# Solution of oligopoly market equilibrium problem using modified newton
method

A. Duttaa,1 and A. K. Dasb,2
_a Department of Mathematics, Jadavpur University, Kolkata, 700 032, India_
_b SQC & OR Unit, Indian Statistical Institute, Kolkata, 700 108, India_
_1 Email<EMAIL_ADDRESS>
_2 Email<EMAIL_ADDRESS>

###### Abstract

The paper aims to find the solution of oligopoly market equilibrium problem
through system of nonlinear equations. We propose modified newton method to
obtain the solution of system of nonlinear equations. We show that our
proposed method has higher order of convergence. Keywords: System of nonlinear
equations, modified newton method, oligopolistic market equilibrium problem.

##  1 Introduction

An oligopolistic market structure in which $n$ firms supply a homogeneous
product in a noncooperative fashion.

Following Murphy et al. [23] the oligopolistic market equilibrium problem is
as follows:

Let there be $n$ firms, which supply a homogeneous product in a noncooperative
fashion. Let $P(\tilde{Q}),\tilde{Q}\geq 0$ denote the inverse demand, where
$\tilde{Q}=\sum_{i=1}^{n}Q_{i},$ $Q_{i}\geq 0$ denote the $i$-th firm’s
supply. Let $c_{i}(Q_{i})$ be the total cost of supplying $Q_{i}$ units. Now
the Nash equilibrium solution is a set of nonnegative output levels
${Q_{1}}^{*},{Q_{2}}^{*},\cdots,{Q_{n}}^{*},$ such that ${Q_{i}}^{*}$ is an
optimal solution to the following problem $\forall i\in\\{1,2\cdots,n\\}:$

$\displaystyle\text{maximize}\
{Q_{i}P(Q_{i}+{\tilde{Q}_{i}}^{*})-c_{i}(Q_{i})}$ (1.1)

where ${\tilde{Q}_{i}}^{*}=\sum_{j\neq i}{Q_{j}}^{*}.$ Murphy et al. show that
if $c_{i}(Q_{i})$ is convex and continuously differentiable $\forall
i\in\\{1,2,\cdots,n\\}$ and the inverse demand function $P(\tilde{Q})$ is
strictly decreasing and continuously differentiable and the industry revenue
curve $\tilde{Q}P(\tilde{Q})$ is concave, then
$({Q_{1}}^{*},{Q_{2}}^{*},\cdots,{Q_{n}}^{*})$ is a Nash equilibrium solution
if and only if

$\displaystyle[P(\tilde{Q}^{*})+{Q_{i}}^{*}P^{\prime}(\tilde{Q}^{*})-{c_{i}}^{\prime}({Q_{i}}^{*})]{Q_{i}}^{*}=0$
(1.2)
$\displaystyle{c_{i}}^{\prime}({Q_{i}}^{*})-P(\tilde{Q}^{*})-{Q_{i}}^{*}P^{\prime}(\tilde{Q}^{*})\geq
0$ (1.3) $\displaystyle{Q_{i}}^{*}\geq 0\ \ \forall i\in\\{1,2,\cdots,n\\}$
(1.4)

where $\tilde{Q}^{*}=\sum_{i=1}^{n}{Q_{i}}^{*},$ which is a nonlinear
complementarity problem with
$f_{i}(z)={c_{i}}^{\prime}({Q_{i}}^{*})-P(\tilde{Q}^{*})-{Q_{i}}^{*}P^{\prime}(\tilde{Q}^{*}),$
and $z_{i}={Q_{i}}^{*}.$
Note that here the functions $c_{i}(Q_{i})$ and $-\tilde{Q}P(\tilde{Q})$ are
convex. So the 1st order derivative of these two functions are increasing
function. Hence the function
$f_{i}(z)={c_{i}}^{\prime}({Q_{i}}^{*})-P(\tilde{Q}^{*})-{Q_{i}}^{*}P^{\prime}(\tilde{Q}^{*})$
is an increasing function.

The nonlinear complementarity problem is identified as an important
mathematical programming problem and has been used as a general framework for
quadratic programming, linear complementarity problems and some equilibrium
problems. A number of applications of nonlinear complementarity problems are
reported in operations research [38], multiple objective programming problem
[17], mathematical economics and engineering. The concept of complementarity
is synonymous with the notion of system equilibrium. The nonlinear
complementarity problem is well studied in the literature on mathematical
programming and arises in a number of applications in operations research,
control theory, mathematical economics, geometry and engineering.

The idea of nonlinear complementarity problem is based on the concept of
linear complementarity problem. For recent study on this problem and
applications see [3], [33], [24], [25], [5], [14], [22] and references
therein. For details of several matrix classes in complementarity theory, see
[12], [16], [32], [27], [20], [19], [4], [6] [29], [7] and references cited
therein. The problem of computing the value vector and optimal stationary
strategies for structured stochastic games for discounted and undiscounded
zero-sum games and quadratic multi-objective programming problem are
formulated as linear complementary problems. For details see [21], [15], [30],
[26] and [35]. The complementarity problems are considered with respect to
principal pivot transforms and pivotal method to its solution point of view.
For details see [1], [34], [13], [28], [31] and [2].

A wide class of problems, which arise in various fields of sciences, can be
studied via the nonlinear system of equations, $f(x)=0$ using various
techniques. Finding solutions of systems of nonlinear equations has an
important role to deal with problems in various fields such as chemical
production processes, engineering design, economic equilibrium, transportation
and applied physics. A number of methods are proposed to solve systems of
equations. Newton and quasi-newton methods are well known iterative methods to
solve nonlinear systems of equations. In recent years, researchers are
interested to solve system of nonlinear equations both analytically and
numerically. Several iterative methods have been developed using different
techniques such as Taylor’s series expansion, quadrature formulas, homotopy
method, interpolation, decomposition and its various modification. For
details, see [39], [36], [11], [37], [9] and [40].
Suppose $f:R\to R,$ a nonlinear function. Then the equation $f(x)=0$ can be
solved by newton method with the iterative process
$x^{k+1}=x^{k}-f^{\prime}(x^{k})^{-1}f(x^{k}).$ Consider the nonlinear
function $g:R^{n}\to R^{n}$ . Then the system of nonlinear equations $g(x)=0$
can be solved by newton method with the iterative process
$x^{k+1}=x^{k}-J_{g}(x^{k})^{-1}g(x^{k}),$ where $J_{g}$ is the jacobian of
the function $g(x)=0.$ Many researchers extended newton method in various way
to get the solution with higher order of convergence . For details see [10],
[18], [8].

##  2 Preliminaries

We begin by introducing some basic notations used in this paper. $R^{n}$
denotes the $n$ dimensional real space, $R^{n}_{+}$ and $R^{n}_{++}$ denote
the nonnegative and positive orthant of $R^{n}.$ We consider vectors and
matrices with real entries. Any vector $x\in R^{n}$ is a column vector and
$x^{T}$ denotes the row transpose of $x.$ $e$ denotes the vector of all $1.$
The nonlinear complementarity problem is to find the vector $z\in R^{n},$ such
that $f(z),z\geq 0,\ z^{T}f(z)=0,$ where $f(z)$ is a nonlinear equation.

##  3 Main Results

Now we show that the nonlinear complementarity problem can be solved by system
of nonlinear equations.

######  Theorem 3.1:

Let $\phi:R\to R$ be any increasing function such that $\phi(0)=0$. Then $z$
solves the complementarity problemLABEL:cp if and only if

$\phi((f_{i}(z)-z_{i})^{2})-\phi(f_{i}(z)|f_{i}(z)|)-\phi(z_{i}|z_{i}|)=0$
(3.5)

###### Proof.

Necessery. For each $i=1,2,\ldots,n$, either $z_{i}=0,f_{i}(z)\geq 0$ or
$f_{i}(z)=0,z_{i}\geq 0$.
If $z_{i}=0,f_{i}(z)\geq 0$ then
$\phi((f_{i}(z)-z_{i})^{2})-\phi(f_{i}(z)|f_{i}(z)|)-\phi(z_{i}|z_{i}|)$
$=\phi((f_{i}(z))^{2})-\phi(f_{i}(z)|f_{i}(z)|)=\phi((f_{i}(z))^{2})-\phi((f_{i}(z))^{2})=0$.
If $f_{i}(z)=0,z_{i}\geq 0$ then
$\phi((f_{i}(z)-z_{i})^{2})-\phi(f_{i}(z)|f_{i}(z)|)-\phi(z_{i}|z_{i}|)$
$=\phi((z_{i})^{2})-\phi(z_{i}|z_{i}|)=\phi((z_{i})^{2})-\phi((z_{i})^{2})=0$.
So the solution of $(1.1)$ satisfies $(1.2)$.
Sufficient. (a) To show that $f(z)\geq 0$ assume the contrary, that is,
$f_{i}(z)<0$ for some $i=1,2,\dots,n$. Then

$0\leq\phi((f_{i}(z)-z_{i})^{2})$
$=\phi(f_{i}(z)|f_{i}(z)|)+\phi(z_{i}|z_{i}|)$
$=\phi(-f_{i}(z)^{2})+\phi(z_{i}|z_{i}|)$ $<\phi(z_{i}|z_{i}|)$

This implies that $\phi(z_{i}|z_{i}|)>0$ $\implies z_{i}|z_{i}|>0$ $\implies
z_{i}>0$ and $\phi((f_{i}(z)-z_{i})^{2})<\phi(z_{i}|z_{i}|)$
$\implies((f_{i}(z)-z_{i})^{2})<z_{i}|z_{i}|$.
But for $z_{i}>0$, $(z_{i})^{2}<(f_{i}(z)-z_{i})^{2}$ [as
$(f_{i}(z)-z_{i})<0,z_{i}>0$ then $(f_{i}(z)-z_{i})<0$,
$|f_{i}(z)-z_{i}|>z_{i}$ so $(f_{i}(z)-z_{i})^{2}>z_{i}^{2}$ ].
So, it contradicts that $f_{i}(z)<0$. So $f_{i}(z)\geq 0$.
(b) To show that $z\geq 0$ interchange the roles of $z_{i}$ and $f_{i}(z)$.
(c)From (a) and (b) we have that $z\geq 0$ and $f(z)\geq 0$. To show
$z^{T}f(z)=0$ assume the contrary $z_{i}>0$ and $f_{i}(z)>0$ for some
$i=1,2,\ldots,n.$
If $f_{i}(z)\geq z_{i}$, then
$\phi((f_{i}(z)-z_{i})^{2})<\phi((f_{i}(z))^{2})+\phi((z_{i})^{2})=\phi(f_{i}(z)|f_{i}(z)|)+\phi(z_{i}|z_{i}|)$.
This contradicts that
$\phi((f_{i}(z)-z_{i})^{2})=\phi(f_{i}(z)|f_{i}(z)|)+\phi(z_{i}|z_{i}|)$
Similarly, for $z_{i}\geq f_{i}(z),$ just interchange the roles of $z_{i}$ and
$f_{i}(z)$ and will get a contradiction.

Hence it is shown that the complementarity problem of finding a $z\in R^{n}$
satisfying $z^{T}f(z)=0,\ \ \ f(z)\geq 0,\ \ \ z\geq 0$ where
$f:R^{n}\rightarrow R^{n}$, a nonlinear equation, is equivalent to the problem
of solving system of n nonlinear equations in n variables.

${}\psi_{i}(z)=\phi((f_{i}(z)-z_{i})^{2})-\phi(f_{i}(z)|f_{i}(z)|)-\phi(z_{i}|z_{i}|)=0\
\ \ \forall i\in\\{1,2,\cdots n\\},$ (3.6)

where $\phi$ is an increasing function defined by $\phi:R\to R$ such that
$\phi(x)=x^{3}.$ Now $\frac{\partial\psi_{i}}{\partial z_{j}}$
$=\phi^{{}^{\prime}}((f_{i}(z)-z_{i})^{2})2(f_{i}(z)-z_{i})(\frac{\partial
f_{i}}{\partial z_{j}}-\delta_{ij})$
$-\phi^{{}^{\prime}}(f_{i}(z)|f_{i}(z)|)2f_{i}(z)sgn(f_{i}(z))\frac{\partial
f_{i}}{\partial z_{j}}$
$-\phi^{{}^{\prime}}(z_{i}|z_{i}|)2z_{i}sgn(z_{i})\delta_{ij}$ where

$sgn(x)=\begin{cases}\ 1&\quad\text{if }x>0\\\ \ -1&\quad\text{if }x<0\\\
\end{cases}$

###  3.1 Modified Newton method to solve system of nonlinear equations

Now we introduce the modified Newton’s method to solve the system of nonlinear
equations. The algorithm of the modified newton’s method is given below.
Algorithm:

Modified Newton Method

Step 0: Give the initial approximation $z_{0}$ and a very small number $e,$
such that $0<e.$ Compute the Jacobian of $\psi(z)$ with respect to the
variable $z,$ which is denoted by $\cal{J}$ and $\psi^{\prime}(z)=\cal{J}.$
Set $k=0.$
Step 1: For $k$-th iteration compute the followings:

$\displaystyle
y^{k}=z^{k}-\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k})]^{-1}\psi(z^{k});$
(3.7) $\displaystyle
x^{k}=z^{k}-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2}]^{-1}$
$\displaystyle[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k});$
$\displaystyle
w^{k}=x^{k}-[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2}]^{-1}$
$\displaystyle[\psi^{\prime}(x^{k})+\psi^{\prime}(y^{k})]\psi(x^{k});$
$\displaystyle
z^{k+1}=w^{k}-[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]^{-1}\psi(w^{k}),$

where $\psi^{\prime}(x)$ is the jacobian of $\psi(x)$ and
sgn$(t_{i}\psi_{i}(z^{k}))=$sgn$(\frac{\partial\psi_{i}}{\partial
z_{i}}(z^{k})),$
sgn$(\lambda_{i})=$sgn$(\frac{\partial\psi_{i}}{\partial z_{i}}(z^{k})),$
sgn$(\mu_{i})=$sgn$(\frac{\partial\psi_{i}}{\partial x_{i}}(x^{k})),$
sgn$(\eta_{i})=$sgn$(\frac{\partial\psi_{i}}{\partial w_{i}}(w^{k})).$
Step 2: Compute the norm of $\psi(z)=0$ which is denoted by $n_{1}.$ Here
$n_{1}$ is defined by
$n_{1}=\|\psi(z)\|=\sqrt{\sum_{i=1}^{n}\\{\psi_{i}(z)\\}^{2}}$.
Step 3: If the norm $n_{1}$ is less than the tolerance $e,$ i.e. $n_{1}<e,$
then the new iteration i.e. $z^{k+1}$ is the required solution of $\psi(z)=0,$
i.e. $\psi(z^{k+1})=\begin{bmatrix}\psi_{1}(z^{k+1})\\\ \psi_{2}(z^{k+1})\\\
\vdots\\\ \psi_{n}(z^{k+1})\end{bmatrix}$ $=\begin{bmatrix}0\\\ 0\\\ \vdots\\\
0\end{bmatrix}.$ Otherwise set $k=k+1$ and go to step $1.$

######  Theorem 3.2:

Let $\psi:R^{n}\to R^{n}$ has a root $z^{*}\in D\subseteq R^{n}$, where $D$ is
an open convex set. Assume that $\psi(z)$ is three times Fre’chet
differentiable in some neighborhood $N$ of the root $(z^{*})$. If for all
$z\in N$,
$diag(t_{i}\psi_{i}(z^{k}),diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2},\\\
diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2}$ and
$diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}$ are nonsingular, then the method
defined by 3.7 is of seventh-order convergence.

###### Proof.

Let $e^{k}=z^{k}-z^{*}.$ Now using Taylor series expansion we have,
$\psi(z^{*})=\psi(z^{k})+\psi^{\prime}(z^{k})(z^{*}-z^{k})+\frac{1}{2}\psi^{\prime\prime}(z^{k})(z^{*}-z^{k})^{2}+o(\|e^{k}\|^{3})$
$\implies\psi(z^{k})=\psi^{\prime}(z^{k})e^{k}-\frac{1}{2}\psi^{\prime\prime}(z^{k})(e^{k})^{2}+o(\|e^{k}\|^{3}).$
Let
$d^{k}=y^{k}-z^{k}=-\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]^{-1}\psi(z^{k}).$
Now using Taylor series expansion we have,
$\psi(y^{k})=\psi(z^{k})+\psi^{\prime}(z^{k})d^{k}+\frac{1}{2}\psi^{\prime\prime}(z^{k})(d^{k})^{2}+o(\|d^{k}\|^{3}).$
Now
$d^{k}=y^{k}-z^{k}=-\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]^{-1}\psi(z^{k})\\\
=-\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]^{-1}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})\\\
$
$=-\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]^{-1}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]e^{k}+\\\
\frac{1}{2}[\psi^{\prime}(z^{k})+diag(t_{i}\psi_{i}(z^{k}))]^{-1}[diag(t_{i}\psi_{i}(z^{k}))]e^{k}+o(\|e^{k}\|^{2})$
$=-\frac{1}{2}e^{k}+o(\|e^{k}\|^{2}),$ as
$diag(t_{i}\psi_{i}(z^{k}))]e^{k}=o(\|e^{k}\|^{2}).$
Hence
$\psi(y^{k})=\psi(z^{k})-\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})=\psi^{\prime}(z^{k})e^{k}-\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})=\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})$
and $\psi^{\prime}(y^{k})=\frac{1}{2}\psi^{\prime}(z^{k})+o(\|e^{k}\|).$
Let $a^{k}=x^{k}-z^{*}.$ Hence
$x^{k}=z^{k}-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k})$
implies that
$a^{k}=e^{k}-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k})$
$\implies
a^{k}=[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}\\{[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]e^{k}-\frac{1}{2}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k})\\}\\\
$
$=[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}\\{[diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]e^{k}+\psi^{\prime}(z^{k})\\}^{2}e^{k}+\psi^{\prime}(y^{k})\\}^{2}e^{k}-\frac{1}{2}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k})\\}=o(\|e^{k}\|^{3}),$
as $diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})e^{k}=o(\|e^{k}\|^{3}).$
Again using Taylor series expansion we have,
$\psi(x^{k})=\psi(z^{k})+\psi^{\prime}(z^{k})c^{k}+\frac{1}{2}\psi^{\prime\prime}(z^{k})(c^{k})^{2}+o(\|c^{k}\|^{3}),$where
$c^{k}=x^{k}-z^{k}=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})]\psi(z^{k})$
$=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\psi^{\prime}(z^{k})+\psi^{\prime}(y^{k})][\psi^{\prime}(z^{k})e^{k}]+o(\|e^{k}\|^{2})$
$=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(z^{k})\\}^{2}e^{k}+\psi^{\prime}(y^{k})\psi^{\prime}(z^{k})e^{k}]+o(\|e^{k}\|^{2})$
$=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+$
$diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]e^{k}+\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+$
$diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\lambda_{i}\\{\psi_{i}(z^{k})\\}^{2})-\psi^{\prime}(y^{k})\psi^{\prime}(z^{k})]e^{k}+o(\|e^{k}\|^{2})=-\frac{1}{2}e^{k}+o(\|e^{k}\|^{2}).$
Hence
$\psi(x^{k})=\psi(z^{k})-\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})=\psi^{\prime}(z^{k})e^{k}-\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})=\frac{1}{2}\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2})$
and $\psi^{\prime}(x^{k})=\frac{1}{2}\psi^{\prime}(z^{k})+o(\|e^{k}\|).$

Let
$b^{k}=w^{k}-z^{*}=x^{k}-z^{*}-\frac{1}{2}[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\psi^{\prime}(x^{k})+\psi^{\prime}(y^{k})]\psi(x^{k})=a^{k}-\frac{1}{2}[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\psi^{\prime}(x^{k})+\psi^{\prime}(y^{k})]\psi(x^{k})$
$\implies
b^{k}=[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}\\{[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]a^{k}-[\psi^{\prime}(x^{k})+\psi^{\prime}(y^{k})]\psi(x^{k})\\}=o(\|e^{k}\|^{5}),$
as $diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})a^{k}=o(\|e^{k}\|^{5}).$ Now using
Taylor series expansion we have,
$\psi(w^{k})=\psi(z^{k})+\psi^{\prime}(z^{k})m^{k}+\frac{1}{2}\psi^{\prime\prime}(z^{k})(m^{k})^{2}+o(\|m^{k}\|^{3}),$
where $m^{k}=w^{k}-z^{k}.$
Let
$n^{k}=w^{k}-x^{k}=-[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\psi^{\prime}(x^{k})+\psi^{\prime}(y^{k})]\psi(x^{k})=-[\\{\psi^{\prime}(x^{k})\\}^{2}+\\{\psi^{\prime}(y^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\frac{1}{2}\psi^{\prime}(z^{k})+\frac{1}{2}\psi^{\prime}(z^{k})][\frac{1}{2}\psi^{\prime}(z^{k})e^{k}]+o(\|e^{k}\|^{2})$
$=-\frac{1}{2}[\frac{1}{4}\\{\psi^{\prime}(z^{k})\\}^{2}+\frac{1}{4}\\{\psi^{\prime}(z^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(z^{k})\\}^{2}e^{k}]+o(\|e^{k}\|^{2})$
$=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(z^{k})\\}^{2}e^{k}]+o(\|e^{k}\|^{2})=-\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[\\{\psi^{\prime}(z^{k})\\}^{2}+diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]e^{k}+\frac{1}{2}[\\{\psi^{\prime}(z^{k})\\}^{2}+\\\
diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})]^{-1}[diag(\mu_{i}\\{\psi_{i}(x^{k})\\}^{2})e^{k}]+o(\|e^{k}\|^{2})=-\frac{1}{2}e^{k}+o(\|e^{k}\|^{2}).$
Hence
$m^{k}=w^{k}-z^{k}=w^{k}-x^{k}+x^{k}-z^{k}=n^{k}+c^{k}=-e^{k}+o(\|e^{k}\|^{2}).$
Therefore
$\psi(w^{k})=\psi(z^{k})-\psi^{\prime}(z^{k})e^{k}+o(\|e^{k}\|^{2}).$
Now
$z^{k+1}-z^{*}=w^{k}-z^{*}-[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]^{-1}\psi(w^{k})$
$\implies
e^{k+1}=b^{k}-[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]^{-1}\psi(w^{k})=[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]^{-1}\\{[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]b^{k}-\psi(w^{k})\\}=[\psi^{\prime}(w^{k})+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}]^{-1}\\{\psi^{\prime}(w^{k})b^{k}+diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}b^{k}-\psi(w^{k})\\}=o(\|e^{k}\|^{7}),$
as $diag(\eta_{i}\\{\psi_{i}(w^{k})\\}^{2}b^{k}=o(\|e^{k}\|^{7})$. Therefore
the introduced modified Newton method has seventh order of convergency.

##  4 Numerical Example

Now consider an oligopoly with five firms, each with a total cost function of
the form:

$c_{i}(Q_{i})=n_{i}Q_{i}+\frac{\beta_{i}}{\beta_{i}+1}{L_{i}}^{\frac{1}{\beta_{i}}}{Q_{i}}^{\frac{\beta_{i}+1}{\beta_{i}}}$
(4.8)

The demand curve is given by:

$\tilde{Q}=5000P^{-1.1},\ \ P(\tilde{Q})=5000^{1/1.1}\tilde{Q}^{-1/1.1}.$
(4.9)

The parameters of the equation 4.8 for the five firms are given below:

Table 1: Value of parameters for five firms firm $i$ | $n_{i}$ | $L_{i}$ | $\beta_{i}$
---|---|---|---
1 | 10 | 5 | 1.2
2 | 8 | 5 | 1.1
3 | 6 | 5 | 1
4 | 4 | 5 | 0.8
5 | 2 | 5 | 0.6

This oligopoly market equilibrium problem is a nonlinear complementarity
problem $z\geq 0,f(z)\geq 0,z^{T}f(z)=0$. The nonlinear complementarity
problem is solved by solving system of nonlinear equations

$\phi((f_{i}(z)-z_{i})^{2})-\phi(f_{i}(z)|f_{i}(z)|)-\phi(z_{i}|z_{i}|)=0$
(4.10)

where $\phi(z)=z^{3}.$
Now we solve the system of nonlinear equations 4.10 by the above proposed
algorithm.

Now to solve this oligopoly problem by modified newton’s method 3.1, first
take the initial point $z_{0}=$ $\left[\begin{array}[]{c}40\\\ 50\\\ 60\\\
55\\\ 45\\\ \end{array}\right].$ Set $e=10^{-7},n_{1}=10^{-9}.$ After $17$
iterations we obtain $z=$ $\left[\begin{array}[]{c}15.4293\\\ 12.4986\\\
9.6635\\\ 7.1651\\\ 5.1326\\\ \end{array}\right],$

##  5 Conclusion

In this study we consider oligopolistic market equilibrium problem and give
solution approach through system of nonlinear equation. An oligopolistic
market equilibrium problem can be reduced to a nonlinear complementarity
problem. We show the equivalency between the nonlinear complementarity problem
and the system of nonlinear equations. We introduce modified Newton method
with higher order of convergence to solve the system of nonlinear equations to
obtain the solution of nonlinear complementarity problem. Although the
Jacobian of the system of nonlinear equations be singular, this method can be
processed with suitable initial point. We show that the rate of convergency of
this method is $7$. Finally a real life oligopolistic market equilibrium
problem is considered to demonstrate the effectiveness of our results and
algorithms.

##  6 Acknowledgment

The author A. Dutta is thankful to the Department of Science and Technology,
Govt. of India, INSPIRE Fellowship Scheme for financial support.

## References

  * [1] AK Das.  Properties of some matrix classes based on principal pivot transform.  Annals of Operations Research, 243, 05 2014.
  * [2] AK Das, R Jana, and Deepmala.  On generalized positive subdefinite matrices and interior point algorithm.  In Frontiers in Optimization: Theory and Applications, pages 3–16. Springer, 2016.
  * [3] AK Das, R Jana, and Deepmala.  Finiteness of criss-cross method in complementarity problem.  In International Conference on Mathematics and Computing, pages 170–180. Springer, 2017.
  * [4] AK Das, R Jana, and Deepmala.  Invex programming problems with equality and inequality constraints.  Transactions of A. Razmadze Mathematical Institute, 172(3):361–371, 2018.
  * [5] AK Das, R Jana, and Deepmala.  Some aspects on solving transportation problem.  arXiv preprint arXiv:1808.05351, 2018.
  * [6] A Dutta and AK Das.  On some properties of $k$-type block matrices in the context of complementarity problem.  arXiv preprint arXiv:2109.09549, 2021.
  * [7] A Dutta, R Jana, and AK Das.  On column competent matrices and linear complementarity problem.  In Proceedings of the Seventh International Conference on Mathematics and Computing, pages 615–625. Springer, 2022.
  * [8] Louis W Ehrlich.  A modified newton method for polynomials.  Communications of the ACM, 10(2):107–108, 1967.
  * [9] Jin-yan Fan.  A modified levenberg-marquardt algorithm for singular system of nonlinear equations.  Journal of Computational Mathematics, pages 625–636, 2003.
  * [10] Herbert HH Homeier.  A modified newton method with cubic convergence: the multivariate case.  Journal of Computational and Applied Mathematics, 169(1):161–169, 2004.
  * [11] Hossein Jafari and Varsha Daftardar-Gejji.  Revised adomian decomposition method for solving a system of nonlinear equations.  Applied Mathematics and Computation, 175(1):1–7, 2006.
  * [12] R Jana, AK Das, and A. Dutta.  On hidden z-matrix and interior point algorithm.  OPSEARCH, 56, 09 2019.
  * [13] R Jana, AK Das, and Vishnu Narayan Mishra.  Iterative descent method for generalized leontief model.  Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 91(2):237–244, 2021.
  * [14] R Jana, AK Das, and S Sinha.  On processability of lemke’s algorithm.  Applications and Applied Mathematics: An International Journal (AAM), 13(2):31, 2018.
  * [15] R Jana, AK Das, and S Sinha.  On semimonotone star matrices and linear complementarity problem.  arXiv preprint arXiv:1808.00281, 2018.
  * [16] R Jana, A. Dutta, and AK Das.  More on hidden z -matrices and linear complementarity problem.  Linear and Multilinear Algebra, 69:1–10, 06 2019.
  * [17] Michael M Kostreva and Malgorzata M Wiecek.  Linear complementarity problems and multiple objective programming.  Mathematical Programming, 60(1-3):349–359, 1993.
  * [18] Jisheng Kou, Yitian Li, and Xiuhua Wang.  A modification of newton method with third-order convergence.  Applied Mathematics and Computation, 181(2):1106–1111, 2006.
  * [19] SR Mohan, S.K. Neogy, and AK Das.  More on positive subdefinite matrices and the linear complementarity problem.  Linear Algebra and Its Applications, 338(1-3):275–285, 2001.
  * [20] SR Mohan, S.K. Neogy, and AK Das.  On the classes of fully copositive and fully semimonotone matrices.  Linear Algebra and its Applications, 323:87–97, 01 2001.
  * [21] S.R. Mohan, S.K. Neogy, and AK Das.  A note on linear complementarity problems and multiple objective programming.  Mathematical Programming. Series A. Series B, 100, 06 2004.
  * [22] Prasenjit Mondal, S Sinha, S.K. Neogy, and AK Das.  On discounted ar–at semi-markov games and its complementarity formulations.  International Journal of Game Theory, 45(3):567–583, 2016.
  * [23] Frederic H Murphy, Hanif D Sherali, and Allen L Soyster.  A mathematical programming approach for determining oligopolistic market equilibrium.  Mathematical Programming, 24(1):92–106, 1982.
  * [24] S.K. Neogy, R Bapat, AK Das, and T Parthasarathy.  Mathematical Programming and Game Theory for Decision Making.  04 2008.
  * [25] S.K. Neogy, Ravindra Bapat, AK Das, and Biswabrata Pradhan.  Optimization models with economic and game theoretic applications.  Annals of Operations Research, 243, 07 2016.
  * [26] S.K. Neogy and AK Das.  Linear complementarity and two classes of structured stochastic games.  Operations Research with Economic and Industrial Applications: Emerging Trends, eds: SR Mohan and SK Neogy, Anamaya Publishers, New Delhi, India, pages 156–180, 2005.
  * [27] S.K. Neogy and AK Das.  On almost type classes of matrices with q-property.  Linear & Multilinear Algebra - LINEAR MULTILINEAR ALGEBRA, 53:243–257, 07 2005.
  * [28] S.K. Neogy and A.K. Das.  Principal pivot transforms of some classes of matrices.  Linear Algebra and its Applications, 400:243–252, 2005.
  * [29] S.K. Neogy and AK Das.  Some properties of generalized positive subdefinite matrices.  SIAM J. Matrix Analysis Applications, 27:988–995, 01 2006.
  * [30] SK Neogy and AK Das.  Generalized monotone maps and complementarity problems.  In Topics in Nonconvex Optimization, pages 27–46. Springer, 2011\.
  * [31] S.K. Neogy and A.K. Das.  On singular n0-matrices and the class q.  Linear Algebra and its Applications, 434(3):813–819, 2011.
  * [32] S.K. Neogy and AK Das.  On weak generalized positive subdefinite matrices and the linear complementarity problem.  Linear and Multilinear Algebra, 61, 07 2013.
  * [33] S.K. Neogy, AK Das, and R. Bapat.  Modeling, computation and optimization.  06 2022.
  * [34] S.K. Neogy, AK Das, and Abhijit Gupta.  Generalized principal pivot transforms, complementarity theory and their applications in stochastic games.  Optimization Letters, 6:339–356, 02 2012.
  * [35] S.K. Neogy, AK Das, S Sinha, and A Gupta.  On a mixture class of stochastic game with ordered field property.  In Mathematical programming and game theory for decision making, pages 451–477. World Scientific, 2008.
  * [36] Muhammad Aslam Noor and Muhammad Waseem.  Some iterative methods for solving a system of nonlinear equations.  Computers & Mathematics with Applications, 57(1):101–106, 2009\.
  * [37] Takeo Ojika, Satoshi Watanabe, and Taketomo Mitsui.  Deflation algorithm for the multiple roots of a system of nonlinear equations.  Journal of mathematical analysis and applications, 96(2):463–479, 1983.
  * [38] Jong-Shi Pang.  Complementarity problems.  In Handbook of global optimization, pages 271–338. Springer, 1995\.
  * [39] K Sayevand and H Jafari.  On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth-and fifth-order convergence.  Applied Mathematical Modelling, 40(2):1467–1476, 2016.
  * [40] Kiyotaka Yamamura, Hitomi Kawata, and Ai Tokue.  Interval solution of nonlinear equations using linear programming.  BIT Numerical Mathematics, 38(1):186–199, 1998.


# Genuine Electronic Structure and Superconducting Gap Structure in
(Ba0.6K0.4)Fe2As2 Superconductor

Yongqing Cai1,2,♯, Jianwei Huang1,♯, Taimin Miao1,2, Dingsong Wu1,2, Qiang
Gao1,2, Cong Li1,2, Yu Xu1,2, Junjie Jia1,2, Qingyan Wang1,2,3, Yuan
Huang1,2,3, Guodong Liu1,2,3, Fengfeng Zhang4, Shenjin Zhang4, Feng Yang4,
Zhimin Wang4, Qinjun Peng4, Zuyan Xu4, Lin Zhao1,2,3∗ and X. J. Zhou1,2,3,5,∗
1National Lab for Superconductivity, Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3 Songshan Lake Materials Laboratory, Dongguan 523808, China
4Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
5Beijing Academy of Quantum Information Sciences, Beijing 100193, China
♯These authors contribute equally to the present work.
∗Corresponding author<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
(March 24, 2021)

The electronic structure and superconducting gap structure are prerequisites
to establish microscopic theories in understanding the superconductivity
mechanism of iron-based superconductors. However, even for the most
extensively studied optimally-doped (Ba0.6K0.4)Fe2As2, there remain
outstanding controversies on its electronic structure and superconducting gap
structure. Here we resolve these issues by carrying out high-resolution angle-
resolved photoemission spectroscopy (ARPES) measurements on the optimally-
doped (Ba0.6K0.4)Fe2As2 superconductor using both Helium lamp and laser light
sources. Our results indicate the ”flat band” feature observed around the
Brillouin zone center in the superconducting state originates from the
combined effect of the superconductivity-induced band back-bending and the
folding of a band from the zone corner to the center. We found direct evidence
of the band folding between the zone corner and the center in both the normal
and superconducting state. Our resolution of the origin of the flat band makes
it possible to assign the three hole-like bands around the zone center and
determine their superconducting gap correctly. Around the zone corner, we
observe a tiny electron-like band and an M-shaped band simultaneously in both
the normal and superconducting states. The obtained gap size for the bands
around the zone corner ($\sim$5.5 meV) is significantly smaller than all the
previous ARPES measurements. Our results establish a new superconducting gap
structure around the zone corner and resolve a number of prominent
controversies concerning the electronic structure and superconducting gap
structure in the optimally-doped (Ba0.6K0.4)Fe2As2. They provide new insights
in examining and establishing theories in understanding superconductivity
mechanism in iron-based superconductors.

Since the discovery of superconductivity in iron-based superconductors in 2008
HHosono2008YKamihara ; ZXZhao2008ZARen ; MKWu2008FCHsu ; DJohrendt2008MRotter
; CQJin2008XCWang , the superconductivity mechanism remains under hot debate
in spite of great experimental and theoretical efforts HHosono2009KIshida ;
RLGreene2010JPaglione ; IIMazin2011Hirschfeld ; AVChubukov2017RMFernandes .
Understanding the electronic structure and superconducting gap structure is
the prerequisite to establish microscopic theories to understand the
superconductivity mechanism of the iron-based superconductors. Angle-resolved
photoemission spectroscopy (ARPES) has provided key information in studying
the iron-based superconductorsHDing2011PRichard ; DLFeng2013ZRYe ;
XJZhou2015XLiu ; XJZhou2018LZhao ; ZXShen2020JASobota ; XJZhou2021LZhao .
However, even for the prototypical (Ba,K)Fe2As2 superconductor that has been
most extensively studied by ARPESXJZhou2008LZhao ; HDing2008NLWang ;
HDing2009PRichard ; MZHasan2008LWray ; SVBorisenko2009DVEvtushinsky ;
SVBorisenko2009AKoitzsch ; SVBorisenko2009VBZabolotnyyNature ;
DLFeng2010YZhang ; HDing2011YMXu ; NLWang2011HDing ; SShin2011TShimojima ;
SVBorisenko2011DVEvtushinsky ; SShin2012WMalaeb ; KIshizaka2012TShimojima ;
SVBorisenko2014DVEvtushinsky ; SShin2014WMalaeb ; HDing2014PZhang ;
SShin2017TShimojima ; XJZhou2019JWhuang , there remain outstanding
controversies on its electronic structure and superconducting gap structure.
First, regarding the electronic structure around the $\Gamma$ point, a
prominent issue concerns the origin of the ”flat band” in the superconducting
state. The ARPES measurements revealed that a flat band appears near the
$\Gamma$ point in the superconducting state of the optimally-doped
(Ba0.6K0.4)Fe2As2 (Fig. 1b)XJZhou2008LZhao ; DLFeng2010YZhang ;
SVBorisenko2011DVEvtushinsky ; HDing2014PZhang ; SShin2017TShimojima . This
flat band has been attributed to the formation of
BogoliubonsSVBorisenko2011DVEvtushinsky (Fig. 1d), appearance of the in-gap
stateHDing2014PZhang (Fig. 1e), the superconductivity-induced back-bending of
a bandDLFeng2010YZhang (Fig. 1f), or the folding of the band at M to
$\Gamma$SShin2017TShimojima (Fig. 1g). The resolution of this controversy is
directly related to the assignment of the three hole-like bands around
$\Gamma$ and their superconducting gap determination. Second, there has been a
long-time debate on the Fermi surface topology and electronic structure around
the M point for the optimally-doped (Ba0.6K0.4)Fe2As2. It has been
controversial on whether the Fermi surface around the M point consists of only
a tiny electron pocketXJZhou2008LZhao , two sizable electron
pocketsNLWang2011HDing , or propeller-shaped Fermi surface with one tiny
electron pocket at M and four hole-like pockets around
MSVBorisenko2009VBZabolotnyyNature ; SVBorisenko2009DVEvtushinsky . Third,
there are contradicting results on the superconducting gap structure even for
the same optimally-doped (Ba0.6K0.4)Fe2As2. For the three hole-like bands
around the $\Gamma$ point, most ARPES measurements report Fermi surface sheet-
dependent superconducting gapXJZhou2008LZhao ; HDing2008NLWang ;
MZHasan2008LWray ; SVBorisenko2009DVEvtushinsky ; DLFeng2010YZhang ;
HDing2011YMXu ; XJZhou2019JWhuang  while some ARPES measurements found that
the superconducting gap is independent of the Fermi surface sheets with much
smaller magnitudeSShin2011TShimojima ; SShin2012WMalaeb . Regarding the
superconducting gap structure around the M point, because of the controversy
of the Fermi surface topology, it remains unsettledXJZhou2008LZhao ;
HDing2008NLWang ; MZHasan2008LWray ; SVBorisenko2009DVEvtushinsky ;
TTakahashi2009KNakayama ; SShin2017TShimojima .

In this paper, we carried out high-resolution ARPES measurements on the
optimally-doped (Ba0.6K0.4)Fe2As2 to resolve the detailed electronic structure
and superconducting gap structure by using both Helium lamp and laser light
sources. We observe a clear ”flat band” feature around the $\Gamma$ point in
the superconducting state and reveal its origin. We found direct evidence of
the band folding between M and $\Gamma$. Near the $\Gamma$ point, our
resolution of the origin of the flat band makes it possible to assign the
three hole-like bands and determine their superconducting gap correctly. Near
the M point, we observe both a tiny electron-like band $\delta$ and an
M-shaped $\varepsilon$ band simultaneously in the normal and superconducting
states. This makes it possible to establish a new superconducting gap
structure around the M point. Our results resolve a number of significant
controversies concerning the genuine electronic structure and superconducting
gap structure in the prototypical iron-based superconductor, the optimally-
doped (Ba0.6K0.4)Fe2As2.

The optimally-doped (Ba0.6K0.4)Fe2As2 single crystals were grown using a FeAs
flux method with excess FeAs as flux GFChen2008JLLuo . It shows a
superconducting transition temperature TC of $\sim$ 38 K with a narrow
transition width of 0.5 K from the magnetic susceptibility
measurementXJZhou2019JWhuang . High resolution angle-resolved photoemission
measurements were carried out on our two lab-based ARPES
systemsXJZhou2008GDLiu ; XJZhou2018 . One is equipped with Helium discharge
lamp as the light source which can provide a photon energy of _h_ $\nu$ =
21.21818 eV (Helium I), and a Scienta DA30L electron energy analyzer. It can
cover a large momentum space so that we can get the overall Fermi surface and
band structures at both the Brillouin zone center and the zone corners. The
energy resolution was set at 10 meV for the Fermi surface mapping and at 4 meV
for the band structure measurements, and the angular resolution is $\sim$ 0.3
∘. The other system is equipped with the 6.994 eV vacuum-ultra-violet (VUV)
laser and angle-resolved time-of-flight electron energy analyzer (ARToF 10k by
Scienta Omicron). It can cover two-dimensional momentum space simultaneously
and has much weaker non-linearity effect so that the intrinsic signal can be
measured. The energy resolution was set at 1 meV and the angular resolution
was 0.3 ∘ corresponding to 0.004 ${\AA}^{-1}$ momentum resolution at the
photon energy of 6.994 eV. The detailed Fermi surface and band structure
around the $\Gamma$ point were measured in this system. All the samples were
cleaved _in situ_ at 18 K and measured in ultrahigh vacuum with a base
pressure better than 5$\times$10-11 mbar. The Fermi level is referenced by
measuring on a clean polycrystalline gold that is electrically connected to
the sample, as well as the normal state measurement of the sample. We have
performed measurements on multiple samples and the results are reproducible.

Figure 1a shows the overall Fermi surface mapping of the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor measured at 18 K in the superconducting state
using Helium discharge lamp. Two hole-like Fermi pockets around $\Gamma$ and
strong spots around M are observed, which is consistent with the previous
resultsXJZhou2008LZhao ; MZHasan2008LWray ; SVBorisenko2009VBZabolotnyyNature
. Fig. 1b shows the band structure measured at 18 K along the $\Gamma$-M1
direction crossing the $\Gamma$ point (Cut 1 in Fig. 1a). In addition to the
two hole-like bands, the inner and outer ones, a prominent feature is the
”flat band” observed around the $\Gamma$ point, as marked by the dashed red
rectangular frame in Fig. 1b. Fig. 1c shows the corresponding photoemission
spectra (energy distribution curves, EDCs) where the position of the flat band
is marked and plotted in Fig. 1b. It is found that this flat band can cross
the inner hole-like band and extend to the outer hole-like band. Such a flat
band has been commonly observed in the previous measurementsXJZhou2008LZhao ;
DLFeng2010YZhang ; SVBorisenko2011DVEvtushinsky ; HDing2014PZhang ;
SShin2017TShimojima . However, the origin of this flat band remains under
debate and a number of possible scenarios have been proposed. First, the
abnormal strong intensity of the flat band near the $\Gamma$ point is
attributed to the fusion of Bogoliubov dispersions (Bogoliubons) which is due
to comparability of all relevant energy scales—electronic band energy, pairing
energy, and energy of a mode, as schematically shown in Fig.
1dSVBorisenko2011DVEvtushinsky . The second possibility is associated with the
formation of ”in-gap” state inside the superconducting gap likely due to
disorder scatteringHDing2014PZhang . In this case, the flat band is considered
to be due to the Bogoliubov back-bending of the inner hole-like band (Fig.
1e)HDing2014PZhang . The third possibility is shown in Fig. 1f. The inner
hole-like band consists of two degenerate bands ($\alpha$ and $\beta$) in the
normal state; they exhibit different superconducting gap size in the
superconducting stateDLFeng2010YZhang . The Bogoliubov back-bending from the
$\beta$ band, which shows a big superconducting gap, forms the spectral
enhancement and flat band. The fourth possibility is the folded electronic
structure from the M point due to suggestion of an antiferroic electronic
instability, coexisting with the superconductivity in the nonmagnetic state of
(Ba0.6K0.4)Fe2As2 superconductorSShin2017TShimojima . The flat band crossing
with the $\alpha$ band is from folding of the flat band around the M point
with a folding vector (0,0)-($\pm\pi$,$\pm\pi$), as shown in Fig. 1g. Above
four possibilities all can produce the spectral weight enhancement and flat
band features, but the underlying physics is different. The clarification on
the origin of the flat band near the $\Gamma$ point is important not only for
arriving at the correct electronic structure and related superconducting gap
determination, but also for understanding the underlying physics in
(Ba0.6K0.4)Fe2As2 superconductor.

In order to understand the origin of the flat band, we also carried out high
resolution laser-based ARPES measurements on the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor. Fig. 2a shows the Fermi surface mapping
measured on the fresh sample at 13 K in the superconducting state. For
comparison, Fig. 2b shows the Fermi surface mapping on the aged sample
measured under the same condition. In both cases, the two hole-like Fermi
surface sheets are clearly observed. Figs. 2c-2d show the band structure
measured along the same $\Gamma$-M1 direction crossing the $\Gamma$ point for
the fresh and aged samples. The two hole-like bands are clearly observed
although their intensity is suppressed in the aged sample (Fig. 2d). In
addition, a flat band is present in both the fresh and aged samples at
$\sim$15 meV and spans a wide momentum space between the $\Gamma$ point and
the outer hole-like band crossing the inner hole-like band. Such a flat band
can be more clearly seen in the corresponding second derivative images shown
in Figs. 2e-2f. Figs. 2g-2h show photoemission spectra (EDCs) for the band
structure of the fresh sample in Fig. 2c while the EDCs for the aged sample
(Fig. 2d) are shown in Fig. 2i. For the fresh sample, the EDCs at the Fermi
momenta (kβ and kγ in Figs. 2g-2h) exhibit sharp superconductivity-induced
coherence peaks. Such coherence peaks are strongly suppressed in the aged
sample (Fig. 2i). In Figs. 2g-2i, the position of the flat band in the EDCs is
marked by the red tips and the obtained dispersion of the flat band in the
fresh and aged samples is plotted in Fig. 2e and Fig. 2f, respectively. Figs.
2j-2m compare directly the EDCs for the fresh and aged samples at several
typical momenta. While the sharp coherence peaks in the fresh sample become
strongly suppressed in the aged sample (Fig. 2k and Fig. 2m), the flat band is
quite similar in the fresh and aged samples both in its intensity and energy
position (Figs. 2j-2m). In the aged sample, the intensity of the main bands
and the flat band becomes comparable (Fig. 2k and Fig. 2m), facilitating the
identification of the flat band and understanding its origin. Our results
indicate that the sample aging mainly affects the superconducting coherence
peaks. We note that some previous laser ARPES measurement results of the
optimally-doped (Ba0.6K0.4)Fe2As2 superconductorSShin2011TShimojima ;
SShin2017TShimojima  are similar to the results we obtained on the aged
samples.

In both our Helium lamp (Fig. 1) and laser (Fig. 2) ARPES measurements, we
find that the flat band around the $\Gamma$ point spans a wide momentum space
crossing the inner hole-like band. These clearly indicate that the Bogoliubov
back-bending of the inner hole-like band alone can not account for the
formation of such a flat band, as proposed in Fig. 1d, Fig. 1e and Fig. 1f. In
our Helium lamp ARPES measurements in Fig. 1b, we observe clear
superconductivity-induced coherence peak (Fig. 1c). In this case, we do not
observe the ”in-gap” state feature for the inner hole-like band as reported
beforeHDing2014PZhang . In our laser ARPES measurements on the fresh sample in
Fig. 2c, when the inner hole-like band is dominant and sharp superconducting
coherence peaks are observed (Figs. 2g-2h), we do not observe any signature of
the ”in-gap” state either. In our laser ARPES measurements on the aged sample
in Fig. 2d, when the intensity of the inner hole-like band and flat band is
comparable (Figs. 2j-2m), the observed band looks similar to that reported
before where a part of the band near the Fermi level seems to be separated
from the rest of the inner hole-like bandHDing2014PZhang . Careful examination
of the band in Fig. 2d and its comparison with that in Fig. 2c indicates that,
even in this case, there is no formation of the ”in-gap” state. First, because
the flat band crosses the inner band (Fig. 2d), the flat band can not be
produced from the back-bending of the inner $\beta$ band. Second, the crossing
of the inner band and the flat band makes the overlapping area stronger (Fig.
2d); this produces spectral dips in the EDCs around the Fermi momenta of the
inner band (Fig. 2i), making the part of the inner band appear isolated from
the rest of the band.

In order to resolve the controversies on the electronic
structureXJZhou2008LZhao ; NLWang2011HDing ; SVBorisenko2009VBZabolotnyyNature
; SVBorisenko2009DVEvtushinsky  and the associated superconducting gap
structureXJZhou2008LZhao ; HDing2008NLWang ; MZHasan2008LWray ;
SVBorisenko2009DVEvtushinsky ; DLFeng2010YZhang ; HDing2011YMXu ;
XJZhou2019JWhuang  in the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor, in
particular to examine possible band folding between $\Gamma$ and
MSShin2017TShimojima , we carried out detailed ARPES measurements around the M
point in both the normal and superconducting states. Fig. 3 shows the band
structure measured crossing the M2 point along two high-symmetry $\Gamma$-M1
(Figs. 3a-3c at 45 K in the normal state and Figs. 3d-3f at 18 K in the
superconducting state) and $\Gamma$-X (Figs. 3g-3i at 45 K in the normal state
and Figs. 3j-3l at 18 K in the superconducting state) directions. In the
measured band structures along the $\Gamma$-M1 direction (Figs. 3a-3f), two
bands are observed. One is a tiny electron-like band (named $\delta$ in Fig.
3) which lies with its bottom just touching the Fermi level in the normal
state (marked by the cyan curve in Fig. 3c). It forms a short section of flat
band in the superconducting state (marked by the cyan curve in Fig. 3f). The
other band is the M-shaped $\varepsilon$ band as marked by the red curves in
Fig. 3c and Fig. 3f. In the measured band structures along the $\Gamma$-X
direction (Figs. 3g-3l), these two coexisting bands are also observed. The
tiny electron-like $\delta$ band in the normal state (marked by the cyan curve
in Fig. 3i) turns into a section of flat band in the superconducting state
(marked by the cyan curve in Fig. 3l). But in this case, the $\varepsilon$
band shows up as a section of electron-like band in both the normal and
superconducting states, as marked by the red curves in Fig. 3i and Fig. 3l.
The strong spots around the M point in the Fermi surface mapping (Fig. 1a),
which are located along the $\Gamma$-M directions, can be understood because
they originate from the M-shaped $\varepsilon$ band which appears only along
the $\Gamma$-M directions (Fig. 3c and Fig. 3f), not along the $\Gamma$-X
direction (Fig. 3i and Fig. 3l).

Figure 3m and Fig. 3n show photoemission spectra (EDCs) measured in the normal
and superconducting states at several typical momentum points along the
$\Gamma$-M1 and $\Gamma$-X directions, respectively. It is found that the
normal state of the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor is quite
incoherent. Except for a small peak in the EDCs at the M point, no
quasiparticle peaks are observed in the other EDCs in the normal state. This
is consistent with the previous observation of incoherent state around the
$\Gamma$ point in the normal stateXJZhou2019JWhuang . Upon entering the
superconducting state, both the $\delta$ band and the $\varepsilon$ band form
sharp superconducting coherence peaks (Fig. 3m and Fig. 3n). In particular,
sharp coherence peaks form along the M-shaped $\varepsilon$ band (Fig. 3c)
over a wide momentum space (Fig. 3m). The electronic states around the M point
are dramatically driven to transform from the incoherent normal state into a
highly coherent superconducting state.

The simultaneous observation of both the $\delta$ band and the $\varepsilon$
band, in both the normal and superconducting states, makes it possible to make
a proper band assignment and a precise superconducting gap determination. Fig.
3o and Fig. 3p show the symmetrized EDCs at the M2 point in the normal (black
curves) and superconducting (red curves) states from the bands measured along
the $\Gamma$-M1 (Fig. 3a and Fig. 3d) and $\Gamma$-X (Fig. 3g and Fig. 3j)
directions. Two peaks are observed below the Fermi level in both cases: the
one closer to the Fermi level corresponds to the $\delta$ band while the other
one with a higher binding energy corresponds to the $\varepsilon$ band. For
the tiny electron-like $\delta$ band, the observed band is nearly flat in the
superconducting state (Fig. 3f and Fig. 3l). It gives a superconducting gap
size of 5.5 $\pm$ 0.5 meV for the $\delta$ band. Regarding the superconducting
gap for the $\varepsilon$ band, first, this band is already observed in the
normal state with the band position at the M point (k4 in Fig. 3a and Fig. 3c)
at $\sim$13 meV (marked by black arrows in Fig. 3o and Fig. 3p) and the tip
position of the M-shaped band (k6 in Fig. 3a and Fig. 3c) at $\sim$11 meV
(marked by black arrows in Fig. 3q). The M-shaped $\varepsilon$ band (Figs.
3a-3c) does not cross the Fermi level in the normal state. In the
superconducting state, although strong superconducting coherence peaks form on
the $\varepsilon$ band, the overall energy position of the band shows only a
slight shift compared to that in the normal state, as seen from the EDCs at
both the M point (k4) and the tip position of the M-shaped band (k6) in Figs.
3o-3q. This indicates that the gap opening on the $\varepsilon$ band must be
significantly smaller than the energy position of the M point (13 meV) and the
tip position of the M-shaped band (11 meV). If there is any gap opening on the
$\varepsilon$ band, the estimated gap size should be smaller than 6 meV
(estimated from the peak position of the EDCs at $k_{4}$ (Figs. 3o and 3p) and
$k_{6}$ (Fig. 3q) in the normal state (13 meV for $k_{4}$ and 11 meV for
$k_{6}$) and superconducting state (14 meV for $k_{4}$ and 11 meV for
$k_{6}$)). We note that the superconducting gap we obtained for both the
$\delta$ band ($\sim$5.5 meV) and the $\varepsilon$ ($<$ 6 meV) band near the
M point in the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor is
significantly smaller than the values (10-12 meV) reported in the previous
studiesXJZhou2008LZhao ; HDing2008NLWang ; MZHasan2008LWray ;
TTakahashi2009KNakayama ; SShin2017TShimojima . Since we can observe the
$\delta$ and $\varepsilon$ bands simultaneously both in the normal state and
in the superconducting state, we can make a proper assignment of the bands and
keep track on their positions in both the normal and superconducting states.
Therefore, the gap size $\sim$5.5 meV we obtained for the $\delta$ band is
reliable and represents the intrinsic superconducting gap structure near the M
point. When the band structure in the normal state is not measured or only one
band is observed near the M point in the superconducting state, the energy
position of the $\varepsilon$ band may be mistakenly taken as the
superconducting gap size, either at the M point or at the tip position of the
M-shaped band.

In order to check on the possible folding of bands from M to $\Gamma$ in the
superconducting state of the optimally-doped (Ba0.6K0.4)Fe2As2
superconductorSShin2017TShimojima , we directly compare the band structures
along the same directions crossing $\Gamma$ and M. Figs. 4a-4d show the
comparison of the two bands measured along the $\Gamma$-M1 direction while
Figs. 4e-4h show the two bands measured along the $\Gamma$-X direction. For
each band measurement, we have identified the observed bands and marked them
in the second derivative images (Figs. 4c, 4d, 4g and 4h). Fig. 4i directly
compares EDCs at two typical momentum points around the $\Gamma$ point
($\Gamma$k1, $\Gamma$k2 in Fig. 4a and $\Gamma$k3, $\Gamma$k4 in Fig. 4e) with
those at the two equivalent momentum points around the M point (Mk1, Mk2 in
Fig. 4b and Mk3, Mk4 in Fig. 4f). The features corresponding to various bands
are marked in the EDCs. In the band structure measured along the $\Gamma$-M1
direction crossing the $\Gamma$ point (Fig. 4a and Fig. 4c), in addition to
the regular hole-like $\alpha$/$\beta$ and $\gamma$ bands, we can identify
another two bands around the $\Gamma$ point, $\delta^{\prime}$ band (dashed
thin cyan line in Fig. 4c) and $\varepsilon^{\prime}$ band (dashed thin red
line in Fig. 4c). The $\varepsilon^{\prime}$ band crosses the $\alpha$/$\beta$
band and extends to the region between the $\alpha$/$\beta$ band and the
$\gamma$ band, as seen directly from the measured band structure (Fig. 4a and
Fig. 4c) and the EDC at $\Gamma$k2 in Fig. 4i. The shape and position of the
$\delta^{\prime}$ and $\varepsilon^{\prime}$ bands around the $\Gamma$ point
show a strong resemblance to the $\delta$ and M-shaped $\varepsilon$ bands
around the M point in Fig. 4d. In the band structure measured along the
$\Gamma$-X direction crossing the $\Gamma$ point (Fig. 4e and Fig. 4g), in
addition to the hole-like $\alpha$/$\beta$ and $\gamma$ bands, we can also
identify another two bands around the $\Gamma$ point, $\delta^{\prime}$ band
(dashed thin cyan line in Fig. 4g) and $\varepsilon^{\prime}$ band (dashed
thin red line in Fig. 4g). The shape and position of the $\delta^{\prime}$ and
$\varepsilon^{\prime}$ bands around the $\Gamma$ point also exhibit a strong
similarity to the $\delta$ and $\varepsilon$ bands around the M point in Fig.
4h. Note that, when the band at Mk4 is absent (Fig. 4f, Fig. 4h and the EDC at
Mk4 in Fig. 4i), it also becomes absent at $\Gamma$k4 (Fig. 4e, Fig. 4g and
the EDC at $\Gamma$k4 in Fig. 4i). All these results indicate that there is a
band folding from M to $\Gamma$ in the superconducting state of the optimally-
doped (Ba0.6K0.4)Fe2As2 superconductor.

We find that the band folding alone from M to $\Gamma$ can not fully account
for the flat band feature around the $\Gamma$ point. First, if the flat band
near the $\Gamma$ point comes only from the band folding of the $\varepsilon$
band around the M point, one would expect that the band position of the flat
band at $\Gamma$ is exactly the same as the position of the $\varepsilon$ band
at the M point. But we find that these two positions do not exactly match;
there is a feature in the EDCs at $\Gamma$ which has a higher binding energy
than the EDC peak position at M, as seen in the EDCs at ($\Gamma$k1,Mk1) and
($\Gamma$k3,Mk3) in Fig. 4i. Second, as seen in Fig. 4c and Fig. 4g, around
the $\Gamma$ point, there are additional features observed below the folded
bands, $\varepsilon^{\prime}$. Third, a careful examination of the EDCs around
the $\Gamma$ point indicates that they are composed of two features, as marked
by the black and red tips in Fig. 1c. All these observations indicate that the
flat band around the $\Gamma$ point consists of two components, one is the
folded band from the $\varepsilon$ band around M, the other is from the
Bogoliubov back-bending of the $\alpha$/$\beta$ band as shown in Fig. 4c and
Fig. 4g. There are three hole-like bands around the Brillouin zone center,
$\alpha$, $\beta$ and $\gamma$, that exhibit distinct kz
dependenceDLFeng2011YZhang ; HDing2011YMXu . For the photon energy of 21.218
eV, the corresponding kz is close to zero, and the $\alpha$ and $\beta$ bands
are nearly degenerateDLFeng2011YZhang ; HDing2011YMXu . In the superconducting
state, these two bands may have different superconducting gap size; the
$\beta$ band shows a larger superconducting gap than the $\alpha$ band as seen
in Fig. 4c and Fig. 4g, consistent with the previous resultDLFeng2011YZhang .

As we have shown above, there is band folding from M to $\Gamma$ in the
superconducting state of the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor.
If such a band folding exists, it is natural to ask whether there is also band
folding from $\Gamma$ to M, and furthermore, whether such a band folding also
occurs in the normal state. Fig. 5 directly compares the band structures along
the same directions crossing $\Gamma$ and M in the normal and superconducting
states. Figs. 5a-5d show the comparison of the two bands measured along the
$\Gamma$-M1 direction in the normal and superconducting states while Figs.
5e-5h show the two bands measured along the $\Gamma$-X direction. In all these
four cases, we can observe signatures of bands around M that are folded from
the bands around $\Gamma$. The band folding of the $\gamma$ band around
$\Gamma$ is particularly clear as the folded band around the M point can be
clearly seen, as marked by the red arrows in Fig. 5b, Fig. 5d, Fig. 5f and
Fig. 5h. These results have provided further evidence of the band folding
between $\Gamma$ and M in both the normal state and superconducting state in
the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor.

We have shown that the flat band feature around $\Gamma$ consists of two
components: the folded $\varepsilon$ band from M ($\varepsilon^{\prime}$ in
Fig. 4c and Fig. 4g) and the back-bending band of the $\beta$ band ($\beta$
band in Fig. 4c and Fig. 4g). In the measured band structures around the
$\Gamma$ point using Helium lamp _h_ $\nu$=21.218 eV (Fig. 1b, Fig. 4a, Fig.
4c, Fig. 4e and Fig. 4g), the $\beta$ back-bending component of the flat band
is stronger than the folded component. On the other hand, in the measured band
structures around the $\Gamma$ point using laser _h_ $\nu$=6.994 eV (Figs.
2c-2f), the folded component of the flat band is dominant. Fig. 6a shows the
dispersions of the back-bending component of the flat band obtained from the
Helium lamp measurement (Fig. 4a) and laser measurement (Fig. 2d). For
comparison, the dispersion of the $\varepsilon$ band from Helium lamp
measurement (Fig. 4b) is also plotted in Fig. 6a. These dispersions, measured
along the $\Gamma$-M1 direction, are M-shaped and can span a wide momentum
space crossing the $\alpha$/$\beta$ band. They agree with each other in both
the shape and the positions of energy and momentum. This further confirms that
this component of the flat band around $\Gamma$ originates from the folding of
the $\varepsilon$ band around M.

Based on the above results, we can now establish a coherent picture to
understand the observed electronic structure of the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor both near the $\Gamma$ region and the M
region. The Fermi surface measured by Helium lamp _h_ $\nu$=21.218 eV (kz = 0)
consists of two hole-like Fermi surface sheets around $\Gamma$ and a tiny
electron-like pocket at M (Fig. 6b). When there is a band folding between the
$\Gamma$ and M points, the Fermi surface sheets will be folded between the two
points (Fig. 6c). Whether the folded Fermi surface sheets can be seen depends
on the folding strength. In the normal state without band folding, the band
structure measured along the $\Gamma$-M direction consists of three hole-like
bands around $\Gamma$ ($\alpha$, $\beta$ and $\gamma$, with $\alpha$ and
$\beta$ being degenerate in this case), as well as a tiny electron-like band
($\delta$) and an M-shaped $\varepsilon$ band (Fig. 6d). The $\varepsilon$
band is rather incoherent in the normal state. Upon entering the
superconducting state, superconducting gap opens and superconducting coherence
peak forms and Bogoliubov back-bending bands appear on all these bands (Fig.
6e). Around the $\Gamma$ point, the initially degenerate $\alpha$ and $\beta$
bands open superconducting gap with different sizes; the gap size of the
$\beta$ band is larger than that of the $\alpha$ band. Around the M point, the
electron-like $\delta$ band produces a section of a flat band below the Fermi
level and the initially incoherent M-shaped $\varepsilon$ band becomes rather
coherent in the superconducting state (Fig. 6e). When the band folding is
considered in the superconducting state, the original hole-like bands around
$\Gamma$ are folded to the M region while the original bands around the M
point are folded to the $\Gamma$ region (Fig. 6f). In this way, the flat band
feature observed around $\Gamma$ in the superconducting state can be
understood as arising from the combination of the folded $\varepsilon$ band
from M and the back-bending of the $\beta$ band. In the meantime, two coherent
bands are observed simultaneously around the M point. The folded $\varepsilon$
band around $\Gamma$ can be more clearly observed in the superconducting state
because the incoherent $\varepsilon$ band around M in the normal state becomes
rather coherent in the superconducting state.

The clarification of the electronic structures in the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor is critical to make correct band assignment
and precise determination of the related superconducting gap. As we have now
understood, the degenerate $\alpha$ and $\beta$ bands in the normal state open
superconducting gaps of different sizes. The EDC at the Fermi momentum from
the measured band (Fig. 7a) is shown in Fig. 7b which consists of two peaks.
The extracted gap size from the symmetrized EDC (Fig. 7d) is 12 meV for the
$\beta$ band and 7 meV for the $\alpha$ band. The superconducting gap of the
$\gamma$ band obtained from Fig. 7c and Fig. 7e is 5 meV. It has been found
that the superconducting gap of the hole-like bands around the $\Gamma$ point
is highly isotropicXJZhou2019JWhuang . Combining the superconducting gap
results measured around the M point (Fig. 3o and Fig. 3p), we establish an
overall superconducting gap structure for the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor shown in Fig. 7f. The superconducting gap
structure around the M point is distinct from all the previous
measurementsXJZhou2008LZhao ; HDing2008NLWang ; MZHasan2008LWray ;
TTakahashi2009KNakayama ; SShin2017TShimojima .

Our results prove that there is a band folding effect between the Brillouin
zone center and the corners with a wave vector of (0,0)-($\pm\pi$,$\pm\pi$) in
both the normal state and superconducting state of the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductor. The band folding from the zone corner to the
zone center in the superconducting state was also reported before, which is
attributed to some unknown electronic instabilitySShin2017TShimojima . In the
parent and underdoped BaFe2As2, there is a spin-density wave (SDW) order which
plays an important role in determining the electronic structureXHChen2009HChen
. ARPES experiments have clearly shown a folding effect between the $\Gamma$
and M due to the SDW order DLFeng2009LXYang ; XJZhou2009GDLiu ; ZXShen2009MYi
; AKaminski2010TKondo . However, in the optimally-doped (Ba0.6K0.4)Fe2As2
superconductor, the SDW order is completely suppressed and can not explain the
band structure folding between $\Gamma$ and M we observed. Recently, scanning
tunneling spectroscopy revealed the presence of regions with a
$\sqrt{2}\times\sqrt{2}$ buckling reconstruction in the optimally-doped
(Ba0.6K0.4)Fe2As2 superconductorSHPan2019ALi . Whether the band folding we
observed can be due to this reconstruction or some other origins asks for
further investigations.

In summary, we have carried out high-resolution ARPES measurements on the
optimally-doped (Ba0.6K0.4)Fe2As2 to resolve the detailed electronic structure
and superconducting gap structure. By using both Helium lamp and laser light
sources, we observe a clear ”flat band” feature around the $\Gamma$ point in
the superconducting state. Our results indicate that this flat band originates
from the combined effect of both the superconductivity-induced band back-
bending of the $\beta$ band and the folding of the $\varepsilon$ band from M
to $\Gamma$. Direct evidence of the band folding between M and $\Gamma$ is
observed in both the normal state and superconducting state. Our resolution of
the origin of the flat band around the $\Gamma$ point makes it possible to
assign the three hole-like bands around $\Gamma$ and determine their
superconducting gap properly. Near the M point, we observe both a tiny
electron-like band $\delta$ and an M-shaped $\varepsilon$ band simultaneously
in normal and superconducting states. This makes it possible to correctly
determine the superconducting gap around the M point. The obtained
superconducting gap for the bands around the M point ($\sim$5.5 meV) is
significantly different from the previous measurements. Our results resolve a
number of prominent controversies concerning the electronic structure and
superconducting gap structure in the prototypical iron-based superconductor,
the optimally-doped (Ba0.6K0.4)Fe2As2. They provide key information in
examining and establishing theories in understanding superconductivity
mechanism in iron-based superconductors.

## References

  * (1) Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono.  Iron-based layered superconductor La[O1-xFx]FeAs (x=0.05-0.12) with Tc = 26 K.  J. Am. Chem. Soc., 130(11):3296–7, 2008.
  * (2) Z. A. Ren, W. Lu, J. Yang, W. Yi, X. L. Shen, Z. Cai, G. C. Che, X. L. Dong, L. L. Sun, F. Zhou, and Z. X. Zhao.  Superconductivity at 55 K in iron-based F-doped layered quaternary compound Sm[O1-xFx]FeAs.  Chin. Phys. Lett., 25(6):2215–6, 2008.
  * (3) F. C. Hsu, J. Y. Luo, K. W. Yeh, T. K. Chen, T. W. Huang, P. M. Wu, Y. C. Lee, Y. L. Huang, Y. Y. Chu, D. C. Yan, and M. K. Wu.  Superconductivity in the PbO-type structure $\alpha$-FeSe.  Proc. Natl. Acad. Sci. USA, 105(38):14262–4, 2008.
  * (4) M. Rotter, M. Pangerl, M. Tegel, and D. Johrendt.  Superconductivity and crystal structures of Ba1-xKxFe2As2 (x=0-1).  Angew. Chem. Int. Ed. Engl., 47(41):7949–52, 2008.
  * (5) X. C. Wang, Q. Q. Liu, Y. X. Lv, W. B. Gao, L. X. Yang, R. C. Yu, F. Y. Li, and C. Q. Jin.  The superconductivity at 18 K in LiFeAs system.  Solid State Commun., 148(11-12):538–540, 2008.
  * (6) K. Ishida, Y. Nakai, and H. Hosono.  To what extent iron-pnictide new superconductors have been clarified: a progress report.  J. Phys. Soc. JPN, 78(6):062001, 2009.
  * (7) J. Paglione and R. L. Greene.  High-temperature superconductivity in iron-based materials.  Nat. Phys., 6(9):645–58, 2010.
  * (8) P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin.  Gap symmetry and structure of Fe-based superconductors.  Rep. Prog. Phys., 74(12):124508, 2011.
  * (9) R. M. Fernandes and A. V. Chubukov.  Low-energy microscopic models for iron-based superconductors: a review.  Rep. Prog. Phys., 80(1):014503, 2017.
  * (10) P. Richard, T. Sato, K. Nakayama, T. Takahashi, and H. Ding.  Fe-based superconductors: an angle-resolved photoemission spectroscopy perspective.  Rep. Prog. Phys., 74(12):124512, 2011.
  * (11) Z. R. Ye, Y. Zhang, B. P. Xie, and D. L. Feng.  Angle-resolved photoemission spectroscopy study on iron-based superconductors.  Chin. Phys. B, 22(8):087407, 2013.
  * (12) X. Liu, L. Zhao, S. L. He, J. F. He, D. F. Liu, D. X. Mou, B. Shen, Y. Hu, J. W. Huang, and X. J. Zhou.  Electronic structure and superconductivity of FeSe-related superconductors.  J. Phys.: Condens. Mat., 27(18):183201, 2015.
  * (13) L. Zhao, G. D. Liu, and X. J. Zhou.  Angle-resolved photoemission studies on iron based high temperature superconductors.  Acta Phys. Sin-CH ED, 67(20):207413, 2018.
  * (14) A. J. Sobota, Y. He, and Z. X. Shen.  Electronic structure of quantum materials studied by angle-resolved photoemission spectroscopy.  arXiv:2008.02378, 2020.
  * (15) L. Zhao, G. D. Liu, and X. J. Zhou.  Angle-resolved photoemission spectroscopy studies on the electronic structure and superconductivity mechanism for high temperature superconductors.  Acta Phys. Sin-CH ED, 70(1):017406, 2021.
  * (16) L. Zhao, H. Y. Liu, W. T. Zhang, J. Q. Meng, X. W. Jia, G. D. Liu, X. L. Dong, G. F. Chen, J. L. Luo, N. L. Wang, W. Lu, G. L. Wang, Y. Zhou, Y. Zhu, X. Y. Wang, Z. Y. Xu, C. T. Chen, and X. J. Zhou.  Multiple nodeless superconducting gaps in Ba0.6K0.4Fe2As2 superconductor from angle-resolved photoemission spectroscopy.  Chin. Phys. Lett., 25(12):4402–5, 2008.
  * (17) H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, and N. L. Wang.  Observation of Fermi-surface–dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2.  Europhs. Lett., 83(4):47001, 2008.
  * (18) P. Richard, T. Sato, K. Nakayama, S. Souma, T. Takahashi, Y. M. Xu, G. F. Chen, J. L. Luo, N. L. Wang, and H. Ding.  Angle-resolved photoemission spectroscopy of the Fe-based Ba0.6K0.4Fe2As2 high temperature superconductor: evidence for an orbital selective electron-mode coupling.  Phys. Rev. Lett., 102(4):047003, 2009.
  * (19) L. Wray, D. Qian, D. Hsieh, Y. Xia, L. Li, J. G. Checkelsky, A. Pasupathy, K. K. Gomes, C. V. Parker, A. V. Fedorov, G. F. Chen, J. L. Luo, A. Yazdani, N. P. Ong, N. L. Wang, and M. Z. Hasan.  Momentum dependence of superconducting gap, strong-coupling dispersion kink, and tightly bound Cooper pairs in the high Tc (Sr,Ba)${}_{1-x}(K,Na)_{x}Fe_{2}As_{2}$ superconductors.  Phys. Rev. B, 78(18):184508, 2008.
  * (20) D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy, A. Koitzsch, M. Knupfer, B. Buechner, M. S. Viazovska, G. L. Sun, V. Hinkov, A. V. Boris, C. T. Lin, B. Keimer, A. Varykhalov, A. A. Kordyuk, and S. V. Borisenko.  Momentum dependence of the superconducting gap in Ba1-xKxFe2As2.  Phys. Rev. B, 79(5):054517, 2009.
  * (21) A. Koitzsch, D. S. Inosov, D. V. Evtushinsky, V. B. Zabolotnyy, A. A. Kordyuk, A. Kondrat, C. Hess, M. Knupfer, B. Buechner, G. L. Sun, V. Hinkov, C. T. Lin, A. Varykhalov, and S. V. Borisenko.  Temperature and doping-dependent renormalization effects of the low energy electronic structure of Ba1-xKxFe2As2 single crystals.  Phy. Rev. Lett., 102(16):167001, 2009.
  * (22) V. B. Zabolotnyy, D. S. Inosov, D. V. Evtushinsky, A. Koitzsch, A. A. Kordyuk, G. L. Sun, J. T. Park, D. Haug, V. Hinkov, A. V. Boris, C. T. Lin, M. Knupfer, A. N. Yaresko, B. Buechner, A. Varykhalov, R. Follath, and S. V. Borisenko.  (pi, pi) electronic order in iron arsenide superconductors.  Nature, 457(7229):569–72, 2009.
  * (23) Y. Zhang, L. X. Yang, F. Chen, B. Zhou, X. F. Wang, X. H. Chen, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, J. P. Hu, B. P. Xie, and D. L. Feng.  Out-of-plane momentum and symmetry-dependent energy gap of the pnictide Ba0.6K0.4Fe2As2 superconductor revealed by angle-resolved photoemission spectroscopy.  Phy. Rev. Lett., 105(11):117003, 2010.
  * (24) Y. M. Xu, Y. B. Huang, X. Y. Cui, E. Razzoli, M. Radovic, M. Shi, G. F. Chen, P. Zheng, N. L. Wang, C. L. Zhang, P. C. Dai, J. P. Hu, Z. Wang, and H. Ding.  Observation of a ubiquitous three-dimensional superconducting gap function in optimally doped Ba0.6K0.4Fe2As2.  Nat. Phys., 7(3):198–202, 2011.
  * (25) H. Ding, K. Nakayama, P. Richard, S. Souma, T. Sato, T. Takahashi, M. Neupane, Y. M. Xu, Z. H. Pan, A. V. Fedorov, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, and N. L. Wang.  Electronic structure of optimally doped pnictide Ba0.6K0.4Fe2As2: a comprehensive angle-resolved photoemission spectroscopy investigation.  J. Phys.: Condens. Mat., 23(13):135701, 2011.
  * (26) T. Shimojima, F. Sakaguchi, K. Ishizaka, Y. Ishida, T. Kiss, M. Okawa, T. Togashi, C. T. Chen, S. Watanabe, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, K. Ohgushi, S. Kasahara, T. Terashima, T. Shibauchi, Y. Matsuda, A. Chainani, and S. Shin.  Orbital-independent superconducting gaps in iron pnictides.  Science, 332(6029):564–7, 2011.
  * (27) D. V. Evtushinsky, T. K. Kim, A. A. Kordyuk, V. B. Zabolotnyy, B. Buechner, A. V. Boris, D. L. Sun, C. T. Lin, H. Q. Luo, Z. S. Wang, H. H. Wen, R. Follath, and S. V. Borisenko.  Fusion of bogoliubons in Ba0.6K0.4Fe2As2 and similarity of energy scales in high temperature superconductors.  arXiv:0808.2185, 2011.
  * (28) W. Malaeb, T. Shimojima, Y. Ishida, K. Okazaki, Y. Ota, K. Ohgushi, K. Kihou, T. Saito, C. H. Lee, S. Ishida, M. Nakajima, S. Uchida, H. Fukazawa, Y. Kohori, A. Iyo, H. Eisaki, C. T. Chen, S. Watanabe, H. Ikeda, and S. Shin.  Abrupt change in the energy gap of superconducting Ba1-xKxFe2As2 single crystals with hole doping.  Phys. Rev. B, 86(16):165117, 2012.
  * (29) T. Shimojima, W. Malaeb, K. Ohgushi, A. Chainani, S. Shin, S. Ishida, M. Nakajima, S. Uchida, T. Saito, H. Fukazawa, Y. Kohori, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, and K. Ishizaka.  Anomalous two peak structure in the angle-resolved photoemission spectra of Ba1-xKxFe2As2.  arXiv:1206.3163, 2012.
  * (30) D. V. Evtushinsky, V. B. Zabolotnyy, T. K. Kim, A. A. Kordyuk, A. N. Yaresko, J. Maletz, S. Aswartham, S. Wurmehl, A. V. Boris, D. L. Sun, C. T. Lin, B. Shen, H. H. Wen, A. Varykhalov, R. Follath, B. Buechner, and S. V. Borisenko.  Strong electron pairing at the iron 3dxz,yz orbitals in hole-doped BaFe2As2 superconductors revealed by angle-resolved photoemission spectroscopy.  Phys. Rev. B, 89(6):064514, 2014.
  * (31) W. Malaeb, T. Shimojima, Y. Ishida, T. Kondo, K. Okazaki, Y. Ota, K. Ohgushi, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, S. Ishida, M. Nakajima, S. Uchida, H. Fukazawa, T. Saito, Y. Kohori, and S. Shin.  Evidence of a universal relation between electron-mode coupling and Tc in Ba1-xKxFe2As2 superconductor from laser angle-resolved photoemission spectroscopy.  Phys. Rev. B, 90(19):195124, 2014.
  * (32) P. Zhang, P. Richard, T. Qian, X. Shi, J. Ma, L. K. Zeng, X. P. Wang, E. Rienks, C. L. Zhang, Pengcheng Dai, Y. Z. You, Z. Y. Weng, X. X. Wu, J. P. Hu, and H. Ding.  Observation of momentum-confined in-gap impurity state in Ba0.6K0.4Fe2As2: evidence for antiphase $s_{\pm}$ pairing.  Phys. Rev. X, 4(3):031001, 2014.
  * (33) T. Shimojima, W. Malaeb, A. Nakamura, T. Kondo, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, S. Ishida, M. Nakajima, S. I. Uchida, K. Ohgushi, K. Ishizaka, and S. Shin.  Antiferroic electronic structure in the nonmagnetic superconducting state of the iron-based superconductors.  Sci. Adv., 3(8):e1700466, 2017.
  * (34) J. W. Huang, L. Zhao, C. Li, Q. Gao, J. Liu, Y. Hu, Y. Xu, Y. Q. Cai, D. S. Wu, Y. Ding, C. Hu, H. X. Zhou, X. L. Dong, G. D. Liu, Q. Y. Wang, S. J. Zhang, Z. M. Wang, F. F. Zhang, F. Yang, Q. J. Peng, Z. Y. Xu, C. T. Chen, and X. J. Zhou.  Emergence of superconductivity from fully incoherent normal state in an iron-based superconductor (Ba0.6K0.4)Fe2As2.  Sci. Bull., 64(1):11–19, 2019.
  * (35) K. Nakayama, T. Sato, P. Richard, Y. M. Xu, Y. Sekiba, S. Souma, G. F. Chen, J. L. Luo, N. L. Wang, H. Ding, and T. Takahashi.  Superconducting gap symmetry of Ba0.6K${{}_{0}.4}$Fe2As2 studied by angle-resolved photoemission spectroscopy.  Europhs. Lett., 85(6):67002, 2009.
  * (36) G. F. Chen, Z. Li, J. Dong, G. Li, W. Z. Hu, X. D. Zhang, X. H. Song, P. Zheng, N. L. Wang, and J. L. Luo.  Transport and anisotropy in single-crystalline SrFe2As2 and A0.6K0.4Fe2As2(A=Sr, Ba) superconductors.  Phys. Rev. B, 78(22):224512, 2008.
  * (37) G. D. Liu, G. L. Wang, Y. Zhu, H. B. Zhang, G. C. Zhang, X. Y. Wang, Y. Zhou, W. T. Zhang, H. Y. Liu, L. Zhao, J. Q. Meng, X. L. Dong, C. T. Chen, Z. Y. Xu, and X. J. Zhou.  Development of a vacuum ultraviolet laser-based angle-resolved photoemission system with a superhigh energy resolution better than 1 meV.  Rev. Sci. Instrum., 79(2):023105–11, 2008.
  * (38) X. J. Zhou, S. L. He, G. D. Liu, L. Zhao, L. Yu, and W. T. Zhang.  New developments in laser-based photoemission spectroscopy and its scientific applications: a key issues review.  Rep. Prog. Phys., 81(6):062101, 2018.
  * (39) Y. Zhang, F. Chen, C. He, B. Zhou, B. P. Xie, C. Fang, W. F. Tsai, X. H. Chen, H. Hayashi, J. Jiang, H. Iwasawa, K. Shimada, H. Namatame, M. Taniguchi, J. P. Hu, and D. L. Feng.  Orbital characters of bands in the iron-based superconductor BaFe1.85Co0.15As2.  Phys. Rev. B, 83(5):054510, 2011.
  * (40) H. Chen, Y. Ren, Y. Qiu, W. Bao, R. H. Liu, G. Wu, T. Wu, Y. L. Xie, X. F. Wang, Q. Huang, and X. H. Chen.  Coexistence of the spin-density wave and superconductivity in Ba1-xKxFe2As2.  Europhs. Lett., 85(1):17006, 2009.
  * (41) L. X. Yang, Y. Zhang, H. W. Ou, J. F. Zhao, D. W. Shen, B. Zhou, J. Wei, F. Chen, M. Xu, C. He, Y. Chen, Z. D. Wang, X. F. Wang, T. Wu, G. Wu, X. H. Chen, M. Arita, K. Shimada, M. Taniguchi, Z. Y. Lu, T. Xiang, and D. L. Feng.  Electronic structure and unusual exchange splitting in the spin-density-wave state of the BaFe2As2 parent compound of iron-based superconductors.  Phys. Rev. Lett., 102(10):107002, 2009.
  * (42) G. D. Liu, H. Y. Liu, L. Zhao, W. T. Zhang, X. W. Jia, J. Q. Meng, X. L. Dong, J. Zhang, G. F. Chen, G. L. Wang, Y. Zhou, Y. Zhu, X. Y. Wang, Z. Y. Xu, C. T. Chen, and X. J. Zhou.  Band-structure reorganization across the magnetic transition in BaFe2As2 seen via high-resolution angle-resolved photoemission.  Phys. Rev. B, 80(13):134519, 2009.
  * (43) M. Yi, D. H. Lu, J. G. Analytis, J. H. Chu, S. K. Mo, R. H. He, M. Hashimoto, R. G. Moore, I. I. Mazin, D. J. Singh, Z. Hussain, I. R. Fisher, and Z. X. Shen.  Unconventional electronic reconstruction in undoped (Ba,Sr)Fe2As2 across the spin density wave transition.  Phys. Rev. B, 80(17):174510, 2009.
  * (44) T. Kondo, R. M. Fernandes, R. Khasanov, Chang Liu, A. D. Palczewski, N. Ni, M. Shi, A. Bostwick, E. Rotenberg, J. Schmalian, S. L. Bud’ko, P. C. Canfield, and A. Kaminski.  Unexpected Fermi-surface nesting in the pnictide parent compounds BaFe2As2 and CaFe2As2 revealed by angle-resolved photoemission spectroscopy.  Phys. Rev. B, 81(6):060507, 2010.
  * (45) A. Li, J. X. Yin, J. H. Wang, Z. Wu, J. H. Ma, A. S. Sefat, B. C. Sales, D. G. Mandrus, M. A. McGuire, R. Y. Jin, C. L. Zhang, P. C. Dai, B. Lv, C. W. Chu, X. J. Liang, P. H. Hor, C. S. Ting, and S. H. Pan.  Surface terminations and layer-resolved tunneling spectroscopy of the 122 iron pnictide superconductors.  Phys. Rev. B, 99(13):134520, 2019.

Acknowledgement
We are thankful for financial support from the National Key Research and
Development Program of China (Grants No. 2016YFA0300300, 2017YFA0302900,
2018YFA0704200 and 2019YFA0308000), the National Natural Science Foundation of
China (Grants No. 11888101, 11922414, 11874405, 62022089), the Strategic
Priority Research Program (B) of the Chinese Academy of Sciences (Grants No.
XDB25000000, XDB33000000), the Youth Innovation Promotion Association of CAS
(Grants No. 2017013, 2019007), and the Research Program of Beijing Academy of
Quantum Information Sciences (Grant No. Y18G06).

Author Contributions
X.J.Z., L.Z., Y.Q.C. and J.W.H. proposed and designed the research. J.W.H.
contributed in sample growth. Y.Q.C., J.W.H., T.M.M., D.S.W., Q.G., C.L.,
Y.X., J.J.J, Q.Y.W., Y.H., G.D.L., F.F.Z., S.J.Z., F.Y., Z.M.W., Q.J.P.,
Z.Y.X., L.Z. and X.J.Z. contributed to the development and maintenance of the
ARPES systems and related software development. Y.Q.C. carried out the ARPES
experiment with J.W.H.. Y.Q.C., L.Z. and X.J.Z. analyzed the data. Y.Q.C.,
L.Z. and X.J.Z. wrote the paper. All authors participated in discussion and
comment on the paper.

Additional information
Supplementary information is available in the online version of the paper.
Correspondence and requests for materials should be addressed to L. Zhao, and
X. J. Zhou.

Figure 1: Flat band (FB) observed near the $\Gamma$ point in the
superconducting state of the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor
in Helium lamp ARPES measurement and its possible origins. (a) Fermi surface
mapping measured at 18 K with _h_ $\nu$=21.218 eV Helium lamp. It is obtained
by integrating the spectral weight within a $\pm$ 1 meV energy window relative
to the Fermi level EF. The major polarization vector (E) is marked at the
bottom-left corner. The first Brillouin zone is represented by a black square.
For convenience, the equivalent four zone corners are named as $M_{1}$,
$M_{2}$, $M_{3}$ and $M_{4}$. (b) Band structure measured along the
$\Gamma$-$M_{1}$ direction crossing the $\Gamma$ point; the location of the
momentum Cut 1 is marked in (a) by black line. There is a ”flat band” formed
near the $\Gamma$ point, as marked in the red rectangular frame. (c)
Photoemission spectra (EDCs) from the band structure in (b). Red tips mark the
peak position of the flat band that is also plotted in (b) by red circles. The
peak position of the Bogoliubov back-bending band of the inner hole-like band
is marked by black tips. (d-g) shows schematic of four possible scenarios that
are proposed to understand the origin of the flat band. Three hole-like bands
and the flat band are marked as pink, black, blue and red lines, respectively.
(d) The flat band is from the Bogoliubons of the $\beta$ band in the
superconducting state. (e) Formation of the ”in-gap” state and the flat band
is from the Bogoliubov back-bending band of the $\alpha$ bandHDing2014PZhang .
(f) The flat band is from the Bogoliubov back-bending band of the $\beta$ band
in the superconducting stateDLFeng2010YZhang . In this case, $\alpha$ and
$\beta$ bands are degenerate in the normal state but have different
superconducting gaps in the superconducting state. (g) The flat band is from
the band folding of the structure near M ($\pi$,$\pi$)
pointSShin2017TShimojima .  Figure 2: Flat band observed near the $\Gamma$
point in fresh and aged (Ba0.6K0.4)Fe2As2 in the superconducting state in
laser ARPES measurement. (a-b) Fermi surface mapping of fresh (a) and aged (b)
(Ba0.6K0.4)Fe2As2 at 13 K with _h_ $\nu$=6.994 eV laser. They are obtained by
integrating the spectral weight within a $\pm$ 5 meV energy window relative to
the Fermi level EF. The polarization vector (E) is marked at the bottom-left
corner. The data for the fresh sample (a) is obtained right after the
cleaving. After that, the sample was heated up to high temperature up to 70 K
and cooled down to 13 K again. The data for the aged sample (b) is then
measured more than 24 hours after cleaving. (c-d) Band structure measured
along the Cut 1 (c) and Cut 2 (d), respectively. The location of Cut 1 and Cut
2 is shown in (a) and (b) by black lines and they are along the same
$\Gamma$-$M_{1}$ direction in the momentum space. (e-f) The second derivative
images of (c) and (d) with respect to the energy. (g-h) Photoemission spectra
(EDCs) from the band structure in (c). Red tips mark the peak position of the
flat band that is also plotted in (e) by black circles. (i) EDCs from the band
structure in (d). Red tips mark the peak position of the flat band that is
also plotted in (f) by black circles. (j-m) Comparison of EDCs between the
fresh and aged samples at four typical momenta: k1 (j), kβ (k), k2 (l), kγ
(m). The location of the four momenta is marked by arrows in (c) and (d).
Figure 3: Detailed band structure of (Ba0.6K0.4)Fe2As2 near the M point in
both normal and superconducting states measured by _h_ $\nu$=21.218 eV Helium
lamp. (a) Band structure measured along $\Gamma$-$M_{1}$ direction crossing
$M_{2}$ point at 45 K in the normal state. The location of the momentum cut is
shown by the red line in the inset of (f). (b) Band structure of (a) divided
by the corresponding Fermi distribution function. (c) Second derivative image
of (b) with respect to the energy. (d-f) Same as (a-c) but measured at 18 K in
the superconducting state. (g-i) Same as (a-c) but measured along $\Gamma$-X
direction crossing $M_{2}$ point at 45 K in the normal state. The location of
the momentum cut is shown by the red line in the inset of (l). (j-l) Same as
(g-i) but measured at 18 K in the superconducting state. The observed two
bands, $\delta$ and $\varepsilon$, are marked by cyan and red lines,
respectively, in the second derivative images of (c), (f), (i) and (l). (m)
EDCs from the band structures (a) and (d) at several momenta measured at 45 K
(black lines) and 18 K (red lines). The location of the momentum points is
marked by red tips in (a). (n) Same as (m) but from the band structures (g)
and (j). (o) Symmetrized EDCs at 45 K (black line) and 18 K (red line) from
the original EDCs for the momentum $k_{4}$ in (m). (p) Symmetrized EDCs at 45
K (black line) and 18 K (red line) from the original EDCs for the momentum
$k_{4}$ in (n). (q) Symmetrized EDCs at 45 K (black line) and 18 K (red line)
from the original EDCs for the momentum $k_{6}$ in (m).  Figure 4: Detailed
comparison of band structures of (Ba0.6K0.4)Fe2As2 between $\Gamma$ and M
measured at 18 K in the superconducting state. (a-b) Band structures measured
along the same $\Gamma$-$M_{1}$ direction crossing the $\Gamma$ point (a) and
$M_{2}$ point (b). (c-d) Corresponding second derivative images of (a) and (b)
with respect to the energy. The location of the momentum cuts is marked in the
inset of (c) and (d). (e-h) Same as (a-d) but measured along the $\Gamma$-X
direction. The location of the momentum cuts is marked in the inset of (g) and
(h). The observed $\alpha$, $\beta$ and $\gamma$ bands at the $\Gamma$ point
are marked by pink, black and blue thick dashed lines, respectively, while the
bands $\delta$ and $\varepsilon$ at the M point are marked by cyan and red
thick dashed lines. The $\delta^{\prime}$ and $\varepsilon^{\prime}$ bands
marked by cyan and red thin dashed lines at the $\Gamma$ point in (c) and (g)
represent the folded bands from the $\delta$ and $\varepsilon$ bands at the M
point. (i) Comparison of EDCs at the corresponding momenta around $\Gamma$ and
M. The location of the four pairs of momenta, (${\Gamma}k_{1}$, $Mk_{1}$),
(${\Gamma}k_{2}$, $Mk_{2}$), (${\Gamma}k_{3}$, $Mk_{3}$) and (${\Gamma}k_{4}$,
$Mk_{4}$), is marked in (a), (b), (e) and (f). The three features, labelled as
star, tip and triangle, correspond to $\beta$, $\varepsilon$ and $\delta$
bands, respectively.  Figure 5: Signatures of band folding from $\Gamma$ to
the M in (Ba0.6K0.4)Fe2As2 both in the normal state and superconducting state.
(a-b) Band structures measured along the same $\Gamma$-$M_{1}$ direction
crossing the $\Gamma$ point (a) and $M_{2}$ point (b) at 45 K in the normal
state. These are second derivative images with respect to momentum. (c-d) Same
as (a-b) but measured at 18 K in the superconducting state. (e-h) Same as
(a-d) but measured along the $\Gamma$-X direction. The location of the
momentum cuts is marked in the inset. The dashed red lines in the bands near
the M point represent the position of the bands folded from the $\Gamma$ point
(solid red lines).  Figure 6: Formation of the flat band in (Ba0.6K0.4)Fe2As2
at the $\Gamma$ point in the superconducting state from the Bogoliubov back-
bending of the $\beta$ band and the folding of the $\varepsilon$ band at M to
$\Gamma$. (a) Quantitative comparison of the flat band at the $\Gamma$ point
with the $\varepsilon$ band at the M point. The black circles represent the
flat band dispersion at $\Gamma$ extracted from Helium lamp measurement (_h_
$\nu$=21.218 eV) in Fig. 4a, and the blue triangles represent the flat band
dispersion at the $\Gamma$ point from laser measurement (_h_ $\nu$=6.994 eV)
in Fig. 2d. The red squares represent the $\varepsilon$ band dispersion at the
M point from Helium lamp measurement (_h_ $\nu$=21.218 eV) in Fig. 3e. (b-c)
Schematic Fermi surface of (Ba0.6K0.4)Fe2As2 at kz = 0 before (b) and after
(c) $\Gamma$-M band folding. Since $\alpha$ and $\beta$ bands are degenerate,
there are two hole-like pockets around the $\Gamma$ point and a small electron
pocket around the M point. Four strong spots are illustrated by red dashed
lines but the corresponding M-shaped bands lie below the Fermi level. The
folded Fermi pockets are marked by solid thin lines in (c). (d) Schematic band
structure of (Ba0.6K0.4)Fe2As2 in the normal state above Tc at kz = 0. (e)
Corresponding band structure in the superconducting state below Tc. (f) Band
structure in the superconducting state after $\Gamma$-M folding. The folded
bands are shown by the dashed lines.  Figure 7: The overall superconducting
gap structure of the optimally-doped (Ba0.6K0.4)Fe2As2 superconductor. (a)
Band structure measured at 18 K by _h_ $\nu$=21.218 eV Helium lamp along the
$\Gamma$-$M_{1}$ direction crossing the $\Gamma$ point. The location of the
momentum cut is marked by a black line in the inset. (b-c) EDCs at
$k_{\alpha/\beta}$ (b) and at $k_{\gamma}$ (c). (d-e) Corresponding
symmetrized EDCs at $k_{\alpha,\beta}$ (d) and at $k_{\gamma}$ (e). The
location of the $k_{\alpha/\beta}$ and $k_{\gamma}$ is marked by arrows in
(a). (f) Three-dimensional plot of the superconducting gap on the three hole-
like pockets around $\Gamma$ and the tiny electron-like pockets around M. The
corresponding Fermi surface is shown at the bottom.