# The $G$-index, $G$-coindex and $G$-weight of a real moment-angle complex

Akatsuki Kizu  Department of Mathematics, Kyoto University, Kyoto, 606-8502,
Japan<EMAIL_ADDRESS>
###### Abstract.

A real moment-angle complex is a CW complex defined by the combinatoric
information of a simplicial complex, and has a standard action of $2$-torus.
For each subtorus $G$ of $2$-torus, we describe invariants corcerning this
action, the $G$-index, $G$-coindex and $G$-weight, combinatorially.

##  1\. Introduction and the main theorems

Let $K$ be a simplicial complex on the vertex set $[m]=\\{1,\dots,m\\}$. A
subset $I\subset[m]$ is called a face of $K$ if $I\in K$. Otherwise, it is
called a non-face of K. We say that a subset $I\subset[m]$ is a minimal non-
face of $K$ if it is a non-face of $K$ and every proper subset of $I$ is a
face of $K$. The dimension of $I$ is defined as 1 less than its order. If one
point subset $\\{i\\}\subset[m]$ is a non-face of $K$, it is called a ghost
vertex of $K$. In this paper, we assume that every simplicial complex does not
have ghost vertices.

A real moment-angle complex associated with $K$ is defined as a subcomplex of
$(D^{1})^{m}$

$\mathbb{R}Z_{K}:=\bigcup_{\sigma\in K}(D^{1},S^{0})^{\sigma},$

where
$(D^{1},S^{0})^{\sigma}:=\\{(x_{1},\dots,x_{m})\in(D^{1})^{m}\mid\text{$x_{i}\in
S^{0}$ if $i\notin\sigma$}\\}$. As seen in the paper of Davis and
Januszukiewicz [6], $\mathbb{R}Z_{K}$ is one of the key objects and has been
deeply studied in toric topology. Since $\mathbb{R}Z_{K}$ is constructed by
using the combinatoric information of $K$, it is natural to expect that
homotopy invariants of $\mathbb{R}Z_{K}$ could be described in terms of the
combinatorics of $K$. For example, the equivariant cohomology is studied in
the paper of Davis and Januszukiewicz [6]. The cohomology ring is studied in
[2] in the case that $\mathbb{R}Z_{K}$ is a topological manifold. The homotopy
type is studied in [11] when $K$ is flag, that is, every minimal non-face of
$K$ consists of two vertices. In [14], it is shown that $\mathbb{R}Z_{K}$
plays an important role to understand the homotopy type of a polyhedral
product, a generalized object of a real moment-angle complex, by using a
structure which is called fat-wedge filtration.

By construction, there is a standard action of $2$-torus $(\mathbb{Z}/2)^{m}$
on $\mathbb{R}Z_{K}$. Specifically, for
$g=(g_{1},\dots,g_{m})\in(\mathbb{Z}/2)^{m}$ and
$x=(x_{1},\dots,x_{m})\in\mathbb{R}Z_{K}$, we set

$g\cdot x:=\left(g_{1}x_{1},\dots,g_{m}x_{m}\right),$

where we regard $\mathbb{Z}/2=\\{1,-1\\}$ and $D^{1}=[-1,1]$. Since $K$ has no
ghost vertex, it can be easily seen that this action is always not free.
However, some subtori of $(\mathbb{Z}/2)^{m}$ may act on $\mathbb{R}Z_{K}$
freely. The quotient space of $\mathbb{R}Z_{K}$ by the free action of a
subtorus of $(\mathbb{Z}/2)^{m}$, called partial quotient in [1], has been
studied in the many context [8, 9, 12]. In particular, small cover which is
introduced in [6] is a classical example of partial quotient and has been
deeply studied [3, 15].

Let $G$ be a non-trivial discrete group and $X$ be a free $G$-space. We define
the topological invariant $G$-index as

$\mathrm{ind}_{G}(X):=\min\\{n\geq 0\mid\text{there is a $G$-map $X\to
E_{n}G$}\\},$

where $E_{n}G$ is the $(n+1)$-fold join of $G$ with a diagonal $G$-action. We
also consider the similar notion $G$-coindex. Let $X$ be a $G$-space. Define

$\mathrm{coind}_{G}(X):=\min\\{n\geq 0\mid\text{there is a $G$-map $E_{n}G\to
X$}\\}.$

These invariants were introduced in the paper of Conner and Floyd [4]. Note
that they call $G$-index what we call $G$-coindex in [4], and vice versa. As
will be seen later, there is an inequality
$\mathrm{coind}_{G}(X)\leq\mathrm{ind}_{G}(X)$, which is a generalization of
the Borsuk-Ulam theorem. The $G$-index has a close relationship with
Lusternik-Shnirelmann category; see [16, Section 5.3]. The $G$-index and
$G$-coindex are so hard to compute in general. We also introduce the
cohomological invariant $G$-weight. Recall that there is a fibration
$EG\times_{G}X\to BG$ for a $G$-space $X$. Define

$\mathrm{wgt}_{G}(X):=\max\\{n\geq 0\mid\text{$H^{k}(BG)\to
H^{k}(EG\times_{G}X)$ is injective for all $k\leq n$ }\\}.$

This invariant is essentially the same as the ideal-valued index in the paper
of Fadell and Husseini [7]. As will be seen later, there are inequalities
$\mathrm{coind}_{G}(X)\leq\mathrm{wgt}_{G}(X)\leq\mathrm{ind}_{G}(X)$ when $G$
is a $p$-torus. Thus the $G$-weight is helpful to evaluate the $G$-index and
$G$-coindex. However, the $G$-weight is still hard to compute, not as hard as
the $G$-index and $G$-coindex. We can see that the $G$-weight is used in [13,
18, 19]. We will see some properties of these invariants in Section 2.

Our goal in this paper is to describe the $G$-index, $G$-coindex and
$G$-weight of $\mathbb{R}Z_{K}$ in terms of the combinatorics of $K$ for a
given subgroup $G<(\mathbb{Z}/2)^{m}$. Let us introduce the combinatorial
notion $\delta$-number $\delta(K)$ of a simplicial complex $K$ defined as the
minimum number of the dimension of non-faces of $K$, that is,

$\delta(K):=\min\left\\{|I|-1\mid\text{$I$ is a non-face of $K$}\right\\}.$

If $K$ is a full simplex, then we set $\delta(K)=\infty$.

We now can state the main theorems. For a non-empty subset $I\subset[m]$, we
define the full subcomplex of $K$ over $I$ as

$K_{I}:=\\{\sigma\subset I\mid\sigma\in K\\}.$

For each $g=(g_{1},\dots,g_{m})\in G$, let us set the support of $g$ as the
subset of $[m]$

$\mathrm{supp}(g):=\\{i\in[m]~{}|~{}g_{i}=-1\\},$

and set $\mathrm{supp}(G):=\bigcup_{g\in G}\mathrm{supp}(g)$.

######  Theorem 1.1.

Let $G<(\mathbb{Z}/2)^{m}$ be a subgroup of rank one.

  1. (i)

If the action of $G$ on $\mathbb{R}Z_{K}$ is free, then there is an equality

$\mathrm{ind}_{G}(\mathbb{R}Z_{K})=\delta(K_{\mathrm{supp}(G)}).$

  2. (ii)

There are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=\delta(K_{\mathrm{supp}(G)}),$

where the action of $G$ on $\mathbb{R}Z_{K}$ is not necessarily free.

######  Theorem 1.2.

Let $G<(\mathbb{Z}/2)^{m}$ be a non-trivial subgroup.

  1. (i)

If the action of $G$ on $\mathbb{R}Z_{K}$ is free, then there are inequalities

$\max_{g\in
G\setminus\\{1\\}}\delta(K_{\mathrm{supp}(g)})\leq\mathrm{ind}_{G}(\mathbb{R}Z_{K})\leq\dim
K+1.$

  2. (ii)

There are inequalities

$\delta(K_{\mathrm{supp}(G)})\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{K})\leq\min_{g\in
G\setminus\\{1\\}}\delta(K_{\mathrm{supp}(g)}),$

where the action of $G$ on $\mathbb{R}Z_{K}$ is not necessarily free.

These theorems say that the $G$-index, $G$-coindex and $G$-weight for a real
moment-angle complex can be evaluated by using a $\delta$-number and
especially be computed completely when the rank of $G$ is one. We will give
proofs of the main theorems in Section 3.

In Section 4, we study the upper bound of the $G$-index. As shown in Section
2, if $X$ is a free $G$-complex, then its $G$-index is smaller than or equal
to its dimension, that is, $\mathrm{ind}_{G}(X)\leq\dim X$. Thus if $G$ acts
on $\mathbb{R}Z_{K}$ freely, then there is an inequality

$\mathrm{ind}_{G}(\mathbb{R}Z_{K})\leq\dim\mathbb{R}Z_{K}=\dim K+1.$

We find some sufficient condition for making the upper bound smaller. A pair
of simplices $(\sigma,\tau)$ of $K$ is said to be an elementary collapsing
pair if $\tau\subset\sigma$ and $\sigma$ is the only simplex including $\tau$
properly. When $(\sigma,\tau)$ is an elementary collapsing pair of $K$,
removing $\sigma$ and $\tau$ from $K$ is called an elementary collapse,
denoted by $K\searrow_{(\sigma,\tau)}K^{\prime}$, where
$K^{\prime}:=K\setminus\\{\sigma,\tau\\}$. A sequence of elementary collapses
is simply said to be a collapse.

######  Proposition 1.3.

Let $K$ be a simplicial complex and suppose that $G<(\mathbb{Z}/2)^{m}$ acts
on $\mathbb{R}Z_{K}$ freely. If $K$ collapses to some simplicial complex $L$
such that $\dim L<\dim K$, then there is an inequality

$\mathrm{ind}_{G}(\mathbb{R}Z_{K})\leq\dim K.$

Proposition 1.3 means that we can make the upper bound smaller by one under
the condition above. We will see that this modification of the upper bound is
best possible by giving some example.

In Section 5, we also study the condition that the $G$-coindex and $G$-weight
of a real moment-angle complex equals to the lower bound.

######  Corollary 1.4.

Let $G<(\mathbb{Z}/2)^{m}$ be a non-trivial subgroup. If
$\delta(K_{\mathrm{supp}(G)})=1$, then there are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=1.$

######  Corollary 1.5.

Let $G<(\mathbb{Z}/2)^{m}$ be a non-trivial subgroup. If each minimal non-face
of $K_{\mathrm{supp}(G)}$ has the same order and $\mathrm{supp}(g_{0})\notin
K$ for some $g_{0}\in G$, then there are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=\delta(K_{\mathrm{supp}(G)}).$

##  2\. Properties of the $G$-index, $G$-coindex and $G$-weight

In this section, we show some properties of the $G$-index, $G$-coindex and
$G$-weight. Throughout this section, let $G$ be a $p$-torus
$(\mathbb{Z}/p)^{m}$, where $p$ is a prime number. We first observe a
cohomological property of the canonical map $i_{n}\colon B_{n}G\to BG$.

######  Lemma 2.1.

The map $i_{n}^{*}\colon H^{*}(BG;\mathbb{Z}/p)\to H^{*}(B_{n}G;\mathbb{Z}/p)$
is injective for $*\leq n$ and not injective for $*=n+1$.

###### proof.

There is a homotopy fibration $E_{n}G\to B_{n}G\xrightarrow{i_{n}}BG$. Since
$E_{n}G$ is homotopy equivalent to a wedge of $n$-spheres, it is
$(n-1)$-connected. Then the Serre exact sequence for the above homotopy
fibration implies the first statement. On the other hand, it is well known
that

$H^{*}(B\mathbb{Z}/p;\mathbb{Z}/p)=\left\\{\begin{array}[]{ll}\mathbb{Z}/p[x]&(p=2)\\\\[3.0pt]
\Lambda_{\mathbb{Z}/p}(x)\otimes\mathbb{Z}/p[y]&(p\neq 2),\end{array}\right.$

where $\deg x=1$ and $\deg y=2$, implying that $H^{n+1}(BG;\mathbb{Z}/p)\neq
0$. However, since $B_{n}G$ is an $n$-dimensional complex,
$H^{n+1}(B_{n}G;\mathbb{Z}/p)=0$. Thus the second statement is proved. ∎

Let a map $p_{1}\colon EG\times_{G}E_{n}G\to BG$ be the first projection and a
map $p_{2}\colon EG\times_{G}E_{n}G\to B_{n}G$ be the second projection.

######  Lemma 2.2.

The map $p_{2}\colon EG\times_{G}E_{n}G\to B_{n}G$ is a homotopy equivalence
and a diagram

$\textstyle{EG\times_{G}E_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\textstyle{B_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG}$

is homotopy commutative.

###### proof.

The map $p_{2}\colon EG\times_{G}E_{n}G\to B_{n}G$ is a homotopy equivalence
since its fiber $EG$ is contractible. The homotopy commutative diagram in the
statement is obtained by taking the quotient of a diagram of $G$-maps

$\textstyle{EG\times
E_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{EG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{EG}$

which commutes up to $G$-homotopy by the fact that any two $G$-maps from some
$G$-space to $EG$ are $G$-homotopic as in [10, Lemma 5.1]. Thus the proof is
completed. ∎

Let us show fundamental properties of the $G$-index, $G$-coindex and
$G$-weight.

######  Proposition 2.3.

Let $X$ be a $G$-space.

  1. (i)

There is an inequality

$\mathrm{coind}_{G}(X)\leq\mathrm{wgt}_{G}(X).$

  2. (ii)

If $G$ acts on $X$ freely, then there is an inequality

$\mathrm{wgt}_{G}(X)\leq\mathrm{ind}_{G}(X).$

###### proof.

(i) Let $n:=\mathrm{coind}_{G}(X)$. Then there is a $G$-map $\phi\colon
E_{n}G\to X$. By Lemma 2.2, we get a homotopy commutative diagram

$\textstyle{B_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}}$$\textstyle{EG\times_{G}E_{n}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{id}\times_{G}\phi}$$\scriptstyle{p_{1}}$$\textstyle{EG\times_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG}$

which induces a commutative diagram

$\textstyle{H^{k}(B_{n}G;\mathbb{Z}/p)}$$\textstyle{H^{k}(EG\times_{G}X;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{k}(BG;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}^{*}}$$\textstyle{H^{k}(BG;\mathbb{Z}/p).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By the first statement of Lemma 2.1, the map $H^{k}(BG;\mathbb{Z}/p)\to
H^{k}(EG\times_{G}X;\mathbb{Z}/p)$ must be injective for $k\leq n$. Thus we
obtain the inequality $\mathrm{wgt}_{G}(X)\geq n=\mathrm{coind}_{G}(X)$.

(ii) Let $m:=\mathrm{ind}_{G}(X)$. Then there is a $G$-map $\psi\colon X\to
E_{m}G$. By Lemma 2.2, we get a homotopy commutative diagram

$\textstyle{EG\times_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times_{G}\psi}$$\textstyle{EG\times_{G}E_{m}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{\simeq}$$\scriptstyle{p_{1}}$$\textstyle{B_{m}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG}$

which induces a commutative diagram

$\textstyle{H^{k}(B_{m}G;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{k}(EG\times_{G}X;\mathbb{Z}/p)}$$\textstyle{H^{k}(BG;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n}^{*}}$$\textstyle{H^{k}(BG;\mathbb{Z}/p).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By the second statement of Lemma 2.1, the map $H^{k}(BG;\mathbb{Z}/p)\to
H^{k}(EG\times_{G}X;\mathbb{Z}/p)$ cannot be injective for $k=m+1$. Thus we
also obtain the inequality $\mathrm{wgt}_{G}(X)\leq m=\mathrm{ind}_{G}(X)$. ∎

######  Proposition 2.4.

Let $X$ and $Y$ be $G$-spaces, and suppose that there is a $G$-map $f\colon
X\to Y$.

  1. (i)

There are inequalities

$\mathrm{coind}_{G}(X)\leq\mathrm{coind}_{G}(Y)\quad\text{and}\quad\mathrm{wgt}_{G}(X)\leq\mathrm{wgt}_{G}(Y).$

  2. (ii)

If $G$ acts on $X$ and $Y$ freely, then there is an inequality

$\mathrm{ind}_{G}(X)\leq\mathrm{ind}_{G}(Y).$

###### proof.

(i) Let $n:=\mathrm{coind}_{G}(X)$, and take a $G$-map $\phi\colon E_{n}G\to
X$. Then the composition $f\circ\phi\colon E_{n}G\to X\to Y$ is also a
$G$-map. Thus we get the inequality $\mathrm{coind}_{G}(Y)\geq
n=\mathrm{coind}_{G}(X)$. Let $l:=\mathrm{wgt}_{G}(X)$. Since the map $f\colon
X\to Y$ is a $G$-map, it induces a commutative diagram

$\textstyle{EG\times_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\times_{G}f}$$\textstyle{EG\times_{G}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG.}$

Then we get a commutative diagram

$\textstyle{H^{k}(EG\times_{G}Y;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id}\times_{G}f)^{*}}$$\textstyle{H^{k}(EG\times_{G}X;\mathbb{Z}/p)}$$\textstyle{H^{k}(BG;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{k}(BG;\mathbb{Z}/p).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Since the map $H^{k}(BG;\mathbb{Z}/p)\to H^{k}(EG\times_{G}X;\mathbb{Z}/p)$ is
injective for $k\leq l$, the map $H^{k}(BG;\mathbb{Z}/p)\to
H^{k}(EG\times_{G}Y;\mathbb{Z}/p)$ must also be injective for $k\leq l$. Thus
we get the inequality $\mathrm{wgt}_{G}(Y)\geq l=\mathrm{wgt}_{G}(X)$.

(ii) Let $m:=\mathrm{ind}_{G}(Y)$, and take a $G$-map $\psi\colon Y\to
E_{m}G$. Then the composition $\psi\circ f\colon X\to Y\to E_{m}G$ is also a
$G$-map. Thus we get the inequality $\mathrm{ind}_{G}(X)\leq
m=\mathrm{ind}_{G}(Y)$. ∎

######  Proposition 2.5.

Let $X$ be a $G$-space, and $K<G$ be a non-trivial subgroup.

  1. (i)

There are inequalities

$\mathrm{coind}_{G}(X)\leq\mathrm{coind}_{K}(X)\quad\text{and}\quad\mathrm{wgt}_{G}(X)\leq\mathrm{wgt}_{K}(X).$

  2. (ii)

If $G$ acts on $X$ and $Y$ freely, then there is an inequality

$\mathrm{ind}_{K}(X)\leq\mathrm{ind}_{G}(X).$

###### proof.

(i) Let $n:=\mathrm{coind}_{G}(X)$, and take a $G$-map $\phi\colon E_{n}G\to
X$. Then the composition $\phi\circ E_{n}i\colon E_{n}K\to E_{n}G\to X$ is a
$K$-map where the map $E_{n}i\colon E_{n}K\to E_{n}G$ is a $K$-map induced by
the inclusion homomorphism $i\colon K\to G$. Thus we get the inequality
$\mathrm{coind}_{K}(X)\geq n=\mathrm{coind}_{G}(X)$. Let
$l:=\mathrm{wgt}_{G}(X)$. Note that since any subgroup of
$G=(\mathbb{Z}/p)^{m}$ is a direct summand, there is a retraction $r\colon
G\to K$ which induces a commutative diagram

$\textstyle{EG\times_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\times_{G}\mathrm{id}}$$\textstyle{EK\times_{K}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Br}$$\textstyle{BK.}$

Then we get a commutative diagram

$\textstyle{H^{k}(EK\times_{K}X;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(r\times_{G}\mathrm{id})^{*}}$$\textstyle{H^{k}(EG\times_{G}X;\mathbb{Z}/p)}$$\textstyle{H^{k}(BH;\mathbb{Z}/p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(Br)^{*}}$$\textstyle{H^{k}(BG;\mathbb{Z}/p).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note that the map $(Br)^{*}\colon H^{k}(BK;\mathbb{Z}/p)\to
H^{k}(BG;\mathbb{Z}/p)$ is injective since the homomorphism $r\colon G\to K$
is a retraction. Since the map $H^{k}(BG;\mathbb{Z}/p)\to
H^{k}(EG\times_{G}X;\mathbb{Z}/p)$ is injective for $k\leq l$, the map
$H^{k}(BH;\mathbb{Z}/p)\to H^{k}(EK\times_{K}X;\mathbb{Z}/p)$ must be
injective for $k\leq l$. Thus we get the inequality $\mathrm{wgt}_{K}(X)\geq
l=\mathrm{wgt}_{G}(X)$.

(ii) Let $m:=\mathrm{ind}_{G}(X)$, and take a $G$-map $\psi\colon X\to
E_{m}G$. Then the composition $E_{m}r\circ\psi\colon X\to E_{m}G\to E_{m}K$ is
a $K$-map. Thus we get the inequality $\mathrm{ind}_{K}(X)\leq
m=\mathrm{ind}_{G}(X)$.

∎

######  Proposition 2.6.

Let $X$ be a $G$-space.

  1. (i)

There are equalities

$\mathrm{coind}_{G}(E_{n}G)=\mathrm{ind}_{G}(E_{n}G)=n.$

  2. (ii)

If $X$ is $(n-1)$-connected, then there is an inequality

$\mathrm{coind}_{G}(X)\geq n.$

  3. (iii)

If $X$ is a free $G$-complex, then there is an inequality

$\mathrm{ind}_{G}(X)\leq\dim X.$

###### proof.

This proposition follows from the proof of [16, Proposition 6.2.4]. ∎

##  3\. Proof of the main theorems

In this section, we give proofs of the main theorems. We first consider the
case that $K$ is a full simplex $\Delta^{m-1}$.

######  Lemma 3.1.

Let $G<(\mathbb{Z}/2)^{m}$ be a non-trivial subgroup. Then there are
equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{\Delta^{m-1}})=\mathrm{wgt}_{G}(\mathbb{R}Z_{\Delta^{m-1}})=\infty.$

###### proof.

Since $\mathbb{R}Z_{\Delta^{m-1}}=(D^{1})^{m}$ is contractible, there are
inequalities

$n\leq\mathrm{coind}_{G}(\mathbb{R}Z_{\Delta^{m-1}})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{\Delta^{m-1}})$

for any $n\geq 0$ by Proposition 2.3 and Proposition 2.6. ∎

######  Lemma 3.2.

Let $K$ be a simplicial complex on the vertex set $[m]$. If $K$ is
$q$-neighborly, then $\mathbb{R}Z_{K}$ is $q$-connected.

###### proof.

Since $K$ is $q$-neighborly, the $(q+1)$-skeleton of $\mathbb{R}Z_{K}$ equals
to the $(q+1)$-skeleton of $\mathbb{R}Z_{\Delta^{m-1}}=(D^{1})^{m}$ which is
homotopy equivalent to a wedge of $(q+1)$-spheres. Thus by the cellular
approximation theorem, we get

$\pi_{i}\left(\mathbb{R}Z_{K}\right)=\pi_{i}\left((D^{1})^{m}\right)=0\quad\text{for
$i\leq q$},$

completing the proof. ∎

######  Corollary 3.3.

Let $K$ be a simplicial complex on the vertex set $[m]$ and
$G<(\mathbb{Z}/2)^{m}$ be a non-trivial subgroup. Then there is an inequality

$\delta(K)\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K}).$

###### proof.

Let $q:=\delta(K)$. Since $K$ is $(q-1)$-neighborly, $\mathbb{R}Z_{K}$ is
$(q-1)$-connected by Lemma 3.2. Thus the proof is completed by Proposition
2.6. ∎

######  Lemma 3.4.

Let $K$ be a simplicial complex on the vertex set $[m]$ Suppose that $K$ is
not a full simplex, and $G=\mathbb{Z}/2$ acts on $\mathbb{R}Z_{K}$ diagonally.
Then there are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=\mathrm{ind}_{G}(\mathbb{R}Z_{K})=\delta(K).$

###### proof.

We first note that the diagonal action of $G$ on $\mathbb{R}Z_{K}$ is free
since $K$ is not a full simplex. It suffices to show that there are
inequalities

$\delta(K)\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{K})\leq\mathrm{ind}_{G}(\mathbb{R}Z_{K})\leq\delta(K).$

The first inequality follows from Corollary 3.3. The second and the third
inequalities hold by Proposition 2.3. If we set $q:=\delta(K)$, then there is
a minimal non-face $I=\\{i_{1},\dots,i_{q+1}\\}\subset[m]$ of $K$ which yields
a $G$-map

$\mathbb{R}Z_{K}\to\mathbb{R}Z_{K_{I}}=\mathbb{R}Z_{\partial\Delta^{q}}=S^{q}\quad(x_{1},\dots,x_{m})\mapsto(x_{i_{1}},\dots,x_{i_{q+1}}),$

implying the last inequality by Propositions 2.4 and 2.6. ∎

###### proof of Theorems 1.1 and 1.2.

Let us consider the inclusion
$\mathbb{R}Z_{K_{\mathrm{supp}(G)}}\to\mathbb{R}Z_{K}$ and the projection
$\mathbb{R}Z_{K}\to\mathbb{R}Z_{K_{\mathrm{supp}(G)}}$. Since these maps are
$G$-maps, the $G$-index, $G$-coindex and $G$-weight of $\mathbb{R}Z_{K}$ are
the same as those of $\mathbb{R}Z_{K_{\mathrm{supp}(G)}}$ respectively by
Proposition 2.4. Thus we may assume that $K_{\mathrm{supp}(G)}=K$, or
equivalently, $\mathrm{supp}(G)=[m]$.

First, we consider the case that $\mathrm{rank}\,G=1$. Then the action of $G$
on $\mathbb{R}Z_{K}$ is diagonal since $\mathrm{supp}(G)=[m]$. If the diagonal
action of $G$ is not free, then $K$ must be a full simplex $\Delta^{m-1}$. By
Lemma 3.1, there are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=\infty=\delta(K).$

If the action of $G$ on $\mathbb{R}Z_{K}$ is free, then $K$ is not a full
simplex. By Lemma 3.4, there are equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=\mathrm{ind}_{G}(\mathbb{R}Z_{K})=\delta(K).$

Thus the proof of the $\mathrm{rank}\,G=1$ case is completed.

Next, let $G<(\mathbb{Z}/2)^{m}$ be a subgroup whose rank is not necessarily
one. For each non-trivial element $g\in G$, there are inequalities

$\delta(K)\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{\langle
g\rangle}(\mathbb{R}Z_{K})=\delta(K_{\mathrm{supp}(g)})$

by Proposition 2.5 and Corollary 3.3. If the action of $G$ on
$\mathbb{R}Z_{K}$ is free, then there are an inequalities

$\delta(K_{\mathrm{supp}(g)})=\mathrm{ind}_{\langle
g\rangle}(\mathbb{R}Z_{K})\leq\mathrm{ind}_{G}(\mathbb{R}Z_{K})\leq\dim\mathbb{R}Z_{K}=\dim
K+1.$

by Proposition 2.5 and Lemma 2.6. The proof is completed. ∎

##  4\. Upper bound of the $G$-index

In this section, we prove Proposition 1.3 by constructing a
$(\mathbb{Z}/2)^{m}$-subcomplex $X$ of $\mathbb{R}Z_{K}$, which is a
$(\mathbb{Z}/2)^{m}$-equivariant deformation retract. If $\dim X\leq\dim K$,
then we have an inequality

$\text{ind}_{G}(\mathbb{R}Z_{K})\leq\text{ind}_{G}(X)\leq\dim X\leq\dim K$

by Proposition 2.6, implying Proposition 1.3.

Let $I=[0,1]$ be a unit interval. Since a pair
$(I^{q},I^{q-1}\times\\{0\\}\cup\partial I^{q-1}\times I)$ is an NDR pair, we
can choose a deformation retraction $\rho\colon I^{q}\to
I^{q-1}\times\\{0\\}\cup\partial I^{q-1}\times I$ for each $q=1,\dots,m$. Let
$\rho_{j}(x)$ denote the $j$-th entry of $\rho(x)$. For each $j=1,\dots,q$, we
define a map $r_{j}\colon(D^{1})^{q}\to D^{1}$ as

$r_{j}(x_{1},\dots,x_{q})=\begin{cases}\rho_{j}(|x_{1}|,\dots,|x_{q}|)&(x_{j}\geq
0)\\\ -\rho_{j}(|x_{1}|,\dots,|x_{q}|)&(x_{j}\leq 0),\end{cases}$

which is well-defined since $\rho_{j}(|x_{1}|,\dots,|x_{q}|)=|x_{j}|=0$ when
$|x_{j}|=0$. Let us define a map $r\colon(D^{1})^{q}\to(D^{1})^{q}$ as

$r(x):=(r_{1}(x),\dots,r_{q}(x)),$

which is $(\mathbb{Z}/2)^{m}$-equivariant by construction.

######  Lemma 4.1.

The image of the map $r$ is a space

$Z:=\left(\bigcup_{j=1}^{q}(D^{1})^{j-1}\times\\{0\\}\times(D^{1})^{q-j}\right)\cup\left(\bigcup_{j=1}^{q-1}(D^{1})^{j-1}\times
S^{0}\times(D^{1})^{q-j}\right),$

and a map $(D^{1})^{q}\to Z~{}(x\mapsto r(x))$ is a deformation retraction.

###### proof.

Let $x=(x_{1},\dots,x_{q})$ be any element of $(D^{1})^{q}$. We first note
that the image of the map $\rho$ is

$I^{q-1}\times\\{0\\}\cup\partial I^{q-1}\times
I=\left(\bigcup_{j=1}^{q}I^{j-1}\times\\{0\\}\times
I^{q-j}\right)\cup\left(\bigcup_{j=1}^{q-1}I^{j-1}\times\\{1\\}\times
I^{q-j}\right).$

If $\rho(|x_{1}|,\dots,|x_{q}|)\in\bigcup_{j=1}^{q}I^{j-1}\times\\{0\\}\times
I^{q-j}$, then $r_{j}(x)=0$ for some $j=1,\dots,q$ since
$\rho_{j}(|x_{1}|,\dots,|x_{q}|)=0$ for some $j=1,\dots,q$, implying that

$r(x)\in\bigcup_{j=1}^{q}(D^{1})^{j-1}\times\\{0\\}\times(D^{1})^{q-j}.$

If
$\rho(|x_{1}|,\dots,|x_{q}|)\in\bigcup_{j=1}^{q-1}I^{j-1}\times\\{1\\}\times
I^{q-j}$, then $r_{j}(x)=1$ for some $j=1,\dots,q-1$ since
$\rho_{j}(|x_{1}|,\dots,|x_{q}|)=1$ for some $j=1,\dots,q-1$, implying that

$r(x)\in\bigcup_{j=1}^{q-1}(D^{1})^{j-1}\times S^{0}\times(D^{1})^{q-j}.$

Thus it is proved that $r(x)\in Z$. Conversely, if $x=(x_{1},\dots,x_{q})\in
Z$, then $(|x_{1}|,\dots,|x_{q}|)\in I^{q-1}\times\\{0\\}\cup\partial
I^{q-1}\times I$, implying that
$\rho(|x_{1}|,\dots,|x_{q}|)=(|x_{1}|,\dots,|x_{q}|)$ since the map
$\rho\colon I^{q}\to I^{q-1}\times\\{0\\}\cup\partial I^{q-1}\times I$ is a
retraction. Thus for each $j=1,\dots,q$,

$r_{j}(x)=\begin{cases}\rho_{j}(|x_{1}|,\dots,|x_{q}|)=|x_{j}|=x_{j}&(x_{j}\geq
0)\\\ -\rho_{j}(|x_{1}|,\dots,|x_{q}|)=-|x_{j}|=x_{j}&(x_{j}\leq
0),\end{cases}$

so that $r(x)=x$. Therefore the image of the map $r$ is $Z$, and the map $r$
is identical on $Z$. Let a map $H\colon I^{q}\times I\to I^{q}$ be a homotopy
for the deformation retraction $\rho$, and denote the $j$-th entry of $H(x,t)$
by $H_{j}(x,t)$. Then we define a homotopy $\tilde{H}\colon(D^{1})^{q}\times
I\to(D^{1})^{q}$ as

$\tilde{H}(x,t):=(\tilde{H}_{1}(x,t),\dots,\tilde{H}_{q}(x,t)),$

where the map $\tilde{H}_{j}\colon(D^{1})^{q}\times I\to D^{1}$ is defined as

$\tilde{H}_{j}(x,t)=\begin{cases}H_{j}(|x_{1}|,\dots,|x_{q}|,t)&(x_{j}\geq
0)\\\ -H_{j}(|x_{1}|,\dots,|x_{q}|,t)&(x_{j}\leq 0)\end{cases}$

for each $j=1,\dots,q$. By construction, the homotopy $\tilde{H}$ implies that
the map $(D^{1})^{q}\to Z~{}(x\mapsto r(x))$ is a deformation retraction. ∎

Let us also denote the map $(D^{1})^{q}\to Z$ in Lemma 4.1 by $r$.

######  Lemma 4.2.

Let $K$ be a simplicial complex and $K\searrow_{(\sigma,\tau)}K^{\prime}$ be
an elementary collapse. Then there is a $(\mathbb{Z}/2)^{m}$-equivariant
deformation retraction

$\tilde{r}\colon\mathbb{R}Z_{K}\to Y\cup\mathbb{R}Z_{K^{\prime}}$

where $Y$ is a $(\mathbb{Z}/2)^{m}$-subcomplex of $\mathbb{R}Z_{K}$ with $\dim
Y=\dim\sigma$.

###### proof.

We may assume that $\sigma=\\{1,\dots,q\\}$ and $\tau=\\{1,\dots,q-1\\}$ and
consider a map $r\times\mathrm{id}\colon(D^{1})^{q}\times(S^{0})^{m-q}\to
Z\times(S^{0})^{m-q}$. The target space $Z\times(S^{0})^{m-q}$ equals to

$\displaystyle\left(\bigcup_{j=1}^{q}(D^{1})^{j-1}\times\\{0\\}\times(D^{1})^{q-j}\times(S^{0})^{m-q}\right)\cup\left(\bigcup_{j=1}^{q-1}(D^{1})^{j-1}\times
S^{0}\times(D^{1})^{q-j}\times(S^{0})^{m-q}\right)$ $\displaystyle=$
$\displaystyle\left(\bigcup_{j=1}^{q}(D^{1})^{j-1}\times\\{0\\}\times(D^{1})^{q-j}\times(S^{0})^{m-q}\right)\cup\bigcup_{{\sigma}^{\prime}<\sigma,~{}{\sigma}^{\prime}\neq\tau}(D^{1},S^{0})^{{\sigma}^{\prime}}.$

Note that since $K^{\prime}=K\setminus\\{\sigma,\tau\\}$, the space
$(D^{1},S^{0})^{{\sigma}^{\prime}}$ is a subspace of
$\mathbb{R}Z_{K^{\prime}}$ for any face $\sigma^{\prime}<\sigma$ except for
$\tau$. Thus if we set
$Y:=\bigcup_{j=1}^{q}(D^{1})^{j-1}\times\\{0\\}\times(D^{1})^{q-j}\times(S^{0})^{m-q}$,
then $Y$ is a $(\mathbb{Z}/2)^{m}$-complex of $\mathbb{R}Z_{K}$ with $\dim
Y=q-1=\dim\sigma$, and we get a well-defined map

$(D^{1},S^{0})^{\sigma}\xrightarrow{r\times\mathrm{id}}Y\cup\bigcup_{{\sigma}^{\prime}<\sigma,~{}{\sigma}^{\prime}\neq\tau}(D^{1},S^{0})^{{\sigma}^{\prime}}\xrightarrow{\mathrm{inclusion}}Y\cup\mathbb{R}Z_{K^{\prime}},$

which is identical on
$\mathbb{R}Z_{K^{\prime}}\cap(D^{1},S^{0})^{\sigma}=\bigcup_{{\sigma}^{\prime}<\sigma,~{}{\sigma}^{\prime}\neq\tau}(D^{1},S^{0})^{{\sigma}^{\prime}}\subset
Z$ since the map $r$ is identical on $Z$. Therefore we get a map

$\tilde{r}\colon\mathbb{R}Z_{K}=\mathbb{R}Z_{K^{\prime}}\cup(D^{1},S^{0})^{\sigma}\to
Y\cup\mathbb{R}Z_{K^{\prime}},$

which is a $(\mathbb{Z}/2)^{m}$-equivariant deformation retraction by
construction. Thus the lemma is proved. ∎

###### proof of Proposition 1.3.

By assumption, there is a sequence of elementary collapses
$K=K_{0}\searrow_{(\sigma_{1},\tau_{1})}K_{1}\searrow_{(\sigma_{2},\tau_{2})}\cdots\searrow_{(\sigma_{l},\tau_{l})}K_{l}=L$.
For each $i=1,\dots,l$, by Lemma 4.2, there is a deformation retraction

$\tilde{r}_{i}\colon\mathbb{R}Z_{K_{i-1}}\to Y_{i}\cup\mathbb{R}Z_{K_{i}}$

where $Y_{i}$ is a $(\mathbb{Z}/2)^{m}$-subcomplex of $\mathbb{R}Z_{K_{i-1}}$.
Let us consider a map

$\mathbb{R}Z_{K_{i-1}}\xrightarrow{\tilde{r}_{i}}Y_{i}\cup\mathbb{R}Z_{K_{i}}\xrightarrow{\mathrm{inclusion}}\bigcup_{p=1}^{i-1}Y_{p}\cup
Y_{i}\cup\mathbb{R}Z_{K_{i}}=\bigcup_{p=1}^{i}Y_{p}\cup\mathbb{R}Z_{K_{i}}.$

We show that this composition is identical on a space
$\bigcup_{p=1}^{i-1}Y_{p}\cap\mathbb{R}Z_{K_{i-1}}$. It suffices to show that
the space $\bigcup_{p=1}^{i-1}Y_{p}\cap\mathbb{R}Z_{K_{i-1}}$ is a subset of a
space $Y_{i}\cup\mathbb{R}Z_{K_{i}}$ since the map $\tilde{r}_{i}$ is
identical on $Y_{i}\cup\mathbb{R}Z_{K_{i}}$. Let $x$ be an arbitrary element
of $\bigcup_{p=1}^{i-1}Y_{p}\cap\mathbb{R}Z_{K_{i-1}}$. Then $x\in Y_{p}$ for
some $p=1,\dots,i-1$, which means that the $j$-th entry of $x$ is zero for
some $j\in\sigma_{p}$. Thus if
$x\in\mathbb{R}Z_{K_{i-1}}\setminus\mathbb{R}Z_{K_{i}}$, then it is necessary
that $x\in(D^{1},S^{0})^{\sigma_{i}}$, implying that $x\in Y_{i}$ since
$j\in\sigma_{p}\cap\sigma_{i}$. Therefore the above argument shows that $x$ is
in $Y_{i}\cup\mathbb{R}Z_{K_{i}}$. Thus we get a well-defined map

$\bigcup_{p=1}^{i-1}Y_{p}\cup\mathbb{R}Z_{K_{i-1}}\to\bigcup_{p=1}^{i}Y_{p}\cup\mathbb{R}Z_{K_{i}},$

which is a $(\mathbb{Z}/2)^{m}$-equivariant deformation retraction by
construction. Then the composition of maps

$\mathbb{R}Z_{K}\to
Y_{1}\cup\mathbb{R}Z_{K_{1}}\to\cdots\to\bigcup_{p=1}^{l}Y_{p}\cup\mathbb{R}Z_{K_{l}}$

is a $(\mathbb{Z}/2)^{m}$-equivariant deformation retraction. Thus if we set
$X:=\bigcup_{p=1}^{l}Y_{p}\cup\mathbb{R}Z_{K_{l}}$, then there is an
inequality

$\dim X=\max_{1\leq p\leq l}\left\\{\dim\mathbb{R}Z_{K_{l}},\dim
Y_{p}\right\\}\leq\dim K$

since there are inequalities $\dim K_{l}=\dim L<\dim K$ and $\dim
Y_{p}=\dim\sigma_{p}\leq\dim K$ for all $p=1,\dots,l$. Therefore we get
inequalities

$\text{ind}_{G}(\mathbb{R}Z_{K})\leq\text{ind}_{G}(X)\leq\dim X\leq\dim K$

by Proposition 2.4, completing the proof. ∎

######  Example 4.3.

Let $K$ be a cone of the boundary of an $(m-2)$-simplex $\Delta^{m-2}$ and
$G=\mathbb{Z}/2$ be a group acting on $\mathbb{R}Z_{K}$ diagonally. Then $\dim
K=m-1$ and $K$ can collapse to a point. However, Theorem 1.1 implies that

$\text{ind}_{G}(\mathbb{R}Z_{K})=\delta(K)=m-1=\dim K.$

Thus this example shows that the upper bound in Proposition 1.3 is best
possible.

##  5\. Lower bound of the $G$-coindex and the flag number

In this section, we discuss a lower bound for the $G$-coindex. The goal is to
prove Corollaries 1.4 and 1.5.

######  Lemma 5.1.

Let $g,h$ be non-trivial elements of $G<(\mathbb{Z}/2)^{m}$, and take elements
$i\in\mathrm{supp}(g)$ and $j\in\mathrm{supp}(h)$. If
$j\notin\mathrm{supp}(g)$ and $i\notin\mathrm{supp}(h)$, then
$i,j\in\mathrm{supp}(gh)$.

###### proof.

By assumption, the $i$-th and $j$-th entry of $g$ are $-1$ and $1$
respectively, and the $i$-th and $j$-th entry of $h$ are $1$ and $-1$
respectively. Thus both of the $i$-th and $j$-th entry of $gh$ are $-1$, so
that $i,j\in\mathrm{supp}(gh)$. ∎

###### proof of Corollary 1.4.

Suppose that $\delta(K_{\text{supp}(G)})=1$. Then there are two elements
$i,j\in\text{supp}(G)$ such that $\\{i,j\\}\notin K_{\text{supp}(G)}$. If we
show that $\\{i,j\\}\in\text{supp}(g_{0})$ for some $g_{0}\in
G\setminus\\{1\\}$, then $\\{i,j\\}\notin K_{\text{supp}(g_{0})}$ since
$\\{i,j\\}\notin K_{\text{supp}(G)}$ and $K_{\text{supp}(g_{0})}\subset
K_{\text{supp}(G)}$, that is, $\delta(K_{\text{supp}(g_{0})})=1$. By Theorem
1.2, there are inequalities

$\begin{split}1=\delta(K_{\mathrm{supp}(G)})&\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{K})\\\
&\leq\min_{g\in
G\setminus\\{1\\}}\delta(K_{\mathrm{supp}(g)})\leq\delta(K_{\mathrm{supp}(g_{0})})=1,\end{split}$

so that we obtain equalities
$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}(\mathbb{R}Z_{K})=1$.
Thus it suffices to show that there is some element $g_{0}\in G$ such that
$i,j\in\text{supp}(g_{0})$. Since $\text{supp}(G)=\bigcup_{g\in
G}\text{supp}(g)$, we can choose elements $g_{i},g_{j}\in G\setminus\\{1\\}$
such that $i\in\text{supp}(g_{i}),j\in\text{supp}(g_{j})$. If
$j\in\text{supp}(g_{i})$ or $i\in\text{supp}(g_{j})$, then
$i,j\in\text{supp}(g_{i})$ or $i,j\in\text{supp}(g_{j})$ respectively. If
$j\notin\text{supp}(g_{i})$ and $i\notin\text{supp}(g_{j})$, then
$i,j\in\text{supp}(g_{i}g_{j})$ by Lemma 5.1. Thus the proof is completed. ∎

To prove Corollary 1.5, let us define the combinatorial notion flag number
$\text{flag}(K)$ of a simplicial complex $K$ as the maximum number of the
dimension of minimal non-faces of $K$, that is,

$\mathrm{flag}(K):=\max\\{|I|-1\mid\text{$I$ is a minimal non-face of $K$}\\}$

If $K$ is a full simplex, then we set $\mathrm{flag}(K)=\infty$. It is easy to
see that a simplicial complex $K$ is flag if and only if $K$ is a full simplex
or its flag number equals to 1.

######  Lemma 5.2.

Let $K$ be a simplicial complex on the vertex set $[m]$ and $I\subset[m]$ be a
non-face of $K$. Then there are inequalities

$\delta(K)\leq\delta(K_{I})\quad\text{and}\quad\mathrm{flag}(K_{I})\leq\mathrm{flag}(K).$

###### proof.

We first note that $K_{I}$ is not a full simplex by assumption. Let
$q:=\delta(K_{I})$. Then there is a non-face $J$ of $K_{I}$ with $|J|=q+1$. By
definition of full subcomplex, $J$ is also a non-face of $K$. Thus there is an
inequality

$\delta(K)\leq q=\delta(K_{I}).$

Let $q^{\prime}:=\mathrm{flag}(K)$ and take an arbitrary minimal non-face
$J^{\prime}$ of $K_{I}$. Then $J^{\prime}$ is also a minimal non-face of $K$,
so that $|J^{\prime}|-1\leq q^{\prime}$ by definition of flag number. Thus
there is an inequality

$\mathrm{flag}(K_{I})\leq q^{\prime}=\mathrm{flag}(K).$

The proof is completed. ∎

######  Corollary 5.3.

Let $K$ be a simplicial complex. Then there is an inequality

$\delta(K)\leq\mathrm{flag}(K).$

###### proof.

If $K$ is a full simplex, then this corollary is trivial since both
$\delta(K)$ and $\mathrm{flag}(K)$ are $\infty$. If $K$ is not a full simplex,
then there is a minimal non-face $I$ of $K$. Then $K_{I}$ is the boundary of a
full simplex. By definition, the $\delta$-number and the flag number of the
boundary of a full simplex have the same number, so that there are
inequalities

$\delta(K)\leq\delta(K_{I})=\mathrm{flag}(K_{I})\leq\mathrm{flag}(K)$

by Lemma 5.2, completing the proof. ∎

######  Proposition 5.4.

Let $K$ be a simplicial complex. Then the following are equivalent.

  1. (i)

$\delta(K)=\mathrm{flag}(K)$.

  2. (ii)

Each minimal non-face of $K$ has the same order.

###### proof.

If $K$ is a full simplex, then this proposition is trivial, so we may assume
that $K$ is not a full simplex. Suppose that $\delta(K)=\mathrm{flag}(K)=q$
and take an arbitrary minimal non-face $I$ of $K$. Since $\delta(K)=q$, every
non-face of $K$ has an order larger than or equals to $q+1$, so that there is
an inequality

$|I|\geq q+1.$

Since $\mathrm{flag}(K)=q$, every minimal non-face of $K$ has an order smaller
than or equals to $q+1$, so that there is also an equality

$|I|\leq q+1.$

Therefore each minimal non-face of $K$ has the same order $q+1$. Conversely,
suppose that each minimal non-face of $K$ has the same order $q+1$. Since
there is no non-face of $K$ with an order smaller than $q+1$, there is an
inequality

$\delta(K)\geq q.$

On the other hand, since every minimal non-face $I$ of $K$ satisfies an
inequality $|I|-1\leq q$, there is also an inequality

$\mathrm{flag}(K)\leq q.$

By Corollary 5.3, we get inequalities

$q\leq\delta(K)\leq\mathrm{flag}(K)\leq q.$

Thus the equality $\delta(K)=\mathrm{flag}(K)$ holds, completing the proof. ∎

###### proof of Corollary 1.5.

Since there is some element $g_{0}\in G\setminus\\{1\\}$ such that
$\text{supp}(g_{0})\notin K$, $K_{\text{supp}(G)}$ cannot be a full simplex.
By assumption, there is an equality

$\delta(K_{\text{supp}(G)})=\mathrm{flag}(K_{\text{supp}(G)})$

by Proposition 5.4. Since $K_{\text{supp}(g_{0})}$ is not a full simplex,
there are inequalities

$\delta(K_{\text{supp}(G)})\leq\delta(K_{\text{supp}(g_{0})})\leq\mathrm{flag}(K_{\text{supp}(g_{0})})\leq\mathrm{flag}(K_{\text{supp}(G)})$

by Proposition 5.3 and Lemma 5.2. Thus we get an equality

$\delta(K_{\text{supp}(G)})=\delta(K_{\text{supp}(g_{0})}).$

Finally, let us consider inequalities

$\delta(K_{\mathrm{supp}(G)})\leq\mathrm{coind}_{G}(\mathbb{R}Z_{K})\leq\mathrm{wgt}_{G}(\mathbb{R}Z_{K})\leq\min_{g\in
G\setminus\\{1\\}}\delta(K_{\mathrm{supp}(g)})\leq\delta(K_{\mathrm{supp}(g_{0})})$

in Theorem 1.2. Since the left and right sides are equal, we get the
equalities

$\mathrm{coind}_{G}(\mathbb{R}Z_{K})=\mathrm{wgt}_{G}=\delta(K_{\mathrm{supp}(G)}),$

completing the proof. ∎

## Acknowledgement

The author is deeply grateful to Daisuke Kishimoto for much helpful and
valuable advice.

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# Experimental and theoretical study of the correlated compound YbCdSn:
Evidence for large magnetoresistance and mass enhancement

Antu Laha  Department of Physics, Indian Institute of Technology, Kanpur
208016, India  P. Rambabu  Department of Physics, Indian Institute of
Technology Hyderabad, Kandi, Medak 502 285, Telengana, India  Department of
Pure and Applied Physics, Guru Ghasidas Vishwavidyalaya, Koni, Bilaspur
495009, India  V. Kanchana  Department of Physics, Indian Institute of
Technology Hyderabad, Kandi, Medak 502 285, Telengana, India  L. Petit
Science and Technology Facilities Council, Daresbury Laboratory, Daresbury WA4
4AD, United Kingdom  Z. Szotek  Science and Technology Facilities Council,
Daresbury Laboratory, Daresbury WA4 4AD, United Kingdom  Z. Hossain
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur
208016, India

###### Abstract

The unusual features of topological semimetals arise from its nontrivial band
structure. The impact of strong electron correlations on the topological
states remains largely unexplored in real materials. Here, we report the
magnetotransport properties of YbCdSn single crystals. We found two
fundamental experimental evidences of electron correlations through magnetic
susceptibility and specific heat. The electron correlations in this compound
lead to an intermediate valence state and enhance the effective mass of the
charge carriers. This correlated state exhibits large nonsaturating
magnetoresistance, low carrier density, magnetic field induced metal-
semiconductor-like crossover and a plateau in resistivity at low temperatures.
This compound also shows a cusp-like magnetoconductivity at low magnetic field
which indicates the presence of weak antilocalization effect. Our band
structure calculations of Yb2+ state predict YbCdSn to be a topological nodal-
line semimetal.

##  I Introduction

Strongly correlated electron systems and topological semimetals are two
separate active frontiers of modern condensed matter research but their
coexistence is sparse. The low-energy electronic excitations in topological
semimetals are characterized by Dirac-like linear band dispersion, yielding a
variety of exotic transport phenomena such as extremely large nonsaturating
magnetoresistance (MR), chiral anomaly induced negative longitudinal MR and
anomalous Hall effect Armitage  _et al._ (2018); Yan and Felser (2017); Hasan
and Kane (2010). On the other side, strong electron correlations give rise to
several other remarkable features like valence fluctuations, Kondo effect and
heavy fermion behaviour Stewart (1984); Hewson and Kondo (2009); Lawrence  _et
al._ (1981); Laha and Hossain (2018).

Since the topological states are robust against perturbations and local
disorder, the appearance of electron correlations makes this states more
intriguing. For example, the electron correlations in the perovskite oxide
CaIrO3 tune the Dirac line node to the proximity of Fermi level which yields
highly mobile electrons Fujioka  _et al._ (2019); Yamada  _et al._ (2019).
Moreover, correlation induced unconventional mass enhancement of charge
carriers was observed at very high magnetic field around the Dirac nodal loop
in the topological semimetal ZrSiS Pezzini  _et al._ (2017).

The interplay between topology and electron correlations can be better
visualized in rare-earth based topological materials where couplings between
the $f$-electrons and conduction electrons are prominent. In these systems,
strong correlation renormalizes the nontrivial electronic bands which exhibit
strong $f$-character and enhance the effective mass of the charge carriers
Dzero  _et al._ (2010). The $d$-$f$ hybridization opens a bulk band gap at low
temperature due to electron correlations which lead to a topological Kondo
insulator state in SmB6 Ruan  _et al._ (2014); Zhang  _et al._ (2013).
Moreover, theory predicted few correlated topological phases with gapless
excitations in rare-earth based compounds such as Weyl heavy fermion state and
Weyl-Kondo semimetal phase Lai  _et al._ (2018); Xu  _et al._ (2017). Among
them, experimental evidence was found in YbPtBi, where Weyl fermion state is
pronounced at high temperatures while $f$-electrons are well localized,
whereas strong Kondo interaction renormalizes the bands and leads to a heavy
fermion state at low temperatures Guo  _et al._ (2018).

In the present work, we report the presence of electron correlations in YbCdSn
through magnetic susceptibility and specific heat measurements. This compound
hosts an intermediate valence state in which significant mass enhancement of
the charge carriers is observed. The magnetotransport studies reveal several
novel transport properties such as large nonsaturating magnetoresistance, low
carrier density, magnetic field induced metal-semiconductor-like crossover and
resistivity plateau at low temperatures. Weak antilocalization effect is also
observed at low magnetic fields and low temperatures. The main objective of
the electronic structure calculations, based on a first-principles
methodology, is to establish the energy ground state and electronic
configuration (valence) of the system and with that inspect details of the
corresponding density of states (DOS) and band structure, in the vicinity of
the Fermi energy, in search of a possible topological nodal-line semimetal
state in YbCdSn.

##  II Methods

###  II.1 Experimental details

Single crystals of YbCdSn were synthesized using cadmium flux Laha  _et al._
(2019, 2020). Yb ingot (99.99$\%$, Alfa Aesar), Cd shots (99.99$\%$, Alfa
Aesar) and Sn pieces (99.99$\%$, Alfa Aesar) in molar ratio of 1:47:1 were
mixed in an alumina crucible, then the crucible was sealed into a quartz tube
under partial pressure of argon gas. The content was heated to 1000∘C, kept
for 6 hours at that temperature, and then cooled to 500∘C at a rate of
3∘C/hour Laha  _et al._ (2019, 2020). Needle-like single crystals were
extracted from the flux by centrifuging. The crystals are air sensitive in
nature and they require handling in inert atmosphere. The crystal structure
and the phase purity were determined by x-ray diffraction (XRD) technique
using Cu-Kα radiation in a PANalytical X′Pert PRO diffractometer.
Magnetotransport measurements were carried out in a physical property
measurement system (PPMS, Quantum Design) via standard four-probe method.
Magnetic susceptibility and specific heat were also measured in the same PPMS
using the vibrating sample magnetometer and relaxation method respectively.

###  II.2 Computational details

Two different approaches, both based on density functional theory (DFT)
Hohenberg and Kohn (1964); Kohn and Sham (1965), are combined here to study
the electronic structure of YbCdSn. The first one, used mainly for exploring
the correlated nature of the YbCdSn ground state and the corresponding valence
state of Yb ion, is the self-interaction corrected local spin density
approximation (SIC-LSDA) to exchange-correlation energy functional (see
APPENDIX for details) Perdew and Zunger (1981); Temmerman  _et al._ (2009);
Petit  _et al._ , implemented in the linear muffin-tin orbitals method with
the atomic sphere approximation (LMTO-ASA) Andersen (1975), where the
polyhedral Wigner Seitz cell is approximated by slightly overlapping atom
centered spheres, with a total volume equal to the actual crystal volume. The
second approach makes use of the generalized gradient approximation (GGA) to
the exchange-correlation energy functional, with an empirical effective
Hubbard U parameter (GGA+U), added to account for the Yb $f$-electron
correlations, as implemented in the Vienna ab-initio simulation package
(VASP). It is employed mostly for optimizing the YbCdSn crystal structure and
its band structure calculation.

The first-principles band structure calculations are performed using the
projector augmented wave (PAW) method from VASP package Blöchl (1994); Kresse
and Furthmüller (1996a, b); Kresse and Hafner (1993); Kresse and Joubert
(1999). The generalized gradient approximation of Perdew, Burke and Ernzerhof
(GGA-PBE) Perdew  _et al._ (1996) is used to describe the exchange and
correlation effects in the system. The strong correlation effects of rare
earth Yb-$f$ states are treated by applying an effective Hubbard U value of
6.0 eV to Yb-$f$ states, using Dudarev’s approach Dudarev  _et al._ (1998).
The cutoff value of plane wave kinetic energy is set as 516 eV and a k-mesh of
$8\times 8\times 12$ grid is chosen using Monkhorst-Pack scheme Monkhorst and
Pack (1976), while the Brillouin zone integration is done using Gaussian
broadening Elsässer  _et al._ (1994) of 0.05 eV in the calculations. The
equilibrium of the system is achieved with a force convergence of 10-3 eV/Å
and the self-consistent energy criteria is set as 10-6 eV/cell. For the
surface state calculations, firstly tight-binding Hamiltonian is obtained
based on maximally localized Wannier functions using Wannier90 package Pizzi
_et al._ (2020). The orbital projections chosen for Yb, Cd and Sn atoms
respectively are $s,p,d,f$; $s,p$; and $s,p$. Based on this tight-binding
model, the iterative Green’s function method Wu  _et al._ (2018); Sancho  _et
al._ (1985) which is implemented in WannierTools Wu  _et al._ (2018) package
is used to investigate the topological properties such as surface state
spectrum, Fermi arc and Z2 indices.

##  III Crystal structure

YbCdSn hosts a similar hexagonal crystal structure (space group $P$-$62m$)
like the family of nodal-line semimetals RCdX (R=Ca,Yb; X=Ge,Sn) Emmanouilidou
_et al._ (2017); Laha  _et al._ (2020, 2019). The crystal structure and the
Brillouin zone are shown in Fig.1(a), where three Yb, three Cd, two Sn1, and
one Sn2 atoms are located at 3$g$ ($x$, 0, 1/2), 3$f$ ($x$, 0, 0), 2$c$ (1/3,
2/3, 0), and l$b$ (0, 0, 1/2) positions respectively. The Rietveld structural
refinement of the powder XRD data was done using the FULLPROF software package
Rodríguez-Carvajal (1993) as shown in Fig.1(b). No impurity phase was observed
within our experimental resolution. The Rietveld refinement yields a hexagonal
structure with space group $P$-$62m$ (No. 189) and the refined lattice
parameters are $a$ = $b$ = 7.5937(7) Å and $c$ = 4.6790(2) Å.

Figure 1: (a) Crystal structure and Brillouin zone of YbCdSn. (b) Powder x-ray
diffraction pattern of crushed YbCdSn single crystals, recorded at room
temperature. The observed intensity (red scattered points), Rietveld
refinement fit (solid black line), difference between the experimentally
observed and calculated intensities (solid blue line) and Bragg peak positions
(vertical green bars) are shown.

##  IV Experimental results

###  IV.1 Magnetic susceptibility

To find out the magnetic ground state of Yb in YbCdSn, we measured temperature
dependent zero field cooled (ZFC) magnetic susceptibility ($\chi$) of the
grown single crystal along the crystallographic $c$-axis ($B\parallel c$). The
inverse magnetic susceptibility $1/\chi$ fits to the modified Curie-Weiss law,
$\chi(T)=\chi_{0}+C/(T-\theta)$ in the temperature range 100 K-300 K where
$C$, $\theta$, $\chi_{0}$ are Curie constant, paramagnetic Curie temperature
and the temperature independent part of magnetic susceptibility respectively
[Fig.2(a)]. The effective magnetic moment of Yb ions estimated from Curie
constant is $\mu_{eff}=2.61\mu_{B}$ which lies in between the values for Yb2+
state ($0\mu_{B}$) and Yb3+ state ($4.54\mu_{B}$) and the nature of inverse
magnetic susceptibility curve is typical for intermediate valence compounds
Matsumoto  _et al._ (2011); Laha  _et al._ (2019).

For more insight, we use two-level ionic interconfiguration fluctuations model
(ICF) to describe the magnetic susceptibility data. According to ICF model,
the overall $\chi(T)$ is given by Hirst (1970); Sales and Wohlleben (1975);
Franz  _et al._ (1980)

$\chi(T)=\frac{N}{3k_{B}}\Big{[}\frac{\mu_{n}^{2}~{}\nu(T)+\mu_{n-1}^{2}(1-\nu(T))}{T^{*}}\Big{]}+f\frac{C}{T-\theta}+\chi_{0}$
(1)

with

$\nu(T)=\frac{2J_{n}+1}{(2J_{n}+1)+(2J_{n-1}+1)~{}\textrm{exp}(-E_{ex}/k_{B}T^{*})}$
(2)

where $T^{*}=(T+T_{sf})$. Here, $\nu(T)$ is the fractional occupation of
ground state, $\mu_{n}$ and $\mu_{n-1}$ are the effective moments in
$4f^{n}$(Yb2+) and $4f^{n-1}$(Yb3+) states, $(2J_{n}+1)$ and $(2J_{n-1}+1)$
are the degeneracies of the corresponding energy states of $E_{n}$ and
$E_{n-1}$, $E_{ex}$ = ($E_{n}$ \- $E_{n-1}$) is the interconfigurational
excitation energy, $T_{sf}$ is the spin fluctuation temperature associated
with the valence fluctuation and $f$ is the fraction of the contribution
originating from stable Yb3+ ions. In the equation 1, first term represents
valence fluctuation part, second term is for stable Yb3+ state and $\chi_{0}$
is the temperature independent part. The fitting of equation 1 gives a good
agreement with the experimental data (red solid line in Fig.2(b)). The fitting
parameters are comparable with several other Yb-based intermediate valence
compounds Adroja  _et al._ (1990); Kowalczyk  _et al._ (2010); Kaczorowski
_et al._ (2016). Therefore, the result indicates that YbCdSn exhibits a
strongly correlated electron phenomenon which is valence fluctuation.

Figure 2: (a) The Curie-Weiss law fitting (solid blue line) to the inverse
magnetic susceptibility (zero field cooled) in the temperature range 100 K-300
K ($B\parallel c$). (b) The solid red line fits to the two-level ionic
interconfiguration fluctuations model (Eq.(1)). (c) Temperature dependence of
the specific heat from 4 K to 30 K. (d) Linear fitting to the curve $C_{P}/T$
vs. $T^{2}$ in the temperature range 4 K-7 K.

###  IV.2 Specific heat

To get further evidence for electron correlation, we measured temperature
dependent specific heat ($C_{P}$) as shown in Fig.2(c). The specific heat
follows the expression $C_{P}=\gamma T+\beta T^{3}$ in the temperature range 4
K-7 K. The estimated value of $\gamma$ from the linear fit to the curve
$C_{P}/T$ vs. $T^{2}$ is 28 mJ mol-1 K-2 [ Fig.2(d)]. This value is larger
than that observed in non-correlated systems but it is comparable to that
observed in intermediate valence compounds such as YbAl3 ($\gamma$ = 58 mJ
mol-1 K-2), YbFe2Al10 ($\gamma$ = 35 mJ mol-1 K-2) and YbFe4Sb12 ($\gamma$ =
140 mJ mol-1 K-2) Ebihara  _et al._ (2000); Lv  _et al._ (2017); Dilley  _et
al._ (1998). The large $\gamma$-value in these compounds indicates the
enhancement of effective mass due to the interaction between $4f$ and
conduction electrons.

Figure 3: (a) Hall resistivity ($\rho_{yx}$) as a function of $B$. (b) Global
fitting of Hall conductivity ($\sigma_{xy}$) and longitudinal conductivity
($\sigma_{xx}$) at 4 K using two band model (equation 3 and 4). (c)
Temperature dependence of electrical resistivity ($\rho_{xx}$). (d) Electrical
resistivity as a function of $T$ for various $B$ up to 12 T. (e) First
derivative of resistivity $d\rho_{xx}/dT$ as a function of $T$. (f) The two
characteristic temperatures $T_{i}$ and $T_{m}$ as a function of $B$. ($B\perp
I$ and $I\parallel c$).

###  IV.3 Hall resistivity and longitudinal resistivity

To determine the mobility and the density of the charge carriers in YbCdSn, we
measured magnetic field dependent Hall resistivity ($\rho_{yx}$) at 4 K. The
Hall resistivity is presented in Fig.3(a) after removing the MR contribution
using the expression $\rho_{yx}=[\rho_{yx}(B)-\rho_{yx}(-B)]/2$. The
$\rho_{yx}$ shows linear field dependence behavior up to a high magnetic field
of 12 T. In order to probe the electron-hole compensation in this compound, we
consider the semiclassical two band model Hurd (1972),

$\sigma_{xx}=e\Big{[}\frac{n_{h}\mu_{h}}{(1+\mu_{e}B)^{2}}+\frac{n_{e}\mu_{e}}{(1+\mu_{h}B)^{2}}\Big{]},$
(3)
$\sigma_{xy}=eB\Big{[}\frac{n_{h}\mu_{h}^{2}}{(1+\mu_{h}B)^{2}}-\frac{n_{e}\mu_{e}^{2}}{(1+\mu_{e}B)^{2}}\Big{]},$
(4)

where $n_{h}$ ($n_{e}$) and $\mu_{h}$ ($\mu_{e}$) are the hole (electron)
density and mobility respectively. The longitudinal conductivity and the Hall
conductivity are obtained using the expressions
$\sigma_{xx}=\rho_{xx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$ and
$\sigma_{xy}=\rho_{yx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$ respectively.
Simultaneous fitting of the $\sigma_{xx}$ and $\sigma_{xy}$ [Fig.3(b)] yields
$n_{h}=7.08\times 10^{19}$ cm-3, $n_{e}=5.96\times 10^{19}$ cm-3 and
$\mu_{h}=4.90\times 10^{2}$ cm2V-1s-1, $\mu_{e}=6.63\times 10^{2}$ cm2V-1s-1.
The $\sigma_{xx}$ deviates from the semiclassical two band model at low
magnetic field due to the influence of weak antilocalization effect which is
discussed later. The ratio $n_{e}/n_{h}$ is about 0.84 which indicates nearly
perfect electron-hole compensation Hou  _et al._ (2015a). Our theoretically
calculated hole and electron densities ($n_{h}=2.45\times 10^{19}$ cm-3,
$n_{e}=1.61\times 10^{19}$ cm-3) are comparable with the experimental values
and also support the electron-hole compensation.

Now, we present the temperature dependence of electrical resistivity
($\rho_{xx}$) of YbCdSn under various magnetic fields as shown in Fig.3(d).
The $\rho_{xx}$ decreases with decreasing temperature down to 7 K and then
saturates with a residual resistivity of $35.2~{}\mu\Omega$ cm [Fig.3(c)].
Above a certain magnetic field ($\geq 3$ T), the $\rho_{xx}$ shows a field
induced metal-semiconductor-like crossover and a resistivity plateau at low
temperatures, which become more prominent at higher magnetic field. These
features were observed in its sister compound CaCdSn and several other
topological semimetals Laha  _et al._ (2020, 2019); Du  _et al._ (2005); Tafti
_et al._ (2015); Wang  _et al._ (2016); Li  _et al._ (2016); Zeng  _et al._
(2016); Gao  _et al._ (2017); Ali  _et al._ (2016); Du  _et al._ (2018). To
analyze these features, we plot first derivative of resistivity
($d\rho_{xx}/dT$) as a function of temperature as shown in Fig.3(e). From this
figure, two characteristic temperatures can be clearly identified - (i) the
temperature associated with metal-semiconductor-like crossover ($T_{m}$) where
$d\rho_{xx}/dT=0$, (ii) the inflection point in resistivity ($T_{i}$) where
$d\rho_{xx}/dT$ is minimum. The resistivity plateau starts to appear just
below $T_{i}$. Magnetic field dependent variations of $T_{m}$ and $T_{i}$ are
shown in Fig.3(f).

###  IV.4 Magnetoresistance

The magnetoresistance of YbCdSn as a function of $B$ is shown in Fig.4(a) for
various $T$. A large non-saturating MR of $2.46\times 10^{2}\%$ is observed at
4 K and 12 T. The MR decreases significantly as we increase the temperature
and reaches to a lower value of 4.4$\%$ at 100 K and 12 T but still no sign of
saturation is observed. The possible origin of large MR in this compound is
electron-hole compensation effect Wang  _et al._ (2015); Luo  _et al._ (2015);
Emmanouilidou  _et al._ (2017). To describe the MR, we also use Kohler’s rule
that is MR = $\alpha(B/\rho_{0})^{m}$, where $\alpha$ is a constant. In
Fig.4(b), the MR data are plotted as a function of $(B/\rho_{0})$ but the MR
data at various temperatures are not merging into a single curve which
indicates a clear violation of Kohler’s scaling. Violation of Kohler’s scaling
can be observed due to various reasons including the existence of multiple
carriers and different scattering rates of various carriers McKenzie  _et al._
(1998); Wang  _et al._ (2017); Xu  _et al._ (2015); Sun  _et al._ (2014).

Figure 4: (a) Magnetic field dependence of magnetoresistance
[MR=$(\rho_{xx}(B)-\rho_{xx}(0))/\rho_{xx}(0)$] for various $T$ ($B\perp I$
and $I\parallel c$). (b) MR as a function of $B/\rho_{0}\rightarrow$ Kohler’s
plot. (c) Magnetoconductivity [$\Delta\sigma=\sigma(B)-\sigma(0)$] as a
function of $B$. The red solid line fits to the Eq.5 [HLN formula]. (d) Solid
blue line shows $-ln(B)$ fit.

Furthermore, a cusp-like feature is observed in the MR at low magnetic field
which is the reminiscent of weak antilocalization (WAL) effect. This effect
was generally observed in 2D-materials but recently it was also found in 3D
topological materials Shrestha  _et al._ (2017); Xu  _et al._ (2014); Hou  _et
al._ (2015b); Laha  _et al._ (2019). To understand the WAL effect in our
system, we analyzed the magnetoconductivity (MC) data using two different
available models. One is well known Hikami-Larkin-Nagaoka (HLN) model Hikami
_et al._ (1980). According to this model, the magnetoconductivity for WAL
effect can be expressed as

$\Delta\sigma=\frac{\alpha
e^{2}}{2\pi^{2}\hbar}\Bigg{[}\Psi\Bigg{(}\frac{1}{2}+\frac{\hbar}{4el_{\phi}^{2}B}\Bigg{)}-ln\Bigg{(}\frac{\hbar}{4el_{\phi}^{2}B}\Bigg{)}\Bigg{]},$
(5)

where $\Psi$ is the digamma function and $l_{\phi}$ is the phase coherence
length. The parameter $\alpha$ gives an estimation for number of conduction
channel participating in transport and $\alpha$ is found to be -1/2 per
conduction channel in 2D electron systems Chen  _et al._ (2010); He  _et al._
(2011). We calculate the MC [$\Delta\sigma=\sigma(B)-\sigma(0)$] using the
expression $\sigma=\rho_{xx}/(\rho_{xx}^{2}+\rho_{yx}^{2})$, where $\sigma(B)$
and $\sigma(0)$ are the conductivity at finite magnetic field and at zero
magnetic field respectively. The MC data fits well to the HLN formula in the
field range -1.5 T $\leq B\leq$ 1.5 T as shown in Fig.4(c). The fitting yields
$\alpha\sim-10^{4}$ and $l_{\phi}\sim 67$ nm. Such a large value of $\alpha$
was also found in other 3D materials due to the presence of large number of
conduction channels in the bulk Shrestha  _et al._ (2017); Xu  _et al._
(2014); Hou  _et al._ (2015b); Laha  _et al._ (2019).

Recently another model was proposed to describe the WAL effect in nodal-line
semimetals. According to this model, the MC at low field varies as $-ln(B)$
for a torus-shaped Fermi surface Chen  _et al._ (2019). A bad fitting of our
MC data to this model suggests that our system does not belong to the above
mentioned particular case [see Fig.4(d)].

##  V Theoretical results

###  V.1 Ground state configuration and density of states

Figure 5:  Total energy as a function of volume for YbCdSn with respectively
Yb2+ and Yb3+ configurations. The energy minima for the respective curves
define their theoretical volumes. Figure 6: The spin decomposed total
densities of states of YbCdSn with respectively a) Yb2+ and b) Yb3+
configurations. Displayed are also partial densities of states for some of the
electron states of Yb, Cd and Sn. The spin-up DOS is plotted on the positive
side of the y-axis, and the spin-down DOS on its negative side. Energy equal
to 0 eV marks the Fermi energy

In the lanthanide compounds the ground-state valence depends on the lanthanide
ion as well as its chemical environment Petit  _et al._ . Lanthanide ions are
usually found in trivalent state, although in some compounds Yb-ions can be
found in divalent and intermediate/mixed valence states Strange  _et al._
(1999); Temmerman  _et al._ (1999); Svane  _et al._ (2000). Here for YbCdSn,
we realize two relevant valence configurations, namely Yb2+ with all 14
$f$-electrons treated as localized, and Yb3+ with 13 localized $f$-electrons
and one $f$-electron band-like, allowed to hybridize with the other valence
electrons. For these two cases we performed the self-consistent SIC-LSDA total
energy calculations as a function of volume and calculated also the
corresponding electronic structure in terms of densities of states. The
results of these SIC-LSD calculations are shown in Figs. 5 and 6. As seen in
Fig. 5, the global energy minimum occurs in the fully localized, Yb2+,
scenario, at a volume of 231.2 Å3, only slightly below the experimental value.
The Yb3+ configuration is energetically less favourable by about 2 eV and has
a spin magnetic moment of -0.65 $\mu_{B}$, while the Yb2+ scenario has zero
spin magnetic moment. The total energy versus volume behaviour of the LSDA
configuration, with all $f$-electrons treated as itinerant, has also been
calculated, but turns out to be energetically unfavourable by about 55 eV and
is therefore not shown in Fig. 5. Under pressure, with decreasing volume, a
localization/delocalization transition from Yb2+ to Yb3+ is seen to occur at a
rather considerable pressure of 54 GPa. Although this transition would seem
rather difficult to realize in practice, the valence fluctuations between the
Yb2+ and Yb3+ configurations could not be excluded, in line with the proposed
intermediate-valent state of this system.

The spin-decomposed total densities of states of YbCdSn for both the Yb2+ and
Yb3+ configurations are plotted respectively in Figs. 6a and 6b, in the
vicinity of the Fermi energy. For both configurations, respectively 14 and 13
localized $f$ band states are not shown in the figure, and only a single band-
like $f$ state for the Yb3+ configuration is seen to sit right at the Fermi
energy in the negative, spin-down panel of the figure. For the Yb2+
configuration, the total density of states per spin is equal to 0.34
states/unit cell/eV, in agreement with a semimetallic character, while for the
Yb3+ configuration it is equal to 2.67 states/unit cell/eV for the spin-up
component and 99.87 states/unit cell/eV for the spin-down component.

One should mention here that because the SIC-LSDA approach is still a one-
electron ground state theory, with the screening/relaxation effects neglected,
it does not give accurate removal energies of the localized $f$ states which,
as a result, are usually situated at too low an energy (here around -10 eV and
therefore are not shown in Figs. 6a and 6b). Of course, the position of the
delocalized $f$-peak in the DOS for the Yb3+ configuration is also affected by
the neglect of the screening/relaxation effects which, if included, would
shift it slightly above the Fermi energy (according to a $\Delta$SCF-like
estimate Temmerman  _et al._ (1993)). Thus the $f$ character would still be
present at the Fermi energy, through hybridization with the other conduction
electron states, but its contribution to the spin-down density of states at
the Fermi energy would be considerably reduced, making the spin-down DOS
closer to the value of the spin-up DOS. Depending on the actual position of
the delocalized $f$-peak relative to the Fermi level, the DOS of Yb3+
configuration could be interpreted as either metallic or heavy Fermion-like.

###  V.2 Electronic structure

Figure 7: Band structure of YbCdSn along $M-\Gamma-K$ path without SOC in (a),
with SOC in (b). The character bands $Cd-s$ and $Sn-p$ are given without SOC
in (c,d) and with SOC in (e,f) respectively. The number of bands that cross EF
are marked as 1, 2 and 3 for without SOC case in (a). Figure 8: (a) The (001)
projected surface states of YbCdSn without SOC and (b) the corresponding Fermi
arc in $k_{z}$=0 plane. (c) The Fermi surface of YbCdSn in 3D-Brillouin Zone
without SOC.

The bulk band structure of YbCdSn with and without spin-orbit coupling (SOC)
calculated for the Yb2+ ground state, as established by the SIC-LSDA self-
consistent calculations, is shown in Fig.7. The valence (red) and conduction
(blue) bands cross each other along $\Gamma-M$ direction at 0.03 eV and
$\Gamma-K$ direction at 0.01 eV as shown in Fig.7(a). These bands are
constructed by $Yb-d$, $Cd-s$ and $Sn-p$ states and present an inverted band
structure at $\Gamma$-M and $\Gamma$-K directions which is supported by
character bands of $Cd-s$ and $Sn-p$ orbitals as shown in Fig.7(c,d). The
presence of more than one Dirac points around the same energy level at
different high symmetry points indicates the nodal line, and hence the two
nontrivial bands, having a constant energy, form a nodal loop above the Fermi
level ($E_{F}$). The (001) projected surface state spectrum of YbCdSn with
(Cd,Sn) termination is shown in Fig.8(a). From this, we can clearly see a
nodal line (NL) and surface states (SS) apart from bulk states. The bulk bands
touch at Dirac points and surface states merge into bulk states at Dirac
points at the same time Wang and Wang (2017) which can be seen from Fig.8(a).
The same is confirmed from the Fermi arc calculation in $k_{x}$-$k_{y}$ plane
which also shows the nodal line pointed by black arrow around $\Gamma$-point
as shown in Fig.8(b) and the other rings may be due to the presence of surface
states far from $E_{F}$ as shown in Fig.8(a). The three dimensional Fermi
surface of YbCdSn without SOC is shown in Fig.8(c), where one can observe a
hexagonal shaped nodal line around $\Gamma$ in $k_{x}$-$k_{y}$ plane
connecting $M$ and $K$ points. Here the green spherical pocket at the zone
centre corresponds to 1 and 2 bands that crossed EF and the red pockets
correspond to band number 3 as shown in Fig.7(a). With the inclusion of SOC as
shown in Fig.7(b), the degeneracy lifted bands are seen in solid and dashed
red and blue lines and the band hybridization opens a gap of 28 $meV$along
$\Gamma-K$ direction and tiny gap of 7 $meV$along $\Gamma-M$ direction and the
band hybridization is due to $Cd-s$ and $Sn-p$ characters which are shown in
Fig.7(e,f). Now the system becomes topologically nontrivial with the inclusion
of SOC, which is confirmed by the Z2 topological indices ($\upsilon_{0}$;
$\upsilon_{1}$$\upsilon_{2}$$\upsilon_{3}$) = (1; 000). The value
$\upsilon_{0}$=1 clearly tells us the nontrivial topology of the system with
the inclusion of SOC.

##  VI Discussion and conclusion

We next turn to discuss the novel transport phenomena and the mass enhancement
of the charge carries in the correlated state of YbCdSn. Nearly equal electron
and hole densities suggest that electron-hole compensation is responsible for
large MR in this compound Emmanouilidou  _et al._ (2017). The large
$\gamma$-value indicates an enhancement of effective mass which is further
supported by electronic structure calculation. The calculated effective mass
from the electronic structure is around 25.5$m_{e}$ along $A-L$ and $K-H$
directions. Mass enhancement was also observed in the intermediate valence
compound SmB6, where band inversion between opposite parity $4f$ and $5d$
states makes it topologically nontrivial and correlations induced bulk
insulating gap formation leads to a topological Kondo insulator state Ruan
_et al._ (2014); Lu  _et al._ (2013). In the Weyl semimetal YbPtBi, Kondo
coupling strengthens at low temperatures and modifies the dispersion of the
nontrivial bands in the vicinity of the topologically protected Weyl points
which enhances the effective mass whereas Weyl fermion state dominates at high
temperature Guo  _et al._ (2018). We have also estimated the Wilson ratio in
order to get some idea about the strength of electron correlations. The Wilson
ratio is expressed as $R_{W}=(\pi^{2}K_{B}^{2}\chi_{0})/(\gamma\mu_{eff})$
Wilson (1975); Falkowski and Kowalczyk (2018). The estimated value of $R_{W}$
is 0.68 for YbCdSn, where $\chi_{0}=1.8\times 10^{-3}$ emu/mol,
$\mu_{eff}=4.54\mu_{B}$ and $\gamma$= 28 mJ mol-1 K-2. This value of $R_{W}$
falls in the range of valence fluctuating compounds and indicates significant
electron correlations Kishimoto  _et al._ (2003); Falkowski and Kowalczyk
(2018).

In conclusion, our experimental studies reveal that YbCdSn hosts an
intermediate valence state. The specific heat indicates an enhancement of
effective mass in this state. More intriguing thing is that the correlated
phase of this compound shows unusual magnetotransport properties such as large
nonsaturating MR of $\sim 2.5\times 10^{2}$ % at 4 K and 12 T, low carrier
density, field induced metal-semiconductor-like crossover and a plateau in
resistivity at low temperatures. Moreover, a cusp-like feature in
magnetoconductivity at low field indicates the presence of WAL effect in this
system. On the theory side, our first-principles SIC-LSDA calculations predict
a divalent state for Yb ion and the corresponding density of states appears to
support the semimetalic behaviour. With applying a considerable pressure,
achieved by decreasing volume, a transition from Yb2+ to Yb3+ was seen to
occur at a very small volume, seemingly difficult to realize in practice but
not excluding valence fluctuations between the two configurations. As for the
band structure calculations, they predict a topological nodal-line semimetalic
state of YbCdSn for the ground state divalent Yb ion. To confirm this
prediction, and describe valence fluctuations, further experiments such as
ARPES and quantum oscillations measurements, as well as calculations
accounting for dynamical correlations, would be required.

##  VII Acknowledgements

Research support from IIT Kanpur, IIT Hyderabad and SERB India (Grant No.
CRG/2018/000220) are gratefully acknowledged. This work made use of
computational support by CoSeC, the Computational Science Centre for Research
Communities, through CCP9.

##  VIII Appendix: SIC-LSD formalism

The basis of the SIC-LSD formalism is an orbital dependent DFT energy
functional, obtained from the LSDA total energy functional by subtracting a
spurious self-interaction of each occupied electron state, thus making it
self-interaction free. The self-interaction error is inherent in LSDA due to
local approximation applied to the exchange-correlation energy functional and
to a large extent is responsible for the failure of LSDA (and even GGA) to
describe strong electron correlations. The self-interaction correction (SIC)
constitutes a negative energy contribution gained by an electron upon
localization which competes with the band formation energy gained by the
electron if allowed to delocalize and hybridize with the available conduction
electron states. Hence, in this formulation, one distinguishes between
localized and itinerant states but treats them on equal footing through
expanding them in the same set of basis functions in an ab initio manner. The
decision whether a state is treated as localized or extended is based on a
delicate energy balance between band formation and localization. One has to
explore a variety of possible configurations consisting of different
distributions of localized and itinerant states and minimize the SIC-LSDA
total energy functional with respect to all those configurations which leads
to a number of local minima of the same functional, meaning that their total
energies may be compared. The configuration with the lowest total energy
defines the ground state energy and the ensuing valence. If no localized
states are assumed, then the SIC-LSDA energy functional coincides with the
LSDA energy functional, namely the LSDA functional is also a local minimum of
the SIC-LSDA energy functional. The SIC-LSDA formalism still considers the
electronic structure of the solid to be built from individual one-electron
states, but offers an alternative description to the Bloch picture, namely in
terms of periodic arrays of localized atom-centred states, i.e., the Heitler-
London picture in terms of exponentially decaying Wannier functions.
Nevertheless, there always exist states that will never benefit from SIC,
retaining their itinerant character of the Bloch form and moving in the LSDA
effective potential. This is the case for the non-$f$ conduction electron
states in the lanthanide systems. In the SIC-LSD method, the eigenvalue
problem is solved in the space of Bloch states, but a transformation to the
Wannier representation is made at every step of the self-consistency process
to calculate the localized orbitals and the corresponding charges to evaluate
the SIC potentials of the states that are truly localized. These repeated
transformations between Bloch and Wannier representations constitute the major
difference between LSDA and SIC-LSDA. Since the main effect of SIC is to
reduce hybridization of localized electron states with the valence band, the
technical difficulties of minimizing the SIC-LSDA functional in solids are
often circumvented by introducing an empirical effective Coulomb interaction
parameter U for the orbitals that are meant to be localized, which leads to
the DFT+U approach used here for the band structure calculations of YbCdSn. Of
course, this way the major advantage of the SIC-LSDA formalism, namely
providing a dual picture of coexisting localized and band-like $f$-electrons
and allowing one to determine valencies of the constituent elements in the
solids, is lost.

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