\section*{Supplementary Note}
\if0
\subsection{Suppression of the superstructure in the Bi-O layer}
We performed the X-ray diffraction (XRD) measurement on \ce{Bi_{1.7}Pb_{0.5}Sr_{1.9}CaCu2O_{8+$\delta$}} (Pb-Bi2212) which is the very sample measured in the ARPES experiment in order to check whether the superstructural modulation of \ce{Bi_{2}Sr_{2}CaCu2O_{8+$\delta$}} (Bi2212) was suppressed by substituting Pb for Bi \cite{MUSOLINO20031}. We used a High Luminance In-plane Type X-ray Diffractometer SmartLab 9kW (Rigaku) at room temperature.
The absence of the superstructure modulation was confirmed by the absence of the satellite peaks which indicate the superstructure modulation around the main (020)${}_\text{o}$ and (040)${}_\text{o}$ peaks. Here, the orthorhombic notation for unit cell size of $\sqrt{2}a \times \sqrt{2}a$ is employed.

\begin{figure}[h]
\includegraphics[width = 11cm, bb = 0 0 820 537]{20181114_S1SA.png
\caption{\label{fig:S1} In-plane XRD for Pb-Bi2212. Black arrows indicate the (020)${}_\text{o}$ and (040)${}_\text{o}$ Bragg peaks in Pb-Bi2212. There are no satellite peaks which reflect the superstructure modulation around the main peaks. Extrinsic four peaks marked by * are supposed to come from other elements of the mechanical strain device.}
\end{figure}
\fi

\subsection{Strain estimated from the lattice parameters }
After the ARPES measurements, we carried out XRD measurements in order to estimate the magnitude of the tensile strain applied to the sample by the screw shown in Fig. \ref{fig:S2}(c). We estimated the strain from the anisotropy of the $a$ and $b$ lattice parameters because it is difficult to measure the strain directly because of the plasticity of the device. As shown in Fig. \ref{fig:S2}(a), we focused on the (200) and (020) Bragg peaks of Pb-Bi2212 and each peak was fitted using a single Gaussian. Figure \ref{fig:S2}(b) shows the $a$ and $b$ lattice parameters estimated as $a = 3.814$  \AA \, and $b = 3.812$ \AA. The anisotropy of the crystal structure is $(a-b)/(a+b) < 0.03$ \%. The anisotropy of the electronic structure  induced by the distortion $\delta_\text{crystal} = 7 \times (a-b)/(a+b) < 0.2$ \% expected from Eq. (S4.3) is much smaller than the asymmetry parameter $\delta$ of order 1\% which we observed in the ARPES measurements shown in Fig. 3. Therefore, the enhancement of the nematicity in the pseudogap state cannot be explained due merely to the anisotropy of the crystal structure.
A picture and a schematic figures of the strain device are shown in Fig \ref{fig:S2}(c).

\begin{figure}[h]
\includegraphics[width = 15cm, bb = 0 0 722 677]{20181114_S1SA.png
\caption{\label{fig:S2} Strength of the tensile strain estimated from lattice parameter. (a) X-ray diffraction pattern around the (200) and (020) peaks of Pb-Bi2212. Data were fitted using a single Gaussian. (b) Lattice parameters of $a$ and $b$ estimated from the Bragg peaks using Bragg's law.  The difference between $a$ and $b$ was less than 0.03 \%. (c) Sample holder designed to apply strain to the sample by tightening the stainless screw. The top view is shown on the right.}
\end{figure}


\clearpage
\subsection{Matrix-element effect}

In vacuum, the vector potential $\vec{A}(\vec{r},t)$ satisfies the wave equation,
\begin{equation}
\nabla ^2 \vec{A}(\vec{r},t) = \frac{1}{c^2} \frac{\partial ^2 \vec{A}(\vec{r},t)}{\partial t^2}. \label{eq:2.5.1}
\end{equation}
One of the solutions of this equation is
\begin{equation}
\vec{A}(\vec{r},t) = \vec{\varepsilon}e^{i(\vec{k}\cdot\vec{r}-\omega t)}, \label{eq:2.5.2}
\end{equation}
where $\vec{\varepsilon}$ denotes the polarization (the oscillating direction of the electric field), and $\vec{k}$ and $\omega$ are the wave vector and the frequency of light, respectively ($\omega = c k$). The probability of photoemission is given by
\begin{equation}
w_{\text{i} \rightarrow \text{f}} \propto \left|\braket{f|H'|i}\right|^2,\label{eq:2.5.3}
\end{equation}
where
\begin{equation}
H' = \frac{e}{m}\vec{A}(\vec{r})\cdot \vec{p} = \frac{e}{m}\vec{\varepsilon}e^{i\vec{k}\cdot\vec{r}}\cdot \vec{p},\label{eq:2.5.4}
\end{equation}
according to Fermi's golden rule. In ARPES measurement, we often use vacuum-ultra violet, whose photon energy is about 100 eV i.e., the wave length is about 100 \AA { } which is much longer than the distance between neighboring atoms in the material, so that one obtains
\begin{equation}
e^{i\vec{k}\cdot\vec{r}} = 1 + i\vec{k}\cdot\vec{r} + \cdots  \simeq 1  .\label{eq:2.5.5}
\end{equation}
Then the matrix element in eq (\ref{eq:2.5.3}) can be expressed as
\begin{eqnarray}
\braket{f|H'|i} &\simeq& \braket{f|\frac{e}{m}\vec{\varepsilon}\cdot \vec{p}|i}\\ \label{eq:2.5.6}
&=& \frac{e}{i\hbar} \braket{f|\vec{\varepsilon}\cdot [\vec{x},H_0]|i}\\
&=& \frac{e}{i\hbar} \braket{f|\vec{\varepsilon}\cdot \vec{x}H_0^\dagger - H_0\vec{\varepsilon}\cdot \vec{x}|i}\\
&=& \frac{e}{i\hbar}(E_\text{f}-E_\text{i}) \braket{f|\vec{\varepsilon}\cdot \vec{x}|i},\label{eq:2.5.9}
\end{eqnarray}
using the Hermite property of $H_0$ and the commutation relationship
\begin{equation}
[\vec{x},H_0]=\frac{i\hbar \vec{p}}{m}.\label{eq:2.5.10}
\end{equation}
In summary, the transition probability can be approximated by
\begin{equation}
w_{\text{i} \rightarrow \text{f}} \propto \left|\braket{f|\vec{\varepsilon}\cdot \vec{x}|i}\right|^2,\label{eq:2.5.11}
\end{equation}
and the matrix element can be described by that of the dipole transition.\par
The $k$-dependence of the intensity in $k$-space obtained by ARPES can be estimated by calculating the matrix element (eq. (\ref{eq:2.5.11})) in the case that $\ket{i} = \ket{d_{x^2-y^2}}$, $\vec{E} \propto (+1,-1,0)$, which represents our experimental condition, and $\ket{f} = e^{i\vec{k}\cdot\vec{r}}$ was assumed for simplicity. Thus, we obtain
\begin{eqnarray}
|\braket{f|\vec{E}\cdot \vec{r}|i}|^2 &=& \left|\braket{e^{i\vec{k}\cdot\vec{r}}|\vec{E}\cdot \vec{r}|d_{x^2-y^2}}\right|^2\\
&\propto& \left|\int d^2\vec{r} (x^2-y^2)(x-y)\cos(k_xx+k_yy)\right|^2\\
&& +\left|\int d^2\vec{r} (x^2-y^2)(x-y)\sin(k_xx+k_yy)\right|^2.
\end{eqnarray}
Figure \ref{fig:2.5.1} is the result of the $k$-dependence of the transition probability in the first Brillouin zone. 
\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm,bb = 0 0 634 524]{ME4}
\caption{Intensity plot in the first Brillouin zone for $\ket{i} = \ket{d_{x^2-y^2}}$, $\vec{E} \propto (+1,-1,0)$, and $\ket{f} = e^{i\vec{k}\cdot\vec{r}}$ is assumed.}
\label{fig:2.5.1}
\end{center}
\end{figure}

\clearpage

\subsection{Energy dependence of the FWHM of the momentum distribution curves in the normal, pseudogap, and superconducting states}

Figures S3 - S5 show the MDC for the off-nodal cut and  the FWHM of the two MDC peaks as functions of binding energy for the superconducting, pseudogap, and normal states, respectively. The figures show that the FWHM at $E-E_\text{F}$ = -30 meV is small compared to other binding energies, because around $E-E_\text{F}$ = -10 meV the superconducting gap opens, and in the higher binding energy region, where the quasi-particle scattering rate increases.

\begin{figure}[h]
\includegraphics[width = 15cm, bb = 0 0 610 579]{20181005_S3SA.png
\caption{\label{fig:S3} Energy dependence of the FWHM of MDC from ARPES spectra taken in the superconducting state (5 K). (a) FWHM of the MDCs shown in (b) at off-nodal k points where the cut shown in the inset crosses the Fermi surface. The solid curves are fitted ones using a quadratic function. The MDC is broad around -10 meV due to the superconducting gap and around -50 meV because of the higher scattering rate of quasi-particles. (b) MDCs at $E-E_\text{F} =$ -30 meV fitted to  two Lorentzians. (c) Band dispersion along the cut shown in the inset.}
\end{figure}

\begin{figure}[h]
\includegraphics[width = 15cm, bb = 0 0 608 574]{20181005_S4SA.png
\caption{\label{fig:S4} The same as Fig. S4 except for  taken in the pseudogap state (100 K). }
\end{figure}

\begin{figure}[h]
\includegraphics[width = 15cm, bb = 0 0 501 459]{20181005_S5SA.png
\caption{\label{fig:S5} The same as Fig. S4 except for taken in the normal state (200 K).}
\end{figure}

\clearpage
\subsection{Effect of strain on hopping parameters}

One may suspect that the anisotropy arises only from the effect of the uniaxial strain and that the temperature dependence of the anisotropy is caused by thermal expansion. Such a possibility can be excluded by the following consideration: 
Here, we consider a transfer integral between atomic orbitals of different atoms {\it i} and {\it j}. When the axis connecting both atoms is taken as the $z$-axis, the transfer integral between the i and j atoms $-t_{ij}(0,0,1) = \braket{i|h|j}$ is given by the Slater-Koster parameter $(l_il_j\mu)$, where $l_A$ is the azimuthal quantum number of atom $A$ $(A=a, b)$ and $\mu$ is the magnetic quantum number of atoms $i$ and $j$.  In the following, $l_A = 0, 1, 2,...$ and $\mu = 0, \pm1, \pm2,...$ are noted as  {\it s, p, d},... and $\sigma, \pi, \delta,...$, respectively.\par
As for the transfer integral between the $2p$ orbital of O atom and the $3d$ orbital of Cu atom, there is approximate relationship between the Slater-Koster parameter and the distance between two atoms $d$ \cite{Harrison}
\begin{equation}
(pd \mu ) \propto d^{-3.5}. \label{eq:3.2.3}
\end{equation}
Hopping between nearest-neighbor Cu atoms can be regarded as consisting of two hopping processes from Cu to O and from O to another Cu and, therefore, the nearest-neighbor hopping parameter $t$ in eq (1) as a function of $d$ is given by
\begin{equation}
t { \propto (pd \mu )^2} \propto d^{-7}. \label{eq:3.2.4}
\end{equation}
If we assume that $t$ depends only on $d$ for simplicity, one can obtain
\begin{equation}
\frac{\Delta t}{t} \simeq -7\frac{\Delta d}{d}. \label{eq:3.2.5}
\end{equation}
If the anisotropy of $t$ which we observed is only due to the uniaxial strain, the anisotropy parameter $\delta \left(= \frac{\Delta t}{t} \right)$ should change monotonically because the atomic distance changes monotonically with temperature by thermal expansion. However, the anisotropy parameter $\delta$ observed in our measurement was enhanced only in the pseudogap state. This means that the stress works to align the nematic domain properly and ensures that the anisotropy does not come only from the distortion of the sample.

\clearpage



\input{Bi2212_nematicity_supplemenal.bbl}

\end{document}

\section{\label{sec:level1}Introduction}

\section*{Introduction}
The pseudogap in cuprate superconductors is characterized by the suppression of the density of states around the Fermi level above the superconducting transition temperature $T_\text{c}$.
Broadly speaking, two possible scenarios for the pseudogap have been discussed, that is, a precursor to the superconducting state and distinct order which competes with superconductivity \cite{Keimer2015From-quantum-ma}.
The latter scenario has been put forward by several measurements which detect distinct orders with spontaneous symmetry breaking such as charge order and time-reversal symmetry breaking at or below the pseudogap temperature $T^*$ \cite{Ghiringhelli821,Chang2012Direct-obs,Comin1335,Le-Tacon2013Inelastic-,Comin390,PhysRevLett.96.197001,PhysRevB.89.094523,PhysRevLett.118.097003,0953-8984-26-50-505701,PhysRevLett.88.137005,Hinkov597,Lawler2010Intra-unit,Sato2017Thermodyna,Daou2010Broken-rot}. 

Recently, nematicity that the electronic structure preserves the translational symmetry but breaks the rotational symmetry of the underlying crystal lattice has been found to exist in the pseudogap state of the cuprate superconductors \cite{PhysRevLett.88.137005,Hinkov597,Lawler2010Intra-unit,Sato2017Thermodyna,Daou2010Broken-rot,PhysRevB.92.224502,Wu2017Spontaneou}. 
As a possible mechanism, it has been proposed that the nematic order arises from fluctuations of stripe order \cite{Kivelson1998Electronic, RevModPhys.75.1201} or from the instability of the Fermi surface (so-called Pomeranchuk instability) \cite{PhysRevB.73.214517, PhysRevB.64.195109, PhysRevB.72.024502,PhysRevLett.85.5162}. 
From experimental perspectives, the nematicity in the cuprate superconductors was first pointed out by the transport measurements of lightly-doped \ce{La_{2-$x$}Sr_$x$CuO4 } and \ce{YBa2Cu3O_$y$} (YBCO) \cite{PhysRevLett.88.137005}. 
Anisotropic signals in the spin excitation measured by neutron scattering was observed for untwinned underdoped YBCO \cite{Hinkov597}. 
Inequivalence between the oxygen atoms in the $a$ and $b$ directions was detected by scanning tunneling spectroscopy measurements for underdoped \ce{Bi2Sr2CaCu2O_{8+$\delta$}} (Bi2212) \cite{Lawler2010Intra-unit}. 
Nernst effect and magnetic torque measurements on underdoped and optimally doped YBCO showed a systematic temperature dependence of the nematicity \cite{Daou2010Broken-rot,Sato2017Thermodyna}. 
The onset temperature of the in-plane anisotropic signals in the Nernst coefficient and that in the magnetic susceptibility coincide with $T^*$. 
Thus the pseudogap state can be considered as a distinct thermodynamic phase characterized by nematicity.
Although the orthorhombic distortion of the \ce{CuO2} plane caused by the Cu-O chains along the $b$-axis of the untwinned YBCO samples already breaks the four-fold rotational symmetry even above $T^*$, the orthorhombicity is considered to help to form a microscopic nematic domain in one particular direction and enables us to detect nematicity in macroscopic measurements.
That is, this weak orthorhombicity of the \ce{CuO2} plane for nematicity induced by the Cu-O chain has the same role as a weak external magnetic field for ferromagnets \cite{Vojta2009Lattice-sy}. 

Motivated by those previous studies on nematicity in cuprate superconductors, we have performed angle-resolved photoemission spectroscopy (ARPES) measurements on \ce{Bi_{1.7}Pb_{0.5}Sr_{1.9}CaCu2O_{8+$\delta$}} (Pb-Bi2212) by applying a uniaxial strain in the Cu-O bond direction in order to detect nematicity below $T^*$. 
While most of the studies on nematicity in the cuprate superconductors have been done on YBCO, Pb-Bi2212, whose \ce{CuO2} plane has the tetragonal ($C_4$) symmetry, is a more appropriate material to study nematicity than YBCO.
Owing to the presence of a natural cleavage plane between the BiO layers and rich information accumulated from the previous ARPES studies, Bi2212 is an ideal material for ARPES experiment to investigate novel phenomena in cuprates \cite{RevModPhys.75.473}. 
In the present work, we analyze the electronic structure without the assumption that the symmetry of the electronic structure is the same as the symmetry of the \ce{CuO2} lattice in order to shed light on the possibly lowered symmetry of the electronic structure. 
Furthermore, using ARPES, one can investigate the behavior of the nematicity not only in the pseudogap state but also in the superconducting state, where transport and magnetic measurements cannot be performed due to the zero resistivity and the Meissner effect, respectively \cite{Sato2017Thermodyna,Daou2010Broken-rot,PhysRevB.92.224502}.


\section*{Results}



\begin{figure}[t]
\includegraphics[width = 8.5cm, bb = 0 0 606 835]{20181123_1SA.png}


  \caption{Energy distribution curves (EDCs) of Pb-Bi2212 at various temperatures. The measurements were performed with decreasing temperature for (A)-(D) and increasing temperature for (E)-(G). Each EDC around the X (Y) point has been integrated from $k_x = -1.1 $ to $-0.8$ (integrated from $k_y = 0.8$ to $1.1$). The difference curves between the intensities of EDCs around the X and Y points are also shown. In the pseudogap state ($T =$ 100 K and 125 K), the spectral intensities between the X and Y points are clearly different while they are identical in the normal ($T =$ 200 K and 210 K) and superconducting states ($T =$ 5 K, 25 K and 50 K). Inset shows the experimental geometry that ensures the spectroscopic equivalence of the $a$- and $b$-axis directions.}
    \label{fig:EDC}
\end{figure}

\begin{figure*}[t]
\includegraphics[width=17cm, bb = 0 0 1920 1007]{20181005_2SA.png}

\caption{\label{fig:Mapping} Constant energy-surface mapping of the ARPES spectra of Pb-Bi2212  taken at various temperatures with decreasing temperature (A)-(D) and increasing temperature (E)-(G). The constant energy is $E$ = - 30 meV and the intensity is integrated within $\pm$ 10 meV of $E$. Red circles show the constant energy surface determined from the peaks of the momentum distribution curves (MDCs). Gray curves are the fitted curves using the tight-binding model of Eq. (2).}


\end{figure*}


In Fig. \ref{fig:EDC}, energy distribution curves (EDCs) around $\vec{k} = (-\pi,0)$ (X point) and $(0,\pi)$ (Y point) at various temperatures are shown. The data were obtained from one sample but from two cleavages: We performed ARPES measurements with decreasing temperature at $T =$ 5 K, 50 K, 100 K, and 200 K on one cleaved surface and with increasing temperature at $T =$ 25 K, 125 K, and 210 K on the other cleaved surface. In both measurements, the tensile strain was applied in the $x$-direction ($||a$-axis). In the normal state (200 K), the line shapes of the EDCs were almost identical between the X and Y points, as  expected from the four-fold rotational symmetry of the \ce{CuO2} plane. With decreasing temperature, the line shapes of the EDCs around the X and Y points became different below $T^*$ in the pseudogap state (100 K) and then below $T_\text{c}$  identical again in the superconducting state (5 K and 50 K).  In the heating series, $T$ = 25 K $\rightarrow$ 125 K $\rightarrow$ 210 K, the spectral changes of the cooling series was reproduced as shown in Fig. 1.
Let us focus on the pseudogap state, where the difference of the line shapes of the EDCs at the X and Y points can be seen.  In Bi2212, it is well known that the Cu-O band is split into the anti-bonding and bonding bands due to the bilayer structure \cite{RevModPhys.75.473}. From a systematic temperature-dependent study of the dispersions near $\vec{k} = (\pi,0)$ in the overdoped sample ($T_\text{c}=$ 87 K and 91 K) whose doping level is close to and the same as the present samples, we consider that the binding energies of the bottoms of the anti-bonding band and bonding band are located around 25 meV  and 110 meV below the Fermi level, respectively, in the pseudogap state. Thus, we conclude that the intensity of the bonding (anti-bonding) band is higher around the X (Y) point than around the Y (X) point, reflecting the in-plane anisotropy of the electronic structure in the pseudogap state. Here, we would like to emphasize that this inequivalence between the X and Y points is not caused by matrix-element effect because the $x$- and $y$-directions are equivalent in our measurement geometry.


In Fig. \ref{fig:Mapping}, the intensity map of the constant energy surface at $E-E_\text{F} =$ -30 meV at various temperatures are displayed. The intensity around $k_y=-k_x$ almost vanishes due to matrix-element effect (See Supplementary Information for the matrix-element effect). From systematic photon energy-dependent-studies of the intensities from the anti-bonding and bonding bands around $\vec{k} = (\pi,0)$, the intensity of the anti-bonding band is stronger than that of the bonding band for $h\nu =$ 60 eV \cite{PhysRevLett.89.077003,PhysRevLett.90.207001} and, therefore, we believe that the constant energy surfaces shown in Fig. \ref{fig:Mapping} largely arise from the anti-bonding band.

The band structure near the Fermi level of the cuprate superconductor can be fitted using the tight-binding model
\begin{eqnarray}
 \varepsilon (k_x,k_y)- \mu = \varepsilon_0 &-&2t(\cos k_x + \cos k_y) \nonumber\\
 -4 t' \cos k_x \cos k_y &-&2t'' (\cos 2k_x + \cos 2k_y)  , \label{eq:3.2.1}
\end{eqnarray}
where $t$, $t'$, and $t''$ are the nearest-neighbor, second-nearest-neighbor, and third-nearest-neighbor hopping parameters, respectively, and $\mu$ is the chemical potential \cite{PhysRevB.95.075109}. The model has four-fold rotational ($C_4$) symmetry. 
In order to examine the possibility of the rotational symmetry breaking, we introduce an anisotropy parameter $\delta$ which represents the orthrombicity of the hopping parameters ($t$ and $t''$) \cite{PhysRevB.73.214517} as
\begin{eqnarray}
&&\varepsilon (k_x,k_y) - \mu=\varepsilon_0 -2t[(1-\delta)\cos k_x +(1+\delta) \cos k_y] \nonumber\\
&& -4 t' \cos k_x \cos k_y -2t'' [(1-\delta)\cos 2k_x +(1+\delta) \cos 2k_y] . \nonumber\\ \label{eq:3.2.2}
\end{eqnarray}
By fitting the constant energy surface using this formula, we estimated its deviation from the $C_4$ symmetry through finite $\delta$ values.
As shown in Fig. \ref{fig:Mapping}, constant energy surfaces have been determined from the peak positions of the momentum distribution curves (MDCs) and fitted to the tight-binding model Eq. (\ref{eq:3.2.2}). In the same way, we also evaluated the anisotropy of the constant energy surfaces at $E-E_\text{F} =$ -10 meV and -50 meV although the MDCs are relatively broad due to the superconducting gap opening for $E - E_\text{F} =$ -10 meV and the higher scattering rate for $E - E_\text{F} =$ -50 meV. (See Supplementary Information for the energy dependence of the line width of the MDCs). 

The anisotropy parameter values deduced from the series of measurement with decreasing temperature were generally smaller than those deduced from the series of measurement with increasing temperature by a factor of 0.5. Therefore, the former values have been multiplied by 0.5 and are plotted in Fig. \ref{fig:delta}. This procedure would be justified because the uniaxial strain to the sample may change after cleaving the sample depending on the degree of alignment of nematic domains under different magnitudes of the uniaxial strain. Figure 3 indicates that, in the normal state ($T > T_\text{c}$), the anisotropy parameter $\delta$ was almost zero, which was expected from the four-fold rotational symmetry of the \ce{CuO2} plane. In the pseudogap state, $\delta$ became finite in both measurements. The sign of $\delta$ indicates that the hopping parameters along the tensile strain direction became small in the pseudogap state. The sign of $\delta$ is also the same as the anisotropic Fermi surface of \ce{YBa2Cu4O8} under the strain from the Cu-O chains \cite{PhysRevB.80.100505}. Thus, the uniaxial strain served as an external perturbation to align a majority of the nematic domains in one direction (See also Supplementary Information for the effect of strain on hopping parameters). 

Figure 3 also shows that $\delta$ was suppressed in the superconducting state. One may suspect that the temperature dependence of $\delta$ is inconsistent with the elongation of the spin excitation peak along the perpendicular direction to the Cu-O chain observed by the neutron scattering measurement which was monotonically enhanced with decreasing temperature even into the superconducting state \cite{Hinkov597}. However, the doping level in that work was much lower than that in the present study and a quantum phase transition was proposed at the hole concentration of  $ \simeq 0.75$ between the neutron study and the present study. Therefore, it is reasonable to consider that the nematicity observed in the present work is a different type from that detected by the neutron scattering measurement, whose onset temperature was much lower than $T^*$.
In the normal and pseudogap states, our result is consistent with the previous magnetic and transport measurements on YBCO in that the nematicity becomes finite below $T^*$\cite{Sato2017Thermodyna,Daou2010Broken-rot}. As for superconducting state, the magnetic and transport measurement do not give information, while our result may indicate competition between nematicity and superconductivity.

\section*{Discussion}

From the theoretical side, dynamical mean-field theory (DMFT) combined with the fluctuation exchange (FLEX) approximation has shown that a Pomeranchuk instability where the  anisotropy of the electronic structure becomes finite, appears in the overdoped region \cite{PhysRevB.95.075109}. The calculation showed nematicity in the overdoped regime and indicated the coexistence of nematicity and superconductivity \cite{PhysRevB.95.075109}. In contrast, according to a cellular dynamical mean-field theory (CDMFT) study, four-fold rotational symmetry breaking was shown in the underdoped pseudogap regime rather than in the overdoped regime \cite{PhysRevB.86.094522,PhysRevB.82.180511}. 
According to mean-field model calculations, the Pomerachuk instability competes with superconductivity \cite{doi:10.1143/JPSJ.69.2151,PhysRevB.75.155117}. Thus, it is not clear at present whether the Pomeranchuk instability consistently explains our result or not. 

\begin{figure}[t]
\includegraphics[width = 9cm, bb = 0 0 737 530]{Delta_180705_2.png
\caption{\label{fig:delta} Temperature dependence of nematicity represented by $t$. The deduced anisotropy parameter $\delta$ is defined by Eq. (2). The error bars are standard errors of the fitting parameter at each temperature.}
\end{figure}

Nematicity was derived also from fluctuations of charge-density wave (CDW) or stripes \cite{Nie03062014,PhysRevB.92.174505}. In YBCO, CDW resides inside the pseudogap phase and competes with superconductivity \cite{Ghiringhelli821,Chang2012Direct-obs,Comin1335,Le-Tacon2013Inelastic-}. Recently, CDW in Bi2212 was also observed by resonant x-ray scattering experiments and found to compete with superconductivity \cite{PhysRevB.89.220511,Silva-Neto2014Ubiquitous} as in the case of YBCO. More recent work on Bi2212 showed that the CDW signal survived up to the pseudogap temperature $T^*$ although the hole concentration of that work was in the underdoped region ($T_\text{c} =$ 40 K)  \cite{Chaix2017Dispersive}.  From the fact that CDW in Bi2212 competes with superconductivity and possibly survives up to $T^*$ including our case, it is also likely that the nematicity observed in our measurements arises from CDW fluctuations.
Although at present it is difficult to identify the microscopic origin of the nematicity, our result provides evidence that the pseudogap state shows nematicity and competes with superconductivity. The apparent competition between the nematicity and the superconductivity may possibly be caused by the $d$-wave symmetry of the superconducting gap, whose magnitude is identical between the $k_x$ and $k_y$ directions.


However, there are still unsolved issues regarding the pseudogap, that is, how the nematicity is related to the other proposed orders inside the pseudogap phase, e.g., CDW \cite{Ghiringhelli821,Chang2012Direct-obs,Comin1335,Le-Tacon2013Inelastic-,Comin390} and loop current order \cite{PhysRevLett.96.197001,PhysRevB.89.094523,PhysRevLett.118.097003,0953-8984-26-50-505701} which correspond to translational and time-reversal symmetry breaking, respectively and how the $\vec{Q}=0$ nematic order opens pseudogap. It is difficult to discuss the origin of nematicity  and further work is necessary to identify the nature of nematicity.

\section*{Materials and Methods}
Pb-Bi2212 single crystals were grown by the floating-zone method. 
The hole concentration was optimimal to slightly overdoped one ($T_\text{c} =$ 91 K) after annealing the samples in a \ce{N2} flow, which allowed us to measure samples above and below $T^*$ ($\sim$160 K) rather easily compared to the underdoped samples whose $T^*$'s are too high \cite{Vishik06112012}. 
We measured Pb-doped samples in order to suppress the superstructure modulation present in the BiO layers of Bi2212 and shifted by  $\vec{k} = \pm(0.21\pi ,0.21\pi)$ caused by the diffraction of photoelectrons \cite{RevModPhys.75.473}. The Umklapp replicas make the spectral intensity of the Fermi surface complicated, especially around $\vec{k} = (\pi,0)$ \cite{PhysRevB.64.180505}. \par
YBCO has Cu-O chains in addition to the \ce{CuO2} plane, which helps nematic domains to form in one particular direction, while Pb-Bi2212 has no Cu-O chains and the in-plane crystal structure is tetragonal.  
Therefore, we applied a tensile strain to the sample in the Cu-O bond direction in attempt to align nematic domains in one direction using a device similar to that used for Fe-based superconductors, a picture and schematic figure of which is shown in Supplementary Fig. S2(c) \cite{PhysRevLett.115.027005}.
The strain was applied at room temperature and the stressed sample was cooled down to measurement temperatures before cleaving {\it{in situ}} (See Supplementary Information for the strain estimated from the lattice parameters by x-ray diffraction measurement). Such an operation was necessary to obtain anisotropic signals from spectroscopic data because the size of the nematic domains is not necessarily larger than the beam size; otherwise, one can measure only ``averaged isotropic'' signals \cite{Yi26042011,PhysRevB.90.121111}.\par

ARPES measurements were carried out at the undulator beamline BL5U of UVSOR using an MBS A-1 analyzer. The photon energy $h\nu$ was fixed at 60 eV. The energy resolution was set at 30 meV. The linear polarization of the incident light was chosen perpendicular to the analyzer slit and the tilt axis parallel to the analyzer slit, which realizes the unique experimental configuration that preserves the equivalence of the $a$- and $b$-axis directions with respect to the light polarization $\vec{E}$ and the strain direction, as shown in the inset of Fig. \ref{fig:EDC}. This setting ensures that the difference of the spectroscopic data between $a$- and $b$-axis directions does not originate from  matrix-element effects but from intrinsic inequivalence in the electronic structure. The measurements were performed in the normal, pseudogap, and superconducting states in two processes, that is, with increasing temperature and decreasing temperature in order to check the reproductivity. The samples were cleaved {\it in situ} under the pressure better than $2\times 10^{-8}$ Pa.
\section*{Acknowledgements}
Informative discussion with T. Shibauchi and K. Ishida is gratefully acknowledged.  A part of this work was conducted at Advanced Characterization Nanotechnology Platform of the University of Tokyo, supported by "Nanotechnology Platform" of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. ARPES experiments were performed at UVSOR (Proposal Nos. 28-813 and 29-821). This work was supported by KAKENHI Grant No. 15H02109.
\section*{Author contributions} S.N., M.H., K.K., K.H., C.L., and M.S. performed ARPES measurements with the assistance of S.I. and K.T., S.N. analyzed the data. D.S., Y.Y., and E.H. synthesized and characterized single crystals. S.N. and A.F. wrote the manuscript with suggestions by M.H., D.S., and all other coauthors. A.F. was responsible for overall project direction and planning.
\section*{Competing interests} The authors declare that they have no competing interests. 
\section*{Data and materials availability} All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.



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