2 boundaries. ∎

######  Theorem 7.7.

A flat plane in an even simply connected Moebius–Kantor complex is either a
flat with the root distribution of $\Pi_{t}$, or a union of strips of height 1
and rank 2.

###### Proof.

Let $\Delta$ be an even Moebius–Kantor complex and $\Pi$ be a flat in
$\Delta$. The previous results show that the conclusions hold if $\Pi$
contains $t$ (see Lemma 7.1) or $t^{\prime}$ (see Lemma 7.6). Otherwise, $\Pi$
contains neither $t$ nor $t^{\prime}$. This implies the following statement:

  1. If $\tau$ is a trapezoid of base of length 3 and top root of rank 3/2, then the bottom roots are of rank 3/2.

By connectedness, this statement in turn implies that the roots of rank 3/2
are aligned in $\Pi$. There exists a unique configuration with this property.
Since the roots of rank 3/2 are aligned in $\Pi$ and $\Delta$ is a
Moebius–Kantor complex, the two transverse directions must be rank 2. Thus,
$\Pi$ is a union of strips of height 1 and rank 2. ∎

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