# Combinatorial meaning of the number of the even parts in a partition of $n$
into distinct parts

Jiyou Li
School of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai, China, 200240
Email<EMAIL_ADDRESS>
Sicheng Zhao
School of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai, China, 200240
Email<EMAIL_ADDRESS>
###### Abstract

In a recent paper [2], Andrews and Merca investigated the number of even parts
in all partitions of $n$ into distinct parts, which arise naturally from the
Euler-Glaisher bijective proof. They obtained new combinatorial
interpretations for this number by using generating functions. We obtain a
direct combinatorial proof in this note.

##  1 Introduction

For a positive integer $n$, its partition $\lambda$ is a multi-set of positive
integers $\lambda=\\{\lambda_{1},\cdots,\lambda_{k}\\}$ such that
$\sum\limits_{i=1}^{k}\lambda_{i}=n$. Each number $\lambda_{i}$ is called a
part of $\lambda$. For simplicity, subscripts are used to denote the
multiplicity of a part in a partition. For example, $\\{5_{2},2_{2},1\\}$
denotes $\\{5,5,2,2,1\\}$. Also, for two partitions
$\lambda=\\{\lambda_{1},\cdots,\lambda_{k}\\}$ and
$\mu=\\{\mu_{1},\cdots,\mu_{t}\\}$, $\lambda+\mu$ is defined to be
$\\{\lambda_{1},\cdots,\lambda_{k}\\}\bigcup\\{\mu_{1},\cdots,\mu_{t}\\}$ as a
new partition.

A partition is called odd if all of its parts are odd and distinct if all of
its parts are distinct. One of the first and most beautiful results in
partition theory is that the number of odd partitions of $n$ is equal to the
number of distinct partitions of $n$. This is well known as Euler’s partition
identity. A combinatorial proof by doubling and halving naturally shows the
correspondence and it is regarded as a typical example of elegant
combinatorial proofs.

Precisely, recall the Glaisher’s map $\varphi$ from the set of distinct
partitions to the set of odd partitions is constructed as follows. For any
distinct partition
$\lambda=\\{2^{k_{1}}(2s_{1}-1),\cdots,2^{k_{m}}(2s_{m}-1)\\}$ with distinct
pairs $(k_{i},s_{i})$ for $1\leq i\leq m$, $\varphi$ maps it to
$\varphi(\lambda)=\\{(2s_{1}-1)_{2^{k_{1}}}\\}+\cdots+\\{(2s_{m}-1)_{2^{k_{m}}}\\}$.
For example, the Glaisher’s map $\varphi$ sends the partition
$\\{12,7,6,5,3\\}$ to the partition $\\{7,5,3_{7}\\}$. This elegant map plays
an important role in our proof (Section 3).

In [2], Andrews and Merca investigated the number of even parts in all
partitions of $n$ into distinct parts and they obtained new combinatorial
interpretations for this number by using generating functions. They asked if
there is any direct combinatorial proof.

Two elegant combinatorial proofs were given by Ballantine and Merca [3], and
Li and Wang [5] independently. In this paper, we provide a new combinatorial
proof.

###### Definition 1

Let $n$ be a non-negative integer. Let $N(n)$ be the number of partitions of
$n$ into distinct parts and $N_{\hat{m}}(n)$ be the number of partitions of
$n$ into distinct parts avoiding $m$.

For convenience, $N(n)$ and $N_{\hat{m}}(n)$ are defined to be zero when
$n<0$. For simplicity, a distinct partition of $n$ is the abbreviation for a
partition of $n$ into distinct parts and similarly an odd partition of $n$ is
the abbreviation for a partition of $n$ with all parts odd.

###### Definition 2

Let $n$ be a non-negative integer. Define $a(n)$ to be the number of even
parts in all distinct partitions of $n$.

For instance, since $6=5+1=4+2=3+2+1$, one has $a(6)=4$.

###### Definition 3

Let $n$ be a non-negative integer. We define:

1\. $B_{o}(n)$ to be the set of partitions of $n$ into an odd number of parts
in which the set of even part has only one element;

2\. $B_{e}(n)$ to be the set of partitions of $n$ into an even number of parts
in which the set of even part has only one element;

3\. $b_{o}(n)=|B_{o}(n)|$, $b_{e}(n)=|B_{e}(n)|$, $b(n)=b_{o}(n)-b_{e}(n)$.

Clearly,
$B_{o}(6)=\\{\\{6\\},\\{4,1_{2}\\},\\{3,2,1\\},\\{2_{3}\\},\\{2,1_{4}\\}\\}$,
$B_{e}(6)=\\{\\{2_{2},1_{2}\\}\\}$ and thus $b(6)=4$.

###### Definition 4

Let $n$ be a non-negative integer. We define:

1\. $C_{o}(n)$ to be the set of partitions of $n$ in which exactly one part is
repeated and this part is odd;

2\. $C_{e}(n)$ to be the set of partitions of $n$ in which exactly one part is
repeated and this part is even;

3\. $c_{o}(n)=|C_{o}(n)|$, $c_{e}(n)=|C_{e}(n)|$, $c(n)=c_{o}(n)-c_{e}(n).$

Similarly, one checks that $c_{o}(6)=5$, $c_{e}(6)=1$ and thus $c(6)=4$.

Beck conjectured that the number of partitions of $n$ in which the set of even
parts has exactly one element equals the number of partitions of $n$ in which
exactly one part is repeated, namely $b_{o}(n)+b_{e}(n)=c_{o}(n)+c_{e}(n)$.
The conjecture was proved in [1] and [4]. Furthermore, the following two
theorems in [2] given by Andrews and Merca provided more insights on this
identity.

###### Theorem 1

Suppose that $a(n),b(n),c(n)$ are defined as above. Then for $n\geq 1$,
$a(n)=(-1)^{n}b(n)=c(n)$.

###### Theorem 2

Suppose that $b_{o}(n),b_{e}(n),c_{o}(n),c_{e}(n)$ are defined as above, then
we have

1\. $b_{o}(n)=$ $\left\\{\begin{array}[]{cc}c_{o}(n),&if\ n\ is\ even,\\\
c_{e}(n),&if\ n\ is\ odd;\\\ \end{array}\right.$

2\. $b_{e}(n)=$ $\left\\{\begin{array}[]{cc}c_{e}(n),&if\ n\ is\ even,\\\
c_{o}(n),&if\ n\ is\ odd.\\\ \end{array}\right.$

We have the following example.

$\displaystyle B_{o}(7)=\\{\\{3,2_{2}\\},\\{2_{2},1_{3}\\}\\},$ $\displaystyle
B_{e}(7)=\\{\\{6,1\\},\\{5,2\\},\\{4,3\\},\\{4,1_{3}\\},\\{3,2,1_{2}\\},\\{2_{3},1\\},\\{2,1_{5}\\}\\},$
$\displaystyle
C_{o}(7)=\\{\\{3_{2},1\\},\\{5,1_{2}\\},\\{3,2,1_{2}\\},\\{4,1_{3}\\},\\{3,1_{4}\\},\\{2,1_{5}\\},\\{1_{7}\\}\\},$
$\displaystyle C_{e}(7)=\\{\\{2_{3},1\\},\\{3,2_{2}\\}\\}.$

The distinct partitions of $7$ are
$\\{7\\},\\{6,1\\},\\{5,2\\},\\{4,3\\},\\{4,2,1\\}$, so that
$a(7)=0+1+1+1+2=5=-b(7)=c(7)$. Thus the theorems hold for $n=7$.

This paper is organized as follows. We will first give proofs of Theorem 1 and
2 without using generating functions in Section 2 and direct combinatorial
proofs in Section 3.

##  2 Proof of the main results

###### Lemma 1

Suppose that $N(n)$ and $N_{\hat{m}}(n)$ are defined as above. Then
$N_{\hat{m}}(n)=\sum\limits_{i\geq 0}(-1)^{i}N(n-im)$.

Proof  We first divide the distinct partitions of $n$ into two parts: the ones
including $m$ and the other $N_{\hat{m}}(n)$ ones. Observing the one-to-one
correspondence between distinct partitions of $n$ including $m$ as a part and
distinct partitions of $n-m$ avoiding $m$, we obtain the identity
$N_{\hat{m}}(n)=N(n)-N_{\hat{m}}(n-m)$. The lemma then follows directly by
induction.

###### Definition 5

Let $n$ be a non-negative integer. We define:

1\. $B_{o}^{{}^{\prime}}(n)$ to be the set of partitions of $n$ in which the
set of even parts has only one element, and this element is repeated for an
odd number of times;

2\. $B_{e}^{{}^{\prime}}(n)$ to be the set of partitions of $n$ in which the
set of even parts has only one element, and this element is repeated for an
even number of times;

3\. $b_{o}^{{}^{\prime}}(n)=|B_{o}^{{}^{\prime}}(n)|$,
$b_{e}^{{}^{\prime}}(n)=|B_{e}^{{}^{\prime}}(n)|$,
$b^{{}^{\prime}}(n)=b_{o}^{{}^{\prime}}(n)-b_{e}^{{}^{\prime}}(n)$.

Obviously
$B_{o}^{{}^{\prime}}(6)=\\{\\{6\\},\\{4,1_{2}\\},\\{3,2,1\\},\\{2_{3}\\},\\{2,1_{4}\\}\\}$,
$B_{e}^{{}^{\prime}}(6)=\\{\\{2_{2}1_{2}\\}\\}$ and thus $b^{\prime}(6)=4$.

The new sets $B_{o}^{{}^{\prime}}(n)$ and $B_{e}^{{}^{\prime}}(n)$ are closely
related to $B_{o}(n)$ and $B_{e}(n)$ due to the following lemma.

###### Lemma 2

Let $B_{o}(n),B_{e}(n),B_{o}^{{}^{\prime}}(n),B_{e}^{{}^{\prime}}(n)$ be
defined as above. Then,

1\. $B_{o}(n)=$ $\left\\{\begin{array}[]{cc}B_{o}^{{}^{\prime}}(n),&if\ n\ is\
even,\\\ B_{e}^{{}^{\prime}}(n),&if\ n\ is\ odd;\\\ \end{array}\right.$

2\. $B_{e}(n)=$ $\left\\{\begin{array}[]{cc}B_{e}^{{}^{\prime}}(n),&if\ n\ is\
even,\\\ B_{o}^{{}^{\prime}}(n),&if\ n\ is\ odd.\\\ \end{array}\right.$

Proof  First we prove that $B_{o}(n)=B_{o}^{{}^{\prime}}(n)$ if $n$ is even.
For a partition $\lambda=\\{(2k)_{t},2a_{1}-1,\cdots,2a_{m}-1\\}\in B_{o}(n)$,
the identity $n=t\cdot 2k+\sum_{i=1}^{m}(2a_{i}-1)$ requires $m$ to be even
and the definition of $B_{o}(n)$ requires $m+t$ to be odd. Thus $t$ is odd,
and $\lambda\in B_{o}^{{}^{\prime}}(n)$. On the other hand, for a partition
$\lambda=\\{(2k)_{2s-1},2a_{1}-1,\cdots,2a_{m}-1\\}\in
B_{o}^{{}^{\prime}}(n)$, the identity $n=(2s-1)\cdot
2k+\sum_{i=1}^{m}(2a_{i}-1)$ requires $m$ to be even. Thus $m+(2s-1)$ is odd,
and $\lambda\in B_{o}(n)$. We conclude that $B_{o}(n)=B_{o}^{{}^{\prime}}(n)$
for even $n$.

The remainder of the proof is similar and we omit it.

Proof of Theorem 1 and 2:  By Lemma 2, Theorem 1 and 2 are equivalent to the
following equalities.

$b_{o}^{{}^{\prime}}(n)=c_{o}(n),$ $b_{e}^{{}^{\prime}}(n)=c_{e}(n),$
$a(n)=b^{{}^{\prime}}(n)=c(n).$

We first express
$b_{o}^{{}^{\prime}}(n),b_{e}^{{}^{\prime}}(n),c_{o}(n),c_{e}(n)$ in terms of
$N(n)$ and $N_{\hat{m}}(n)$ to show the relations among them.

For a partition $\lambda=\\{(2k)_{2s-1},2a_{1}-1,\cdots,2a_{m}-1\\}\in
B_{o}^{{}^{\prime}}(n)$ with $k,s,a_{i}$ being positive integers, we write
$\lambda=\\{(2k)_{2s-1}\\}+\\{2a_{1}-1,\cdots,2a_{m}-1\\}$. It implies that

$b_{o}^{{}^{\prime}}(n)=\sum_{s\geq 1}\sum_{k\geq 1}N(n-(2s-1)\cdot 2k).$

Similarly,

$b_{e}^{{}^{\prime}}(n)=\sum_{s\geq 1}\sum_{k\geq 1}N(n-2s\cdot 2k).$

For any partition $\lambda=\\{(2s-1)_{t},b_{1},\cdots,b_{m}\\}\in C_{o}(n)$
with $t\geq 2$ and $b_{i}\neq 2s-1$ being distinct positive integers, we have
$\lambda=\\{(2s-1)_{t}\\}+\\{b_{1},\cdots,b_{m}\\}$. One obtains that

$c_{o}(n)=\sum_{t\geq 2}\sum_{s\geq 1}N_{\hat{2s-1}}(n-t\cdot(2s-1)).$

Similarly,

$c_{e}(n)=\sum_{t\geq 2}\sum_{s\geq 1}N_{\hat{2s}}(n-t\cdot 2s).$

Applying Lemma 1 to $c_{o}(n)$, we have

$c_{o}(n)=\sum_{t\geq 2}\sum_{s\geq 1}\sum_{i\geq
0}(-1)^{i}N(n-(t+i)\cdot(2s-1)).$

For fixed $2s-1$, the coefficient of $N(n-l\cdot(2s-1))(l\geq 2)$ is
$\sum\limits_{2\leq t\leq l}(-1)^{l-t}$. It equals 1 when $l$ is even and 0
when $l$ is odd. Thus

$c_{o}(n)=\sum_{s\geq 1}\sum_{k\geq
1}N(n-2k\cdot(2s-1))=b_{o}^{{}^{\prime}}(n).$

Similarly, for $c_{e}(n)$, we have

$c_{e}(n)=\sum_{s\geq 1}\sum_{k\geq 1}N(n-2k\cdot
2s))=b_{e}^{{}^{\prime}}(n).$

Now we come to $a(n)$. By definition,

$b^{{}^{\prime}}(n)=b_{o}^{{}^{\prime}}(n)-b_{e}^{{}^{\prime}}(n)=\sum_{t\geq
1}\sum_{k\geq 1}(-1)^{t-1}N(n-t\cdot 2k)=\sum_{k\geq 1}N_{\hat{2k}}(n-2k).$

Note that $\sum\limits_{k\geq 1}N_{\hat{2k}}(n-2k)$ equals the number of pairs
(a distinct partition of $n$, an even part in it). By definition,
$b^{{}^{\prime}}(n)=a(n)$ and the proof is complete.

##  3 A Combinatorial Proof

Since our proof of Lemma 2 is combinatorial, it suffices to establish the
following equalities combinatorially.

$b_{o}^{{}^{\prime}}(n)=c_{o}(n),$ (1) $b_{e}^{{}^{\prime}}(n)=c_{e}(n),$ (2)
$a(n)=b^{{}^{\prime}}(n)=c(n).$ (3)

Let $\varphi$ be any bijection constructed combinatorially from the set of odd
partitions to the set of distinct partitions. For instance, $\varphi$ might be
chosen to be the Glaisher’s map.

We now prove (1) first. Let
$\lambda=\\{(2k)_{2s-1},2a_{1}-1,\cdots,2a_{m}-1\\}\in
B_{o}^{{}^{\prime}}(n)$. We shall construct a map $\theta$ sending $\lambda$
to a partition in $C_{o}(n)$. Let $D=\varphi(\\{2a_{1}-1,\cdots,2a_{m}-1\\})$
be the image of the partition $\\{2a_{1}-1,\cdots,2a_{m}-1\\}$ under the map
$\varphi$. Note that $D$ is a distinct partition of $n-(2s-1)\cdot 2k$. We
then send $\\{(2k)_{2s-1}\\}+D$ to the target partition $\theta(\lambda)$
according the following rules:

$\left\\{\begin{array}[]{cc}\\{(2s-1)_{2k}\\}+D,&\text{if}\ 2s-1\notin D;\\\
\\{(2s-1)_{2k+1}\\}+D\backslash\\{2s-1\\},&\text{otherwise}.\\\
\end{array}\right.$

Here $D\backslash\\{2s-1\\}$ denotes the multi-set (namely, the partition)
obtained by removing exactly one part $2s-1$. Obviously both of the two forms
represent partitions in $C_{o}(n)$. One checks that this $\theta$ is
injective, since two partitions must contain the same parts in $2k$ (and its
multiplicity $2s-1$) and $D$ if they are sent into the same partition.

For example, let $\varphi$ be the Glaisher’s map. Let
$\lambda=\\{3,2_{3},1_{2}\\}$ be a partition of 11. Since
$\varphi(\\{3,1_{2}\\})=\\{3,2\\}$ and $3$ appears as a part,
$\theta(\lambda)=\\{3_{3},2\\}$. Let $\lambda=\\{5,3_{2},2\\}$ be a partition
of 13. Since $\varphi(\\{5,3_{2}\\})=\\{6,5\\}$ and $1$ does not appear as a
part, $\theta(\lambda)=\\{6,5,1_{2}\\}$.

Conversely, for a partition $\lambda=\\{(2s-1)_{t},b_{1},\cdots,b_{m}\\}\in
C_{o}(n)$ with $t\geq 2$ and $b_{i}\neq 2s-1$ distinct, we send it to
$\\{t_{2s-1},\varphi^{-1}(\\{b_{1},\cdots,b_{m}\\})\\}$ if $t$ is even and
$\\{(t-1)_{2s-1},\varphi^{-1}(\\{2s-1,b_{1},\cdots,b_{m}\\})\\}$ if $t$ is
odd. This map is clearly injective, and the above examples also work here.
This finishes the combinatorial proof of (1).

The combinatorial proof of (2) is almost the same and is left to the
interested readers.

Now we turn to prove (3). By definition we may rewrite (3) as
$b_{o}^{{}^{\prime}}(n)=a(n)+b_{e}^{{}^{\prime}}(n)$. Let $A(n)$ be the set of
the pairs of a distinct partition of $n$ and an even part in it. That is,
$A(n)=\\{(\lambda,2k)|\lambda\ is\ a\ distinct\ partition\ of\ n,\
2k\in\lambda\\}$. Clearly, $|A(n)|=a(n)$. We shall construct a map $\theta$
sending any partition $\lambda\in B_{o}^{{}^{\prime}}(n)$ to a partition in
$B_{e}^{{}^{\prime}}(n)$, or an element in $A(n)$. Let
$\lambda=\\{(2k)_{2s-1},2a_{1}-1,\cdots,2a_{m}-1\\}\in
B_{o}^{{}^{\prime}}(n)$. Let $D=\varphi(\\{2a_{1}-1,\cdots,2a_{m}-1\\})$ be
the image of the partition under the map $\varphi$. We send $\lambda$ to
$\theta(\lambda)$ according the following rules:

$\left\\{\begin{array}[]{cc}\\{(2k)_{2s}\\}+\varphi^{-1}(D\backslash\\{2k\\}),&\text{if}\
2k\in D;\\\ \\{(2k)_{2s-2}\\}+\varphi^{-1}(D+\\{2k\\}),&\text{if}\ 2k\notin
D,\ s\not=1;\\\ (\\{2k+D\\},2k)&\text{otherwise.}\end{array}\right.$ (4)

One checks that the first two forms in (4) represent all partitions in
$B_{e}^{{}^{\prime}}(n)$ and the third one is in $A(n)$. For two different
partitions of $n$ in $B_{o}^{{}^{\prime}}(n)$, if they are sent to the same
partition in $B_{e}^{{}^{\prime}}(n)$, then they should correspond to the same
even number $2k$ and the same image $D$, and thus also have the same
multiplicity $2s-1$; if they are sent to the same element in $A(n)$, then they
should be the same in partition ${2k+D}$ and corresponding even number $2k$,
and thus be the same. Hence the map $\theta$ is injective.

For example, let $\varphi$ be the Glaisher’s map and for the partition
$\lambda=\\{5,2_{3},1_{3}\\}$ of 14, since
$\varphi(\\{5,1_{3}\\})=\\{5,2,1\\}$ includes $2$ as a part,
$\theta(\lambda)=\\{2_{4}\\}+\varphi^{-1}(\\{5,1\\})=\\{5,2_{4},1\\}$.

For the partition $\lambda=\\{6_{3},3_{2}\\}$ of 24, since
$\varphi(\\{3_{2}\\})=\\{6\\}$ does not include $6$ as a part,
$\\{6_{2}\\}+\varphi^{-1}(\\{6,6\\})=\\{6_{2},3_{4}\\}$.

For the partition $\lambda=\\{5,3_{2},2\\}$ of 13, since
$\varphi(\\{5,3_{2}\\})=\\{6,5\\}$ does not include $2$ as a part and the
multiplicity of $2$ in $\\{5,3_{2},2\\}$ is 1,
$\theta(\lambda)=(\\{6,5,2\\},2)$.

The converse map $\theta^{-1}$ can be obtained easily by starting from the
forms in (4) and inverting the steps we took before. The details are left to
the readers.

## References

  * [1] G.E. Andrews, Euler’s partition identity and two problems of George Beck, Math.Student 86, 115–119 (2017)
  * [2] G.E. Andrews and M. Merca, On the number of even parts in all partitions of $n$ into distinct parts, Annals of Combinatorics, 24, 47–54 (2020)
  * [3] C. Ballantine and M. Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of $n$ into distinct parts, Ramanujan Journal 54, 107–112 (2021)
  * [4] C. Ballantine and R. Bielak, Combinatorial proofs of two Euler-Type identities due to Andrews, Annals of combinatorics, 23, 511–525 (2019)
  * [5] R. Li and Andrew Y.Z. Wang, On the combinatorics of the number of even parts in all partitions with distinct parts, Ramanujan Journal 56, 721–727 (2021)


# Correlation between plastic rearrangements and local structure in a
cyclically driven glass

Saheli Mitra  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France.  Susana Marín-Aguilar<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France.  Soft Condensed Matter, Debye Institute for Nanomaterials
Science, Department of Physics, Utrecht University, Princetonplein 1, 3584 CC
Utrecht, The Netherlands.  Srikanth Sastry  Jawaharlal Nehru Center for
Advanced Scientific Research, Jakkur Campus, Bengaluru 560064, India.  Frank
Smallenburg<EMAIL_ADDRESS>Université Paris-
Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.
Giuseppe Foffi<EMAIL_ADDRESS>Université Paris-
Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.

###### Abstract

The correlation between local structure and the propensity for structural
rearrangements has been widely investigated in glass forming liquids and
glasses.
In this paper we use the excess two-body entropy $S_{2}$ and tetrahedrality
$n_{\mathrm{tet}}$ as the per-particle local structural order parameters to
explore such correlations in a three-dimensional model glass subjected to
cyclic shear deformation. We first show that for both liquid configurations
and the corresponding inherent structures, local ordering increases upon
lowering temperature, signaled by a decrease in the two-body entropy and an
increase in tetrahedrality. When the inherent structures, or glasses, are
periodically sheared athermally, they eventually reach absorbing states for
small shear amplitudes, which do not change from one cycle to the next. Large
strain amplitudes result in the the formation of shear bands, within which
particle motion is diffusive. We show that in the steady state, there is a
clear difference in the local structural environment of particles that will be
part of plastic rearrangements during the next shear cycle and that of
particles that are immobile. In particular, particles with higher $S_{2}$ and
lower $n_{\mathrm{tet}}$ are more likely to go through rearrangements
irrespective of the average energies of the configurations and strain
amplitude. For high shear, we find very distinctive local order outside the
mobile shear band region, where almost $30\%$ of the particles are involved in
icosahedral clusters, contrasting strongly with the fraction of $<5\%$ found
inside the shear band.

##  I Introduction

The mechanism involved in how a liquid loses its fluidity upon decreasing
temperature or increasing density has been the subject of intense research
investigations for several decades now Berthier and Biroli (2011). One
interesting feature that emerges in liquids close to the glass transition is
dynamical heterogeneity, where over time scales corresponding to the
$\alpha$-relaxation time, some regions of the systems have significantly
higher mobility than others. A growing body of literature now supports the
existence of a correlation between this heterogeneity and the local structure
Royall and Williams (2015); Schoenholz  _et al._ (2016); Tanaka  _et al._
(2019); Boattini  _et al._ (2020); Paret  _et al._ (2020). One of the
pioneering ideas in analysing the local structure on glasses was proposed by
Charles Frank in 1952 indicating the possible prevalence of icosahedral
clusters in the glassy regime Frank (1952). Later, many studies corroborated
the presence of locally favored structures in the glassy regime related to the
slowdown of dynamics, with icosahedral clusters being the focus of attention
in several studies Jónsson and Andersen (1988); Kondo and Tsumuraya (1991);
Steinhardt  _et al._ (1981); Jónsson and Andersen (1988); Hirata  _et al._
(2013); Ding  _et al._ (2014); Marín-Aguilar  _et al._ (2019, 2021). Along
similar lines, there have been proposals of simpler tetrahedron-based order
parameters capable of capturing the changes in local structure and
dynamicsTong and Tanaka (2018); Xia  _et al._ (2015); Anikeenko and Medvedev
(2007); Charbonneau  _et al._ (2012); Hirata  _et al._ (2013); Marín-Aguilar
_et al._ (2020). In particular, a recent development was the introduction of
the concept of tetrahedrality of the local structure $n_{\mathrm{tet}}$, which
measures the number of tetrahedral clusters each particle is involved in
Marín-Aguilar  _et al._ (2020), based on the notion that, at least for some
glass formers, most of the locally favoured structures (including icosahedral)
can be decomposed into tetrahedra. It was found that the particles with higher
values of $n_{\mathrm{tet}}$ are strongly correlated with slower dynamics.
These studies are based strictly on local structures whose ability to fully
predict glassy dynamics has been questioned Berthier and Jack (2007). In
general, while a full understanding of the dynamics may require information
beyond the static local environment, local structures represent an important
starting point as testified by the success of machine learning based
approaches that can be trained to predict the local mobility of different
regions in a glassy fluid Schoenholz  _et al._ (2016); Bapst  _et al._ (2020);
Boattini  _et al._ (2021). Moreover, the relevance of icosahedral and
polytetrahedral local structure has been recently highlighted by novel
unsupervised machine learning approaches, such as community inference Paret
_et al._ (2020) and neural-network-based auto-encodersBoattini  _et al._
(2020).
Apart from the heterogeneity in supercooled liquids, local structure can play
an important role in the plastic rearrangements occurring in a sheared solid.
When shearing a crystalline solid, rearrangement events are normally expected
to occur in the vicinity of crystal defects. This idea can be extended to
sheared glasses, where certain spots in the system may be “weaker” than
others, and hence more prone to plastic rearrangement. These “soft
spots”Widmer-Cooper  _et al._ (2008); Chen  _et al._ (2011); Manning and Liu
(2011) or “shear transformation zones”Falk and Langer (1998); Argon (1979) can
be seen as analogues of crystal defects in glassy materials. In recent years,
a variety of approaches have been explored to predict such “soft spots”,
including e.g. machine learning Cubuk  _et al._ (2015), local vibrational
modes Manning and Liu (2011), local yield stress Patinet  _et al._ (2016). A
summary of the efficiency of the different methods is reported in a recent
paper by Richard et al. Richard  _et al._ (2020), in which the authors compare
the performance of a variety of methods to predict plasticity due to uniform
shear deformation. For a $2d$ glass former, they demonstrated that
rearrangements are indeed deeply encoded in the structure Richard  _et al._
(2020); Bhaumik  _et al._ (2021). At the experimental level, soft colloidal
glasses under thermal cycling display high correlation between local
rearrangements and local two-body entropyYang  _et al._ (2016).
In the aforementioned works, the role of local structure in amorphous systems
undergoing cyclic mechanical deformation has not been studied in detail.
Cyclic shear deformation is a commonly used technique to test mechanical
properties of materialsRogers  _et al._ (2011); Koumakis  _et al._ (2013);
Gibaud  _et al._ (2016), memory effects Keim and Nagel (2011); Fiocco  _et
al._ (2014), self organizationRoyer and Chaikin (2015), and annealing of glass
Das  _et al._ (2018). When a glass is deformed in the quasi-static limit at
zero temperature Maloney and Lemaître (2006), compared to uniform shearing,
the cyclic shear displays a sharp yielding transition from an absorbing to a
diffusive state at a critical strain amplitude $\gamma_{y}$ Leishangthem  _et
al._ (2017). Below this critical threshold
$\mathrm{\gamma}_{\mathrm{max}}<\gamma_{y}$, after several training cycles of
deformation, the system reaches a steady state. In this state, the plastic
rearrangements during a deformation cycle are fully reversibleFiocco  _et al._
(2014); Adhikari and Sastry (2018); Priezjev (2016a) and localized. Above
yielding, rearrangements are connected to large displacements and are
irreversible. As a consequence plastic events become spatially organized,
leading to the appearance of a shear band Parmar  _et al._ (2019); Vasisht and
Del Gado (2020).
In this paper we use a binary repulsive $3d$ glass former to explore the
structural properties of particles undergoing plastic rearrangements due to
cyclic shear deformation. We analyze how the local structural order of the
system is correlated with these rearrangements. In particular, we compute the
per-particle two-body excess entropy $S_{2}$ and tetrahedrality
$n_{\mathrm{tet}}$, the latter not having been explored yet in the context of
plastic rearrangements. Our analysis focuses on the steady state and we look
for structural differences between the local environments for subsets of
particles that exhibit the largest and the smallest displacements in
successive cycles.
The paper is organized as follows: In section II, we discuss the glass former
model in detail and we describe the simulation methods. In section III we
introduce the structural descriptors. Thereafter, in section IV, we present
our results in the following order. We first report (i) the variation of mean
local ordering in liquid and inherent structures (IS) using the
$\left<n_{\mathrm{tet}}\right>$ and $\left<S_{\mathrm{2}}\right>$. (ii) We
correlate the liquid structures and the corresponding IS at different
temperatures. We then sample a high and a low temperature glass and shear them
athermally for many deformation cycles with strain amplitudes
$\mathrm{\gamma}_{\mathrm{max}}$. (iii) In the steady state we compute the
mean values of our structural order parameters as
$\mathrm{\gamma}_{\mathrm{max}}$ varies in a range across the yielding
amplitude $\gamma_{y}$. (iv) From one cycle to the next we classify mobile and
immobile particles with larger and smaller rearrangements respectively by
computing the non-affine displacements $D^{2}_{min}$ Falk and Langer (1998).
(v) In terms of $n_{\mathrm{tet}}$ and $S_{2}$ we examine whether there is a
difference between the mean local order of the mobile and immobile particles.
(vi) Later, we use the topological cluster classification algorithm (TCC)
Malins  _et al._ (2013) to point out the different cluster association between
the two classes. (vii) Finally, we explore the structure inside and outside
the shear band for the sheared glasses above the yielding transition.
Our results show that there is a clear structural difference in local
arrangements of the particles involved in the largest and smallest plastic
rearrangements. Similarly, above yielding, we find that particles inside and
outside the shear band have strongly different local environments.

Figure 1:  Schematic representation of an Athermal Quasi Static (AQS)
deformation move. In the top row we present a cartoon of the configuration and
in the bottom one a simplified potential energy landscape. An initial
configuration in its energy minima (left column) is deformed with small strain
steps (central column). In this specific case, the system is not at a minimum
anymore and the following minimization leads it into a new minima that
corresponds to a new configuration (right column).

##  II Methods

###  II.1 Glass-former model and Simulation

As a model glass system, we explore the behavior of purely repulsive Wahnström
(WH) model Wahnström (1991). It consists of a $50:50$ mixture of particles
with additive diameters interacting through a Lennard-Jones (LJ) potential,

$V_{\alpha\beta}(r)=\begin{cases}4\epsilon_{\alpha\beta}\left[\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{12}-\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{6}\right]-V_{r_{c}},&r<r_{c}{}{}{}\\\
0,&r\geq r_{c}\end{cases}$ (1)

where $\alpha$ and $\beta$ denote the type of particle (A or B), the cut-off
distance is defined as $r_{c}=2^{\frac{1}{6}}\sigma_{\alpha\beta}$ and
$V_{r_{c}}$ is the value of the LJ potential evaluated at $r_{c}$. With this
choice of the cut-off, only the repulsive part of the potential is retained.
The potential parameters are defined with respect to type $A$:
$\sigma_{BB}/\sigma_{AA}=1.2$, $\sigma_{AB}/\sigma_{AA}=1.1$ and
$\epsilon_{AA}=\epsilon_{BB}=\epsilon_{AB}$. We fix the total packing fraction
to $\phi=\frac{\pi}{6V}(N_{A}\sigma_{AA}^{3}+N_{B}\sigma_{BB}^{3})=0.58$,
where $V$ is the simulation box volume and the total number of particles
$N=N_{A}+N_{B}=64000$.

We perform molecular dynamics simulations using LAMMPS Plimpton (1993) with a
fixed time step size of $dt=0.005\tau$, where
$\tau=\sqrt{m\sigma_{AA}^{2}/\epsilon_{AA}}$ is the time unit, with $m$ the
particle mass. We use reduced LJ units where $\sigma_{AA}$ is the unit for
distance and $\epsilon_{AA}$ the energy unit. We prepare a set of equilibrated
configurations at temperature $T$ by running $NVT$ molecular dynamics for
$2\times 10^{5}$ (high $T$) - $2\times 10^{6}$ (low $T$) time steps followed
by constant energy ($NVE$) relaxation. From these equilibrated samples, we
obtain the corresponding inherent structures (IS) by minimizing the potential
energy, using conjugate gradient minimization, with a tolerance $10^{-16}$.
The IS configurations are the "glass" configurations which we subject to shear
deformation, which we associate with the temperature of the liquid simulations
from which they are generated. The range of $T\in[0.7,2.0]$ for this WH system
is sufficient to show the dynamical slowdown from high to low temperature as
expected in a typical glass-former.

Next, to study the plastic re-arrangements in glasses, we focus on one high
temperature ($T=1.5$) and one low temperature ($T=0.7$) glass, both quenched
to the IS and sheared using the athermal quasi static shear (AQS) protocol
Maloney and Lemaitre (2004); Maloney and Lemaître (2006). In AQS, the $xz$
plane of a chosen IS configuration is sheared in small strain steps
$d\gamma=0.0002$ in one direction and each deformation step is followed by an
energy minimization. A schematic representation of an AQS step is presented in
Fig. 1. This is continued until the strain values reach a specified amplitude
$\gamma_{max}$ in either direction, at which the strain direction is reversed.
Therefore, one complete cycle of deformation has the following sequence:

$\displaystyle\gamma=0\rightarrow\gamma_{max}\rightarrow
0\rightarrow-\gamma_{max}\rightarrow 0.$ (2)

Hence, each cycle consists of $4*\mathrm{\gamma}_{\mathrm{max}}/d\gamma$
steps.

For each value of the parameter $\mathrm{\gamma}_{\mathrm{max}}$, we deform
the system for $100-600$ cycles until a steady state is reached. The energy of
stroboscopic configurations, sampled at the end of each cycle with $\gamma=0$,
will depend on $\mathrm{\gamma}_{\mathrm{max}}$. At
$\mathrm{\gamma}_{\mathrm{max}}$ below the yielding transition,
$\mathrm{\gamma}_{\mathrm{max}}<\gamma_{y}$ where $\gamma_{y}$ is the yielding
amplitud of the system, the energies attain a constant value in the steady
state. For ($\mathrm{\gamma}_{\mathrm{max}}>\gamma_{y}$), the energies
fluctuate around a mean value in the steady state. In our WH system,
$\gamma_{y}\approx 0.06$. For higher amplitudes than $\gamma_{y}$ the system
yields, becoming diffusive.

###  II.2 Local structure observables

####  II.2.1 Local entropy

To characterize local structure at the particle level, it is useful to first
consider a structural descriptor that is known to correlate with dynamical
slowdown below the onset temperature. For this purpose, we consider the local
entropy $S_{2}$, defined in terms of pairwise structural correlations. In
general, the entropy $S$ can be expanded in terms of multi-particle
correlation functions as $S=S_{1}+S_{2}+S_{3}+..$, where $S_{1}$ is the ideal
gas value, $S_{2}$ is the two-body excess entropy which can be calculated in
terms of the pair distribution function, $S_{3}$ involves three-body
correlations, etc. Wallace (1987). The two-body term $S_{2}$Dyre (2018);
Rosenfeld (1977, 1999) has been widely used in the context of transport
coefficients and plasticity. In general, it measures the loss of entropy due
to positional correlations with a lower value of $S_{2}$ corresponding to a
more ordered structure. Even though $S_{2}$ has been introduced long ago
Nettleton and Green (1958); Baranyai and Evans (1989), it is a useful concept
that continues to be widely used Truskett  _et al._ (2000); Tanaka  _et al._
(2010); Anikeenko and Medvedev (2007); Hallett  _et al._ (2018). As a matter
of fact, recently, scaling relations have been reported between $S_{2}$ and
diffusivity in glassy systems Bell  _et al._ (2020) and relaxation rates in
cyclic sheared systems Galloway  _et al._ (2020). In order to look at this
quantity on a single-particle level, the two-body entropy of particle $i$ can
be defined as:

$\displaystyle S^{i}_{2}=-2\pi\rho
k_{B}\int_{0}^{\infty}[g_{m}^{i}(r)\log\left(g_{m}^{i}(r)\right)-g_{m}^{i}(r)+1]r^{2}dr,$
(3)

where $g_{m}^{i}(r)$ is the mollified radial distribution function Piaggi and
Parrinello (2017); Hallett  _et al._ (2018),

$\displaystyle g^{i}_{m}(r)=\frac{1}{4\pi N\rho r^{2}}\sum_{i\neq
j}\frac{1}{2\pi\sigma^{2}}\exp{[-(r-r_{ij})^{2}/(2\sigma^{2})]}.$ (4)

with $r_{ij}$ the distance between the $i^{th}$ and $j^{th}$ particle and
$\sigma$ a parameter setting the width of the Gaussian kernel. This per-
particle radial distribution function is well behaved thanks to the broadening
of the neighbor position from a delta peak to a narrow Gaussian. The parameter
$\sigma$ is chosen such that the global average of $g_{m}^{i}(r)$ is close to
the true radial distribution function $g(r)$ and we have a smooth integral to
compute $S_{2}$. For our WH system, we choose $\sigma=0.09$ and for each
particle the integration in Eq. 3 is taken from zero to $r_{max}=5.0$. With
this approach we obtain the per particle excess entropy that we call simply
$S_{2}$ in the rest of the paper. When averaged over all particles in the
system, and over different configurations, we obtain the average two-body
entropy $\langle S_{2}\rangle$.

By its nature, $S_{2}^{i}$ provides a measure for the local radial ordering of
neighbors around particle $i$: many neighbors at the same absolute distance
$r$, as one might find in highly symmetric local environments, will result in
a low value of $S_{2}^{i}$. Hence, a more negative value of $S_{2}$ indicates
higher local order.

####  II.2.2 Tetrahedrality

The second descriptor we characterize is the tetrahedrality of the local
structure, $n_{\mathrm{tet}}$, which measures, for each particle, the number
of tetrahedra it is involved in Marín-Aguilar  _et al._ (2020). As some of the
clusters that have been found in supercooled liquids, such as the icosahedral
cluster, can be decomposed in a large number of tetrahedron, the
tetrahedrality is a good descriptor of the local environment of a particle
that correlates strongly with glassy dynamics in hard-sphere-like systems
Marín-Aguilar  _et al._ (2020); Boattini  _et al._ (2020).

In order to quantify the tetrahedrality, we first identify the neighbors of
the particles through a modified Voronoi construction identical to the one
done in the Topological Cluster Classification algorithm Malins  _et al._
(2013). We define a tetrahedral cluster as a cluster formed by four particles
which are all considered nearest neighbors of each other according to the
Voronoi construction. For an individual particle, $n_{\mathrm{tet}}$ is than
simply the number of distinct tetrahedral clusters that include this particle.
Finally, we can also characterize the global structure of a system through the
average of the per-particle tetrahedrality $\left<n_{\mathrm{tet}}\right>$
Marín-Aguilar  _et al._ (2020). A higher value of $n_{\mathrm{tet}}$ generally
is related to more compact and long-lived clusters such as the icosahedral
cluster.

##  III Results

###  III.1 Liquid and Inherent Structures

We begin our study by exploring the behavior of both
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ for both
liquid and inherent structures at different temperatures. In Fig. 2, we show
the dependence of both structural quantities for temperatures between
$T=0.7-2.0$. In the liquid, upon lowering the temperature, the average local
order increases as it can be deduced by the increase in the value of
$\left<n_{\mathrm{tet}}\right>$ and decrease of $\left<S_{\mathrm{2}}\right>$.
As expected, for all temperatures, the inherent structure presents a higher
degree of ordering that translates into a larger
$\left<n_{\mathrm{tet}}\right>$ and smaller $\left<S_{\mathrm{2}}\right>$ in
comparison to the liquid. However, for the IS, these quantities become
essentially independent of $T$ at sufficiently high temperatures. This trend
is consistent with the previous observation of an increase of the number of
locally favored structures (LFS) Hallett  _et al._ (2018) when the dynamical
slowdown start occurring and the aforementioned temperature can be identified
as the onset temperature at which the dynamics start to be dominated by the
potential energy landscape Sastry  _et al._ (1998). Above this temperature,
the dynamics are typical of a high-temperature liquid and only marginally
influenced by the energy landscape. As a consequence, in this regime, the
system samples several independent minima of the IS and
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ are
insensitive to temperature variations. This means that the system explores a
large range of energy basins with no specific bias toward a particular
energetic level. Below the onset temperature, the system starts to explore
preferentially basins with a low energy that decrease with temperature.

Figure 2: a) Two-body entropy $\left<S_{\mathrm{2}}\right>$ as a function of
temperature $T$ for liquid and inherent structure. b) Tetrahedrality
$\left<n_{\mathrm{tet}}\right>$ as a function of $T$ for liquid and inherent
structure. Figure 3: Density plot between inherent structure and liquid of a)
$S_{2}$ and b) $n_{\mathrm{tet}}$ at a temperature $T=0.7$. c) Spearman’s
correlation between the structural descriptors of liquid and inherent
structures as a function of temperature $T$.

In the low temperature regime, the dynamics of the liquid start to be
dominated by the landscape and the system tends to spend increasing time
within individual basins corresponding to the IS. Consequently, an increased
correlation may be expected between structural parameters for the liquid and
the corresponding IS. To verify this, we inspect the relation between the
values of our observables before and after quenching. In Fig. 3 a) and b) we
show the $2d$ density plot of the $S_{2}$ and $n_{\mathrm{tet}}$ values at
temperature $T=0.7$. We can observe a clear correlation between the
observables in the liquid state and the IS. Note that in Fig. 3 b) there is an
increase of correlation at a value of $n_{\mathrm{tet}}=20$, corresponding to
particles located in the center of icosahedral clusters. Hence, we can deduce
that some specific structural motifs remain unchanged when liquid
configurations are mapped to the IS.
Clearly, this correlation might be temperature dependent. In order to have a
broader picture of the correlation between the two sets of configurations
(liquid and IS) at different temperatures, we calculate Spearman’s rank order
correlation Spearman (1987) between the values of $S_{2}$ and
$n_{\mathrm{tet}}$ per particles in the liquid and the values in the
corresponding IS. The results presented in Fig. 3 c) show that the correlation
is stronger at lower temperatures, reaching a correlation of approximately 0.6
at $T=0.7$ for both $S_{2}$ and $n_{\mathrm{tet}}$, demonstrating that the
two-body entropy and tetrahedrality of the inherent structures are largely
governed by the structure of the unquenched liquid as we observed in Fig. 3 a)
and b). As expected, above the onset temperature, the correlation becomes
negligible. This indicates that the local structure of a high temperature
liquid is mostly insensitive to the underlying IS.

###  III.2 Deformed glassy structures

Now that we have characterized the static structure of our system using these
descriptors, we can ask ourselves how these quantities evolve under cyclical
shear deformation. To this end, we take the IS from the equilibrated
configurations at a low and a high temperature of $T=0.7$ and $T=1.5$ and
cyclically shear them with a range of strain amplitudes $\gamma_{max}$. In
particular, we distinguish two regimes of interest:
I. Below yielding ($\gamma_{max}<\gamma_{y}$): For small amplitudes of shear
below yielding, the systems can anneal and, in the steady state, the system
can visit lower energy minimum basins. In our system from $\gamma_{max}=0.02$
to $\gamma_{max}=0.06$, we have indeed more annealed absorbing states: the
steady-state energy is lower than the one of the initial configuration, in
agreement with what has been observed in the literature Leishangthem  _et al._
(2017); Bhaumik  _et al._ (2019) . If the initial configuration corresponds to
a high-temperature liquid, the annealing effect is more pronounced in
comparison to low-temperature initial configurations Bhaumik  _et al._ (2019).
In the steady state, the local plastic rearrangements during a cycle of
deformation repeat from one cycle to the next Fiocco  _et al._ (2014).
Therefore, from one cycle to another, there is no appreciable net displacement
when considering the configurations stroboscopically.
II. Above yielding ($\gamma_{max}>\gamma_{y}$): In this case, the system
becomes diffusive. In the steady state, the plastic rearrangements during a
cycle lead to a net displacement of particles. For a large enough system, like
in our case, $N=64000$, it is possible to observe the formation of shear bands
of highly diffusive particles Fiocco  _et al._ (2013); Parmar  _et al._
(2019); Priezjev (2020). In the case of $XZ$ shearing, the shear band consists
of a slab of particles parallel either to the $YZ$ or the $XY$ plane. These
particles on average present net displacements, between two consecutive
cycles, larger than the rest of the particles. Interestingly, the region
inside the shear band is characterized by a lower local density and a higher
local energy Parmar  _et al._ (2019).

Figure 4:  Behavior of a) $\left<S_{\mathrm{2}}\right>$ and b)
$\left<n_{\mathrm{tet}}\right>$ in the steady state as a function of
$\gamma_{max}$, for cyclically sheared inherent states of equilibrium
configurations obtained at $T=0.7$ (blue) and $T=1.5$ (red). The vertical
dashed line shows the yielding amplitude $\gamma_{y}$.

Now that we have established the macroscopic effects of shear on our system,
we examine the impact of the shearing amplitude on the local structure. To
this end, we first we compute $\left<S_{\mathrm{2}}\right>$ and
$\left<n_{\mathrm{tet}}\right>$ in the steady state for strain amplitudes
$\gamma_{max}\in[0.02,0.09]$, i.e. both below and above $\gamma_{y}\sim 0.06$.
In Fig. 4, we report the behaviour of the mean value of these local
descriptors as a function of $\gamma_{max}$, for temperatures $T=1.5$ and
$T=0.7$. Note that $\gamma_{max}=0.0$ is actually the value for the initial IS
configuration. The high temperature glass in the range
$\gamma_{max}\in[0.0,0.06]$ displays noticeable annealing and, as a
consequence, an increase in local order reflected by a lower
$\left<S_{\mathrm{2}}\right>$ and a higher $\left<n_{\mathrm{tet}}\right>$. As
we cross $\gamma_{y}$, the diffusive state is reached, and the average local
order decreases significantly. This behaviour implies that remarkably, at
yielding, a maximum local order is attained. Moreover, the transition across
$\gamma_{y}$ manifests in a cusp-like variation for both quantities. For
$T=0.7$, no strong annealing is observed below yielding and
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ remain
almost constant. Upon crossing $\gamma_{y}$, the two observables display a
discontinuous jump. Above yielding, the structural observables are the same
for both temperatures, as has previously been observed for the energy Fiocco
_et al._ (2013). The cusp for the high temperature and the jump for the low
one is compatible with the existence of a critical temperature that divides
two different regimes or yielding behaviour. Bhaumik  _et al._ (2019).

To examine the effects of these changes in local structure on the dynamics of
individual particles, we need to quantify how much a given particle moves
during a displacement cycle. When a system is sheared, the motion of the
particles can be separated into affine and non-affine displacements Priezjev
(2016b). One way of detecting and characterizing the local re-arrangements
that the particles undergo during the cyclic deformations is through their
non-affine displacement $D^{2}_{i}$ as introduced in Ref. Falk and Langer,
1998. We employ a modified version of $D^{2}_{i}$ where we calculate the non-
affine displacement as a function of the accumulated deformation $\gamma$
instead of time:

$\displaystyle
D^{2}_{i}(\gamma)=\frac{1}{N_{i}}\sum_{j=0}^{N_{i}}[(\mathbf{r}_{j}(\gamma)-\mathbf{r}_{i}(\gamma))-\Gamma(\mathbf{r}_{j}(0)-\mathbf{r}_{i}(0))]^{2},$
(5)

where $\mathbf{r}_{i}(0)$ is the position of particle $i$ at the beginning of
a deformation cycle and $\mathbf{r}_{i}(\gamma)$ is its position in the
deformed box at a total accumulated deformation of $\gamma$. The sum in
equation 5 is over the $N_{i}$ nearest neighbors of particle $i$, taken to be
the neighbors within a cut off $r_{cut}=1.4\sigma_{ij}$, which corresponds to
the first minimum of $g(r)$ of the full system. The matrix $\Gamma$ is such
that it minimizes the non-affine displacement giving a measure of local strain
Falk and Langer (1998). This definition best maps deviation from affine
displacement at a local level Priezjev (2016b). In the case of a cyclic
deformation below yielding, the system returns back to the same configuration
each cycle once the steady state has been reached. Consequently, if we
consider configurations at the beginning ($\gamma=0$) and at the end (after
$4\gamma_{max}/d\gamma$ deformation steps) of the cycle, we would observe
$D^{2}_{min}(4\gamma_{max})\approx 0$. However, we are interested in
identifying which particles went through maximum re-arrangements during the
deformation, regardless of reversibility. Therefore, we recorded the maximum
value $\max(D^{2}_{min})$ for each particle in a cycle of deformation.

Figure 5: Distribution of $\max(D^{2}_{min})$ in the steady state at shear
amplitude a) below yielding at $\gamma_{max}=0.06<\gamma_{y}$ and b) above
yielding at $\gamma_{max}=0.08>\gamma_{y}$. Above yielding we have longer
tail. In both cases, the temperature of the initial configuration was $T=0.7$.

In Fig. 5, we plot the distribution of $\max(D^{2}_{min})$ for two strain
amplitudes below and above yielding. It is evident that, above yielding, a
long tail is observed in the distribution. This tail is due to the particles
in the shear band that present the maximum degree of plastic rearrangement.

The calculation of $\max(D^{2}_{min})$ enables us to identify particles with
high displacements, corresponding to those undergoing plastic rearrangements.
At this stage it is natural to ask, is there any structural difference between
the particles that undergo strong rearrangements and the particles that are
immobile? In other words, at the end of a cycle, are particles with high and
low $\max(D^{2}_{min})$ values characterized by a different structures at the
beginning of a cycle? These are the questions that we will address in the
remaining part of the paper.

Figure 6: Distributions of a) $S_{2}$ and b) $n_{tet}$ for unsheared
configurations in the steady state at temperature $T=0.7$ and shear amplitude
$\gamma_{max}=0.06$. The black line corresponding to the overall distribution.
Distributions are shown for immobile (blue) and mobile (red) particles,
corresponding to the lowest and highest 5% of displacements
$\max(D^{2}_{min})$ during a shear cycle, respectively.

To answer these questions, we turn our attention again to the structural local
descriptors $S_{2}$ and $n_{tet}$. In the steady state, we compute both
descriptors for each particle in the stroboscopic configurations at a given
strain amplitude. Subsequently, we subject the system to a complete
deformation cycle during which we compute the values of $\max(D^{2}_{min})$
for each particle. As in previous studies on dynamical heterogeneities Kob
_et al._ (1997), we define a threshold based on the displacement of the
individual particles that allows us to classify the particles as immobile or
mobile if their $\max(D^{2}_{min})$ values are among the smallest or the
largest respectively. In particular, we choose the top $5\%$ as mobile
particles and bottom $5\%$ for the immobile ones. The resulting distributions
are shown in Fig. 6. Here, we present the case of $T=0.7$ and
$\gamma_{max}=0.06$. However, the trend is similar in all the state points we
investigated. From $P(S_{2})$, we see that immobile particles have
significantly lower $\left<S_{\mathrm{2}}\right>$ implying a stronger local
order in comparison to the mobile particles. This trend is confirmed by the
behaviour of $P(n_{tet})$ which shows that immobile particles are much more
likely to be found in highly tetrahedral environments. Interestingly, for the
immobile particles, we observe a strong peak at $n_{tet}=18$ and $n_{tet}=20$,
the first corresponding to clusters formed by rings of 5 particles leading to
high tetrahedrality and the second to particles at the center of an
icosahedral cluster. This peak is essentially absent in the mobile particles,
indicating that icosahedral caging has a strong effect on the local
rearrangements in this system. The appearance of this cluster has been
investigated on many occasions in supercooled liquid systems as they represent
the fingerprint of geometrical frustration introduced by FrankFrank (1952) and
have been found to be strongly linked to slow dynamics in a wide variety of
systems Jónsson and Andersen (1988); Kondo and Tsumuraya (1991); Steinhardt
_et al._ (1981); Jónsson and Andersen (1988); Leocmach and Tanaka (2012);
Hirata  _et al._ (2013); Ding  _et al._ (2014); Marín-Aguilar  _et al._
(2019); Vasisht  _et al._ (2020).

Figure 7: Behavior of a) $\left<S_{\mathrm{2}}\right>$ b) $n_{\mathrm{tet}}$
as a function of $\gamma_{max}$ in the steady state, for the $5\%$ most mobile
(red circle) and $5\%$ most immobile particles (blue squares). Filled symbols
correspond to $T=1.5$ and open symbols to $T=0.7$.

We now turn our attention to the average structural characteristics of the
mobile and immobile particles. In Fig. 7, we plot
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ for immobile
and mobile particles as a function of strain amplitude $\gamma_{max}$ and for
both high ($T=1.5$) and low ($T=0.7$) temperatures. For the immobile
particles, below yielding, $\left<n_{\mathrm{tet}}\right>$ increases and
$\left<S_{\mathrm{2}}\right>$ decreases. These trends are more pronounced for
$T=1.5$ and can be ascribed to the stronger tendency to anneal and the
consequent increase of local order, as we observed in Fig. 4. Above yielding,
the immobile particles are present outside the shear band. We find that upon
crossing the yielding amplitude ($\gamma_{max}>0.06$), both
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ flattens out
for the high temperature case. For the mobile particles, as expected, the
values of $\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$
are consistent with a less structured local structure. However, we notice that
their structural features change very little between the two temperatures. In
fact, both $\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$
remain almost constant below and above yielding. Although the origin of this
effect is not clear, it suggests that the most mobile particles are
characterized by a similar local structure, which is independent of
temperature, shear amplitude, and even the presence of a shear band. Equally
interestingly, even though they do not participate in the shear bands in which
strain is localized, the structure around immobile particles is significantly
affected by shear band formation.
So far, we have established that the particles characterised by different
propensity to re-arrange are embedded, on average, in different local
environment as reflected by the local descriptors considered here. However, we
have not discussed the actual geometrical shape of the local cluster around a
particle and how it is connected with plastic rearrangement. To this end, we
classify the local arrangement of particles by using the Topological Cluster
Classification algorithm (TCC)Malins  _et al._ (2013). This algorithm detects
predefined specific local clusters of different sizes. The detection is based
on the connections between nearest neighbors, found via a modified Voronoi
construction. As a first step, simple small clusters from 3 up to 7 particles
are identified. Subsequently, more complex clusters are found as combinations
of the simple ones. Note that each particle can simultaneously be part of
multiple clusters. For the $5\%$ most mobile and immobile particles we examine
all possible clusters detected by TCC and we calculate the fraction of
particles involved in a certain type cluster for each case. In Fig. 9 a) we
show the results for mobile (red) and immobile (blue) particles for the case
of $T=0.7$ and $\gamma_{max}=0.06$. Here, we report only the cluster types
that show significant differences between the two communities. In particular,
we highlight the $13A$ cluster corresponding to an icosahedral cluster for
which the largest difference between mobile and immobile particles is noticed.
Similarly, another cluster that presents large differences, but less
pronounced than the $13A$ case, is the $10B$ cluster which corresponds to a
defective icosahedral cluster. It appears that these two clusters play an
important role in determining whether a given particle will rearrange during a
shear cycle. Note that the two of them in particular can be decomposed in
several tetrahedral clusters characterised a particularly high value of
$n_{\mathrm{tet}}$ value. For example, for a particle located in the center of
an icosahedral cluster, its $n_{\mathrm{tet}}=20$ as each of the faces of the
icosahedral form a tetrahedral cluster with the central particle. In the
following, we will focus mainly on these two clusters.
In particular, in Fig. 8(b-e), we show the fraction of particles involved in
the $10B$ and $13A$ cluster for each of the communities as a function of
$\gamma_{max}$ as they present the largest difference. For the immobile
particles at $T=0.7$, the population of $10B$ clusters is essentially
constant, while the icosahedral cluster displays a noticeable maximum at the
yielding transition. For $T=1.5$, below yielding, there is a more pronounced
effect as the fraction of immobile particles involved into these clusters
grows with the amplitude $\gamma_{max}$. Above yielding, this effect flattens
out as no noticeable increase is observed. For mobile particles at both
temperatures, it appears that the variation in the population of the two
clusters is negligible. Similar results are found for other TCC clusters that
differ in prevalence between the immobile and mobile particles, such as the
ones shown in Fig. 9 a). Essentially, there are no appreciable changes in the
structure of the mobile particles with different amplitudes. In contrast, the
immobile ones show significantly different behavior across the yielding
transition.

These trends are consistent with the variation of
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ in Fig. 7
and seems to confirm that the local environment in which the particles are
embedded plays an important role. In particular, it is interesting to note
that despite the immobile particles displaying a limited activity in terms of
plastic rearrangements, there is a more remarkable variation of their local
structure upon the change of amplitude of the deformation.

From Fig. 6 b) we see that a large fraction of $5\%$ most immobile particles
are characterized by $n_{tet}=20$, which is the number of tetrahedra that one
would expect in an icosahedral cluster. We confirm this idea by inspecting
Fig. 8 c) and e), where we notice that the $5\%$ most mobile and $5\%$ least
mobile particles on average show significant difference regarding to the
icosahedral ordering.

Figure 8: TCC analysis for mobile (red) and immobile (blue) particles. In a)
for $T=0.7$ and $\gamma_{max}=0.06$ we report the fraction of these mobile and
immobile particles in a particular type of cluster, where $13A$ is the
icosahedral cluster. Next we plot these fractions against $\gamma_{max}$ for
two particular classes b) $10B$ (defective icosahedral cluster) and c) $13A$
(icosahedral cluster) at a temperature of $T=0.7$ and the corresponding d)
$10B$ and e) $13A$ at $T=1.5$.

###  III.3 Glassy structure and the shear band

As discussed in the introduction, above yielding, the dynamics become
heterogeneous and localized in space and, as a consequence, shear band
formation is expected. There are numerous hints that the properties of the
structure inside the shear band is different from what is observed outside. It
has been observed, for a similar model to the one discussed here, that the
density inside the shear band is lower Parmar  _et al._ (2019) and that it may
present a certain degree of hyperuniformity Mitra  _et al._ (2021). The
question we want to answer next is whether its structure is different and if
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ capture the
differences inside and outside the shear band.

To detect the shear band, we bin the system along the axis perpendicular to
the plane of the shear band. In this case this corresponds to the $z$-axis.
Subsequently, we compute the mean square displacement (MSD) of the particles
between the $n^{th}$ and $(n+1)^{th}$ cycle as a function of $z$ Parmar  _et
al._ (2019). The position of the shear band can be clearly identified as the
region where the particles have a larger MSD. The result, for the case of
$T=0.7$, is presented in Fig. 9 a). Along the same axis, we measure
$\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$ and show the
results in Fig. 9 b) and c) respectively. Both these structural features
indicate strong structural differences between the portions of the system
inside and outside of the shear band. On average, outside the shear band we
observe higher structural order.

Figure 9: Shear band captured by the different properties a) mean-squared
displacement, b) $S_{2}$, c) $n_{\mathrm{tet}}$ and d) fraction of particles
involved in icosahedral clusters $N_{Ico}/N$ as a function of the
perpendicular axis to the shear band. All corresponding to sheared system at a
$\gamma_{max}=0.08$ at a temperature $T=0.7$.

In Fig. 9 d), we show the distribution of the icosahedral clusters along the
direction perpendicular to the shear band. We find a clear localization, where
$30\%$ of the particles outside the shear band are involved in an icosahedral
cluster, whereas inside the shear band this fraction is much lower. This
observation suggests that the lower density that has been observed inside the
shear band Parmar  _et al._ (2019) is related to a pronounced structural
signature. In particular, the systems prefers to localize a great number of
icosahedral structures, that are know to be more energetically favourable,
outside the more mobile region. This localization is not found below yielding,
where the icosahedral clusters are homogeneously distributed inside the
simulation box.

##  IV Conclusions

Predicting plastic re-arrangements based on local structures information is a
challenging domain of research (see Ref. Richard  _et al._ , 2020 and
reference therein) that has attracted a lot of interest in recent years Xu
_et al._ (2021); Fan and Ma (2021); Shimada  _et al._ (2021); Kriuchevksyi
_et al._ (2021). In this paper, we have studied the local structural
rearrangements of particles undergoing cyclic shear deformation in a model
glass former focusing on local tetrahedrality $n_{\mathrm{tet}}$ that we
introduced previously Marín-Aguilar  _et al._ (2020) and the well known two-
body excess entropy $S_{2}$. High values of $n_{\mathrm{tet}}$ and low values
of $S_{2}$ indicate a higher degree of local order. In particular, we consider
two temperatures, above and below the onset temperatures, for which different
levels of initial average order is observed. The inherent structure
configurations were sheared following a cyclic athermal quasistatic
deformation protocol that revealed the typical phenomenology of the weakly
(high temperature) and deeply (low temperature) annealed glass Bhaumik  _et
al._ (2019). We then investigated the links between local structure and local
rearrangements at different strain amplitudes $\mathrm{\gamma}_{\mathrm{max}}$
below and above yielding.
Below yielding, the system anneals, resulting in an increase of local
structural order until the system reaches a periodic state and no longer
diffuses. In contrast, above yielding both the low-temperature and high-
temperature glasses become diffusive and exhibit shear banding. Moreover,
above yielding, we found that both systems present on average the same values
of $\left<S_{\mathrm{2}}\right>$ and $\left<n_{\mathrm{tet}}\right>$
irrespective of its preparation history. This is compatible with the
phenomenology observed before in several different systems Fiocco  _et al._
(2013); Regev  _et al._ (2013); Keim and Arratia (2014); Priezjev (2016a);
Leishangthem  _et al._ (2017); Ozawa  _et al._ (2018); Bhaumik  _et al._
(2019).
During the deformation cycle in steady states below yielding, particles
rearrange reversibly, meaning that at the end of the cycle they come back to
their original positions. In contrast, above yielding the rearrangements are
irreversible and particle, inside shear band, display diffusive motion. By
investigating the local structure around the 5% most immobile and most mobile
particles during a given shear cycle, we found that the immobile and mobile
particles have on average different local order at the beginning of the cycle.
In particular, particles with lower $n_{\mathrm{tet}}$ and higher $S_{2}$ are
more likely to rearrange, an observation which holds for all investigated
$\mathrm{\gamma}_{\mathrm{max}}$ and temperatures. Moreover, the two
temperatures explored here correspond to glass that would have a brittle (low
temperature) and ductile (high temperature) mechanical response Ozawa  _et
al._ (2018); Bhaumik  _et al._ (2019) corresponding to different levels of
annealing. Interestingly, for the low temperature, the average local
structures do not display significant evolution below yielding. However, if we
consider only the immobile particles, we observe a clear evolution, for
example, in the number of tetrahedra. This indicates that the main effect of
annealing below yielding is the emergence of a population of particles with
high local order, which are highly stable against further rearrangement.
In line with these results, we closely investigated the structure inside and
outside shear band. We showed that both of our local structure descriptors are
spatially correlated to such shear band. In particular, outside it, the local
structure is more ordered presenting higher values of $n_{\mathrm{tet}}$ and
lower of $S_{2}$. Interestingly, we found that outside the shear band more
than $30\%$ of the particles are involved in icosahedral clusters whereas
inside it this percentage is essentially negligible. In other words, the
population of highly ordered particles that emerges upon annealing persists
even beyond yielding, but remains localized outside the shear band, where they
may have a stabilizing influence on their surroundings. Our results confirm
that, as for the case of continuous shearing Richard  _et al._ (2020), certain
aspects of the response to cyclic shear deformation can be correlated to
structural features.

###### Acknowledgements.

The authors would like to thank Davide Fiocco for the original production of
the potential energy cartoons in Fig. 1. S.M., S.S and G.F. gratefully
acknowledge IFCPAR/CEFIPRA for support through project no. 5704-1. S.M.-A.
acknowledges support from the Consejo Nacional de Ciencia y Tecnología
(CONACyT scholarship No. 340015/471710). SS acknowledges support through the
JC Bose Fellowship (JBR/2020/000015) SERB, DST (India).

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