establish the $2$-equivalence of all these qubit strategies. In other words,
they transform, via Eq. 14, the state of Eq. 112 and the observables of Eq.
113, respectively, to the reference state of Eq. 114a and reference
observables of LABEL:MDS2meas for all $a,x,b,y$. Note that in arriving at Eq.
114a, we have also made use of the fact that the sign of
$\zeta=\cos(\omega_{B}+\gamma_{B}-\phi-\omega_{A})$ follows that of of
$\sin{2\theta}\sin{\alpha}\sin{2\beta}$.

Finally, a straightforward application of Theorem 2.5 using the just-
established $2$-equivalence and the fact that $\vec{P}_{\text{Q,3}}$ is
extremal in $\mathcal{Q}$ (see the end of Section E.1) completes the proof of
the theorem. ∎

##  Appendix F Quantum correlation giving maximal CHSH value in
$\mathcal{Q}_{1}$ is a self-test

To find the maximal CHSH violation for correlations in this Class, one may
follow the procedure given in Section D.1 or Section E.1 by considering
general qubit unitary matrices of unit determinant connecting the different
measurement bases. For simplicity, however, we note that in the maximization
of $\mathcal{S}_{\text{\tiny CHSH}}$, it suffices to consider extreme points
in $\mathcal{Q}$. By Fact 2.3, it thus suffices to consider observables of the
form of Eq. 13. Consider now the Bell operator [Braunstein  _et
al._(1992)Braunstein, Mann, and Revzen] corresponding to the Bell inequality
of Eq. 5

$\begin{split}\mathcal{B}&=-A_{0}\otimes B_{0}-A_{1}\otimes B_{0}-A_{0}\otimes
B_{1}+A_{1}\otimes B_{1}.\end{split}$ (116)

The maximal value of $\mathcal{S}_{\text{\tiny CHSH}}$ is obtained by
measuring the observables of Eq. 13 on the eigenstate $\ket{\psi}$ of
$\mathcal{B}$ giving the largest eigenvalue.

For correlations from Class 1, however, we also have the constraint
$P(0,0|0,0)=|\braket{00}{\psi}|^{2}=0$. Thus, to find the maximal CHSH value
achievable by $\vec{P}\in\mathcal{Q}_{1}$, we may consider $\ket{\psi}$ to be
an eigenstate of the principle $3\times 3$ sub-matrix $\mathcal{B}^{\prime}$
of $\mathcal{B}$ related to the span of {$\ket{10},\ket{01},\ket{11}$}, i.e.,

$\mathcal{B}^{\prime}=\begin{bmatrix}\tau&S_{\alpha}S_{\beta}&(1-C_{\beta})S_{\alpha}\\\
S_{\alpha}s_{\beta}&\tau&(1-C_{\alpha})S_{\beta}\\\
(1-C_{\beta})S_{\alpha}&(1-C_{\alpha})s_{\beta}&-\tau\\\ \end{bmatrix}.$ (117)

where $\tau:=1+C_{\alpha}+C_{\beta}-C_{\alpha}C_{\beta}$,
$C_{\alpha}=\cos\alpha$, and $S_{\beta}=\sin\beta$, etc.

To compute the eigenvalue of $\mathcal{B}^{\prime}$, note that its
characteristic polynomial is

$-\lambda^{3}+\tau\lambda^{2}+4\lambda-4C_{\alpha}^{2}S_{\beta}^{2}+4C_{\alpha}(C_{\beta}-1)-4C_{\beta}(C_{\beta}+1),$
(118)

where $\lambda$ is the eigenvalue. Using $\mathsf{Mathematica}$, we find that
the largest value of $\mathcal{S}_{\text{\tiny CHSH}}$, i.e., the largest real
value of $\lambda$ is $\xi_{3}\approx 2.6353$ with the unique parameters
$C_{\alpha}=C_{\beta}=\xi_{1}\approx 0.2862$, and the corresponding
eigenvector is

$\ket{\psi}=\frac{\sin{\theta}}{\sqrt{2}}\left(\frac{\sin{\alpha}}{|{\sin{\alpha}}|}\ket{0}\\!\ket{1}+\frac{\sin{\beta}}{|{\sin{\beta}}|}\ket{1}\\!\ket{0}\right)+\cos{\theta}\ket{1}\\!\ket{1},$
(119)

where $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$ are, respectively, the smallest
positive roots of the cubic polynomials given in Eq. 46.

######  Theorem F.1.

If the CHSH value of $\xi_{3}$ is observed alongside $P(0,0|0,0)=0$, then the
underlying state and measurements are equivalent to those given in Eq. 45 up
to local isometries, i.e., $\vec{P}_{\text{Q,4}}$ self-tests this particular
quantum realization.

###### Proof.

Since the only remaining freedom is the sign of $\sin{\alpha}$ and
$\sin{\beta}$, it can be easily seen that the local unitary $\sigma_{z}$ can
be used to generate the required sign. Therefore, the correlation
$\vec{P}_{\text{Q,4}}$ which is the only one giving the maximal value of
$\mathcal{S}_{\text{\tiny CHSH}}$ in Class 1 is an extreme point of
$\mathcal{Q}$ and any quantum strategy realizing this correlation is
$2$-equivalent to the reference state and measurements given in Eq.
45.131313Strictly, the current proof only applies to all quantum measurements
of the form given in Eq. 13. However, even more general measurements involving
complex state and observables, such as those considered in the proof in
Section D.1 or Section E.1 may be shown to be $2$-equivalent to the reference
strategy in a similar manner.  An application of Theorem 2.5 then completes
the proof. ∎

##  Appendix G Robustness results

Here, we provide the remaining plots that illustrate the level of robustness
of the self-testing results established for $\vec{P}_{\text{Q}}$ of Eq. 31
from Class 3b, $\vec{P}_{\text{Q,2}}$ of Eq. 35 from Class 2a,
$\vec{P}_{\text{Q,3}}$ of Eq. 38 from Class 2b, $\vec{P}_{\text{Cabello}}$ of
Eq. 42 from Class 2c, and $\vec{P}_{\text{Q,4}}$ of Eq. 47 from Class 1. In
other words, for any given value $\mathcal{S}_{\text{\tiny CHSH}}$ of the
(suboptimal) CHSH Bell violation and the allowed tolerance $\varepsilon$ from
the zero probabilities, we show the best numerically determined lower bound on
(1) the fidelity with respect to the reference state, Eq. 74, and (2) an
appropriate figure of merit, Eq. 81, that quantifies the similarity with the
target measurements. Further details about these figures of merit can be found
in Appendix C.

Figure 5:  Plots illustrating the robustness of the self-testing of Eq. 45
based on the correlation $\vec{P}_{\text{Q,4}}$ of Eq. 47 from Class 1. From
left to right, we show, respectively, as a function of the Bell value
$\mathcal{S}_{\text{\tiny CHSH}}$, a lower bound on Eq. 74 for self-testing
the reference two-qubit state and a lower bound on Eq. 81 for self-testing
both parties’ observables.

Figure 6: Plots illustrating the robustness of the self-testing of Eq. 27 and
Eq. 29 based on the correlation $\vec{P}_{\text{Q}}$ of Eq. 31 from Class 3b.
From top to bottom, the plots show, as a function of the Bell value
$\mathcal{S}_{\text{\tiny CHSH}}$, a lower bound on Eq. 74 for self-testing
the reference two-qubit state and a lower bound on Eq. 81 for self-testing
Alice’s and Bob’s observables. For the significance of the dashed horizontal
line and other details related to the plots, see the caption of Fig. 1.

Figure 7: Plots illustrating the robustness of the self-testing of Eq. 34
based on the correlation $\vec{P}_{\text{Q,2}}$ of Eq. 35 from Class 2a. From
top to bottom, the plots show, as a function of the Bell value
$\mathcal{S}_{\text{\tiny CHSH}}$, a lower bound on Eq. 74 for self-testing a
Bell state and a lower bound on Eq. 81 for self-testing Alice’s and Bob’s
observables. For the significance of the dashed horizontal line and other
details related to the plots, see the caption of Fig. 1.

Figure 8:  Plots illustrating the robustness of the self-testing of Eq. 23 and
Eq. 37 based on the correlation $\vec{P}_{\text{Q,3}}$ of Eq. 38 from Class
2b. From top to bottom, the plots show as a function of the Bell value
$\mathcal{S}_{\text{\tiny CHSH}}$, a lower bound on Eq. 74 for self-testing
the reference two-qubit state and a lower bound on Eq. 81 for self-testing
Alice’s and Bob’s observables. For the significance of the dashed horizontal
line, and other details related to the plots, see the caption of Fig. 1.

Figure 9:  Plots illustrating the robustness of the self-testing of Eq. 44
with Eq. 43b based on the correlation $\vec{P}_{\text{Cabello}}$ of Eq. 42
from Class 2c. From top to bottom, the plots show as a function of the Bell
value $\mathcal{S}_{\text{\tiny CHSH}}$, a lower bound on Eq. 74 for self-
testing the reference two-qubit state and a lower bound on Eq. 81 for self-
testing Alice’s and Bob’s observables. For the significance of the dashed
horizontal line and other details related to the plots, see the caption of
Fig. 1.

##  Appendix H Non-exposed quantum extreme points on the no-signaling boundary

To arrive at the non-exposed property of the various example correlations
given in Section 3, we apply the method described in Appendix H of Ref. [Goh
_et al._(2018)Goh, Kaniewski, Wolfe, Vértesi, Wu, Cai, Liang, and Scarani],
i.e., we show that for each of the self-testing $\vec{P}$ described below,
there is no Bell function $\vec{B}$ such that it is the unique maximizer. For
this matter, it suffices to show that for any Bell function where $\vec{P}$ is
a maximizer, a local extreme point gives also the same Bell value.

To this end, consider a vector $\vec{M}$ with components

$M(a,b|x,y)=M^{A}_{a|x}\otimes M^{B}_{b|y}.$ (120)

Then a Bell operator [Braunstein  _et al._(1992)Braunstein, Mann, and Revzen]
can be written as

$\mathcal{B}=\vec{B}\cdot\vec{M},$ (121)

where $\vec{B}$ is a vector of real numbers associated with an arbitrary Bell
function.

Now, let $\vec{B}$ be a Bell function maximized by the self-testing
correlation $\vec{P}$. Since the corresponding reference state
$\ket{\widetilde{\psi}}$ is an eigenstate of the Bell operator $\mathcal{B}$,
it follows from Eq. 121 that any state vector $\ket{\phi}$ orthogonal to
$\ket{\widetilde{\psi}}$, i.e., satisfying
$\braket{\phi}{\widetilde{\psi}}=0$, will result in a vector

$\vec{T}\equiv\bra{\phi}\vec{M}\ket{\psi}$ (122)

orthogonal to $\vec{B}$, giving $\vec{B}\cdot\vec{T}=0$. Note also from Eq.
120, Eq. 122, and the completeness relations of POVM that
$\sum_{a,b}T(a,b|x,y)=0$ for all $x$ and $y$.

In our work, all $\ket{\widetilde{\psi}}$ are two-qubit states, so one can
always find three $\ket{\phi}$ orthogonal to each reference state. For
concreteness, we label all the sixteen local deterministic extreme points by
$j=0,1,\ldots,15$ such that

$P_{d,j}(a,b|x,y)=\delta_{a,(j\bmod{2})x\oplus\lfloor\frac{j\bmod{4}}{2}\rfloor}\delta_{b,\lfloor\frac{j\bmod{8}}{4}\rfloor
y\oplus\lfloor\frac{j}{8}\rfloor}.$ (123)

Then, if the optimum value of the following linear program

$\begin{split}\max_{\vec{B}\in\mathbb{R}^{8}}\quad&\vec{B}\cdot\vec{P}\\\ {\rm
s.t.}\quad&\vec{B}\cdot\vec{T}_{i}=0\quad\text{for}\quad i\in\\{1,2,3\\}\\\
&\vec{B}\cdot\vec{P}_{d,j}\leq 1\quad\text{for}\quad
j=0,1,\ldots,15,\end{split}$ (124a) is $1$ and this bound is saturated by at
least one of the $\vec{P}_{d,j}$, we obtain a proof that the self-testing
correlation $\vec{P}$ is non-exposed. To certify that the maximum value of Eq.
124a is upper bounded by 1, we make use of weak duality [Boyd and
Vandenberghe(2004)], namely the fact that any feasible solution of the linear
program dual to Eq. 124a, i.e., $\begin{split}\min_{y_{j}\geq
0,z_{i}\in\mathbb{R}}\quad&\sum_{j=0}^{15}y_{j}\\\ {\rm
s.t.}\quad&\sum_{j=0}^{15}y_{j}P_{d,j}+\sum_{i}z_{i}\vec{T}_{i}=\vec{P}\\\
&\text{for}\quad i\in\\{1,2,3\\}\text{ and }j=0,1,\ldots,15.\end{split}$
(124b)

always provides an upper bound to the optimum value of Eq. 124a. Note further
that in Eq. 124, each $\vec{T}_{i}$ is obtained form a vector $\ket{\phi_{i}}$
orthogonal to $\ket{\widetilde{\psi}}$ via Eq. 122. To complete the proof, it
suffices to consider any subset $\\{\ket{\phi_{i}}\\}$ such that the a dual
feasible solution gives exactly $1$.

In the following, we provide the $\\{\ket{\phi_{i}}\\}$, $\\{\vec{T}_{i}\\}$
and the corresponding dual variables $\\{y_{j}\\}$, $\\{z_{i}\\}$ needed for
the proof of the non-exposed nature of $\vec{P}_{\text{Q}}$,
$\vec{P}_{\text{Q,2}}$, $\vec{P}_{\text{Q,3}}$ and $\vec{P}_{\text{Cabello}}$.

###  H.1 Class 3b

For the self-testing correlation $\vec{P}_{\text{Q}}$ of Eq. 31b, which gives
the maximal CHSH Bell violation in $\mathcal{Q}_{3b}$, the corresponding
reference state and measurements are provided in Eq. 27 and Eq. 29. To certify
the non-exposed nature of $\vec{P}_{\text{Q}}$, it suffices to consider
$\ket{\phi_{1}}=\ket{0}\\!\ket{0}$ as the state orthogonal to the reference
state of Eq. 27a, which gives

$\begin{split}\vec{T}_{1}=\,\,\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$0$&$-t_{1}$&$t_{1}$\\\
&$1$&$0$&$0$&$0$&$0$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$-t_{2}$&$0$&$-t_{2}$&$0$\\\
&$1$&$t_{2}$&$0$&$t_{2}-t_{1}$&$t_{1}$\\\
\hline\cr\end{tabular}\leavevmode\nobreak\ ,\end{split}$ (125)

where

$\displaystyle\begin{split}t_{1}&=\tfrac{1}{6}\left(\sqrt[3]{3\sqrt{33}-17}-\tfrac{2}{\sqrt[3]{3\sqrt{33}-17}}+4\right)\approx
0.2282\\\
t_{2}&=\tfrac{1}{6}\left(\sqrt[3]{19-3\sqrt{33}}+\tfrac{4}{\sqrt[3]{19-3\sqrt{33}}}-2\right)\approx
0.4196\end{split}$ (126)

The optimized $\vec{B}$ has only the non-zero components
$B(a,b|1,0)=1\quad\forall\,\,a,b$.

For the dual problem of Eq. 124b, the assignment

$\displaystyle y_{3}$
$\displaystyle=\tfrac{1}{6}\left(\sqrt[3]{2\left(3\sqrt{33}+13\right)}-\tfrac{4\
2^{2/3}}{\sqrt[3]{3\sqrt{33}+13}}-1\right)\approx 0.1478,$ $\displaystyle
y_{8}$
$\displaystyle=\tfrac{1}{3}\left(\tfrac{\sqrt[3]{11\left(3\sqrt{33}-11\right)}}{2^{2/3}}-\tfrac{22^{2/3}}{\sqrt[3]{3\sqrt{33}-11}}+2\right)\approx
0.1041$ $\displaystyle y_{12}$ $\displaystyle=\tfrac{1}{2}-y_{8},\quad
y_{6}=y_{3}-z_{1}t_{1},$ $\displaystyle y_{9}$
$\displaystyle=1-y_{3}-y_{6}-y_{8}-y_{12},\quad z_{1}=2y_{3},$ (127)

and zero otherwise, is easily verified to be a feasible solution with a value
of $1$, thus showing that $\vec{P}_{\text{Q}}$ is not exposed.

###  H.2 Class 2a

For the self-testing correlation $\vec{P}_{\text{Q,2}}$ of Eq. 35, which gives
the maximal CHSH violation in $\mathcal{Q}_{2a}$, the corresponding reference
state and measurements are, respectively, $\ket{\Phi^{+}_{2}}$ and those
provided in Eq. 34. To certify the non-exposed nature of
$\vec{P}_{\text{Q,2}}$, we consider the singlet state
$\ket{\phi_{1}}=\frac{1}{\sqrt{2}}(\ket{0}\\!\ket{1}-\ket{1}\\!\ket{0})$ as
the state orthogonal to $\ket{\Phi^{+}_{2}}$, which gives

$\begin{split}\tfrac{8}{\sqrt{3}}\,\vec{T}_{1}&=\,\,\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$0$&$-1$&$1$\\\
&$1$&$0$&$0$&$1$&$-1$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$-1$&$1$&$-1$&$1$\\\
&$1$&$1$&$-1$&$1$&$-1$\\\ \hline\cr\end{tabular}\leavevmode\nobreak\
.\end{split}$ (128)

The optimized $\vec{B}$ has only the non-zero components
$B(1,b|0,0)=B(0,b|0,1)=1\quad\forall\,\,b$.

For the dual problem of Eq. 124b, the assignment

$y_{3}=y_{6}=y_{9}=y_{12}=\tfrac{1}{4},\quad z_{1}=\tfrac{1}{\sqrt{3}},$ (129)

and zero otherwise, is easily verified to be a feasible solution with a value
of $1$, thus showing that $\vec{P}_{\text{Q,2}}$ is not exposed.

###  H.3 Class 2b

For the self-testing correlation $\vec{P}_{\text{Q,3}}$ of Eq. 38, which gives
the maximal CHSH violation in $\mathcal{Q}_{2b}$, the corresponding reference
state and measurements are provided in Eq. 23 and Eq. 37. To certify the non-
exposed nature of $\vec{P}_{\text{Q,3}}$, it suffices to consider the
following states orthogonal to the reference state of Eq. 23 with Eq. 37,

$\begin{split}\ket{\phi_{1}}&=\ket{0}\\!\ket{0},\quad\ket{\phi_{2}}=\sin{\alpha}\ket{0}\\!\ket{1}+\cos{\alpha}\ket{1}\\!\ket{1},\end{split}$
(130)

where $\alpha=\frac{\pi}{6}$. In this case,

$\begin{split}8\sqrt{\tfrac{2}{3}}\,\vec{T}_{1}&=\,\,\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$0$&$-2$&$2$\\\
&$1$&$0$&$0$&$0$&$0$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$-2$&$0$&$-3$&$1$\\\
&$1$&$2$&$0$&$1$&$1$\\\ \hline\cr\end{tabular}\leavevmode\nobreak\ ,\\\
8\sqrt{\tfrac{2}{3}}\,\vec{T}_{2}&=\,\,\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$0$&$0$&$0$\\\
&$2$&$2$&$-2$&$0$&$0$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$1$&$-3$&$0$&$-2$\\\
&$1$&$1$&$1$&$0$&$2$\\\ \hline\cr\end{tabular}\leavevmode\nobreak\
,\end{split}$ (131)

and the optimized $\vec{B}$ takes the form of

$\begin{split}\vec{B}&=\scalebox{0.92}{\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$-b_{1}$&$b_{2}$&$0$&$1-b_{1}-b_{2}$\\\
&$1$&$0$&$b_{2}$&$0$&$b_{1}$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$b_{2}$&$0$&$b_{3}$&$b_{1}$\\\
&$1$&$2b_{1}+b_{2}$&$0$&$0$&$0$\\\ \hline\cr\end{tabular}}\leavevmode\nobreak\
,\end{split}$ (132)

where $b_{1}\approx 0.2197$, $b_{2}\approx 0.3410$, and $b_{3}\approx 0.6590$.

For the dual program in Eq. 124b, the assignment

$\begin{split}y_{2}&=y_{3}=y_{12}=y_{15}=\tfrac{1}{4},\quad
z_{1}=z_{2}=\tfrac{1}{\sqrt{6}},\end{split}$ (133)

and zero otherwise, is easily verified to be a feasible solution with a value
of $1$, thus showing that $\vec{P}_{\text{Q,3}}$ is not exposed.

###  H.4 Class 2c

For the self-testing correlation $\vec{P}_{\text{Cabello}}$ of Eq. 42, which
gives the maximal CHSH violation in $\mathcal{Q}_{2c}$, the corresponding
reference state and measurements are provided in Eq. 44. To certify the non-
exposed nature of $\vec{P}_{\text{Cabello}}$, it suffices to consider the
following states orthogonal to the reference state of Eq. 44a

$\begin{split}\ket{\phi_{1}}&=k_{2}\ket{0}\\!\ket{0}-k_{1}\ket{0}\\!\ket{1}-k_{3}\ket{1}\\!\ket{0}+k_{2}\ket{1}\\!\ket{1},\\\
\ket{\phi_{2}}&=k_{3}\ket{0}\\!\ket{0}-k_{2}\ket{0}\\!\ket{1}+k_{2}\ket{1}\\!\ket{0}-k_{1}\ket{1}\\!\ket{1}.\end{split}$
(134)

The exact analytic expression of $\vec{T}_{1}$ and $\vec{T}_{2}$ can be
computed from Eq. 122, Eq. 44, and Eq. 134. Below, we provide their
approximate form for ease of reference:

$\begin{split}\vec{T}_{1}&=\scalebox{0.92}{\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$-t_{1}$&$t_{2}$&$t_{3}$\\\
&$1$&$0$&$t_{1}$&$t_{4}$&$-t_{2}-t_{3}-t_{4}$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$t_{5}$&$t_{6}$&$t_{7}$&$0$\\\
&$1$&$-t_{5}$&$-t_{6}$&$t_{8}$&$-t_{7}-t_{8}$\\\
\hline\cr\end{tabular}}\leavevmode\nobreak\ ,\\\
\vec{T}_{2}&=\scalebox{0.92}{\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$-u_{1}$&$u_{2}$&$u_{3}$\\\
&$1$&$u_{1}$&$0$&$u_{4}$&$-u_{2}-u_{3}-u_{4}$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$u_{5}$&$-u_{6}$&$u_{7}$&$0$\\\
&$1$&$u_{6}$&$-u_{5}$&$u_{8}$&$-u_{7}-u_{8}$\\\
\hline\cr\end{tabular}}\leavevmode\nobreak\ ,\\\ \end{split}$ (135)

where $t_{1}\approx 0.1507$, $t_{2}\approx 0.1319$, $t_{3}\approx-0.2826$,
$t_{4}\approx 0.2891$, $t_{5}=0.3202$, $t_{6}=0.0909$, $t_{7}\approx 0.4112$,
and $t_{8}\approx 0.0098$ for $\vec{T}_{1}$, while $u_{1}\approx 0.4214$,
$u_{2}\approx-0.2950$, $u_{3}\approx-0.1264$, $u_{4}\approx 0.2770$,
$u_{5}\approx 0.0871$, $u_{6}\approx 0.3342$, $u_{7}\approx-0.2471$,
$u_{8}\approx 0.2291$ for $\vec{T}_{2}$.

The optimized $\vec{B}$ takes the form of

$\begin{split}\vec{B}&=\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$b_{1}$&$0$&$0$&$b_{2}$\\\
&$1$&$1$&$0$&$0$&$0$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$0$&$1$&$0$&$-b_{2}$\\\
&$1$&$-b_{3}$&$0$&$b_{4}$&$b_{5}$\\\
\hline\cr\end{tabular}\leavevmode\nobreak\ ,\end{split}$ (136)

where $b_{1}\approx 0.6918$, $b_{2}\approx 0.4170$, $b_{3}\approx 0.2005$,
$b_{4}\approx 0.1349$, and $b_{5}\approx 0.0917$.

For the dual program in Eq. 124b, the assignment

$\displaystyle y_{3}$
$\displaystyle=\tfrac{1}{6}\left(-\sqrt[3]{53-6\sqrt{78}}-\tfrac{1}{\sqrt[3]{53-6\sqrt{78}}}+7\right)\approx
0.3427,$ $\displaystyle y_{12}$
$\displaystyle=-\tfrac{\sqrt[3]{9-\sqrt{78}}}{3^{2/3}}-\tfrac{1}{\sqrt[3]{3\left(9-\sqrt{78}\right)}}+2\approx
0.4786,$ $\displaystyle y_{15}$ $\displaystyle=1-y_{3}-y_{12}\approx 0.1787,$
$\displaystyle z_{1}$
$\displaystyle=-\sqrt{\tfrac{\zeta_{1}-\frac{133727}{\zeta_{1}}-145}{48}}\approx-0.6414,$
(137) $\displaystyle z_{2}$
$\displaystyle=\tfrac{1}{6}\left(\zeta_{2}-\tfrac{23}{\zeta_{2}}-11\right)\approx
0.3855,$ $\displaystyle\zeta_{1}$
$\displaystyle=\sqrt[3]{6827808\sqrt{78}+35282447},$ $\displaystyle\zeta_{2}$
$\displaystyle=\sqrt[3]{186\sqrt{78}+1639},$

and zero otherwise, is easily verified to be a feasible solution with a value
of $1$, thus showing that $\vec{P}_{\text{Cabello}}$ is not exposed.

###  H.5 Class 1

For the self-testing correlation $\vec{P}_{\text{Q,4}}$ of Eq. 47, which gives
the maximal CHSH violation in $\mathcal{Q}_{1}$, the corresponding reference
state and measurements are provided in Eq. 45. To certify the non-exposed
nature of $\vec{P}_{\text{Q,4}}$, it suffices to consider the following states
orthogonal to the state of Eq. 45a

$\begin{split}\ket{\phi_{1}}&=\ket{0}\\!\ket{0},\\\
\ket{\phi_{2}}&=\cos\theta\left(\frac{\ket{0}\\!\ket{1}+\ket{1}\\!\ket{0}}{\sqrt{2}}\right)-\sin\theta\ket{1}\\!\ket{1}.\end{split}$
(138)

The exact analytic expression of $\vec{T}_{1}$ and $\vec{T}_{2}$ can be
computed from Eq. 122, Eq. 44, and Eq. 134. Below, we provide their
approximate form for ease of reference:

$\displaystyle\vec{T}_{1}$
$\displaystyle=\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$0$&$t_{1}$&$-t_{1}$\\\
&$1$&$0$&$0$&$0$&$0$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$t_{1}$&$0$&$t_{2}$&$t_{3}$\\\
&$1$&$-t_{1}$&$0$&$t_{3}$&$-t_{2}-2t_{3}$\\\
\hline\cr\end{tabular}\leavevmode\nobreak\ ,$ (145) $\displaystyle\vec{T}_{2}$
$\displaystyle=\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$0$&$u_{1}$&$u_{2}$&$u_{3}$\\\
&$1$&$u_{1}$&$-2u_{1}$&$u_{4}$&$-u_{234}$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$u_{2}$&$u_{4}$&$u_{5}$&$u_{6}$\\\
&$1$&$u_{3}$&$-u_{234}$&$u_{6}$&$-u_{5}-2u_{6}$\\\
\hline\cr\end{tabular}\leavevmode\nobreak\ ,$ (152)

where $t_{1}\approx 0.3298$, $t_{2}\approx 0.4766$, and $t_{3}\approx-0.1468$,
for $\vec{T}_{1}$, while $u_{1}\approx 0.1111$, $u_{234}\equiv
u_{2}+u_{3}+u_{4}$, $u_{2}\approx 0.0397$, $u_{3}\approx 0.0715$,
$u_{4}\approx-0.3113$, $u_{5}\approx-0.1429$, and $u_{6}\approx-0.1288$ for
$\vec{T}_{2}$.

The optimized $\vec{B}$ takes the form of

$\begin{split}\vec{B}&=\begin{tabular}[]{cc|cc|cc|}&&\lx@intercol\hfil$x=0$\hfil\lx@intercol\vrule\lx@intercol
&\lx@intercol\hfil$x=1$\hfil\lx@intercol\vrule\lx@intercol \\\
&&$0$&$1$\nobreak\leavevmode&$0$&$1$\\\
\hline\cr\hbox{\multirowsetup$y=0$}&$0$&$1$&$1+b_{1}$&$-b_{1}$&$0$\\\
&$1$&$1$&$1$&$0$&$b_{2}$\\\
\hline\cr\hbox{\multirowsetup$y=1$}&$0$&$0$&$0$&$0$&$b_{4}$\\\
&$1$&$0$&$b_{3}$&$b_{5}$&$0$\\\ \hline\cr\end{tabular}\leavevmode\nobreak\
,\end{split}$ (153)

where $b_{1}\approx 0.4850$, $b_{2}\approx-0.1944$, $b_{3}\approx-0.6795$,
$b_{4}\approx-0.7872$, and $b_{5}\approx-0.3025$.

For the dual program in Eq. 124b, let $\zeta_{1}\approx
0.4450,\zeta_{2}\approx 0.7530$, and $\zeta_{3}\approx 0.0677$ be the
respective smallest positive roots of the following cubic polynomials:
$q_{1}(x)=x^{3}-x^{2}-2x+1,q_{2}(x)=x^{3}-7x^{2}+14x-7,$ and
$q_{3}(x)=x^{3}-32x^{2}-116x+8.$ The assignment

$\displaystyle y_{3}$ $\displaystyle=y_{12}=\zeta_{1},$ $\displaystyle y_{15}$
$\displaystyle=1-2y_{3}\approx 0.1099,$ $\displaystyle z_{1}$
$\displaystyle=-\sqrt{\zeta_{2}}\approx-0.8678,$ $\displaystyle z_{2}$
$\displaystyle=\sqrt{\zeta_{3}}\approx 0.2602,$ (154)

and zero otherwise, is easily verified to be a feasible solution with a value
of $1$, thus showing that $\vec{P}_{\text{Q,4}}$ is not exposed.

##  Appendix I Proof of Corollary 4.3

From the proof of Corollary 4.2, we see that in maximizing the value of any
Bell inequality in the CHSH Bell scenario by $\ket{\Phi^{+}_{d}}$, it suffices
to consider correlations $\vec{P}\in\mathcal{M}_{d}$ taking the form of Eq.
52. Let $\vec{B}$ be the Bell coefficients associated with the CHSH inequality
of Eq. 5, then

$\displaystyle\max_{\vec{P}\in\mathcal{M}_{d}}\vec{B}\cdot\vec{P}$
$\displaystyle=$
$\displaystyle\max_{\vec{P}^{\prime}_{i}\in\mathcal{M}_{2},\vec{P}^{\prime\prime}_{j}\in\mathcal{L},k\geq
0,\frac{d-k}{2}\in\mathbb{Z}^{+}}\frac{2}{d}\sum_{i=1}^{\frac{d-k}{2}}\vec{B}\cdot\vec{P}^{\prime}_{i}+\frac{1}{d}\sum_{j=1}^{k}\vec{B}\cdot\vec{P}^{\prime\prime}_{j}$
$\displaystyle\leq$ $\displaystyle\max_{k\geq
0,\frac{d-k}{2}\in\mathbb{Z}^{+}}\frac{2}{d}\sum_{i=1}^{\frac{d-k}{2}}\max_{\vec{P}^{\prime}_{i}\in\mathcal{M}_{2}}\vec{B}\cdot\vec{P}^{\prime}_{i}+\frac{1}{d}\sum_{j=1}^{k}\max_{\vec{P}^{\prime\prime}_{j}\in\mathcal{L}}\vec{B}\cdot\vec{P}^{\prime\prime}_{j}$
$\displaystyle=$ $\displaystyle\max_{k\geq
0,\frac{d-k}{2}\in\mathbb{Z}^{+}}\frac{2}{d}\sum_{i=1}^{\frac{d-k}{2}}2\sqrt{2}+\frac{1}{d}\sum_{j=1}^{k}2$
$\displaystyle=$ $\displaystyle\left\\{\begin{aligned} \leavevmode\nobreak\
&2\sqrt{2}\left(\frac{d-1}{d}\right)+\frac{2}{d},&&d=\text{odd},\\\
\leavevmode\nobreak\ &2\sqrt{2},&&d=\text{even}.\end{aligned}\right.$ (155)

These upper bounds are indeed attainable [see Eq. (5) of [Liang and
Doherty(2006)]], thus completing the proof that the maximal CHSH violation by
$\ket{\Phi^{+}_{d}}$ is given in the last line of Appendix I.

##  Appendix J Proof of Lemma 4.5

Let us begin by noting that the transposition operation preserves the
hermiticity of a matrix. Then, from spectral theorem, we may write :

$\begin{split}\begin{cases}&E_{0|0}^{\text{\,\tiny
T}}=\sum_{i=0}^{r^{0}}\lambda_{i}\ket{e_{i}}\\!\\!\bra{e_{i}},\quad
0<\lambda_{i}\leq 1,\\\ &E_{1|0}^{\text{\,\tiny
T}}=\mathbb{1}_{d}-E_{0|0}^{\text{\,\tiny T}},\\\ \end{cases}\\\
\begin{cases}&F_{0|0}=\sum_{i=0}^{r^{1}}\lambda^{\prime}_{i}\ket{f_{i}}\\!\\!\bra{f_{i}},\quad
0<\lambda^{\prime}_{i}\leq 1,\\\
&F_{1|0}=\mathbb{1}_{d}-F_{0|0}.\end{cases}\end{split}$ (156)

where $\lambda_{i}$ and $\lambda_{i}^{\prime}$ are, respectively, the
nonvanshing eigenvalues of $E^{\text{\,\tiny T}}_{0|0}$ and $F_{0|0}$. Without
loss of generality, suppose $P(0,0|0,0)=0$, then from Eq. 8, we get

$\frac{1}{d}{\rm tr}[E_{0|0}^{\text{\,\tiny
T}}F_{0|0}]=\frac{1}{d}\sum_{i=0}^{r^{0}}\sum_{j=0}^{r^{1}}\lambda_{i}\lambda^{\prime}_{j}|\langle
e_{i}|f_{j}\rangle|^{2}=0.$ (157)

Notice that the RHS of Eq. 157 is a sum of non-negative terms. The fact that
this sum vanishes means that $|\langle e_{i}|f_{j}\rangle|=0$ for all $i,j$,
which means that $E_{0|0}^{\text{\,\tiny T}}F_{0|0}=0$, i.e.,
$E_{0|0}^{\text{\,\tiny T}}$ and $F_{0|0}$ commute and hence can be
diagonalized in the same basis.

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# Molten Salt Flux Liquid Transport Method for Ultra Clean Single Crystals
UTe2

Dai Aoki1 E-mail<EMAIL_ADDRESS>Tohoku University Tohoku
University Oarai Oarai Ibaraki 311-1313 Ibaraki 311-1313 Japan
Japan

###### Abstract

Various single-crystal growth techniques have been presented for the
unconventional superconductor UTe2. The molten salt flux liquid transport
(MSFLT) method is employed to grow high-quality and large single crystals,
exhibiting a high residual resistivity ratio ($\mbox{RRR}\sim
200\mbox{--}800$). However, the Te self-flux and chemical vapor transport
(CVT) methods produce samples of lower quality. The MSFLT method is a hybrid
approach combining the molten salt flux (MSF) and CVT methods. One significant
advantage is that the materials gradually crystallize at a relatively low
temperature, which is maintained throughout the main process. This may be
crucial for preventing U deficiency and obtaining high-quality large single
crystals of UTe2. Many different single crystals obtained using different
techniques have been characterized by resistivity and specific heat
measurements. The superconducting transition temperature $T_{\rm c}$ decreases
with residual resistivity $\rho_{0}$, followed by the Abrikosov-Gor’kov pair-
breaking theory, and reaches $2.1\,{\rm K}$ for $\rho_{0}\to 0$. The residual
$\gamma$-value of the specific heat for the highest quality sample was only
$3\,{\%}$ of the normal-state $\gamma$-value. The specific heat jump, $\Delta
C_{\rm e}/(\gamma T_{\rm c})$ reached approximately $2.7$ for high-quality
samples, indicating a strong-coupling superconductor. Furthermore, the
magnetic susceptibility for $H\parallel a$-axis in a high-quality single
crystal does not show an up-turn behavior on cooling, which is consistent with
the results of NMR Knight shift and $\mu$SR experiments.

UTe2 has attracted considerable attention because of its unusual
superconducting properties [1, 2, 3]. The large upper critical field $H_{\rm
c2}$ associated with the field-reentrant behavior and multiple superconducting
phases at pressure [4, 5, 6] as well as at high fields for $H\parallel b$-axis
[7, 8, 9], demonstrate spin-triplet superconductivity owing to the spin and
orbital degrees of freedom. The determination of superconducting order
parameters is an important task, and therefore, high-quality single crystals
are required. In the early stages of the discovery of superconductivity with
insufficient sample quality, “partially gapped superconductivity” was
proposed, owing to the large residual density of states in the specific heat
measurements [1]. However, recent high-quality single crystals have revealed
an almost zero residual density of states [3], and fully gapped
superconductivity with spin-triplet $A_{u}$ state has been proposed [10, 11].
Single or double superconducting transition in the specific heat is also an
important issue, because it is related to the breaking of time-reversal
symmetry in the superconducting state and the components of the
superconducting order parameters [12, 13, 14, 15]. In high-quality single
crystals, a single superconducting transition of the specific heat is
established at ambient pressure and zero field [6, 3].

A breakthrough in high-quality single-crystal growth has recently been
achieved [16]. Owing to a new technique called the molten salt flux (MSF)
method, the quality of single crystals UTe2 has been improved significantly.
The superconducting transition temperature increased from $1.6$ to $2.1\,{\rm
K}$, and de Haas-van Alphen (dHvA) oscillations were observed for the first
time [17]. The quasi-two-dimensional Fermi surfaces with heavy quasiparticles
were directly detected using dHvA experiments. These results were confirmed
using different techniques by different groups [18, 19, 20].

In this paper, we present our empirical results on different methods for the
single-crystal growth of UTe2 and their characterization. The MSF method
significantly improves the sample quality, resulting in a high residual
resistivity ratio (RRR) of more than 100. This technique is further improved
by employing the so-called molten salt flux liquid transport (MSFLT) method,
which is a hybrid of the CVT and MSF methods. Over 170 attempts at single-
crystal growth using different techniques have yielded single crystals with
different qualities. These have been characterized using resistivity, specific
heat, and magnetic susceptibility measurements. Resistivity measurements were
performed using the four-probe AC method with a current along $a$-axis.
Specific heat measurements were performed using a relaxation technique down to
$0.34\,{\rm K}$. The magnetic susceptibility was measured using a SQUID
magnetometer down to $1.9\,{\rm K}$ for $H\parallel a$-axis. Single-crystals
of different qualities were also tested through dHvA experiments using a
field-modulation technique.

The resistivity results reveal that $T_{\rm c}$ is dependent on the sample
quality, with the maximum $T_{\rm c}=2.1\,{\rm K}$, and it follows the
Abrikosov-Gor’kov pair-breaking theory. The specific heat results show that
$T_{\rm c}$ decreases with an increase in the residual density of states. The
highest-quality sample showed an almost zero residual density of states and a
large specific heat jump ($\Delta C_{\rm e}/(\gamma T_{\rm c})\sim 2.7$),
revealing strong coupling superconductivity.

The first attempt at single crystal growth involved a self-flux method using
excess Te. The starting materials U and Te, with an atomic ratio of
$\mbox{Te/U}=3.55$, were placed in an alumina crucible. This was sealed in a
tantalum tube in an arc furnace under Ar gas. The tantalum tube was further
sealed under vacuum in a quartz ampoule. According to the U-Te binary alloy
phase diagram [21], UTe2 can grow above 950∘C. The quartz ampoule was heated
in an electrical furnace to 1050∘C and cooled to 955∘C at a cooling rate of
$1.5\,^{\circ}{\rm C/hr}$. At this temperature, excess Te was removed via
centrifugation. Large single crystals of millimeter size were obtained.
Superconductivity is observed at $1.08\,{\rm K}$ defined by zero resistivity
with a low residual resistivity ratio, $\mbox{RRR}=3.6$, despite the absence
of bulk superconductivity found in the specific heat measurements. The self-
flux method is applicable to the growth of large single crystals of UTe2;
however, the sample quality was poor.

The CVT method with an off-stoichiometric amount of starting materials yielded
better-quality samples. The starting materials in the ratio $\mbox{Te/U}\sim
1.5$ and iodine as the transport agent adjusted to a density of $\sim 3\,{\rm
mg/cm^{3}}$ were put in a quartz ampoule sealed under vacuum. The ampoule was
then placed in a horizontal furnace and heated. The temperature gradient with
$\Delta T=60\mbox{--}100\,^{\circ}{\rm C}$ was maintained for 10–14 days. The
starting materials were transported from the high-temperature side to the low-
temperature side, where single crystals were grown. Figures 1(f) and 1(b)
present a schematic of the CVT method and a photograph of single crystals,
respectively. By selecting optimal conditions, a large single crystal of $\sim
1\,{\rm g}$ can be grown, as shown in Fig. 1(c).

Figure 1: (Color online) Photographs of single crystals UTe2 grown by (a)
MSFLT and (b) CVT. (c) A very large single crystal of 0.992 g can be grown by
the CVT method. Schematics of quartz ampoules for the (d) MSF, (f) CVT, and
(g) MSFLT methods, and (e) a photograph of a quartz ampoule for MSFLT method.
(h) Temperature profile for the MSFLT method.

The values of $T_{\rm c}$ depend on the growth temperature of the CVT method.
Figure 2(b) shows the $T_{\rm c}$ determined by specific heat measurements as
a function of the growth temperature on the lower-temperature side in 54
different samples grown using the CVT method. $T_{\rm c}$ varies from $2$ to
$\sim 1.4\,{\rm K}$, which reveals that a lower growth temperature is
preferable for a higher $T_{\rm c}$. A similar trend was reported in Ref.
Ros22.

Figure 2: (Color online) (a) Schematic Te-U phase diagram, which is slightly
different from that in Ref. Mas90. (b) $T_{\rm c}$ vs. growth temperature at
lower temperature side for CVT grown samples. Here, $T_{\rm c}$ is defined by
the specific heat jump. The lines connecting two data points correspond to
double transitions in specific heat.

According to the single-crystal X-ray analysis [23], a low-quality sample
without superconductivity is associated with a slight U deficiency of
approximately $4\,\%$, whereas the superconducting sample exhibits a
stoichiometric ratio of U to Te. Thus, we propose a Te-U equilibrium phase
diagram [24], as shown schematically in Fig. 2(a), which differs slightly from
that reported in Ref. Mas90. Growth at high temperatures may yield small
amounts of U deficiency, whereas a high-quality sample with stoichiometric U
and Te is available through low-temperature growth. Figure 2(a) is also
consistent with the fact that the Te self-flux method always produces a low-
quality sample with U deficiency.

Note that the starting composition for the CVT method must be off-
stoichiometric and U-rich, specifically, $\mbox{Te/U}<2$, to obtain high-
quality single crystals with less U deficiency. In fact, we obtained low-
quality samples using the CVT method with stoichiometric starting materials,
$\mbox{Te/U}=2$, and a temperature gradient $950/850\,^{\circ}{\rm C}$, in
which RRR is approximately 2.5. This was consistent with the results in
previous studies [25, 26].

The MSF method significantly improves the sample quality [16]. Single crystals
can be grown at relatively low temperatures using this method, and the excess
U in the molten salt acts as a reducing agent and suppresses the U vacancies.
A mixture of NaCl and KCl with an equi-atomic ratio was selected as flux,
which has the eutectic melting point of $650\,^{\circ}{\rm C}$. As shown in
Fig. 1(d), the starting materials and flux with an atomic ratio of U : Te :
NaCl : KCl = 1 : 1.65 : 30 : 30 were placed in a long alumina crucible or
directly into a quartz ampoule. The quartz ampoule was then evacuated with
gentle heating at approximately $200\,^{\circ}{\rm C}$ to dehydrate the flux,
and then sealed under vacuum. The quartz ampoule was heated in an electrical
furnace up to $950\,^{\circ}{\rm C}$ and maintained for one day. It was then
slowly cooled to $650\,^{\circ}{\rm C}$ at a rate of
$1\mbox{--}2\,^{\circ}{\rm c}/{\rm hr}$. After dwelling at $650\,^{\circ}{\rm
C}$ for 1 day, the furnace was switched off. Single crystals were grown in
salt at the bottom of the crucible. When the ratio of the starting materials
was selected as $\mbox{Te/U}=1.6\mbox{--}1.7$, the resistivity results showed
$\mbox{RRR}=50\mbox{--}220$ and $T_{\rm c}=2.00\mbox{--}2.08\,{\rm K}$, which
are defined as zero resistivity. The results of the specific heat yielded
$T_{\rm c}=1.96\mbox{--}2.07\,{\rm K}$ and the residual $\gamma$-value scaled
by the value in the normal state, $\gamma_{0}/\gamma_{\rm
N}=0.034\mbox{--}0.16$. A higher Te ratio, $\mbox{Te/U}=1.8$ produces a lower-
quality sample with $\mbox{RRR}=22$ and a lower $T_{\rm c}$ of $1.45\,{\rm K}$
at the specific heat. Single crystals are not obtained with the atomic ratio
of $\mbox{Te/U}=1.5\mbox{--}1.55$.

Most single crystals obtained by the MSF method are rather small and elongated
along the $a$-axis, with a thin flat surface perpendicular to the $c$-axis or
$[011]$ direction in reciprocal space. Even if a thick single crystal is
obtained, several halls may exist in the crystal, in which the layered
crystals are piled up as in “mille-feuille.” This is a disadvantage of the MSF
method compared to that in the CVT method. Another problem is that the
temperature gradually decreases during crystallization. This may lead to
samples of different quality as well as other impurity phases such as U7Te12
and U3Te5 [16] in the crucible.

To obtain larger single crystals of higher quality, we developed a new
technique called molten salt flux liquid transport (MSFLT) method. The
starting materials and salt flux were placed into a long quartz ampoule with a
length of $120\mbox{--}190\,{\rm mm}$ and an inner diameter of $8\,{\rm mm}$.
The quartz ampoule was sealed under vacuum after gentle heating to dehydrate
the salt ( Fig. 1(e) and (g)). The ampoule was gradually heated in a
horizontal furnace and maintained at a temperature gradient
$750/670\,^{\circ}{\rm C}$ for 10–14 days, as shown in Fig. 1(h).

This is a hybrid approach comprising the CVT and MSF methods. Instead of
vapor, liquid flux carries the starting material as a transport agent [27]. An
advantage of the MSFLT method is that the growth temperature is fixed at a low
temperature during the process, which might be favorable for high-quality
single-crystal growth of UTe2 because the lower growth temperature yields a
higher quality sample, as shown in Fig. 2(b). Another advantage of the MSFLT
method is that relatively large, high-quality single crystals can be obtained,
as shown in Fig. 1(a). This was because the growth conditions were stable at a
fixed growth temperature, preventing nucleation in the ampoule during the
process. As the materials were transported, some impurities are expected to
remain at the starting position as in the CVT method.

The crystal-growth conditions for the self-flux, CVT, MSF and MSFLT methods
are summarized in Table 2, along with the results of characterization by
specific heat and resistivity measurements. [tbh] Crystal growth conditions
and results of characterizations in different samples obtained by self-flux,
chemical vapor transport (CVT), molten salt flux (MSF) and molten salt flux
liquid transport (MSFLT) methods, where Te/U is the atomic ratio of the
starting material to Te and U. A mixture of NaCl and KCl with equi-atomic
ratio was used for MSF and MSFLT. The temperature decreased slowly in the
self-flux and MSF methods, whereas a fixed temperature gradient was applied in
the CVT and MSFLT methods. RRR and $\gamma_{0}/\gamma_{\rm N}$ denote the
residual resistivity ratio and residual $\gamma$-value scaled by the normal
state $\gamma$-value, respectively. $T_{\rm c}^{\rho=0}$ is $T_{\rm c}$
defined as the zero resistivity. $T_{\rm c}^{\rm C_{p}}$ is $T_{\rm c}$
defined by the specific heat jump. ∗No superconductivity was observed down to
$1.7\,{\rm K}$ for sample #S4.

Method | Batch | Te/U | Flux | Temperature (∘C) | RRR | $\gamma_{0}/\gamma_{\rm N}$ | $\Delta C_{\rm e}/\gamma T_{\rm c}$ | $T_{\rm c}^{\rho=0}({\rm K})$ | $T_{\rm c}^{\rm C_{p}}({\rm K})$
---|---|---|---|---|---|---|---|---|---
Self flux | #F2 | 3.55 | Te | $1050\\!\to\\!955\,(1.5^{\circ}{\rm C/h})$ | 3.6 | – | – | 1.08 | –
CVT | #C5 | 2.00 | – | 950 / 850 | 2.5 | – | – | – | –
CVT | #C34 | 1.50 | – | 1050 / 990 | 14 | 0.610 | 1.29 | 1.65 | 1.47
CVT | #C89 | 1.40 | – | 780 / 680 | 49 | 0.130 | 2.36 | 2.01 | 1.97
MSF | #S4 | 1.80 | NaCl/KCl | $950\\!\to\\!650\,(1^{\circ}{\rm C/h})$ | 22 | 0.780 | 0.58 | –∗ | 1.45
MSF | #S6 | 1.65 | NaCl/KCl | $950\\!\to\\!650\,(2^{\circ}{\rm C/h})$ | 220 | 0.046 | 2.33 | 2.06 | 2.04
MSFLT | #S56 | 1.50 | NaCl/KCl | 750 / 670 | 179 | 0.124 | 2.25 | 2.06 | 2.01
MSFLT | #S37 | 1.65 | NaCl/KCl | 750 / 670 | 800 | 0.034 | 2.57 | 2.09 | 2.05

Figure 3: (Color online) Electronic specific heat and resistivity at low
temperatures for sample #S37 with $\mbox{RRR}\sim 800$. A sharp single jump in
the specific heat is observed at $T_{\rm c}^{\rm C_{\rm p}}=2.05\,{\rm K}$.
The transition width is $\Delta T_{\rm c}\sim 0.05\,{\rm K}$. The residual
$\gamma$-value scaled by the normal state $\gamma$-value is
$\gamma_{0}/\gamma_{\rm N}=0.034$. Zero resistivity is observed at $T_{\rm
c}^{\rho=0}=2.09\,{\rm K}$. The onset of the superconducting transition is
observed at approximately $2.2\,{\rm K}$. The inset shows the resistivity
curve up to $300\,{\rm K}$. The dotted lines are results of fitting
extrapolated to $0\,{\rm K}$.

Figure 3 shows the specific heat and resistivity measured at low temperatures
for sample #S37 grown using the MSFLT method. The electronic specific heat,
$C_{\rm e}$ was obtained by subtracting the phonon contribution. A sharp
single specific heat jump at $2.05\,{\rm K}$ was defined by the entropy
balance with a transition width $\sim 0.05\,{\rm K}$. A large jump in $\Delta
C_{\rm e}/\gamma T_{\rm c}=2.57$ reveals strong coupling superconductivity.
The data were fitted from $0.34$ to $0.65\,{\rm K}$ using $C_{\rm
e}/T=\gamma_{0}+BT^{2}$, which revealed a very small residual $\gamma$-value
of $\gamma_{0}/\gamma_{\rm N}=0.034$. The resistivity of the current along the
$a$-axis also shows a sharp drop with $T_{\rm c}=2.09\,{\rm K}$ defined by
zero resistivity with an onset at approximately $2.2\,{\rm K}$. The residual
resistivity, $\rho_{0}$, obtained by the $T^{2}$ fitting below $3.5\,{\rm K}$
and RRR are $0.40\,\mu\Omega\\!\cdot\\!{\rm cm}$ and 800, respectively. Here,
the resistivity at $300\,{\rm K}$ is scaled using a previously reported value
[28].

Figure 4: (Color online) (a) $T_{\rm c}$ as a function of RRR or inverse RRR
for different 91 samples grown by MSF/MSFLT (blue circle), CVT (purple square)
and self-flux (red-purple diamond) methods. $T_{\rm c}$ is defined by zero
resistivity. The inset shows the zoomed view at high RRR. The dotted line
denotes the results of Abrikosov-Gor’kov pair breaking theory. (b) $T_{\rm c}$
as a function of the residual $\gamma$-value scaled by the normal state
$\gamma$-value for 95 samples. The inset shows the zoomed view at low residual
$\gamma$-values. The dotted line is a guide to the eyes. (c) $\Delta C_{\rm
e}/\gamma T_{\rm c}$ as a function of the scaled residual $\gamma$-value.

Figure 4(a) shows the relationship between $T_{\rm c}$ defined by zero
resistivity, and the inverse RRR for 91 samples grown using the MSF, MSFLT,
CVT, and self-flux methods. $T_{\rm c}$ clearly depended on the sample quality
and showed almost no change for $\mbox{RRR}\gtrsim 100$. The maximum $T_{\rm
c}$ is $2.1\,{\rm K}$ when $\rho_{0}\to 0$. High-quality samples were mainly
obtained using the MSF or MSFLT methods. The results approximately follow the
Abrikosov-Gor’kov pair-breaking theory [29], as indicated by the dotted line.

The results of the specific heat measurements as bulk properties are shown in
Fig. 4(b). All measurements were performed down to $\sim 0.4\,{\rm K}$, and
the residual $\gamma$-values were extracted by fitting $C_{\rm
e}/T=\gamma_{0}+BT^{2}$ below $0.65\,{\rm K}$. A higher $T_{\rm c}$ indicates
a lower residual $\gamma$-value, reaching $\gamma_{0}/\gamma_{\rm N}\sim
0.03$, which implies that the residual $\gamma$-value is zero for an ideal
crystal. Note that the residual density of states and the residual thermal
conductivity are theoretically discussed in Ref. Min22.

The specific heat jump was also affected by the sample quality. Figure 4(c)
shows $\Delta C_{\rm e}/\gamma T_{\rm c}$ as a function of
$\gamma_{0}/\gamma_{\rm N}$. The value of $\Delta C_{\rm e}/\gamma T_{\rm c}$
increases with a decrease in the residual $\gamma$-value and reaches 2.7 for
$\gamma_{0}/\gamma_{\rm N}\to 0$.

Figure 5: (Color online) Magnetic susceptibilities at low temperatures for
$H\parallel a$-axis at $0.1\,{\rm T}$ in two different quality samples grown
by CVT (#C2, $T_{\rm c}\sim 1.6\,{\rm K}$) and MSF (#S7, $T_{\rm c}\sim
2.0\,{\rm K}$). The drop below $2\,{\rm K}$ for sample #S7 is due to
superconductivity.

The magnetic susceptibility for $H\parallel a$-axis, which corresponds to the
easy-magnetization axis, also differs between high- and low-quality samples.
As shown in Fig. 5, the low-quality sample with $T_{\rm c}=1.6\,{\rm K}$
showed an upturn upon cooling below $10\,{\rm K}$, which had been analyzed by
scaling to the ferromagnetic quantum criticality. However, the high-quality
sample with $T_{\rm c}=2\,{\rm K}$ shows no upturn below $10\,{\rm K}$,
revealing that the upturn behavior is not intrinsic, but due to
disorder/defects or magnetic clusters. These results are consistent with those
obtained by the NMR Knight shift [31] and $\mu$SR experiments [32].

Figure 6: (Color online) (a)dHvA oscillations at $0.06\,{\rm K}$ for the field
direction tilted by $27$ deg from $c$ to $a$-axis in different samples #S37,
#S6 and #S1, and (b)the corresponding FFT spectra. The data are vertically
shifted for clarity. The values of RRR, $T_{\rm c}$ by specific heat, and
$\gamma_{0}/\gamma_{\rm N}$ are $\sim 800$, $2.05\,{\rm K}$ and $0.03$ for
sample #S37, $\sim 220$, $2.04\,{\rm K}$ and $0.05$ for sample #S6, $\sim
100$, $1.99\,{\rm K}$ and $0.12$ for sample #S1, respectively.

The dHvA oscillations are highly sensitive to sample quality, particularly for
frequencies with heavy effective masses. Figure 6 shows the high-quality
sample #S37 with four fundamental dHvA branches, $\alpha_{1}$,
$\alpha_{1}^{\prime}$, $\beta$ and $\alpha_{2}$, whereas the dHvA amplitudes
are strongly suppressed in the lower-quality samples #S6 and #S1. The Dingle
temperatures for branch $\alpha_{1}$ with effective mass of $35\,m_{0}$ were
$0.11$ and $0.15\,{\rm K}$ for samples #S37 and #S6, respectively. The
corresponding mean free paths were $1050$ and $830\,{\rm\AA}$, revealing a
higher quality of sample #S37.

In summary, different growth techniques for UTe2 single crystals were
presented. High-quality single crystals were obtained using MSFLT, where
$T_{\rm c}$ and the residual $\gamma$-value, $\gamma_{0}/\gamma_{\rm N}$,
reached $2.1\,{\rm K}$ and $0.03$, respectively. $T_{\rm c}$ approximately
follows the Abrikosov-Gor’kov pair-breaking theory, and is related to the
residual $\gamma$-values. No upturn was observed in the susceptibility to
$H\parallel a$-axis at low temperatures in the high-quality sample. The MSFLT
method is a hybrid approach between the MSF and CVT methods. This technique,
which is generally called the liquid transport method or horizontal flux
method, can be applied to high-quality single-crystal growth of other
materials [33, 34] by leveraging the fixed growth temperature during this
process.

## Acknowledgements

We thank Y. Ōnuki, H. Sakai, J. Flouquet, G. Knebel, V. Mineev, Y. Haga, G.
Lapertot, J. P. Brison, D. Braithwaite, Y. Homma, D. X. Li, A. Nakamura, Y.
Shimizu and A. Miyake for fruitful discussions and technical support. This
study was supported by KAKENHI (JP19H00646, JP20K20889, JP20H00130,
JP20KK0061, JP22H04933).

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