# Generalized Reed-Muller codes: A new construction of information sets

José Joaquín Bernal  Departmento de Matemáticas
Universidad de Murcia
30100 Murcia, SPAIN
Email<EMAIL_ADDRESS>
###### Abstract

In [2] we show how to construct information sets for Reed-Muller codes only in
terms of their basic parameters. In this work we deal with the corresponding
problem for $q$-ary Generalized Reed-Muller codes of first and second order.
We see that for first-order codes the result for binary Reed-Muller codes is
also valid, while for second-order codes, with $q>2$, we have to manage more
complex defining sets and we show that we get different information sets. We
also present some examples and associated open problems.

##  I Introduction

In 1954, D. E. Muller introduced the family of (binary) Reed-Muller codes [9].
In the same year, I. S. Reed presented a specific decoding algorithm for them
[10]. Later, Kasami, Lin and Peterson generalized this family of codes to
other ground fields [8]. Independently. E. J. Weldon presented the
nonprimitive generalization and the single-variable approach using the
Mattson-Solomon polynomial [11]. The reader may find an extensive explanation
on both generalizations in [4, 6] and [5]. In 1954

In this paper we consider primitive GRM codes seen as affine-invariant codes
in the algebra $\mathbb{F}G$, where $\mathbb{F}$ is the field with $q$
elements and $G$ is the abelian group of the field with $q^{m}$ elements, for
some natural number $m$. Then, as any affine-invariant code, GRM codes have an
associated defining set (see Definition 1 below).

The goal of this paper is to give information sets for first and second order
GRM codes only in terms of their defining sets and ultimately, in terms of
their basic parameters, namely, their order and their length. In [2] we reach
such a description in the case of (binary) Reed-Muller codes; however, for GRM
codes we deal with a more complex structure of their defining sets, specially
in the case of second-order GRM codes.

In Sections II and III we give a basic introduction to abelian codes and we
show the essential tools about cyclic codes seen as two-dimensional cyclic
codes, respectively. Then, in Section III we introduce briefly GRM codes as
affine-invariant codes. In Section V we present our main results, with two
subsections corresponding to first and second-order GRM codes respectively.
Finally, in Section VI we include some examples and in Section VII we give our
conclusions and we mention some related open problems.

##  II Preliminaries

We need to introduce some basic notation and definitions about abelian codes
that will be needed all throughout this paper. For our purposes we only need
to consider two-dimensional abelian codes so we are giving this introduction
just in that case. First, we will always denote by $\mathbb{F}$ the field of
$q$ elements where $q$ is a power of a prime $p$. Then, we denote

$\mathbb{A}(r_{1},r_{2}):=\mathbb{F}[X_{1},X_{2}]/\left\langle
X_{1}^{r_{1}}-1,X_{2}^{r_{2}}-1\right\rangle$

and we consider that any (two-dimensional) abelian code is an ideal in
$\mathbb{A}(r_{1},r_{2})$. We identify the codewords with polynomials
$P(X_{1},X_{2})$ such that every monomial satisfies that the degree of the
indeterminate $X_{i}$ is in ${\mathbb{Z}}_{r_{i}}$. We write $n=r_{1}\cdot
r_{2}$ and we always assume that $r_{i}$ and $q$ are coprime, for $i=1,2$,
that is, we assume that $\mathbb{A}(r_{1},r_{2})$ is a semisimple algebra.

We will base our construction on the study of the so-called defining set of an
abelian code, so let us introduce that definition.

######  Definition 1.

Let ${\mathcal{C}}\leq\mathbb{A}(r_{1},r_{2})$ be an abelian code. Fixed
$\alpha_{i}$ a primitive $r_{i}$-th root of unity in some extension of
$\mathbb{F}$, $i=1,2$, the defining set of ${\mathcal{C}}$ with respect to
$\alpha=\\{\alpha_{1},\alpha_{2}\\}$ is

$\begin{array}[]{rl}D_{\alpha}\left({\mathcal{C}}\right)=&\left\\{(a_{1},a_{2})\in{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}\mid\right.\\\
&\left.P(\alpha_{1}^{a_{1}},\alpha_{l}^{a_{2}})=0\text{ for all
}P\in{\mathcal{C}}\right\\}.\end{array}$

######  Remark 2.

The simply adaptation of the previous definition to the case
$\mathbb{A}(n)=\mathbb{F}[X]/\langle X^{n}-1\rangle$ coincides with the
classical notion of defining set of a cyclic code of length $n$.

It is clear that the defining set of a code depends on the election of the
primitive roots of unity and it can be proven that fixed $\alpha$ any abelian
code is completely determined by its defining set.

In particular, we are working with the description of defining sets as union
of some especial subsets of integers. Let us introduce them.

######  Definition 3.

Let $a$ and $r$ be integers. The $q$-cyclotomic coset of $a$ modulo $r$ is the
set

$C_{q,r}(a)=\left\\{a\cdot q^{i}\mid
i\in{\mathbb{N}}\right\\}\subseteq{\mathbb{Z}}_{r}.$

The following definition gives the generalization of $q$-cyclotomic coset to
the two-dimensional space $\mathbb{A}(r_{1},r_{2})$.

######  Definition 4.

Given $(a_{1},a_{2})\in{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$, its
$q$-orbit modulo $\left(r_{1},r_{2}\right)$ is the set

$Q_{q}(a_{1},a_{2})=\left\\{\left(a_{1}\cdot q^{i},a_{2}\cdot q^{i}\right)\mid
i\in{\mathbb{N}}\right\\}\subseteq{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}.$

It is easy to see that, under our assumption $\gcd(r_{1},q)=\gcd(r_{2},q)=1$,
the $q$-orbits define a partition in
${\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$; moreover, any defining set
is a (disjoint) union of $q$-orbits and conversely, any union of $q$-orbits
define a defining set (with respect to a fixed $\alpha$) and so an abelian
code. Since we will only consider $q$-cyclotomic cosets and $q$-orbits, for
the sake of simplicity, we shall write $C_{r}(a)$ and $Q(a_{1},a_{2})$ without
mention to $q$.

To do the required computations on the elements in
${\mathcal{D}}({\mathcal{C}})$ we will select a subset satisfying the
following definition.

######  Definition 5.

Let $D$ be a union of $q$-orbits modulo $(r_{1},r_{2})$ and
$\overline{D}\subset D$ a complete set of representatives. Then $\overline{D}$
is called a set of restricted representatives if $\overline{D}_{1}$ is a
complete set of representatives of the $q$-cyclotomic cosets modulo $r_{1}$ in
$D_{1}$ ($\overline{D_{1}}$ and $D_{1}$ the projections of $\overline{D}$ and
$D$ onto ${\mathbb{Z}}_{r_{1}}$ respectively).

To finish this section we need to give the formal notion of information set in
our ambient space $\mathbb{A}(r_{1},r_{2})$.

######  Definition 6.

An information set for an abelian code
${\mathcal{C}}\subseteq\mathbb{A}(r_{1},r_{2})$ with dimension $k$ is a set
${\mathcal{I}}\subseteq{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$ such
that $\left|{\mathcal{I}}\right|=k$ and
${\mathcal{C}}_{\mathcal{I}}=\mathbb{F}^{k}$, where
${\mathcal{C}}_{\mathcal{I}}$ denotes the projection of the codewords in
${\mathcal{C}}$ to the positions corresponding with ${\mathcal{I}}$.

The complementary set
$\left({\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}\right)\setminus{\mathcal{I}}$
is called a set of check positions for ${\mathcal{C}}$.

##  III Information sets for cyclic codes seen as two-dimensional abelian
codes

According to the notation that has been introduced in the previous sections we
take a cyclic code, denoted by ${\mathcal{C}}^{*}$, with length $n=r_{1}\cdot
r_{2}$, and assume that $r_{1},r_{2}>1,\gcd(r_{1},r_{2})=1$, $\gcd(r_{i},q)=1$
for $i=1,2$. Then let
$T:{\mathbb{Z}}_{n}\rightarrow{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$
be an isomorphism, and let us denote $T=(T_{1},T_{2})$; that is, $T_{i}(e)$ is
the projection of $T(e)$ onto ${\mathbb{Z}}_{r_{i}}$, for $i=1,2$ and any
$e\in{\mathbb{Z}}_{n}$.

According to Remark 2, let
${\mathcal{D}}^{*}=D_{\alpha}({\mathcal{C}}^{*})\subseteq{\mathbb{Z}}_{n}$ be
the defining set of ${\mathcal{C}}^{*}$ with respect to an arbitrary primitive
$n$-th root of unity $\alpha$. Then, since there exist integers
$\eta_{1},\eta_{2}$ such that $\eta_{1}r_{1}+\eta_{2}r_{2}=1$, we have that
$\alpha_{1}=\alpha^{\eta_{2}r_{2}}$ and $\alpha_{2}=\alpha^{\eta_{1}r_{1}}$
are primitive $r_{1}$-th and $r_{2}$-th roots of unity respectively. Fix $T$
an isomorphism as above and set $T(1)=(\delta_{1},\delta_{2})$; observe that
$\gcd(\delta_{1},r_{1})=1$ and $\gcd(\delta_{2},r_{2})=1$. We define the
abelian code
${\mathcal{C}}={\mathcal{C}}_{({\mathcal{C}}^{*},T)}\leq\mathbb{A}(r_{1},r_{2})$
as the code with defining set
${\mathcal{D}}=D({\mathcal{C}})=T({\mathcal{D}}^{*})$, with respect to
$(\beta_{1},\beta_{2})=(\alpha_{1}^{\delta_{1}^{-1}},\alpha_{2}^{\delta_{2}^{-1}})$.
In this situation, we have that ${\mathcal{C}}$ is the image of
${\mathcal{C}}^{*}$ by the map

$\displaystyle\mathbb{A}(n)$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{A}(r_{1},r_{2})$ $\displaystyle\sum a_{i}X^{i}$
$\displaystyle\hookrightarrow$ $\displaystyle\sum b_{jl}X^{j}Y^{l},$

where $b_{jl}=a_{i}$ if and only if $T(i)=(j,l)$. Therefore, ${\mathcal{I}}$
is an information set for ${\mathcal{C}}$ if and only if
$T^{-1}({\mathcal{I}})$ is an information set for ${\mathcal{C}}^{*}$.
Usually, we omit the reference to the original cyclic code and the isomorphism
$T$ in the notation of the new abelian code, and we will write ${\mathcal{C}}$
instead of ${\mathcal{C}}_{({\mathcal{C}}^{*},T)}$; those references will by
clear by the context.

For the rest of this section we assume that we have fixed a choice of $\alpha$
and $T$ (consequently, the roots $\beta_{1},\beta_{2}$ are also fixed).

######  Definition 7.

Let $\overline{{\mathcal{D}}^{*}}\subseteq{\mathcal{D}}^{*}$ be a complete set
of representatives of the $q$-cyclotomic cosets modulo $n$ in
${\mathcal{D}}^{*}$. Then $\overline{{\mathcal{D}}^{*}}$ is said to be a
suitable set of representatives if $T(\overline{{\mathcal{D}}^{*}})$ is a set
of restricted representatives of the $q$-orbits in $T({\mathcal{D}}^{*})$ (see
Definition 5).

It can be proved that we will always be able to take a suitable set of
representatives in ${\mathcal{D}}^{*}$ (see [2]).

The goal of this section is to give an information set for the abelian code
${\mathcal{C}}\leq\mathbb{A}(r_{1},r_{2})$ just in terms of the defining set
of the original cyclic code, this is established in Theorem 10. To get it we
need to introduce some special subsets in $\overline{{\mathcal{D}}^{*}}$ on
which we will do the computations.

######  Definition 8.

Given $\overline{{\mathcal{D}}^{*}}\subseteq{\mathcal{D}}^{*}$ a suitable set
of representatives, we define $\sim$ the equivalence relation on
$\overline{{\mathcal{D}}^{*}}$ given by the rule

$a\sim b\in\overline{{\mathcal{D}}^{*}}\text{ if and only if }a\equiv b\mod
r_{1}.$ (1)

We denote by ${\mathcal{U}}\subseteq\overline{{\mathcal{D}}^{*}}$ a complete
set of representatives of the equivalence classes given by $\sim$. Then, for
any $u\in{\mathcal{U}}$ we write

$\mathcal{O}(u)=\\{a\in\overline{{\mathcal{D}}^{*}}\mid a\sim u\\}.$

######  Remark 9.

It is easy to see that for any $e\in\overline{{\mathcal{D}}^{*}}$ there exists
a unique $u\in{\mathcal{U}}$ such that $T_{1}(u)=T_{1}(e)$. By abuse of
notation, we will write

$C_{r_{1}}(e)=C_{r_{1}}(T_{1}(e))=C_{r_{1}}(T_{1}(u))=C_{r_{1}}(u).$

Next result gives what we were aiming for in this section.

######  Theorem 10.

In the setting described above, let ${\mathcal{C}}^{*}\leq\mathbb{A}(n)$ be a
cyclic code of length $n$, $\overline{{\mathcal{D}}^{*}}$ a suitable set of
representatives of its defining set, with respect to a given $n$-th root of
unity, and ${\mathcal{U}}$ as in Definition 8. Let
${\mathcal{C}}\leq\mathbb{A}(r_{1},r_{2})$ be the abelian code
${\mathcal{C}}_{({\mathcal{C}}^{*},T)}$. In this situation we consider the
values

$\left\\{M(u)=\frac{1}{|C_{r_{1}}(u)|}\sum_{v\in\mathcal{O}(u)}|C_{n}(v)|\mid
u\in{\mathcal{U}}\right\\}.$

Then we define

$f_{1}=\max\limits_{u\in{\mathcal{U}}}\\{M(u)\\},\qquad
f_{i}=\max\limits_{u\in{\mathcal{U}}}\\{M(u)\mid M(u)<f_{i-1}\\}.$

which yields the sequence $f_{1}>\dots>f_{s}>0=f_{s+1}$. On the other hand,
for any $k=1,\dots,s$ we define

$g_{k}=\sum_{\genfrac{}{}{0.0pt}{2}{u\in{\mathcal{U}}}{M(u)\geq
f_{k}}}|C_{r_{1}}(u)|.$

Then, the set

$\displaystyle\Gamma({\mathcal{C}})=$
$\displaystyle\\{(i_{1},i_{2})\in{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}\mid\text{
there exists }1\leq j\leq s\text{ with }$ $\displaystyle f_{j+1}\leq
i_{2}<f_{j},\text{ and }0\leq i_{1}<g_{j}\\}.$

is a set of check positions for ${\mathcal{C}}$. Furthermore, the set
$T^{-1}\left(\Gamma({\mathcal{C}})\right)$ is a set of check positions for
${\mathcal{C}}^{*}$.

###### Proof.

The reader may check that the proof is constructed by using some results in
[2] and [1]. Furthermore, it can be seen that the proofs do not depend on the
values of $q$.

Specifically, we begin by taking Theorem 20 in [1] which gives us the
description of an information set for an arbitrary abelian code from some
calculations on the cardinalities of the $q$-orbits in its defining set. Then
we use Lemma 18 and Proposition 19 in [1] which relate the cardinalities of
the cyclotomic cosets in the defining set of a cyclic code and the
cardinalities of the $q$-orbits for the corresponding abelian code obtained as
in the beginning of the previous section. Finally, Theorem 20 in [2] complete
the computations we need. ∎

##  IV Generalized Reed-Muller codes

We are seeing GRM codes as affine-invariant codes (and, in particular, as
extended cyclic codes) (see [5]). Let us introduce this family of codes
briefly. The ambient space will be the group algebra $\mathbb{F}G$, where $G$
is the additive group of the field with $q^{m}$ elements, for some
$m\in{\mathbb{N}}$. All throughout we fixed $m$ and we set $n=q^{m}-1$ (this
notation is being consistent with that given in the previous sections).

Since $G$ is an elementary abelian group of order $|G|=q^{m}$ and
$G^{*}=G\setminus\\{0\\}$ is a cyclic group, we may take a generator element
$\langle\alpha\rangle=G^{*}$ (note that $\alpha$ is also a primitive $n$-th
root of unity in the field extension of $\mathbb{F}$ with $q^{m}$ elements).
Then, we write the elements in $\mathbb{F}G$ as

$bX^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{i}}\quad(a_{i},b\in\mathbb{F}).$
(2)

On the other hand, we consider $S_{G}$, the group of automorphisms of $G$,
acting on $\mathbb{F}G$ via

$\tau\left(bX^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{i}}\right)=bX^{\tau(0)}+\sum\limits_{i=0}^{n-1}a_{i}X^{\tau(\alpha^{i})}$

for any $\tau\in S_{G}$. Then, we define the permutation automorphisms group
of a code (ideal) ${\mathcal{C}}\leq\mathbb{F}G$ as

$\rm PAut({\mathcal{C}})=\\{\tau\in
S_{G}\mid\tau({\mathcal{C}})={\mathcal{C}}\\}.$

Now, we can give the definition of affine-invariant code.

######  Definition 11.

Let ${\mathcal{C}}\leq\mathbb{F}G$ be a code:

  1. a)

${\mathcal{C}}$ is an extended cyclic code if for any
$bX^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{i}}\in{\mathcal{C}}$ one has
that $bX^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{(i+1)}}\in{\mathcal{C}}$
and $b+\sum\limits_{i=0}^{n-1}a_{i}=0\in\mathbb{F}$.

  2. b)

${\mathcal{C}}$ is an _affine-invariant_ code if it is an extended cyclic code
and ${\rm GA}(G,G)\subseteq\rm PAut({\mathcal{C}})$ where

${\rm GA}(G,G)=\\{x\mapsto ax+b\mid x\in\mathbb{F}G,a\in G^{*},b\in G\\}.$

For any affine-invariant code ${\mathcal{C}}\leq\mathbb{F}G$ we denote by
${\mathcal{C}}^{*}$ the punctured code at the position $X^{0}$. Then,
${\mathcal{C}}^{*}$ is a cyclic code in the sense that it is the projection to
$\mathbb{F}G^{*}$ of the image of a cyclic code via the map

$\displaystyle\mathbb{A}(n)$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{F}G$ $\displaystyle\sum\limits_{i=0}^{n-1}a_{i}X^{i}$
$\displaystyle\hookrightarrow$
$\displaystyle\left(-\sum\limits_{i=0}^{n-1}a_{i}\right)X^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{i}}.$
(3)

To give our new construction of information sets we will work with the so-
called defining set of an affine-invariant code. Let us note that this notion
is not the same as it was given in Definition 1; we will clarify the
difference below.

######  Definition 12.

Let ${\mathcal{C}}\leq\mathbb{F}G$ be an affine-invariant code and
$\langle\alpha\rangle=G^{*}$. For any $s\in\\{0,\dots,n=q^{m}-1\\}$ we
consider the $\mathbb{F}$-linear map $\phi_{s}:\mathbb{F}G\rightarrow G$ given
by

$\phi_{s}\left(bX^{0}+\sum\limits_{i=0}^{n-1}a_{i}X^{\alpha^{i}}\right)=0^{s}+\sum\limits_{i=0}^{n-1}a_{i}\alpha^{is}$

where we assume $0^{0}=1\in\mathbb{F}$ by convention. Then the set

$D({\mathcal{C}})=\\{i\mid\phi_{i}(x)=0\text{ for all }x\in{\mathcal{C}}\\}$

is called the defining set with respect to $\alpha$ of the affine-invariant
code ${\mathcal{C}}$.

It is clear that $D({\mathcal{C}})$ depends on the election of $\alpha\in G$,
however all throughout this paper it will be fixed and an arbitrary one so,
for the sake of simplicity, we do not include it in the notation.

It is also well known that any affine-invariant code is totally determined by
its defining set (see [5]). On the other hand, it is easy to see that
$D({\mathcal{C}})$ is a union of $q$-cyclotomic cosets modulo $n$; moreover,
since ${\mathcal{C}}$ is an extended-cyclic code, one has that $0\in
D({\mathcal{C}})$. Conversely, any subset of $\\{0,\dots,n\\}$ which is a
union of $q$-cyclotomic cosets and contains $0$ defines an affine-invariant
code in $\mathbb{F}G$.

It essential for us to note that with regard to (IV), we talk about the
defining set of ${\mathcal{C}}^{*}$ when we are making reference to that of
the corresponding cyclic code in $\mathbb{A}(n)$. So, the defining set of
${\mathcal{C}}^{*}$ is $D({\mathcal{C}}^{*})=D({\mathcal{C}})\setminus\\{0\\}$
(according to Definition 1).

######  Remark 13.

In the context of affine-invariant codes, a set
${\mathcal{I}}\subseteq\\{0,\alpha^{0},\dots,\alpha^{n-1}\\}$ is an
information set for a code ${\mathcal{C}}\leq\mathbb{F}G$ with dimension $k$,
if $\left|{\mathcal{I}}\right|=k$ and
${\mathcal{C}}_{\mathcal{I}}=\mathbb{F}^{\left|{\mathcal{I}}\right|}$.

It is also important to note that, since ${\mathcal{C}}^{*}$ is contained in
$\mathbb{F}G^{*}$ we have that any information set for it will be a subset of
$\\{\alpha^{0},\dots,\alpha^{n-1}\\}$. Furthermore, an information set for
${\mathcal{C}}^{*}$ is an information set for ${\mathcal{C}}$ too. However,
note that if $\Gamma$ is a set of check positions for ${\mathcal{C}}^{*}$ then
a set of check positions for ${\mathcal{C}}$ is $\Gamma\cup\\{0\\}$.

Finally, to give the definition of GRM codes we need to introduce the notion
of $q$-weight of a natural number.

######  Definition 14.

For any natural number $k$ its $q$-ary expansion is the sum

$\sum_{r\geq 0}k_{r}q^{r}=k$

with $k_{r}\in\\{0,1,\dots,q-1\\}$. The $q$-weight of $k$ is ${\rm
wt}_{q}(k)=\sum_{r\geq 0}k_{r}$.

######  Definition 15.

Let $0<\rho\leq m(q-1)$ and $\alpha$ a generator of $G^{*}$. The Generalized
Reed-Muller (GRM) code of order $\rho$ and length $q^{m}$, denoted by
$R_{q}(\rho,m)$, is the affine-invariant code in $\mathbb{F}G$ with defining
set (with respect to $\alpha$)

$D(R_{q}(\rho,m))=\\{0\leq i<q^{m}-1\mid{\rm wt}_{q}(i)<m(q-1)-\rho\\}.$

In this paper we are interested in GRM codes of first and second-order
respectively, for them

$D(R_{q}(1,m))=\\{0\leq i<q^{m}-1\mid{\rm wt}_{q}(i)<m(q-1)-1\\}.$
$D(R_{q}(2,m))=\\{0\leq i<q^{m}-1\mid{\rm wt}_{q}(i)<m(q-1)-2\\}.$

######  Remarks 16.

  1. 1.

We denote by $R^{*}(\rho,m)$ the punctured code of $R_{q}(\rho,m)$ at the
position $X^{0}$.

  2. 2.

It is well-known that the dual codes of $R_{q}(1,m)$ and $R_{q}(2,m)$ are
$R_{q}(m(q-1)-2,m)$ and $R_{q}(m(q-1)-3,m)$ respectively (see [7]). This fact
will be important for us in the subsequent section.

##  V Information sets for Generalized Reed-Muller codes

In this main section, we are applying the results of Section III to the
(cyclic) punctured code of first and second-order GRM codes seen as two
dimensional abelian codes. Recall that we are assuming that
$n=q^{m}-1=r_{1}\cdot r_{2},\ r_{1},r_{2}>1,\gcd(r_{1},r_{2})=1$. (Note that
the condition $\gcd(r_{i},q),i=1,2$ holds by definition of $n$). Let use
denote $a=Ord_{r_{1}}(q)$, that is, the order of $q$ modulo $r_{1}$; then, it
is easy to see that $a\mid m$.

Since the given construction of information sets relies on the structure of
the defining sets, it is more convenient to look at the dual codes instead of
the original ones because they have smaller defining sets and so they are more
manageable; that is, for $i=1,2$, instead of considering the code $R_{q}(i,m)$
we work with its dual $R_{q}(m(q-1)-i-1),m)$. Then, Theorem 10 will give us a
set of check positions for $R_{q}(m(q-1)-i-1),m)$ and consequently an
information set for $R_{q}(i,m)$.

Let us denote ${\mathcal{C}}^{*}=R^{*}_{q}(m(q-1)-i-1,m),i=1,2,$ and define

$\displaystyle\Omega(K)$ $\displaystyle=$
$\displaystyle\left\\{0<j<q^{m}-1\mid{\rm wt}_{q}(j)=K\right\\}$

then

$D^{*}=D({\mathcal{C}}^{*})=\bigcup_{i\in
D(R(\rho,m))\setminus\\{0\\}}C_{n}(i)$

where $\rho=m(q-1)-i-1$.

We need to describe completely the structure of $D^{*}$ and firstly we have to
study how we can take a suitable set of representatives and the corresponding
set ${\mathcal{U}}$ (see Definition 8).

As in Section III we assume that we have fixed a choice of $\alpha$, a
primitive $n$-th root of unity, and $T$ an isomorphism from ${\mathbb{Z}}_{n}$
to ${\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$.

###  V-A First-order GRM codes

In this case,

$D(R_{q}(m(q-1)-2,m))=\\{0\leq i<q^{m}-1\mid{\rm wt}_{q}(i)<2\\}$

and so $D^{*}=\Omega(1)$.

######  Lemma 17.

In the case of $R_{q}(m(q-1)-2,m)$ we can take
$\overline{{\mathcal{D}}^{*}}={\mathcal{U}}=\\{1\\}$.

###### Proof.

By definition

$\Omega(1)=\left\\{0<j<q^{m}-1\mid{\rm wt}(j)=1\right\\}=\\{q^{i}\mid 0\leq
i<m\\}$

So, $\Omega(1)=C_{n}(1)$. Then $D^{*}$ has a unique $q$-cyclotomic coset and
the result is straightforward. ∎

Therefore, we have the same situation as for (binary) Reed-Muller codes.
Namely, from Theorem 10 we get our first main result.

######  Theorem 18.

The set $T^{-1}\left(\Gamma\right)$, where

$\Gamma=\left\\{(i_{1},i_{2})\in{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}\mid
0\leq i_{1}<a,0\leq i_{2}<\frac{m}{a}\right\\},$

is a set of check positions for $R^{*}_{q}(m(q-1)-2,m)$.

The reader may check that the value of $q$ has no influence in the proof of
this theorem, so that given in [2, Theorem 34] also works in this context.

######  Corollary 19.

Given $\Gamma$, the set of check positions described in the previous theorem,
the set $\\{0,\alpha^{i}\mid i\in T^{-1}\left(\Gamma\right)\\}\subseteq G$ is
an information set for $R_{q}(1,m)$ and $\\{\alpha^{i}\mid i\notin
T^{-1}\left(\Gamma\right)\\}$ is an information set for $R_{q}(m(q-1)-2,m)$.

###### Proof.

It follows from the last statement of Theorem 10. ∎

The reader may see that we have obtained the same information set that was
given in [2], that is, it is valid for any value of $q\geq 2$.

###  V-B Second-order GRM codes

In this section we will assume $q>2$ as it is a necessary condition for the
results to come until the end of this paper. Therefore, the main results in
this case are not valid for (binary) Reed-Muller codes.

For second-order GRM codes we have

$D(R_{q}(m(q-1)-3,m))=\\{0\leq i<q^{m}-1\mid{\rm wt}_{q}(i)<3\\}$

and then

$D^{*}=\Omega(1)\cup\Omega(2),$

more specifically, if we write $B_{1}=\\{2q^{i}\mid 0\leq i<m\\}$ and
$B_{2}=\left\\{q^{i}+q^{j}\mid 0\leq i<j<m\right\\}$ , we have
$\Omega(2)=B_{1}\cup B_{2}$ and so

$D^{*}=\Omega(1)\cup B_{1}\cup B_{2}.$ (4)

(Note that $B_{1}=C_{n}(2)$.)

Note that in this case we have that the structure of $D^{*}$ is more
sophisticated. So, in order to be able to do the required computations we will
impose the additional condition $r_{1}=q^{a}-1$ for some natural number $a$.
(We will see in Section VI that this additional condition is not a strong
restriction.) Then the restrictions needed are

$\displaystyle n=q^{m}-1=r_{1}\cdot r_{2},\ r_{1},r_{2}>1,\
\gcd(r_{1},r_{2})=1$ (5) $\displaystyle r_{1}=q^{a}-1,\text{ for some
}a\in{\mathbb{N}},$

######  Remark 20.

By definition we have that the order of $q$ modulo $r_{1}$ is precisely $a$.
Note that the notation is consistent with that given at the beginning of this
section. Let us write

$m=ab.$

First, we need the following lemma.

######  Lemma 21.

If $q>2$ ($q$ a power of a prime number $p$) and $r=q^{a}-1$ for some natural
number $a$ then

  1. 1.

$C_{r}(1)\neq C_{r}(2).$

  2. 2.

$|C_{r}(1)|=|C_{r}(2)|=a.$

###### Proof.

To prove statement $1)$ suppose that $2\in C_{r}(1)$. Then, there exists
$\delta\in{\mathbb{N}}$ such that $q^{\delta}\equiv 2$ mod $r$, that is, there
exists $k\in{\mathbb{Z}}$ with $q^{\delta}-2=k\cdot(q^{a}-1)$. But, since $q$
is a power of a primer number greater than $2$, we have that $q^{a}-1$ is even
which implies $q^{\delta}$ even, a contradiction. So, $2\notin C_{r}(1)$ and
since the cyclotomic cosets define a partition we get what we wanted.

To prove statement $2)$ note first that $|C_{r}(1)|=a$ from the definition of
$q$-cyclotomic coset modulo $r$. Now, let $\mu=|C_{r}(2)|$, that is, $\mu$ is
the minimum positive integer such that $2\equiv q^{\mu}\cdot 2$ mod $r$. Let
us denote $I=\\{i\in{\mathbb{Z}}\mid 2\equiv q^{i}\cdot 2\text{ mod }r\\}$.
Then, it is clear that $0,a\in I$. Moreover, given $i,j\in I$ we have that
$2\equiv 2q^{i}$ mod $r$ implies (since $q$ is a unit in ${\mathbb{Z}}_{r}$)
that $2q^{-i}\equiv 2$ mod $r$, $2q^{i+j}\equiv 2q^{i}q^{j}\equiv 2q^{j}\equiv
2$ mod $r$, and for any $h\in{\mathbb{Z}},2q^{ih}\equiv 2q^{i}q^{i(h-1)}\equiv
2q^{i(h-1)}\equiv\dots\equiv 2q^{i}\equiv 2$ mod $r$. Therefore, $I$ is an
ideal in ${\mathbb{Z}}$ so it is a principal ideal. Then, by definition of
$\mu$ it is easy to see that $I$ is the ideal generated by $\mu$. This implies
that $\mu\mid a$. Let $s\in{\mathbb{N}}$ be such that $a=\mu s$, then there
exists $k\in{\mathbb{Z}}$ such that $2(q^{\mu}-1)=k\cdot(q^{a}-1)=$ so

$2(q^{\mu}-1)=k\cdot(q^{\mu}-1)(q^{\mu(s-1)}+q^{\mu(s-2)}+\cdots+q^{\mu}+1)$

and hence

$2=k\cdot(q^{\mu(s-1)}+q^{\mu(s-2)}+\cdots+q^{\mu}+1)$

which is possible only in case $s=1$, that is, $\mu=a$. This finishes the
proof. ∎

Now, we can prove that we will always may assume that the essential values $1$
and $2$ belong to the sets we have to manage in our computations.

######  Proposition 22.

Let $D^{*}$ be the defining set described in (4). Then

  1. 1.

We may take $\overline{D^{*}}\subseteq D^{*}$, a suitable set of
representatives, such that $1,2\in\overline{D^{*}}$. Moreover,

  2. 2.

we may take ${\mathcal{U}}\subseteq{\mathcal{D}}^{*}$, with
$1,2\in{\mathcal{U}}$, and

  3. 3.

$\mathcal{O}(2)=\\{2\\}$.

###### Proof.

Note first that by Lemma 21 we have that $C_{n}(1)\neq C_{n}(2)$. Now, we can
take $1,2$ in a suitable set of representatives,
$\overline{{\mathcal{D}}^{*}}$, if and only if $T(1)$ and $T(2)$ belong to a
restricted set of representatives of the $q$-orbits in $T(D^{*})$ if and only
if $T(1)$ and $T(2)$ do not belong to the same $q$-cyclotomic coset modulo
$r_{1}$; since $r_{1}=q^{a}-1$, this last condition holds again by Lemma 21.
Furthermore, this also implies that $1\not\equiv 2$ mod $r_{1}$ and then we
can take them as elements in ${\mathcal{U}}$.

Finally, as we have seen, $1\notin\mathcal{O}(2)$ and if
$e\in{\mathcal{U}}\setminus\\{1,2\\}$ then $e\in B_{2}$, which implies that
${\rm wt}_{q}(e)=2$ and so $e\notin C_{r_{1}}(2)$. This means that
$e\notin\mathcal{O}(2)$. We can conclude that $\mathcal{O}(2)=\\{2\\}$. ∎

From the previous result we have that we may treat separately the value
$2\in{\mathcal{U}}$. This is a crucial fact in order to be able to use the
computations made in [2], this will be describe explicitly within the proof of
Theorem 24.

To give our main result we only need one more lemma.

######  Lemma 23.

For any election of the set of representatives ${\mathcal{U}}$ one has that

$\left|{\mathcal{U}}\right|=2+\left|\\{C_{r_{1}}(e)\mid e\in{\mathcal{U}}\cap
B_{2}\\}\right|=2+\left\lfloor a/2\right\rfloor.$

###### Proof.

From the previous results we have that $1,2\in{\mathcal{U}}$ and
$\mathcal{O}(2)=\\{2\\}$. On the other hand, note that

$\overline{{\mathcal{D}}^{*}}\setminus\\{1,2\\}\subseteq B_{2}.$

Then, by [2, Lemma 45], we have that
$\left|\left\\{e\in\overline{{\mathcal{D}}^{*}}\setminus\\{1,2\\}\mid
e\notin\mathcal{O}(1)\right\\}\right|=\left\lfloor a/2\right\rfloor$. This
finishes the proof. ∎

######  Theorem 24.

Let ${\mathcal{C}}^{*}=R^{*}_{q}(m(q-1)-3,m)$ and suppose that the conditions
given in (5) hold. Let ${\mathcal{D}}^{*}\subseteq{\mathbb{Z}}_{n}$ be the
defining set of ${\mathcal{C}}^{*}$, with respect to $\alpha$, a primitive
$n$-th root of unity, and let ${\mathcal{C}}\leq\mathbb{A}(r_{1},r_{2})$ be
the abelian code with defining set $D({\mathcal{C}})=T({\mathcal{D}}^{*})$. We
define the sets
$\gamma_{i}\subseteq{\mathbb{Z}}_{r_{1}}\times{\mathbb{Z}}_{r_{2}}$, for
$i=1,2,3$, given by

$\begin{array}[]{rl}\gamma_{1}=&\displaystyle\left\\{(i_{1},i_{2})\mid 0\leq
i_{1}<\frac{a(a-1)}{2}\ \text{ and }\ 0\leq i_{2}<b^{2}\right\\},\\\ &\\\
\gamma_{2}=&\left\\{(i_{1},i_{2})\mid\displaystyle\frac{a(a-1)}{2}\leq
i_{1}<\frac{a(a+1)}{2}\ \text{ and }\right.\\\ &\displaystyle\left.0\leq
i_{2}<\frac{b(b+1)}{2}\right\\},\\\ &\\\
\gamma_{3}=&\displaystyle\left\\{(i_{1},i_{2})\mid\frac{a(a+1)}{2}\leq
i_{1}<\frac{a(a+3)}{2}\ \text{ and }\right.\\\ &\displaystyle\left.0\leq
i_{2}<b\right\\}.\\\ &\end{array}$

Then, the set $T^{-1}\left(\Gamma\right)$ where

$\Gamma=\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}$

is a set of check positions for $R^{*}_{q}(m(q-1)-3,m)$.

###### Proof.

Let $\overline{{\mathcal{D}}^{*}}$ be a suitable set of representatives of the
$q$-cyclotomic cosets modulo $n$ in ${\mathcal{D}}^{*}$ and let
${\mathcal{U}}$ be as in Definition 8, such that $1,2\in{\mathcal{U}}$. We
want to apply Theorem 10 to the code ${\mathcal{C}}$. To do it we need to
compute the value $M(u)$ for any $u\in{\mathcal{U}}$.

In [2] we obtain those values for any $u\in{\mathcal{U}}\setminus\\{2\\}$, to
wit

$M(1)=\frac{b(b+1)}{2}$

and

$M(e)=b^{2},\text{ for any }e\in{\mathcal{U}}\setminus\\{1,2\\}.$

The reader may see that these computations are also valid for $q\neq 2$.

Now, let us compute $M(2)$. By definition

$M(2)=\frac{1}{|C_{r_{1}}(2)|}\sum\limits_{v\in\mathcal{O}(2)}|C_{n}(v)|=\frac{|C_{n}(2)|}{|C_{r_{1}}(2)|}=\frac{m}{a}=b$

where we have used that $|C_{n}(2)|=|C_{n}(1)|$ and
$|C_{r_{1}}(2)|=|C_{r_{1}}(1)|$ (see Lemma 21). So, we have the sequence

$f_{1}=b^{2}>f_{2}=\frac{b(b+1)}{2}>f_{3}=b$

Observe that we have the restricted inequalities because $b>1$ (as $r_{2}>1$).
From this, following the notation in Theorem 10 we have the other sequence

$g_{1}=\sum_{\genfrac{}{}{0.0pt}{2}{u\in{\mathcal{U}}}{M(u)\geq
f_{1}}}|C_{r_{1}}(u)|=\sum\limits_{u\in{\mathcal{U}}\setminus\\{1,2\\}}|C_{r_{1}}(u)|=\frac{a(a-1)}{2}$
$g_{2}=\sum_{\genfrac{}{}{0.0pt}{2}{u\in{\mathcal{U}}}{M(u)\geq
f_{2}}}|C_{r_{1}}(u)|=\frac{a(a-1)}{2}+|C_{r_{1}}(1)|=\frac{a(a+1)}{2}$
$g_{3}=\sum_{\genfrac{}{}{0.0pt}{2}{u\in{\mathcal{U}}}{M(u)\geq
f_{3}}}|C_{r_{1}}(u)|=\frac{a(a+1)}{2}+|C_{r_{1}}(2)|=\frac{a(a+3)}{2}$

with

$g_{1}<g_{2}<g_{3}.$

Finally, a direct application of Theorem 10 gives us what we wanted. ∎

As in the previous section for first-order GRM codes we conclude with the
following corollary.

######  Corollary 25.

Given $\Gamma$, the set of check positions described in the previous theorem,
the set $\\{0,\alpha^{i}\mid i\in T^{-1}\left(\Gamma\right)\\}\subseteq G$ is
an information set for $R_{q}(2,m)$ and $\\{\alpha^{i}\mid i\notin
T^{-1}\left(\Gamma\right)\\}$ is an information set for $R_{q}(m(q-1)-3,m)$.

##  VI Examples

###  VI-A Examples for first-order GRM codes

First, we include in Table I examples of suitable decompositions of
$n=q^{m}-1$, with $3\leq m\leq 10$, and $q=3,5$.

TABLE I: Suitable decompositions for $n=q^{m}-1$ in first-order GRM codes $q$ | $m$ | $r_{1}$ | $r_{2}$ | $q$ | $m$ | $r_{1}$ | $r_{2}$
---|---|---|---|---|---|---|---
3 | 3 | 13 | 2 | 5 | 3 | 31 | 4
.. | 4 | 5 | 16 | .. | 4 | 13 | 48
.. | 5 | 121 | 2 | .. | 5 | 11 | 284
.. | 6 | 7 | 104 | .. | 6 | 7 | 2232
.. | 7 | 1093 | 2 | .. | 7 | 19531 | 4
.. | 8 | 41 | 160 | .. | 8 | 313 | 1284
.. | 9 | 757 | 26 | .. | 9 | 19 | 102796
.. | 10 | 61 | 968 | .. | 10 | 33 | 295928

In the following example we show how to apply the construction of information
sets for first-order GRM.

######  Example 26.

Let $q=3,m=3$ so we look for an information set for $R_{3}(1,3)$. As we have
seen in Table I a suitable decomposition of $n=3^{3}-1=26$ is
$r_{1}=13,r_{2}=2$. Moreover, it is straightforward to see that
$a=Ord_{13}(3)=3$.

Now, we fix $\alpha$ a primitive $26$-th root of unity and the isomorphism
$T:{\mathbb{Z}}_{26}\rightarrow{\mathbb{Z}}_{13}\times{\mathbb{Z}}_{2}$ given
by $T(1)=(1,1)$. Then, by Theorem 18 the set $T^{-1}(\Gamma)$ where

$\Gamma=\\{(i_{1},i_{2})\in{\mathbb{Z}}_{13}\times{\mathbb{Z}}_{2}\mid 0\leq
i_{1}<3,i_{2}=0\\}$

is a set of check positions for $R^{*}_{3}(4,3)$. It is easy to see that
$T^{-1}(\Gamma)=\\{0,2,14\\}$. Moreover, by Corollary 19 the set
$\\{0,1,\alpha^{2},\alpha^{14}\\}$ is an information set for $R(1,3)$.

######  Remark 27.

The reader may check that a different election of $\alpha$ is equivalent to a
different election of $T$. Furthermore, different elections of $T$ may yield
different information sets. So, with this construction we get at most as many
information sets as there are isomorphisms.

###  VI-B Second-order GRM codes

First, let us show that conditions (5) are not too restrictive. As examples,
we take $q=3,5$ and in Table II we show that we can find suitable values
easily for $m\leq 10$. Recall that we need the condition $r_{1}=q^{a}-1$, for
some natural number $a$, so we have included the column with the values $a$
instead the corresponding to $r_{2}$; the reader could obtain $r_{2}$ as
$n/r_{1}$.

TABLE II: Suitable decompositions for $n=q^{m}-1$ in second-order GRM codes $q$ | $m$ | $r_{1}$ | $a$ | $q$ | $m$ | $r_{1}$ | $a$
---|---|---|---|---|---|---|---
3 | 3 | 2 | 1 | 5 | 3 | 4 | 1
.. | 5 | 2 | 1 | .. | 5 | 4 | 1
.. | 6 | 8 | 2 | .. | 7 | 4 | 1
.. | 7 | 2 | 1 | .. | 9 | 4 | 1
.. | 9 | 2 | 1 | .. | 9 | 124 | 3
.. | 9 | 26 | 3 | .. | 10 | 24 | 2
.. | 10 | 8 | 2 |  |  |  |

Now, let us show how the construction works.

######  Example 28.

Let $q=5,m=3$ so we look for an information set for $R_{5}(2,3)$. As we have
seen in Table II a suitable decomposition of $n=5^{3}-1=124$ is
$r_{1}=4,r_{2}=31$, where $a=1$. Now, we fix $\alpha$ a primitive $124$-th
root of unity and the isomorphism
$T:{\mathbb{Z}}_{124}\rightarrow{\mathbb{Z}}_{4}\times{\mathbb{Z}}_{31}$ given
by $T(1)=(1,1)$.

Then, we have the following sets, which were mentioned in Theorem 24:

$\gamma_{1}=\emptyset$
$\gamma_{2}=\\{(i_{1},i_{2})\in{\mathbb{Z}}_{4}\times{\mathbb{Z}}_{31}\mid
i_{1}=0,0\leq i_{2}<6\\}$
$\gamma_{3}=\\{(i_{1},i_{2})\in{\mathbb{Z}}_{4}\times{\mathbb{Z}}_{31}\mid
i_{1}=1,0\leq i_{2}<3\\}$

Then, by Theorem 24 the set $T^{-1}(\Gamma)$ where

$\Gamma=\gamma_{2}\cup\gamma_{3}$

is a set of check positions for $R^{*}_{5}(9,3)$. Moreover, by Corollary 25
the set $\\{0,\alpha^{i}\mid i\in T^{-1}\left(\Gamma\right)\\}$ is an
information set for $R(2,3)$.

Let us note that the same comment that we made in Remark 27 is valid in this
case.

##  VII Conclusions and related open problems

In this work we have dealt with the problem of constructing information sets
for GRM codes of first and second order just in term of their basic
parameters. We have obtained that while for first-order GRM codes we get the
same information sets that was presented in [2] for (binary) Reed-Muller
codes, for second-order GRM codes we get different ones with a more complex
structure.

In relation with this results some open problems arise:

  * •

It seems to be possible to study the approach to the aim of this paper seeing
the punctured cyclic codes as multidimensional cyclic codes, that is, in the
algebra $\mathbb{A}(r_{1},\dots,r_{l})$ with $l>2$. Meanwhile, it is clear
that this drive us to a considerably increase of the complexity of the
computations on the cyclotomic cosets.

  * •

In [3] we show how to apply the permutation decoding algorithm to first-order
GRM codes and we obtain very relevant improvements in the number of errors to
be corrected. So, it becomes a very interesting problem how we can apply that
decoding algorithm to second-order GRM codes with respect to the new
information sets described in the present work.

## Acknowledgments

This work has been supported by the Spanish Government under Grant
PID2020-113206GBI00 funded by MCIN/AEI/10.13039/501100011033 and Fundación
Séneca of Murcia, Project PI/22.

## References

  * [1] J. J. Bernal and J. J. Simón, Information sets from defining sets in abelian codes, IEEE Trans. Inform. Theory, vol. 57, no. 12, pp. 7990-7999, 2011.
  * [2] J. J. Bernal and J. J. Simón, “Information sets from defining sets for Reed-Muller codes of first and second order”, IEEE Trans. Inform. Theory, vol. 64, no. 10, pp. 6484-6497, 2018.
  * [3] J. J. Bernal and J. J. Simón, “Permutation decoding of first-oder Generalized Reed-Muller codes”, submitted.
  * [4] P. Delsarte, J. M. Goethals and F. J. MacWilliams, “On generalizaed Reed-Muller codes and their relative”, Inform. and Control, vol. 16, pp. 403-442, 1970.
  * [5] W. C. Huffman, “Codes and Groups”, in Handbook of Coding Theory, vol II, V. S. Pless, W. C. Huffman and R. A. Brualdi Eds. Amsterdam. North-Holland, 1998.
  * [6] E. F. Assmus and J. D. Key, “Polynomials codes and finite geometries”, in Handbook of Coding Theory, vol II, V. S. Pless, W. C. Huffman and R. A. Brualdi Eds. Amsterdam. North-Holland, 1998.
  * [7] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge, 2003.
  * [8] T. Kasami, S. Lin and W. W. Peterson, “New generalizations of the Reed-Muller codes. Part I: Primitive Codes”, IEEE. Trans. Inform. Theory, vol 14, pp.189-199, 1968.
  * [9] D. E. Muller, “Application of boolean algebra to switching circuit design and to error detection”, IEEE Trans. Comput. vol. 3, pp. 6-12, 1954.
  * [10] I. S. Reed, “A class of multiple-error-correcting codes and the decoding scheme”, IRE Trans. Inform. Theory, IT-4, pp 38-49, 1954.
  * [11] E. J. Weldon, “New generalizations of the Reed-Muller codes. Part II: Nonprimitive codes”, IEEE Trans. Inform. Theory, vol 14, pp. 199-205, 1968.


#  Resonant Inelastic X-Ray Scattering Spectra of Cuprate Superconductors
Predicted by Model of Fractionalized Fermions

Masatoshi Imada Waseda Research Institute for Science and EngineeringWaseda
Research Institute for Science and Engineering Waseda University Waseda
University Okubo Okubo Shinjuku Shinjuku Tokyo Tokyo 169-8555 169-8555 Japan.
Toyota Physical and Chemical Research Institute Japan.
Toyota Physical and Chemical Research Institute Yokomichi Yokomichi Nagakute
Nagakute Aichi Aichi 480-1118 480-1118 Japan. Japan.

###### Abstract

We theoretically analyze the intensity of the resonant inelastic X-ray
scattering (RIXS) based on Ansatz that the electron is fractionalized into two
components. By fitting the angle resolved photoemission spectroscopy (ARPES)
data of a cuprate high-$T_{\rm c}$ superconductor Bi2Sr2CaCu2O8+δ and the
accompanied machine learning results obtained before by Yamaji et al.
(arXiv:1903.08060), we determine the paprameters of the two-component fermion
model which describes the electron fractionalization. The RIXS spectra are
predicted by using this formulation. We find that the intensity is enhanced in
the superconducting phase relative to the normal or pseudogap phase. Since the
enhancement is unusual, we propose this can be used as a critical test whether
the unconventional electron fractionalization captures the essence of the
cuprate superconductivity. This type of combined analyses using multiple
independent spectroscopy data, with the help of theoretical and machine
learning insights, regarded as integrated spectroscopy opens a new route to
study difficult open issues in strongly correlated electron systems.

##  1 Introduction

Since the discovery of copper oxide (cuprate) superconductors in 1986[1],
marvelous progress has been made toward the goal of understanding the
mechanism of the superconductivity and the normal state properties exhibiting
the pseudogap near the Mott insulating phase. Momentum and energy resolved
spectroscopic experiments including the angle resolved photoemission
spectroscopy (ARPES)[2], the quasiparticle interference (QPI) [3] at the
scanning tunnel microscope (STM), and resonant inelastic X-ray spectroscopy
(RIXS) [4], in addition to conventional probes such as optical measurements,
neutron scattering are typical tools to elucidate the momentum/energy resolved
dynamics of strongly correlated electrons. Despite their achievements, the
mechanisms promoting the superconductivity and the pseudogap have not reached
consensus.

On the other hand, ab intio framework of electronic structure calculations
recently developed for strongly correlated electron systems [5] has opened a
route to understand high-temperature strong-coupling superconductivity in
examples of the cuprate [6] and iron-based superconductors [7], where the
calculated phase diagrams have shown quantitative agreements with those of
experiments without adjustable parameters after deriving the low-energy
lattice Hamiltonians with both the kinetic and interaction strengths.

One of the important insights gained from the ab initio studies for the origin
of high-$T_{\rm c}$ superconductivity is the commonly identified strong
effective attraction of electrons caused near the Mott insulator both for the
cuprates and the iron-based superconductors (and in the iron-based case,
proximity of additional high-spin states caused by the Hund’s rule coupling),
in other words, caused in correlation induced bad metals. This attraction is
paradoxically caused by the original strong Coulomb repulsion in the evolution
at the release of carriers from incoherence. This attraction causes severe
competition toward the real space charge inhomogeneity as well including
charge order against superconductivity. It was proposed that the strong
attraction and severe competition may also result in the underlying electronic
bistability at double-well like energy structure as a function of the electron
filling. This naturally accompanies resultant fractionalization of a bare
electron into the conventional quasiparticles and dark fermions [8]. Although
the charge or spin may not necessarily have a fractional number in the present
definition of fractionalized particles unlike the fractional quantum Hall
effect, the fractionalization here implies that the bare electron splinters
into constituents or splits into two or more degrees of freedom, which
essentially behaves as independent fermi particles. The splintered two degrees
of freedom may naturally represent the bistability as well.

From a related but independent perspective, theoretical and numerical studies
were conducted based on the cluster dynamical mean-field studies of the
Hubbard model [9]. They revealed that the superconductivity in the Hubbard
model, which is adiabatically connected to the superconducting state found in
the solution of ab inito Hamiltonians for a hole-doped cuprate compound,
HgBa2CuO4 [6], has a marvelous property: Each of the normal and anomalous
self-energies has a prominent peak in the imaginary part at the same energy in
the $d$-wave superconducting state, while their peak contributions strictly
cancel in the single-particle spectral function $A(k,\omega)$ so that the
anomalous peak does not show up, consistently with the known ARPES
measurement. Furthermore, the peak in the anomalous self-energy is identified
as the real origin of the high-$T_{\rm c}$ superconductivity in the Kramers-
Kronig analysis.

This seemingly puzzling cancellation supports the idea that the underdoped
region of the cuprates exhibits the fractionalization consistently with the
above insight from the ab initio study of the effective Hamiltonians. This is
because that if an electron fractionalizes into the conventional quasiparticle
and a dark fermion, the resultant two-component fermion model (TCFM)
remarkably shows exact cancellation described above, whereas the single-
component fermions in the absence of the strong correlation effect do not show
such a cancellation. The prominent peaks in the imaginary part of the normal
and anomalous self-energies are understood from the pole of the dark fermion
Green’s function generating the self-energy pole of the quasiparticle Green’s
function (the imaginary part of which, namely the spectral function
$A(k,\omega)={\rm Im}G(k,\omega)$ is measurable by ARPES) through the
hybridization of the dark fermion and the visible quasiparticle. In fact, the
emergence of the pseudogap in the normal state is consistently understood from
the remaining pole in the normal self-energy, which does not cancel with the
anomalous self-energy part any more, because the anomalous part becomes
vanishing in the normal phase. This pole in the pseudogap state has a
structure similar to that proposed from the resonating valence bond picture
for the pseudogap phase [10].

Furthermore, the machine learning study on the ARPES data in the
superconducting states of Bi2Sr2Cu2O6+δ (Bi2201) and Bi2Sr2CaCu2O8+δ (Bi2212)
has succeeded in inferring and extracting the normal and anomalous self-
energies separately purely from the experimental data quite independently of
the above model studies and has shown consistency with the peak formation and
the cancellation. This consistency again supports the fractionalization [11].

However, because this cancellation has a strong impact and poses a severe
constraint on the mechanism of superconductivity, it is desired to gain
further insights whether the fractionalization has consequences in other
experimental probes especially in momentum-energy resolved spectroscopic
measurements to deepen understanding from different angle of viewpoint.

For decades, the strong correlations of electrons hamper the full and unified
understanding of the electron dynamics and resultant physical phenomena even
with recent revolutionary improved each spectroscopic methods such as ARPES,
QPI and RIXS. However, we may have hope to overcome the difficulty, if
different and independent experimental tools are combined as integrated
spectroscopy. The purpose of this paper is to initiate such an attempt of the
integrated spectroscopy, by proposing an exemplified combination of the ARPES
experimental data [12] together with results obtained by the machine learning
analyses [11], with the RIXS measurement using the simple TCFM based on the
fractionalization of electrons. Especially in this paper, I focus on the RIXS
spectra to gain insight into this enigmatic issue by utilizing the already
analyzed ARPES data [11]. RIXS is a spectroscopic method of measuring the
electron-hole excitation, which may also show a signal of the
fractionalization as we discuss in this paper.

So far we do not have sufficient and comprehensive data of RIXS measurement,
which is rapidly developing with remarkably improved resolution and here I
predict the unique qualitative behavior and dynamics of excitons in the RIXS
particularly on the charge dynamics if the fractionalization is valid and the
consequential TCFM essentially captures the exciton dynamics: I show that the
RIXS intensity weight shows distinct enhancement in the superconducting than
normal states if the fractionalization is evolved in the normal state.

In Sec.2, we introduce the TCFM. The parameters of the model are determined so
as to reproduce the ARPES data for Bi2212 [12] and its analysis by the machine
learning [11]. In Sec.3, theoretical formulation for RIXS quantities in the
experimental measurements is summarized. In Sec.4, theoretical predictions on
the RIXS spectra based on the TCFM derived in Sec.2 are presented. Sec.5 is
devoted to discussions including them on the mechanism of the enhancement. In
Sec.6, I conclude the paper with future outlook.

##  2 Two-Component Fermion Model (TCFM)

The TCFM for the cuprate superconductors is represented by the Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k,\sigma}[\epsilon_{c}(k)c_{k,\sigma}^{\dagger}c_{k,\sigma}+\epsilon_{d}(k)d_{k,\sigma}^{\dagger}d_{k,\sigma}$
$\displaystyle+$
$\displaystyle\Lambda(k)(c_{k,\sigma}^{\dagger}d_{k,\sigma}+{\rm H.c.})$
$\displaystyle+$
$\displaystyle(\Delta_{c}(k)c_{k,\sigma}^{\dagger}c_{-k,-\sigma}^{\dagger}+\Delta_{d}(k)d_{k,\sigma}^{\dagger}d_{-k,-\sigma}^{\dagger}+{\rm
H.c})].$

Here, the dark fermion represented by the annihilation operator $d$ with the
dispersion $\epsilon_{d}(k)$ is hybridizing with the bare electron or
quasiparticle ($c$ being its annihilation operator) with the dispersion
$\epsilon_{c}(k)$ at the momentum $k$ in a form of a noninteracting
Hamiltonian, where the hybridization is represented by $\Lambda(k)$ term. In
this paper, we employ a square lattice with simple dispersion for
$\epsilon_{c}(k)$ and $\epsilon_{d}(k)$ as

$\displaystyle\epsilon_{c}(k)=-(2t_{c1}(\cos k_{x}+\cos k_{y})+4t_{c2}\cos
k_{x}\cos k_{y})+\mu_{c},$ $\displaystyle\epsilon_{d}(k)=-(2t_{d1}(\cos
k_{x}+\cos k_{y})+4t_{d2}\cos k_{x}\cos k_{y})+\mu_{d},$ (2)

where $t_{c1}$ ($t_{d1}$) and $t_{c2}$ ($t_{d2}$) represent the nearest-
neighbor and next-nearest-neighbor hoppings of $c$ ($d$) fermion,
respectively. The terms proportional to $\Delta_{c}(k)$ and $\Delta_{d}(k)$
represent the anomalous part in the mean-field approximation emerging in the
superconducting state. Here we assume the simple $d$-wave gap

$\displaystyle\Delta_{c}(k)=\frac{\Delta_{c0}}{2}(\cos k_{x}-\cos k_{y}),$
$\displaystyle\Delta_{d}(k)=\frac{\Delta_{d0}}{2}(\cos k_{x}-\cos k_{y}),$ (3)

with temperature dependent constants $\Delta_{c0}$ and $\Delta_{d0}$. Note
that the pseudogap is the consequence of the fractionalization of an electron
into the quasiparticle $c$ and the dark fermion $d$ as we describe below and
only the low-energy part near the Fermi level has mapping between the $c$ and
$d$ fermions and the original electron by exhausting the degrees of freedom in
the Hilbert space [8]. The higher energy part far from the Fermi level
contained in the upper and lower Hubbard bands is not taken into account in
this simple TCFM.

In the normal state ($\Delta_{c}(k)=\Delta_{d}(k)=0$), the direct
hybridization gap at each momentum in the normal state is given by

$\displaystyle\Delta_{\rm
HG}(k)=\sqrt{(\epsilon_{c}(k)-\epsilon_{d}(k))^{2}+4\Lambda(k)^{2}}.$ (4)

representing the pseudogap. After introducing a phenomenological damping
$\gamma(\omega)$, Green’s function in Nambu representation is obtained from
the 4x4 matrix inversion as

$\displaystyle G(k,\omega)=(\omega-H-iI\gamma(k,\omega))^{-1},$ (5)

with $H$ given by Eq.(LABEL:TCfermionAnomalous) and $I$ being the 4 by 4
identity matrix. Then normal Green’s function for $c$ is given from the (1,1)
component in the form

$\displaystyle
G_{c}(k,\omega)=\frac{1}{\omega-\epsilon_{c}(k)-\Sigma(k,\omega)},$ (6)

with the self-energy

$\displaystyle\Sigma(k,\omega)=\frac{\Lambda(k)^{2}}{\omega-\epsilon_{d}(k)},$
(7)

where the pseudogap is represented by the suppressed density of sates caused
by the pole of the self-energy at $\omega=\epsilon_{d}(k)$.

In the superconducting state, the normal part of the single-particle Green’s
function is given by

$\displaystyle G_{c}(k,\omega)=\frac{1}{\omega-\epsilon_{c}(k)-\Sigma^{\rm
nor}(k,\omega)-W(k,\omega)},$ (8)

with

$\displaystyle W(k,\omega)=\frac{\Sigma^{\rm
ano}(k,\omega)^{2}}{\omega+\epsilon_{c}(k)+\Sigma^{\rm nor}(k,-\omega)^{*}},$
(9)

where

$\displaystyle\Sigma^{\rm
nor}(k,\omega)=\frac{\Lambda(k)^{2}(\omega+\epsilon_{d}(k))}{\omega^{2}-\epsilon_{d}(k)^{2}-\Delta_{d}(k)^{2}},$
(10)

and

$\displaystyle\Sigma^{\rm
ano}(k,\omega)=\Delta_{c}(k)+\frac{\Lambda(k)^{2}\Delta_{d}(k)}{\omega^{2}-\epsilon_{d}(k)^{2}-\Delta_{d}(k)^{2}}.$
(11)

Now the pole position of $\Sigma^{\rm nor}$ at $\omega=\epsilon_{d}(k)$ in the
normal state (expected to generate the pseudogap) (see Eq.(7)) is modified to
$\omega=\pm\sqrt{\epsilon_{d}(k)^{2}+\Delta_{d}(k)^{2}}$. The residue of this
pole in $\Sigma^{\rm nor}$ cancels with that of $W$ at the same $\omega$,
which results in the cancellation of the poles in the normal and anomalous
contributions in $G_{c}$ [9].

We employ the spectral function $A(k,\omega)$ given in the ARPES data reported
by Kondo et al. [12]. for Bi2212 and the successfully derived self-energies by
machine learning [11]. The obtained spectral function and the self-energy are
fitted by the TCFM. Hereafter, we take the energy unit by eV. In this unit, we
obtain the parameters $t_{c1}=0.1953$, $t_{c2}=-0.0762$, $t_{d1}=-0.0100$,
$t_{d2}=-0.0036$, $\mu_{c}=0.2875$, $\mu_{d}=0.0405$, $\Delta_{c0}=0$,
$\Delta_{d0}=0.0570$, and $\Lambda(k)=\Lambda_{0}+\Lambda_{1}(\cos
k_{x}+1)(\cos k_{y}+1)$ with $\Lambda_{0}=0.0658$ and $\Lambda_{1}=-0.014$.
These satisfy the Fermi surface position including $(\pi,0.14\pi)$ as the
antinodal (AN) point and $(0.37\pi,0.37\pi)$ as the nodal (N) point in the
ARPES measurement [12]. $\Lambda_{0}$ and $\Lambda_{0}+\Lambda_{1}$
essentially play the role of determining the hybridization at the AN and N
points respectively. They also essentially reproduce the dispersion across the
Fermi surface reported in Ref. Chen2019.

The broadening factor $\gamma(k,\omega)$ is taken to have the $k$-independent
form

$\displaystyle\gamma(\omega)=a|\omega|+b$ (12)

with $a=0.15$ and $b=0.002$ to reproduce the broadening of both the peak and
hump in the measured ARPES data analyzed below. This expression actually has
the marginal Fermi liquid form [14]. With these parameters, the spectral
function $A(k,\omega)$ is illustrated in Fig. 1. The peak, dip and hump
positions and the superconducting gap amplitude at the AN Fermi surface point
$(\pi,0.14\pi)$ are seen as shown in Fig. 1(a). Here, we can confirm that the
peak, dip and hump energies, relative peak heights and peak widths in the
ARPES data in the negative $\omega$ side for Bi2212 at the antinodal point
well reproduce those measured in ARPES [12] and its machine learning result
[11] by employing fitting values of the TCFM above. In the normal state shown
in Fig. 1(b), the gap structure identified as the pseudogap is essentially
consistent with the ARPES data slightly above above $T_{\rm c}$ [15]. In the
actual ARPES data, there exists presumable background effect as well as broad
$\omega$ insensitive structure partly because of the extrinsic experimental
conditions, which does not exist in the TCFM and we ignored it. The obtained
normal and anomalous self-energies of the obtained TCFM in the superconducting
phase indeed have prominent peaks in the imaginary part, ${\rm Im}\Sigma^{\rm
nor}$ and ${\rm Im}W$ at $\sim\pm 0.07$ eV as is seen in Fig. 2 and their peak
cancels in the Green’s function completely consistent with the machine
learning result. Although an ambiguity exists, the fitting is better for
$\Delta_{c0}=0$ rather than $\Delta_{c0}$ dominant superconductivity to
reproduce the self-energy structure estimated in the machine learning [11],
implying that the superconductivity is primarily triggered by the dark fermion
contribution. Note that the $d$-wave superconducting gap becomes nonzero even
when $\Delta_{c0}=0$, because nonzero $\Delta_{d}$ generates the gap in the
$c$ component as well through the momentum diagonal hybridization
$\Lambda(k)$. It should be noted that the pseudogap has an $s$-wave like full
gap extending from the AN to N points as we see from the $k$ dependence of
$\Lambda(k)$, consistently again with the previous model study [9].

Figure 1:  (a) Spectral function $A(k,\omega)$ obtained from TCFM in the
superconducting phase to reproduce the ARPES and the machine learning data.
(b) $A(k,\omega)$ in the normal (pseudogap) state for the same parameters as
(a) except for the choice $\Delta_{d0}=\Delta_{c0}=0$.  Figure 2:  (a) ${\rm
Im}\Sigma(k,\omega)$ and (b) ${\rm Im}W(k,\omega)$ obtained from TCFM showing
prominent either negative ((a)) or positive((b)) peaks at $\omega\sim\pm 0.07$
eV. The peaks at $\sim\pm 0.07$ eV cancels in ${\rm Im}\Sigma(k,\omega)+{\rm
Im}W(k,\omega)$ as is plotted in (c).

##  3 Formulation for RIXS spectra

Figure 3:  Illustration of the electron dynamics in RIXS measurement. (a)
Excitation of a core electron to the conduction band by X-ray absorption. (b)
Dynamics of the excited electron under the potential of the core hole. (c)
Decay of a valence electron and recombination with the core hole accompanied
by reemission of X-ray. (d) Dynamical evolution of electron-hole pair
(exciton).

To proceed to studies by the integrated spectroscopy, we discuss now the
consequence for the RIXS. The process of the RIXS dynamics is illustrated in
Fig. 3, where the X-ray excites a core electron to the conduction band and the
subsequent decay of the valence electron to the core hole by X-ray emission
makes an exciton (electron-hole pair), which evolves in time until the pair
recombination. This RIXS dynamics intensity is expressed by

$\displaystyle I_{\rm RIXS}(Q,\omega,\omega_{0};\sigma,\rho)$
$\displaystyle\propto$
$\displaystyle\sum_{l}|B_{li}(Q,\omega_{0};\sigma,\rho)|^{2}\delta(E_{l}-\omega),$
(13) $\displaystyle B_{li}(Q,\omega_{0};\sigma,\rho)=\sum_{m,j}e^{iQ\cdot
R_{m}}\chi_{\rho,\sigma}\langle l|c_{m\sigma}|j\rangle\langle
j|(\omega_{0}-E_{j}+i\Gamma)^{-1}|j\rangle\langle
j|c^{\dagger}_{m\rho}|i(k=0)\rangle,$ (14)

where $|i(k=0)\rangle$ is the ground state and $\chi_{\rho,\sigma}$ is the
spin ($\rho$ and $\sigma$) dependent matrix representing the dipole transition
amplitude. The inverse of the core hole lifetime is denoted by $\Gamma$. Here,
$\Omega$ is the energy of incident X-ray in Fig. 3, but we subtract $E_{\rm
F}-E_{\rm c}$ in the expression as $\omega_{0}=\Omega-(E_{\rm F}-E_{\rm c})$
to measure energy from $E_{\rm F}$, where $E_{\rm F}$ is the Fermi energy and
$E_{\rm c}$ is the core hole level. Here, $c_{m}^{\dagger}$ is the local
electron creation operator at the $m$-th site while $\omega_{0}=E_{j}$ is the
excitation energy of an eigenstate $|j\rangle$ of the Hamiltonian
$H_{m}=H+V_{m}$ for the system with one electron excited from the core to the
conduction band at the $m$-th lattice site by the X-ray injection to the
initial ground state. The excitation energy $E_{j}$ of the conduction electron
is measured not from the core level but from the Fermi energy. The particle-
hole (exciton) state $|l\rangle$ is characterized by the energy $E_{l}=\omega$
and the total momentum $Q$. In the present TCFM, the created hole energy
becomes $E_{h}=E_{l}-E_{j}$ (we take the hole creation energy positive).
Although we ignore the attractive potential of electron and hole in the
exciton state in the TCFM, its effect can be approximately taken into account
just by shifting $\omega$ with the amount of the exciton binding energy
$E_{\rm B}$ by interpreting as $\omega\rightarrow\omega-E_{\rm B}$. Here, the
annihilation and creation of the core electron are not explicitly expressed in
the equations and the core hole is treated just by the core hole potential
$V_{m}$ at the $m$-th site in the intermediate state and we take $H$ as the
TCFM (Eq.(LABEL:TCfermionAnomalous)).

Since the effect of nonzero $V_{m}$ is limited and qualitative feature does
not change [16], we take $V_{m}=0$. In the translational invariant system with
$V_{m}=0$, Eq.(14) is rewritten as

$\displaystyle B_{li}(Q,\omega_{0};\sigma,\rho)$ $\displaystyle=$
$\displaystyle\sum_{j(k_{e}),k_{h},k_{e}}\delta(Q-(k_{h}-k_{e}))\chi_{\rho,\sigma}\langle
l(Q,k_{e})|c_{k_{h}\sigma}|j(k_{e})\rangle\langle j(k_{e})|$ (15)
$\displaystyle\times(\omega_{0}-E_{j}(k_{e})+i\Gamma)^{-1}|j(k_{e})\rangle\langle
j(k_{e})|c^{\dagger}_{k_{e}\rho}|i(k=0)\rangle,$

where $|j(k_{e})\rangle$ is the one-electron excited state with the specified
momentum $k_{e}$ and $|l(Q,k_{e})\rangle$ is the electron-hole (exciton) state
with the electron momentum $k_{e}$ and the hole momentum $k_{h}=k_{e}+Q$.

In this paper, we consider spin non-flipping contribution $\Delta S=0$, which
reduces to $B_{li}(Q,\omega;\sigma,\rho)\rightarrow
B_{li}(Q,\omega;\uparrow,\uparrow)+B_{li}(Q,\omega;\downarrow,\downarrow)$.
This can be measured in Cu $L_{3}$ or O $K$ edge in RIXS. If $\Gamma$ can be
taken as $\Gamma\rightarrow\infty$, $I_{\rm RIXS}$ reduces to the charge
dynamical structure factor

$\displaystyle N(Q,\omega)=\int dr_{m}dte^{iQ\cdot r_{m}-i\omega t}\langle
n_{0}(0)n_{m}(t)\rangle$ (16)

with the charge operator
$n_{j}(t)=e^{-iH_{m}t}c_{j}^{\dagger}c_{j}e^{iH_{m}t}$. $\langle\cdots\rangle$
represents the ground state average.

The single-particle electronic excitation structure in the unoccupied
conduction states above the Fermi level can be probed by the X-ray absorption
spectra (XAS) using the same X-ray source simultaneously as

$\displaystyle I_{\rm XAS}(\omega)=-\frac{1}{\pi}{\rm Im}\sum_{j,m,k}\langle
0|c_{m\sigma}(\omega-E_{j}(k)+i\Gamma_{\rm
XAS}(k,\omega))^{-1}c^{\dagger}_{m\sigma}|i(k=0)\rangle,$ (17)

where $\Gamma_{\rm XAS}(k,\omega)$ is a momentum-energy dependent damping
constant.

##  4 Results

###  4.1 XAS Spectra

By using the parameters of the TCFM that reproduce the spectral function
measured by ARPES as described above, we calculate the XAS and RIXS spectra
both in the normal (pseudogap) state with $\Delta_{c}=\Delta_{d}=0$ and the
$d$-wave superconducting state and compare these two. These will also be
compared with the single-band case.

For the fitted parameters, the XAS spectra are shown in Fig. 4 for the
pseudogap phase (a) and for the superconducting phase (b). This indicates that
a large intensity of quasiparticle states is found with two peak structures in
the region $0<\omega<0.2$ eV and the broad intensity arising from the ingap
states separated by the $s$-wave like pseudogap from the quasiparticle band is
found in $0.2<\omega<1.5$ eV. Here, the quasiparticle band and the ingap
states are the consequence of the diagonalization of the TCFM and the
pseudogap corresponds to the hybridization gap generated by $\Lambda(k)$.
Because the TCFM ignores the upper Hubbard band, we can interpret that the
high energy part near 1.5 eV partially represents the upper Hubbard weight in
a compromised fashion. In the realistic XAS measurements, the damping
$\Gamma_{\rm XAS}(k,\omega)$ is essentially the same as $\Gamma$ in Eqs.(14)
and (15), which depends on the natural width of the core electron level as is
discussed below and may give broader peaks than those in Fig. 4. Here, we
intend to figure out the intrinsic feature of the excitation spectra of
conduction bands by taking an artificially small $\Gamma_{\rm XAS}$.

Figure 4:  Energy dependence of XAS spectra in the normal (pseudogap) phase
(upper panel, (a)) and $d$-wave superconducting phase with
$\Delta_{d0}=0.057$eV and $\Delta_{c0}$=0 (lower panel, (b)). For simplicity,
we take $\Gamma_{\rm XAS}(k,\omega)=0.01$eV to be $k$ and $\omega$ independent
form.

###  4.2 RIXS Spectra in the Limit of Charge Structure Factor

We next study the RIXS spectra and consider the case where an electron is
excited from the core to the energy above the Fermi level $E_{\rm F}$ (namely
$E_{j}>0$) and a hole is created below $E_{\rm F}$ (accompanying the decay of
an electron to the core hole.) Note that we set $E_{\rm F}=0$. In the
superconducting state, other cases can contribute in principle to the RIXS
intensity through the anomalous component of the Bogoliubov quasiparticle, but
its effect must be limited only for small $\Delta\omega$ less than or
comparable to the superconducting gap in actual experiments and we ignore it.
In the following, we introduce a small smearing factor $\eta=0.005$ eV instead
of $\delta$-function in Eq.(13).

Let us discuss the RIXS spectra first for large $\Gamma$, in the region
essentially the same as the limit $\Gamma\rightarrow\infty$, where the RIXS
intensity is reduced to the charge structure factor (Eq.(16)). More
specifically we take $\Gamma=3.0$ eV and show results in the case of initial
excitation of an electron to the conduction band with $E_{j}$ in Eqs. (13) and
(14) at the peak position of XAS spectra. Namely, $E_{j}=0.076$ eV for the
pseudogap phase and $E_{j}=0.032$ eV for the superconducting phase, because
experimentally the excitation to the energy having the largest density of
states makes the signal detection easier. However, the RIXS intensity does not
sensitively depend on $E_{j}$, if it is small $<0.2$ eV. We will examine the
incident-energy dependence later. Figure 5 shows the RIXS intensity for the
momentum transfer $Q=(\pi,0)$, which has a prominent peak around $\omega\sim
0.26$ eV followed by a long tail extended to $\sim 1.5$ eV with a shoulder at
$1.4$ eV in the pseudogap state shown in (a). In the superconducting phase in
(b), the peak at 0.26 eV splits into two peaks ascribed to the superconducting
gap formation. This splitting scales with the superconducting gap amplitude of
the dark fermion $\Delta_{d0}$ at the AN point. In addition, the shoulder at
$\sim 1.4$ eV becomes more prominent and develops a small peak in the
superconducting state. The shoulder around 1.4 eV mainly comes from the
excitation of electron by the X-ray to the region $(\pi,\pi)$ followed by the
creation of a hole around $(0,\pi)$, which results in the exciton momentum
$Q=(\pi,0)$.

Figure 5:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(\pi,0)$ for
$\Gamma=3.0$ eV in (a) normal (pseudogap) phase (upper panel), (b) $d$-wave
superconducting phase for $\Delta_{d0}=0.057$ eV and $\Delta_{c0}$=0. The
incident energy $\omega_{0}$ is 0.076 eV for the pseudogap case and 0.032 eV
for the superconducting case.

Even when the dark fermion is absent (namely by switching off the
hybridization $\Lambda(k)$), the RIXS intensity is qualitatively similar as
that seen in Fig. 5(a) in the normal state (not shown), which means that the
pseudogap does not have an appreciable effect in the normal state, if the
incident energy is small and $\Gamma$ is large as in this case. The same
applies to other $Q$. We will discuss this issue later when we discuss the
case of small $\Gamma$, which shows qualitative difference from the single-
band case.

For the momentum transfer $Q=(0,0)$, the spectra is shown in Fig. 6. The
intensity is concentrated as a low-energy peak arising from the transition
between the quasiparticle and ingap states representing the direct gap
corresponding to the infrared structure in the optical conductivity. The peak
energy corresponds to the averaged direct $s$-wave-like psudogap energy scale.

Figure 6:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(0,0)$ for
$\Gamma=3.0$ eV in (a) normal (pseudogap) phase (upper panel), (b) $d$-wave
superconducting phase for $\Delta_{d0}=0.057$ eV and $\Delta_{c0}$=0. The
incident energies are the same as Fig. 5.

The case of the momentum transfer $Q=(\pi,\pi)$ is shown in Fig. 7, where the
tail found for $Q=(\pi,0)$ is replaced by a broad prominent plateau-like
structure in the region $0.5<\omega<1.5$ eV and a more complicated splitting
of the peaks than the case $Q=(\pi,0)$ around 0.14 eV is seen in the
superconducting phase. The plateau contains all the contribution including the
transitions from $(0,0)$ to $(\pi,\pi)$ and the peak splitting reflects the
transitions between the states including gapped AN points namely between the
$(0,\pi)$ and $(\pi,0)$ regions as well as the transitions between the N
points around $(\pm\pi/2,\pm\pi/2)$. Inspections of intermediate $Q$ points in
the Brillouin zone show continuous evolution of the typical spectral features
shown here from $(0,0)$, and $(\pi,0)$ through $(\pi,\pi)$.

Figure 7:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(\pi,\pi)$
for $\Gamma=3.0$ eV in (a) normal (pseudogap) phase (upper panel), (b)
$d$-wave superconducting phase for $\Delta_{d0}=0.057$ eV and $\Delta_{c0}$=0.
The incident energies are the same as Fig. 5.

###  4.3 Beyond Charge Structure Factor

Since experimental $\Gamma$ does not satisfy the large $\Gamma$ condition, we
next study the case of small $\Gamma$ to make the prediction for the
experiment more relevant. Actually, RIXS spectra at small $\Gamma$ are
expected to contain richer information of the exciton dynamics, because the
momenta of electron and hole can separately be resolved and extracted. See
also Ref.Tohyama2018 for theoretical studies on the $\Gamma$ dependence of
RIXS spectra, where the more prominent peak at large energy for the larger
$\Gamma$ is similar. Below we study the case $\Delta_{d0}=0.05$ instead of
$\Delta_{d0}=0.057$ for simplicity to understand the qualitative trend and see
the dependence on the energy of the excited conduction electron $E_{j}$
regardless of the XAS peak energy.

Since $\Gamma$ for the RIXS measurement using Cu $L_{3}$ edge is estimated to
be around 0.28 eV [18], we next discuss the case $\Gamma=0.3$ eV. Figure 8
shows the RIXS intensity at $Q=(\pi,0)$ for the incident energy 0.3 eV. It has
a peak at low energies $\sim 0.3$ eV similarly to the case of $\Gamma=3.0$ eV.
and the increment $\Delta I_{\rm RIXS}=I_{\rm RIXS}^{\rm SC}-I_{\rm RIXS}^{\rm
PG}$ from the pseudogap to the superconducting phase is appreciable basically
only around 0.3 eV, where the peak splitting by the superconducting gap
formation makes the positive-negative-positive peak structure in $\Delta
I_{\rm RIXS}$ with total weight nearly unchanged between $I_{\rm RIXS}^{\rm
SC}$ and $I_{\rm RIXS}^{\rm PG}$.

Figure 8:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(\pi,0)$ for
$\Gamma=0.3$ eV and the incident energy $\omega_{0}=0.3$ eV in (a) normal
(pseudogap) phase (upper panel), (b) $d$-wave superconducting phase and (c)
their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0.05$ eV and
$\Delta_{c0}$=0.

However, if the incident energy increases and the electron is excited around
the top of the ingap state ($\sim 1.37$ eV) by X-ray in the present TCFM, the
RIXS intensity shows a sharp peak around the incident energy and it is clearly
seen that the RIXS weight is enhanced in total in the superconducting state
relative to the normal (pseudogap) state at around the incident energy as one
can see in Fig. 9. The enhancement by the superconductivity with the gap size
0.05 eV at such a high energy $>1$ eV is a remarkable and nontrivial property
of this fractionalized two-component model. For the single-band case, such an
enhancement does not occur as one sees in Fig. 10.

Figure 9:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(\pi,0)$ for
$\Gamma=0.3$ eV and the incident energy $\omega_{0}=1.37$ eV in (a) normal
(pseudogap) phase (upper panel), (b) $d$-wave superconducting phase and (c)
their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0.05$ eV and
$\Delta_{c0}$=0.  Figure 10:  Energy dependence of RIXS spectra $I_{\rm RIXS}$
at $Q=(\pi,0)$ for $\Gamma=0.3$ eV and the incident energy $\omega_{0}=1.37$
eV for the single-band case, where $\Lambda(k)$ is switched off to 0. (a)
normal (pseudogap) phase (upper panel), (b) $d$-wave superconducting phase and
(c) their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0$ and
$\Delta_{c0}=0.05$.

This enhancement disappears for small $Q$, but even at $(0.5\pi,0)$, a small
enhancement is still seen as in Fig. 11. This is helpful if the available $Q$
is limited in the experiments. When $Q$ is in the diagonal direction, around
$(\pi,\pi)$, though the peak is prominent around the incident energy as in Fig
12, the enhancement becomes less clear, while the enhancement is tiny but
visible for smaller $Q$ such as $Q=(0.5\pi,0.5\pi)$(not shown).

Figure 11:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(0.5\pi,0)$
for $\Gamma=0.3$ eV and the incident energy $\omega_{0}=1.37$ eV in (a) normal
(pseudogap) phase (upper panel), (b) $d$-wave superconducting phase and (c)
their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0.05$eV and
$\Delta_{c0}$=0.  Figure 12:  Energy dependence of RIXS spectra $I_{\rm RIXS}$
at $Q=(\pi,\pi)$ for $\Gamma=0.3$ eV and the incident energy $\omega_{0}=1.37$
eV in (a) normal (pseudogap) phase (upper panel), (b) $d$-wave superconducting
phase and (c) their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0.05$eV
and $\Delta_{c0}$=0.

This enhancement becomes relatively even more prominent in the superconducting
phase, when $\Gamma$ is further decreased: When we reduce $\Gamma$ to 0.07 eV,
which is similar to the value $\sim 0.072$ eV for oxygen $K$ edge absorption
for $1s\rightarrow 2p$ transition [19], Fig. 13 shows more prominent
enhancement by the superconductivity at the incident energy 1.37 eV than the
case of larger $\Gamma$, in the ratio to the peak intensity in the pseudogap
state at $Q=(\pi,0)$. The reason for the larger enhancement in small $\Gamma$
is that the combination of $(\pi,0)$ hole and $(\pi,\pi)$ electron is
selectively picked up for $Q=(\pi,0)$ without broadening, where the effects of
the fractionalization is larger as we discuss in the next section. This
selective enhancement is broadened and smeared if $\Gamma$ gets large and a
small enhancement can be seen even when $\omega_{0}$ is at the peak of XAS. To
detect the enhancement easily, however, smaller $\Gamma$ with the tuned
$\omega_{0}$ at the energy of the ingap top at $(\pi,\pi)$ is better.

Figure 13:  Energy dependence of RIXS spectra $I_{\rm RIXS}$ at $Q=(\pi,\pi)$
for $\Gamma=0.07$ eV and the incident energy $\omega_{0}=1.37$ eV in (a)
normal (pseudogap) phase (upper panel), (b) $d$-wave superconducting phase and
(c) their difference $\Delta I_{\rm RIXS}$ for $\Delta_{d0}=0.05$eV and
$\Delta_{c0}$=0.

##  5 Discussion

The mechanism of the enhancement, we propose, is the following: For the
excited particle at $\omega_{0}=E_{j}$, if $E_{j}$ corresponds to the region
near the top of the ingap dispersion, (note that in the present case
$E_{j}=1.37$ eV corresponds to the ingap band top at $(\pi,\pi)$), the
dominant contribution is always the $c$ component, which does not change
between the pseudogap and superconducting state as one easily expects. On the
other hand, the dominant contribution for the hole in the partner of the
excited exciton is found around the AN point including $(\pm\pi,0)$ and
$(0,\pm\pi)$. The two bands below the Fermi level contributing to the hole
creation around the AN point are dominantly $d$-hole like in one band and
$c$-electron like in the other in terms of TCFM. Only the $c$-electron like
state below the Fermi level contributes to annihilate an electron to decay to
the core hole. With increasing the pairing for $d$ fermion, the total weight
of the $c$ component for the wave functions with the energy below the Fermi
level increases toward 1.0 as one sees in Fig. 14, which causes the increase
in the RIXS weight at $(\pi,0)$ in the superconducting state for the incident
energy at around the top of the ingap state at $(\pi,\pi)$. On the contrary,
the weight of the particle component for $c$ fermion in the single-component
systems (for instance in the case $\Lambda(k)=0$), does not change in total
and the RIXS weight stays the same between the normal (pseudogap) and
superconducting states. The absence of appreciable enhancement in the
superconducting phase for $(\pi,\pi)$ even at large $\omega$ (Fig. 12) is also
consistent with the fact that the peak at large $\omega$ is mainly contributed
from the transition between $(0,0)$ and $(\pi,\pi)$, where the states at these
momenta mainly consist of the $c$ component and does not change between the
normal (pseudogap) and the superconducting states.

Figure 14:  Weight of $c$ component in the wave functions with negative
energies in TCFM at the AN point. Dashed and dash-dotted curves are the weight
of lower and higher energy states at the energy $<0$, respectively. Solid
curve is the total weight.

Therefore, when the band top of the conduction electron (either the ingap or
the upper Hubbard) is excited in the superconducting state for the above
choice of $Q$ and $\omega_{0}$, the enhancement of the RIXS weight is a clear
indication of the electron fractionalization representable by the TCFM.

The fractionalization implies that substantial degrees of freedom of electrons
are hidden in the experimental measurements, which is detected as the
incoherent part of electrons in transport and optical measurements. The
physical entity of the fractionalized part, namely the dark fermion, is an
open issue to be clarified in the future, but the enhancement if observed
indicates that the hidden (dark) particle or hole does not directly make an
exciton with the visible electron or hole degrees of freedom. It also
indicates that the hidden part has hybridization with the normal electron (or
quasiparticle) and it also dominantly develops the superconducting gap over
the conventional electron degrees of freedom. In this understanding, the real
visible superconducting gap in the $c$ part is drugged and driven by the gap
formation of the dark fermion, $d$.

Bi2212 is a bilayer system, and in principle it has a bonding and a
antibonding band. One might think that these two components may equally show
the enhancement. However, in this case, the superconducting transition is
driven by the band near the fermi level and the increase in the weight or
restructuring of this band does not occur. So, the enhancement if it occurs
can also exclude the contribution of this trivial two-component system from
the contribution of the present fractionalization.

##  6 Summary and Outlook

We have studied the RIXS intensity and have shown that the intensity is
enhanced in the superconducting phase from the normal (pseudogap) phase for
the appropriate choices of the incident energy $\omega_{0}$ and the momentum
transfer $Q$, if the electron fractionalization captures the essence of low-
energy charge excitation based on the two-component fermion description. The
parameters for the two-component description are derived and fitted from the
ARPES measurement of Bi2212 cuprate superconductor combined with the machine
learning study to infer the structure of the normal and anomalous self-
energies of the single-particle Green’s function. The predicted RIXS spectra
can be measured and critically tested in the experiment with the present
improved accuracy for the O $K$ and Cu $L_{3}$ edges, by picking up the spin
non-flip charge excitations to examine the validity of the present type of
fractionalization. Since the single-band description of the superconducting
phase does not show such enhancement, the presence or absence of the
enhancement can indeed be used as a stringent test for the electron
fractionalization in the cuprate superconductors.

If the line shape of the RIXS intensity is clarified with high resolution, the
self-energy of the exciton can be extracted beyond the TCFM. This will serve
to clarify the entity of the dark fermion as well as the dynamics of the
exciton, which is useful to directly extract the superconducting mechanism.

Combined analyses of independent spectroscopic measurements with the help of
the theoretical and computational analyses including the newly developed
machine learning attempted in the present study will provide us in more
general perspective with a powerful way to overcome the limitation and
uncertainty we are facing with in conventional analyses based on single and
independent probe measurement for strongly correlated electron systems for
decades. The approach exemplified here may open the new direction of research
as “integrated spectroscopy”.

Acknowledgements The author thanks useful discussions with Atsushi Fujimori
and Di-Jing Huang. This work was financially supported by Grant-in-Aids for
Scientific Research (JSPS KAKENHI) (No. 16H06345) from Ministry of Education,
Culture, Sports, Science and Technology (MEXT), Japan. This work was also
supported in part by the projects conducted under MEXT Japan named as
“Priority Issue on Post-K computer” and “Program for Promoting Research on the
Supercomputer Fugaku” in the subproject, “Basic Science for Emergence and
Functionality in Quantum Matter: Innovative Strongly-Correlated Electron
Science by Integration of Fugaku and Frontier Experiments”. We also thank the
support by the RIKEN Advanced Institute for Computational Science through the
HPCI System Research project (hp190145, and hp200132) supported by MEXT.

## References

  * [1] J. G. Bednorz and K. A. Müller, Z. Phys. 64, 189 (1986).
  * [2] A. Damascelli, Z. Hussain, and Z. X. Shen, Rev. Mod. Phys. 75, 473 (2003).
  * [3] J. E. Hoffman, K. McElroy, D.-H. Lee, K. M. Lang, H. Eisaki, S. Uchida, and J. C. Davis, Science 297, 1148 (2002).
  * [4] A. Kotani and S. Shin, Rev. Mod. Phys. 73, 203 (2001).
  * [5] M. Imada and T. Miyake, J. Phys. Soc. Jpn. 79, 112001 (2010).
  * [6] T. Ohgoe, M. Hirayama, T. Misawa, K. Ido, Y. Yamaji, and M. Imada, Phys. Rev. B 101, 045124 (2020).
  * [7] T. Misawa and M. Imada, Nat. Commun. 5, 5738 (2014).
  * [8] M. Imada and T. J. Suzuki, J. Phys. Soc. Jpn. 88, 024701 (2019).
  * [9] S. Sakai, M. Civelli, and M. Imada, Phys. Rev. Lett. 116, 057003 (2016).
  * [10] K. Y. Yang, T. M. Rice, and F. C. Zhang, Phys. Rev. B 73, 174501 (2006).
  * [11] Y. Yamaji, T. Yoshida, A. Fujimori, and M. Imada, arXiv:1903.08060 (2012).
  * [12] T. Kondo, Y. Hamaya, A. D. Palczewski, T. Takeuchi, J. Wen, Z. Xu, G. Gu, J. Schmalian, and A. Kaminski, Nature Physics 7, 21 (2011).
  * [13] S.-D. Chen, M. Hashimoto, Y. He, D. Song, K.-J. Xu, J.-F. He, T. P. Devereaux, H. Eisaki, D.-H. Lu, J. Zaanen, and Z.-X. Shen, Science 366, 1099 (2019).
  * [14] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1990).
  * [15] T. Kondo, W. Malaeb, Y. Ishida, T. Sasagawa, H. Sakamoto, T. Takeuchi, T. Tohyama, and S. Shin, Nat. Commun. 6, 7699 (2015).
  * [16] Y. Shi, A. J. A. James, E. Demler, and I. Klich, Phys. Rev. B 96, 155101 (2017).
  * [17] T. Tohyama and K. Tsutsui, Int. J. Mod. Phys. B 32, 1840017 (2018).
  * [18] M. O. Krause and J. H. Oliver, J. Phys. Chem. Ref. Data 8, 329 (1979).
  * [19] A. Menzel, S. Benzaid, M. O. Krause, and C. D. Caldwell, Phys. Rev. A 54, R991 (1996).