$\mathbf{n}=(n_{1},n_{2},n_{3},n_{4},n_{5})=\text{Tr}X^{\dagger}(-\sigma_{2}\tau_{2},-\sigma_{2}\tau_{1},\sigma_{1},\sigma_{2}\tau_{3},\sigma_{3})X$.
More specifically, one can label the generator of $SO(5)$ as $L_{\alpha\beta}$
with $\alpha<\beta$ and $\alpha,\beta=1,2,3,4,5$. One can check that
$\Gamma=\sigma_{0}\tau_{3},\sigma_{3}\tau_{2},\sigma_{0}\tau_{1},\sigma_{1}\tau_{2},\sigma_{3}\tau_{1},\sigma_{0}\tau_{2},\sigma_{1}\tau_{1},\sigma_{3}\tau_{3},\sigma_{2},\sigma_{1}\tau_{3}$
corresponds to
$L_{12},L_{13},L_{14},L_{15},L_{23},L_{24},L_{25},L_{34},L_{35},L_{45}$ up to
a sign convention. Here $L_{\alpha\beta}$ generates an $SO(2)$ rotation in the
$(n_{\alpha},n_{\beta})$ subspace. $\mathbf{n}\cdot\mathbf{n}$ is invariant
under the $SO(5)$ rotation and $\mathcal{L}$ has the $SO(5)$ symmetry with a
finite $\lambda_{0}$ as long as $\lambda=\lambda^{\prime}=0$. There is also an
anti-unitary symmetry $R\mathcal{T}$, so the final symmetry is $SO(5)\rtimes
Z_{2}^{T}$444$RT$ and $U(1)_{c}$ do not commute..

####  IX.3.2 $O(4)\rtimes Z_{2}^{T}$ symmetry at $\lambda=\lambda^{\prime}$

Consider the high symmetry line with $\lambda=\lambda^{\prime}\neq 0$ and
$g\neq 0,\lambda_{0}\neq 0$. The interaction term can be rewritten as:

$\mathcal{L}_{int}=g\text{Tr}X^{\dagger}X+(\lambda+\lambda_{0})(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2})+\lambda_{0}n_{5}^{2}.$
(71)

Then one still has an $SO(4)$ symmetry which rotates the vector
$(n_{1},n_{2},n_{3},n_{4})$. $SO(4)$ has 4 generators, corresponding to
$X\rightarrow e^{i\Gamma\theta}X$ with
$\Gamma=\sigma_{0}\tau_{3},\sigma_{3}\tau_{2},\sigma_{0}\tau_{1},\sigma_{3}\tau_{1},\sigma_{0}\tau_{2},\sigma_{3}\tau_{3}$.
Meanwhile, there is a $Z_{2}$ action from $X\rightarrow
e^{i\sigma_{1}\tau_{3}\frac{\pi}{2}}X$, which acts as
$(n_{1},n_{2},n_{3},n_{4},n_{5})\rightarrow(n_{1},n_{2},n_{3},-n_{4},-n_{5})$,
an improper rotation in the $(n_{1},n_{2},n_{3},n_{4})$ space. Together we
have a $O(4)$ symmetry. Including the anti-unitary symmetry $R\mathcal{T}$,
together the symmetry becomes $O(4)\rtimes Z_{2}^{T}$.

####  IX.3.3 Duality under $\lambda\leftrightarrow\lambda^{\prime}$

A special operation: $X\rightarrow
e^{i\tau_{2}\frac{\pi}{4}}e^{i\sigma_{3}\tau_{2}\frac{\pi}{4}}X$ corresponds
to $\Phi_{1}\rightarrow-i\tau_{2}\Phi_{1}^{*},\Phi_{2}\rightarrow\Phi_{2}$.
This belongs to a special element in the $SO(4)$ group discussed in the
previous subsection for $\lambda=\lambda^{\prime}$. This operation maps
$(n_{1},n_{2},n_{3},n_{4},n_{5})\rightarrow(n_{3},n_{4},-n_{1},-n_{2},n_{5})$.
It flips $A_{c;\mu}\leftrightarrow A_{r;\mu}$, but leaves the action invariant
at the special line $\lambda=\lambda^{\prime}$. When
$\lambda\neq\lambda^{\prime}$, it is no longer a symmetry. Instead, it induces
a duality which maps one critical theory with $(\lambda,\lambda^{\prime})$ to
a different critical theory with $(\lambda^{\prime},\lambda)$. As shown below,
the CSL-SC fixed point and the CSL-CDWxy fixed point on square lattice are
related by this duality. This duality constrains the renormalization group
(RG) flow in the $(\lambda,\lambda^{\prime})$ space to be symmetric under the
reflection $\lambda\leftrightarrow\lambda^{\prime}$. This duality precisely
corresponds to the duality between Eq. 44 and Eq. 60 with $A_{c}$ and $A_{r}$
exchanged.

###  IX.4 Various fixed points in the $SU(2)_{-1}$ $2\Phi$ theory

We now discuss possible fixed points in the $SU(2)_{-1}$ $2\Phi$ theory. There
is one obvious relevant direction tuned by $r$. When $r>0$, the CSL phase with
$\Phi$ gapped occurs. When $r<0$, $\Phi$ condenses and higgses the $SU(2)$
gauge field, leading to a symmetry breaking phase. The exact symmetry breaking
pattern depends on the quartic term and there are various different fixed
points in the $g,\lambda_{0},\lambda,\lambda^{\prime}$ space. $g,\lambda_{0}$
terms are $SO(5)$ invariant and presumably flow to a fixed point value. They
do not select the symmetry breaking pattern after $r<0$. Therefore one focuses
on the $(\lambda,\lambda^{\prime})$ space.

A schematic phase diagram and fixed points are shown in Fig. 6. First consider
the triangular or kagome lattice. Then $C_{6}$ rotation guarantees
$\lambda^{\prime}=0$. In this case, CSL-SC QCP must be the fixed point at
$(\lambda,\lambda^{\prime})=(\lambda^{*}_{+},0)$ with $\lambda^{*}_{+}>0$,
shown as the blue point in Fig. 6(a). Similarly there is a fixed point at
$\lambda<0$ for CSL-CDW transition, shown as light blue point in Fig. 6(a).
The red point at $(\lambda,\lambda^{\prime})=(0,0)$ is then a tri-critical
point on triangular/kagome lattice.

Next square lattice. We argue that the CSL-SC transition still corresponds to
the same fixed point as on triangular lattice for the following two reasons:
(I) The CSL phase and SC phase on square and triangular lattice are the same.
So it is natural to expect that the QCP on these two lattices are also the
same. (II) As discussed in the $U(1)_{1}$ $2\psi$ theory for the CSL-SC
transition, there is an $SO(3)\times O(2)$ symmetry, which is possible only
when $\lambda^{\prime}=0$. Now that the blue point is identified as the CSL-SC
transition, we use the duality map $\lambda\leftrightarrow\lambda^{\prime}$ to
obtain another fixed point at $(\lambda,\lambda^{\prime})=(0,\lambda^{*}_{+})$
(the orange point in Fig. 6(a)), which corresponds to the CSL-CDWxy transition
on square lattice. There should be a tri-critical point between CSL, SC and
CDWxy, though not necessarily at the $(\lambda,\lambda^{\prime})=(0,0)$ point.
The tri-critical point, if exists, most naturally occurs somewhere along the
$\lambda=\lambda^{\prime}$ line marked as the pink point. Note that the RG
flow constrained by the $\lambda\leftrightarrow\lambda^{\prime}$ duality fixes
this fixed point to be along the $\lambda=\lambda^{\prime}$ line, so it has an
$O(4)$ symmetry rotating $(n_{1},n_{2},n_{3},n_{4})$.

The light blue point is believed to describe the CSL-CDW transition on
triangular/kagome lattice. According to the analysis in Sec. VIII.3, it should
also be the tri-critical point between CSL, CDWz, CDWxy on square lattice.
Using the $\lambda\leftrightarrow\lambda^{\prime}$ duality, we know there is
another fixed point labeled as the grey point. This should be the tri-critical
point between CSL, SC, CDWx (see Sec. VIII.4). The RG flow suggests a fixed
point along the line $\lambda=\lambda^{\prime}<0$. This yellow point is
exactly the CSL-CDWz critical point discussed in Sec. VIII.2. This QCP has an
additional symmetry relating $(n_{1},n_{2})$ to $(n_{3},n_{4})$. The $SU(2)$
theory enables one to make a stronger statement that there is an $O(4)$
symmetry rotating $(n_{1},n_{2},n_{3},n_{4})$ and a self-duality symmetry from
$\lambda\leftrightarrow\lambda^{\prime}$ for the CSL-CDWz QCP.

The $SU(2)$ theory offers a unified framework for all of the critical points
and tri-critical points discussed in previous sections using bosonic holons or
Dirac fermions in $U(1)$ theories. The blue and orange points are known to be
dual to the $U(1)_{1}$ 2$\psi$ theory. The blue, light blue, orange, yellow
and grey points can all be described by $U(1)_{-2}$ 2$\varphi$ theory in Eq.
44 and Eq. 60 by tuning the easy-plane anisotropy term $\lambda$. The red and
pink point, however, are not captured by the simple $U(1)_{-2}$ 2$\varphi$
theory. $SU(2)$ theory proves to be the most convenient way to describe these
two tri-critical points.

Although not shown in Fig. 6(a), the SC-CDWxy transition on square lattice and
SC-CDW transition on triangular/kagome lattice are in the same universality
class as the famous DQCP between Neel order and VBS order with or without
easy-plane anisotropy respectively. The DQCP can be viewed as descendant from
the tri-critical points (the red and pink fixed point). Let us start from the
red fixed point at $\lambda=\lambda^{\prime}=0$ with a $SO(5)$ symmetry. After
$r<0$, the CSL phase is higgsed and the theory should be reduced to a non-
linear sigma model in terms of $\vec{n}=(n_{1},n_{2},n_{3},n_{4},n_{5})$. It
can be derived that the $SU(2)_{-1}$ Chern-Simons term leads to a Wess-Zumino-
Witten (WZW) term with level $k=1$ for the non-linear sigma model in terms of
$\vec{n}$Benini  _et al._ (2017), which exactly corresponds to the isotropic
DQCPWang  _et al._ (2017). With an easy-plane anisotropy, the $SO(5)$ non-
linear sigma model with WZW term reduces to an $O(4)$ non-linear sigma model
with $\theta=\pi$, corresponding to the easy-plane DQCP between CDWxy and SC.
This will be discussed in more details in Sec. XI.

The CDWxy-SC and CDWxy-CDWz transitions should be first order because there
are only three independent order parameters and there is no non-trivial Wess-
Zumino-Witten term or $\theta$ term for order parameters living on $S^{2}$
manifold in $2+1$ dimension.

Figure 6: (a)Fixed points of the $SU(2)$ theory with $N_{b}=2$ bosons. On
triangular and kagome lattice, $\lambda^{\prime}=0$ is enforced by lattice
symmetry. There is a duality transformation
$\lambda\leftrightarrow\lambda^{\prime}$. The red point has $SO(5)$ symmetry.
Pink and bright yellow points have $O(4)$ symmetry.(b,c) plot phase diagram on
square and triangular lattice, respectively. The critical line or point has
the same color as the fixed points in renormalization group(RG) flow of (a).
Note that fixed points in the RG flow diagram may correspond to phase
boundaries (line) or the intersection points of $3$ phases in (b,c) depending
on the lattice and symmetries. The green line in (b,c) separating
superconducting and CDW phases are described by isotropic or easy-plane DQCP
not displayed in the flow diagram (a). The dashed line represents first-order
transitions.

##  X CSL-SC and CSL-CDWxy transition: duality between $SU(2)_{-1}$ $2\Phi$
and $U(1)_{-2}$ $2\varphi$ theory

We have shown that the blue and the orange fixed point in Fig. 6 describe the
CSL-SC and CSL-CDWxy transition. Sec. VI and Sec. VII also discussed $U(1)$
theories with $2\varphi$ or $2\psi$ for the same CSL-SC transition. The same
is true for the CSL-CDWxy transition discussed in Sec. VIII. It is natural to
expect these three theories are dual to each other at the blue and orange
fixed points in Fig. 6(a). The boson-fermion duality between the two $U(1)$
theories have already been demonstrated. Here we discuss the duality between
the $SU(2)$ $2\Phi$ and $U(1)$ $2\varphi$ theories. The duality between the
$SU(2)$ $2\Phi$ and $U(1)$ $2\psi$ was already proposed previouslyAharony
(2016); Hsin and Seiberg (2016); Benini  _et al._ (2017). One interesting
property about the CSL-SC and CSL-CDWxy transition is the enlarged $SO(3)$
symmetry among $(\varphi^{\dagger}\sigma_{3}\varphi,\mathcal{M}_{a(b)})$,
which is not obvious in the $U(1)$ $2\varphi$ theory. This section provides an
understanding of this $SO(3)$ symmetry in the $U(1)$ $2\varphi$ theory. The
CSL-CDWxy-CDWz tri-critical point also has $SO(3)$ symmetry, simply from fine
tuning to $\lambda=0$. An enlarged $SO(3)$ theory in the $U(1)$ $2\varphi$
theory with easy-plane anisotropy $\lambda>0$ in Eq. 44 or Eq. 60 is more
nontrivial.

Starting from the $SU(2)_{-1}$ $2\Phi$ theory defined in Eq. 63, one obtains a
$U(1)_{-2}$ $2\varphi$ theory by higgsing the $SU(2)$ gauge field down to
$U(1)$ generated by $\tau_{3}$. There are two different types of the Higgs
term:

(I) $-\vec{m}\cdot\Phi^{\dagger}\vec{\sigma}\tau_{3}\Phi$,

(II)
$\vec{m}\cdot(\text{Re}\Phi^{T}\sigma_{1}\tau_{1}\Phi,\text{Im}\Phi^{T}\sigma_{1}\tau_{1}\Phi,\Phi^{\dagger}\tau_{3}\Phi)$,

where $\vec{m}$ is a three dimensional unit vector. These two groups are
related by the duality transformation $X\rightarrow
e^{i\tau_{2}\frac{\pi}{4}}e^{i\sigma_{3}\tau_{2}\frac{\pi}{4}}X$ or
equivalently
$\Phi_{1}\rightarrow-i\tau_{2}\Phi_{1}^{*},\Phi_{2}\rightarrow\Phi_{2}$ (see
Appendix. B). The type I $U(1)$ and type II $U(1)$ ansatz in Sec. IV are
obtained with $\vec{m}=(0,0,1)$ in the two groups respectively. In the first
group, other values of $\vec{m}$ can be generated from $\vec{m}=(0,0,1)$ with
an $SO(3)$ rotation generated by $L_{34},L_{35},L_{45}$, which rotate in the
subspace of $(n_{3},n_{4},n_{5})$. On the other hand, $\vec{m}$ in the second
group is generated from $\vec{m}=(0,0,1)$ with a $SO(3)$ rotation in the
subspace of $(n_{1},n_{2},n_{5})$.

The $SU(2)_{-1}$ theory will flow to the $U(1)$ $2\varphi$ theory in Eq. 44
and Eq. 60 by adding the Higgs term in the two groups respectively,
corresponding to the CSL-SC and CSL-CDWxy transition. The $SU(2)$ theories at
the fixed point $(\lambda,\lambda^{\prime})=(\lambda^{*}_{+},0)$ and
$(\lambda,\lambda^{\prime})=(0,\lambda^{*}_{+})$ are dual to Eq. 44 and Eq. 60
respectively. There is a manifold of $U(1)$ theories specified by the vector
$\vec{m}$ in the Higgs term rotated by $SO(3)$ symmetry.

We discuss CSL-SC transition as an example. CSL-CDWxy is in parallel as
related by the $\lambda\leftrightarrow\lambda^{\prime}$ duality. For the CSL-
SC transition, we propose the following duality:

$\displaystyle\mathcal{L}_{SU(2)}=|(\partial_{\mu}-ia^{s}_{\mu}\tau^{s}-i\frac{1}{2}A_{c;\mu}\tau_{0}\sigma_{0})\Phi_{a}|^{2}-r|\Phi|^{2}+\frac{1}{4\pi}Tr[a\wedge
da+\frac{2}{3}ia\wedge a\wedge a]-\frac{1}{8\pi}A_{c}dA_{c}$
$\displaystyle~{}~{}~{}-g|\Phi^{\dagger}\Phi|^{2}-\lambda_{0}\mathbf{n}\cdot\mathbf{n}+\lambda(n_{1}^{2}+n_{2}^{2})$
$\displaystyle\leftrightarrow\mathcal{L}_{SU(2)}-h\Phi^{\dagger}\vec{m}\cdot\vec{\sigma}\tau_{3}\Phi$
$\displaystyle\leftrightarrow\mathcal{L}_{U(1),\vec{m}}=|(\partial_{\mu}-ia_{\mu}-i\frac{1}{2}A_{c;\mu}\sigma_{3})\varphi|^{2}-r|\varphi|^{2}+\frac{2}{4\pi}ada-\frac{1}{8\pi}A_{c}dA_{c}-\tilde{g}(|\varphi|^{2})^{2}+\tilde{\lambda}|\varphi_{1}|^{2}|\varphi_{2}|^{2}.$

$\varphi=(\varphi_{1},\varphi_{2})^{T}$ descends from
$\Phi=(\Phi_{1},\Phi_{2})^{T}$ after adding the Higgs term
$-h\Phi^{\dagger}\vec{m}\cdot\vec{\sigma}\tau_{3}\Phi$ on energetic grounds.
For example, if $\vec{m}=(0,0,1)$, we have $\varphi_{1}=\Phi_{1;1}$ and
$\varphi_{2}=\Phi_{2;2}^{*}$. The probing field $A_{r}$ is omitted because the
$U(1)_{r}$ is not explicit in the $U(1)$ theory for a generic $\vec{m}$,
unless $\vec{m}=(0,0,\pm 1)$.

Note that $L_{U(1),\vec{m}}$ is gauge equivalent to $L_{U(1),-\vec{m}}$ after
a gauge transformation $\Phi\rightarrow i\tau_{1}\Phi$. Therefore the real
manifold of $U(1)_{-2}$ $2\varphi$ theories specified by a unit vector
$\vec{m}$, is RP${}^{2}=S^{2}/Z_{2}$ after one mods out the equivalence
between $\vec{m}$ and $-\vec{m}$. For any $\vec{m}$, $L_{U(1),\vec{m}}$
reduces to the $U(1)_{-2}$ $2\varphi$ theory, albeit the symmetry actions are
different as discussed in the following. The implication of the duality is
that all of these seemingly gauge nonequivalent theories flow to the same IR
fixed point, which is also the
$(\lambda,\lambda^{\prime})=(\lambda^{*}_{+},0)$ fixed point of the $SU(2)$
theory without Higgs term.

The $SU(2)$ $2\Phi$ theory at the $(\lambda,\lambda^{\prime})=(\lambda^{*},0)$
fixed point has a $SO(3)\times O(2)$ symmetry. The $SO(3)$ symmetry is also
transparent in the $U(1)_{1}$ $2\psi$ theory. Lattice symmetry like
$T_{1},T_{2},C_{4}$ (or $C_{6}$ on triangular/kagome lattice) are special
elements in the $SO(3)$. In contrast, the $L_{U(1),\vec{m}}$ theory above does
not have the $SO(3)$ symmetry and the lattice symmetry explicitly for a
generic $\vec{m}$. So how does one understand the symmetry in the $U(1)$
theory? The answer is that the $SO(3)$ symmetry needs to act non-locally in
the sense that it leaves the partition function invariant but not the action.
It needs to transform the field $\Phi_{\vec{m}}$ in $L_{U(1),\vec{m}}$ to
another field $\Phi_{\mathbf{R}\cdot\vec{m}}=U\Phi_{\vec{m}}$ in a different
theory $L_{U(1),\mathbf{R}\cdot\vec{m}}$, where $\mathbf{R}\in SO(3)$ is
generated accordingly from $U\in SU(2)$. 555Note that $\vec{m}$ is defined in
the northern hemisphere of $S^{2}$. If $\mathbf{R}\cdot\vec{m}$ is in the
southern hemisphere, we can map it back to $L_{U(1),-\mathbf{R}\cdot\vec{m}}$
by simply an additional transformation
$\Phi_{-\mathbf{R}\cdot\vec{m}}=i\tau_{1}\Phi_{\mathbf{R}\cdot\vec{m}}$. In
the low energy regime of the $U(1)$ theory, one uses the CP1 field variable
$(\varphi_{1;\vec{m}},\varphi^{*}_{2;\vec{m}})$ for each $\vec{m}$, which are
obtained by projecting to the two components of lowest energy from the term
$-\Phi^{\dagger}\vec{m}\cdot\vec{\sigma}\Phi$. After an $SO(3)$ rotation,
$(\varphi_{1;\vec{m}},\varphi^{*}_{2;\vec{m}})^{T}$ maps to
$(\varphi_{1;\pm\mathbf{R}\cdot\vec{m}},\varphi^{*}_{2;\pm\mathbf{R}\cdot\vec{m}})^{T}$.

Let us illustrate this procedure using one example with $\vec{m}=(0,0,1)$ and
consider the translation $T_{1}:-i\sigma_{1}$ in the original $SU(2)$ theory.
Now, it will first transform $\vec{m}$ to $-\vec{m}$ with
$\Phi_{-\vec{m}}=-i\sigma_{1}\Phi_{\vec{m}}$. Then we map it back to $\vec{m}$
using
$\Phi_{\vec{m}}=i\tau_{1}\Phi_{-\vec{m}}=\sigma_{1}\tau_{1}\Phi_{\vec{m}}$.
For this particular example, the translation symmetry $T_{1}$ acts locally in
the sense that it maps $\Phi$ to the same $U(1)$ theory specified by
$\vec{m}=(0,0,1)$. Focusing on the low energy degree
$(\varphi_{1},\varphi_{2}^{*})=(\Phi_{1;1},\Phi_{2,2})$, $T_{1}$ acts simply
as $\sigma_{1}$. In the similar way one can see that $T_{2}$ acts as
$-i\sigma_{2},\vec{m}=(0,0,1)\rightarrow\vec{m}=(0,0,1)$. However, consider a
different $U(1)$ theory labeled by $\vec{m}=(1,1,0)$, then $T_{1}$ needs to
act non locally and it maps $(\varphi_{1},\varphi_{2}^{*})$ to the field of a
different $U(1)$ theory labeled by $\vec{m}^{\prime}=(1,-1,0)$.

On square lattice, all of the lattice symmetries map $\vec{m}=(0,0,1)$ to
$\vec{m}=(0,0,1)$. Therefore for the $U(1)$ theory with $\vec{m}=(0,0,1)$,
lattice symmetries act locally. Thus the $U(1)$ theory can be regularized by a
lattice model, as derived from the parton mean field theory in Sec. VI. A
generic $SO(3)$ rotation still acts non-locally, but it does not correspond to
a microscopic lattice symmetry. In contrast, on triangular lattice, the
$C_{6}$ symmetry transforms $\vec{m}=(0,0,1)$ to $\vec{m}=(1,0,0)$ and then to
$\vec{m}=(0,1,0)$, following the rule in Table. 2, since $C_{6}$ relates
$(n_{3},n_{4},n_{5})$ to each other. The $\varphi^{\dagger}\sigma_{z}\varphi$
in the $U(1)$ theory with these three different $\vec{m}$ correspond to these
three CDW orders. As a result, $C_{6}$ needs to act non-locally in the $U(1)$
theory with a generic $\vec{m}$, unless $\vec{m}=\frac{1}{\sqrt{3}}(1,1,1)$.
But $T_{1}$ needs to act non-locally for $\vec{m}=\frac{1}{\sqrt{3}}(1,1,1)$.
Therefore, on triangular/kagome lattice, one can not find any $\vec{m}$ such
that the microscopic lattice symmetries can act locally. This is consistent
with the observation that there is no type I $U(1)$ ansatz on
triangular/kagome lattice discussed in Sec. IV. It is impossible to derive the
$U(1)$ $2\varphi$ theory for the CSL-SC transition starting from a parton mean
field construction on triangular/kagome lattice, because otherwise the lattice
symmetry should act locally in the resulting theory.

A similar discussion can be made for the CSL-CDWxy transition following the
$\lambda\leftrightarrow\lambda^{\prime}$ duality. At the fixed point
$(\lambda,\lambda^{\prime})=(0,\lambda^{*}_{+})$, we expect the duality
$\mathcal{L}_{SU(2)}\leftrightarrow\mathcal{L}_{SU(2)}-h\vec{m}\cdot(\text{Re}\Phi^{T}\sigma_{1}\tau_{1}\Phi,\text{Im}\Phi^{T}\sigma_{1}\tau_{1}\Phi,\Phi^{\dagger}\tau_{3}\Phi)$,
which again leads to a manifold of $U(1)_{-2}$ $2\varphi$ theory specified by
$\vec{m}$. The symmetry action is different. For example, the
$\varphi^{\dagger}\sigma_{z}\varphi$ operator in the resulting $U(1)$ theory
correspond to the order parameter $(n_{1},n_{2},n_{5})$ now for
$\vec{m}=(1,0,0),(0,1,0),(0,0,1)$ respectively. The $SO(3)$ symmetry still
needs to act non-locally, mapping one $\vec{m}$ to a different $\vec{m}$.
However, in this case all lattice symmetries on square, triangular and kagome
lattice act locally for $\vec{m}=(0,0,1)$, since microscopically the SC order
$(n_{1},n_{2})$ are very different from CDWz order $n_{5}$ and no microscopic
symmetry can rotate $n_{5}$ to mix with $(n_{1},n_{2})$. Hence the $U(1)$
$2\varphi$ theory for the CSL-CDWxy transition can be derived from parton mean
field theory on square, triangular and kagome lattice.

##  XI DQCP between CDW and SC

We have discussed CSL-SC and CSL-CDW transitions on square and
triangular/kagome lattice. It is then interesting to ask whether there can be
a direct transition between SC and CDW phase. Both phases are symmetry
breaking phases without fractionalization, so a direct transition needs to be
beyond Landau framework although nearby phases are conventional phases. In
this section we show that the SC-CDWxy on square lattice and SC-CDW transition
on triangular lattice are in the same universality class as the DQCP of easy
plane or isotropic Neel to VBS transitionSenthil  _et al._ (2004). The key is
that the topological superconductor can be understood as from condensation of
skyrmion. The CDWxy or CDW order live on the manifold $S^{1}$ or $S^{2}$ and
the CDW phase is a Chern insulator with $C=2$. Similar to the quantum Hall
ferromagnetism, the skyrmion defect of the $S^{2}$ order carries physical
charge $2e$ and can be identified as a bosonic Cooper pair. Condensation of
these skyrmions will disorder the CDW order and lead to a superconductor at
the same time.

Skyrmion superconductor has been discussed from a quantum spin Hall insulator
(QSHI)Grover and Senthil (2008); Liu  _et al._ (2019). There the resulting
superconductor is topologically trivial. In contrast, the skyrmion
superconductor in our case still inherits the chiral central charge $c=2$ from
the $C=2$ Chern insulator and is thus topologically equivalent to a $d+id$
superconductor. Nevertheless the CDW-SC transition in the bulk is the same as
the QSHI-SC transition if we ignore the edge physics. Skyrmion superconductor
was also proposed from a spin polarized Chern insulator with $C=2$ in moiré
systemsZhang  _et al._ (2019). The physics is similar to our case and the
skyrmion superconductor there, if possible, should also be topological with
chiral central charge $c=2$.

Technically, the easiest way to describe this transition is to use CP1
representation of the $SO(3)$ CDW order parameter. To obtain the topological
superconductor,the CP1 boson needs to be in a bosonic integer quantum Hall
insulator phase, instead of a trivial insulator. We start from the type II
$U(1)$ ansatz and the low energy boson field is
$\varphi=(\varphi_{1},\varphi_{2})^{T}$ with
$\varphi^{\dagger}\vec{\sigma}\varphi$ represents the CDW order. Following the
discussion in Sec. VII, we let these bosons enter a bIQHE phase to provide a
term $-\frac{2}{4\pi}ada$ term, which cancels the Chern-Simons term from the
fermionic spinons. Finally there is no Chern-Simons term for $a_{\mu}$
anymore. Now one has gapped bosonic holon $\varphi$ excitations and the ground
state is a Superconductor whose order parameter is the monopole of the gauge
field $a$. Then consider a transition between the bIQHE phase and the
superfluid phase for the boson $\varphi$, which leads to the SC to CDW
transition for the physical system. The superfluid transition from bIQHE for
bosons is the same as that from a trivial insulator and is simply captured by
condensation of $\varphi$, so the SC-CDW transition is described by the
following critical theory:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-ia_{\mu}-i\frac{1}{2}\sigma_{3}A_{r;\mu})\varphi|^{2}-\frac{1}{2\pi}A_{c}da$
$\displaystyle~{}~{}~{}-r|\varphi|^{2}-g(|\varphi|^{2})^{2}+\lambda|\varphi_{1}|^{2}|\varphi_{2}|^{2}$
(73)

When $r>0$, $\varphi$ is gapped and one obtains the SC phase. When $r<0$ and
$\lambda>0$, $\varphi$ condenses with easy-plane anisotropy, describing the
Chern insulator phase with CDWxy order. This is exactly the same critical
theory for the easy plane DQCP between Neel and VBS order on square
latticeSenthil  _et al._ (2004). If $\lambda=0$, then this is the isotropic
DQCP. When $\lambda>0$, naively the above action can describe the transition
between SC and CDWz, but this transition should be first order. The above
theory also has a self duality symmetry, which will be discussed in the next
section.

##  XII Tri-critical point and self-duality

We have provided critical theories for CSL-SC, CSL-CDW and SC-CDW
transitions.This section discusses the tri-critical point at the intersection
of these three phases. For square lattice we only consider the CDWxy order.
For triangular/kagome lattice we consider the isotropic CDW order with $SO(3)$
symmetry. The tri-critical points on square and triangular lattice correspond
to the pink and red fixed points in Fig. 6 for the $SU(2)$ theory. There is a
self-duality symmetry from the $\lambda\leftrightarrow\lambda^{\prime}$
transformation in the $SU(2)$ theory. Here we provide alternative theories
with two $U(1)$ gauge fields coupled to both $2\varphi$ and $2\psi$.

We discuss the CSL-CDWxy-SC tri-critical point on square lattice first. We
start from the type II $U(1)$ ansatz for the CSL phase.As discussed
previously, when the holons go through a plateau transition into IQHE, the
system goes into an SC phase. While if the holons simply condense with
$\langle\varphi\rangle\propto(1,1)^{T}$, the condensation pattern breaks
lattice symmetry and results in a CDWxy state. We see there are $2$ different
tuning parameters controlling transition into either SC or CDW. Hence we
arrive at a tri-critical point among CSL,CDW and SC.To formulate the
tricritical theory, we use two Dirac fermions to describe the CSL-SC
transition as in Eq. 49. Meanwhile we also keep track the bosonic holons
$\varphi$ and consider the mass of $\varphi$ as another tuning parameter. 666
This is a bit unusual as the $2$ relevant tuning parameters are expressed in
different low energy degrees of freedom, namely masses of $\psi,\varphi$
respectively.  Putting them together, the following critical theory emerges:

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}(-i\gamma_{\mu}\partial_{\mu}-b_{\mu}\gamma_{\mu}+\frac{1}{2}A_{r;\mu}\sigma_{3}\gamma_{\mu})\psi+m\bar{\psi}_{i}\psi_{i}-\frac{1}{4\pi}(\frac{1}{2}A_{c}+a)d(\frac{1}{2}A_{c}+a)+\frac{1}{2\pi}bd(\frac{1}{2}A_{c}+a)+\frac{1}{16\pi}A_{r}dA_{r}$
$\displaystyle~{}~{}~{}+|(\partial_{\mu}-ia_{\mu}-i\frac{1}{2}\sigma_{0}A_{c;\mu}-i\frac{1}{2}\sigma_{3}A_{r;\mu})\varphi|^{2}-r|\varphi|^{2}-g(|\varphi|^{2})^{2}+\lambda|\varphi_{1}|^{2}|\varphi_{2}|^{2}$
$\displaystyle~{}~{}~{}+\frac{2}{4\pi}ada-\frac{1}{8\pi}A_{c}dA_{c}-\mathcal{L}_{int}$
(74)

where the first line describes the plateau transition of the bosonic holon and
the second line the condensation of the holon. The third line is from the
integration of the fermionic spinons and the stacking of the $\nu=-2$ IQHE
phase. Interactions between $\varphi$ and $\psi$ $\mathcal{L}_{int}$ will be
specified in the next equation.

With a simplification $a_{\mu}\rightarrow a_{\mu}-\frac{1}{2}A_{c;\mu}$, the
tri-critical point theory is cast into:

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}(-i\gamma_{\mu}\partial_{\mu}-b_{\mu}\gamma_{\mu}+\frac{1}{2}A_{r;\mu}\sigma_{3}\gamma_{\mu})\psi+m\bar{\psi}_{i}\psi_{i}+|(\partial_{\mu}-ia_{\mu}-i\frac{1}{2}\sigma_{3}A_{r;\mu})\varphi|^{2}-r|\varphi|^{2}-g(|\varphi|^{2})^{2}+\lambda|\varphi_{1}|^{2}|\varphi_{2}|^{2}$
$\displaystyle~{}~{}~{}+\frac{1}{4\pi}ada+\frac{1}{2\pi}(b-A_{c})da+\frac{1}{16\pi}A_{r}dA_{r}-g_{0}(\bar{\psi}\psi)(\varphi^{\dagger}\varphi)-\tilde{g}(\bar{\psi}\vec{\sigma}\psi)\cdot(\varphi^{\dagger}\vec{\sigma}\varphi)-\tilde{\lambda}(\bar{\psi}\sigma_{3}\psi)(\varphi^{\dagger}\sigma_{3}\varphi)$
(75)

where $\varphi=(\varphi_{1},\varphi_{2})^{T}$ and
$\psi=(\psi_{1},\psi_{2})^{T}$.

We have also included the interaction terms between $\psi$ and $\varphi$.
$\lambda$ and $\tilde{\lambda}$ are easy-plane anisotropy terms. When $m<0$,
one can integrate $\psi$ and then integrate $b_{\mu}$, after which we recover
Eq. 60 for the CSL-CDWxy transition. When $r>0$, $\varphi$ is gapped and one
can integrate $a_{\mu}$ and get the $U(1)_{1}$ $2\psi$ theory for the CSL-SC
transition in Eq. 56. In summary, $m<0,r>0$ is the CSL phase, $m>0,r>0$ is the
SC phase. $r<0$ is the CDW phase regardless of the sign of $m$. $r=m=0$
corresponds to the tri-critical point. These are summarized in Fig. 7(b). When
$\lambda=\tilde{\lambda}=0$, this is the CSL-SC-CDW tri-critical point on
triangular lattice, shown in Fig. 7(a). When $\lambda>0,\tilde{\lambda}>0$,
this is the CSL-SC-CDWxy tri-critical point. In this theory,
$\varphi^{\dagger}\vec{\sigma}\varphi\sim\bar{\psi}\vec{\sigma}\psi$
correspond to the CDW order $(n_{3},n_{4},n_{5})$.

In the $SU(2)$ theory, these two tri-critical points have a self-duality which
exchanges $A_{c}$ and $A_{r}$. Thus we believe the above tri-critical theory
is also self dual to itself except the exchange between $A_{c}$ and $A_{r}$.
One can derive this self-duality in the following way on square lattice. On
square lattice, starting from the type I $U(1)$ ansatz, then the CSL-SC
transition is described by the $U(1)_{-2}$ $2\varphi$ theory. Alternatively
one could let the holon $\varphi$ goes through a plateau transition and get a
CDWxy order. Putting them together, the following tri-critical theory reads:

$\displaystyle\mathcal{L}$
$\displaystyle=\bar{\psi}(-i\gamma_{\mu}\partial_{\mu}-b_{\mu}\gamma_{\mu}-\frac{1}{2}A_{c;\mu}\sigma_{3}\gamma_{\mu})\psi+m\bar{\psi}_{i}\psi_{i}+|(\partial_{\mu}-ia_{\mu}-i\frac{1}{2}\sigma_{3}A_{c;\mu})\varphi|^{2}-r|\varphi|^{2}-g(|\varphi|^{2})^{2}+\lambda|\varphi_{1}|^{2}|\varphi_{2}|^{2}$
$\displaystyle~{}~{}~{}+\frac{1}{4\pi}ada+\frac{1}{2\pi}(b-A_{r})da+\frac{1}{8\pi}A_{r}dA_{r}-\frac{1}{16\pi}A_{c}dA_{c}-g_{0}(\bar{\psi}\psi)(\varphi^{\dagger}\varphi)-\tilde{g}(\bar{\psi}\vec{\sigma}\psi)\cdot(\varphi^{\dagger}\vec{\sigma}\varphi)-\tilde{\lambda}(\bar{\psi}\sigma_{3}\psi)(\varphi^{\dagger}\sigma_{3}\varphi)$
(76)

where $\varphi=(\varphi_{1},\varphi_{2})^{T}$ and
$\psi=(\psi_{1},\psi_{2})^{T}$.

When $r>0$ and $\varphi$ is gapped, we recover Eq. 62 after integrating $a$.
When $m<0$, we recover Eq. 44 after integrating $\psi$ and $b$. $m<0,r>0$ is
the CSL phase. $m>0,r>0$ is the CDWxy order. $r<0$ is the SC phase regardless
of the sign of $m$. $m=r=0$ is the tri-critical point. This time
$\varphi^{\dagger}\vec{\sigma}\varphi\sim\bar{\psi}\vec{\sigma}\psi$
correspond to $(n_{1},n_{2},n_{5})$. In the above we assume the easy plane
anisotropy terms $\lambda,\tilde{\lambda}$ so the tri-critical point is
between CSL-SC-CDWxy. We believe the $\lambda=\tilde{\lambda}=0$ corresponds
to the CSL-SC-CDW tri-critical point on triangular/kagome lattice, though now
$\varphi$ transforms non-locally under lattice symmetry. This means that both
tri-critical points on square and triangular/kagome lattice have the self-
duality at $r=m=0$ of Eq. 75 and Eq. 76.

Both theories only have explicit $U(1)_{c}\times U(1)_{r}\rtimes Z_{2}$
symmetry with $Z_{2}$ coming from either translation $T_{1}$ or $T_{2}$. In
Eq. 75, one can see the mixed anomaly that $dA_{c}=2\pi$ carries charge $1/2$
under $A_{r}$ because one needs to attach $\varphi$ to cancel the charge under
$a$. Similarly $dA_{r}=2\pi$ carries charge $1/2$ under $A_{c}$. This mixed
anomaly is crucial for the DQCP between SC and CDW and it already exits at the
tri-critical point. The self-duality relates order parameter $(n_{1},n_{2})$
to $(n_{3},n_{4})$ and suggests an enlarged $O(4)$ symmetry when there is
easy-plane anisotropy terms $\lambda,\lambda^{\prime}$. When
$\lambda=\lambda^{\prime}=0$, there is explicit SO(3)$\times$ O(2) symmetry in
either theory. In Eq. 75 the SO(3) rotates $(n_{3},n_{4},n_{5})$, while in Eq.
76 the SO(3) rotates $(n_{1},n_{2},n_{5})$. The self-duality then implies a
$SO(5)$ symmetry. Nevertheless, $O(4)$ and $SO(5)$ symmetries are not explicit
in the above form. To explicitly see the symmetry, we still need to use the
$SU(2)$ theory.

We plot phase diagrams for Eq. 75 and Eq. 76 in Fig. 7. The tri-critical
critical theories here are different from that of the $SU(2)$ theory in Sec.
IX. Let us take the square lattice as an example. In the $SU(2)$ theory,
bosonic holons are in either trivial insulator and superfluid phases. SC and
CDW correspond to different condensation patterns of the superfluid phase of
the bosonic holon $\Phi$. In contrast, here we start from the $U(1)$ ansatz of
the CSL. If we start from the type I $U(1)$ ansatz, then CSL, SC and CDWxy
correspond to trivial insulator, superfluid and bIQHE insulator of the bosonic
holons $\varphi$. If we start from type II $U(1)$ ansatz, CSL, CDWxy and SC
correspond to trivial insulator, superfluid and bIQHE insulator of bosonic
holon $\varphi$. One can clearly see the duality between type I and type II
$U(1)$ with exchange of SC and CDWxy. For triangular and kagome lattice, we
only have type II ansatz and can only derive Eq. 75 from parton construction.
But we believe it has a dual theory as Eq. 76, just the lattice symmetry needs
to act non-locally as discussed in Sec. X.

Figure 7: The phase diagram summarizing the transitions between CSL,d+id SC
and CDW Chern insulator on triangular (a) and square lattices (b,c). (b,c) are
described by Eq (75) and Eq (76),respectively.

##  XIII Honeycomb lattice: $U(1)$ $1\varphi$ theory and absence of symmetry
breaking order

We have shown various critical theories and proximate phases nearby a chiral
spin liquid phase on square, triangular and kagome lattice. In contrast, on
honeycomb lattice, we do not expect a direct CSL-SC transition because there
are two electrons per unit cell in the CSL phase of a Mott insulator. On
honeycomb lattice, the natural mean field ansatz of the CSL phase is the
Haldane model with $T_{1}T_{2}=T_{2}T_{1}$. As a result there is only one
single bosonic holon mode $\varphi$, which leads to a critical theory:

$\mathcal{L}=|(\partial_{\mu}-ia_{\mu})\varphi|^{2}-r|\varphi|^{2}-g|\varphi|^{4}+\frac{2}{4\pi}ada-\frac{1}{2\pi}A_{c}da$
(77)

In this case the $r<0$ side is a symmetry Chern insulator with $C=2$, simply
described by the Haldane model. The critical theory is known to be dual to a
$U(1)$ theory with one Dirac fermion, an $SU(2)$ theory with one boson and an
$SU(2)$ theory with one Dirac fermionLee  _et al._ (2018); Samajdar  _et al._
(2019). There is also an emergent $SO(3)$ symmetry which rotates
$(\nabla\times\mathbf{a},\text{Re}\mathcal{(}M\varphi^{*}),\text{Im}(\mathcal{M}\varphi^{*}))$.
They represent the density fluctuation and the Cooper pair creation operators
which transform trivially under lattice symmetries. Gapless charge mode is
expected at the QCP. There is no other symmetry breaking order parameter
fluctuating at the QCP. One can clearly see that the difference of the square,
triangular and kagome lattice arises from the projective translation symmetry
of the semion, which guarantees the existence of two bosonic modes in the
$U(1)$ $2\varphi$ theory.

##  XIV Experimental signatures

We discuss the possible experimental realization and signatures of the CSL-SC
transition. Recently CSL was observed numerically in triangular lattice
Hubbard model in the intermediate regime of U/tSzasz  _et al._ (2020). The
next phase close to the CSL may be a superconductorArovas  _et al._ (2021) and
a CSL-SC transition as described by our paper can be naturally realized by
reducing $U/t$. Bandwidth tuned metal insulator transition has recently been
observed in moiré superlattice based on transition metal dichalcogenide(TMD)Li
_et al._ (2021). Although a CSL phase was not reported in the current
experiment, it may be found at lower temperature or a different parameter
regime. Therefore it may be interesting to search for CSL-SC transition in
moiré materials. Here we will provide experimental predictions in the critical
regime, focusing on the transport at the CSL-SC critical point. One can access
the conductivity tensor from the $U(1)$ $2\varphi$ theory, $U(1)$ $2\psi$
theory or $SU(2)$ $2\Phi$ theory, and we mainly use the two $U(1)$ theories.

First, let us start with the $U(1)$ $2\varphi$ theory in Eq. 44. We will
ignore $A_{r}$ as the transport measured in the experiments is associated only
with $A_{c}$. In the original action a term $-\frac{1}{8\pi}A_{c}dA_{c}$ was
added coming from the stacking of $\nu=-2$ IQHE phase, which will be ignored
because the $\nu=-2$ IQHE phase is stacked only for simplicity and does not
really exist. $\varphi_{1}$ couples to $a+\frac{1}{2}A$ and $\varphi_{2}$
couples to $a-\frac{1}{2}A$. Translation symmetry acts as
$T_{1}:\varphi_{1}\rightarrow\varphi_{2}^{*},\varphi_{2}\rightarrow\varphi_{1}^{*}$.
Integrating $\varphi_{1},\varphi_{2}$, we get:

$\displaystyle\mathcal{L}_{eff}$
$\displaystyle=\mathcal{L}_{a}+\frac{1}{2}\sum_{\omega,\mathbf{q}}\frac{1}{4}(A_{x}(\omega,\mathbf{q}),A_{y}(\omega,\mathbf{q}))(\Pi_{11}(\omega,\mathbf{q})+\Pi_{22}(\omega,\mathbf{q})-\Pi_{12}(\omega,\mathbf{q})-\Pi_{21}(\omega,\mathbf{q}))\begin{pmatrix}A_{x}(-\omega,-\mathbf{q})\\\
A_{y}(-\omega,-\mathbf{q})\end{pmatrix}$ (78)

where $\Pi_{11}$ and $\Pi_{22}$ are from the conductivity tensor of
$\varphi_{1}$ and $\varphi_{2}$. $\Pi_{12},\Pi_{21}$ encode the drag
conductivity between $\varphi_{1}$ and $\varphi_{2}$. Note that under $T_{1}$,
we have $\mathbf{A}\rightarrow\mathbf{A},\mathbf{a}\rightarrow-\mathbf{a}$, so
that there is no crossing term between $A$ and $a$. $\mathcal{L}_{a}$ includes
the terms for $a_{\mu}$, which is not interesting for physical transport.

We simply define
$\Pi(\omega,\mathbf{q})=\frac{1}{4}(\Pi_{11}(\omega,\mathbf{q})+\Pi_{22}(\omega,\mathbf{q})-\Pi_{12}(\omega,\mathbf{q})-\Pi_{21}(\omega,\mathbf{q}))$.
It encodes the conductivity tensor of the physical electron:

$\Pi(\omega,\mathbf{q})=\frac{e^{2}}{\hbar}\frac{1}{2\pi}(-i\omega)\begin{pmatrix}\sigma_{0}&\sigma_{xy}\\\
-\sigma_{xy}&\sigma_{0}\end{pmatrix}$ (79)

$\sigma_{0}$ and $\sigma_{xy}$ are universal numbers. $\sigma_{0}$ is the
universal conductivity usually present in 2+1 d CFT for bosons. There is no
symmetry to forbid the Hall conductivity $\sigma_{xy}$. However, note that
$\langle da\rangle=0$ on average and it does not couple to the physical gauge
field $A_{\mu}$ due to symmetry $T_{1}$. So we expect that $\sigma_{xy}$
should be very small even if it exist.

We can also derive the conductivity tensor from the $U(1)$ theory with two
Dirac fermions in Eq. 56. Again we ignore $A_{r}$ and add back a term
$\frac{1}{8\pi}A_{c}dA_{c}$. Integration of $\psi$ leads to

$\displaystyle\mathcal{L}_{eff}$
$\displaystyle=\frac{1}{2}\mathbf{b}^{T}(\omega,\mathbf{q})\frac{-i\omega}{2\pi}\big{(}\sigma_{\psi}I+(\sigma_{\psi}^{xy}-1)\epsilon\big{)}\mathbf{b}(-\omega,-\mathbf{q})-\frac{1}{4}\mathbf{A}^{T}(\omega,\mathbf{q})\frac{-i\omega}{2\pi}\epsilon\mathbf{A}(-\omega,-\mathbf{q})$
$\displaystyle~{}~{}~{}+\frac{1}{2}\mathbf{A}^{T}(\omega,\mathbf{q})\frac{-i\omega}{2\pi}\epsilon\mathbf{b}(-\omega,-\mathbf{q})+\frac{1}{2}\mathbf{b}^{T}(\omega,\mathbf{q})\frac{-i\omega}{2\pi}\epsilon\mathbf{A}(-\omega,-\mathbf{q})$
(80)

where $\mathbf{A}^{T}=(A_{x},A_{y})$ and $\mathbf{b}^{T}=(b_{x},b_{y})$. $I$
and $\epsilon$ are $2\times 2$ matrix. $I$ is the identity matrix.
$\epsilon=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$.

Integration of $b_{\mu}$ leads to

$\mathcal{L}_{eff}=\frac{1}{2}\sum_{\omega,\mathbf{q}}(A_{x}(\omega,\mathbf{q}),A_{y}(\omega,\mathbf{q}))\frac{-i\omega}{2\pi}\sigma(\omega,\mathbf{q})\begin{pmatrix}A_{x}(-\omega,-\mathbf{q})\\\
A_{y}(-\omega,-\mathbf{q})\end{pmatrix}$ (81)

with

$\displaystyle\sigma(\omega,\mathbf{q})$
$\displaystyle=\big{(}\sigma_{\psi}I+(\sigma_{\psi}^{xy}-1)\epsilon\big{)}^{-1}-\frac{1}{2}\epsilon$
$\displaystyle=\frac{\sigma_{\psi}}{\sigma_{\psi}^{2}+(1-\sigma^{xy}_{\psi})^{2}}I+\big{(}\frac{1-\sigma^{xy}_{\psi}}{\sigma_{\psi}^{2}+(1-\sigma^{xy}_{\psi})^{2}}-\frac{1}{2}\big{)}\epsilon$
(82)

Compared to Eq. 79, we have the following constraint for the two dual
theories, where similar results are obtained in ref Shyta  _et al._ (2022):

$\displaystyle\sigma_{0}$
$\displaystyle=\frac{\sigma_{\psi}}{\sigma_{\psi}^{2}+(1-\sigma^{xy}_{\psi})^{2}}$
$\displaystyle\sigma_{xy}$
$\displaystyle=\frac{1-\sigma^{xy}_{\psi}}{\sigma_{\psi}^{2}+(1-\sigma^{xy}_{\psi})^{2}}-\frac{1}{2}$
(83)

We have argued that $\sigma_{xy}$ should be very small from the $U(1)$
$2\varphi$ theory. This will impose non-trivial constraint for the theory with
Dirac fermion.

In addition to a universal conductivity, there also should be quasi long range
fluctuation of the CDW order at the CSL-SC critical point. The CDW fluctuation
can persist to the superconductor phase and the CDW can be stabilized in the
vortex core of the superconductor phase. This can be tested by X-ray
scattering and scanning tunneling microscope (STM) experiments.

We have restricted to the transition tuned by bandwidth with the density fixed
at integer filling. For the chemical potential tuned transition, we can still
use the $SU(2)$ theory in Sec. IX. But now there is a term
$i\Phi^{*}\partial_{t}\Phi$ and we have $z=2$. The fixed point should still be
decided by the quartic terms $(\lambda,\lambda^{\prime})$. On triangular
lattice, the CSL-SC transition is still fixed at the $\lambda^{\prime}=0$ axis
by the $C_{6}$ symmetry. If we assume that the chemical potential tuned CSL-SC
transition is the same on square and triangular lattice, then the transition
on square lattice is at the same fixed point and still has SO(3)$\times O(2)$
symmetry. The $\lambda\leftrightarrow\lambda^{\prime}$ duality is still there
and the structure of the fixed points should remain the same as the $z=1$
case. We should still have a dual $U(1)$ theory with $2\varphi$ by adding
Higgs terms to the $SU(2)$ theory, though now it is also $z=2$. There is no
obvious theory with Dirac fermion. We no longer expect a universal
conductivity, but the intertwinement of the SC and CDW order in the critical
regime should still exist.

##  XV Conclusion

In summary we present critical theories for transitions between chiral spin
liquid, topological superconductor and CDW Chern insulator. In the CSL to SC
transition, the are also CDW orders transforming under an emergent $SO(3)$
symmetry. Such an intertwinement of an additional symmetry breaking order is
guaranteed by the LSM theorem at odd electron filling per unit cell. We
present the critical theories in three forms: $U(1)$ theory with two bosons,
$U(1)$ theory with two Dirac fermions and $SU(2)$ theory with two bosons. Our
work demonstrates the duality between these three theories and possible
experimental realizations of these interesting CFTs. In the $SU(2)$ theory,
there are several fixed points decided by the quartic terms
$\lambda,\lambda^{\prime}$, corresponding to bi-critical and tri-critical
points with $SO(5),O(4)$ or $SO(3)\times O(2)$ global symmetry. There is also
a duality transformation $\lambda\leftrightarrow\lambda^{\prime}$ which
exchanges the easy-plane CDW order and the SC order. We offer a new
perspective to understand the $d+id$ superconductor as from skyrmion
condensation of the CDW order. The CDW-SC transitions are in the same
universality classes as the usual Neel to VBS DQCP, but now both of them are
proximate to a CSL phase. The CDW-SC DQCP theories, along with the enlarged
symmetry and self-duality, are simply descendants of unified tri-critical
theories. We also discuss possible experimental realizations and detection of
the CSL-SC transition.

##  XVI Acknowledgement

We thank Ashvin Vishwanath, Cenke Xu and Maissam Barkeshli for useful
discussions. X-Y Song thanks Chong Wang and Max Metlitski for helpful
discussions. YHZ is supported by a startup grant from Johns Hopkins
University. X-Y S is supported by the Gordon and Betty Moore Foundation EPiQS
Initiative through Grant No. GBMF8684 at the MIT.

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  * Note (5) Note that $\mathaccentV{vec}17Em$ is defined in the northern hemisphere of $S^{2}$. If $\mathbf{R}\cdot\mathaccentV{vec}17Em$ is in the southern hemisphere, we can map it back to $L_{U(1),-\mathbf{R}\cdot\mathaccentV{vec}17E{m}}$ by simply an additional transformation $\Phi_{-\mathbf{R}\cdot\mathaccentV{vec}17Em}=i\tau_{1}\Phi_{\mathbf{R}\cdot\mathaccentV{vec}17Em}$.
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##  Appendix A $SU(2),U(1)$ mean-field states for holons and symmetries

Here we list the solution for the $SU(2)$ mean-field ansatz for holons and the
projective symmetry group of the holons.

###  A.1 Square lattice

The unit cell for the mean-field for holons eq (III.1) is enlarged to contain
$2$ sites $A=(0,0),B=(1,0)$ connected by a horizontal bond. It translates into
$k$ space form in terms of $Z=(z_{A,1},z_{B,1})^{T}$:

$\displaystyle H_{k}=\left(\begin{array}[]{cc}\sin k_{2}&-\sin
k_{1}+i\eta(\cos(k_{1}+k_{2})+\cos(k_{2}-k_{1})\\\ -\sin
k_{1}-i\eta(\cos(k_{1}+k_{2})+\cos(k_{2}-k_{1}))&-\sin
k_{2}\end{array}\right).$ (86)

The Dirac nodes at $(0,0(\pi))$ are gapped out by a mass of order $2\eta$. For
$z_{2}$ it differs by a minus sign.

The holon dynamics at low-energy is dominated by the fluctuations around the
lowest energy states of $H_{k}$, located at $Q_{1(2)}=(\pi/2,\pm\pi/2)$.The
eigenvectors for $z_{1},z_{2}$ are

$\displaystyle
Q_{1}:(z_{1,A},z_{1,B})=(\cos\frac{\pi}{8},-\sin\frac{\pi}{8}),(z_{2,A},z_{2,B})=(\sin\frac{\pi}{8},\cos\frac{\pi}{8}),$
$\displaystyle
Q_{2}:(z_{1,A},z_{1,B})=(-\sin\frac{\pi}{8},\cos\frac{\pi}{8}),(z_{2,A},z_{2,B})=(\cos\frac{\pi}{8},\sin\frac{\pi}{8})$
(87)

at $Q_{1,2}$, respectively.

The symmetry actions on spinons and holons are projective, albeit the
composite action for electrons is a faithful representation. The actions on
spinons read:

$\displaystyle
T_{1,2}:(f_{r,\uparrow}^{\dagger},f_{r,\downarrow})^{T}\rightarrow
i\epsilon_{r}\tau^{2}(f_{r+{\widehat{r}}1(2),\uparrow}^{\dagger},f_{r+{\widehat{r}}1(2),\downarrow})^{T},$
$\displaystyle
C_{4}:(f_{r,\uparrow}^{\dagger},f_{r,\downarrow})^{T}\rightarrow
i\epsilon_{r}\tau^{2}(f_{C(r),\uparrow}^{\dagger},f_{C(r)),\downarrow})^{T},$
$\displaystyle
R_{1}\mathcal{T}:(f_{r,\uparrow}^{\dagger},f_{r,\downarrow})^{T}\rightarrow\mathcal{K}i\epsilon_{r}\tau^{2}(f_{R_{1}(r),\downarrow}^{\dagger},f_{R_{1}(r)),\uparrow})^{T},$
(88)

where $\mathcal{K}$ denoting anti-unitary operations and $\tau^{1,2,3}$
matrices are Pauli matrices acting on the spinor indices. The transformation
for holons follows from those of spinons by requiring the electron operators
transform faithfully under the symmetries, i.e. $\Psi_{r,f}\rightarrow
g_{R}(r)\Psi_{R(r),f},Z_{r,b}\rightarrow g_{R}(r)^{-1}Z_{R(r),b}$ for a
symmetry operation $R$ with a site-dependent gauge transform $g_{R}(r)$.

The lattice translation along $r_{1}$ direction $T_{1}$ gives a momentum boost
of $(0,\pi)$, i.e.

$\displaystyle T_{1}:z_{1(2),r}\rightarrow
i(-1)^{r_{2}}z_{1(2),r+{\widehat{r}}_{1}}$
$\displaystyle(\Phi_{1},\Phi_{2})^{T}\rightarrow-i\sigma_{1}(\Phi_{1},\Phi_{2})^{T}$
(89)

where we use $\sigma_{a}$ to label the Pauli matrix that rotates
$\Phi=(\Phi_{1},\Phi_{2})$. Translation along $r_{2}$ just sends
$z_{1(2),r}\rightarrow z_{1(2),r+{\widehat{r}}_{2}}$ and at low-energy:

$\displaystyle
T_{1}:(\Phi_{1},\Phi_{2})^{T}\rightarrow-i\sigma_{1}(\Phi_{1},\Phi_{2})^{T},$
$\displaystyle
T_{2}:(\Phi_{1},\Phi_{2})^{T}\rightarrow-i\sigma_{3}(\Phi_{1},\Phi_{2})^{T},$
(90)

For $C_{4}$ rotation around a site e.g. at $B$ sublattice with coordinate
$(1,0)$, one choice of transform that leave the ansatz invariant reads

$\displaystyle z_{1,r}\rightarrow
g(C4(r))z_{1,C4(r)},g(r)=\begin{cases}-(-1)^{r_{1}}&if\mod(r_{2},2)=0\\\
1&if\mod(r_{2},2)=1\end{cases}.$ (91)

The transform for $z_{2}^{*}$ differs by an overall minus sign to make the
rotation of $\Phi$ free of $SU(2)$ gauge transform.

It sends the low energy field as:

$\displaystyle
C_{4}:(\Phi_{1},\Phi_{2}))^{T}\rightarrow-\sigma_{3}e^{i\frac{\pi}{4}\sigma_{2}}(\Phi_{1},\Phi_{2}))^{T}.$
(92)

For $R_{1}\mathcal{T},R_{2}\mathcal{T}$, the holon fields are sent to
(identical for $z_{1},z_{2}$)

$\displaystyle
R_{1,2}\mathcal{T}:z_{r}\rightarrow(-1)^{r_{2,1}}\mathcal{K}z_{R_{1,2}T(r)},(\Phi_{1},\Phi_{2})\rightarrow-(\Phi_{1},\Phi_{2}).$
(93)

To simplify our notation, we make a redefinition
$(\Phi_{1},\Phi_{2})^{T}\rightarrow
e^{-i\sigma_{1}\pi/4}(\Phi_{1},\Phi_{2})^{T}$, so that the symmetry
transformation now is: $T_{1}:-i\sigma_{1}$, $T_{2}:i\sigma_{2}$,
$C_{4}:\sigma_{2}e^{i\frac{\pi}{4}\sigma_{3}},R_{1,2}\mathcal{T}:i\sigma_{1}$.

###  A.2 Triangular lattices

As a starting point, we consider the $U(1)$ CSL mean field ansatz. We define
the coordinate to be $\mathbf{r}=x\mathbf{a_{1}}+y\mathbf{a_{2}}$.
$\mathbf{a_{1}}$ is along x direction, $\mathbf{a_{2}}$ is along $120^{\circ}$
direction.

$\displaystyle H_{\Psi}$
$\displaystyle=t_{f}\Psi^{\dagger}(\mathbf{r}+{\widehat{r}_{1}})ie^{i(\frac{\pi}{2}+\theta)\tau_{3}}\Psi(\mathbf{r})+h.c.$
$\displaystyle+t_{f}\Psi^{\dagger}(\mathbf{r}+{\widehat{r}_{2}})(-1)^{r_{1}}ie^{i(\frac{\pi}{2}+\theta)\tau_{3}}\Psi(\mathbf{r})+h.c.$
$\displaystyle+t_{f}\Psi^{\dagger}(\mathbf{r}+{\widehat{r}_{1}}+{\widehat{r}_{2}})(-1)^{r}_{1}ie^{-i\theta\tau_{3}}\Psi(\mathbf{r})+h.c.$
(94)

We then do a gauge transformation $\Psi(\mathbf{r})\rightarrow
e^{i\frac{\pi}{2}(r_{1}+r_{2})\tau_{3}}\Psi(\mathbf{r})$, and get the mean-
field eq (III.2).

The mean-field composes of $2$-site unit cells denoted as sublattice $A,B$. In
the basis $(Z_{A}(\mathbf{k}),Z_{B}(\mathbf{k}))$, we get

$\displaystyle H_{Z}$
$\displaystyle=(Z_{A}^{\dagger}(\mathbf{k}),Z_{B}^{\dagger}(\mathbf{k}))h(\mathbf{k})\begin{pmatrix}Z_{A}(\mathbf{k})\\\
Z_{B}(\mathbf{k})\end{pmatrix}$ (95)

where

$\displaystyle h(\mathbf{k})$
$\displaystyle=\begin{pmatrix}2t_{b}\cos\theta\sin(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})&2t_{b}\cos\theta\sin
k_{x}-2it_{b}\cos\theta\cos(\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})\\\
2t_{b}\cos\theta\sin
k_{x}+2it_{b}\cos\theta\cos(\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})&-2t_{b}\cos\theta\sin(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})\end{pmatrix}$
$\displaystyle+\begin{pmatrix}-2t_{b}\sin\theta\tau_{3}\cos(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})&-2t_{b}\sin\theta\cos
k_{x}\tau_{3}+2it_{b}\sin\theta\sin(\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})\\\
-2t_{b}\sin\theta\cos
k_{x}\tau_{3}-2it_{b}\sin\theta\sin(\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})&2t_{b}\sin\theta\tau_{3}\cos(-\frac{1}{2}k_{x}+\frac{\sqrt{3}}{2}k_{y})\end{pmatrix}$
(96)

When taking $\theta=0$, the mean-field is invariant under $SU(2)$, with
symmetry transformation:

$\displaystyle T_{1}:\
\Psi(\mathbf{r})\rightarrow(-1)^{r_{2}}\Psi(\mathbf{r}+{\widehat{r}_{1}})$
$\displaystyle T_{2}:\
\Psi(\mathbf{r})\rightarrow\Psi(\mathbf{r}+{\widehat{r}_{2}})$ $\displaystyle
C_{6}:\
\Psi(\mathbf{r})\rightarrow\begin{cases}i(i)^{r_{2}}\Psi(C_{6}(r))&Mod(r_{2},2)=1\\\
-(-1)^{r_{1}}(i)^{r_{2}}\Psi(C_{6}(r))&Mod(r_{2},2)=0\end{cases}$
$\displaystyle R\mathcal{T}:\
\Psi(\vec{r})\rightarrow\begin{cases}(-1)^{r_{1}}\Psi(R(r))&Mod(r_{2},4)=0,1\\\
-(-1)^{r_{1}}\Psi(R(r))&Mod(r_{2},4)=2,3\end{cases}$ (97)

The holons hop in the same ansatz as the spinons, with the k space Hamiltonian
reads,

$\displaystyle H_{Z}(\mathbf{k})=\sin k_{2}\eta^{3}-\sin
k_{1}\eta^{1}-\cos(k_{1}+k_{2})\eta^{2},$ (98)

where $k_{1,2}$ are coordinates in reciprocal space spanned by
$\mathbf{b}_{1,2}$, that satisfies
$\mathbf{b}_{i}\cdot\mathbf{a}_{j}=2\pi\delta_{ij}$. $\eta$ rotates $A,B$
sublattices.

There are $2$ holon minima at $Q_{1,2}=(\pi/2,\pm\pi/2)$, with states of
minimal energy denoted as $\Phi=(\Phi_{1},\Phi_{2})$, at $Q_{1,2}$,
respectively. They transform as

$\displaystyle T_{1}:\Phi\rightarrow-i\sigma_{1}\Phi,$ $\displaystyle
T_{2}:\Phi\rightarrow-i\sigma_{3}\Phi,$ $\displaystyle C_{6}:\Phi\rightarrow
e^{i\frac{\pi}{3}}e^{-i\frac{\sigma_{3}+\sigma_{2}+\sigma_{1}}{\sqrt{3}}\frac{\pi}{3}}\Phi,$
$\displaystyle
R\mathcal{T}:\Phi\rightarrow\frac{e^{-i\frac{\pi}{12}}}{\sqrt{2}}(1-i\sigma_{1})\Phi$
(99)

where $\sigma$ rotates two valleys $Q_{1,2}$.

For the $U(1)$ ansatz on triangular lattices discussed in section IV, the
projective symmetry group for the spinons reads:

$\displaystyle
T_{1}:\psi_{r}\rightarrow(-1)^{r_{2}}\psi_{r+{\widehat{r}}_{1}},$
$\displaystyle T_{2}:\psi_{r}\rightarrow\psi_{r+{\widehat{r}}_{2}},$
$\displaystyle
C_{6}:\psi_{r}\rightarrow\begin{cases}\tau_{1}\psi_{C_{6}r}&(C_{6}r)_{2}\mod
2=0\\\ -(-1)^{(C_{6}r)_{x}}i\tau_{1}\psi_{C_{6}r}&(C_{6}r)_{2}\mod
2=1\end{cases},$ (100)

those for the holons in terms of $(z_{1},z_{2}^{*})^{T}$ are the same for the
spinon symmetry transforms.

There are $2$ holon minima at $Q_{1,2}=(\pi/2,\pm\pi/2)$ for each holon
species, respectivelySong  _et al._ (2021). The flux is translation invariant
but breaks naive $C_{6}$ rotation defined by eq (A.2). Note compared to
$SU(2)$ invariant ansatz, the additional hopping for small $\theta$ projects
to the lowest-energy holon states to be $0$.

Denote the states of minimal energy as $\Phi=(\Phi_{1},\Phi_{2})$, at
$Q_{1,2}$. They transform as

$\displaystyle T_{1}:\Phi\rightarrow-i\sigma_{1}\Phi,$ $\displaystyle
T_{2}:\Phi\rightarrow-i\sigma_{3}\Phi,$ $\displaystyle C_{6}:\Phi\rightarrow
e^{i\frac{\pi}{3}}\tau_{1}e^{-i\frac{\sigma_{3}+\sigma_{2}+\sigma_{1}}{\sqrt{3}}\frac{\pi}{3}}\Phi,$
$\displaystyle
R\mathcal{T}:\Phi\rightarrow\frac{e^{-i\frac{\pi}{12}}}{\sqrt{2}}(1-i\sigma_{1})\Phi$
(101)

where $\sigma$ rotates two valleys $Q_{1,2}$.

###  A.3 Kagome lattices

The projective symmetry group on the spinons act as

$\displaystyle
T_{1}:f_{i}\rightarrow(-1)^{R_{1}+R_{2}}f_{i+{\widehat{R}}_{1}}$
$\displaystyle T_{2}:f_{i}\rightarrow f_{i+{\widehat{R}}_{2}}$ $\displaystyle
C_{6}:f_{i}\rightarrow G(C_{6}(i))f_{C_{6}(i)},(G(C_{6}(i))\textrm{in fig
\ref{fig:kagomecsl}(a)})$ $\displaystyle R\mathcal{T}:f_{i}\rightarrow
G_{R}(R(i))f_{R(i)},(G_{R}(R(i))\textrm{in fig \ref{fig:kagomecsl}(b)})$ (102)

Figure 8:  (a)The gauge transform for $C_{6}$ where the red dot denotes $-1$
gauge transform and no point denotes trivial gauge transform. With the
$2\times 2$ cell enclosed by the parallelogram, the gauge transform carries a
$(\pi,\pi)$ momentum, i.e. gauge transform is invariant upon translation by
$4$ units of primitive lattice vectors. (b) Gauge transform for
$R\mathcal{T}$, with a $2\times 2$ unit cell. (d) The electron pairing(spin
singlet) and hopping amplitude when condensing holons at low-energy fields
$\Phi_{1;1}=\Phi_{2;1}=\Phi_{1;2}=-\Phi_{2;2}=1$, which makes
$\Phi^{T}\tau_{2}\sigma_{2}\Phi=-2$.The corresponding BCS Hamiltonian is
translation invariant, though pairing breaks inversion.

For the holons $z$ they hop in a similar ansatz as for spinons from eq (24),
with the ansatz differ by a negative sign (for purely imaginary hopping) for
$z_{1,2}$. This results in a degenerate line of lowest-energy states in the
Brillouin zone. To simplify the case, we focus on a degenerate pair of
momentum points $Q_{1,2}=(0,\pm\pi/4)$ and do not require the states at
$Q_{1,2}$ to the lowest energy, since we are concerned about the symmetry
properties of the resulting electron Hamiltonian, not energetics. The holons
condense at states at $Q_{1,2}$ with a condensation value of $\Phi_{1,2;i}$
for $z_{i}$, respectively.Note that the lowest-energy state wavefunctions of
$z_{1,2}$ are related by $\psi_{1}(Q_{i})=\psi_{2}(Q_{3-i})^{*}$. For a
generic CSL ansatz with NNN hopping, the lowest-energy states for holons
reads,

$\displaystyle\psi_{1}(Q_{1})=(e^{-5i\pi/12},\sqrt{2}e^{i\pi/6},i,-e^{-5i\pi/12},0,1),$
$\displaystyle\psi_{1}(Q_{2})=(-e^{-5i\pi/12},0,i,-e^{-5i\pi/12},-\sqrt{2}e^{i\pi/6},1),$
(103)

independent of NNN hopping.

The PSG on low-energy holon fields at $Q_{1,2}$, arranged in the form
$\Phi=(\Phi_{1;1},\Phi_{2;2}^{*},\Phi_{2;1},\Phi_{1;2}^{*})$, since
$(z_{1},z_{2}^{*})$ transform as a vector under the $SU(2)$ gauge group, reads
as

$\displaystyle T_{1}:\Phi\rightarrow i\sigma_{2}\Phi$ $\displaystyle
T_{2}:\Phi\rightarrow i\sigma_{3}\Phi$ $\displaystyle C_{6}:\Phi\rightarrow
e^{i\pi/6}e^{i\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{\sqrt{3}}\pi/3}\Phi$
$\displaystyle R\mathcal{T}:\Phi\rightarrow
e^{i\frac{\pi}{4}}\frac{-\sigma_{3}+\sigma_{1}}{\sqrt{2}}\Phi$ (104)

##  Appendix B Details in the $SU(2)$ theory

Here we list more details about the $SU(2)$ $2\Phi$ theory in Sec. IX and its
global symmetry. First, let us consider the square lattice. Because the
$\tilde{C}_{4}$ rotation: $\Phi\rightarrow-
ie^{i\frac{\pi}{4}\sigma_{3}\tau_{0}}\Phi$, the most generic action is:

$\displaystyle\mathcal{L}$
$\displaystyle=\sum_{a=1,2}|(\partial_{\mu}-ia^{s}_{\mu}\tau^{s}-iA^{c}_{\mu}\sigma_{0}\tau_{0}-iA^{r}_{\mu}\sigma_{3}\tau_{0})\Phi_{a}|^{2}+r|\Phi|^{2}+\frac{1}{4\pi}Tr[a\wedge
da+\frac{2}{3}ia\wedge a\wedge a]-\mathcal{L}_{int},$ $\displaystyle L_{int}=$
$\displaystyle
g|\Phi|^{4}+\lambda_{\tau}\sum_{i}|\Phi^{\dagger}\tau^{i}\Phi|^{2}+\lambda_{\sigma}\sum_{i}|\Phi^{\dagger}\sigma_{i}\Phi|^{2}+\lambda_{\sigma
3}|\Phi^{\dagger}\sigma_{3}\Phi|^{2}+\lambda_{m}\sum_{i,j}|\Phi^{\dagger}\tau^{i}\sigma_{j}\Phi|^{2}+\lambda_{m3}\sum_{i}|\Phi^{\dagger}\tau^{i}\sigma_{3}\Phi|^{2}$
(105)

where $a^{s},s=1,2,3$ is an $SU(2)$ gauge field and $\sum_{i}=\sum_{i=1,2,3}$.
$A_{c}$ is the physical probing field. We have used the symmetry transforms
above to simplify the possible quartic interaction. Using identities of

$\displaystyle\sum_{i}|\Phi^{\dagger}\tau^{i}\Phi|^{2}=\sum_{i}|\Phi^{\dagger}\sigma_{i}\Phi|^{2},$
$\displaystyle\sum_{i,j}|\Phi^{\dagger}\tau^{i}\sigma_{j}\Phi|^{2}=-2\sum_{i}|\Phi^{\dagger}\sigma_{i}\Phi|^{2}+3|\Phi^{\dagger}\Phi|^{2},$
(106)

, we further simplify the quartic terms to

$\displaystyle\mathcal{L}_{int}=$ $\displaystyle
g|\Phi^{\dagger}\Phi|^{2}+\lambda_{1}\sum_{i}|\Phi^{\dagger}\sigma_{i}\Phi|^{2}+\lambda_{2}\sum_{i}|\Phi^{\dagger}\tau^{i}\sigma_{3}\Phi|^{2}+\lambda_{3}|\Phi^{\dagger}\sigma_{3}\Phi|^{2}$
(107)

With some algebra, we can show that:

$\sum_{i}|\Phi^{\dagger}\tau^{i}\sigma_{3}\Phi|^{2}=(\Phi^{T}\sigma_{2}\tau_{2}\Phi)^{*}(\Phi^{T}\sigma_{2}\tau_{2}\Phi)+(\Phi^{\dagger}\sigma_{3}\Phi)^{2}$
(108)

Then in terms of
$\mathbf{n}=(n_{1},n_{2},n_{3},n_{4},n_{5})=(\text{Re}\Phi^{T}\sigma_{2}\tau_{2}\Phi,\text{Im}\Phi^{T}\sigma_{2}\tau_{2}\Phi,\Phi^{\dagger}\sigma_{1}\Phi,\Phi^{\dagger}\sigma_{2}\Phi,\Phi^{\dagger}\sigma_{3}\Phi)$,
we can rewrite the quartic terms to be:

$\mathcal{L}_{int}=g|\Phi^{\dagger}\Phi|^{2}+\lambda_{1}(n_{3}^{2}+n_{4}^{2}+n_{5}^{2})+\lambda_{2}(n_{1}^{2}+n_{2}^{2}+n_{5}^{2})+\lambda_{3}n_{5}^{2}$
(109)

Note that the $\tilde{C}_{4}$ rotates $n_{3}\rightarrow
n_{4},n_{4}\rightarrow-n_{3}$ and thus forbids terms like
$n_{3}^{2}-n_{4}^{2}$ and $n_{3}n_{4}$.

We will group the interaction into the form:

$\mathcal{L}_{int}=g|\Phi^{\dagger}\Phi|^{2}+\lambda_{0}\mathbf{n}\cdot\mathbf{n}+\lambda(n_{1}^{2}+n_{2}^{2})+\lambda^{\prime}(n_{3}^{2}+n_{4}^{2})$
(110)

On triangular and kagome lattice, because $(n_{3},n_{4},n_{5})$ can be rotated
to each other by $C_{6}$, we must have $\lambda^{\prime}=0$. When $\lambda$ or
$\lambda^{\prime}$ vanishes, there is a $SO(3)\times O(2)$ global symmetry. At
special line $\lambda=\lambda^{\prime}$, there is a $O(4)$ symmetry. When
$\lambda=\lambda^{\prime}=0$, the symmetry is further enlarged to be SO(5). To
see these enlarged symmetries, it is more convenient to use the $4\times 2$
matrix field $X$ introduced in Sec. IX.3.

###  B.1 Action in terms of the matrix field $X$

We have $X=\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}$ with

$X_{a}=\frac{1}{\sqrt{2}}\begin{pmatrix}\Phi_{a;1}&\Phi_{a;2}\\\
-\Phi_{a;2}^{*}&\Phi_{a;1}^{*}\end{pmatrix}$ (111)

under the constraint

$X_{a}^{*}=\tau_{2}X_{a}\tau_{2}$ (112)

One can derive these equations:

$\text{Tr}X_{a}^{\dagger}X_{b}=\frac{1}{2}(\Phi_{a}^{\dagger}\Phi_{b}+\Phi_{b}^{\dagger}\Phi_{a}),\
\
\text{Tr}X_{a}^{\dagger}\tau_{3}X_{b}=\frac{1}{2}(\Phi_{a}^{\dagger}\Phi_{b}-\Phi_{b}^{\dagger}\Phi_{a})$
(113)

$\text{Tr}X_{a}^{\dagger}(-i\tau_{1})X_{b}=\text{Im}\Phi_{a}^{T}i\tau_{y}\Phi_{b},\
\
\text{Tr}X_{a}^{\dagger}(-i\tau_{2})X_{b}=\text{Re}\Phi_{a}^{T}i\tau_{y}\Phi_{b}$
(114)

Then we can derive:

$\Phi^{\dagger}\sigma_{0}\tau_{0}\Phi=\text{Tr}X^{\dagger}X,\ \
\text{Tr}X^{\dagger}\sigma_{3}X=\Phi^{\dagger}\sigma_{3}\Phi$ (115)

$\text{Tr}X^{\dagger}\sigma_{1}X=\Phi^{\dagger}\sigma_{1}\Phi,\ \
\text{Tr}X^{\dagger}\sigma_{2}\tau_{3}X=\Phi^{\dagger}\sigma_{2}\Phi$ (116)

$\text{Tr}X^{\dagger}(-\sigma_{2}\tau_{1})X=\text{Im}\Phi_{1}^{T}\sigma_{2}\tau_{2}\Phi,\
\
\text{Tr}X^{\dagger}(-\sigma_{2}\tau_{2})X=\text{Re}\Phi_{1}^{T}\sigma_{2}\tau_{2}\Phi$
(117)

$\text{Tr}X^{\dagger}\sigma_{2}X=0,\ \
\text{Tr}X^{\dagger}\sigma_{0,1,3}\tau_{3}X=0,\ \
\text{Tr}X^{\dagger}\sigma_{0,1,3}\tau_{1,2}X$ (118)

The five symmetry breaking order parameters now can be rewritten as:
$(n_{1},n_{2},n_{3},n_{4},n_{5})=(\text{Re}\Phi^{T}\sigma_{2}\tau_{2}\Phi,\text{Im}\Phi^{T}\sigma_{2}\tau_{2}\Phi,\Phi^{\dagger}\sigma_{1}\Phi,\Phi^{\dagger}\sigma_{2}\Phi,\Phi^{\dagger}\sigma_{3}\Phi)=\text{Tr}X^{\dagger}(-\sigma_{2}\tau_{2},-\sigma_{2}\tau_{1},\sigma_{1},\sigma_{2}\tau_{3},\sigma_{3})X$,

$SU(2)$ gauge transformation acts as $X\rightarrow XU_{g}^{\dagger}$,
$a_{\mu}\rightarrow
U_{g}a_{\mu}U_{g}^{\dagger}-iU_{g}\partial_{\mu}U_{g}^{\dagger}$. The critical
theory at $(\lambda=\lambda^{\prime}=\lambda_{3}=0$ can be rewritten as:

$\displaystyle\mathcal{L}=\text{Tr}(\partial_{\mu}X^{\dagger}+ia_{\mu}X^{\dagger})(\partial_{\mu}X-iXa_{\mu})+r\text{Tr}X^{\dagger}X+\frac{1}{4\pi}\text{Tr}[a\wedge
da+\frac{2}{3}ia\wedge a\wedge a]-\mathcal{L}_{int}$ (119)

where $a_{\mu}$ is the abbreviation of $a_{\mu}^{s}\tau_{s}$ with $s=1,2,3$.

The interaction term is now:

$\mathcal{L}_{int}=g\text{Tr}X^{\dagger}X+\lambda_{0}\mathbf{n}\cdot\mathbf{n}+\lambda(n_{1}^{2}+n_{2}^{2})+\lambda^{\prime}(n_{3}^{2}+n_{4}^{2})$
(120)

###  B.2 Higgs term

We also discuss the Higgs term needed to reach the $U(1)$ theory from $SU(2)$
theory.

It is easy to derive:

$\text{Tr}X_{a}^{\dagger}\tau_{3}X_{b}\tau_{3}=\frac{1}{2}(\Phi_{a}^{\dagger}\tau_{3}\Phi_{b}+\Phi_{b}^{\dagger}\tau_{3}\Phi_{a})$
(121)

Then we obtain:

$\Phi^{\dagger}\tau_{3}\Phi=\text{Tr}X^{\dagger}\tau_{3}X\tau_{3},\ \
\Phi^{\dagger}\sigma_{3}\tau_{3}\Phi=\text{Tr}X^{\dagger}\sigma_{3}\tau_{3}X\tau_{3}$
(122)
$\Phi^{\dagger}\sigma_{1}\tau_{3}\Phi=\text{Tr}X^{\dagger}\sigma_{1}\tau_{3}X\tau_{3},\
\
\Phi^{\dagger}\sigma_{2}\tau_{3}\Phi=\text{Tr}X^{\dagger}\sigma_{2}X\tau_{3}$
(123)

$\text{Re}\Phi^{T}\sigma_{1}\tau_{1}\Phi=-\text{Tr}X^{\dagger}\sigma_{1}\tau_{1}X\tau_{3},\
\
\text{Im}\Phi^{T}\sigma_{1}\tau_{1}\Phi=\text{Tr}X^{\dagger}\sigma_{1}\tau_{2}X\tau_{3}$
(124)

Under the duality transformation $X\rightarrow
e^{i\tau_{2}\frac{\pi}{4}}e^{i\sigma_{3}\tau_{2}\frac{\pi}{4}}X$,
$\text{Tr}X^{\dagger}\sigma_{3}\tau_{3}X\tau_{3}\rightarrow-\text{Tr}X^{\dagger}\tau_{3}X\tau_{3}$,
$\text{Tr}X^{\dagger}\sigma_{1}\tau_{3}X\tau_{3}\rightarrow\text{Tr}X^{\dagger}\sigma_{1}\tau_{1}X\tau_{3}$,
$\text{Tr}X^{\dagger}\sigma_{2}X\tau_{3}\rightarrow-\text{Tr}X^{\dagger}\sigma_{1}\tau_{2}X\tau_{3}$

##  Appendix C Monopole quantum numbers in $U(1)_{-2}$ 2$\varphi$ theory for
CSL-SC transition

Monopoles in the $U(1)_{-2}$ Chern-Simons theory with $2\varphi$ are dressed
with $2$ units of gauge charges to be gauge-invariant. We consider

$\displaystyle\varphi^{T}\epsilon\sigma_{i}\varphi\mathcal{M}^{\dagger}(i=1,2,3)$
(125)

where $\epsilon=i\sigma_{2}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ is the
antisymmetric tensor and is used to make symmetric combinations of two gauge
charges $\varphi$. The dressed charges $\varphi^{T}\epsilon\sigma_{i}\varphi$
transform identically as $\varphi^{\dagger}\sigma_{i}\varphi$, which carry the
quantum numbers of order parameters $(n_{1},n_{2},n_{5})$ . To identify
monopoles that transform as the remaining two order parameters
$(n_{3},n_{4})$, we consider the composite operator which is singlet under the
$SO(3)$ symmetry transforming $(n_{1},n_{2},n_{5})$:

$\displaystyle\mathcal{M}_{a}^{\dagger}=(\varphi^{\dagger}\vec{\sigma}_{i}\varphi)\cdot(\varphi^{T}\epsilon\vec{\sigma}\varphi)\mathcal{M}^{\dagger}.$
(126)

Since the dressed gauge charges now preserve symmetries, we need only to
concern about the Berry phase or symmetry transforms of the bare flux
$\mathcal{M}$. The berry phase for rotation symmetries can be inferred from
the atomic limit of the holonsSong  _et al._ (2020). $q$ units of gauge
charges located at the rotation center contribute to the angular momentum of
the monopoles from the Ahronov-Bohm effects, i.e. $l_{M}=q$. We remark that
the atomic limit of holons that preserves space group symmetries and gauge-
invariant (total gauge charge $0$) is not unique, e.g. the vacuum state is a
trivial example. This is due to the fact that we are dealing with an effective
low-energy theory after integrating out chern bands from the fermionic
spinons. We show in fig 9 a symmetric atomic limit for $U(1)_{-2}$ theories on
square lattices that describe CSL-SC and CSL-CDW transitions. Monopoles carry
CDW, $d+id$ SC order parameters in these two cases, respectively. Close and
open circles represent $\pm$ gauge charges, respectively, and they sum to zero
as required by gauge invariance.

We now elaborate on the case for CSL-SC transition listed in table 4.

Note that $\mathcal{M}_{a}^{\dagger}$ changes to an anti-monopole under
translations or $C_{4}$ as the symmetry actions exchanges
$\varphi_{1}\leftrightarrow\varphi_{2}^{*}$ in section IV, with opposite
charge of $a$. There is hence a phase ambiguity for the symmetry actions that
send $\mathcal{M}_{a}^{\dagger}\rightarrow\mathcal{M}_{a}$ from a $U(1)$ phase
attachment to $\mathcal{M}_{a}$. We fix the phase by fixing e.g. the $T_{1}$
action as simply sending $\mathcal{M}_{a}^{\dagger}\rightarrow\mathcal{M}_{a}$
with a trivial sign. The relative sign of symmetry actions among
$T_{1,2},C_{4}$ is meaningful and can be further determined by finding the
atomic limit of the holon state that obeys the symmetries in table 4, as shown
in fig 9.

The translation

$T_{2}=C_{4A}C_{4B}^{-1}T_{1},$ (127)

where $C_{4A,4B}$ is the four-fold rotation around the plaquette center of
$A,B$ plaquettes, i.e. occupied by $\varphi_{1,2}$ charges, respectively in
fig 9(a). Since $\varphi_{1,2}$ carry $1$ unit of charge for $a$, the holons
with charge $\pm 1$ act as a source of angular momentum of $\pm 1$ for the
monopole. The monopole hence obtains a factor $-1$ from $C_{4A}C_{4B}^{-1}$ in
eq (127). We thus obtain $T_{2}$ action. For site-center rotation $C_{4}$, it
is related to translation by

$C_{4}=T_{1}C_{4A}.$ (128)

$C_{4A}$ contributes a factor of $i$ for the monopoles and we arrive at
$C_{4}$ action for monopoles in table 4.

Figure 9: The holon center that determines monopole $\mathcal{M}_{a}$ symmetry
transforms in CSL-SC transition on square lattices.Close, open circles
represent holons with $\pm$ gauge charge, respectively, with $\cdot,\times$
sign indicating a holon of the $\varphi_{1,2}$ species.

For the action $R_{1}\mathcal{T}$ on monopoles, since it exchanges
$\varphi_{1},\varphi_{2}^{*}$ with an additional anti-unitary time reversal
action, it sends a monopole to itself up to an arbitrary phase factor.
Therefore we obtain the symmetry actions for gauge-invariant operators in
$U(1)_{-2}$ theory.