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# Lowest Order Constrained Variational calculation for Polarized Liquid
${}^{3}\mathrm{He}$ at Finite Temperature

G.H. Bordbar111Corresponding author 222E-Mail<EMAIL_ADDRESS>M.J. Karimi and J. Vahedi  Department of Physics, Shiraz University, Shiraz
71454, Iran

###### Abstract

We have investigated some of the thermodynamic properties of spin polarized
liquid ${}^{3}\mathrm{He}$ at finite temperature using the lowest order
constrained variational method. For this system, the free energy, entropy and
pressure are calculated for different values of the density, temperature and
polarization. We have also presented the dependence of specific heat,
saturation density and incompressibility on the temperature and polarization.

##  I Introduction

The spin polarized liquid ${}^{3}\mathrm{He}$ at finite temperature is an
interesting many-body system which obeys the Fermi-Dirac statistics. Several
theoretical techniques and experimental method have been used for
investigating the properties of unpolarized liquid $\mathrm{{}^{3}He}$ Wi ; Ke
; Vbf ; Nd ; Lt ; Ty ; Er ; Sv . Recently, we have studied the unpolarized
liquid $\mathrm{{}^{3}He}$ at finite temperature and the spin polarized liquid
$\mathrm{{}^{3}He}$ at zero temperature Gh1 ; Gh2 ; Gh3 . In this
calculations, we have applied the lowest order constrained variational method
based on the cluster expansion of the energy functional. This method is a
powerful microscopic technique used in the many-body calculations of dense
matter Gh1 ; Gh2 ; Gh3 ; Gh4 ; Gh5 ; Gh6 ; Gh7 ; Gh8 ; Gh9 ; Gh10 .

In the present work, we use the lowest order constrained variational method to
investigate some of the thermodynamic properties of spin polarized liquid
$\mathrm{{}^{3}He}$ at finite temperature. In our calculations, we employ the
Lennard-Jones LJ  and Aziz Aziz  potentials.

##  II Method

We consider a system consisting of $N$ interacting $\mathrm{{}^{3}He}$ atoms
with $N^{+}$ spin up and $N^{-}$ spin down atoms. These atoms have a different
chemical potential ($\mu$) and different Fermi-Dirac distribution function Fet
as follows,

$n^{\pm}(k)=\frac{1}{exp[\beta(\frac{\hbar^{2}k^{2}}{2m}-\mu^{\pm})]+1},$ (1)

where $\beta=\frac{1}{k_{B}T}$ and $T$ is the temperature. We define the total
number density by $\rho$ and the spin asymmetry parameter by $\xi$ ($\xi=0$ is
unpolarized and $\xi=1$ is fully polarized cases),

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\rho^{+}+\rho^{-},$
$\displaystyle\xi$ $\displaystyle=$ $\displaystyle\frac{N^{+}-N^{-}}{N}.$ (2)

Thus, to determine the chemical potential for every $\rho$, $T$ and $\xi$, we
solve the following equation by numerical method,

$\rho^{\pm}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}n^{\pm}(k)k^{2}dk.$ (3)

We calculate the energy using the lowest order constrained variational method
based on the cluster expansion of the energy. We consider up to the two-body
energy term in cluster expansion,

$E=E_{1}+E_{2}.$ (4)

For this system, the one-body energy per particle, $E_{1}$, is given by

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle E_{1}^{+}+E_{1}^{-}$ (5)
$\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{4m\pi^{2}\rho}\left[\int_{0}^{\infty}n^{+}(k)k^{4}dk+\int_{0}^{\infty}n^{-}(k)k^{4}dk\right].$

The two-body energy per particle, $E_{2}$, is defined

$E_{2}=\frac{1}{2N}\sum_{i,j}\langle ij\mid W(12)\mid ij-ji\rangle,$ (6)

where

$W(r_{12})=\frac{\hbar^{2}}{m}(\nabla{f}(r_{12}))^{2}+f^{2}(r_{12})V(r_{12}).$
(7)

In the above equation, $V(r_{12})$ is the two body potential between helium
atoms and $f(r_{12})$ is the two-body correlation function. By considering the
$|i\rangle$ as a plane wave, we have derived the following relation for the
two-body energy per particle,

$E_{2}=\frac{1}{2}\rho\int
d\vec{r}\left[1-\frac{1}{4\pi^{4}\rho^{2}}\left[(\gamma^{+}(r))^{2}+(\gamma^{-}(r))^{2}\right]\right]W(r),$
(8)

where

$\gamma^{\pm}(r)=\int_{0}^{\infty}\frac{\sin(kr)}{kr}n^{\pm}(k)k^{2}dk.$ (9)

Now, we minimize the two-body energy Eq. (8) with respect to the variations in
the two-body correlation function subject to the normalization constraint
Clark ; Feen ,

$\frac{1}{N}\sum_{i,j}\langle ij|h^{2}(r_{12})-f^{2}(r_{12})|ij-ji\rangle=1.$
(10)

The normalization constraint is conveniently re-written in the integral form
as

$\rho\int d\vec{r}[h^{2}(r)-f^{2}(r)]\Gamma(r)=1,$ (11)

where

$\Gamma(r)=\left[1-\frac{1}{4\pi^{4}\rho^{2}}\left[(\gamma^{+}(r))^{2}+(\gamma^{-}(r))^{2}\right]\right].$
(12)

and $h(r)$ is the Pauli function,

$h(r)=[\Gamma(r)]^{-\frac{1}{2}}.$ (13)

The minimization of the two-body Energy $E_{2}$ gives the following Euler-
Lagrange differential equation for the two-body correlation function

$f^{{}^{\prime\prime}}(r)+f^{{}^{\prime}}(r)\frac{\Gamma^{{}^{\prime}}(r)}{\Gamma(r)}-\frac{m}{\hbar^{2}}\left(\lambda+V(r)\right)f(r)=0.$
(14)

The lagrange multiplier $\lambda$ imposes by normalization constraint. The
two-body correlation function calculated by numerically integrating Eq. (14)
and then the energy per particle of system can be obtained. Finally the
Helmholtz free energy per particle can be determined using the following
equation,

$F=E-TS.$ (15)

$S$ is the entropy per particle Fet  which is given by

$S=S^{+}+S^{-},$ (16)

where

$S^{\pm}=-\frac{1}{2\pi^{2}\rho}\int_{0}^{\infty}\left[n^{\pm}(k)\ln(n^{\pm}(k))+(1-n^{\pm}(k)\ln(1-n^{\pm}(k))\right]k^{2}dk.$
(17)

By calculating $E$, $S$ and $F$, we can obtain the thermodynamic properties of
the spin polarized liquid $\mathrm{{}^{3}He}$ at finite temperature.

##  III Results

The two-body correlation function at $T=1.0K$ and $T=4.0K$ for different
values of the spin asymmetry parameter ($\xi$) are shown in Fig. 1. This
figure shows that at low temperatures, the correlation function increases more
rapidly and reaches the limiting value ($f(r)=1$) at larger interatomic
distances ($r$). It is shown that by increasing $\xi$, the correlation
function reaches the limiting value at lower $r$. It is seen that our results
for the correlation function with the Lennard-Jones and Aziz potentials are
identical.

Our results for the free energy of liquid $\mathrm{{}^{3}He}$ with the
Lennard-Jones and Aziz potentials at different spin asymmetry parameters
($\xi$) are presented in Fig. 2. We can see that at low densities
($\rho<0.011A^{-3}$), the free energies with the Lennard-Jones and Aziz
potentials are similar. However, at higher densities, the difference between
these free energies are noticeable. Fig. 2 shows that the free energy
increases by increasing spin asymmetry parameter and this variation becomes
higher at high temperatures. We see that the free energy doesn’t show a bound
state above a certain value of temperature (called flashing temperature,
$T_{f}$ ). We have found $T_{f}$ for different values of $\xi$ which given in
Table 1. From Fig. 2, it is also seen that for each value of spin asymmetry
parameter, the density of saturation point (minimum point of the free energy)
decreases by increasing temperature. We see that the saturation density
($\rho_{0}$) increases by increasing spin asymmetry parameter for each value
of temperature. Our results for the saturation density are given in Table 2.
Table 1 (Table 2) shows that for all values of $\xi$ the flashing temperature
(the saturation density) with the Aziz potential is lower than that of the
Lennard-Jones potential.

The entropy of liquid $\mathrm{{}^{3}He}$ is shown as a function of density
for two different temperatures and various asymmetry parameters in Fig. 3. It
is seen that the entropy decreases by increasing both density and spin
asymmetry parameter and increases by increasing temperature. We have found
that for each temperature, the difference between entropy of fully polarized
and unpolarized liquid $\mathrm{{}^{3}He}$ decreases for higher values of
density, especially at low temperatures.

In Figs. 4-6, we have shown the free energy, entropy and the specific heat of
liquid $\mathrm{{}^{3}He}$ as a function of temperature for different values
of $\xi$ at $\rho=0.0166A^{-3}$. In these figures, the experimental results Wi
for the unpolarized liquid $\mathrm{{}^{3}He}$ are also shown for comparison.
Fig. 4 indicates a good agreement between our results for the free energy with
the Lennard-Jones potential and the experimental results. From Figs. 4-6, it
is seen that the free energy (entropy and specific heat) decreases (increase)
by increasing temperature. In addition, we see that for the free energy,
entropy and the specific heat of liquid $\mathrm{{}^{3}He}$, the differences
between fully polarized and unpolarized cases increase by increasing
temperature.

The isothermal pressure is obtained from the free energy using the following
equation,

$P(\rho,T)=\rho^{2}\frac{\partial{F(\rho,T)}}{\partial{\rho}}.$ (18)

In Fig. 7, we have plotted the isothermal pressure as a function of density
for the various temperatures and $\xi$. It is seen that the equation of state
with the Aziz potential is stiffer than that of the Lennard-Jones potential.
This figure shows that at low temperatures we have a liquid-gas phase
equilibrium for the liquid $\mathrm{{}^{3}He}$. We know that at the critical
temperature $(T_{c})$ there is no liquid-gas phase equilibrium. Our results of
the critical temperature for different values of $\xi$ are presented in Table
3. We see that the critical temperature increases by increasing $\xi$. For all
$\xi$ it is seen that the $(T_{c})$ with the Aziz potential is lower than that
of the Leonard-Jones potential.

For liquid $\mathrm{{}^{3}He}$, the saturation incompressibility at saturation
density is given by,

$K_{0}(T)=9\left(\rho^{2}\frac{\partial^{2}F(\rho,T)}{\partial{\rho^{2}}}\right)_{\rho_{0}}.$
(19)

In Table 2, our calculated values of the saturation incompressibility are
presented for different values of $\xi$ and $T$. It is seen that for each
value of $\xi$, the incompressibility decreases by increasing temperature and
at each temperature, it increases by increasing $\xi$. From Table 2 we see
that for all values of spin asymmetry parameter our results for the $K_{0}$
with the Aziz potential are lower than those of the Leonard-Jones potential.

##  IV Summary and Conclusion

We have considered a system consisting of the Helium atoms
(${}^{3}\mathrm{He}$) with an asymmetrical spin configuration. For this
system, we have calculated some of the thermodynamic properties using the
lowest order constrained variational method with the Lennard-Jones and Aziz
potentials. Our calculations lead to the following conclusions for the liquid
${}^{3}\mathrm{He}$:

  * •

At high temperatures, the correlation function reaches the limiting value for
the smaller values of the interatomic distances.

  * •

The results of the correlation function with the Lennard-Jones and Aziz
potentials are very similar.

  * •

The free energy with the the Lennard-Jones and Aziz potentials are identical
for small values of the density.

  * •

The free energy, saturation density and incompressibility decrease by
increasing temperature.

  * •

The free energy, saturation density, incompressibility and critical
temperature increase by increasing spin asymmetry parameter.

  * •

The entropy, specific heat and flashing temperature decrease by increasing
spin asymmetry parameter.

  * •

The entropy decreases by increasing density and increases by increasing
temperature.

  * •

The equation of state with the Aziz potential is stiffer than that of the
Lennard-Jones potential.

  * •

For all values of spin asymmetry parameter our results for the flashing
temperature, saturation density, critical temperature and saturation
incompressibility with the Aziz potential are lower than those of the Leonard-
Jones potential.

###### Acknowledgements.

Financial support from the Shiraz University research council is gratefully
acknowledged.

## References

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Table 1: The flashing temperature ($T_{f}$) for different values of $\xi$.

Lennard- Jones |  Aziz |
---|---|---

$\xi$ |  $T_{f}(K)$ |  $T_{f}(K)$
---|---|---
0.0 | 2.8 | 2.6
0.4 | 2.7 | 2.5
0.8 | 2.6 | 2.3
1.0 | 2.2 | 1.9

Table 2: The saturation density ($\rho_{0}$) and the saturation
incompressibility ($K_{0}$) of liquid $\mathrm{{}^{3}He}$ for different values
of $\xi$ and temperature.

Lenard- Jones |  Aziz |
---|---|---

$\xi$ |  $T(K)$ |  $\rho_{0}(A^{-3})$ |  $K_{0}(K)$ |  $\rho_{0}(A^{-3})$ |  $K_{0}(K)$
---|---|---|---|---|---
0.0 | 0.5 | 0.0117 | 185 | 0.011 | 174
0.0 | 1.0 | 0.0116 | 163 | 0.0109 | 157
0.0 | 1.5 | 0.0113 | 135 | 0.0106 | 121
0.4 | 0.5 | 0.0117 | 187 | 0.011 | 183
0.4 | 1.0 | 0.0116 | 172 | 0.0109 | 162
0.4 | 1.5 | 0.0114 | 145 | 0.0107 | 132
0.8 | 0.5 | 0.0120 | 237 | 0.0113 | 212
0.8 | 1.0 | 0.0120 | 232 | 0.0113 | 206
0.8 | 1.5 | 0.0119 | 203 | 0.0112 | 179
1.0 | 0.5 | 0.0128 | 275 | 0.0119 | 260
1.0 | 1.0 | 0.0128 | 263 | 0.0119 | 246
1.0 | 1.5 | 0.0127 | 212 | 0.0118 | 204

Table 3: The critical temperature ($T_{c}$) for different values of $\xi$.

Lennard- Jones |  Aziz |
---|---|---

$\xi$ |  $T_{c}(K)$ |  $T_{c}(K)$
---|---|---
0.0 | 5.3 | 5.1
0.4 | 5.5 | 5.3
0.8 | 6.2 | 5.8
1.0 | 6.5 | 6.1

Figure 1: The correlation function versus the interatomic distance ($r$) with
the Leonard-Jones (full curves) and Aziz (dotted curves) potentials at
$\rho=0.0166A^{-3}$ for $\xi=0.0$ (a), $\xi=0.4$ (b), $\xi=0.8$ (c) and
$\xi=1.0$ (d).

Figure 2: The free energy of liquid ${}^{3}\mathrm{He}$ versus density at
different values of temperature ($T=1.0,2.0,3.0,4.0,5.0K$)with the Leonard-
Jones (full curves) and Aziz (dotted curves) potentials for $\xi=0.0$ (a),
$\xi=0.4$ (b), $\xi=0.8$ (c) and $\xi=1.0$ (d).  Figure 3: The entropy of
liquid ${}^{3}\mathrm{He}$ versus density at $T=1.0K$ and $T=4.0K$ for
different values of $\xi$.  Figure 4: The free energy of liquid
${}^{3}\mathrm{He}$ versus temperature for $\rho=0.0166A^{-3}$ at different
values of $\xi$ with the Leonard-Jones (full curves) and Aziz (dotted curves)
potentials. The experimental results (dashed curve) for unpolarized liquid
${}^{3}\mathrm{He}$ Wi  are also shown for comparison.  Figure 5: The entropy
of liquid ${}^{3}\mathrm{He}$ versus temperature for $\rho=0.0166A^{-3}$ at
different values of $\xi$. Figure 6:  As Fig. 4, but for the specific heat.

Figure 7: The pressure versus density at different values of temperature
($T=1.0,3.0,5.0,6.0K$) with the Leonard-Jones (full curves) and Aziz (dotted
curves) potentials for $\xi=0.0$ (a), $\xi=0.4$ (b), $\xi=0.8$ (c) and
$\xi=1.0$ (d).


\arrow["{u_{T \d n_j}}"', from=1-1, to=2-2]
\end{tikzcd}\]

Immediate from <ref>, observing that $\Opalg(T \d n_j) \iso \Opalg(T)$ under $A$ by <cit.>.

§ STRICTIFICATION FOR S

Recall that , their functors, and tight transformations form a 2-category $\VDbl$ <cit.>. An equivalence of  is an equivalence in $\VDbl$.

For every  $\X$ admitting restrictions, there is an equivalent  $\X'$, with the same underlying category of tight-cells, admitting a strictly functorial choice of restrictions.

We define a  $\X'$ as follows.

* The category of objects and tight-cells is given by that of $\X$.

* A loose-cell from $A$ to $D$ is a diagram $A \xto g B \xlto p C \xfrom f D$ in $\X$.

* A 2-cell with frame

\[\begin{tikzcd}[column sep=7em]
{A_0} & {A_1} & \cdots & {A_{n - 1}} & {A_n} \\
{B_0} &&&& {B_n}
\arrow["{(f_1, p_1, g_1)}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["{(f_2, p_2, g_2)}"', "\shortmid"{marking}, from=1-3, to=1-2]
\arrow["{(f_{n - 1}, p_{n - 1}, g_{n - 1})}"', "\shortmid"{marking}, from=1-4, to=1-3]
\arrow["{(f_n, p_n, g_n)}"', "\shortmid"{marking}, from=1-5, to=1-4]
\arrow["{(f', p', g')}", "\shortmid"{marking}, from=2-5, to=2-1]
\arrow["f"', from=1-1, to=2-1]
\arrow["g", from=1-5, to=2-5]
\end{tikzcd}\]

is a 2-cell with the following frame in $\X$.

\[\begin{tikzcd}[column sep=7em]
{A_0} & {A_1} & \cdots & {A_{n - 1}} & {A_n} \\
{B_0} &&&& {B_n}
\arrow["{p_1(f_1, g_1)}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["{p_2(f_2, g_2)}"', "\shortmid"{marking}, from=1-3, to=1-2]
\arrow["{p_{n - 1}(f_{n - 1}, g_{n - 1})}"', "\shortmid"{marking}, from=1-4, to=1-3]
\arrow["{p_n(f_n, g_n)}"', "\shortmid"{marking}, from=1-5, to=1-4]
\arrow["{p'(f', g')}", "\shortmid"{marking}, from=2-5, to=2-1]
\arrow["f"', from=1-1, to=2-1]
\arrow["g", from=1-5, to=2-5]
\end{tikzcd}\]

* Composition of 2-cells is given by that in $\X$.

* The identity of a loose-cell is given by that in $\X$.

The associativity and unitality laws for $\X'$ follow from those for $\X$. There is a canonical functor $\X' \to \X$ sending each loose-cell $(f, p, g)$ to $p(f, g)$, which is the identity on the tight category and  on 2-cells. It is trivially surjective on loose-cells, since each loose-cell $p \colon A \lto D$ is the image of the diagram $A =\!= A \xlto p D =\!= D$, and thus the canonical functor is an equivalence of .

It remains to show that $\X'$ admits a strict choice of restrictions. To do so, observe that the cartesian 2-cell in $\X$ on the left below exhibits a cartesian 2-cell in $\X'$ on the right below.

\[
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJBXzAiXSxbMSwwLCJBX24iXSxbMCwxLCJCXzAiXSxbMSwxLCJCX24iXSxbMywyLCJwKGYsIGcpIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn19fV0sWzAsMiwiZiciLDJdLFsxLDMsImcnIl0sWzEsMCwicChmJ2YsIGcnZykiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifX19XSxbNSw2LCJcXGNhcnQiLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
\begin{tikzcd}[column sep=huge]
{A_0} & {A_n} \\
{B_0} & {B_n}
\arrow["{p(f, g)}", "\shortmid"{marking}, from=2-2, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f'}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{g'}", from=1-2, to=2-2]
\arrow["{p(f'f, g'g)}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["\cart"{description}, draw=none, from=0, to=1]
\end{tikzcd}
\hspace{4em}
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJBXzAiXSxbMSwwLCJBX24iXSxbMCwxLCJCXzAiXSxbMSwxLCJCX24iXSxbMywyLCIoZiwgcCwgZykiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifX19XSxbMCwyLCJmJyIsMl0sWzEsMywiZyciXSxbMSwwLCIoZiBmJywgcCwgZyBnJykiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJiYXJyZWQifX19XSxbNSw2LCJcXGNhcnQiLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
\begin{tikzcd}[column sep=huge]
{A_0} & {A_n} \\
{B_0} & {B_n}
\arrow["{(f, p, g)}", "\shortmid"{marking}, from=2-2, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, "{f'}"', from=1-1, to=2-1]
\arrow[""{name=1, anchor=center, inner sep=0}, "{g'}", from=1-2, to=2-2]
\arrow["{(f f', p, g g')}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["\cart"{description}, draw=none, from=0, to=1]
\end{tikzcd}
\]

It is clear that this choice of restrictions is strictly functorial, being inherited from composition of tight-cells in $\X$.

One might expect the definition of a strict  to include also a condition on the loose-identities,  that composition of loose-cells is strictly unital. However, as the following lemma shows, we may always choose composites in a given  in such a way to make this hold.

Every opcartesian 2-cell,

\[\begin{tikzcd}
{A_0} & \cdots & {A_n} \\
{A_0} && {A_n}
\arrow["{q_m}"', "\shortmid"{marking}, from=1-3, to=1-2]
\arrow["{q_1}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["q", "\shortmid"{marking}, from=2-3, to=2-1]
\arrow[""{name=0, anchor=center, inner sep=0}, Rightarrow, no head, from=1-3, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, Rightarrow, no head, from=1-1, to=2-1]
\arrow["\opcart"{description}, draw=none, from=1, to=0]
\end{tikzcd}\]

for which the object $A_i$ admits a loose-identity (for some fixed $0 \leq i \leq n$), factors through an opcartesian 2-cell as follows.

\[\begin{tikzcd}[column sep=large]
{A_0} & \cdots & {A_i} & {A_i} & \cdots & {A_n} \\
{A_0} & \cdots & {A_i} & {A_i} & \cdots & {A_n} \\
{A_0} &&&&& {A_n}
\arrow["{q_m}"{description}, from=2-6, to=2-5]
\arrow["q", "\shortmid"{marking}, from=3-6, to=3-1]
\arrow[""{name=0, anchor=center, inner sep=0}, Rightarrow, no head, from=2-6, to=3-6]
\arrow[""{name=1, anchor=center, inner sep=0}, Rightarrow, no head, from=2-1, to=3-1]
\arrow["{q_1}"{description}, from=2-2, to=2-1]
\arrow["{A(1, 1)}"{description}, from=2-4, to=2-3]
\arrow["{q_i}"{description}, from=2-3, to=2-2]
\arrow["{q_{i + 1}}"{description}, from=2-5, to=2-4]
\arrow[Rightarrow, no head, from=1-4, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, Rightarrow, no head, from=1-3, to=2-3]
\arrow[""{name=3, anchor=center, inner sep=0}, Rightarrow, no head, from=1-4, to=2-4]
\arrow[""{name=4, anchor=center, inner sep=0}, Rightarrow, no head, from=1-1, to=2-1]
\arrow[""{name=5, anchor=center, inner sep=0}, Rightarrow, no head, from=1-6, to=2-6]
\arrow["{q_m}"', "\shortmid"{marking}, from=1-6, to=1-5]
\arrow["{q_{i + 1}}"', "\shortmid"{marking}, from=1-5, to=1-4]
\arrow["{q_i}"', "\shortmid"{marking}, from=1-3, to=1-2]
\arrow["{q_1}"', "\shortmid"{marking}, from=1-2, to=1-1]
\arrow["\opcart"{description}, draw=none, from=1, to=0]
\arrow["{=}"{description}, draw=none, from=4, to=2]
\arrow["{=}"{description}, draw=none, from=3, to=5]
\arrow["\opcart"{description}, draw=none, from=2, to=3]
\end{tikzcd}\]

That the given opcartesian 2-cell factors through some 2-cell follows immediately from the universal property of the opcartesian 2-cell defining $A(1, 1)$. That the 2-cell is opcartesian follows from opcartesianness of the given 2-cell.

We conclude by observing that <ref> gives a full strictification result for representable s.

For every  admitting companions and conjoints, there is an equivalent strict double category admitting a strictly functorial choice of companions and conjoints.

To every  $\X$, there is an equivalent strict double category $\X'$ <cit.>. Applying <ref> then produces an equivalent  $\X''$ admitting a strictly functorial choice of restrictions. By <cit.>, a (strictly functorial) choice of restrictions in a  is equivalent to a (strictly functorial) choice of companions and conjoints. It remains to show that $\X''$ is actually a strict double category. The loose-composite of $(f, p, g)$ and $(f', p', g')$ in $\X''$ is represented by the loose-composite of $p(f, g)$ and $p'(f', g')$ in $\X'$, and so if loose-composition is strict in $\X'$, it is also strict in $\X''$.

Michael Shulman has suggested an alternative approach to the strictification of pseudo double categories with companions and conjoints <cit.>, in which one modifies the tight category – equipping each tight-cell with a specified companion and conjoint – rather than modifying the loose bicategory. However, such a strictification $\X'$ is equivalent to the original pseudo double category $\X$ only in a weak sense, in which the tight 2-categories $\tX$ and $\u{\X'}$ are merely biequivalent rather than being 2-equivalent, so that $\X$ and $\X'$ are not equivalent as objects of $\VDbl$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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