# Grothendieck ring of pairs of quasi-projective varieties 111Math. Subject
Class.: 18F30. Keywords: complex quasi-projective varieties, Grothendieck
rings, lambda-structure, power structure.

S.M. Gusein-Zade The work of the first author (Sections 3 and 4) was supported
by the grant 21-11-00080 of the Russian Science Foundation. Address: Moscow
State University, Faculty of Mechanics and Mathematics, Moscow Center for
Fundamental and Applied Mathematics, GSP-1, Moscow, 119991, Russia & National
Research University “Higher School of Economics”, Usacheva street 6, Moscow,
119048, Russia. E-mail<EMAIL_ADDRESS>I. Luengo  The last two authors were
partially supported by a competitive Spanish national grant MTM
PID2020-114750GB-C32. Address: ICMAT (CSIC-UAM-UC3M-UCM), Dept. of Algebra,
Geometry and Topology, Complutense University of Madrid, Plaza de Ciencias 3,
Madrid, 28040, Spain. E-mail<EMAIL_ADDRESS>A. Melle-Hernández Address:
Instituto de Matemática Interdisciplinar (IMI), Dept. of Algebra, Geometry and
Topology, Complutense University of Madrid, Plaza de Ciencias 3, Madrid,
28040, Spain. E-mail<EMAIL_ADDRESS>
###### Abstract

We define a Grothendieck ring of pairs of complex quasi-projective varieties
(that is a variety and a subvariety). We describe $\lambda$-structures and a
power structure on/over this ring. We show that the conjectual symmetric power
of the projective line with several orbifold points described by A. Fonarev is
consistent with the symmetric power of this line with points as a pair of
varieties.

##  1 Introduction

We consider the Grothendieck ring $K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$
of pairs of complex quasi-projective varieties and define natural
$\lambda$-structures and a power structure on/over it. One motivation for that
came from the intention to define the Grothendieck ring of (quasi-projective)
orbifolds (and to study $\lambda$-structures and power structures on/over it).
The simplest (non-trivial) orbifold is a complex cuve $C$ with several
orbifold points $a_{1}$, …, $a_{s}$ which can be considered as the pair
$(C,\\{a_{1},\ldots,a_{s}\\})$ of varieties (of dimension 1 and 0
respectively). Submanifolds in affine or projective spaces also can be
considered as pairs.

Grothendieck rings of varieties (with additional structures) often are endowed
by $\lambda$-structures. Natural $\lambda$-structures on them are defined by
analogues of the Kapranov zeta function (the generating series of the classes
of the symmetric products) and of the generating series of the classes of the
configuration spaces. We consider natural analogues of these series for pairs
of varieties. A $\lambda$-structure on a ring defines a power structure over
it (see [2]). Power structures over Grothendieck rings of varieties with
additional structures turn out to be useful to write down and/or to prove
statements about generating series of classes of some varieties: see, e.g.,
[3], [4], [6]. We show that two natural $\lambda$-structures on $K_{0}^{\rm
pairs}({\rm{Var}_{\mathbb{C}}})$ define one and the same power structure over
it and give its geometric description.

##  2 The Grothendieck ring of pairs of varieties

A pair of (complex quasi-projective) varieties is a pair $(X,Y)$ consisting of
a complex quasi-projective variety $X$ and a Zariski locally closed subvariety
$Y$ in it. The variety $X$ will be called the ambient variety, $Y$ will be
called the subvariety of the pair.

The Grothendieck semiring $S_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$ of
pairs of complex quasi-projective varieties is the semigroup generated by the
isomorphism classes $[(X,Y)]$ of pairs modulo the relation

$[(X,Y)]=[(Z,Y\cap Z)]+[(X\setminus Z,Y\cap(X\setminus Z))]$

for a Zariski closed subvariety $Z\subset X$; the multiplication in
$S_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$ is defined by

$[(X_{1},Y_{1})]\widehat{\times}[(X_{2},Y_{2})]=[(X_{1}\times
X_{2},(X_{1}\times Y_{2})\cup(Y_{1}\times X_{2}))]\,.$

The Grothendieck ring $K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$ (of pairs
of complex quasi-projective varieties) is the (abelian) group generated by the
isomorphism classes of pairs with the same relation and the same
multiplication.

The unit $1$ in this ring (and also in the semiring) is represented by the
pair $({\rm pt},\emptyset)$ where ${\rm pt}$ is a one point set (${\rm
Spec\,}{\mathbb{C}}$).

##  3 $\lambda$-structures on $K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$

A $\lambda$-structure on a ring $R$ (often called a pre-$\lambda$-structure)
is defined by a series $\lambda_{a}(t)\in 1+t\cdot R[[t]]$ given for all $a\in
R$ such that $\lambda_{a}(t)=1+at+~{}terms~{}of~{}higher~{}order$,
$\lambda_{a+b}(t)=\lambda_{a}(t)\cdot\lambda_{b}(t)$ (see, e. g., [5]).

Let $(X,Y)$ be a pair of varieies. The $n$th symmetric product of $(X,Y)$ is
$S^{n}(X,Y)=(X,Y)^{n}/S_{n}$, that is the pair consisting of the symmetric
product $S^{n}X$ of $X$ with the subvariety of (unordered) $n$-tuples
$\\{x_{1},\ldots,x_{n}\\}$ of points of $X$ such that (at least) one of these
points belongs to $Y$. (Here and below the Cartesian power $(X,Y)*n$ of pair
means $(X,Y)\widehat{times}\ldots\widehat{times}(X,Y)$, $n$ times.) Example:
$S^{n}1=1$, where $1=[({\rm pt},\emptyset)]$. ($(X,Y)^{0}=({\rm
pt},\emptyset)$ and therefore $[S^{0}(X,Y)]=1$.)

The $n$th (unordered) configuration space of $(X,Y)$ is
$\Lambda^{n}(X,Y)=\left((X,Y)^{n}\setminus\Delta\right)/S_{n}$, where $\Delta$
is the big diagonal consisting of $n$-tuples of points with at least two
coinciding ones. In other words $\Lambda^{n}(X,Y)$ is the pair consisting of
the usual configuration space $\Lambda^{n}X=(X\setminus\Delta)/S_{n})$ of $X$
with the subspace of (unordered) $n$-tuples $\\{x_{1},\ldots,x_{n}\\}$ of
different points of $X$ such that (at least) one of them belongs to $Y$.
Example: $\Lambda^{n}1=0$ for $n\geq 2$.

###### Definition 1

The Kapranov zeta function of $[(X,Y)]$ is the series

$\zeta_{X,Y}(t)=1+\sum_{n=1}^{\infty}[S^{n}(X,Y)]\cdot t^{n}\in 1+t\cdot
K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})[[t]]\,.$

Example: $\zeta_{1}(t)=1+t+t^{2}+\ldots=(1-t)^{-1}$.

###### Statement 1

The Kapranov zeta function $\zeta_{X,Y}(t)$ defines a $\lambda$-structure on
$K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})[[t]]$.

Proof. We has to show that, for two pairs $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$,
one has

$\zeta_{X_{1}\sqcup X_{2},Y_{1}\sqcup
Y_{2}}(t)=\zeta_{X_{1},Y_{1}}(t)\cdot\zeta_{X_{2},Y_{2}}(t)\,.$ (1)

The coefficient of $t^{n}$ in the LHS of (1) is the pair consisting of the
space of unordered $n$-tuples $\\{x_{1},\ldots,x_{n}\\}$ of points of
$X_{1}\sqcup X_{2}$ (that is $S^{n}(X_{1}\sqcup X_{2})$) with the subvariety
of tuples $K=\\{x_{1},\ldots,x_{n}\\}$ such that at least one of the pointx
$x_{i}$ belongs to $Y_{1}\sqcup Y_{2}$. This means that this space of pairs is
the union over $s$, $0\leq s\leq n$, of the spaces of pairs consisting of the
products $S^{s}X_{1}\times S^{n-s}X_{2}$ with the subvarieties of them
consisting of an (unordered) $s$-tuple $K_{1}$ of points of $X_{1}$ and an
(unordered) $(n-s)$-tuple $K_{2}$ of points of $X_{2}$ such that at least one
point of one of them belongs to the corresponding $Y_{i}$, $i=1\text{ or }2$.
The coefficient of $t^{n}$ in the RHS of (1) is
$\left[\bigsqcup_{s=0}^{n}S^{s}(X_{1},Y_{1})\widehat{\times}S^{n-s}(X_{2},Y_{2})\right]$
and has the same description as the LHS. $\square$

Let

$\lambda_{X,Y}(t)=1+\sum_{n=1}^{\infty}[\Lambda^{n}(X,Y)]\cdot t^{n}\in
1+t\cdot K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})[[t]]$

be the generating series of the classes of the configuration spaces
$\Lambda^{n}(X,Y)$. Example: $\lambda_{1}(t)=1+t$.

###### Statement 2

The series $\lambda_{X,Y}(t)$ defines a $\lambda$-structure on $K_{0}^{\rm
pairs}({\rm{Var}_{\mathbb{C}}})[[t]]$.

The proof is literally the same as the one of Statement 1 with the only
difference that the tuples under consideration should consist of different
points.

##  4 Example: the symmetric product of the projective line with several
distinguished points

Let us consider the pair $(\mathbb{C}\mathbb{P}^{1},\\{a_{1},\ldots,a_{s}\\})$
($a_{i}\neq a_{j}$ for $i\neq j$) consisting of the projective line and an
$s$-point subset of it. We shall describe its $n$th symmetric product.

The $n$the symmetric product of the projective line $\mathbb{C}\mathbb{P}^{1}$
with the coordinates $(u:v)$ is the $n$-dimensional projective space
$\mathbb{C}\mathbb{P}^{n}$. An isomorphism between
$S^{n}(\mathbb{C}\mathbb{P}^{1})$ and $\mathbb{C}\mathbb{P}^{n}$ can be
established by the following map. Let $(p_{0}:p_{1}:\ldots:p_{n})$ be
coordinates on $\mathbb{C}\mathbb{P}^{n}$. A point of
$S^{n}(\mathbb{C}\mathbb{P}^{1})$ is represented by an $n$-tuple of points
$\\{x_{1},\ldots,x_{n})$ of $\mathbb{C}\mathbb{P}^{1}$, $x_{i}=(u_{i}:v_{j})$.
The point in $\mathbb{C}\mathbb{P}^{n}$ corresponding to this $n$-tuple is

$\displaystyle p_{0}$ $\displaystyle=$ $\displaystyle+u_{1}u_{2}\ldots u_{n},$
$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle-\sum_{1\leq i\leq
n}u_{1}u_{2}\ldots\widehat{u_{i}}v_{i}\ldots u_{n},$ $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle+\sum_{1\leq i<j\leq
n}u_{1}u_{2}\ldots\widehat{u_{i}}v_{i}\ldots\widehat{u_{j}}v_{j}\ldots u_{n},$
$\displaystyle\ldots\ldots\ldots$ $\displaystyle p_{n}$ $\displaystyle=$
$\displaystyle\pm\ v_{1}v_{2}\ldots v_{n},$

where (as usual) $\widehat{\cdot}$ means that the corresponding term is
excluded. (This means that $z_{i}=u_{i}/v_{i}$ are the roots of the polynomial
$P(z)=p_{0}+p_{1}z+p_{2}z^{2}+\ldots+p_{n}z^{n}$.) Let $a_{i}=(x_{i}:1)$ The
subset $Y$ in the pair
$S_{n}(\mathbb{C}\mathbb{P}^{1},\\{a_{1},\ldots,a_{s}\\})=(\mathbb{C}\mathbb{P}^{n},Y)$
consists of the points $(p_{0}:p_{1}:\ldots:p_{n})\in\mathbb{C}\mathbb{P}^{n}$
such that one of the roots of the polynomial
$p_{0}+p_{1}z+p_{2}z^{2}+\ldots+p_{n}z^{n}$ is one of $x_{i}$. This means that
$Y$ is the union over $i=1,\ldots,s$ of the subsets of the points
$(p_{0}:p_{1}:\ldots:p_{n})\in\mathbb{C}\mathbb{P}^{n}$ such that
$p_{0}+p_{1}x_{i}+p_{2}x_{i}^{2}+\ldots+p_{n}x_{i}^{n}=0$. These are $s$
hyperplanes in $\mathbb{C}\mathbb{P}^{n}$ in general position. This gives a
partial explanation of the (conjectual) description of the orbifold
$S^{n}(\mathbb{C}\mathbb{P}^{1},\\{a_{1},\ldots,a_{s}\\})$ with $s$ points of
order $2$ in [1].

##  5 The geometric power structure over the ring $K_{0}^{\rm
pairs}({\rm{Var}_{\mathbb{C}}})[[t]]$

A power structure over a ring $R$ is a method to give sense to an expression
of the form $(A(t))^{m}$, where $A(t)=1+a_{1}t+a_{2}t^{2}+\ldots\in 1+t\cdot
R[[t]]$ (that is $a_{i}\in R$) and $m\in R$, as a series from $1+t\cdot
R[[t]]$ (in other words it is a map from $(1+t\cdot R[[t]])\times R$ to
$1+t\cdot R[[t]]$: $(A(t),m)\mapsto(A(t))^{m}$ so that all the usual
properties of the exponential function hold the main of them (those listed in
[2]; later some more requirements (obvious in the case under consideration)
were added: [4]) being

  1. 1)

$\left(A(t)\right)^{0}=1$;

  2. 2)

$\left(A(t)\right)^{1}=A(t)$;

  3. 3)

$\left(A^{(1)}(t)\cdot
A^{(2)}(t)\right)^{m}=\left(A^{(1)}(t)\right)^{m}\cdot\left(A^{(2)}(t)\right)^{m}$;

  4. 4)

$\left(A(t)\right)^{m_{1}+m_{2}}=\left(A(t)\right)^{m_{1}}\cdot\left(A(t)\right)^{m_{2}}$;

  5. 5)

$\left(A(t)\right)^{m_{1}m_{2}}=\left(\left(A(t)\right)^{m_{2}}\right)^{m_{1}}$.

A $\lambda$-structure on a ring $R$ defines a power structure over it. On the
other hand, in general, there are many $\lambda$-structures corresponding to
one and the same power structure.

A power structure over the ring $K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$
is called effective if, in fact, it is defined over the semiring $S_{0}^{\rm
pairs}({\rm{Var}_{\mathbb{C}}})$. This means that, if the coefficients $a_{i}$
of the series $A(t)$ and the exponent $m$ are the classes of pairs of
varieties (not differences of such classes), then all the coefficients of the
series $(A(t))^{m}$ are also represented by classes of pairs.

###### Theorem 1

The $\lambda$-structures on the ring $K_{0}^{\rm
pairs}({\rm{Var}_{\mathbb{C}}})[[t]]$ described in Section 3 define one and
the same power structure over it, which is effective.

Proof. To prove that, we shall give a geometric description of this power
structure. This means that we shall describe the coefficient at $t^{k}$ in the
series $(A(t))^{[(M,N)]}$, where
$A(t)=1+\sum_{k=1}^{\infty}[(A_{k},B_{k})]\cdot t^{k}$, as the class of a pair
of quasi-projective varieties. This will mean that the power structure is
effective (after it is proved that the description really gives a power
structure). The fact that this power structure corresponds both to the
Kapranov zeta function $\zeta_{X,Y}(t)$ an to the generating series
$\lambda_{X,Y}(t)$ of the classes of the configuration spaces will follow from
the equations

$\displaystyle\zeta_{X,Y}(t)$ $\displaystyle=$
$\displaystyle(1-t)^{-[(X,Y)]}\quad\text{and}$ (2)
$\displaystyle\lambda_{X,Y}(t)$ $\displaystyle=$
$\displaystyle(1+t)^{[(X,Y)]}$ (3)

(in the sense of the power structure).

The description is as follows. Let
${\mathcal{A}}^{*}=\bigsqcup_{i=1}^{\infty}A_{i}$,
${\mathcal{A}}:={\mathcal{A}}^{*}\cup\\{{\rm pt}\\}$,
${\mathcal{B}}={\mathcal{B}}^{*}=\bigsqcup_{i=1}^{\infty}B_{i}$. Let $i$ be
the tautological (integer valued) function on ${\mathcal{A}}$ and on
${\mathcal{B}}$ which sends, for $i>0$, $A_{i}$ and $B_{i}$ to
$i\in{\mathbb{Z}}$ and the point ${\rm pt}\in{\mathcal{A}}$ to $0$.

The coefficient at $t^{n}$ in the series ${A}(t)^{[(M,N)]}$ is represented by
the following pair of varieties. The ambient space is the configuration space
of pairs $(K,\varphi)$, where $K$ is a finite subset of the variety $M$ and
$\varphi$ is a map from $K$ to ${\mathcal{A}}^{*}$ such that
$\sum\limits_{x\in K}i(\varphi(x))=n$. The subvariety of the pair is the set
of points of the ambient space such that either (at least) one of the points
of $K$ lies in $N$ or the image of (at least) one point of $K$ is in $B^{*}$.

To describe such a configuration space as a quasi-projective variety one can
write it as

$\sum_{{\mathbf{k}}:\,\sum{i}\,k_{i}=n}\left[\left(\left((\prod_{i}(M,N)^{k_{i}})\setminus\Delta\right)\widehat{\times}\prod_{i}(A_{i},B_{i})^{k_{i}}\right)\left/\prod_{i}S_{k_{i}}\right.\right]\,,$
(4)

where ${\mathbf{k}}=\\{k_{1},k_{2},\ldots\\}$, $k_{i}\in{\mathbb{Z}}_{\geq 0}$
and $\Delta$ is the “large diagonal” in $(M,N)^{\Sigma k_{i}}$ which consists
of $(\sum k_{i})$-tuples of points of $M$ with at least two coinciding ones;
the permutation group $S_{k_{i}}$ acts by permuting corresponding $k_{i}$
factors in
$\prod\limits_{i}(M,N)^{k_{i}}\supset(\prod\limits_{i}(M,N)^{k_{i}})\setminus\Delta$
and the pairs $(A_{i},B_{i})$ in $(A_{i},B_{i})^{k_{i}}$ simultaneously (the
connection between this formula and the description above is clear).

To prove that the described construction really gives a power structure over
$K_{0}^{\rm pairs}({\rm{Var}_{\mathbb{C}}})$, one has to verify the conditions
1) – 5). The first two are obvious. The ambient spaces in the summands in
Equation (4) coincide with the corresponding summands for the expression
$\left(1+\sum_{i}[A_{i}]t^{i}\right)^{[M]}$ in [2, Equation (1)] (i.e. for the
(usual) power structure over the Grothendieck ring
$K_{0}({\rm{Var}_{\mathbb{C}}})$ of complex quasi-projective varieties).
Therefore the corresponding equalities for them can be found in [2]. Thus it
is necessary to verify the conditions 3) – 5) only for the subvarieties of the
pairs.

Condition 3)
($A^{(j)}(t)=1+\sum\limits_{i=1}^{\infty}[(A^{(j)}_{i},B^{(j)}_{i})]t^{i}$,
$j=1,2$, $m=[(M,N)]$). The map (an isomorphism) from the right hand side to
the left hand side on the ambient spaces is the following. If
$A(t)=A^{(1)}(t)\widehat{times}A^{(2)}(t)=1+\sum_{i=1}^{\infty}(A_{i},B_{i})$,
then ${\mathcal{A}}={\mathcal{A}}^{(1)}\times{\mathcal{A}}^{(2)}$. If $K_{1}$
and $K_{2}$ are finite subsets of $M$ with maps $\psi_{1}$ and $\psi_{2}$ to
$A^{(1)}$ and $A^{(1)}$ respectively (these data define a point from the union
of the coefficients of the right hand side), then the corresponding (finite)
subset $K$ is the union $K_{1}\cup K_{2}$ with the map $\psi$ to $A^{*}$
defined by $\psi(x)=(\psi_{1}(x),\psi_{2}(x))$, where, if $x\notin K_{1}$
(respectively if $x\notin K_{2}$), one assumes $\psi_{1}(x)={\rm pt}$
($\psi_{2}(x)={\rm pt}$ respectively). If a point of $K$ lies in $N$, but
either a point of $K_{1}$ or a point of $K_{2}$ lies there. If all the points
of $K$ are not in $N$, then one of its points maps to
${\mathcal{B}}=({\mathcal{A}}^{(1)}\times{\mathcal{B}}^{(2)})\cup({\mathcal{B}}^{(1)}\times{\mathcal{A}}^{(2)})$,
then either its image under $\psi_{1}$ lies in ${\mathcal{B}}^{(1)}$ or its
image under $\psi_{2}$ lies in ${\mathcal{B}}^{(2)}$. This means that the
corresponding point of the union of the coefficients of $(A(t))^{[(M,N)]}$
lies in the image of the subspace in the product of the unions of coefficients
of $(A^{(1)}(t))^{[(M,N)]}$ and of $(A^{(2)}(t))^{[(M,N)]}$ and vice versa.

Condition 4) ($m_{1}=[(M_{1},N_{1})]$, $m_{2}=[(M_{2},N_{2})]$). The map (an
isomorphism) from the right hand side to the left hand side on the ambient
spaces is the following. If $K_{1}$ (respectively $K_{2}$) is a finite subset
of $M_{1}$ (respectively of $M_{2}$) with maps
$\psi_{1}:K_{1}\to{\mathcal{A}}^{*}$ and
$\psi_{2}:K_{1}=2\to{\mathcal{A}}^{*}$ (these data define a point from the
union of the coefficients of the right hand side), then $K\subset M_{1}\cup
M_{2}$ is the union of (disjoint) $K_{1}$ and $K_{2}$ with the natural map to
${\mathcal{A}}^{*}$. If a point of $K$ lies in $N_{1}\cup N_{2}$, then either
a point of $K_{1}$ lies in $N_{1}$ or a point of $K_{2}$ lies in $N_{2}$. If
all the points of $K$ are outside of $N_{1}\cup N_{2}$, but the image of one
of them lies in ${\mathcal{B}}$, then either of a point of $K_{1}$ or a point
of $K_{2}$ maps to ${\mathcal{B}}$. This means that the corresponding point of
the union of the coefficients of $(A(t))^{[(M_{1}\cup M_{2},N_{1}\cup
N_{2})]}$ lies in the image of the subspace in the product of the unions of
coefficients of $(A(t))^{[(M_{1},N_{1})]}$ and of $(A(t))^{[(M_{2},N_{2})]}$
and vice versa.

Condition 5) ($m_{1}=[(M_{1},N_{1})]$, $m_{2}=[(M_{2},N_{2})]$). The map (an
isomorphism) from the left hand side to the right hand side on the ambient
spaces (in this case it is more convenient to describe the map in this
direction) is the following. If $K$ is a finite subset of $M_{1}\times M_{2}$
with a map $\psi$ to ${\mathcal{A}}^{*}$ (these data define a point from the
union of the coefficients of the left hand side), then the corresponding
(finite) subset $K_{1}$ of $M_{1}$ is the projection $\pi_{1}(K)$ to the first
factor. For each point $y$ from $K_{1}$ the corresponding point of the union
of the coefficients of $\left(A(t)\right)^{[M_{2}]}$ (i.e. the point to which
$y$ is mapped) is represented by the subset $\pi_{2}\pi_{1}^{-1}(y)$ of
$M_{2}$ with the map to ${\mathcal{A}}^{*}$ being the corresponding
restriction. If a point $x\in K$ lies in $M_{1}\times N_{2}\cup M_{2}\times
N_{1}$, then either its projection to $M_{1}$ lies in $N_{1}$ or its
projection to $M_{2}$ lies in $N_{2}$. If this is not the case, but $\psi(x)$
lies in ${\mathcal{B}}$, then the corresponding point $\pi_{1}(x)\in M_{2}$ is
mapped to the subspace in the union of the coefficients of
$(A(t))^{[(M_{2},N_{2})]}$. This means that the corresponding point of the
union of the coefficients of
$\left((A(t))^{[(M_{2},N_{2})]})\right)^{[(M_{1},N_{1})]}$ lies in the image
of the subspace of the union of coefficients of
$(A(t))^{[(M_{1},N_{1})]\widehat{\times}[(M_{1},N_{1})]}$ and vice versa.
$\square$

## References

  * [1] A. Fonarev. Derived category of moduli of parabolic bundles on $\mathbb{P}^{1}$. arXiv:2305.18058.
  * [2] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. A power structure over the Grothendieck ring of varieties. Math. Res. Lett. 11 (2004), 49–57.
  * [3] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points. Michigan Math. J. v.54 (2006), no.2, 353–359.
  * [4] S.M. Gusein-Zade, I. Luengo, A. Melle-Hernández. On the power structure over the Grothendieck ring of varieties and its applications. Tr. Mat. Inst. Steklova 258 (2007), 58–69; translated in Proc. Steklov Inst. Math. 258 (2007), no.1, 53–64.
  * [5] D. Knutson. $\lambda$-rings and the representation theory of the symmetric group. Lecture Notes in Mathematics, 308, Springer-Verlag, Berlin – New York, 1973.
  * [6] A. Morrison, J. Shen. Hodge numbers of generalized Kummer schemes via relative power structures. Mosc. Math. J., v.21 (2021), no.4, 807–830.


# A public key infrastructure for compliant spectrum use by wireless devices

Stefan Ruehrup
ITS Services
ASFINAG Maut Service GmbH
Vienna, Austria

###### Abstract

A multitude of wireless devices for radio local area networks (RLAN) use
frequency bands that are shared with other radio services or applications.
European market surveillance activities on 5 GHz RLAN indicate a growing
number of non-compliant devices that do not fulfill all radio regulation
requirements and may cause interference.

This paper describes a concept and method for mitigating the use of non-
compliant equipment based on a public key infrastructure. If every RLAN device
holds a security certificate, which is revoked when non-compliance is found,
then the connection establishment between devices can be made dependent on
compliance. The aim is that legitimate equipment rejects connections to non-
compliant devices, so that non-compliant devices will become isolated.

###### Index Terms:

PKI, spectrum sharing, non-compliance, enforcement

##  I Introduction

Wireless local area networks (WLANs) are one of the most popular means to
provide Internet connectivity to mobile and portable devices. WLAN devices use
frequency bands assigned to Wireless Access Systems and Radio Local Area
Networks (WAS/RLAN) [4]. These frequency bands are shared with other radio
services or applications, and certain mitigation techniques are required by
regulation, such as Dynamic Frequency Selection (DFS) to protect radars.

European market surveillance activities conducted in 2018 [1] indicated a
growing number of non-compliant 5 GHz RLAN devices that do not fulfill all
requirements of the radio equipment directive. If non-compliant equipment is
brought to the market and used, it may cause harmful interference to other
radio services and applications.

In a concrete case of interference, the search for its source might become as
difficult as finding a needle in a haystack. First, because there are lots of
devices, since WLANs are present in most offices and households. Second, these
devices are not alway turned on, but rather used sporadically. Therefore it is
difficult to detect and locate a single non-compliant device. This problem
serves as motivation for this paper. We will abstract from the concrete
example and use in the following only the terms “compliant” and “non-
compliant” devices for radio equipment that does not fulfill the requirements
of the radio regulation (which is different from compliance to communication
standards).

In order to mitigate this problem, the main idea and method described in this
paper is to establish a “trusted” environment using digital certificates, in
which all compliant equipment operates. Non-compliant equipment, once
identified, is not part of or removed from this environment an becomes
isolated.

Such environment can be built based on public key infrastructures (PKI). A PKI
is a hierarchy of digital certificates, where each certificate consist of a
public key and a signature by a higher level certificate authority. Using for
verifying digital signatures – a valid signature indicates a valid
certificate. Assume that wireless devices (access points, tablets etc.) hold
such certificates and use them create digital signatures on data exchanged
during or after connection establishment. Then, a device can verify whether
the signature and thus the certificate of its counterpart is valid and
maintain the connection, otherwise it drops the connection. Once a device
model is found to be non-compliant, the certificate for this model is revoked,
and this device model cannot connect to other legitimate devices any longer.
This way, the legitimate devices reject connections and non-compliant devices
become isolated (see Fig. 1).

Figure 1: Isolation of non-compliant devices: Compliant devices with valid
certificates communicate to each other, but reject connections to devices
without valid certificate.

This paper describes the overall concept and architecture of using PKIs and
trust lists, which link PKIs together. One option for implementation is the
following: A PKI issuing certificates to devices can be maintained by a vendor
while a trust list of root certificates links the PKIs together. The overall
trust list could be maintained by an industry association or a government
entity.

This paper is structured as follows: In Section II we review related work. In
Section III we describe the overall architecture. In Section IV we discuss
implementation aspects. Section V concludes the paper.

##  II Related Work

Public key infrastructures (PKIs) and trust lists are established mechanisms
to setup and maintain a trust environment with respect to cybersecurity, see
[10] for a survey. They are used in the Internet to established secure
connections, and Internet browsers usually have the capability to import
certificates and validate signatures. PKIs also provide the basis for digital
signatures. There are standards and recommendations for protocols, data
formats and crypto-algorithms. The most common standard for certificates is
the ITU recommendation X.509 [7]. Formats for trust lists and certificate
revocation lists (CRLs) are specified, e.g., in [9] and [8].

Trust lists are also used in the context of digital signatures. As an example,
status information of certification service providers in the EU are maintained
in trusted lists defined in Commission Decision 2009/767/EC and its
implementing act [3], based on the technical specification in ETSI TS 119 612
[5]. These trusted lists are maintained by the Member States.

This paper uses the general concepts of PKIs and trust lists to maintain a
trust environment for equipment compliance. The basis is the ability to
validate digital signature with the help of a certificate as depicted in Fig.
2.

Figure 2: Validation of a signature: The receiver validates the signed message
by decrypting the signature with the help of the public key of the sender. The
public key of the sender is signed by the CA and can be validated if it is
part of the trust list.

Compliance of WLAN equipment has been investigated by Europan market
surveillance in [2] and [1]. The most prominent issue was found to be the
detection of radars through the dynamic frequency selection (DFS) requirement.
WiFi Alliance published a best practice document on this issue[12]. Technical
issues of radar detection, especially when in transmission, are studied in
[11]. These market studies and technical studies serve only as background and
motivation for this paper. In the following we will not elaborate on the
technical issues, but only distinguish between “compliant” and “non-compliant”
devices.

##  III Concept and Overall Architecture

The overall architecture consists of a public key infrastructure with a
Certification Authority (CA) maintained by a manufacturer or vendor (or a
group of vendors). This CA issues certificates, i.e. it signs public keys of
individual devices. These digital certificates are stored on each individual
devices. A device by another manufacturer can verify the certificate if it
knows the public key of the issuing certification authority (of the other
vendor). Though each device has an individual certificate, it is not necessary
to know all other device certificates, but only the higher level CA
certificate that is necessary to validate a digital signature. That is
achieved by collecting all CA certificates in a certificate trust list (CTL),
that is distributed regularly to all devices as depicted in Figure 3.
Certificates of non-compliant devices are put on a certificate revocation list
(CRL). Regular distribution of CTL and CRL to all devices ensure that only
compliant devices participate in the trust environment. A regular update does
not mean a daily update, but a longer time duration that depends on the
frequency of CTL changes (i.e. when new device models are registered) and CRL
changes (i.e. when non-compliance cases are found).

Figure 3: Overall concept of the public key infrastructure

The necessary steps are as follows:

  1. 1.

Digital certificates are issued by a Certification Authority (CA) and stored
in each device at manufacturing time.

  2. 2.

All CA certificates across all vendors are collected in the certificate trust
list (CTL). This trust list is provided to RLAN devices, e.g. as part of
regular software updates.

  3. 3.

Upon connection establishment between two RLAN devices, one device is able to
verify the certificate of the other device. This could be performed in a
request-response fashion after association or part of the network
authentication. It could also be separately queried via protocols and data
elements for device monitoring. If a valid signature and thus a valid
certificate is found, then the

  4. 4.

If a certain model of RLAN equipment is found to be non-compliant by market
surveillance authority, then its certificate is revoked and added to the
revocation list (CTL). Once a devices obtains the updated CTL, it will reject
the non-compliant device. As a consequence, the non-compliant model is limited
in operation.

###  III-A Administration of CRL and CTL

CTL and CRL are managed on an administrative level, which is independent of
individual vendors. This could be an independent organisation, a (govenment)
agency or an industry association. CTL and CRL are public, they do not need to
be encrypted, but digitally signed. Thus, market surveillance authorities can
check whether non-compliance cases are correctly handled.

Certificates in the CTL and CRL can be organised on two levels, on
manufacturer level and on device type level:

Option 1: Each manufacturer maintains only one root certificate. The
certificate trust list (CTL) holds one certificate for each manufacturer. In
this case the CTL is only updated when a new manufacturer is registered, but
on the level of the CTL a certificate revocation affects all devices of a
manufacturer.

Option 2: Each manufacturer maintains a root certificate and derived
certificates for each device type or model. The certificate trust list (CTL)
holds one certificate per device type or model. In this case the CTL is larger
and needs more frequent updates when new equipment is brought to the market.
However, on the level of the CTL, the certificate of a single non-compliant
device type or model can be revoked without affecting other equipment of the
same manufacturer.

Since the device itself is subject to conformance, option 2 might be a
preferred option.

Figure 4: Validation of certificates: The receiver of a signed message with a
certificate validates the signature and the certificate chain via the CTL.

###  III-B Vendor Level

On the vendor level the certificate distribution and the device’s mechanism
for certificate exchange are implemented.

Connections of compliant devices to other wireless devices should only b
established, when the certificate of the counterpart is valid as depicted in
Fig. 5, with the aim to isolate non-compliant devices. Non-compliant devices,
however, do not follow the rules by definition. Therefore, two non-compliant
devices may or may not communicate with each other.

The majority of radio local area networks consist of access points and
stations. Usually, station authenticates itself to the access point, i.e. it
is performed in one direction. For the overall concept in this paper, however,
it is beneficial that both devices check the certificate of their
counterparts, i.e. certificates are validated in both directions.

Thus, devices need to have the following capabilities:

  * •

Store private key and (public) certificate as well as CTL and CRL

  * •

Produce digital signatures using the own private key and present signature and
certificate to another querying device.

  * •

Validate a signature and a certificate from another device and verify the
certificate chain using CTL and CRL

  * •

Securely update CTL and CRL

These capabilities are common for systems participating in a PKI. The exchange
of digital signature and certificate might be part of the connection
establishment. However, in order to avoid modification of low level
authentication, it would be beneficial to use a separate protocol and check
the certificate validity after the devices are associated.

Figure 5: Connections are rejected by the compliant device if the counterpart
does not hold a valid certificate. If both devices are non-compliant, no
result can be stated.

##  IV Discussion

While public key infrastructures are already used in many application areas,
the use of a PKI for an environment of compliance has specific aspects which
we discuss in the following:

  * •

The PKI operation is independent of the product certification process. Vendors
can bring equipment to the market, and can provide their equipment with valid
certificates.

  * •

Non-compliant models, if identified by market surveillance, can be blacklisted
through the certificate revocation list (CRL). Then all devices of a certain
model (type) can be blocked from obtaining new certificates.

  * •

The CTL is essential for certificate validation. Its size depends on which
certificates are stored in it: manufacturer’s certificates, device type
certificates, or individual device certificates. While manufacturer’s
certificate would not allow isolation of certain non-compliant models, it is
unrealistic to maintain millions of device certificates. Device types or
models seem a reasonable choice.

  * •

Blacklisting does not immediately turn off a non-compliant device. It might
still operate, but other equipment will restrict its communication.

  * •

If digital certificates on the CTL and CRL are bound to device types or models
(and not to manufacturers), then blacklisting affects all devices of this
model. A single device could only be blacklisted through certificate
revocation by the manufacturer’s certification authority.

  * •

A simple re-registration of a non-compliant device under a new model number
needs to be prevented. Otherwise, an effective blacklisting can be
circumvented.

The concept does not require any specific addition to existing public key
infrastructures. However, it needs to be implemented on all equipment,
including compliant equipment. Therefore it is beneficial to re-use existing
protocols and data element standards that are implemented on many devices. As
an example, many devices are able to establish TLS connections and the
handling of X.509 certificates is part of their implementation.

##  V Conclusion

This paper describes a concept and method for using public key infrastructures
to establish a trust environment for wireless devices with respect to
compliance to radio regulation. The concept distinguishes between a vendor
level and and administrative level. On the administrative level, non-compliant
communication equipment can be blacklisted through a certificate revocation
list. On the vendor level, the distribution of certificates and the device’s
rejection of communication links to non-compliant equipment without valid
certificate is implemented. The aim of rejecting wireless communication to
non-compliant equipment is the isolation or limitation of using non-compliant
air interfaces. The proposed public key infrastructure does not enforce
compliance, but it can be regarded as a tool to manage an environment of
compliant wireless devices. While non-compliance needs to be detected by the
responsible authority, the PKI helps to isolate non-compliant equipment, with
the aim of limiting its use.

## References

  * [1] ADCO RED Report on the 9th joint cross-border RED market surveillance campaign (2018) on WLAN 5GHz. 20 March 2019. https://ec.europa.eu/docsroom/documents/35443
  * [2] ADCO R&TTE Report on the 5th Joint Cross-Border R&TTE Market Surveillance Campaign (2013) on WLAN 5 GHz. 23 April 2015. https://ec.europa.eu/docsroom/documents/9922
  * [3] Commission Implementing Decision of 14 October 2013 amending Decision 2009/767/EC as regards the establishment, maintenance and publication of trusted lists of certification service providers supervised/accredited by Member States
  * [4] ECC Decision of 09 July 2004 on the harmonised use of the 5 GHz frequency bands for the implementation of Wireless Access Systems including Radio Local Area Networks (WAS/RLANs), ECC/DEC/(04)08, 30 October 2009. https://www.ecodocdb.dk/document/381
  * [5] ETSI TS 119 612 V2.2.1 (2016-04) Electronic Signatures and Infrastructures (ESI); Trusted Lists
  * [6] Directive 2014/53/EU of the European Parliament and of the Council of 16 April 2014 on the harmonisation of the laws of the Member States relating to the making available on the market of radio equipment and repealing Directive 1999/5/EC. http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:32014L0053
  * [7] ITU Recommendation X.509 Information technology - Open Systems Interconnection - The Directory: Public-key and attribute certificate frameworks. https://www.itu.int/rec/T-REC-X.509/en
  * [8] IETF RFC 5280 Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile. https://tools.ietf.org/html/rfc5280
  * [9] IETF RFC 5914. Trust Anchor Format. https://tools.ietf.org/html/rfc5914
  * [10] Shahrzade Mazaher, Per Røe: A survey of state of the art in Public Key Infrastructure. Norsk Regnesentral Rapport nr. 995, August 2003. http://publications.nr.no/3749/Mazaher_-_A_Survey_of_State_of_the_Art_in_Public_Key_Infrast.pdf
  * [11] H. Safavi-Naeini, S. Roy, S. Ashrafi: Spectrum Sharing of Radar and Wi-Fi Networks: The Sensing/Throughput Tradeoff. IEEE Transactions on Cognitive Communications and Networking, 1(4), p. 372-382, December 2015.
  * [12] The Wi-Fi Alliance Spectrum & Regulatory Committee. Sharing in the 5 GHz Band - DFS Best Practices. 10 October 2007.


††thanks: Equal contribution
Present address: Fakultät für Physik, Ludwig-Maximilians Universität München,
Geschwister-Scholl-Platz 1, 80539 München, Germany††thanks: Equal
contribution††thanks: Present address: Wuhan National High Magnetic Field
Center and School of Physics, Huazhong University of Science and Technology,
Wuhan 430074, China

# Glass-like thermal conductivity and narrow insulating gap of EuTiO3

Alexandre Jaoui  JEIP, USR 3573 CNRS, Collège de France, PSL Research
University, 11, Place Marcelin Berthelot, 75231 Paris Cedex 05, France
Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France  Shan Jiang  Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France  Xiaokang Li  Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France  Yasuhide Tomioka  National Institute of Advanced Industrial Science
and Technology (AIST), Tsukuba 305-8565, Japan  Isao H. Inoue  National
Institute of Advanced Industrial Science and Technology (AIST), Tsukuba
305-8565, Japan  Johannes Engelmayer  II. Physikalisches Institut, Universität
zu Köln, 50937 Köln, Germany  Rohit Sharma  II. Physikalisches Institut,
Universität zu Köln, 50937 Köln, Germany  Lara Pätzold  II. Physikalisches
Institut, Universität zu Köln, 50937 Köln, Germany  Thomas Lorenz  II.
Physikalisches Institut, Universität zu Köln, 50937 Köln, Germany  Benoît
Fauqué  JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11,
Place Marcelin Berthelot, 75231 Paris Cedex 05, France  Kamran Behnia
Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France

###### Abstract

Crystals and glasses differ by the amplitude and the temperature dependence of
their thermal conductivity. However, there are crystals known to display
glass-like thermal conductivity. Here, we show that EuTiO3, a quantum
paraelectric known to order antiferromagnetically at 5.5 K, is one such
system. The temperature dependence of resistivity and Seebeck coefficient
yield an insulating band gap of $\sim 0.22$ eV. Thermal conductivity is
drastically reduced. Its amplitude and temperature dependence are akin to what
is seen in amorphous silica. Comparison with non-magnetic perovskite solids,
SrTiO3, KTaO3, and EuCoO3, shows that what impedes heat transport are $4f$
spins at Eu2+ sites, which couple to phonons well above the ordering
temperature. Thus, in this case, superexchange and valence fluctuations, not
magnetic frustration, are the drivers of the glass-like thermal conductivity.

##  I Introduction

In most insulating crystals, the flow of heat can be understood by considering
the response of a gas of phonons to a temperature gradient. This picture,
first drawn by Peierls [1], using a linearized Boltzmann equation, is
remarkably successful in describing the thermal conductivity, $\kappa$, of
most insulators [2, 3]. It explains why at an intermediate temperature,
$\kappa$ peaks [4], separating a high-temperature decrease by warming (due to
anharmonicity), and a low-temperature decrease by cooling (due to phonon
depopulation). In amorphous solids, on the other hand, there is no such peak
in $\kappa(T)$ and heat diffuses thanks to off-diagonal coupling across
harmonic branches [5, 6, 7, 8].

Some crystals, however, display glass-like thermal conductivity [9, 10, 11,
12, 13, 14, 15]. The thermal conductivity of these materials can be very low
and/or feature a monotonic temperature dependence (lacking the $T^{-1}$
decrease at high temperature). They are sought after, since a low lattice
thermal conductivity would lead to a large thermoelectric figure of merit
[16].

EuTiO3 (ETO) is a perovskite with a cubic structure at room temperature [17].
Its electric permittivity increases upon cooling without giving rise to a
ferroelectric instability. This quantum paraelectric behavior [18] is akin to
what has been seen in other ABO3 compounds such as SrTiO3 (STO) and KTaO3
(KTO). In contrast to them, however, it magnetically orders at T${}_{N}=5.5$
K, with an antiferromagnetic alignment of the nearest neighbor Eu2+ spins [19,
20]. Like STO, but at higher temperatures, it goes also through an
antiferrodistortive transition where adjacent TiO6 octahedra rotate in
opposite directions [21]. Upon doping, either with oxygen vacancies [22] or La
substitution of Ti [23], it becomes a dilute metal [22, 23, 24], but without a
superconducting ground state.

Here, we show that the temperature dependence of thermal conductivity in ETO
is glass-like and its amplitude, over a broad temperature range starting from
room temperature and extending to cryogenic temperatures, is lower than in STO
and KTO. The drastic attenuation of thermal conductivity occurs well above the
Néel temperature and when the magnetic entropy is saturated to its expected
value. This points to an unusual version of spin-phonon coupling such as the
phonon-paramagnon hybridization postulated by Bussmann-Holder et al. [17].

Evidence for spin-lattice coupling in ETO has been around for more than two
decades [20]. However, it was restricted to temperatures comparable to the
magnetic ordering. According to our findings, the phonon mean free path is
affected by spins at Eu sites even at high temperature when there is neither a
magnetic order nor a field-dependent entropy.

Monitoring the temperature dependence of electrical conductivity and
thermopower in insulating EuTiO3, we find an activation gap of 0.11 eV,
indicating that the intrinsic transport band gap in EuTiO3 is as low as
$\sim$0.2 eV, much smaller than the gaps found by optical studies, but close
to what is expected by Ab Initio calculations [25, 26].

The narrow gap between the chemical potential and Eu2+ energy level and the
random orientation of large magnetic moments in the paramagnetic state emerge
as principal suspects shortening the lifetime of heat carrying phonons at
elevated temperatures in the paramagnetic state.

In contrast to many other crystals displaying glass-like conductivity, ETO is
not a spin liquid candidate, but a simple $G$-type antiferromagnet and there
is no magnetic frustration. In this context, the glassy behavior could be
framed in a picture of off-diagonal coupling across harmonic modes [7, 15],
which in this case would be phonons and paramagnons.

Figure 1: Electrical resistivity, thermopower and activation gap of EuTiO3:
a) Resistivity, $\rho$, as a function of temperature in two single crystals of
EuTiO3 in a semi-logarithmic plot. The inset shows the unit cell (in the cubic
phase). b) Arrhenius plot of the same data: $ln\rho$ vs. $T^{-1}$. The solid
blue line corresponds to $\Delta=0.11eV$. The inset shows the temperature
variation of $\Delta=-k_{B}T^{2}\frac{\partial ln\rho}{\partial T}$, with an
arrow pointing to $T_{AFD}$, the temperature of antiferrodistortive
transition. c) Seebeck coefficient, $S$ as a function of temperature. The data
is restricted to $T>120K$, below which measurement becomes problematic. The
inset shows $S$ as a function of $T^{-1}$. The solid blue lines corresponds to
$\Delta=0.1eV$. The arrow points to $T_{AFD}$.

##  II Results

###  II.1 Activation and band gap

Fig.1a shows the temperature dependence of electrical resistivity $\rho$ in
as-grown EuTiO3 (ETO) single crystals. One can see that $\rho$ increases by
seven orders of magnitude upon cooling from 360 K to 50 K.

An Arrhenius activation behavior becomes visible by plotting ln$\rho$ as a
function of the inverse of temperature (Fig.1b). The activation gap is $\sim
0.11$ eV. The Hall resistivity (see the supplement [27]) shows also an
Arrhenius behavior with a comparable $\Delta$. At low temperature, resistivity
begins to saturate when the Hall carrier density is as low as $10^{10}cm^{-3}$
(See the supplement [27]). This implies an extremely low level of extrinsic
donors . An activated behavior in longitudinal and Hall resistivities of as-
grown ETO crystals was previously reported by Engelmayer et al. [22], whose
study focused on the emergence of metallicity in oxygen-deficient ETO.

This activated behavior is also confirmed by our measurements of the Seebeck
coefficient (See Fig.1(c)) and and the temperature dependence of the electric
permittivity [22].

The thermoelectric power has a negative sign and an amplitude in the range of
mV/K, typical of a narrow gap semiconductor. The Seebeck coefficient in an
intrinsic semiconductor is expected to be
$\approx\frac{k_{B}}{e}\frac{\Delta}{k_{B}T}$ [28]. Thus, by plotting $S$ as a
function of $T^{-1}$ (see the inset), one expects to see a straight line whose
slope yields $\Delta$. As seen in the inset of Fig.1c, this is indeed what our
data yields, with $\Delta\approx 0.1eV$. Note that this temperature dependence
is qualitatively distinct from what is expected in extrinsic semiconductors
[29] as seen, for example, in the case of Nb-doped STO [30].

Thus, the temperature dependence of electric and thermoelectric conductivity
both point to a similar energy gap between the chemical potential and the
conduction band. The Hall and the Seebeck coefficients are both negative,
indicating that carriers are electron-like and thermally excited to live in
the conduction band originating from Ti$-d$ orbitals.

Both $\rho$ and $S$ show an anomaly near 260 K, which we identify as the
temperature of the structural transition in ETO [17, 31]. Like STO [32], this
transition makes ETO tetragonal [33]. We cannot rule out a very small
difference in the amplitude of the activation gap between the cubic and the
tetragonal phases (See the inset of Fig1b). We also detected an unexpected and
reproducible hysteresis near this phase transition. Taken at its face value,
this indicates that this structural phase transition is first order (see the
supplement).

Assuming that the chemical potential is at halfway between the chemical
potential and the valence band leads us to conclude that the band gap of ETO
is $\approx 0.22$ eV. This is remarkably smaller than the 3.2 eV gap of STO
[34], but only slightly smaller than what a recent DFT calculation [26] found
(0.27-0.33 eV). According to another and earlier theoretical study [25], the
magnitude of the band gap in ETO depends on the amplitude of the Hubbard $U$,
and a realistic $U$ (5-6 eV) would lead to a band gap of 0.2-0.4 eV. Thus, the
narrow gap detected by our transport studies is close to what is intrinsically
expected in this solid. Optical probes, however, have detected a larger gap
(0.8-0.9 eV) [35, 36]. Such larger energy scales are possible indications that
the Density of States (DOS) near the Eu$-f$ level is not featureless [26].

Let us keep in mind the contrast between ETO on one hand and STO and KTO on
the other hand. The first has a valence band originating from Eu$-f$ orbitals
close to the chemical potential, while the two others have a valence Band
emanating from O$-p$ orbitals and much further away from the chemical
potential.

###  II.2 Thermal conductivity

Fig. 2 shows the temperature dependence of thermal conductivity, $\kappa$, of
two ETO crystals from slightly above room temperature down to dilution
refrigerator temperatures. We measured several crystals and found that the
thermal conductivity of all lies somewhere between the two cases shown in this
figure. Moreover, we could not detect a correlation between the amplitude of
$\kappa$ and a weak variation of the saturation magnetization observed across
various ETO samples (see the supplement [27]).

Figure 2: Thermal conductivity of two EuTiO3 crystals:  a) Thermal
conductivity, $\kappa$, as a function of temperature in two EuTiO3 crystals in
a log-log plot. The arrow points to a minimum near the Néel temperature. The
inset shows a linear plot, with an arrow pointing to $T_{AFD}$.

In contrast to typical crystalline insulators [4], $\kappa$ does not show a
prominent peak. As seen in the inset, there is a clear anomaly at $T_{AFD}$
and below this temperature $\kappa$ rises by ten percent increase with
cooling. However, there is no sign of a kinetic regime with $\kappa\propto
T^{-1}$, as seen in many other insulators [37, 38].

###  II.3 Crystals, glasses and glass-like crystals

In order to put our observation in a proper context, we compare our data with
what has been reported in the case of SiO2, which shows a spectacular
difference in the thermal conductivity in its crystalline and amorphous
structures [39]. As seen in Fig.3, $\kappa$ in amorphous silica monotonically
decreases with cooling, in contrast to crystalline quartz, which has a
prominent peak. At any given temperature, the crystal conducts heat at least
an order of magnitude more than the glass [4]. Not only the thermal
conductivity of ETO is similar to silica in temperature dependence, but also
in the cryogenic temperature range surrounding the Néel temperature
($T_{N}\simeq 5.5$K), the ETO crystalline samples conduct heat less than
amorphous silica. The order of magnitude of thermal conductivity and its
temperature dependence in ETO is comparable with other crystalline solids
displaying a glass-like thermal conductivity, such as Tb2Ti2O7 [11], Tb3Ga5O12
[40], Na4Ir3O8 [41], Pr2Ir2O7 [42] and La0.2Nd0.4Pb0.4MnO4 [9]. In order to
allow a direct comparison, Fig.3 includes the data for Tb2Ti2O7 [11], the
compound with the lowest thermal conductivity among these frustrated magnets.
Note that in contrast to other members of this club of materials, EuTO3 is a
G-type antiferromagnet and is not frustrated.

As a further comparison, we also include $\kappa$ of EuCoO3 [43], which
displays a temperature dependence typical of a crystalline insulator. EuCoO3
is a perovskite like ETO, but it has an orthorhombic symmetry, and the valence
is Eu3+ with only 6 electrons in the 4f shell. According to Hund’s rules, this
causes a vanishing magnetic moment ($J=L-S=0$), in agreement with the
experimentally observed van Vleck susceptibility [44], drastically different
from the large local moments of $7\,\mu_{B}/$Eu2+ in ETO.

Figure 3: Comparison with crystals, glasses and glass-like crystals: Thermal
conductivity as a function of temperature in crystalline quartz, in vitreous
silica [39], in EuTiO3 and in the frustrated magnet Tb2Ti2O7 [11]. The latter
two crystalline compounds conduct heat like a glass rather than like a
crystal. Also shown is the crystal-like thermal conductivity of non-magnetic
EuCoO3 [43].

Figure 4: Thermal conductivity of three quantum para-electric solids: Thermal
conductivity of the two EuTiO3 samples in a linear scale compared with thermal
conductivity of SrTiO3 [45] and KTaO3 [46, 12]. Note the drastic reduction of
thermal conductivity in the paramagnetic EuTiO3. The visible sample dependence
of the data for each material is much smaller than the differnce between the
three compounds.

###  II.4 ETO compared to its non-magnetic sisters

Fig.4 compares the thermal conductivity of ETO and two other ABO3 perovskites.
SrTiO3 (STO) and KTaO3 (KTO) are also quantum paraelectric, but not magnetic,
solids. Our new data on these two materials is in good agreement with previous
studies of heat transport in STO [47, 45, 48] and in KTO [47, 46, 48]. In both
cases, there is also a visible sample dependence, which is more pronounced
near the peak. However, this sample dependence is much smaller than the
difference between the three compounds. At room temperature, this difference
is small, yet visible: $\kappa(300K)$ is $\approx 9$ W/(K.m) in ETO, $\approx
11$ W/(K.m) in STO and $\approx 17$ W/(K.m) in KTO. Much more drastic is the
difference in the temperature dependence between the three sister compounds.
The enhancement with cooling observed in the other perovskites is absent in
ETO. This difference extends over the full temperature range above the
magnetic ordering.

Figure 5: Specific heat in three quantum paraelectric solids: a) Temperature
dependence of the total specific heat in KTaO3, SrTiO3 and EuTiO3 plotted as
$C/T$ vs. $T$. The inset is an enlarged view of the same data shown as
$\frac{C_{p}}{T^{3}}$ vs. $T$. Note the presence of Einstein modes in all
three solids and the additional magnetic specific heat in ETO emerging below
15 K. b) Evolution of $C_{mag}/T$ of ETO for different magnetic fields up to
10 T. The inset shows the corresponding magnetic entropy $S_{mag}=\int
C_{mag}/T\,dT$ of ETO and the maximum magnetic entropy $N_{A}k_{B}\ln(2S+1)$
of a spin 7/2 system is indicated by the dashed line.

It is instructive to scrutinize the specific heat of the three sister
compounds. Fig. 5(a) shows that $C/T$ of STO, ETO and KTO approach each other
towards room temperature and reach $C\approx 100\,$J/(mol.K). With five atoms,
one expects the specific heat saturating to the Dulong-Petit value of
$5\,N_{A}k_{B}=125\,$J/(mol.K), which is indeed what happens to STO, heated to
1800 K [49], a remarkably high temperature, broadly compatible with the
highest energy scale in the phonon spectrum of these solids ($\approx$ 100
meV) [31, 50]. A systematic difference in the specific heat evolves at lower
temperatures. As is shown in the inset of Fig. 5(a), all three solids show a
peak in the temperature dependence of $\frac{C}{T^{3}}$. In the case of STO,
this peak is known to be caused by the presence of Einstein modes [51, 52].
Similar peaks are visible for KTO and for ETO, and the position of this peak
shifts with the increase in the molar mass. In STO (184 g/mol), it occurs at
30 K, in ETO (248 g/mol) at 25 K, and in KTO (267 g/mol) at 12 K. In the case
of ETO, a distinct additional contribution shoots up below 15 K, which signals
strong magnetic fluctuations above the Néel ordering temperature of the Eu2+
spins.

###  II.5 Field dependence

As reported previously [53], the specific heat in ETO displays a strong
dependence on magnetic field. In order to separate the magnetic $C_{mag}$ and
the phononic $C_{phon}$ contribution, we subtracted from the total specific
heat of EuTiO3 the measured specific heat of SrTiO3 (with a temperature re-
scaled by a factor of 0.83 such that the positions of the $C/T^{3}$ maxima of
both materials match). Note that this scaling factor is close to the ratio of
the molar masses $\sqrt{M_{\mathrm{STO}}/M_{\mathrm{ETO}}}=0.86$, the expected
ratio of their Debye temperatures. Figure 5(b) shows the magnetic heat
capacity $C_{mag}/T$ of EuTiO3 for different magnetic fields from 0 up to
$10\,$T applied along a $[100]_{\mathrm{c}}$ direction of the cubic room
temperature phase. The sharp zero-field anomaly signals the antiferromagnetic
order at $T_{\mathrm{N}}=5.5\,$K that gets strongly broadened already in a
field of $1\,$T. This reflects the magnetic-field-induced switching to the
polarized magnetic state, which sets in around 1.2 T in ETO (see, e.g., the
supplement [27] or [54]) and, consequently, the $C_{mag}/T$ data in larger
fields no longer signal a spontaneous magnetic ordering transition, but a
continuous evolution of magnetic entropy as it is the case for ferromagnets in
an external magnetic field. In fact, the $C_{mag}/T$ data of EuTiO3 strongly
resembles the corresponding data obtained on the Eu2+-based ferromagnet EuC2,
which orders at $T_{\mathrm{C}}=14$ K [55]. The magnetic entropy obtained by
integrating $C_{mag}/T$ is displayed in the inset of Fig. 5(b) and reveals
that the full magnetic entropy $N_{A}k_{B}\ln(2S+1)$ of a spin 7/2 system is
reached above about $15\,$K for zero field and also for 1 T, whereas for
fields above 2 T the entropy evolution drastically broadens and extends to
much higher temperatures.

The field dependence of $\kappa$, shown in Fig. 6, further points to an
intricate coupling between Eu2+ spins and heat-carrying phonons. Thermal
conductivity begins to depend on magnetic field below 15 K. Interestingly, as
seen in the inset of Fig. 5, this is the temperature below which there is a
significant magnetic entropy.

Above $T_{N}$, magnetic field slightly amplifies $\kappa$, indicating a
weakening of the spin-induced localisation of phonons. This field-induced
amplification of $\kappa$ in ETO is modest, in contrast with to the thirty-
fold field-induced amplification of the ultra-low thermal conductivity in
Tb2Ti2O7 [11].

Below 3 K, well below the Néel temperature, the field-dependence becomes non-
monotonic (See Fig. 6. We leave the quantitative understanding of the field
dependence of $\kappa$ in EuTiO3 as a subject of study for future
investigations.

Figure 6: Field dependence of thermal conductivity in EuTiO3 a) $\kappa$ vs. T
from 2 K to 10 K for B = 0 T to 9 T. b) Normalised $\kappa$ vs. B for T = 2.4
K to 15.6 K for B$//$[001].

###  II.6 Thermal diffusivity and phonon mean-free-path

Figure 7: Thermal Diffusivity of EuTiO3 and the mean-free-path of phonons:
Temperature dependence of thermal diffusivity, $D$, by taking the ratio of
thermal conductivity and the specific heat per volume in sample C1. Note the
drastic drop across the magnetic transition. The insets shows the Temperature
dependence of the mean free path in the paramagnetic phase, assuming that all
phonons have the same scattering time and velocity for the same sample. The
red solid line represents the inverse of the wavevector of the thermally
excited acoustic phonons.

Replacing Sr by Eu reduces the thermal conductivity in a wide temperature
range and enhances the specific heat below 15 K. Therefore, the ratio of
thermal conductivity (in W/(K.m)) to specific heat per volume (in J/(K. m3)),
i.e. the thermal diffusivity, $D$, (in m2/s) is drastically modified. It is
plotted in Fig. 7. The most striking feature of $D(T)$ is its two-orders-of-
magnitude drop at the Néel temperature. Within the entry to the
antiferromagnetically ordered phase, at 5.5 K, it becomes exceptionally low.
Its minimum, 0.03 mm2/s, is almost two orders of magnitude lower than the
thermal diffusivity of a typical glass [56]. This low thermal diffusivity, a
consequence of the combination of an unusually low lattice thermal
conductivity and a very large magnetic entropy may find applications in heat
management in a cryogenic context.

The thermal conductivity of an insulator can be written as:

$\kappa=\sum_{s,\bf{q}}C_{s}(\bf{q})v^{2}_{s}(\bf{q})\tau_{s}(\bf{q})$ (1)

The index $s$ refers to different modes and $\bf{q}$ is the wave-vector.
$C_{s}$, $v_{s}$ and $\tau_{s}$ are, specific heat, velocity and scattering
time. There are modes contributing to the total specific heat
($C=\sum_{s}C_{s}$ ), but not to thermal conductivity, because of their
negligible velocity. Theoretically [17], paramagnons in the paramagnetic state
have no dispersion and therefore do not carry heat. They can, however, reduce
the phonon thermal transport. In ETO, phonons not only dominate thermal
conductivity, but also, at least down to 15 K, the specific heat. Therefore,
one can simply write: $\kappa=1/3Cv_{s}\ell_{ph}$. This neglects the $q$
dependence of the scattering time and assumes that all modes have the same
velocity. Taking $v_{s}$= 6.8 km/s, the measured sound velocity in STO [57],
and in reasonable agreement with the dispersion of acoustic branches in ETO
[31], one can estimate $\ell_{ph}$, shown in the inset of Fig. 7. Comparing it
to the inverse of the wave-vector of thermally excited phonons:
$q_{s}=\frac{k_{B}T}{\hbar v_{s}}$, one finds that below 20 K,
$q_{s}\ell_{ph}\cong 1$ , reminiscent of the Anderson localization. There is a
drop at 15K, below which specific heat is no more phonon dominated.

##  III Discussion

Evidence for a coupling between spin and lattice degrees of freedom in this
compound was first reported by Katsufuji and Takagi [20], who found that when
the spins order at 5.5 K, the electric permittivity of ETO drops by 7 percent
and this drop is suppressed by the application of a magnetic field of the
order of 3 T. This magneto-electric effect implies coupling between Eu2+ spins
and the soft mode governing the electric permittivity. Reuvekamp et al. [58]
have found a quantitative agreement between the amplitude of the magneto-
electric effect and the low-temperature magnetostriction of the system.

Our main finding is that lattice-spin coupling drastically attenuates the
propagation of heat in ETO, even at temperatures where magnetic ordering is
absent and magnetic entropy is practically saturated at its maximum value
$k_{B}\ln(2S+1)$/spin. This implies that even when the spins are randomly
oriented, heat-carrying phonons couple to Eu2+ states and their large magnetic
moments (6.9-7 $\mu_{B}$). According to ref. [26], without incorporating the
loss of spin symmetry, the DFT calculations cannot explain the absence of
metallicity and the finite band gap of the paramagnetic phase.

Figure 8: Crystal structure of EuTiO3 in its paramagnetic phase: Random
magnetic moments at Eu sites are located between TiO6 octahedra, which are
tilted off each other. Superexchange interaction between these spins involves
Ti and oxygen atoms. This can impede phonons to have a well-defined wave-
vector over long distances.

The random orientation of magnetic moments at Eu sites (See Fig. 8) is the
most plausible source of phonon localization. The superexchange interaction
between Eu spins occurs through Ti ions [59]. The inter-atomic force constant
can depend on the relative orientation of spins. The calculated phonon
frequencies for parallel or anti-parallel alignment of adjacent spins are not
the same [60], which implies that phonons cannot keep a well-defined
dispersion in presence of random spin orientation.

The narrow energy separation between the Eu2+ energy level and the chemical
potential may also play a role. In some Eu-based metals, the thermoelectric
response has been linked to the temperature dependence of Eu valence [61].
Remarkably, a theoretical study [62] has concluded that the contribution of
optical phonons to the overall lattice thermal conductivity is unusually large
in this lattice structure. A coupling between heat-carrying optical phonons
and Eu valence may be the source of observed $\kappa$ attenuation.

Inelastic neutron scattering is a promising probe of this physics. In the case
of Tb3Ga5O12, for example, it documented the coupling between spin and lattice
[63]). A recent study on STO [64] has found evidence for an unusual
hybridization between acoustic and optic phonon branches. No neutron
scattering data is presently available to compare ETO with STO.

Finally, let us note that the glass-like thermal conductivity of EuTiO3, in
contrast to spin-liquid crystals, occurs in a solid with a simple G-type anti-
ferromagnetic ground state [19]. A formal theoretical treatment may be
achieved by complementing the picture drawn by Eq.1 with an ‘off-diagonal’
coupling between different vibrational states [7, 8, 15]:

$\kappa_{od}=\sum_{ss^{\prime},\bf{q}}^{s\neq
s^{\prime}}C_{ss^{\prime}}(\bf{q})v^{2}_{ss^{\prime}}(\bf{q})\tau_{ss^{\prime}}(\bf{q})$
(2)

This equation was conceived for non-magnetic crystals, in which harmonic
coupling occurs across phonon branches [7, 15]. It can be extended to magnetic
modes.

##  IV Acknowledgements

We thank Annette Bussman-Holder, Yo Machida, Igor Mazin and Nicola Spaldin for
stimulating discussions. This work was supported in France by JEIP-Collège de
France, and by the Agence Nationale de la Recherche (ANR-18-CE92-0020-01;
ANR-19-CE30-0014-04) and a grant by the Ile de France regional council. In
Germany, it was supported by the DFG (German Research Foundation) via Project
No. LO 818/6-1. I.H.I.In Japna, it was supported by the Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant No. 18H03686 and the Japan Science
and Technology Agency (JST) CREST Grant No. JPMJCR19K2.

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Supplementary Materials for _Glass-like thermal conductivity and narrow
insulating gap of EuTiO 3 _

##  S1 Crystal growth

EuTiO3 single crystals were grown in two different labs. In Cologne, the
floating-zone technique was used as described in Ref. [22]. In order to
prevent the formation of the Eu2Ti2O7 pyrochlore phase, the growth was
performed in Argon atmosphere and without any preliminary powder reaction,
similar to Ref. [65]. Powders of Eu2O3, TiO, and TiO2 were mixed in a ratio to
reach EuTiO3-δ, pressed to a rod, and installed into the floating-zone
furnace. In a series of growth attempts, it turned out that it was necessary
to mix the above materials with a nominal oxygen deficiency of $\delta=0.05$.
This allowed to compensate possible off-stoichiometries of the starting
materials or an oxygen capture during the growth process. Single crystallinity
of the grown crystal was confirmed by Laue images and phase purity was
verified by X-ray powder diffraction.

In Tsukuba, similarly to the above description, EuTiO3 crystals were prepared
by the floating zone method [23].

##  S2 Magnetization

Figure S1(a) displays magnetization data M(T) of a EuTiO3 single crystal for
different magnetic fields between 10 and 500 mT applied along a
$[100]_{\mathrm{c}}$ direction of the cubic room temperature phase. At the
lowest fields, the antiferromagnetic ordering causes a maximum in $M(T)$. This
peak becomes a kink at an intermediate field of 500 mT. The corresponding
inverse susceptibility follows a straight line over the entire temperature
range from 300 K down to about 7 K (See panels (b) and (c)). The Curie-Weiss
fit yields an effective magnet moment of
$\mu_{eff}=7.8\,\mu_{\mathrm{B}}$/f.u. and a positive Weiss temperature of
$\theta_{\mathrm{W}}=3.55\,$K, which signals a net ferromagnetic coupling.
These values agree with the literature. The positive $\theta_{\mathrm{W}}$ has
been explained by the fact that in EuTiO3 the antiferromagnetic coupling of
the Eu2+ spins to 6 nearest neighbors (NN) is overcompensated by a larger
ferromagnetic coupling to 12 next nearest neighbors[19, 54, 66, 36, 53].

Figure S1: (a) Magnetization of EuTiO3 as a function of temperature for
different magnetic fields applied along the cubic $[100]_{\mathrm{c}}$
direction. (b) Inverse susceptibility $\chi^{-1}=\mu_{0}H/M$ at 500 mT with a
Curie–Weiss fit (line) for $T>20$ K (c) Detailed view of the low-temperature
regime, revealing the Weiss temperature $\theta_{\mathrm{W}}=3.55$ K. (d)
Magnetization curve measured at 2 K, which almost reaches the saturation
magnetization expected for a spin 7/2 system (dashed line) (e) Magnetization
curves measured on 2 different pristine EuTiO3 single crystals (JE93, JE111),
which both were insulating, in comparison to 2 metallic EuTiO3 samples, which
were weakly electron doped by vacuum annealing [22]. Insets show enlarged
views of the same data in the region of (f) the saturation magnetization and
of (g) a spin-flop transition occurring at $\approx 0.2\,$T when the field is
applied along a cubic [100]c direction (3 samples), but remains absent for
$H[110]_{c}$ (doped sample JE93).

Note that there is no magnetic frustration resulting from the ferromagnetic
and antiferromagnetic couplings because both couplings support the G-type
antiferromagnetic order. Nevertheless, a comparatively weak magnetic field of
about 1.2 T is sufficient to induce a polarized magnetic state. This is
because only the weaker NN antiferromagnetic coupling needs to be overcome. As
shown in Fig. S1(d), the saturation magnetization $M_{\mathrm{sat}}\simeq
6.73\,\mu_{\mathrm{B}}$/f.u. reaches about 96 % of the expected spin-only
value of $M_{\mathrm{sat}}\simeq g\mu_{\mathrm{B}}S_{z}=7\mu_{\mathrm{B}}$ for
Eu2+ with half-filled $4f$ shell. An analogous 4 % reduction is obtained from
the Curie-Weiss fit in the paramagnetic phase for the Curie constant, which
results in a 2 % reduction of the effective magnetic moment
$\mu_{eff}=7.8\,\mu_{\mathrm{B}}$/f.u. in comparison to the expected
$\mu_{eff}=g\sqrt{S(S+1)}\mu_{\mathrm{B}}=7.94\,\mu_{\mathrm{B}}$ for $S=7/2$.

As is shown in Fig S1(e) this weakly reduced saturation magnetization has been
observed in various pristine insulating samples that were obtained from
different growth attempts. In contrast, however, other EuTiO3 samples that
were cut from the same mother crystals, but were made conducting via vacuum
annealing at high temperature, see e.g. [22], reached the expected saturation
magnetization of $7\mu_{\mathrm{B}}$/Eu2+. As it is known that the vacuum
annealing of STO and ETO causes electron doping which is traced back to a
reduction of the oxygen content, the different saturation magnetizations of
pristine versus doped ETO can be interpreted as follows: appearently, the
pristine ETO samples have a weak oxygen surplus that results in a
corresponding partial oxidization of Eu2+ to Eu3+ which has a vanishing
magnetic moment thereby reducing the overall magnetization. Vice versa, with
decreasing oxygen content, the magnetization of pristine ETO should first
increase but remain insulating until all Eu3+ ions are reduced to Eu2+ and
only with further oxygen reduction a metal-insulator transition sets in via
electron doping into the 3d band of Ti [22].

##  S3 Methods for measuring thermal conductivity

We measured thermal conductivity with a standard one-heater-two-thermometers
method. Above 2K, the measurements were carried out with a home-built setup in
a Quantum Design PPMS. Down to 100mK, the measurements were realized in a
Dilution refrigerator system. The sensors used for the measurement of ETO were
Cernox chips Cx-1030 and RuO2 thermometers. Above 20 K and up to room
temperature, type E thermocouples were used. Close attention was paid to the
overlap between different sets of experimental data. The thermometers were
directly glued to the samples with Dupont 4922N silver paste. These sensors
were also thermally isolated from the copper sample holder by long manganin
wires with very low thermal conductance. The ETO/STO samples were connected to
a heat sink (copper finger) with Dupont 4922N silver paste on one side and to
a RuO2-resistor (used as a heater) on the other side. A heating current was
applied by flowing a current $I$ through the heater from a DC current source
(Keithley 6220). We evaluated the heating power from the relation $I\times V$
with $V$: the electric voltage measured across the heater by a digital
multimeter (Keithley 2000). $\kappa$ was controlled to be independent of the
applied thermal gradient $\Delta T$ by changing $\Delta T/T$. $(\Delta
T/T)_{max}$ was kept under $10\%$. Finally, calibration of the sensors was
realized during each experiment and upon verification.

##  S4 Sample dependence

Figure S2: (a) Comparison of thermal conductivity data $\kappa(T)$ of EuTiO3,
which were measured on different samples, which stem from different sources
(T1 versus C1, C2, C3) (b) Magnetization data measured on the same samples
apart from C1, that got broken in the meanwhile. Note that the field axis was
corrected for demagnetization effects to account for the rather different
sample geometries.

As discussed above, the magnetization data of pristine ETO crystals indicate
the presence of a certain mixture of Eu2+ and Eu3+ valencies, and it is known
from many intermetallic materials that Eu (or other rare-earth) valence
instabilities can drastically influence the magnetic and/or transport
properties [61]. Therefore, we studied several EuTiO3 crystals in order to
check whether there is some correlation between the surprisingly low, glass-
like thermal conductivity and the reduced saturation magnetization. As can be
seen from Fig. S2(a), the overall variation of the $\kappa(T)$ curves obtained
on the different samples is covered by the data obtained on samples C1 and T1,
which are discussed in the main article. The magnetization data of the
pristine samples C2 and T1 are essentially identical to each other and agree
with the data of the other pristine samples shown in Fig.S1. While this is not
surprising for C1, which is another piece from the same mother crystal JE111,
this was not necessarily to be expected for T1, which was grown independently
under somewhat different conditions (see above) and it is evident from panel
(a) that overall the $\kappa(T)$ curve of T1 is lying above those measured on
the samples C1, C2, and C3. On the other hand, the magnetization of C3 is the
only one which essentially reaches $7\mu_{\mathrm{B}}$. This can be traced
back to the fact this sample was vacuum annealed for 1 h at 700∘C. Resistivity
measurements before and after annealing did not show a measurable difference,
whereas the enhanced magnetization indicates a significant variation of Eu
valence towards almost pure Eu2+. Nevertheless, the thermal conductivity of
this particular C3 sample remains as low and glassy as those of all the ETO
samples. Thus, we conclude that the weak Eu2+/3+ off-stoichiometry is not the
main mechanism that induces the unusual glass-like phonon heat transport in
the ETO crystals.

Figure S3: Temperature profiles of resistivity in two EuTiO3 crystals (samples
A1 and A2). Inset displays ln $\rho$ vs. T-1. Dotted line is calculated using
D = 101 meV.

We also confirmed the activated behavior of the resistivity in several samples
measured in three different laboratories. Figure S3 indicates temperature
profiles of resistivity, $\rho$ of two EuTiO3 crystals prepared by the
floating zone method [23]. Tiny anomalies can be discerned at near 260 K,
which is the transition between cubic and tetragonal structure, consistent
with the discussion in Sec. II. A of the main paper. The inset shows ln$\rho$
vs. T-1 of these crystals. The estimated value of the activation gap, $\Delta$
is comparable to that of samples T1 and T2 in Figure 1(b).

##  S5 The Hall coefficient

Figure S4: The Hall carrier density as a function of the inverse of a
temperature in an Arrhenius plot.

As shown in Fig.S4, we found that the Hall carrier density displays an
Arrhenius behavior from 180 K down to 70 K. A deviation starts below the
latter temperature, when the resistance was in the range of $10^{8}$ ohms
rendering the data unreliable. The corresponding carrier density was
$10^{10}$cm -3. At very low temperature, the saturated Hall number corresponds
to a carrier density of $\approx 10^{9}$cm -3.

##  S6 Hysteresis near the structural transition

The structural transition in ETO is believed to be similar to the structural
transition in STO an antiferrodistortive transition in which adjacent
octahedra rotate in opposite orientations [17]. In the case of STO, this
transition is known as a rare case of second order structural phase
transition. However, in our study of resistivity, we found that the anomaly at
$\approx$ 260 K presents a hysteresis, which was found to be reproducible. The
data is shown in Fig S5. This may point to the first-order nature of this
structural transition in ETO.

Figure S5: a) Temperature dependence of resistivity near the structural
transition for upward and downward sweeps either with a continuous ramp or by
steps. b) The temperature derivative of resistivity for an upward and a
downward sweeps with discrete steps.