# Cosine and Sine addition and subtraction law with an automorphism

Youssef Aserrar and Elhoucien Elqorachi  Ibn Zohr University, Faculty of
sciences, Department of mathematics,
Agadir, Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: January 1, 2020)

###### Abstract.

Let $S$ be a semigroup. Our main results is that we describe the complex-
valued solutions of the following functional equations

$g(x\sigma(y))=g(x)g(y)+f(x)f(y),\ x,y\in S,$
$f(x\sigma(y))=f(x)g(y)+f(y)g(x),\ x,y\in S,$

and

$f(x\sigma(y))=f(x)g(y)-f(y)g(x),\ x,y\in S,$

where $\sigma:S\rightarrow S$ is an automorphism that need not be involutive.
As a consequence we show that the first two equations are equivalent to their
variants. We also give some applications.

###### Key words and phrases:

Functional equation, semigroup, addition law, automorphism.

###### 1991 Mathematics Subject Classification:

39B52, 39B32

##  1\. Introduction

Let $S$ be a semigroup and $\sigma:S\rightarrow S$ an automorphism, i.e
$\sigma(xy)=\sigma(x)\sigma(y)$ for all $x,y\in S$. The cosine subtraction
formula, sine addition formula and the sine subtraction formula are
respectively the functional equations

$g(x\sigma(y))=g(x)g(y)+f(x)f(y),\quad x,y\in S,$ (1.1)
$f(x\sigma(y))=f(x)g(y)+f(y)g(x),\quad x,y\in S,$ (1.2)
$f(x\sigma(y))=f(x)g(y)-f(y)g(x),\quad x,y\in S,$ (1.3)

for unknown functions $f,g:S\rightarrow\mathbb{C}$. These functional equations
generalizes respectively the trigonometric identities

$\cos\left(x-y\right)=\cos(x)\cos(y)+\sin(x)\sin(y),\quad x,y\in\mathbb{R},$
$\sin\left(x+y\right)=\sin(x)\cos(y)+\sin(y)\cos(x),\quad x,y\in\mathbb{R},$
$\sin\left(x-y\right)=\sin(x)\cos(y)-\sin(y)\cos(x),\quad x,y\in\mathbb{R},$

and have been investigated by many authors. In the case of an involutive
automorphism $\sigma$ , i.e an automorphism satisfying $\sigma(\sigma(x))=x$
for all $x\in S$, equation (1.1) was solved on abelian groups by Vincze [13],
and on general groups by Chung et al. [7]. Poulsen and Stetkær [11] described
the continuous solutions of (1.1), (1.2) and (1.3) on topological groups. In
[3, 4] Ajebbar and Elqorachi obtained the solutions of (1.1), (1.2) and (1.3)
on semigroups generated by their squares. The solutions of (1.2) with
$\sigma=id$ are also described in [9, Theorem 3.1] on a semigroup not
necessarily generated by its squares. In [8], Ebanks solved (1.1) and (1.3) on
general monoids. Recently the authors [5, 6] solved (1.1), (1.2) and (1.3) on
semigroups. We also refer to [1, Section 3.2.3], [2, Chapter 13] and [12,
Chapter 4] for further contextual and historical discussions.

The purpose of this paper is to solve the functional equations (1.1), (1.2)
and (1.3) on a semigroup $S$, where $\sigma:S\rightarrow S$ is an automorphism
not necessarily involutive, our results are natural extensions of previous
results about the solutions of (1.1), (1.2) and (1.3).

The character involutive of the automorphism $\sigma$ is used in the proofs of
[12, Theorem 4.12, Theorem 4.16], [8, Theorem 4.1, Corollary 4.3] and [6,
Theorem 4.2, Theorem 5.1]. The present paper shows that this condition is not
crucial for the giving proofs even in the setting of semigroups.

The contributions of the present work to the knowledge about solutions of
(1.1), (1.2) and (1.3) are the following:
(1) By the help of Theorem 2.2 and Theorem 2.3 (See section 2) we find all
complex-valued solutions of (1.1), (1.2) and (1.3) on a semigroup $S$ in terms
of multiplicative functions and solutions $\phi$ of the special sine addition
law

$\phi(xy)=\phi(x)\chi(y)+\phi(y)\chi(x),\ \ x,y\in S,$ (1.4)

where $\chi:S\rightarrow\mathbb{C}$ is a multiplicative function. The new
result here is that the automorphism $\sigma$ is not involutive. That makes
the exposition more involved and explains why our proofs are longer than those
of previous papers about the same functional equations.
(2) We consider the variants of (1.1) and (1.2) respectively

$g(\sigma(y)x)=g(x)g(y)+f(x)f(y),\quad x,y\in S,$ (1.5)
$f(\sigma(y)x)=f(x)g(y)+f(y)g(x),\quad x,y\in S.$ (1.6)

We show that (1.5) is equivalent to (1.1), and (1.6) is equivalent to (1.2).
(3) As an application, we determine the complex-valued solutions of the new
functional equations

$g(x+\beta y)=g(x)g(y)+f(x)f(y),\quad x,y\in\mathbb{R},$ $f(x+\beta
y)=f(x)g(y)+f(y)g(x),\quad x,y\in\mathbb{R},$

where $\beta\in\mathbb{R}\backslash\\{0,-1,1\\}$. Obviously these equations
generalizes the functional equations

$g(x\pm y)=g(x)g(y)+f(x)f(y),\quad x,y\in\mathbb{R},$ $f(x\pm
y)=f(x)g(y)+f(y)g(x),\quad x,y\in\mathbb{R},$

which have been studied by many authors (See for example [10, Corollary
3.56.a, Corollary 3.56.c] and [12, Corollary 4.17]).

The outline of the paper is as follows: In the next section we give some
notations and terminology. The complete solution of (1.1) is given in section
3. In section 4 we solve the functional equation (1.2). The sine subtraction
formula (1.3) is solved in section 5. Section 6 contains some applications.

##  2\. Notations and terminology

In this section we give some notations and notions that are essential in our
discussion. Throughout this paper $S$ denotes a semigroup. That is a set
equipped with an associative binary operation. A multiplicative function on
$S$ is a function $\mu:S\rightarrow\mathbb{C}$ satisfying
$\mu(xy)=\mu(x)\mu(y)$ for all $x,y\in S$. A function
$f:S\rightarrow\mathbb{C}$ is central if $f(xy)=f(yx)$ for all $x,y\in S$, and
$f$ is abelian if $f$ is central and $f(xyz)=f(xzy)$ for all $x,y,z\in S$. We
define the set $S^{2}:=\\{xy\ |\ x,y\in S\\}$. Let $\sigma:S\rightarrow S$ be
an automorphism, for any function $f:S\rightarrow\mathbb{C}$ we define the
function $f^{*}:=f\circ\sigma$. A topological semigroup is a pair, consisting
of a semigroup $S$ and a topology on $S$, such that the product $(x,y)\mapsto
xy$ is continuous from $S\times S$ to $S$, when $S$ is given the product
topology. If $S$ is a topological semigroup, let $C(S)$ denote the set of
continuous functions mapping $S$ into $\mathbb{C}$.

######  Notation 2.1.

Let $\chi$ be a non-zero multiplicative function. The sympbol $\phi_{\chi}$
shall denote a solution of the sepcial sine addition law (1.4). i.e

$\phi_{\chi}(xy)=\phi_{\chi}(x)\chi(y)+\phi_{\chi}(y)\chi(x),\ \ x,y\in S.$

The following are respectively [8, Theorem 3.2] and [9, Theorem 3.1], but for
brivety some formulas of solution are expressed with the use of Notation 2.1.

######  Theorem 2.2.

The solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation

$g(xy)=g(x)g(y)-f(x)f(y),\ \ x,y\in S,$

are the following pairs:

$g=f=0$.

$g=\dfrac{\delta^{-1}\chi_{1}+\delta\chi_{2}}{\delta^{-1}+\delta}$ and
$f=\dfrac{\chi_{1}-\chi_{2}}{i\left(\delta^{-1}+\delta\right)}$, where
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different multiplicative
functions and $\delta\in\mathbb{C}\backslash\left\\{0,i,-i\right\\}$.

$f$ is any non-zero function such that $f=0$ on $S^{2}$ and $g=\pm f$.

$g=\chi\pm\phi_{\chi}$ and $f=\phi_{\chi}$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function.

######  Theorem 2.3.

The solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation

$f(xy)=f(x)g(y)+f(y)g(x),\ \ x,y\in S,$

with $f\neq 0$ can be listed as follows:

$f=c\left(\chi_{1}-\chi_{2}\right)$ and $g=\dfrac{\chi_{1}+\chi_{2}}{2}$,
where $\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different
multiplicative functions and $c\in\mathbb{C}\backslash\left\\{0\right\\}$.

$f$ is any non-zero function such that $f=0$ on $S^{2}$ and $g=0$.

$f=\phi_{\chi}$ and $g=\chi$, where $\chi:S\rightarrow\mathbb{C}$ is a non-
zero multiplicative function.

The following lemma will be used throughout the paper without explicit
mentionning.

######  Lemma 2.4.

Let $f:S\rightarrow\mathbb{C}$ be a non-zero function satisfying

$f(x\sigma(y))=\beta f(x)f(y),\quad\text{for all}\quad x,y\in S,$ (2.1)

where $\beta\in\mathbb{C}\backslash\\{0\\}$ is a constant. Then there exists a
non-zero multiplicative function $\chi:S\rightarrow\mathbb{C}$ such that
$\beta f=\chi$ and $\chi^{*}=\chi$.

###### Proof.

By using the associativity of the semigroup operation we compute
$f(x\sigma(y)\sigma(z))$ using the identity (2.1) first as
$f((x\sigma(y))\sigma(z))$ and then as $f(x(\sigma(y)\sigma(z)))$ and compare
the results to obtain

$\beta^{2}f(x)f(y)f(z)=\beta f(x)f(yz),\quad\text{for all}\quad x,y,z\in S.$
(2.2)

Since $f\neq 0$ and $\beta\neq 0$ Eq. (2.2) can be written as

$f(yz)=\beta f(y)f(z),\quad\text{for all}\quad y,z\in S.$ (2.3)

This implies that the function $\chi:=\beta f$ is multiplicative. That is
$f=\dfrac{1}{\beta}\chi$, and then Eq. (2.1) becomes

$\dfrac{1}{\beta}\chi(x)\chi^{*}(y)=\dfrac{1}{\beta}\chi(x)\chi(y).$ (2.4)

Since $\chi\neq 0$ and $\beta\neq 0$, we deduce from (2.4) that
$\chi^{*}=\chi$. This completes the proof of Lemma 2.4. ∎

##  3\. The cosine subtraction formula (1.1)

The most recent result on the cosine subtraction formula (1.1), namely

$g(x\sigma(y)=g(x)g(y)+f(x)f(y),\quad x,y\in S,$

on semigroups is [6, Theorem 4.2]. By using similar computations to those of
[6, Theorem 4.2] we will solve (1.1) on general semigroups, but here $\sigma$
is not assumed to be involutive. It should be mentioned that in the case of an
involutive automorphism $\sigma$, the general solution of (1.1) on monoids can
be found in [8, Theorem 4.1].

######  Lemma 3.1.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (1.1), and suppose that
$f$ and $g$ are linearly independent. Then $g^{*}=g$ and $f^{*}=f$ or
$f^{*}=-f$.

###### Proof.

By using the associativity of the semigroup operation we compute
$g(x\sigma(y)\sigma(z))$ by the help of Eq. (1.1) first as
$g((x\sigma(y))\sigma(z))$ and then as $g(x(\sigma(y)\sigma(z)))$ and compare
the results. We obtain after some rearrangement that

$f(x)\left[f(yz)-f(y)g(z)\right]+g(x)\left[g(yz)-g(y)g(z)\right]=f(z)f(x\sigma(y)).$
(3.1)

Since $f\neq 0$, there exists $z_{0}\in S$ such that $f(z_{0})\neq 0$ and then

$f(x)h(y)+g(x)k(y)=f(x\sigma(y)),$ (3.2)

where

$h(y)=\dfrac{f(yz_{0})-f(y)g(z_{0})}{f(z_{0})},$

and

$k(y)=\dfrac{g(yz_{0})-g(y)g(z_{0})}{f(z_{0})}.$

By using (1.1) and the fact that $\sigma$ is a bijection, we obtain

$k=c_{1}g+c_{2}f,$ (3.3)

for some constants $c_{1},c_{2}\in\mathbb{C}$. Substituting (3.2) into (3.1),
we obtain

$\displaystyle\begin{split}f(x)\left[f(yz)-f(y)g(z)\right]+g(x)\left[g(yz)-g(y)g(z)\right]\\\
=f(x)f(z)h(y)+g(x)f(z)k(y).\end{split}$ (3.4)

Since $f$ and $g$ are linearly independent we deduce from (3.4) that

$g(yz)=g(y)g(z)+f(z)k(y),$ (3.5)

and

$f(yz)=f(y)g(z)+f(z)h(y).$ (3.6)

Substituting (3.3) into (3.5), we get

$g(yz)=\left[g(z)+c_{1}f(z)\right]g(y)+c_{2}f(z)f(y).$ (3.7)

So, by applying (3.7) to the pair $(y,\sigma(z))$, we obtain

$g(y\sigma(z))=\left[g^{*}(z)+c_{1}f^{*}(z)\right]g(y)+c_{2}f^{*}(z)f(y).$
(3.8)

By comparing (3.7) and (1.1), and using the linear independence of $f$ and
$g$, we get

$g=g^{*}+c_{1}f^{*},$ (3.9) $f=c_{2}f^{*}.$ (3.10)

Since $f\neq 0$, we deduce from (3.10) that $c_{2}\neq 0$.
First case : $c_{1}=0$. That is $g^{*}=g$. By applying Eq. (1.1) to the pair
$(\sigma(x),y)$ we find that

$g^{*}(xy)=g^{*}(x)g(y)+f^{*}(x)f(y).$

So since $f^{*}=\dfrac{1}{c_{2}}f$ and $g^{*}=g$, we get that for all $x,y\in
S$

$g(xy)=g(x)g(y)+\dfrac{1}{c_{2}}f(x)f(y).$ (3.11)

Now if we apply Eq. (3.11) to the pair $(x,\sigma(y))$, we get

$g(x\sigma(y))=g(x)g(y)+\dfrac{1}{c_{2}^{2}}f(x)f(y).$ (3.12)

Comparing (1.1) and (3.12) and using that $f\neq 0$, we deduce that
$c_{2}^{2}=1$. So $c_{2}=\pm 1$, and then $f=f^{*}$ or $f=-f^{*}$.
Second case : $c_{1}\neq 0$. We get from (3.9) and (3.10) that
$g^{*}=g-\dfrac{c_{1}}{c_{2}}f$, and then by applying Eq. (1.1) to the pair
$(\sigma(x),y)$ we get

$g(xy)-\dfrac{c_{1}}{c_{2}}f(xy)=g(x)g(y)-\dfrac{c_{1}}{c_{2}}f(x)g(y)+\dfrac{1}{c_{2}}f(x)f(y).$
(3.13)

Then by using Eq. (3.7), we get from (3.13) after some rearrangement that

$f(xy)=f(y)\left(c_{2}g(x)+\left(\dfrac{c_{2}^{2}-1}{c_{1}}\right)f(x)\right)+g(y)f(x).$
(3.14)

Comparing Eq. (3.6) and Eq. (3.14), and using that $f\neq 0$ we deduce that

$h=c_{2}g+\left(\dfrac{c_{2}^{2}-1}{c_{1}}\right)f.$ (3.15)

Taking (3.15) and (3.3) into account Eq. (3.2) becomes

$f(x\sigma(y))=g(x)\left(c_{2}f(y)+c_{1}g(y)\right)+f(x)\left(\left(\dfrac{c_{2}^{2}-1}{c_{1}}\right)f(y)+c_{2}g(y)\right).$
(3.16)

Now, if we apply Eq. (3.14) to the pair $(x,\sigma(y))$ we get

$f(x\sigma(y))=f(x)\left(g(y)+\left(\dfrac{c_{2}^{2}-c_{1}^{2}-1}{c_{1}c_{2}}\right)f(y)\right)+g(x)f(y).$
(3.17)

Comparing Eq. (3.16) and Eq. (3.17) and using the linear independence of $f$
and $g$, we get

$g=\left(\dfrac{1-c_{2}}{c_{1}}\right)f.$

This is a contradiction since $f$ and $g$ are linearly independent. So this
case does not occur. This completes the proof of Lemma 3.1. ∎

######  Remark 3.2.

The result of Lemma 3.1 is also true for the variant (1.5) of equation (1.1).

###### Proof.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (1.5) such that $f$ and
$g$ are linearly independent, if we compute $g(\sigma(yz)x)$ in two different
ways we get by using the linear independent of $f$ and $g$ that

$f(\sigma(z)x)=f(x)h(z)+g(x)\left[a_{1}g(z)+a_{2}f(z)\right],$ (3.18)
$f(yz)=g(y)f(z)+f(y)h(z),$ (3.19)
$g(yz)=g(z)\left[g(y)+a_{1}f(y)\right]+a_{2}f(y)f(z),$ (3.20)

for some constants $a_{1},a_{2}\in\mathbb{C}$ and $h$ is a function. If we
apply Eq. (3.20) to $(\sigma(y),z)$ and compare the preceding equation with
Eq. (1.5) we obtain since $f$ and $g$ are linearly independent that
$g=g^{*}+a_{1}f^{*}$ and $f=a_{2}f^{*}$. $f\neq 0$ implies that $a_{2}\neq 0$.
First case : $a_{1}=0$. In this case $g^{*}=g$, so if we apply Eq. (1.5) to
$(\sigma(x),y)$ and then apply the preceding equation to $(x,\sigma(y))$ we
obtain

$g(\sigma(y)x)=g(x)g(y)+\dfrac{1}{a_{2}^{2}}f(x)f(y).$ (3.21)

Comparing Eq. (3.21) with (1.5) and using that $f\neq 0$ we deduce that
$a_{2}^{2}=1$. That is $a_{2}=\pm 1$, so $f=f^{*}$ or $f=-f^{*}$.
Second case : $a_{1}\neq 0$. If we apply Eq. (1.5) to $(\sigma(x),y)$ and
using Eq. (3.20) we find that

$f(yx)=f(y)\left(a_{2}g(x)+\left(\dfrac{a_{2}^{2}-1}{a_{1}}\right)f(x)\right)+g(y)f(x).$
(3.22)

Comparing Eq. (3.19) and Eq. (3.22) and using that $f\neq 0$, we get that
$h=a_{2}g+\dfrac{a_{2}^{2}-1}{a_{1}}f$. Now Eq. (3.18) becomes

$f(\sigma(y)x)=g(x)\left(a_{2}f(y)+a_{1}g(y)\right)+f(x)\left(\left(\dfrac{a_{2}^{2}-1}{a_{1}}\right)f(y)+c_{2}g(y)\right).$
(3.23)

By applying Eq. (3.22) to the pair $(x,\sigma(y))$, we find that

$f(\sigma(y)x)=f(x)\left(g(y)+\left(\dfrac{a_{2}^{2}-a_{1}^{2}-1}{a_{1}a_{2}}\right)f(y)\right)+g(x)f(y).$
(3.24)

Comparing these last two identities and using the linear independence of $f$
and $g$, we get $g=\dfrac{1-a_{2}}{a_{1}}f$ but this is a contradiction since
$f$ and $g$ are linearly independent. This completes the proof of Remark 3.2.
∎

The next result gives the general solution of (1.1) on semigroups.

######  Theorem 3.3.

The solutions $f,g:S\rightarrow\mathbb{C}$ of the functional equation (1.1)
are the following :

$g=0$ and $f=0$.

$g$ is any non-zero function such that $g=0$ on $S^{2}$, and $f=cg$, where
$c\in\\{i,-i\\}$.

$g=\dfrac{1}{1+\alpha^{2}}\chi$ and $f=\dfrac{\alpha}{1+\alpha^{2}}\chi$,
where $\alpha\in\mathbb{C}\backslash\\{i,-i\\}$ is a constant and
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function such that
$\chi^{*}=\chi$.

$g=\dfrac{\delta^{-1}\chi_{1}+\delta\chi_{2}}{\delta^{-1}+\delta}$ and
$f=\dfrac{\chi_{2}-\chi_{1}}{\delta^{-1}+\delta}$, where
$\delta\in\mathbb{C}\backslash\\{0,i,-i\\}$ and
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different multiplicative
functions such that $\chi_{1}^{*}=\chi_{1}$ and $\chi_{2}^{*}=\chi_{2}$.

$g=\dfrac{\chi+\chi^{*}}{2}$ and $f=\dfrac{\chi-\chi^{*}}{2i}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function such that
$\chi^{*}\neq\chi$ and $\chi\circ\sigma^{2}=\chi$.

$f=-i\phi_{\chi}$ and $g=\chi\pm\phi_{\chi}$ where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function such that
$\chi^{*}=\chi$ and $\phi_{\chi}^{*}=\phi_{\chi}$.
Note that $f$ and $g$ are Abelian in each case.

Furthermore, if $S$ is a topological semigroup and $g\in C(S)$, then
$f,\chi,\chi^{*},\chi_{1},\chi_{2},\phi_{\chi}\in C(S)$.

###### Proof.

If $g=0$ then $f=0$. This is case (1). So from now on we assume that $g\neq
0$. Suppose that $g=0$ on $S^{2}$, then we get from equation (1.1) that

$g(x)g(y)+f(x)f(y)=0.$ (3.25)

Since $g\neq 0$ we obtain from equation (3.25) that $f=cg$ where
$c\in\mathbb{C}$ is a constant. Then if we take this into account in equation
(3.25) we get that $(c^{2}+1)g(x)g(y)=0$. This implies that $c^{2}+1=0$
because $g\neq 0$, so $c\in\\{i,-i\\}$. This occurs in part (2) of Theorem
3.3. If $f=0$, then equation (1.1) can be written as follows
$g(x\sigma(y))=g(x)g(y)$. So $g=:\chi$ is multiplicative and $\chi^{*}=\chi$.
This occurs in part (3) of Theorem 3.3 with $\alpha=0$. Now we assume that
$g\neq 0$ on $S^{2}$, $f\neq 0$ and we discuss two cases according to whether
$f$ and $g$ are linearly dependent or not.
First case : $g$ and $f$ are linearly dependent. There exists a constant
$\alpha\in\mathbb{C}$ such that $f=\alpha g$, so equation (1.1) can be written
as

$g(x\sigma(y))=(1+\alpha^{2})g(x)g(y),\quad x,y\in S.$ (3.26)

Since $g\neq 0$ on $S^{2}$ and $f\neq 0$, we deduce from (3.26) that
$\alpha\notin\\{0,i,-i\\}$, and then $\chi:=(1+\alpha^{2})g$ is multiplicative
and $\chi^{*}=\chi$. This occurs in case (3) with $\alpha\neq 0$.
Second case : $g$ and $f$ are linearly independent. According to Lemma 3.1
$g^{*}=g$ and $f=f^{*}$ or $f^{*}=-f$.
Subcase A : $f=f^{*}$. By applying Eq. (1.1) to the pair $(\sigma(x),y)$, we
obtain

$g(xy)=g(x)g(y)+f(x)f(y),\quad x,y\in S.$ (3.27)

Defining $l:=if$, equation (3.27) can be written as follows

$g(xy)=g(x)g(y)-l(x)l(y),\quad x,y\in S.$

According to Theorem 2.2 and taking into account that $f$ and $g$ are linearly
independent, we have the following possibilities :
(i) $g=\dfrac{\delta^{-1}\chi_{1}+\delta\chi_{2}}{\delta^{-1}+\delta}$ and
$l=\dfrac{\chi_{1}-\chi_{2}}{i(\delta^{-1}+\delta)}$, where
$\delta\in\mathbb{C}\backslash\\{0,i,-i\\}$ is a constant and
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two multiplicative functions
such that $\chi_{1}\neq\chi_{2}$. Since $g=g^{*}$, $f=f^{*}$ and $l=if$, we
deduce that $f=\dfrac{\chi_{2}-\chi_{1}}{\delta^{-1}+\delta}$,
$\chi_{1}=\chi_{1}^{*}$ and $\chi_{2}=\chi_{2}^{*}$. This is case (4).
(ii) $g=\chi\pm l$ and $l=\phi_{\chi}$, where $\chi:S\rightarrow\mathbb{C}$ is
a non-zero multiplicative function. Since $f^{*}=f$ and $g^{*}=g$, we see that
$\chi^{*}=\chi$ and $\phi_{\chi}^{*}=\phi_{\chi}$. In addition $l=if$ implies
that $f=-i\phi_{\chi}$. This occurs in part (6).
Subcase B : $f^{*}=-f$. Equation (1.1) can be written as follows

$g(xy)=g(x)g(y)-f(x)f(y),\quad x,y\in S.$

Similarly to the previous case, we get according to Theorem 2.2 and taking
into account that $f$ and $g$ are linearly independent the two cases:
(i) $g=\dfrac{\delta^{-1}\chi_{1}+\delta\chi_{2}}{\delta^{-1}+\delta}$ and
$f=\dfrac{\chi_{1}-\chi_{2}}{i(\delta^{-1}+\delta)}$, where
$\delta\in\mathbb{C}\backslash\\{0,i,-i\\}$ is a constant and
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different multiplicative
functions. Since $f^{*}=-f$ and $g^{*}=g$, we get

$\delta^{-1}(\chi_{1}-\chi_{1}^{*})+\delta(\chi_{2}-\chi_{2}^{*})=0,$ (3.28)
$\chi_{1}+\chi_{1}^{*}=\chi_{2}+\chi_{2}^{*}.$ (3.29)

Since $\chi_{1}\neq\chi_{2}$, we obtain by the help of [12, Corollary 3.19]
that $\chi_{1}=\chi_{2}^{*}$ and $\chi_{2}=\chi_{1}^{*}$. Then (3.28) reduces
to

$\left(\delta^{-1}-\delta\right)\left(\chi_{1}-\chi_{1}^{*}\right)=0.$

This implies that $\delta^{-1}-\delta=0$ since $\chi_{1}\neq\chi_{2}$. That is
$\delta=\pm 1$. This occurs in case (5) with $\chi_{1}=\chi$ and
$\chi_{2}=\chi^{*}$. In addition $\chi_{1}=\chi_{2}^{*}$ implies that
$\chi\circ\sigma^{2}=\chi$.
(ii) $g=\chi\pm f$ and $f=\phi_{\chi}$ where $\chi:S\rightarrow\mathbb{C}$ is
a non-zero multiplicative function. If $g=\chi+f$ we obtain since $f^{*}=-f$
and $g^{*}=g$ that $g=\chi^{*}-f$. Adding and subtracting this from $g=\chi+f$
we get that

$g=\dfrac{\chi+\chi^{*}}{2}\quad\text{and}\quad f=\dfrac{\chi^{*}-\chi}{2}.$

By assumption $f\neq 0$, so $\chi\neq\chi^{*}$. By substituting the forms of
$f$ and $g$ in (1.1) we find that $\chi=\chi^{*}$. This case does not occur.
Now if $g=\chi-f$, we show by the same way that $g=\dfrac{\chi+\chi^{*}}{2}$
and $f=\dfrac{\chi-\chi^{*}}{2}$ which leads by substitution to
$\chi=\chi^{*}$ ($f=0$). This case does not occur.

Conversely we check by elementary computations that the forms (1), (2), (3),
(4), (5) and (6) satisfy (1.1).

Finally, suppose that $S$ is a topological semigroup and $g\in C(S)$. In case
(1), $f=0\in C(S)$. In case (2), $f=\pm ig\in C(S)$. Now if $f\neq 0$, the
continuity of $f$ follows easily from the continuity of $g$ and the functional
equation (1.1). Let $y_{0}\in S$ such that $f(y_{0})\neq 0$, we get from (1.1)
that

$f(x)=\dfrac{g(x\sigma(y_{0}))-g(y_{0})g(x)}{f(y_{0})}\ \text{for}\ x\in S.$

The function $x\mapsto g(x\sigma(y_{0}))$ is continuous, since the right
translation $x\mapsto x\sigma(y_{0})$ from $S$ into $S$ is continuous, so $f$
is continuous as a linear combination of continuous functions. In cases (4)
and (5) we get the continuity of $\chi_{1},\chi_{2},\chi,\chi^{*}$ by the help
of [12, Theorem 3.18]. In case (6), $\phi_{\chi}=if\in C(S)$ and
$\chi=(g\pm\phi_{\chi})\in C(S)$. This completes the proof of Theorem 3.3. ∎

Now we relate the solution of the variant (1.5) to the functional equation
(1.1).

######  Proposition 3.4.

The functional equation (1.1) and its variant (1.5), namely

$g(\sigma(y)x)=g(x)g(y)+f(x)f(y),\quad x,y\in S,$

have the same solutions.

###### Proof.

Theorem 3.3 shows that if $(g,f)$ is a solution of (1.1), then $g$ is abelian,
in particular central. So any solution $(g,f)$ of (1.1) is also a solution of
(1.5). Now let $(g,f)$ be a solution of (1.5). We show that $g$ is central.
First case: $g$ and $f$ are linearly dependent. That is $f=\delta g$ for some
constant $\delta\in\mathbb{C}$. Eq. (1.5) can be written as

$g(\sigma(y)x)=(1+\delta^{2})g(x)g(y),\quad x,y\in S.$

If $\delta\in\left\\{i,-i\right\\}$, then $g=0$ on $S^{2}$, so $g$ is central
and we are done. If $\delta\neq\pm i$, we get that $(1+\delta^{2})g$ is
multiplicative. Then $g$ is central.
Second case: $g$ and $f$ are linearly independent. By Remark 3.2 we have
$g=g^{*}$ and $f=f^{*}$ or $f=-f^{*}$. By applying Eq.(1.5) to $(\sigma(x),y)$
we obtain

$g(yx)=g(x)g(y)\pm f(x)f(y),\quad x,y\in S.$

This implies that $g$ is central. This completes the proof of Proposition 3.4.
∎

##  4\. The sine addition formula (1.2)

In this section we solve the functional equation (1.2) on semigroups. The
following lemma gives some key properties of the solutions of Eq. (1.2). We
use similar computations to those of [6, Theorem 5.1] but of course here
$\sigma$ is not involutive.

######  Lemma 4.1.

Suppose $f,g:S\rightarrow\mathbb{C}$ satisfy Eq. (1.2) such that $f$ and $g$
are linearly independent. Then $f^{*}=f$ and $g^{*}=g$.

###### Proof.

Computing $f(x\sigma(yz))$ in two different ways using equation (1.2), we
obtain after some rearrangement that

$f(x)\left[g(yz)-g(y)g(z)\right]+g(x)\left[f(yz)-f(y)g(z)\right]=f(z)g(x\sigma(y)).$
(4.1)

Since $f\neq 0$, there exists $z_{0}\in S$ such that $f(z_{0})\neq 0$ and
hence

$f(x)h(y)+g(x)k(y)=g(x\sigma(y)),$ (4.2)

where

$h(y)=\dfrac{g(yz_{0})-g(y)g(z_{0})}{f(z_{0})},$

and

$k(y)=\dfrac{f(yz_{0})-f(y)g(z_{0})}{f(z_{0})}.$

By using Eq. (1.2) and the fact that $\sigma$ is a bijection, we get

$k=\alpha f+\beta g,$ (4.3)

for some constants $\alpha,\beta\in\mathbb{C}$. Now by using (4.2), equation
(4.1) becomes

$\displaystyle\begin{split}f(x)\left[g(yz)-g(y)g(z)\right]+g(x)\left[f(yz)-f(y)g(z)\right]\\\
=f(x)f(z)h(y)+g(x)f(z)k(y).\end{split}$ (4.4)

Since $f$ and $g$ are linearly independent we deduce from (4.4) that for all
$y,z\in S$

$g(yz)=g(y)g(z)+f(z)h(y),$ (4.5)

and

$f(yz)=f(y)g(z)+f(z)k(y).$ (4.6)

By using (4.3), equation (4.6) can be written as follows

$f(yz)=\left(g(z)+\alpha f(z)\right)f(y)+\beta f(z)g(y).$

This implies that

$f(y\sigma(z))=\left(g^{*}(z)+\alpha f^{*}(z)\right)f(y)+\beta f^{*}(z)g(y).$

By comparing this last identitie with Eq. (1.2) and using the linear
independence of $f$ and $g$ we deduce that

$g=g^{*}+\alpha f^{*},$ (4.7) $f=\beta f^{*}.$ (4.8)

Since $f\neq 0$ we get from (4.8) that $\beta\neq 0$, and from (4.7) that
$g^{*}=g-\dfrac{\alpha}{\beta}f$.
So, for all $x,y\in S$ we have

$\displaystyle f(x\sigma(y))$ $\displaystyle=\beta f^{*}(x\sigma(y))=\beta
f(\sigma(x)\sigma(\sigma(y)))$ $\displaystyle=\beta f^{*}(x)g^{*}(y)+\beta
f^{*}(y)g^{*}(x)$
$\displaystyle=f(x)\left[g(y)-\dfrac{\alpha}{\beta}f(y)\right]+f(y)\left[g(x)-\dfrac{\alpha}{\beta}f(x)\right]$
$\displaystyle=f(x)g(y)+f(y)g(x)-\dfrac{2\alpha}{\beta}f(x)f(y)$
$\displaystyle=f(x\sigma(y))-\dfrac{2\alpha}{\beta}f(x)f(y).$

So $\alpha=0$ since $f\neq 0$, and then $g^{*}=g$. Now if we apply Eq. (1.2)
to the pair $(\sigma(x),y)$ and multiplying the preceding equation by $\beta$
we get

$f(xy)=f(x)g(y)+\beta f(y)g(x).$ (4.9)

Computing $f(xyz)$ in two different ways, we obtain from (4.9) by using Eq.
(4.5) after some rearrangement that

$\left(\beta^{2}-\beta\right)g(x)g(y)=\beta f(y)h(x)-f(x)h(y).$ (4.10)

Since $f\neq 0$, we deduce from Eq. (4.10) that

$h=af+bg,$ (4.11)

for some constants $a,b\in\mathbb{C}$. Taking Eq. (4.11) into account Eq.
(4.10) becomes

$\left(\beta^{2}-\beta\right)g(x)g(y)=f(x)\left((a\beta-a)f(y)-bg(y)\right)+b\beta
g(x)f(y).$ (4.12)

Since $f$ and $g$ are linearly independent, we deduce from Eq. (4.12) that

$\left(\beta^{2}-\beta\right)g=b\beta f.$

This implies that $\beta=1$ and $b=0$ since $\beta\neq 0$, $f$ and $g$ are
linearly independent. So $f=f^{*}$. This completes the proof of Lemma 4.1. ∎

######  Remark 4.2.

Similar computations shows that the result of Lemma 4.1 hold for the variant
(1.6) of equation (1.2).

Now we are ready to solve the functional equation (1.2).

######  Theorem 4.3.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (1.2) are the following
pairs:

$f=0$ and $g$ is arbitrary.

$f$ is any non-zero function such that $f=0$ on $S^{2}$, while $g=0$.

$f=\dfrac{1}{2\alpha}\chi$ and $g=\dfrac{1}{2}\chi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function such that
$\chi^{*}=\chi$ and $\alpha\in\mathbb{C}\backslash\\{0\\}$.

$f=c\left(\chi_{1}-\chi_{2}\right)$ and $g=\dfrac{\chi_{1}+\chi_{2}}{2}$,
where $\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different
multiplicative functions such that $\chi_{1}^{*}=\chi_{1}$,
$\chi_{2}^{*}=\chi_{2}$ and $c\in\mathbb{C}\backslash\\{0\\}$.

$f=\phi_{\chi}$ and $g=\chi$ where $\chi:S\rightarrow\mathbb{C}$ is a non-zero
multiplicative function such that $\chi^{*}=\chi$ and
$\phi_{\chi}^{*}=\phi_{\chi}$.
Note that, off the exceptional case (1) $f$ and $g$ are Abelian.

Furthermore, off the exceptional case (1), if $S$ is a topological semigroup
and $f\in C(S)$, then $g,\chi$,$\chi_{1},\chi_{2},\phi_{\chi}\in C(S)$.

###### Proof.

If $f=0$ then $g$ will be arbitrary. This occurs in case (1). From now on we
assume that $f\neq 0$. Suppose that $f=0$ on $S^{2}$. For all $x,y\in S$, we
get from equation (1.2) that

$f(x)g(y)+f(y)g(x)=0.$ (4.13)

Since $f\neq 0$ we deduce from equation (4.13) according to [12, Exercise
1.1(b)] that $g=0$. This occurs in part (2) of Theorem 4.3. Now we assume that
$f\neq 0$ on $S^{2}$ and we discuss two cases according to whether $f$ and $g$
are linearly dependent or not.
First case : $f$ and $g$ are linearly dependent. There exists a constant
$\alpha\in\mathbb{C}$ such that $g=\alpha f$, so equation (1.2) can be written
as follows $f(x\sigma(y))=2\alpha f(x)f(y)$. This implies that $\alpha\neq 0$,
since $f\neq 0$ on $S^{2}$. So the function $\chi:=2\alpha f$ is
multiplicative and $\chi^{*}=\chi$. This is case (3).
Second case : $f$ and $g$ are linearly independent. According to Lemma 4.1 we
have $f=f^{*}$ and $g=g^{*}$. So equation (1.2) becomes

$f(xy)=f(x)g(y)+f(y)g(x).$ (4.14)

According to Theorem 2.3 and taking into account that $f\neq 0$, $g\neq 0$,
$f^{*}=f$ and $g^{*}=g$ we have the following possibilities :
(i) $f=c\left(\chi_{1}-\chi_{2}\right)$ and $g=\dfrac{\chi_{1}+\chi_{2}}{2}$,
for some constant $c\in\mathbb{C}\backslash\\{0\\}$ and
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different multiplicative
functions such that $\chi_{1}^{*}=\chi_{1}$ and $\chi_{2}^{*}=\chi_{2}$. This
is case (4).
(ii) $f=\phi_{\chi}$ and $g=\chi$ where $\chi:S\rightarrow\mathbb{C}$ is a
non-zero multiplicative function such that $\phi_{\chi}^{*}=\phi_{\chi}$ and
$\chi^{*}=\chi$. This occurs in part (5) of Theorem 4.3.

Conversely we check by elementary computations that if $f,g$ have one of the
forms (1)–(5) then $(f,g)$ is a solution of equation (1.2).

For the continuity statements, the continuity of $g$ follows easily from the
continuity of $f$ and the functional equation (1.2). In case (4) we get the
continuity of $\chi_{1}$ and $\chi_{2}$ by the help of [12, Theorem 3.18].
This completes the proof of Theorem 4.3. ∎

At this point of our discussion about solutions of (1.2), a natural question
comes up: Can we derive the solution of the variant (1.6) of (1.2) from
Theorem 4.3 ? the next result gives a positive answer.

######  Proposition 4.4.

The functional equation (1.2) and its variant (1.6), namely

$f(\sigma(y)x)=f(x)g(y)+f(y)g(x),\quad x,y\in S,$

have the same solutions.

###### Proof.

Theorem 4.3 prove that if $(f,g)$ is a solution of Eq. (1.2), then $f$ is
abelian. In particular central. So $(f,g)$ is a solution of the variant (1.6).
Now let $f,g:S\rightarrow\mathbb{C}$ be a solution of (1.6). It suffices to
show that $f$ is central.
First case: $f$ and $g$ are linearly dependent. There exists a constant
$\gamma\in\mathbb{C}$ such that $g=\gamma f$. Equation (1.6) becomes

$f(\sigma(y)x)=2\gamma f(x)f(y),\quad x,y\in S.$

If $\gamma=0$, then $f=0$ on $S^{2}$, so $f$ is central. If $\gamma\neq 0$,
then $2\gamma f$ is multiplicative, and then $f$ is central.
Second case: $f$ and $g$ are linearly independent. According to Ramark 4.2 we
have $f=f^{*}$ and $g=g^{*}$. If we apply Eq. (1.6) to $(\sigma(x),y)$ we get

$f(yx)=f(x)g(y)+f(y)g(x),\quad x,y\in S,$

which implies that $f$ is central. This completes the proof of Proposition
4.4. ∎

##  5\. The sine subtraction formula (1.3)

In this section we solve the functional equation (1.3). The following lemma
will be used later.

######  Lemma 5.1.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (1.3) such that $f$ and
$g$ are linearly independent. Then $f^{*}=-f$ and $g^{*}=g+\beta f$ for some
constant $\beta\in\mathbb{C}$.

###### Proof.

By using similar computations to those of the proof of Lemma 4.1 we find that
for all $y,z\in S$

$g(yz)=g(y)g(z)-f(z)h(y),$ (5.1)

and that

$\displaystyle g=g^{*}+af^{*},$ $\displaystyle f=-bf^{*},$

for some constants $a,b\in\mathbb{C}$ and some function $h$. Since $f\neq 0$
we can see that $b\neq 0$, and then $g^{*}=g-af^{*}=g+\dfrac{a}{b}f$. Choosing
$\beta=\dfrac{a}{b}$, we get $g^{*}=g+\beta f$. Now by letting $x=\sigma(x)$
in (1.3) we get that

$f(xy)=f(x)g(y)+bf(y)g(x)+af(x)f(y),\quad x,y\in S.$

Computing $f(xyz)$ in two different ways and using Eq. (5.1) we get after some
simplifications that

$g(y)\left((a-ab)f(x)+(b-b^{2})g(x)\right)=bf(y)h(x)-f(x)h(y).$ (5.2)

Since $f\neq 0$ we get from Eq. (5.2) that $h=\delta f+\gamma g$ for some
constants $\delta,\gamma\in\mathbb{C}$. Taking this into account Eq. (5.2)
becomes

$g(y)\left((a-ab)f(x)+(b-b^{2})g(x)\right)=f(y)\left((b\delta-\delta)f(x)+b\gamma
g(x)\right)-\gamma g(y)f(x).$

Since $f$ and $g$ are linearly independent we deduce that

$(a-ab)f(x)+(b-b^{2})g(x)=-\gamma f(x).$

Then $b=1$ since $b\neq 0$. That is $f^{*}=-f$. This completes the proof of
Lemma 5.1. ∎

The next theorem generelizes the results about solutions of (1.3) found in
[11, Proposition 3.1], [4, Theorem 5.1], [5, Proposition 3.2] and [8,
Corollary 4.3].

######  Theorem 5.2.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (1.3) are the following
pairs:

  1. (1)

$f=0$ and $g$ is arbitrary.

  2. (2)

$f$ is any non-zero function such that $f=0$ on $S^{2}$ and $g=\alpha f$,
where $\alpha\in\mathbb{C}$.

  3. (3)

$f=c(\chi-\chi^{*})$ and
$g=\dfrac{\chi+\chi^{*}}{2}+c_{1}\dfrac{\chi-\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function such that
$\chi\neq\chi^{*}$, $\chi\circ\sigma^{2}=\chi$, $c_{1}\in\mathbb{C}$ and
$c\in\mathbb{C}\backslash\\{0\\}$.

  4. (4)

$f=\phi_{\chi}$ and $g=\chi+c_{2}\phi_{\chi}$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function such that
$\chi^{*}=\chi$, $\phi_{\chi}^{*}=-\phi_{\chi}$ and $c_{2}\in\mathbb{C}$ is a
constant.

Note that, off the exceptional case (1) $f$ and $g$ are Abelian.

Moreover, off the exceptional case (1), if $S$ is a topological semigroup and
$f\in C(S)$, then $g,\chi$,$\chi^{*},\phi_{\chi}\in C(S)$.

###### Proof.

If $f=0$ it is easy to see that $g$ is arbitrary. This is cas (1). Now we
split the discussion in two cases according to whether $f$ and $g$ are
linearly dependent or not.
First case: $f$ and $g$ are linearly dependent. That is $g=\alpha f$ for some
constant $\alpha\in\mathbb{C}$. So equation (1.3) becomes

$f(x\sigma(y))=\alpha f(x)f(y)-\alpha f(y)f(x)=0,\quad x,y\in S.$

This implies that $f=0$ on $S^{2}$. This occurs in case (2).
Second case: $f$ and $g$ are linearly independent. According to Lemma 5.1, we
have $f^{*}=-f$ and $g^{*}=g+\beta f$, where $\beta\in\mathbb{C}$ is a
constant. Then if we apply Eq. (1.3) to the pair $(\sigma(x),y)$ we obtain

$f(xy)=f(x)g(y)+f(y)g(x)+\beta f(x)f(y),\quad x,y\in S.$

That is

$f(xy)=f(x)\left[g(y)+\dfrac{\beta}{2}f(y)\right]+f(y)\left[g(x)+\dfrac{\beta}{2}f(x)\right],\quad
x,y\in S.$

According to Theorem 2.3 and taking into account that $f$ and $g$ are linearly
indepenedent, we have the following possibilities:
(i) $f=c\left(\chi_{1}-\chi_{2}\right)$ and
$g+\dfrac{\beta}{2}f=\dfrac{\chi_{1}+\chi_{2}}{2}$, for some constant
$c\in\mathbb{C}\backslash\\{0\\}$ and
$\chi_{1},\chi_{2}:S\rightarrow\mathbb{C}$ are two different multiplicative
functions. Since $f=-f^{*}$, we get

$\chi_{1}+\chi_{1}^{*}=\chi_{2}^{*}+\chi_{2}.$

This implies that $\chi_{1}=\chi_{2}^{*}$ and $\chi_{2}=\chi_{1}^{*}$. This
occurs in case (3) with $\chi=\chi_{1}$, $\chi^{*}=\chi_{2}$ and
$c_{1}=\dfrac{-\beta c}{2}$. In addition $\chi_{2}^{*}=\chi_{1}$ implies that
$\chi\circ\sigma^{2}=\chi$.
(ii) $f=\phi_{\chi}$ and $g+\dfrac{\beta}{2}f=\chi$ where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function such that
$\phi_{\chi}^{*}=-\phi_{\chi}$. By applying Eq. (1.3) to the pair
$(\sigma(x),y)$ we obtain

$\phi_{\chi}(xy)=\phi_{\chi}(x)\chi(y)+\phi_{\chi}(y)\chi^{*}(x).$

On the other hand we have

$\phi_{\chi}(xy)=\phi_{\chi}(x)\chi(y)+\phi_{\chi}(y)\chi(x).$

Comparing these last two identities we can see that $\chi=\chi^{*}$ since
$f\neq 0$. This occurs in part (4) of Theorem 5.2 with
$c_{2}=\dfrac{-\beta}{2}$.

For the converse we can check easily that the forms (1)–(4) satisfy Eq. (1.3).
Finally, if $S$ is a topological semigroup, the continuity statements are easy
to verify. This completes the proof of Theorem (5.2). ∎

In the next section we shall apply our theory to two different types of
groups. The first one is abelian and the second one is not.

##  6\. Applications

######  Application 6.1.

Let $S=(\mathbb{R},+)$, let $\beta\in\mathbb{R}\backslash\\{0\\}$ be a fixed
element and let $\sigma(x)=\beta x$ for all $x\in\mathbb{R}$. The functional
equations (1.1) and (1.2) can be written respectively as follows :

$g(x+\beta y)=g(x)g(y)+f(x)f(y),\quad x,y\in\mathbb{R},$ (6.1) $f(x+\beta
y)=f(x)g(y)+f(y)g(x),\quad x,y\in\mathbb{R}.$ (6.2)

We note that equation (6.1) with $\beta=-1$ is [12, Example 4.18], and
equation (6.2) with $\beta=1$ is [12, Example 4.5]. We are interested to
determine the solutions of (6.1) and (6.2) when
$\beta\in\mathbb{R}\backslash\\{0,-1,1\\}$. For this we apply Theorem 3.3 to
Eq. (6.1) and Theorem 4.3 to Eq. (6.2). Let $\chi:S\rightarrow\mathbb{C}$ be a
non-zero multiplicative function such that

$\chi(\beta x)=\chi(x),\ \text{for all}\ x\in\mathbb{R}.$

Since $S$ is a group, then $\chi$ is a character. So we get
$\chi\left((\beta-1)x\right)=1$ for all $x\in\mathbb{R}$. Since $\beta\neq 1$,
we obtain $\chi=1$. By the same way we show that the only non-zero
multiplicative function $\chi$ satisfying $\chi(\beta^{2}x)=\chi(x)$ for all
$x\in\mathbb{R}$ is $\chi=1$ because $\beta\neq\pm 1$. So the special sine
addition law (1.4) becomes

$\phi(x+y)=\phi(x)+\phi(y),\quad x,y\in\mathbb{R}.$

That is $\phi$ additive. In addition if $\phi(\beta x)=\phi(x)$ for all
$x\in\mathbb{R}$, then $\phi=0$ since $\beta\neq 1$.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (6.1) are the following:
1) $f=0$ and $g=0$.
2) $f=\dfrac{\alpha}{1+\alpha^{2}}$ and $g=\dfrac{1}{1+\alpha^{2}}$, where
$\alpha\in\mathbb{C}\backslash\\{i,-i\\}$.
3) $f=0$ and $g=1$.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (6.2) can be listed as
follows:
1) $f=0$ and $g$ is arbitrary.
2) $f=\dfrac{1}{2\alpha}$ and $g=\dfrac{1}{2}$, where
$\alpha\in\mathbb{C}\backslash\\{0\\}$.

######  Application 6.2.

Let $G$ be the $(ax+b)$–group defined by

$G:=\left\\{\left(\begin{matrix}a&b\\\ 0&1\\\ \end{matrix}\right)\mid
a>0,\quad b\in\mathbb{R}\right\\},$

and let $X=\left(\begin{matrix}a&b\\\ 0&1\\\ \end{matrix}\right)$ for all
$a,b\in\mathbb{R}$ such that $a>0$. We consider the following automorphism on
$G$

$\sigma\left(X\right)=\left(\begin{matrix}a&2023b\\\ 0&1\\\
\end{matrix}\right).$

So $\sigma$ is not involutive. According to [12, Example 2.10, Example 3.13],
the continuous additive and the non-zero multiplicative functions on $G$ have
respectively the forms

$A_{c}\left(X\right)=c\log(a),$

and

$\chi_{\lambda}\left(X\right)=a^{\lambda},$

where $c,\lambda\in\mathbb{C}$. We can see that
$\chi_{\lambda}\circ\sigma=\chi_{\lambda}$ and $A_{c}\circ\sigma=A_{c}$, and
it is well known that the non-zero continuous solution $\phi$ of (1.4) on the
group $G$ will be of the form $\phi=\chi_{\lambda}A_{c}$.

The non-zero solutions $f,g\in C(G)$ of Eq. (1.1) are the following:
1) $f\left(X\right)=\dfrac{\alpha a^{\lambda}}{1+\alpha^{2}}$ and
$g\left(X\right)=\dfrac{a^{\lambda}}{1+\alpha^{2}}$, where
$\alpha\in\mathbb{C}\backslash\left\\{0,i,-i\right\\}$ and
$\lambda\in\mathbb{C}$.
2) $f\left(X\right)=-ica^{\lambda}\log(a)$ and $g\left(X\right)=a^{\lambda}\pm
ca^{\lambda}\log(a)$, where $c\in\mathbb{C}\backslash\left\\{0\right\\}$ and
$\lambda\in\mathbb{C}$.

The non-zero solutions $f,g\in C(G)$ of Eq. (1.2) are the following:
1) $f\left(X\right)=\dfrac{a^{\lambda}}{2\alpha}$ and
$g\left(X\right)=\dfrac{a^{\lambda}}{2}$, where
$\alpha\in\mathbb{C}\backslash\left\\{0\right\\}$ and $\lambda\in\mathbb{C}$.
2) $f\left(X\right)=ca^{\lambda}\log(a)$ and $g\left(X\right)=a^{\lambda}$,
where $c\in\mathbb{C}\backslash\left\\{0\right\\}$ and $\lambda\in\mathbb{C}$.

### Declarations

Ethical Approval Not Applicable.

Conmpeting interests None.

Author contributions The authors confirm contribution to the paper as follows:
study conception and design: Y. Aserrar, E. Elqorachi; data collection: Y.
Aserrar; analysis and interpretation of results: Y. Aserrar, E. Elqorachi;
draft manuscript preparation: Y. Aserrar. All authors reviewed the results and
approved the final version of the manuscript.

Funding None.

Availability of data and materials Not applicable.

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††thanks: These authors contribute equally to this work.††thanks: These
authors contribute equally to this work.

# Stripe Symmetry of Short-range Charge Density Waves in Cuprate
Superconductors

Jaewon Choi  Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United
Kingdom  Jiemin Li  Diamond Light Source, Harwell Science and Innovation
Campus, Didcot OX11 0DE, United Kingdom  Beijing National Laboratory for
Condensed Matter Physics and Institute of Physics, Chinese Academy of Science,
Beijing 100190, China  Abhishek Nag  Diamond Light Source, Harwell Science and
Innovation Campus, Didcot OX11 0DE, United Kingdom  Jonathan Pelliciari
Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  National Synchrotron Light Source II, Brookhaven
National Laboratory, Upton, New York 11973, USA  Hannah Robarts  Diamond Light
Source, Harwell Science and Innovation Campus, Didcot OX11 0DE, United Kingdom
H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United
Kingdom  Charles C. Tam  Diamond Light Source, Harwell Science and Innovation
Campus, Didcot OX11 0DE, United Kingdom  H. H. Wills Physics Laboratory,
University of Bristol, Bristol BS8 1TL, United Kingdom  Andrew Walters
Diamond Light Source, Harwell Science and Innovation Campus, Didcot OX11 0DE,
United Kingdom  Stefano Agrestini  Diamond Light Source, Harwell Science and
Innovation Campus, Didcot OX11 0DE, United Kingdom  Mirian García-Fernández
Diamond Light Source, Harwell Science and Innovation Campus, Didcot OX11 0DE,
United Kingdom  Dongjoon Song  National Institute of Advanced Industrial
Science and Technology (AIST), Tsukuba, Ibaraki 305-8560, Japan  Hiroshi
Eisaki  National Institute of Advanced Industrial Science and Technology
(AIST), Tsukuba, Ibaraki 305-8560, Japan  Steven Johnston  Department of
Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee
37996, USA  Institute for Advanced Materials and Manufacturing, The University
of Tennessee, Knoxville, Tennessee 37996, USA  Riccardo Comin  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Hong Ding  Beijing National Laboratory for Condensed Matter
Physics and Institute of Physics, Chinese Academy of Science, Beijing 100190,
China  Ke-Jin Zhou<EMAIL_ADDRESS>Diamond Light Source, Harwell
Science and Innovation Campus, Didcot OX11 0DE, United Kingdom

###### Abstract

The omnipresence of charge density waves (CDWs) across almost all cuprate
families underpins a common organizing principle. However, a longstanding
debate of whether its spatial symmetry is stripe or checkerboard remains
unresolved. While CDWs in lanthanum- and yttrium-based cuprates possess a
stripe symmetry, distinguishing these two scenarios has been challenging for
the short-range CDW in bismuth-based cuprates. Here, we employed high-
resolution resonant inelastic x-ray scattering to uncover the spatial symmetry
of the CDW in Bi2Sr2-xLaxCuO6+δ. Across a wide range of doping and
temperature, anisotropic CDW peaks with elliptical shapes were found in
reciprocal space. Based on Fourier transform analysis of real-space models, we
interpret the results as evidence of unidirectional charge stripes, hosted by
mutually 90∘-rotated anisotropic domains. Our work paves the way for a unified
symmetry and microscopic description of CDW order in cuprates.

INTRODUCTION

High-temperature copper-oxide (or “cuprate”) superconductors are prime
examples of quantum materials that exhibit a unique set of broken-symmetry
phenomena, such as pseudogap, strange metal behavior, nematicity, and various
density-wave states Keimer  _et al._ (2015); Norman (2011). Among these,
charge-density waves (CDWs) — periodic modulations of valence electron density
— have been ubiquitously discovered in both hole- and electron-doped cuprates
Tranquada  _et al._ (1995, 1996); v. Zimmermann  _et al._ (1998); Fujita  _et
al._ (2002); Abbamonte  _et al._ (2005); Wu  _et al._ (2011); Ghiringhelli
_et al._ (2012); Chang  _et al._ (2012); Hanaguri  _et al._ (2004); Hoffman
_et al._ (2002); Howald  _et al._ (2003); Wise  _et al._ (2008); Vershinin
_et al._ (2004); da Silva Neto  _et al._ (2014); Cai  _et al._ (2016); Li  _et
al._ (2021); Comin  _et al._ (2014); da Silva Neto  _et al._ (2015, 2016).
This universality has prompted suggestions of a unified principle underlying
their formation.

Figure 1: Angle-dependent RIXS experiment on charge-density wave (CDW) in La-
Bi2201. a, Schematic illustration of experiment. By changing an in-plane
sample rotation $\phi$, $H$\- and $K$-CDWs are sliced along different
directions, defined by azimuthal angle $\alpha$, in ($H$, $K$) plane. b,c,
Representative RIXS spectra at $\textbf{Q}=(0.25,0)$ with Cu $L_{3}$ (b) and O
$K$ resonance (c). Spectral weight from quasielastic, phonon, and paramagnon
contribution is highlighted by red or cyan, dark-grey and light-grey color,
respectively. d,e, Integrated intensity of quasielastic peaks extracted from
RIXS spectra as a function of $H$ at Cu $L_{3}$\- and O $K$-edge. Solid lines
are Gaussian fits with polynomial background. f,g, Background-subtracted
momentum-dependent profiles of integrated intensity across $H$-CDW (f) and
$K$-CDW (g) at Cu $L_{3}$-edge with different scan angle $\alpha$. The insets
depict scan directions. The number on the right side of each profile
corresponds to fitted full-width half-maximum (FWHM). h, FWHMs of $H$\- and
$K$-CDW peaks measured at both absorption edges are plotted as a function of
$\alpha$. The grey solid line indicates the least square fit to elliptical
equation (See Methods). All data were obtained at $T=20$ K.

However there has been a longstanding controversy, that is, whether its
spatial symmetry, $i.e.$, the spatial distribution of the CDW ordered state,
is bidirectional “checkerboard” or unidirectional “stripe” Comin and
Damascelli (2016). Although the presence of intertwined spin-charge stripe
order is widely perceived, especially in lanthanum (La)-based cuprates
Tranquada  _et al._ (1996), the formation of mutually 90∘-rotated charge
stripe domains due to two equivalent Cu-O bond directions in square-lattice
CuO2 plane leads to experimental results virtually indistinguishable from
those of checkerboard CDW. For example, both CDW symmetries commonly generate
four-fold electronic diffraction peaks along $H$ and $K$ directions in
reciprocal space Comin  _et al._ (2015a, b). Recent resonant x-ray scattering
(RXS) experiments employed uniaxial pressure as an external symmetry-breaking
perturbation to lift the degeneracy of two symmetry candidates, and
successfully revealed a unidirectional nature of the CDW in La- Choi  _et al._
(2022); Simutis  _et al._ (2022) and yttrium (Y)-based cuprates Kim  _et al._
(2018, 2021). For bismuth (Bi)-based compounds, one of three prototypical
material classes in cuprates, the CDW symmetry remains elusive. A plethora of
scanning tunnelling microscopy (STM) studies reported a short-range
checkerboard-like electronic modulation along Cu-O bond directions in real
space with a four-unit-cell periodicity Hoffman  _et al._ (2002); Howald  _et
al._ (2003); Wise  _et al._ (2008); Vershinin  _et al._ (2004); da Silva Neto
_et al._ (2014); Cai  _et al._ (2016); Li  _et al._ (2021). Interestingly,
several STM studies revealed unidirectional nano-domains of electronic
modulation breaking fourfold rotational symmetry Kohsaka  _et al._ (2007);
Lawler  _et al._ (2010); Kohsaka  _et al._ (2012).

Elucidating the CDW symmetry in Bi-based cuprates will therefore be
indispensable for answering the question: is the unidirectional “stripe” CDW
symmetry universal for all different cuprate materials? Since the CDW is
intimately related with superconductivity, this question may be essential for
understanding the interplay of CDW, SC and other broken-symmetry states. For
example, there have been a number of theoretical works suggesting that the
fluctuating stripe may play a significant role in the mechanism of SC Kivelson
_et al._ (1998, 2003); Cvetkovic  _et al._ (2008); Huang  _et al._ (2017); Mai
_et al._ (2022). In addition, a unidirectional pair-density-wave order (PDW),
where a SC order parameter is intertwined with CDW and periodically modulated,
has been proposed as a “primary” order in cuprates Fradkin  _et al._ (2010).
The exact answer to the question of CDW spatial symmetry is pivotal to
differentiate microscopic models of PDW order parameter Agterberg  _et al._
(2020).

Figure 2: Doping and temperature evolution of anisotropic CDW correlation.
a-f, Full-width half-maximum of both $H$-CDW (upper panels) and $K$-CDW (lower
panels) peaks in La-Bi2201 with different doping concentrations as stated.
Solid lines are least square fits to elliptical equation (see Methods). g,h,
Doping concentration $x$ (g) and temperature $T$ dependence (h) of
longitudinal and transverse CDW correlations. The grey dashed line indicates
the superconducting transition temperature $T_{c}$. The colored trends are
guided to the eye.

To address this question, we performed high-resolution resonant inelastic
x-ray scattering (RIXS) measurements on Bi2Sr2-xLaxCuO6+δ (La-Bi2201) to map
out the CDW patterns in a broad range of reciprocal space. Our systematic
studies reveal that the CDW patterns in La-Bi2201 possess an elliptical shape,
elongated along the transverse direction. This anisotropic CDW pattern was
robustly found in a wide range of temperatures and doping concentrations. By
Fourier-transform analysis of real-space models, we concluded that equally
populated 90∘-rotated domains of charge stripe orders lead to anisotropic CDW
peaks. Our results not only provide evidence for the unidirectional stripe
symmetry in Bi-based cuprates, but also suggest that the stripe CDW symmetry
is another universal property shared among different cuprate material
families.

RESULTS

Anisotropic CDW peaks
The CDW order in La-Bi2201 system manifests itself as quasielastic scattering
intensity peaks at four symmetry-equivalent positions in reciprocal space,
($H$, $K$) = ($\pm\delta$, 0) and (0, $\pm\delta$), where the
incommensurability $\delta\sim{1/4}$ reciprocal lattice unit (r.l.u.) is
consistent with the previous reports da Silva Neto  _et al._ (2014); Li  _et
al._ (2020). By rotating the in-plane azimuthal angle $\phi$ with respect to
the CDW peak, we obtained momentum-dependent RIXS spectra across the CDW
reflections at ($\delta$, 0) and (0, $\delta$), hereafter denoted as $H$\- and
$K$-CDW, with different scan angle $\alpha$ as illustrated in Fig. 1a. Here,
$\alpha$ is defined as the angle between the actual scan and the longitudinal
propagating direction. The incoming photon energy was tuned to the resonances
of Cu $L_{3}$-edge ($\sim$931.6 eV) and O $K$-edge ($\sim$528.4 eV) absorption
energy to maximize the sensitivity to CDW. Energy-resolved high-resolution
RIXS has an advantage of disentangling a weak quasielastic short-range CDW
response from the phonon modes as well as the spin and orbital excitations
(Fig. 1b,c), comparing to the energy-integrated RXS Comin  _et al._ (2014). An
enhancement in the integrated quasielastic scattering intensity is clearly
identified near $\textbf{Q}=(0.25,0)$ and $(0,0.25)$ at both Cu and O sites,
confirming its multiorbital nature Li  _et al._ (2020) (Fig. 1d,e).

Figure 3: Model CDW modulation in real and reciprocal space. a-d, Real-space
electron density modulation map $\rho(x,y)$ calculated by assuming equal
population of (a) anisotropic 90∘-rotated stripe domains, (b) 90∘-rotated
anisotropic checkerboard domains, (c) 90∘-rotated isotropic stripe domains,
and (d) isotropic checkerboard domains. e-h, Corresponding CDW peak patterns
in reciprocal space, obtained by direct Fourier transformation of (a-d),
respectively.

To gain an overview of CDW patterns, we show representative CDW scans obtained
at Cu $L_{3}$-edge at selected values of $\alpha$ across $H$-CDW (Fig. 1f) and
$K$-CDW (Fig. 1g) of underdoped La-Bi2201 ($x=0.6$, UD23K). The full-width
half-maximum (FWHM) of $H$-CDW in the transverse direction
($\alpha=90^{\circ}$) is noticeably larger than that in the longitudinal
direction ($\alpha=0^{\circ}$), demonstrating an anisotropic CDW peak
structure (Fig. 1f). Examination of $K$-CDW reveals the same anisotropy with
comparable FWHMs (Fig. 1g). We also found the elongated CDWs present along
both $H$ and $K$ directions at O $K$-edge (see Supplementary Information).

We now scrutinize the analysis on the shape of the individual CDW peaks. Fig.
1h summarises the extracted FWHM of $H$-CDW from both Cu $L_{3}$\- and O
$K$-edges as well as those of $K$-CDW as a function of $\alpha$ from -97.5∘ to
150∘ (Supplementary Figs. 2 and 3). Clearly the data show an oscillatory
behavior as a function of $\alpha$. The least square fit of the data to an
elliptical equation (the grey solid line in Fig. 1h, Methods) returns the
maximum $\sim 0.118$ r.l.u. near $\alpha=90^{\circ}$ and the minimum $\sim
0.066$ r.l.u. near $\alpha=0^{\circ}$. The major and minor axes along 0∘ and
90∘ indicate that the CDW propagation is locked along two orthogonal Cu-O bond
directions. The aspect ratio of 1.8 implies the CDW correlation, $i.e.$,
$\xi=2/\textrm{FWHM}$, in the longitudinal direction
($\xi_{\parallel}\sim$18.7 Å) is almost twice as long as that in the
transverse direction ($\xi_{\perp}\sim$10.4 Å). In Fig. 2c,d, the same set of
data are plotted in polar coordinates. The shape of the individual CDW peak is
best described as an ellipse elongated along the transverse direction,
perpendicular to the direction of CDW propagation. The $K$-CDW ellipse
possesses almost the same size but rotated by $90^{\circ}$, implying that the
CDW pattern in reciprocal space preserves fourfold ($C_{\textrm{4}}$)
symmetry, seemingly reflecting the nature of the underlying CuO2 square-
lattice.

Temperature and doping evolution
We extended the same RIXS study to explore the doping and temperature
dependence. Fig. 2a,b, and Fig. 2e,f exhibit the CDW peaks obtained from a
non-superconducting ($x=0.8$, non-SC) and an optimally-doped ($x=0.4$, OD30K)
La-Bi2201, respectively. Fig. 2g summarises the longitudinal correlation
length, $\xi_{\parallel}$, and the transverse correlation length,
$\xi_{\perp}$, of the samples with different doping concentration $x$.
$\xi_{\parallel}(x)$ reaches its maximum at $x=0.6$. On the other hand,
$\xi_{\perp}(x)$ shows a monotonically decreasing trend. The temperature
dependence of $\xi_{\parallel}$ and $\xi_{\perp}$ in OD30K sample are plotted
in Fig. 2h. $\xi_{\parallel}$ increases much faster below $T_{c}$, whereas
$\xi_{\perp}$ displays a continuous linear trend. These results confirm the
robust existence of the elongated CDW in a wide range of temperature and
doping concentration.

Modelling CDW patterns
The CDW peak patterns in reciprocal space probed by RIXS encode the spatial
symmetry of the CDW order in real space. We thus performed theoretical
simulations by constructing idealized models, that is, the real-space
electronic modulation with either stripe or checkerboard pattern, where both
anisotropic and isotropic domains are considered. For each scenario, two
individual CDW domains are included with a $90^{\circ}$ rotation from each
other to mimic the equally populated orthogonal domains (insets of Fig. 3a-d,
see Supplementary Information). The real-space electron density modulation
maps $\rho(x,y)$ were built by translating the orthogonal CDW domains in the x
and y directions with a random phase shift to represent the actual samples
probed by RIXS (Fig. 3a-d). Fourier transforms of the electron density maps
$\rho(x,y)$ yield a unique CDW reciprocal-space pattern for each scenario
(Fig. 3e-h). For instance, an anisotropic stripe symmetry produces elliptical
CDW diffraction peaks, preserving the C4 symmetry but elongated in the
transverse direction (Fig. 3e). The anisotropic checkerboard domains give rise
to star-shaped CDW peaks elongated along both the transverse and longitudinal
directions (Fig. 3f). On the other hand, isotropic checkerboard and isotropic
stripe phases generate virtually indistinguishable isotropic CDW peaks (Fig.
3g,h). Remarkably, our experimental observations in La-Bi2201 (Fig. 2)
coincide with the spatial symmetry of CDW order having unidirectional stripe,
rather than bidirectional checkerboard. A consistent conclusion is achieved
based on similar Fourier transform analysis on more realistic electron density
maps (Supplementary Fig. 6) Fine (2016).

DISCUSSION

At the first glance, it may look perplexing that we straightforwardly obtain
the unambiguous agreement between experiment and theoretical modeling, while
the CDW symmetry is still debatable despite a considerable number of STM
studies Hoffman  _et al._ (2002); Howald  _et al._ (2003); Hanaguri  _et al._
(2004); Wise  _et al._ (2008); Vershinin  _et al._ (2004). The static
checkerboard-like modulation was proposed for Bi-based cuprates from tunneling
electron conductance maps. In their Fourier-transformed images, fourfold
symmetry-equivalent peaks typically appear at an incommensurate wavevector
position Hoffman  _et al._ (2002); Howald  _et al._ (2003); Wise  _et al._
(2008); Vershinin  _et al._ (2004); da Silva Neto  _et al._ (2014); Cai  _et
al._ (2016); Li  _et al._ (2021). However, the conservation of $C_{4}$
symmetry is not a unique signature of the checkerboard-like modulation pattern
as the coexistence of orthogonal domains of unidirectional stripe-like CDW
also produces fourfold peak pattern. More sophisticated analysis on tunneling
asymmetry at the pseudogap energy scale reveals the presence of four-unit-
cell-wide unidirectional nanodomains breaking $C_{4}$ rotational symmetry
Kohsaka  _et al._ (2007); Lawler  _et al._ (2010); Kohsaka  _et al._ (2012).
This observation can be interpreted as a fluctuating “stripe” with short
correlation. It is worth noting that the nature of short-range CDW order
imposes severe challenges on the assessment of the symmetry based on real-
space images which manifest visually checkerboard-like CDW patterns regardless
of whether its intrinsic symmetry is stripe or checkerboard, as shown by our
modelled density maps (Fig. 3a-d) as well as advanced theoretical studies
Robertson  _et al._ (2006). In this sense, RIXS is more advantageous as it
probes the CDW spatial symmetry directly in reciprocal space, despite the
short-range nature of the CDW order.

Elongated CDW peaks in reciprocal space were also found in Y-based cuprates by
RXS and proposed as evidences of unidirectional stripe-like CDW order Comin
_et al._ (2015a). The interpretation was challenged as the corresponding real-
space correlation function $C(x,y)$, $i.e.$, the direct Fourier transform of
the reciprocal-space CDW pattern, similar to Fig. 3e, appears checkerboard-
like Fine (2016). The calculation of $C(x,y)$ via auto-correlation of the
real-space electron density maps (Fig. 3a-d) confirmed all four scenarios
induce comparable checkerboard-like patterns akin to previous reports
(Supplementary Fig. 5) Fine (2016). Therefore, it is challenging to directly
visualize the symmetry of a short-range CDW from either the electron density
map $\rho(x,y)$ or correlation function $C(x,y)$ in real space. Instead of
examining the shape of CDW peaks, a recent RIXS study under uniaxial pressure
reveals a unidirectional nature of CDW in Y-based cuprates Kim  _et al._
(2021). The anisotropic CDW peaks we observed in La-Bi2201 are in stark
contrast to isotropic CDW peaks reported in La-based Thampy  _et al._ (2013,
2014); Wang  _et al._ (2020, 2022) and Hg-based cuprate materials Campi  _et
al._ (2015). For La1.88Sr0.12CuO4, twinned domains of slanted stripe order
with $\pm 3^{\circ}$ away from Cu-O bond directions result in two CDW peaks
along transverse direction, eventually merging into a single isotropic peak
under mild uniaxial pressure Wang  _et al._ (2022). Despite the isotropic peak
shape, x-ray diffraction employing uniaxial pressure unambiguously verifies
charge stripe order Choi  _et al._ (2022).

Alongside the results from La- and Y-based cuprates, our RIXS measurements on
La-Bi2201 corroborates unidirectional stripes as the fundamental symmetry of
CDW across different cuprate families. This remarkable universality not only
provides a perspective on the unified organizing principle of CDW and its role
in cuprate phase diagram, but also constrains the symmetry in microscopic
models describing CDW and the interplay with other phases.

METHODS

Sample growth and characterization
High-quality single crystals of Bi2Sr2-xLaxCuO6+δ (La-Bi2201) with hole-doping
concentrations of $x=0.8$ (non-SC), $x=0.6$ ($T_{c}=23$ K, UD23K), $x=0.4$
($T_{c}=30$ K, OD30K) were synthesized by the travelling-solvent floating-zone
method. The as-grown samples were annealed at 650∘C in an oxygen atmosphere
for 2 days to improve the homogeneity of the samples. The sample orientation
was characterized by Laue diffraction prior our resonant x-ray inelastic
scattering (RIXS) experiments. Superstructure reflections propagating along
Cu-Cu bond direction, or equivalently along ($H$, $H$) direction, were
identified by the Laue diffraction method. All samples were cleaved by
mechanical exfoliation, before transferring to a loadlock chamber.

X-ray absorption spectroscopy
X-ray absorption spectra of La-Bi2201 samples were collected with a normal
incidence geometry ($\theta=90^{\circ}$) prior to the RIXS experiments. The
total electron yield method was employed by measuring the drain current from
the samples. Supplementary Figs. 1a,b show representative x-ray absorption
spectra of La-Bi2201 $x=0.6$ (UD23K) near Cu $L_{3}$ and O $K$ absorption
edges, respectively. The data exhibit a single resonance peak at an energy of
931.6 eV and 528.4 eV.

High-resolution RIXS experiments
We performed high-resolution RIXS experiments on La-Bi2201 at the I21 RIXS
beamline of Diamond Light Source in the United Kingdom Zhou  _et al._ (2022).
The experimental geometry is schematically illustrated in Fig. 1a of the main
text. Samples were glued on the holder such that its surface normal is the
crystallographic $c$-axis direction. We define a wavevector Q employing
pseudo-tetragonal notation $\textbf{Q}=Ha^{*}+Kb^{*}+Lc^{*}$, or equivalently,
$\textbf{Q}=(H,K,L)$ where $a^{*}=2\pi/a$, $b^{*}=2\pi/b$, $c^{*}=2\pi/c$ are
the reciprocal lattice unit vectors and $H$, $K$, $L$ are the indices in
reciprocal space spanned by $a^{*}$, $b^{*}$, and $c^{*}$. The lattice
parameters of La-Bi2201 are $a=b=3.86$ Å and $c=24.69$ Å. The incoming x-rays
were tuned to the resonant energy of Cu $L_{3}$ and O $K$-edge to maximize the
signal sensitivity to CDW residing in the CuO2 plane. Linear vertical, or
$\sigma$ polarization was used throughout the measurements. The outgoing
x-rays emitted from the La-Bi2201 sample pass through paraboloidal collecting
optics with a large horizontal acceptance angle inside the main sample chamber
to enhance the x-ray photon throughput. The spherical grating optics
implemented on a spectrometer to vertically monochromatise emitted x-rays and
focus to an area detector. The energy resolution of 37.0 meV and 27.3 meV was
achieved for Cu $L_{3}$\- and O $K$-edge, respectively. The scattering angle
$2\theta$ is fixed to 154∘.

RIXS data fitting
RIXS spectra were firstly normalized to the incident photon flux and the
spectral weight of dd excitation integrated over an energy range of 1.2-3 eV.
The position of the quasi-elastic peak, or zero-energy position, was
determined by measuring the amorphous carbon sample. Compared to energy-
integrated resonant elastic x-ray scattering (REXS), the energy-resolved RIXS
has the advantage of filtering out unwanted contributions from inelastic
scatterings due to various low-energy excitations such as phonon, magnon, and
orbital (dd) excitations. To better disentangle quasielastic diffraction
intensity, the low-energy region of RIXS spectra measured at the Cu
$L_{3}$-edge was fitted with a sum of quasielastic, phonon, paramagnon, and
background contributions as follows:

$I_{RIXS}^{Cu}(E)=I_{el}^{Cu}(E)+I_{ph}^{Cu}(E)+I_{mag}^{Cu}(E)+I_{bg}^{Cu}(E).$
(1)

Similarly for the O $K$-edge,

$I_{RIXS}^{O}(E)=I_{el}^{O}(E)+I_{ph}^{O}(E)+I_{bg}^{O}(E).$ (2)

Four terms on the right-hand side of Eq. (1) represent a quasielastic, phonon,
paramagnon, and background contribution on the RIXS intensity, respectively.
The quasielastic peak and phonon excitation are described by Gaussian
functions, whereas the paramagnon excitation is modelled by the response
function of a damped harmonic oscillator multiplied with the Bose factor
Monney  _et al._ (2016), as follows:

$\displaystyle
I_{el}^{Cu,O}(E)=A_{1}exp\bigg{\\{}{\frac{1}{2}\Big{(}\frac{E-A_{2}}{A_{3}}\Big{)}^{2}}\bigg{\\}}$
(3) $\displaystyle
I_{ph}^{Cu,O}(E)=\sum_{i=1}^{n}B_{3i-2}exp\bigg{\\{}{\frac{1}{2}\Big{(}\frac{E-B_{3i-1}}{B_{3i}}\Big{)}^{2}}\bigg{\\}}$
(4) $\displaystyle
I_{mag}^{Cu}(E)=C_{1}\bigg{\\{}\frac{C_{3}E}{(E^{2}-C_{2}^{2})^{2}+E^{2}C_{3}^{2}}\bigg{\\}}\times\bigg{\\{}\frac{1}{1-e^{-E/C_{4}}}\bigg{\\}}$
(5) $\displaystyle I_{bg}^{Cu,O}(E)=D_{1}E^{2}+D_{2}E+D_{3}$ (6)

An integer $n$ indicates the number of phonon modes: $n=1$ (only bond-
stretching mode) for the Cu $L_{3}$-edge and $n=2$ (bond-stretching and bond-
buckling modes) for the O $K$-edge Li  _et al._ (2020). $A_{1},A_{2},A_{3}$,
$B_{1},\cdots,B_{n}$, and $C_{1},C_{2},C_{3}$ are the fitting parameters.
$C_{4}$ was fixed to $k_{B}T$ with the measuring temperature $T$.

Angular-dependent CDW measurement
Taking advantage of broad accessibility to in-plane rotation angle and
accurate control of sample manipulator, we obtained angle- and momentum-
dependent RIXS spectra across both $H$\- and $K$-CDWs. By controlling the
polar angle $\theta$, the azimuthal rotation angle $\phi$, and the tilt angle
$\chi$ using diffcalc, momentum-dependent RIXS scans can be obtained across
the CDW peak with different scan angle $\alpha$. Here, the scan angle $\alpha$
is defined as an angle between the scan direction and the longitudinal scan
direction for each CDW. The representative scan directions are schematically
described in the inset of Figs. 1f,g. For Cu $L_{3}$-edge, the CDW peaks were
measured in a range between -97.5∘ and 150∘. For O $K$-edge, $\alpha$ was
varied from 0∘ to 135∘. Supplementary Figs. 2 and 3 present the momentum-
dependence of integrated quasielastic intensity scans on $H$-CDW taken at
different $\alpha$ as indicated. The momentum-dependence data is fitted with a
combination of Gaussian functions, which represents a CDW peak, and a
composite background.

$\begin{split}I_{CDW}(Q)=G_{1}exp\bigg{\\{}{\frac{1}{2}\Big{(}\frac{Q-G_{2}}{G_{3}}\Big{)}^{2}}\bigg{\\}}+G_{5}exp\big{(}\frac{x-G_{6}}{G_{7}}\big{)}\\\
+G_{8}Q^{2}+G_{9}Q+G_{10}.\end{split}$ (7)

A full-width half-maximum (FWHM) of CDW peak is calculated by
$2\sqrt{2\textrm{ln}2}G_{3}$ for each momentum-dependent profile. In
Supplementary Figs. 2 and 3, the least-square fitted results are shown with
solid lines at different scan angle $\alpha$. The FWHMs of CDW peak are
written in each panel. The angle-dependent FWHM shown in Fig. 1h is fitted
with an elliptical equation stated below:

$FWHM(\alpha)=\frac{R_{1}R_{2}}{\sqrt{\big{\\{}bcos(\alpha+\beta)\big{\\}}^{2}+\big{\\{}bsin(\alpha+\beta)\big{\\}}^{2}}}$
(8)

where $R_{1}$ ($R_{2}$) is the major (minor) axis of the ellipse and $\beta$
is an offset angle. The grey-coloured line presented in Fig. 1h is the result
of least square fitting which gives $R_{1}=0.1184\pm 0.0022$ r.l.u.,
$R_{2}=0.0656\pm 0.0013$ r.l.u., and $\beta=1.2\pm 1.9^{\circ}$. The offset
angle $\beta\sim 0$ indicates that the CDW propagation is locked along two
mutually-orthogonal Cu-O bond directions. For La-Bi2201 $x=0.6$ (UD23K)
samples, the data were obtained for three sample pieces from different
batches.

Acknowledgements
We thank Johan Chang, Stephen M. Hayden, Wei-Sheng Lee, and Qisi Wang for
fruitful discussions. We also acknowledge Thomas Rice, the beamline technician
at I21 beamline, for his technical supports. We thank Gavin Stenning and
Daniel Nye for the Laue diffraction in the Materials Characterisation
Laboratory at the ISIS Neutron and Muon Source. This work was primarily
supported by in-house research program of Diamond Light Source, Ltd. through
beamtime proposals NT21277 and NT29623. J. L. acknowledges Diamond Light
Source, Ltd. and Institute of Physics in Chinese Academy of Science for
providing funding Grant 112111KYSB20170059 for the joint Doctoral Training
under Contact STU0171. H. R. acknowledges funding and support from Engineering
and Physical Sciences Research Council Centre for Doctoral Training through
Grant EP/L015544/01 and EP/R0111141/1. H. D. acknowledges the financial
support from the National Science Foundation of China (Grant 11888101), and
the Ministry of Science and Technology of China (Grant 2016YFA0401000). S. J.
acknowledges support from NSF under Grant DMR-1842056. J. P. acknowledges
financial support by the SNSF Postdoc Mobility Project P400P2_180744.

Author contributions
K. Z. conceived and supervised the project. J. C., J. L., A. N., J. P., H. R.,
C. C. T., A. W., S. A., M. G.-F. and K. Z. performed RIXS experiment. D. S.
and H. E. synthesized and prepared the materials. J. C., J. L., and K. Z.
analyzed the data. J. C. and K. Z. wrote the manuscript with input from all
coauthors.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information is available for this paper. Correspondence and
requests for materials should be addressed to Jaewon Choi and Ke-Jin Zhou.

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##  I Modeling CDW - simplified cases

To understand the physical implication of CDW patterns in reciprocal space, we
constructed electronic density maps using simplified real-space models by
taking into account the following assumptions based on experimental
observations:

  * •

The spatial symmetry of CDW is either stripe or checkerboard. The CDW
propagates along two Cu-O bond directions (either crystallographic a or b
direction) with a four-unit-cell periodicity.

  * •

The shape of domains hosting CDW is either isotropic or anisotropic. The size
of individual domains is proportional to the CDW correlation.

  * •

In the case of stripe CDW, two kinds of CDW domains propagating along either a
or b direction are equally populated, as the system does not energetically
favor one or another. We thus simulate the situation by having mutually 90∘
rotated anisotropic domains with equal populations.

  * •

The real space electron density maps comprise domains distributed with a
random phase shift.

1\. Individual CDW domains

For the first step, a real-space electronic density modulation of individual
CDW domains was modelled. Supplementary Fig. 4 shows individual CDW domains
with (a,b) anisotropic horizontal and vertical stripe, (c,d) anisotropic
horizontal and vertical checkerboard, (e,f) isotropic horizontal and vertical
stripe, and (g) isotropic checkerboard modulations with 4-unit-cell
periodicity, propagating along two orthogonal Cu-O bond directions.

2\. Model Real-space intensity maps
For the next step, an electronic density modulation map $\rho(x,y)$
(1000$\times$1000 unit cells) was modelled in real space by translating the
individual CDW domains into $x$ and $y$ directions with a random phase shift.
Model $\rho(x,y)$ employing individual domains as building blocks allows us to
reveal how the CDWs with different spatial symmetry result in different
macroscopic real-space and reciprocal-space intensity patterns. Supplementary
Figs. 5a-c represent electronic density map $\rho(x,y)$ in real space which
consists equally populated horizontal and vertical domains which are mutually
rotated by 90∘ from each other and are profiled with (a) anisotropic stripe,
(b) anisotropic checkerboard, (c) isotropic stripe, respectively.
Supplementary Fig. 5d corresponds to the case having isotropic checkerboard
domains. The microscopic structure of individual domains is depicted in the
insets for each panel. These maps do not necessarily illustrate an electronic
modulation in a single layer, as random translation inevitably leads to
overlaps of CDW domains. Considering a penetration depth $\lambda>100$ nm for
the Cu $L_{3}$-edge and $\lambda>50$ nm for the O $K$-edge, RIXS measures an
average of more than 40 (Cu) or 20 (O) unit cells stacked along $c$-axis.
Therefore, the real-space density maps shown in Supplementary Fig. 5a-d
provide an excellent model of the electronic modulations in La-Bi2201 probed
by RIXS technique. $\rho(x,y)$ in all four scenarios look very similar to each
other. It is visually impossible to distinguish whether the intrinsic symmetry
of individual CDW domains is stripe or checkerboard, solely based on
$\rho(x,y)$. This difficulty had already been raised by Robertson et al.
(2006) Robertson  _et al._ (2006), especially in highly disordered systems
hosting short-ranged CDW where its correlation length $\xi_{\textrm{CDW}}$ is
comparable to the periodicity, such as in La-Bi2201.

3\. Simulated CDW patterns
Supplementary Figs. 5e-h show intensity maps $S(H,K)$ in reciprocal space
obtained by a direct Fourier transformation of $\rho(x,y)$ shown in
Supplementary Figs. 5a-d. The calculated $S(H,K)$ is equivalent to the
structure factor measured by scattering experiments. $S(H,K)$ from anisotropic
90∘-rotated stripe domains gives four CDW peaks elongated along the transverse
direction (Supplementary Figs. 5a,e), which can be clearly differentiated from
other three cases. Equally-populated anisotropic checkerboard domains result
in four CDW peaks with a cross-like shape (Supplementary Figs. 5b,f).
Interestingly, $S(H,K)$ from both isotropic stripe and checkerboard domains
are almost identical (Supplementary Figs. 5c,d,g,h). This implies that it is
difficult to differentiate the stripe and checkerboard modulation in
reciprocal space, when the CDW domains are isotropic.

4\. Correlation function
Fourier transforms of $S(H,K)$ give a density-density correlation function
$C(x,y)\cong\langle\rho(x+\Delta x,y+\Delta y)\rho(x,y)\rangle$, which is
mathematically equivalent to an auto-correlation of $\rho(x,y)$. Supplementary
Figs. 5i-l present $C(x,y)$ calculated from the auto-correlation of
$\rho(x,y)$ shown in Supplementary Fig. 5a-d. The results from all four
symmetry scenarios provide qualitatively similar checkerboard-like pattern.
This checkerboard-like $C(x,y)$ is consistent with Ref. Fine (2016). We
confirm that neither auto-correlation function $C(x,y)$, nor the electron
density map in real space could provide visually unambiguous clarification on
the symmetry being either stripe or checkerboard.

##  II Modeling CDW - more realistic cases

In this section, we generate more realistic electronic density map $\rho(x,y)$
in which distortions of continuous CDW domains are considered to represent
partially experimental observations by STM Hoffman  _et al._ (2002); Howald
_et al._ (2003); Wise  _et al._ (2008); Vershinin  _et al._ (2004); da Silva
Neto  _et al._ (2014); Cai  _et al._ (2016); Li  _et al._ (2021). The
structure factor $S^{\textrm{CDW}}(H,K)$ in the reciprocal space can be
described as follows:

$S^{\textrm{CDW}}(H,K)=\sum_{i=1}^{4}A_{i}\textrm{exp}\bigg{[}-\bigg{\\{}\frac{(H-Q_{x,i})^{2}}{2\sigma_{x,i}^{2}}+\frac{(K-Q_{y,i})^{2}}{2\sigma_{y,i}^{2}}\bigg{\\}}\bigg{]},$
(9)

Here, $A_{i}$ and $\textbf{Q}_{i}$, represent the CDW order parameter and
wavevector, respectively, are fixed to $A_{i}=1$,
$\textbf{Q}_{i}=(Q_{x,i},Q_{y,i})=(\pm 0.25,0)$ or $(0,\pm 0.25)$ commonly in
all three scenarios. $\sigma_{x,i}$ and $\sigma_{y,i}$ are the width of four
individual CDW peaks along the x and y directions, respectively. We
constructed $S^{\textrm{CDW}}(H,K)$ for the following three different
scenarios as shown in Supplementary Figs. 6a-c:

  1. (a)

Four anisotropic CDW peaks elongated along the transverse direction, where
$\sigma_{y,i}=2\times\sigma_{x,i}$ for $\textbf{Q}_{i}=(\pm 0.25,0)$ and
$\sigma_{x,i}=2\times\sigma_{y,i}$ for $\textbf{Q}_{i}=(0,\pm 0.25)$. The
calculated $S(H,K)$ is shown in Supplementary Fig. 6a, which is similar to
Supplementary Fig. 5e, describing 90∘-rotated anisotropic charge stripe
domains.

  2. (b)

Four isotropic CDW peaks, where $\sigma_{x,i}=\sigma_{y,i}$. The calculated
$S(H,K)$ is shown in Supplementary Fig. 6b, which is similar to Supplementary
Figs. 5g and 5h, describing isotropic stripe or checkerboard domains.

  3. (c)

Four star-like CDW peaks, obtained by the average of $S_{c1}(H,K)$ and
$S_{c2}(H,K)$. $S_{c1}(H,K)$ is identical to $S(H,K)$ in the scenario (a).
$S_{c2}(H,K)$ represents CDW peaks elongated along the longitudinal direction
where $\sigma_{x,i}=2\times\sigma_{y,i}$ for $\textbf{Q}_{i}=(\pm 0.25,0)$ and
$\sigma_{y,i}=2\times\sigma_{x,i}$ for $\textbf{Q}_{i}=(0,\pm 0.25)$. The
calculated $S(H,K)$ is shown in Supplementary Fig. 6c, which is similar to
that in Supplementary Fig. 5f, describing 90∘-rotated anisotropic checkerboard
domains.

Based on the structure factor $S(H,K)$ for the scenario (a)-(c) (Supplementary
Figs. 6a-c), we generated electron density maps $\rho(x,y)$ by a direct
Fourier transformation of $S(H,K)$ employing the following method Fine (2016):

$\rho(x,y)=\sum_{m}\sqrt{S(H_{m},K_{m})}cos(H_{m}x+K_{m}y+\phi)$ (10)

Here, $m$ is an index of each pixel in a discrete $1000\times 1000$ grid of
$S(H,K)$. $\phi$ is the random phase shift. The generated $\rho(x,y)$ map for
each $S(H,K)$ is presented in Supplementary Figs. 6d-f where only $100\times
100$ unit-cell grid is shown for the sake of visibility. They surprisingly
resemble the conductance map in real space obtained by STM Hoffman  _et al._
(2002); Howald  _et al._ (2003); Wise  _et al._ (2008); Vershinin  _et al._
(2004); da Silva Neto  _et al._ (2014); Cai  _et al._ (2016); Li  _et al._
(2021). Similar to Supplementary Figs. 5a-d, it is difficult to find
qualitative differences among these three $\rho(x,y)$ maps. Despite added
randomness, the Fourier-transformed $\rho(x,y)$ shown in Supplementary Figs.
6g-i demonstrate that the symmetry of CDW peak patterns is better revealed in
reciprocal space. These results are qualitatively consistent with those from
our idealized models summarized in Section II.

Supplementary Fig. 1: X-ray absorption spectroscopy on Bi2Sr1.4La0.6CuO6+δ.
a,b Cu $L_{3}$-edge (a) and O $K$-edge (b) x-ray absorption spectroscopy (XAS)
on Bi2Sr1.4La0.6CuO6+δ (La-Bi2201) sample collected with linear-vertical (LV)
polarization of incoming x-ray with normal incident geometry
($\theta=90^{\circ}$).

Supplementary Fig. 2: Charge density wave measured at Cu L3-edge. a-r,
Momentum-dependent profile of integrated quasi-elastic charge-density-wave
(CDW) peak intensity measured at different scan angle $\alpha$ as indicated
with a bold text. The incident photon energy was fixed to the resonance of Cu
$L_{3}$ absorption edge. The solid line is fit to Gaussian function
representing the CDW peak with a composite background (See Methods). The full-
width half-maximum (FWHM) of CDW peak is also presented on each panel.

Supplementary Fig. 3: Charge density wave measured at O K-edge. a-h, Momentum-
dependent profile of integrated quasi-elastic charge-density-wave (CDW) peak
intensity measured at different scan angle $\alpha$ as indicated with a bold
text. The incident photon energy was fixed to the resonance of O $K$
absorption edge. The solid line is fit to Gaussian function representing the
CDW peak with a polynomial background. The full-width half-maximum (FWHM) of
CDW peak is also presented on each panel.

Supplementary Fig. 4: individual CDW domains. a,b, Electronic density
modulation of an anisotropic stripe domain propagating into horizontal $x$ (a)
and vertical $y$ direction (b) with four-unit-cell periodicity. c-g, Same for
horizontally (c) and vertically anisotropic checkerboard domains (d),
isotropic charge stripe domain propagating into horizontal $x$ (e) and
vertical $y$ direction (f), and istoropic checkerboard domain (g). These
domains are used as building blocks for modeling real-space electronic density
map $\rho(x,y)$, illustrated in Supplementary Fig. 5.

Supplementary Fig. 5: Modeling CDW modulation using simplified electron
density maps. a-d, Real-space electronic density modulation map $\rho(x,y)$
generated by equal population of individual charge stripe and checkerboard
domains. The insets represent the microscopic structure of individual domains
for each scenario. e-h, Structure factor $S(H,K)$ in reciprocal space,
calculated by a direct Fourier transform of the corresponding $\rho(x,y)$ in
(a-d), respectively. i-l, Density-density correlation map $C(x,y)$ directly
computed by taking the auto-correlation of real-space map $\rho(x,y)$ in
(a-d).

Supplementary Fig. 6: Modeling CDW modulation for more realistic electron
density maps. a-c, Structure factors $S(H,K)$ calculated for three different
CDW peak patterns in reciprocal space. The parameters used in the calculation
are summarized in Section II. d-f, Model electronic density map $\rho(x,y)$
reconstructed from the panel (a-c), respectively, under the assumption of
random phase shifts. g-i, Fourier transformed intensity map of $\rho(x,y)$ in
(d-f).