#  Doping the chiral spin liquid - topological superconductor or chiral metal?

Xue-Yang Song  Ashvin Vishwanath  Ya-Hui Zhang  Department of Physics, Harvard
University, Cambridge, MA 02138, USA

###### Abstract

We point out that there are two different chiral spin liquid states on the
triangular lattice and discuss the conducting states that are expected on
doping them. These states labeled CSL1 and CSL2 are associated with two
distinct topological orders with different edge states, although they both
spontaneously break time reversal symmetry and exhibit the same quantized spin
Hall conductance. While CSL1 is related to the Kalmeyer-Laughlin state, CSL2
is the $\nu=4$ member of Kitaev’s 16 fold way classification. Both states are
described within the Abrikosov fermion representation of spins, and the effect
of doping can be accessed by introducing charged holons. On doping CSL2,
condensation of charged holons leads to a topological d+id superconductor.
However on doping CSL1, in sharp contrast, two different scenarios can arise:
first, if holons condense, a chiral metal with enlarged unit cell and finite
Hall conductivity is obtained. However, in a second novel scenario, the
internal magnetic flux adjusts with doping and holons form a bosonic integer
quantum Hall (BIQH) state. Remarkably, the latter phase is identical to a
$d+id$ superconductor. In this case the Mott insulator to superconductor
transition is associated with a bosonic variant of the integer quantum Hall
plateau transition. We connect the above two scenarios to two recent numerical
studies of doped chiral spin liquids on triangular lattice. Our work clarifies
the complex relation between topological superconductors, chiral spin liquids
and quantum criticality.

##  I Introduction

Ever since Anderson conceived of the resonating-valence-bond(RVB) liquid as a
precursor state to the superconductor in the cuprates Anderson (1987), the
notion of quantum spin liquids (QSLs) has been inextricably tied to mechanisms
for high-temperature superconductivity. A considerable body of work has
explored doping spin liquid or Mott insulatorsLee  _et al._ (2006); Wen
(1990); Rokhsar (1993); Sigrist  _et al._ (1994); Fabrizio (1996); Senthil and
Fisher (2000); Senthil and Lee (2005); Konik  _et al._ (2006); Balents and
Sachdev (2007); Watanabe  _et al._ (2004); Karakonstantakis  _et al._ (2013);
Chen  _et al._ (2013); White  _et al._ (2015); Venderley and Kim (2019); Li
_et al._ (2015); Kelly  _et al._ (2016); Ye  _et al._ (2016); Jiang  _et al._
(2020); Peng  _et al._ (2020); Gannot  _et al._ (2020). In referring to QSLs,
it is customary to describe them in terms of two features - the emergent gauge
group, eg. $Z_{2}$ and U(1) spin liquids and additionally the statistics of
“spinon” excitations carrying spin 1/2 and gauge charge. Here we will consider
the case of fermionic spinonsWen (2002a, b). A Mott insulator to
superconductor transition upon doping is quite natural when the parent Mott
insulator is a $Z_{2}$ spin liquidLee  _et al._ (2006). Note, a $Z_{2}$ spin
liquid can be described by a parton mean field ansatz with fermionic spinons
in a BCS state. Doped charges are accounted for in terms of bosonic holons,
which can condense. Holon condensation makes the pairing of spinons evolve
into true Cooper pairs of electrons, leading to a superconducting phase. In
contrast, doping a $U(1)$ spin liquid with a spinon Fermi surface naturally
gives us a Fermi liquid phase, rather than a superconductor, because there is
no pairing of spinons in the parent Mott insulatorSenthil (2008).

This brings us to the subject of this paper, the chiral spin
liquid(CSL)Kalmeyer and Laughlin (1987); Wen  _et al._ (1989), and the
conducting states that emerges on doping. Chiral spin liquids spontaneously
break time reversal symmetry and exhibit chiral edge modes. In particular, a
speculative connection between fractional statistics that exists in chiral
spin liquids and two-dimensional superconducting states, so-called anyon
superconductivity, has been studied in the earlier days of high Tc
superconductivityChen  _et al._ (1989); Lee and Fisher (1989); Wen  _et al._
(1989); Lee and Kane (1990); Laughlin (1991). Theoretically, chiral spin
liquids have been found to be the ground state for various spin 1/2 lattice
models on the kagome and triangular latticesBauer  _et al._ (2014); He  _et
al._ (2014); Gong  _et al._ (2014); He and Chen (2015); Szasz  _et al._
(2020); Hu  _et al._ (2016); Wietek  _et al._ (2015) and also in $SU(N)$ model
with $N>2$Hermele and Gurarie (2011); Nataf  _et al._ (2016); Chen  _et al._
(2020); Zhang and Vishwanath (2020). On the experimental side, recent
observation of a quantized thermal Hall effect in a Kitaev materialKasahara
_et al._ (2018) suggests a non-Abelian chiral spin liquid. These examples
motivate us to ask - what is the fate of a chiral spin liquid on doping?
Surprisingly we find that this question is more complicated and potentially
also more interesting than the usual situation of simple $Z_{2}$ spin liquid,
where the conventional RVB theory works well.

The first complexity is that there are actually two different chiral spin
liquids: CSL1 and CSL2. CSL1 is an analog of fractional quantum Hall state in
spin system, as first proposed by Kalmeyer-LaughlinKalmeyer and Laughlin
(1987). In contrast, CSL2 should be thought as a “gauged” chiral
superconductor, whose wavefunction is obtained by Gutzwiller projection of a
spin singlet superconductor with $d_{x^{2}-y^{2}}+id_{xy}$ superconducting
pairing, abbreviated as $d+id$ pairing. Both CSL1 and CSL2 can be conveniently
described using mean field ansatz of Abrikosov fermions, generically not gauge
equivalent to each other. For CSL1, the fermionic spinons are put in a Chern
insulator phase with $C=2$. In contrast, the spinon is in a $d+id$
superconductor ansatz in the CSL2, very similar to a $Z_{2}$ spin liquid in
terms of wavefunction. Hence the conventional RVB theory can be generalized to
this case and a $d+id$ superconductor naturally emerges from doping. A dual
viewpoint involves beginning in the $d+id$ superconductor, and driving a
superconductor-insulator transition by condensing pairs of vortices Balents
_et al._ (1998), leading directly to CSL2.

On the other hand, the wavefunction of CSL1 is a Gutzwiller projection of
Chern insulator. Thus a superconductor is not expected from the RVB picture.
As far as we know, all of the chiral spin liquids found in numerics for spin
rotation invariant spin $1/2$ model are CSL1Bauer  _et al._ (2014); He  _et
al._ (2014); Gong  _et al._ (2014); He and Chen (2015); Szasz  _et al._
(2020); Hu  _et al._ (2016). This strongly suggests CSL1 as a more likely
candidate relevant to realistic spin $1/2$ materials. Then a natural question
that arises is: what should we expect on doping this chiral spin liquid?

The main result of this paper is to propose two different scenarios for doping
CSL1 on the triangular lattice. In the first scenario, a simple generalization
of RVB theory using slave boson condensationLee  _et al._ (2006) predicts a
doped Chern insulator. On triangular lattice, this is a chiral metal with
doubled unit cell and staggered loop current order. It also breaks the $C_{6}$
rotation symmetry completely and has a non-zero electrical Hall conductivity.
This is well described by a mean field theory of electron, which basically
inherits the mean field ansatz of spinons in the Mott insulator. An
unconventional second possibility is a $d+id$ superconductor emerges from a
completely different mechanism, which can not be captured by simple mean field
theory. The doped holon forms a bosonic state with integer quantum
Hall(bosonic-IQH) effect Lu and Vishwanath (2012); Senthil and Levin (2013).
The bosonic holons lead to an opposite Hall conductivity
($\sigma^{b}_{xy}=-2$) compared to the fermionic spinons. The final
wavefunction of electron is a product of a bosonic IQH state and a fermionic
IQH state. This construction, surprisingly, turns out to describe a
superconductor with the same topological property as the $d+id$ superconductor
as we will show below.

Our proposed two scenarios for doping CSL1 are in agreement with with two
recent numerical studies of doped chiral spin liquid on triangular lattice.
Ref. Jiang and Jiang, 2020 finds a $d+id$ superconductor from doping a CSL in
the strong Mott regime described by a $J_{1}-J_{2}-J_{\chi}$ modelGong  _et
al._ (2017); Saadatmand and McCulloch (2017). In contrast, Ref. Zhu  _et al._
, 2020 observes a chiral metal phase upon doping a CSL in the weak Mott regime
Szasz  _et al._ (2020). The CSLs in the parent Mott insulators in the above
two cases are shown to be both the CSL1 because the measured entanglement
spectrum is consistent with the $SU(2)_{1}$ edge theory Gong  _et al._ (2017);
Szasz  _et al._ (2020). Lastly we note that there are other numerical studies
of possible superconductivity on triangular lattice where the parent state is
not a chiral spin liquidVenderley and Kim (2019); Jiang (2019); Gannot  _et
al._ (2020). Further work will be needed to synthesize these observations into
a coherent theory. However it is clear that there are at least two competing
phases upon doping the same CSL, which is consistent with the framework in our
paper.

##  II Two chiral spin liquids on triangular lattice

We adopt a parton construction to decompose the physical electron operator as
a boson carrying its electric charge (called ‘holon‘) and a fermionic spinon
carrying spin indices:

$c_{i,\sigma}=b_{i}^{\dagger}f_{i,\sigma}$ (1)

where $\sigma,i$ denote spin and site, respectively. This formalism introduces
a $U(1)$ gauge redundancy between the holon and spinon, with a $U(1)$ gauge
field $a$. Hence it leads to a gauge constraint $n_{i}^{b}+n_{i}^{f}=1$. Note
that there is an $SU(2)$ version of the parton construction with two slave
bosons per siteLee  _et al._ (2006) reviewed in appendix A, and for our
purpose the above $U(1)$ partons suffice.

At integer filling $n_{i}=1$, the holons are trivially gapped and the mean-
field for spinons is conveniently written in the basis
$\psi_{i}=(f_{i,\uparrow},f_{i,\downarrow}^{\dagger})^{T}$, with Pauli
matrices $\tau^{x,y,z}$ acting in such spinor space.

$\displaystyle\mathcal{H}_{spinon}=\sum_{\langle
ij\rangle}\psi_{j}^{\dagger}u_{i,j}\psi_{i},$ (2)

where one considers only nearest neighbor terms. $u$’s are $2\times 2$
matrices, satisfy $u_{i,j}=u_{j,i}^{\dagger}$ and contain hopping and pairing
as diagonal, off-diagonal elements, respectively. We use coordinates shown at
upper right of fig 2.

There are two kinds of chiral spin liquid variational states that break time
reversal and mirror symmetry on triangular lattices that have been extensively
studied: the $U(1)_{2}$ chiral spin liquid (CSL1) and the projected $d+id$
superconductor (CSL2). Below we review the mean field theory for the two
states and discuss their connection and differences.

###  II.1 CSL1: $U(1)_{2}$ chiral spin liquid

The $U(1)_{2}$ chiral spin liquid, which we study in this paper, is described
by a Chern insulator of the spinons with total Chern number (combining the two
spins) $C=2$. Typical realizations consist of a $\pi$ hopping flux of each
spinon $f_{\sigma}$ through a rhombus of the triangular lattice, with the flux
$\pi/2\pm 3\theta$ around an upward and downward triangle for the spinon. This
staggered-flux ansatz typically gives a gapped dispersion for each spinon
species, forming two Chern bands with $C=\pm 1$; at the point
$\theta=\pm\pi/6$, the dispersion becomes gapless and a Dirac spin liquid
emergesSong  _et al._ (2019). The mean-field ansatz of spinon hopping reads

$\displaystyle u_{r,r+{\widehat{x}}}=ie^{i(\frac{\pi}{2}+\theta)\tau^{z}},$
$\displaystyle
u_{r,r+{\widehat{y}}}=(-1)^{r_{x}}ie^{i(\frac{\pi}{2}+\theta)\tau^{z}},$
$\displaystyle
u_{r,r+{\widehat{x}}+{\widehat{y}}}=(-1)^{r_{x}}ie^{-i\theta\tau^{z}},(\theta\in(-\pi/3,\pi/3))$
(3)

where the $x,y$ coordinates are shown in figure 2. In this gauge choice, the
lower spinon band hits the lowest energy at zero momentum and the highest
energy at momenta $(\pi/3,\pi/3),(4\pi/3,-2\pi/3)$, independent of $\theta$.
See Fig. 1 for the dispersion of the spinon bands for various $\theta$ values.

Figure 1: Dispersion of spinons for $\mathcal{H}_{spinon}$ in eq (II.1).The
band is particle-hole symmetric.(a) The original(reduced) hexagonal(square,
dashed) Brillouin zone and the dispersion plots are along the loop
$\Gamma\rightarrow K\rightarrow K^{\prime}\rightarrow\Gamma$ where ${\bf
Q}=(\pi/3,\pi/3),{\bf Q}^{\prime}=(4\pi/3,-2\pi/3)$ are points of maximal
energy in the valence band. $\Gamma,M_{2}$ are points of lowest energy in
valence band. (b) The dispersion at the DSL gapless point $\theta=\pi/6$.
(c)(d) Dispersions at $\theta=0,0.2$, respectively. The Chern number of
valence (conduction) bands is $C=\pm 1$.

$\mathcal{H}_{spinon}$ as in eq (II.1) is invariant under a projective
translation and six-fold rotation symmetries, while breaking time-reversal and
mirror symmetry except at the special point $\theta=\pm\pi/6$. The projective
symmetry group for the spinons reads:

$\displaystyle T_{x}:\psi_{r}\rightarrow(-1)^{r_{y}}\psi_{r+{\widehat{x}}},$
$\displaystyle T_{y}:\psi_{r}\rightarrow\psi_{r+{\widehat{y}}},$
$\displaystyle
C_{6}:\psi_{r}\rightarrow\begin{cases}\tau^{x}\psi_{C_{6}r}&(C_{6}r)_{y}\mod
2=0\\\ -(-1)^{(C_{6}r)_{x}}i\tau^{x}\psi_{C_{6}r}&(C_{6}r)_{y}\mod
2=1\end{cases},$ (4)

where $C_{6}r$ is the $C_{6}$ rotated position of the site at $r$.

The low-energy physics of the CSL1 is captured by the following action:

$L=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}ad(\alpha_{1}+\alpha_{2})+\frac{1}{2}\frac{A_{s}}{2}d(\alpha_{1}-\alpha_{2})$
(5)

where $A_{s}$ is the external spin field in the convention that its charge
$q_{s}=\frac{1}{2}$ for $S_{z}=\frac{1}{2}$. $\alpha_{1},\alpha_{2}$ are
introduced to represent the IQH of $f_{\uparrow}$ and $f_{\downarrow}$
respectively. $a$ is the internal $U(1)$ gauge field shared by the slave boson
$b$ and the spinon $f_{\sigma}$.

We can also integrate $a$ to lock $\alpha_{1}=-\alpha_{2}=\alpha$, then we get

$L=-\frac{2}{4\pi}\alpha d\alpha+\frac{1}{2\pi}A_{s}d\alpha$ (6)

This is a $U(1)_{2}$ theory which also describes the $\nu=\frac{1}{2}$
Laughlin state. Exploiting the relation between spin $1/2$ and hard core
bosons, one can map the $\nu=\frac{1}{2}$ Laughlin state of boson to a spin
state, which corresponds to CSL1. Thus the CSL1 has the same topological order
as the $\nu=\frac{1}{2}$ Laughlin state. Indeed a model wavefunction using the
Laughlin state was proposed by Kalmeyer-LaughlinKalmeyer and Laughlin (1987).
But we need to emphasize that a general state with any
$\theta\neq\pm\frac{\pi}{6}$ is in the same topological phase as the Kalmeyer-
Laughlin state. There are two anyons in this phase: $I$ and $s$, with $s$ as a
spin $1/2$ semion. The edge state is characterized by chiral central charge
$c=1$.

###  II.2 CSL2: Projected $d+id$ superconductor

The projected $d+id$ superconductor is given by the $u$ matrices below:

$\displaystyle
u_{r,r+{\widehat{x}}}=\frac{1}{\sqrt{\chi^{2}+\eta^{2}}}(\chi\tau^{z}+\eta\tau^{x}),$
$\displaystyle
u_{r,r+{\widehat{y}}}=\frac{1}{\sqrt{\chi^{2}+\eta^{2}}}[\chi\tau^{z}+\eta(\tau^{x}\cos\frac{4\pi}{3}-\tau^{y}\sin\frac{4\pi}{3})],$
$\displaystyle
u_{r,r+{\widehat{x}}+{\widehat{y}}}=\frac{1}{\sqrt{\chi^{2}+\eta^{2}}}[\chi\tau^{z}+\eta(\tau^{x}\cos\frac{2\pi}{3}-\tau^{y}\sin\frac{2\pi}{3})],$
(7)

with $\chi,\eta$ as real parameters and explicit translation invariance. The
pairings for $3$ bonds generated by $C_{3}$ rotation have phases
$0,2\pi/3,4\pi/3$ respectively, the same as that of a $d+id$ superconductor

The topological order depends on the angular momentum of the corresponding
pairing, as classified by Kitaev’s sixteenfold wayKitaev (2006). With $SU(2)$
spin rotation symmetry, the pairing can only be in the even angular momentum
channel, making the gauged $d+id$ superconductor as the simplest chiral spin
liquid in the CSL2 category. There are four anyons labeled by $1,e,m,\epsilon$
in CSL2Moroz  _et al._ (2017). $e,m$ are semions with $\pi/2$ self-statistics
and trivial mutual statistics. $\epsilon$ is a bound state of $e,m$ and has
fermionic statistics. This state corresponds to $\nu=4$ of Kitaev’s
sixteenfold classification of $Z_{2}$ topological orderKitaev (2006) and its
topological order is $U(1)_{2}\times U(1)_{2}$. Its edge theory has chiral
central charge $c=2$. The wavefunction of CSL2 is a Gutzwiller projection of a
$d+id$ superconductor instead of a s wave superconductor.

###  II.3 Relation between two CSLs

We note that these two CSL states generically are not equivalent, since the
invariant gauge group (IGG) for projected $d+id$ states(CSL2) is $Z_{2}$,
while IGG for CSL1 is $U(1)$. Moreover, they differ in the topological orders
and anyon contents as described above.

We note however the following collision between these two ansatz and clarify
its meaning. For CSL2, at the special point when $\eta=\sqrt{2}\chi$, the
Wilson loop around one triangle for projected $d+id$ reads
$\Phi=u_{r,r+{\widehat{x}}}u_{r,r+{\widehat{y}}}u_{r,r-{\widehat{x}}-{\widehat{y}}}\propto
i\tau^{0}$, equivalent to that of the CSL1 at $\theta=0$. Hence the two states
are gauge equivalent at the special point. Specifically, there is an SU(2)
gauge transform $g_{r}$ which rotates the mean field ansatz for CSL1 (II.1) at
$\theta=0$ to the mean field $d+id$ ansatz of CSL2 eq (II.2) at
$\eta=\sqrt{2}\chi$:

$\displaystyle\psi_{r}\rightarrow g_{r}\psi_{r},$ $\displaystyle
g_{r}=\begin{cases}1&r_{x}\textrm{ even, }r_{y}\textrm{ even}\\\
-\frac{\tau^{z}+\sqrt{2}\tau^{x}}{\sqrt{3}}&r_{x}\textrm{ odd, }r_{y}\textrm{
even}\\\
\frac{-(\tau^{z}+\sqrt{2}(\tau^{x}\cos\frac{4\pi}{3}-\tau^{y}\sin\frac{4\pi}{3}))}{\sqrt{3}}&r_{x}\textrm{
even, }r_{y}\textrm{ odd}\\\
\frac{-i(\tau^{z}+\sqrt{2}(\tau^{x}\cos\frac{2\pi}{3}-\tau^{y}\sin\frac{2\pi}{3}))}{\sqrt{3}}&r_{x}\textrm{
odd, }r_{y}\textrm{ odd.}\end{cases}$ (8)

At the special point $\theta=0$, the IGG is actually enlarged to $SU(2)$ and
the low energy theory is $SU(2)_{1}$, which is known to be equivalent to the
$U(1)_{2}$ theory. Hence the special point with $\theta=0$ also belongs to the
CSL1. At the same time, the point $\eta=\sqrt{2}\chi$ does not belong to CSL2,
since the gauge group is enlarged to SU(2). However, on moving away from this
special point, a Higgs condensate develops, lowering the gauge group to Z2.

One can ask what the gauge transformation above accomplishes when $\theta$
shifts away from $0$ for CSL1 ansatz. Then the above rotation eq (II.3) on the
CSL1 does not restore explicitly lattice translation as present in the $d+id$
ansatz (see appendix B). The transformed state at $\theta\neq 0$ has $d+id$
pairing, but breaks translation symmetry and despite the fact that it appears
to be a pairing state, it secretly possesses a U(1) IGG.

Figure 2: A doubled unit cell shown in dotted parallelogram. Spinon hopping
ansatz of the CSL shown in red on the bond inside the dotted parallelogram,
direction in line with arrows on bonds. The relative current strength
$I_{bond\langle ij\rangle}=Im[c^{\dagger}_{i}c_{j}]$ of the chiral metal upon
infinitesimal doping of holes ($\theta=0$) shown schematically by the bond
gray level (the smaller the current is, the grayer the bond looks) with arrows
indicating current directions. The currents are given by holon condensation at
$\Gamma,M_{2}$ momenta such that the kinetic energy of electrons
$-t\sum_{\langle ij\rangle}c_{i}^{\dagger}c_{j}$ is optimized. For doping
holes, holons fully condense at $M_{2}$ and the currents have a _doubled_ unit
cell structure. For doping electrons, currents are opposite to those in the
hole scenario.

##  III Chiral metal

At hole doping level $x$, the average density of the holon and the spinon per
site becomes $n_{i}^{b}=x$ and $n_{i}^{f}=1-x$. One simple possibility is that
slave boson condenses at zero temperature. This is obtained self-consistently
for a mean field ansatz of holons. Assuming an electron hopping term in the
parent Hubbard model of the form $H_{KE}=-t\sum_{\langle
ij\rangle}c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm h.c.}$ with real $t>0$, we can
obtain the mean field theory of the holon as

$\displaystyle\mathcal{H}_{holon}=t\sum_{\langle ij\rangle,\sigma}\langle
f_{i,\sigma}f_{j,\sigma}^{\dagger}\rangle b_{i}^{\dagger}b_{j},$ (9)

where $\langle f_{i,\sigma}f_{j,\sigma}^{\dagger}\rangle$ is computed on the
ground state of the spinon mean field $\mathcal{H}_{spinon}$ and is
proportional to spinon hopping terms on corresponding bonds. Lowest energy
states for $\mathcal{H}_{holon}$ lie at $\Gamma,M_{2}$ momenta, and hence
holon condensation is generically a superposition of such two states. The
favored condensation is given by optimizing the kinetic energy of electrons in
$H_{KE}$,

$E_{KE}=t\sum_{\langle ij\rangle,\sigma}\langle
f_{i,\sigma}f_{j,\sigma}^{\dagger}\rangle\langle b_{i}^{\dagger}\rangle\langle
b_{j}\rangle,$ (10)

where the holon condensed values

$\displaystyle\langle
b_{i}\rangle=\sqrt{2x}[p_{\Gamma}u_{\Gamma}(i_{s})+(1-p_{\Gamma})(-1)^{i_{y}}u_{M2}(i_{s})],$
(11)

where $p_{\Gamma}\in[0,1]$ is the fraction of holons that condense at
$\Gamma$. $u_{\Gamma,M2}(i_{s}=A,B)$ are the Bloch wavefunctions at
$\Gamma,M_{2}$ momenta on two sublattices,

$\displaystyle u_{\Gamma}(A)=0.63(1+i),u_{\Gamma}(B)=0.46,$ $\displaystyle
u_{M2}(A)=0.33(1-i),u_{M2}(B)=0.89,$ (12)

identically for any $\theta$.

We found that for negative hopping in $H_{KE}$, the holons fully condense at
$M_{2}=(0,\pi)$ to optimize the kinetic energy, i.e. $p_{\Gamma}=0$ in eq
(11). This is true for any $\theta\in(-\pi/6,\pi/6)$. For positive hopping (or
equivalently doping electrons), holons fully condense at $\Gamma$,i.e.
$p_{\Gamma}=1$. The wave functions with minimal energy for
$-\mathcal{H}_{holon}$ read,

$\displaystyle\tilde{u}_{\Gamma}(A)=0.33(-1-i),\tilde{u}_{\Gamma}(B)=0.89,$
$\displaystyle\tilde{u}_{M2}(A)=0.63(-1+i),\tilde{u}_{M2}(B)=0.46.$ (13)

Holon condensation leads to the electron operator as

$c_{i,\sigma}=\langle b_{i}^{\dagger}\rangle f_{i,\sigma}.$ (14)

Spinons are away from integer filling upon doping and states at the valence
band top are emptied as Fermi pockets. Electrons, identified as spinons in eq
(14), hence partially fill states in a band and form a metal with $\pi$ flux
and breaks translation, rotation and time reversal, as the spinon Hamiltonian
does. Numerically, we found a nonvanishing bond current pattern of the metal
$I_{bond\langle ij\rangle}\equiv Im[c_{i}^{\dagger}c_{j}]$ shown in figure
2(parameter $\theta=0$), at infinitesimal doping.

For the electron doping scenario, one changes the parton definition to
$c^{\dagger}_{i,\sigma}=b_{i}^{\dagger}f_{i,\sigma}$, which corresponds to a
particle-hole transformation for the electrons, and hence results in a minus
sign multiplying the hopping amplitude $-t$ in $H_{KE}$. To optimize
$-H_{KE}$, holons all condense at $\Gamma$, currents are opposite to the hole
doping scenario shown in fig 2, up to a ${\widehat{r}}_{x}$ translation of $y$
bond currents. We also present the current patterns at $\theta=\pm 0.2,\pm
0.4$ upon infinitesimal hole doping in supplementary fig 8 with holons
condensing fully at $\Gamma$.111In numerics, the relative strength of the
current may change by as much as 0.08 (normalized by the strongest current),
when increasing the density of momenta used for calculations, from a density
of 400 points per unit area to 10000 points per unit area.The current pattern
generically holds a $2\times 2$ unit cell due to holons condensing at both
$\Gamma,M_{2}$ when going beyond a mean-field treatment; only when holons
condense solely at $\Gamma$ or $M_{2}$ does it yield a $2\times 1$ unit cell
pattern.

The electrons have a non-vanishing Hall response as the spinons are in a Chern
band. From Ioffe-Larkin rule, one has for the resistivity tensors
$\rho_{c}=\rho_{b}+\rho_{f}$ where $\rho_{b}=0$ identically since holons have
condensed. At small doping, $\rho_{c}=\rho_{f}$. Note that when holons
condense, an extra term appears for the spinon mean field after substituting
electron operators from eq (14),

$\overline{\mathcal{H}_{spinon}}=\mathcal{H}_{spinon}-\sum_{\langle
ij\rangle,\sigma}t\langle b_{i}\rangle\langle
b_{j}\rangle^{*}f_{i,\sigma}^{\dagger}f_{j,\sigma},$ (15)

which should be small when light doping and we ignored such terms.

For a free fermion system $\sigma_{xy}$ is given by the Kubo formula as

$\sigma_{xy}=\frac{e^{2}}{\hbar}\int_{E_{k}<\epsilon_{f}}\frac{d^{2}k}{4\pi^{2}}2Im[\langle\frac{\partial
u}{\partial k_{1}}|\frac{\partial u}{\partial k_{2}}\rangle],$ (16)

where $u(k_{1},k_{2})$ is the periodic Bloch wavefunction for
$\mathcal{H}_{spinon}$ and the integration is over filled states. See Appendix
D for numerical details on calculating Berry curvatures. Fig. 3 shows the hall
conductivity $\sigma_{xy}$ as one varies hole doping fraction $x$ and hopping
phase $\theta$. It extrapolates to the quantized value of $2e^{2}/h$ at zero
doping. The hall conductivity upon doping electrons, meanwhile, is identical
to that on the hole side of the same doping level $x$.

Near the gapless point $\theta=\pm\pi/6$, one could approximate states near
valence band top as Dirac fermions with a chiral mass, and analytically (see
Appendix D) express the hall conductivity by integrating Berry curvatures of
filled states, i.e. the hall conductivity as a function of
$\Delta=|\pi/6-\theta|,x$ reads

$\sigma_{xy}(x,\Delta)=\frac{4\sqrt{3}\Delta}{\sqrt{12\Delta^{2}+4\sqrt{3}\pi
x}}.$ (17)

Figure 3: The numerical hall conductivity $\sigma_{xy}$ of the chiral metal
upon doping holes at fractions $x$ at selected values $\theta=0,0.2,0.4$,
respectively. The results are expected to be accurate for light doping. The
red line is the analytical expression for $\sigma_{xy}$ (eq (17)) for
$\theta=0.4$, near the gapless point $\theta=\pi/6$. It fits well with the
numerical results at small doping $x<0.1$. $\sigma_{xy}$ in the electron
doping scenario is identical to that of the hole-doping case at same doping
level $x$.

The enlarged unit cell implies a density wave formation in the chiral metal
where states at ${\bf k}$ and ${\bf k}+{\bf G}_{1}$ couple together, where
${\bf G}_{1}=(\pi,0)$ is a reciprocal vector for the reduced Brillouin zone
($2{\bf G}_{1}$ for the original hexagonal Brillouin zone). Fig 4 presents
numerical results on the spectral function $A({\bf k},\omega=0)=2Im[G_{R}({\bf
k},\omega=i0^{+})]$ where $G_{R}$ is the retarded Green’s function, as
measured by angle-resolved photo emission spectroscopy (ARPES). Details of
calculation are shown in appendix E. The duplication of Fermi surfaces
separated by ${\bf G}_{1}$ is a signature of the density wave.

Figure 4: The numerical results for spectral function for the chiral metal at
zero frequency $A({\bf k},\omega=0)=2Im[G_{R}({\bf k},\omega=i0^{+})]$(a.u.),
where the retarded Green’s function $G_{R}({\bf
k},\omega)=\int_{0}^{\infty}\frac{dt}{2\pi}e^{i\omega
t}\langle[c^{\dagger}_{k}(t),c_{k}(0)]\rangle_{GS}$, at $\theta=0$ and doping
level marked at the top. The left(right) panel shows the case for
hole(electron) doping with $p_{\Gamma},1-p_{\Gamma}$ of holons condensing at
$\Gamma,M_{2}$, respectively. The spectral function could be measured as in
angle-resolved photo emission spectroscopy (ARPES), where the absolute momenta
in the original Brillouin zone(black dotted hexagonal) are resolved. The Fermi
surface is duplicated after a translation of ${\bf G_{1}}$(coordinates written
in $\tilde{k}_{x}-\tilde{k}_{y}$ basis) corresponding to the reciprocal vector
of the reduced BZ(white dotted square in the left panel).

In the previous discussion we determine the ansatz of the spinon $f$ and the
wavefunction of the holon $\langle b_{i}\rangle=\mu_{i}$ through mean field
calculation. In practice, one can also apply the variational monte carlo
technique to determine these parameters by studying the following model
wavefunction:

$\Psi(x_{1},...,x_{N})=P_{G}(\prod_{i}\mu_{i}^{1-n_{i}})\text{Slater}[x_{1},...,x_{N}]$
(18)

where Slater is a Slater Determinant according to the mean field ansatz of the
spinon $f$. $P_{G}$ is the usual Gutzwiller projection to forbid the double
occupancy. Here we introduce $\langle b_{i}\rangle=\mu_{i}$ as variational
parameters. In practice one can focus only on a $2\times 2$ unit cell and
hence there are four complex parameters for $\mu_{i}$.

The CSL1 with $\theta=0$ needs special treatment. At this special point, the
spinon ansatz is gauge equivalent to a $d+id$ superconductor. Thus a
translation invariant $d+id$ superconductor is also possible upon doping.
However, there is no reason to expect that a CSL1 state is fine tuned to be at
$\theta=0$. For example, CSL1 found in $J_{1}-J_{2}-J_{\chi}$ modelGong  _et
al._ (2017) is believed to be close to a Dirac spin liquid at $J_{\chi}=0$
line. Thus the spinon ansatz should be a Dirac spin liquid with small chiral
mass, for which $\theta$ should be close to $\frac{\pi}{6}$. For generic
$\theta$, a translation invariant $d+id$ superconductor is not possible in the
picture of holon condensation.

A comparison with doping square lattice CSL is in order: On the square
lattice, there are also two different CSL: CSL1 and CSL2 as on triangular
lattice. But for square lattice, the ansatz of spinons for CSL1 can always be
written in the form of $d+id$ superconductor (without hopping on next nearest
neighbor) (please see Appendix. C). In this case, doping can lead to a
translation invariant $d+id$ or a chiral metal, depending on which one is
energetically favorable. This is different from triangular lattice where only
a special point of CSL1 has spinons in a $d+id$ superconductor ansatz.

##  IV Topological Superconductor from doping the generic Kalmeyer-Laughlin
Spin Liquid CSL1

From the $Z_{2}$ spin liquid, RVB theory predicts a superconducting phase upon
doping. One natural question is: can we also get a superconductor from doping
a chiral spin liquid. For CSL2, this is obvious as the spinons are already in
a BCS state. For the CSL1, the focus here, it is not obvious how to do this
except at the special point $\theta=0$. At a generic point with $\theta\neq
0$, there is no gauge transformation which can make the spinon ansatz into a
translation invariant $d+id$ superconductor. There is a particular gauge in
which spinon are in a translation symmetry breaking superconductor, as shown
in Appendix. B. However, in numerics the $d+id$ superconductor that emerges
from doping a CSL1 is found to be translation invariantJiang and Jiang (2020).
This result therefore cannot be explained by the conventional RVB picture and
requires a new theory. In the following we propose precisely such an exotic
mechanism for obtaining a topological superconductor from doping the Kalmeyer-
Laughlin CSL1.

We first turn to the slave boson theory $c_{i;\sigma}=b_{i}f_{i;\sigma}$ with
the constraint $n_{i}^{b}=n_{i}^{f}$. The slave boson $b$ and the spinon $f$
now have opposite gauge charges. At zero doping, there is an internal flux
$\Phi=\pi$ per unit cell and the effective filling in terms of the magnetic
flux for $b$ and $f$ are $\nu_{b}=-2$ and $\nu_{f}=2$ respectively. For the
CSL, $f$ is in the $C=2$ Chern insulator, while the slave boson $b$ is in a
trivial Mott insulator with one particle per site. When we decrease the
Hubbard $U$ or change the doping, the slave boson $b$ can not be in a Mott
insulator anymore and will delocalize after a Mott transition. The usual story
is that the slave boson will just condense, which in our case leads to a Chern
insulator at zero doping and chiral metal at finite doping. However, as the
slave boson is at filling $\nu_{b}=-2$ in terms of the magnetic flux, there is
another option after the Mott transition: the slave boson can be in a quantum
Hall liquid phase instead of a superfluid phase. The most natural choice is
the bosonic integer quantum Hall (bIQH) state, which has been found
numerically in lattice model with flux $\Phi=\pi$ per unit cellHe  _et al._
(2015). As we show later, the resulting phase turns out to be a topological
superconductor in the same class of $d+id$ superconductor. The easiest way to
see the superconductivity is through the Ioffe-Larkin rule: the resistivity
tensor of electron is $\rho_{c}=\rho_{b}+\rho_{f}$, where $\rho_{b},\rho_{f}$
are the resistivity tensor for $b$ and $f$. As $b$ and $f$ are in quantum Hall
phase with opposite Hall conductivity, $\rho_{c}=\begin{pmatrix}0&-2\\\
2&0\end{pmatrix}+\begin{pmatrix}0&2\\\ -2&0\end{pmatrix}=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}$, which descrbes a superconductor.

A superconductor phase obviously should survive when changing the doping. To
keep it in a superconductor phase, we need both $b$ and $f$ in the quantum
Hall phase even after doping. At finite doping $x$, it turns out that the
internal magnetic flux needs to adjust itself to:

$\Phi(x)=\frac{\langle da\rangle}{2\pi}=\frac{1-x}{2}.$ (19)

In the language of PSG this means that $b$ and $f$ are obeying a magnetic
translation symmetry: $T_{1}T_{2}=T_{2}T_{1}e^{\pm i2\pi\Phi(x)}$, where $\pm$
arises because $b$ and $f$ carry opposite gauge charges. The electron $c=bf$,
however, still satisfies the usual translation symmetry because the phase
factors from $b$ and $f$ cancel each other. We believe that the final electron
phase is translation invariant.

The lock of the internal flux to the additional doping keeps the spinon $f$ in
the $C=2$ Chern insulator phase and $b$ in the bIQH phase according to the
streda formula at arbitrary doping level $x$. The final wavefunction is a
product of two quantum Hall wavefunctions:

$\Psi({\bf x}_{1},...,{\bf x}_{N})=P_{G}\Psi_{bIQH}({\bf x}_{1},...,{\bf
x}_{N})\Psi_{fIQH}({\bf x}_{1},...,{\bf x}_{N})$ (20)

where $\Psi_{bIQH}$ is the wavefunction of the bIQH state at $\nu=-2$ and
$\Psi_{fIQH}$ is the wavefunction of a fermionic IQH state with $\nu=2$.
$P_{G}$ is the usual Gutzwiller projection to forbid double occupancy. A
concrete wavefunction of $\Psi_{bIQH}$ based on a mean field ansatz can be
found in Appendix. F.3.

One can see that this model wavefunction is quite distinct from the usual RVB
wavefunction. In our wavefunction, none of $b$ and $f$ have pairing structure,
but the resulting phase is a superconductor. In the following we study the
topological property of this phase more carefully. The low energy theory is

$L=L_{b,IQH}+L_{f,IQH}$ (21)

We have

$\displaystyle
L_{f,IQH}=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}$
$\displaystyle+\frac{1}{2\pi}ad(\alpha_{1}+\alpha_{2})+\frac{1}{2\pi}\frac{A_{s}}{2}d(\alpha_{1}-\alpha_{2})$
(22)

and

$L_{b,IQH}=\frac{1}{4\pi}\beta_{1}d\beta_{2}+\frac{1}{4\pi}\beta_{2}d\beta_{1}+\frac{1}{2\pi}(A_{c}-a)d(\beta_{1}+\beta_{2})$
(23)

where we have used the fact that the $K$ matrix for bosonic IQH with
$\sigma_{xy}=-2$ is $K=\left(\begin{array}[]{cc}0&-1\\\
-1&0\end{array}\right)$ and charge vector is $q=(1,1)^{T}$. The IQH with
$\sigma_{xy}=2$ for fermion is described by $K=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$ and charge vector $q=(1,1)^{T}$. $\frac{A_{s}}{2}$ is
the spin gauge field. Here we use the convention that $q_{s}=\pm\frac{1}{2}$
for spin up and down. $A_{c}$ is the external EM field and we assign charge to
the slave boson $b$. $a$ is the internal gauge field shared by the slave boson
$b$ and spinon $f_{\sigma}$.

We can simplify the action by integrating out $a$ first, which locks
$\beta_{2}=\alpha_{1}+\alpha_{2}-\beta_{1}$. After substitution, we get

$\displaystyle L$
$\displaystyle=-\frac{2}{4\pi}\beta_{1}d\beta_{1}-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}$
$\displaystyle+\frac{1}{2\pi}A_{c}d(\alpha_{1}+\alpha_{2})+\frac{1}{2\pi}\frac{A_{s}}{2}d(\alpha_{1}-\alpha_{2})+\frac{1}{2\pi}\beta_{1}d(\alpha_{1}+\alpha_{2})$
(24)

Relabeling $\alpha_{c}=\frac{\alpha_{1}+\alpha_{2}}{2}$,
$\alpha_{s}=\frac{\alpha_{1}-\alpha_{2}}{2}$,
$\beta=-\beta_{1}+\frac{\alpha_{1}+\alpha_{2}}{2}$, we can rewrite the action
as

$L=-\frac{2}{4\pi}\beta
d\beta-\frac{2}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s}+\frac{2}{2\pi}A_{c}d\alpha_{c}$
(25)

We need to emphasize that the quantization of the charge is different from the
usual Chern-Simons theory. We have the following dictionary:
$q_{\beta}=-q_{\beta_{1}}$, $q_{c}=q_{1}+q_{2}+q_{\beta_{1}}$ and
$q_{s}=q_{1}-q_{2}$, where $q_{\beta_{1}},q_{1},q_{2}$ are the charges of
$\beta_{1},\alpha_{1},\alpha_{2}$. We require $q_{\beta_{1}},q_{1},q_{2}$ to
be integers to satisfy the usual quantization rule of the Chern-Simons theory.
The excitation is labeled by $l=(q_{\beta},q_{c},q_{s})$.

Given that there is no Chern-Simons term for $\alpha_{c}$, it represents a
gapless charge mode and we have a superconductor. $\alpha_{c}$ represents the
Goldstone mode and its charge $q_{c}$ labels the vortex. As the smallest
$q_{c}=q_{1}+q_{2}+q_{\beta_{1}}$ is $\pm 1$, the elementary charge of the
superconductor is $Q=2\frac{d\alpha_{c}}{2\pi}=2$ and we conclude that it is a
charge $2e$ superconductor.

The other excitations are listed in Table. 1. There are in total four
different anyons $1,e,m,\epsilon$, where $e$ and $m$ are two semions and
$\epsilon=em$ is a fermion. For each excitaion, we can define two charges. The
first one is the spin $S_{z}=\frac{1}{2}q_{s}$. The second one is the vortex
charge $V=q_{c}$. The statistics of the anyon is
$\theta=\frac{\pi}{2}(q_{s}^{2}+q_{\beta}^{2})$. We find that the two semions
$e,m$ always bind with vortex and cost infinite energy. The only finite energy
excitation is the fermion $\epsilon$, which can be identified as a Bogoliubov
fermion. $\epsilon$ has a mutual statistics $\theta_{\epsilon
e}=\theta_{\epsilon m}=\pi$ with $e,m$, consistent with the braiding of
bogolibov fermion around an elementary vortex. The excitations and their
topological properties match that of the $d+id$ superconductor. There is also
a spin quantum Hall conductivity $\frac{1}{2}\frac{1}{4\pi}A_{s}dA_{s}$, in
agreement with the $d+id$ superconductorRead and Green (2000); Senthil  _et
al._ (1999). In summary, we believe the topological property and the symmetry
quantum number of the state defined in Eq. 20 is the same as the $d+id$
superconductor.

| $(q_{\beta_{1}},q_{1},q_{2})$ | $(q_{\beta},q_{c},q_{s})$ | $\theta$ | $S_{z}$ | $V$
---|---|---|---|---|---
$I$ | $(0,0,0)$ | $(0,0,0)$ | $0$ | $0$ | $0$
$e$ | $(0,1,0)$ | $(0,1,1)$ | $\frac{\pi}{2}$ | $\frac{1}{2}$ | $1$
$m$ | $(-1,0,0)$ | $(1,-1,0)$ | $\frac{\pi}{2}$ | $0$ | $-1$
$\epsilon$ | $(-1,1,0)$ | $(1,0,1)$ | $\pi$ | $\frac{1}{2}$ | $0$
Table 1: List of anyons in the topological superconductor. $V$ is the vortex
charge. $S_{z}$ is the spin quantum number and $\theta$ is the self
statistics. We have $V=q_{c}$, $S_{z}=\frac{1}{2}q_{s}$ and
$\theta=\frac{\pi}{2}(q_{\beta}^{2}+q_{s}^{2})$.

We comment on similarity and difference from our construction here and a
previous theory on doped Dirac spin liquid on Kagome latticeKo  _et al._
(2009). Both our proposal and that of Ref. Ko  _et al._ , 2009 needs internal
magnetic flux adjusting to the doping. The superconductor proposed in Ref. Ko
_et al._ , 2009 is not a BCS superconductor and it needs time reversal to be
broken only after doping. In our case, there is already an internal magnetic
flux breaking time reversal in the Mott insulator and the proposed
superconductor is in the same class as a $d+id$ superconductor from BCS
theory. Hence our proposal may be easier to be realized in realistic models.
Actually, a recent numerical study confirms the existence of a $d+id$
superconductorJiang and Jiang (2020) from doping a CSL1, consistent with our
theory.

##  V Deconfined superconductor-insulator transition

In this section we discuss the critical point between the chiral spin liquid
Mott insulator and the chiral metal or the topological superconductor. The
transition can be tuned by either doping $x$ or via bandwidth control by
changing the Hubbard $U$ as shown schematically in Fig 5. At finite doping,
holons may remain in a Mott insulator, i.e., localized by disorder and hence
the CSL phase extends to $x\neq 0$ in fig 5. Usually the bandwidth controlled
or doping controlled Mott transition is described by the condensation of the
holonsSenthil (2008). The new feature in our theory is the possibility of an
unconventional route: the transition into a bIQH insulator for the holons
actually closes the Mott gap and makes the electrons into a topological
superconductor phase.

The different electron phases associate with different phases of holons, as
shown in fig 6. Therefore we can reduce the critical theory to the transition
between different bosonic phases for the slave boson, similar to the theory in
Ref. Senthil, 2008. The transitions marked by black arrows in the fig 6(a)
from a trivial gapped state or an IQH state to a superfluid, are described by
bose condensation. The final transition for the corresponding electron phases
in Fig. 6(b) needs to further include the gauge field. At zero doping, the
chiral metal is really a Chern insulator. In this section we focus on the
superconductor insulator transition marked by the red arrow in Fig. 6.

A key component of this superconductor insulator transition is the plateau
transition for the slave boson. This transition marked by dotted red arrows
from a Mott insulator to bIQH has been studied in Ref. Grover and Vishwanath,
2012; Lu and Lee, 2014; Barkeshli and McGreevy, 2014. Using a parton
construction with $6$ fermionic parton fields (see Appendix F), this
transition from Mott to bIQH state happens when one particular parton
$\psi_{1}$ band’s Chern number changes from $C=1$ to $C=-1$, while other
partons remain in some Chern bands. Hence the transition can be described by
$2$ Dirac cones $\chi_{1,2}$ in the $\psi_{1}$ bands, whose mass sign changes
induce changes in Chern number. We will present the critical theory and
analyze its universal property in this section.

Figure 5: The schematic phase diagram of the system as a function of
interaction strength $U$ and doping level $x$ (positive,negative $x$ denotes
doping electrons, holes, respectively). As one decreases $U$ or dopes the
system, the CSL may become a chiral metal(left) or $d+id$ superconductor
(SC)(right), depending on energetics. For the left figure, at exactly $x=0$
and small $U$, the system transits to a Chern insulator(CI) with Hall
conductivity $\sigma_{xy}=2$.

Figure 6: The possible phases of holons and the corresponding physical phases.
The transition of holons from Mott insulators or boson integer quantum hall
(bIQH) to superfluid is described by Bose condenstion in $(2+1)D$. The direct
transition from Mott insulators to bIQH marked by dotted arrows is studied in
sec V.

###  V.1 Critical action and phase diagram

The critical action reads from Appendix F

$\displaystyle\mathcal{L}_{cri}=\sum_{i=1,2}\bar{\chi}_{i}[\eta_{\mu}(i\partial_{\mu}+A_{c,\mu}+\gamma_{\mu}-a_{\mu})\chi_{i}+m_{i}\bar{\chi}_{i}\chi_{i}$
$\displaystyle+\frac{1}{4\pi}(\beta d\beta-2\gamma
d\beta-\beta_{2}d\beta_{2}+2(A_{c}+\gamma-a)d\beta_{2})$
$\displaystyle-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}$
$\displaystyle+\frac{1}{2\pi}(\frac{1}{2}A_{s}+a)d\alpha_{1}+\frac{1}{2\pi}(-\frac{1}{2}A_{s}+a)d\alpha_{2},$
(26)

where $\eta_{\mu}$’s are Pauli matrices acting on the Dirac spinor indices and
we ignored the Maxwell terms for $A_{c}+\gamma-a$ coupled to the Dirac
fermions. At the critical point, the $\chi_{i}$’s are gapless ($m_{i}=0$). The
masses $m_{1}=m_{2}$ are enforced by the projective translation symmetry of
the fermions. Properly integrating out $\chi_{i}$ when the masses are nonzero
and $\chi_{i}$ filling a band with chern number $\pm 1$, requires a dual field
$\beta_{1}$ and gives $\frac{\mp
1}{4\pi}\beta_{1}d\beta_{1}+\frac{1}{2\pi}\beta_{1}d(A_{c}+\gamma-a)$,
respectively.

When $m_{1}=m_{2}>0$, $\psi_{1}$ fills a band with $C_{1}=1$ and the theory
describes a $d+id$ superconductor. Integrating out fields like
$a,\chi_{i},\gamma$ gives an action (appendix F)

$\displaystyle\mathcal{L}_{d+id}=\frac{1}{4\pi}(-2\alpha_{c}d\alpha_{c}+4\beta_{2}d\alpha_{c}-2\beta_{2}d\beta_{2}+2A_{c}d\alpha_{c})$
$\displaystyle-\frac{2}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s},$
(27)

with $\alpha_{c,s}$ related to $\alpha_{1,2}$ as in section IV. If one
identifies $\tilde{\beta}=\beta_{2}-\alpha_{c}$, the action is identical to
the $d+id$ superconductor action in eq (25).

When $m_{1}=m_{2}<0$,$\psi_{1}$ fills a band with $C_{1}=-1$ and the theory
describes a chiral spin liquid (CSL1). Integrating out $\chi_{i}$ and
$\gamma,\beta_{2}$ gives an action as eq (6)(appendix F), describing the
response of CSL1 with the charge mode gapped.

###  V.2 Universal properties of the critical point

To obtain the transport properties of the critical theory, one integrates out
$\alpha_{c}$ ($\alpha_{c,s}$ as defined in sec IV) in the critical action
(V.1), writing $\Gamma=\gamma-a$, and the critical theory reads,

$\displaystyle\mathcal{L}_{cri}=\sum_{i=1,2}\bar{\chi}_{i}[\eta_{\mu}(i\partial_{\mu}+A_{c,\mu}+\Gamma_{\mu})]\chi_{i}$
$\displaystyle-\frac{1}{4\pi}\Gamma
d\Gamma+\frac{2}{4\pi}A_{c}d\Gamma+\frac{1}{4\pi}A_{c}dA_{c}$
$\displaystyle-\frac{2}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s}.$
(28)

The physical currents $J=\delta L_{cri}/\delta A_{c}$ are

$\displaystyle 2\pi J_{x}=2\pi\mathcal{J}_{x}-E_{y}-e_{y}$ $\displaystyle 2\pi
J_{y}=2\pi\mathcal{J}_{y}+E_{x}+e_{x},$ (29)

where we denote
$\epsilon^{\mu\nu\rho}\partial_{\nu}A_{c,\rho}=(B,-E_{y},E_{x})$. In addition,
we have $\epsilon^{\mu\nu\rho}\partial_{\nu}\Gamma_{\rho}=(b,-e_{y},e_{x})$
and $\mathcal{J}_{\nu}$ as the current for $\chi$. The equations of motion for
$\Gamma$ are

$\displaystyle 2\pi\mathcal{J}_{x}+e_{y}-E_{y}=0,$ $\displaystyle
2\pi\mathcal{J}_{y}-e_{x}+E_{x}=0.$ (30)

Combining the above relations and the assumed universal conductivity for
$\chi$ at the critical point (QED-3),
$\mathcal{J}_{\nu}=\sigma_{\chi}(E_{\nu}+e_{\nu})$, one gets the conductivity
from eq (V.2) as

$\displaystyle
2\pi\sigma_{c}=\left(\begin{array}[]{cc}\frac{4\tilde{\sigma}_{\chi}}{\tilde{\sigma}_{\chi}^{2}+1}&\frac{-2(1-\tilde{\sigma}_{\chi}^{2})}{\tilde{\sigma}_{\chi}^{2}+1}\\\
\frac{2(1-\tilde{\sigma}_{\chi}^{2})}{\tilde{\sigma}_{\chi}^{2}+1}&\frac{4\tilde{\sigma}_{\chi}}{\tilde{\sigma}_{\chi}^{2}+1}\end{array}\right),$
(33)

where $\tilde{\sigma}_{\chi}=2\pi\sigma_{\chi}$.

In summary, the deconfined superconductor-insulator critical point should have
a non-zero universal conductivity shown in Eq. 33. Given that both the
superconductor and the CSL insulator have zero Hall conductivity, a non-zero
Hall effect only at the critical point is quite remarkable. There is also a
spin quantum Hall effect $\frac{1}{8\pi}A_{s}dA_{s}$, which remains constant
across the transition.

##  VI Conclusions and Outlook

In summary, we propose two different phases from doping a $U(1)_{2}$ chiral
spin liquid on triangular lattice. One possibility is a chiral metal from
simple slave boson condensation picture. Another more exotic possibility is a
topological superconductor obtained by putting both the slave boson and the
spinon in a quantum Hall phase. We argue that these two scenarios are
consistent with two recent numerical studies of doped CSL in two different
parameter regimes. Our theory may be relevant to TMD heterobilayer where a
spin 1/2 Hubbard model can be simulated on a triangular moiré superlatticeWu
_et al._ (2018); Tang  _et al._ (2019); Regan  _et al._ (2020); Wang  _et al._
(2019). It has also been proposed that a generalized CSL with different
topological order is possible in a $SU(4)$ model on moiré bilayerZhang and
Vishwanath (2020). In future it is interesting to extend our theory of doped
CSL to the case with $SU(N)$ spin.

From a quantum criticality perspective, we propose an unconventional
superconductor to Mott insulator transition, as described by a plateau
transition of the charged holons. If we start from a $d+id$ superconductor,
there are two different Mott transitions towards two different chiral spin
liquid insulators. We also find a deconfined critical point between a
translation symmetry breaking Chern insulator and a translation invariant
topological superconductor. We hope to study this kind of deconfined critical
points in more detail in the future.

##  VII Acknowledgement

We would like to thank Ehud Altman, Olexei Motrunich and Zheng Zhu for useful
discussions. We acknowledge funding from a Simons Investigator award (AV) and
the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the
Simons Foundation (651440, AV). This research is funded in part by the Gordon
and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8683 to AV.

##  Appendix A $SU(2)$ and $U(1)$ slave boson formalism

We remark that a full treatment of the parton representation for electrons
have an $SU(2)$ gauge symmetry, which is given by two species of spin-$0$
holons $h_{i}=(b_{1,i},b_{2,i})^{T}$ and spinons
$\psi_{i}=(\psi_{i,\uparrow},\psi_{i,\downarrow}^{\dagger})^{T}$ to write the
electron operators as

$\displaystyle
c_{i,\uparrow}=\frac{1}{\sqrt{2}}h_{i}^{\dagger}\psi_{i}=\frac{1}{\sqrt{2}}(b_{1,i}^{\dagger}f_{i,\uparrow}+b_{2,i}^{\dagger}f_{i,\downarrow}^{\dagger}),$
$\displaystyle
c_{i,\downarrow}=\frac{1}{\sqrt{2}}h_{i}^{\dagger}(i\tau^{y}\psi_{i}^{*})=\frac{1}{\sqrt{2}}(b_{1,i}^{\dagger}f_{i,\downarrow}-b_{2,i}^{\dagger}f_{i,\uparrow}^{\dagger}),$
(34)

where both $\psi_{i}$ and $\bar{\psi}_{i}=i\tau^{y}\psi_{i}^{*}$ transform as
doublets for $SU(2)$ group: $\psi_{i}\rightarrow U\psi_{i},(U\in SU(2))$ and
$\bar{\psi}_{i}\rightarrow U\bar{\psi}_{i},(U\in SU(2))$. The physical
electrons stay invariant under such $SU(2)$ transform given the holon doublet
transform accordingly as $h_{i}\rightarrow Uh_{i}$. The gauge-invariant
condition for physical Hilbert space (excluding double occupancy) constrain
each site to contain an $SU(2)$ singlet, i.e.

$h_{i}^{\dagger}{\mathbf{\tau}}h_{i}+\psi_{i}^{\dagger}{\mathbf{\tau}}\psi_{i}|\Psi_{phys}\rangle=0,$
(35)

which for the $\tau^{z}$ component reads

$b_{1,i}^{\dagger}b_{1,i}-b_{2,i}^{\dagger}b_{2,i}+\sum_{s}f_{i,s}^{\dagger}f_{i,s}=1.$
(36)

In the main text parton construction, we effectively fix the gauge for the
holon sector as $b_{i,2}=0$ identically and leave out one holon $b_{i,1}$ as
$b$ in the main text. This is justified since given the $SU(2)$ gauge group,
one could rotate a state to one with $b_{i,2}=0$ without changing the physical
state. The price is a reduced gauge group $U(1)$ away from half-filling, and
that upon transforming the mean field for spinons with $SU(2)$ rotations, the
state changes physically upon doping. For instance, in eq (II.3) one
transforms the spinon ansatz from a $U(1)_{2}$ CSL1 to $d+id$ (CSL2),
equivalent at half-filling, and yet upon doping holes, the state becomes a
chiral metal and $d+id$ superconductor after condensing $b_{1}$, respectively.
Conversely, one could fix the gauge for spinons and then different boson
condensation leads to different physical states. In the paper we adopt the
first approach, allowing only $b_{1}$ to condense, and hence different spinon
ansatz, equivalent at half-filling, give different physical states upon
doping. Therefore at $\theta=0$ on triangular lattice CSLs, upon doping,
either a $d+id$ or a chiral metal could emerge upon condensing different holon
states. This is also true on square lattice (see appendix C), where a $d+id$
ansatz with zero diagonal hopping is equivalent to a chiral spin liquid (DSL
with a chiral mass) through an $SU(2)$ transform. Hence on square lattice
doping a CSL gives either $d+id$ superconductor or a chiral metal.

##  Appendix B Competing translation breaking pairing state on doping CSL1

At noted in the main text, at $\theta=0$, the $U(1)_{2}$ CSL1 is equivalent to
a projected $d+id$ spin liquid on triangular lattice. Hence upon doping the
$U(1)_{2}$ CSL1 at $\theta=0$, the resulting states are a competition between
$d+id$ and chiral metal in sec III. When $\theta$ shifts from zero, however,
one could still perform the gauge transform eq (II.3) on $U(1)_{2}$ CSL1
towards a $d+id$ pairing ansatz, which could be the competing pairing state
with chiral metal upon doping the CSL1. Next we analyze the ansatz for such
pairing state.

The CSL1 ansatz eq (II.1) becomes $\tilde{u}_{ij}=g_{j}^{\dagger}u_{i,j}g_{i}$
after the transform,

$\displaystyle\tilde{u}_{r,r+{\widehat{x}}}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cc}e^{i\theta}&\sqrt{2}e^{-i\theta}\\\
\sqrt{2}e^{i\theta}&-e^{-i\theta}\end{array}\right),(r_{x}\mod 2=0)$ (39)
$\displaystyle\tilde{u}_{r,r+{\widehat{x}}}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cc}e^{i\theta}&\sqrt{2}e^{i\theta}\\\
\sqrt{2}e^{-i\theta}&-e^{-i\theta}\end{array}\right),(r_{x}\mod 2=1)$ (42)
$\displaystyle\tilde{u}_{r,r+{\widehat{y}}}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cc}e^{i\theta}&\sqrt{2}e^{-i\theta-i\frac{2\pi}{3}}\\\
\sqrt{2}e^{i\theta+i\frac{2\pi}{3}}&-e^{-i\theta}\end{array}\right),(r_{x}\mod
2=0)$ (45)
$\displaystyle\tilde{u}_{r,r+{\widehat{y}}}=\frac{1}{3}\left(\begin{array}[]{cc}-i(e^{i\theta}+2e^{-i\theta+i\frac{2\pi}{3}})&\sqrt{2}i(e^{-i\theta+i\frac{2\pi}{3}}-e^{i\theta})\\\
\sqrt{2}i(-e^{i\theta-i\frac{2\pi}{3}}+e^{-i\theta})&-i(e^{-i\theta}-2e^{i\theta-i\frac{2\pi}{3}})\end{array}\right),(r_{x}\mod
2=1)$ (48)
$\displaystyle\tilde{u}_{r,r+{\widehat{y}}+{\widehat{x}}}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{cc}e^{-i\theta}&\sqrt{2}e^{i\theta+i\frac{2\pi}{3}}\\\
\sqrt{2}e^{-i\theta-i\frac{2\pi}{3}}&-e^{i\theta}\end{array}\right),(r_{x}\mod
2=0)$ (51)
$\displaystyle\tilde{u}_{r,r+{\widehat{y}}+{\widehat{x}}}=\frac{1}{3}\left(\begin{array}[]{cc}i(e^{-i\theta}+2e^{i\theta-i\frac{2\pi}{3}})&\sqrt{2}i(-e^{i\theta-i\frac{2\pi}{3}}+e^{-i\theta})\\\
\sqrt{2}i(e^{-i\theta+i\frac{2\pi}{3}}-e^{i\theta})&i(e^{i\theta}+2e^{-i\theta+i\frac{2\pi}{3}})\end{array}\right).(r_{x}\mod
2=1)$ (54)

One sees that at $\theta=0$, the above mean-field ansatz is identical to
$d+id$ states in eq (II.2) at $\eta=\sqrt{2}\chi$, while $\theta\neq 0$, the
pairing has $d+id$ symmetry yet the state breaks lattice translation, distinct
from the conventional translation invariant $d+id$ states.

##  Appendix C Chiral spin liquids on square lattice

For square lattice, under an appropriate gauge choice, CSL mean-field ansatz
in the spinor basis $\psi$ reads,

$\displaystyle
u_{r,r+{\widehat{x}}}=\cos(\theta)\tau^{z}+\sin(\theta)\tau^{y},$
$\displaystyle
u_{r,r+{\widehat{y}}}=\cos(\theta)\tau^{z}-\sin(\theta)\tau^{y},$
$\displaystyle
u_{r,r+{\widehat{x}}+{\widehat{y}}}=\chi\cos(\theta)\tau^{z}+\eta\sin(\theta)\tau^{x},$
$\displaystyle
u_{r,r-{\widehat{x}}+{\widehat{y}}}=\chi\cos(\theta)\tau^{z}-\eta\sin(\theta)\tau^{x},(\chi,\eta\in
R)$ (55)

where the IGG is reduced to $Z_{2}$ when $\theta\neq n\pi/4,(n\in Z)$ and
$\chi\neq 0$. This can be seen by calculating Wilson loop $\Phi$ around a
square or triangular loop on the square lattice,

$\displaystyle\Phi_{\square}=u_{r,r+{\widehat{x}}}u_{r,r+{\widehat{y}}}u_{r,r-{\widehat{x}}}u_{r,r-{\widehat{y}}},$
$\displaystyle=\cos 4\theta-i\sin 4\theta\tau^{x},$
$\displaystyle\Phi_{\triangle}=u_{r,r+{\widehat{x}}}u_{r,r+{\widehat{y}}}u_{r,r-{\widehat{x}}-{\widehat{y}}}=\chi(\cos
2\theta\cos\theta\tau^{z}$ $\displaystyle+\sin
2\theta\cos\theta\tau^{y})+\eta(\cos 2\theta\sin\theta\tau^{x}-i\sin
2\theta\sin\theta).$ (56)

We note that the $d+id$ ansatz contain two different chiral spin liquids: (I)
When $\theta=n\frac{\pi}{4}$ or $\chi=0$ with arbitrary $\theta$, the IGG is
$U(1)$ and we get a CSL1 described by the $U(1)_{2}$ theory; (II) For a
generic ansatz with diagonal hopping, the IGG is $Z_{2}$ and we have a CSL2
phase in the $\nu=4$ of the Kitaev’s 16 fold way.

We note that $d+id$ ansatz _contains_ all square $U(1)$ DSL with a chiral mass
states (CSL1). The ansatz for $U(1)$ CSL1 is a staggered flux one with
diagonal hopping,

$\displaystyle
u_{r,r+{\widehat{x}}}=\cos(\theta)\tau^{z}+i(-1)^{r_{x}+r_{y}}\sin(\theta)\tau^{0},$
$\displaystyle
u_{r,r+{\widehat{y}}}=\cos(\theta)\tau^{z}-i(-1)^{r_{x}+r_{y}}\sin(\theta)\tau^{0},$
$\displaystyle
u_{r,r+{\widehat{x}}+{\widehat{y}}}=(-1)^{r_{x}+r_{y}}\eta\sin(\theta)\tau^{z},$
$\displaystyle
u_{r,r-{\widehat{x}}+{\widehat{y}}}=-(-1)^{r_{x}+r_{y}}\eta\sin(\theta)\tau^{z},(\eta\in
R)$ (57)

where IGG is $U(1)$ generated by $\tau^{z}$. By a gauge transform, such ansatz
is equivalent to $d+id$ with $\chi=0$ in eq (C). The gauge transform reads,

$\displaystyle\psi_{r}\rightarrow\frac{1}{\sqrt{2}}(1+(-1)^{r_{x}+r_{y}}i\tau^{y})\psi_{r}.$
(58)

Hence a generic CSL1 phase on square lattice can be written in the $d+id$
ansatz. After doping a CSL1, we can get a translation invariant $d+id$
superconductor from the simple holon condensation picture, in contrast to
triangular lattice.

##  Appendix D Numerical procedure for Berry curvatures and Dirac fermions
near $\theta=\pi/6$

We present some numerical details for Berry curvatures and analytical
derivation of Dirac Hamiltonian near $\theta=\pi/6$ gapless point.

Berry connections for Bloch wavefunctions $u({\bf k})$ for the electrons can
be approximated numerically as $A_{\epsilon}d{\bf\epsilon}=Arg\langle u({\bf
k}+{\bf\epsilon})|u({\bf k})\rangle$ where ${\bf\epsilon}$ is a small vector
increment in momentum space and Arg gives the phase of the complex number. We
mesh the $kx-ky$ plane with increments
${\bf\epsilon}_{1}=(\delta,0)$,${\bf\epsilon}_{2}=(0,\delta)$, where
$k_{x},k_{y}$ are measured by the reciprocal vectors for basis vector
${\widehat{x}},{\widehat{y}}$ in the main text, shown in fig 7 (b) and
$\delta$ at the order of $0.01$ radian. The total Berry curvature on one
elementary plaquette of the momentum mesh is given by (shown in Fig 7 (a))

$\displaystyle\mathcal{F}({\bf k})d^{2}k=Arg\langle u({\bf
k}+{\bf\epsilon}_{1})|u({\bf k})\rangle$ $\displaystyle+Arg\langle u({\bf
k}+{\bf\epsilon}_{1}+{\bf\epsilon}_{2})|u({\bf k}+{\bf\epsilon}_{1})\rangle$
$\displaystyle+Arg\langle u({\bf k}+{\bf\epsilon}_{2})|u({\bf
k}+{\bf\epsilon}_{1}+{\bf\epsilon}_{2})\rangle$ $\displaystyle+Arg\langle
u({\bf k})|u({\bf k}+{\bf\epsilon}_{2})\rangle.$ (59)

Note that for bands with nonzero Chern number, Berry curvature on some
plaquettes in k-space may have discontinuity of $\pm 2\pi$, since total Berry
curvature of one band in the above discretized formula is always zero. One
manually makes up for the $2\pi$ discontinuities and then sum the Berry
curvature totals $\mathcal{F}({\bf k})d^{2}k$ over filled ${\bf k}$ states for
hall conductivity of the spinons.

Figure 7: (a)The numerical procedure to calculate Berry curvature of Bloch
wavefunction in k-space. One circles around an elementary plaquette (dashed
diamond) and adds up the phase of overlap of wavefunctions at the ends of each
bond. (b) The reciprocal vectors for ${\widehat{x}},{\widehat{y}}$ lattice
vectors used in main text and the orthogonal coordinates used for Dirac
hamiltonian eq (61) $\tilde{k}_{x},\tilde{k}_{y}$.

At $\theta$ close to $\pm\pi/6$, in the _spinon_ Hamiltonian, Dirac cones with
a small mass emerge near half-filling.

We derive the Dirac hamiltonian near the gapless point. The spinon hamiltonian
eq (II.1) in k-space reads

$\displaystyle\mathcal{H}_{spinon}=\sum_{k}\psi_{k}^{\dagger}(-2\cos(\theta+k_{y})\tau^{z}$
$\displaystyle-2\cos(\theta+k_{x})\tau^{x}+2\cos(\theta-
k_{x}-k_{y})\tau^{y})\psi_{k},$ (60)

where $\psi_{k}=(f_{\uparrow,k},f_{\downarrow,-k}^{\dagger})^{T}$. The k-space
hamiltonian can be Taylor expanded near the gapless point, e.g.
$\theta=\pi/6,(k_{x},k_{y})=(\pi/3,\pi/3)$ and we transform to orthogonal
momentum coordinates shown in fig 7(b)
$\tilde{k}_{x}=\frac{\sqrt{3}}{2}k_{x},\tilde{k}_{y}=k_{y}+k_{x}/2$,

$\displaystyle\mathcal{H}_{Dirac}$
$\displaystyle=2[(\frac{2}{\sqrt{3}}\tau^{x}-\frac{1}{\sqrt{3}}\tau^{z}-\frac{1}{\sqrt{3}}\tau^{y})\tilde{k}_{x}+(\tau^{z}-\tau^{y})\tilde{k}_{y}$
(61) $\displaystyle+(\theta-\frac{\pi}{6})(\tau^{z}+\tau^{x}+\tau^{y})],$

with the matrices multiplying
$\tilde{k}_{x},\tilde{k}_{y},\sqrt{\frac{3}{2}}(\theta-\pi/6)$ obeying the
anti-commutation relations of gamma matrices for Dirac hamiltonian in $(2+1)D$
up to an overall factor $2\sqrt{2}$. Hence indeed the band can be approximated
by a Dirac fermion with a small mass. Another Dirac cone appears at
$(\pi/3,-2\pi/3)$ in k-space with the same chirality. Note the velocity is
uniform, $v=2\sqrt{2}$ and the mass $m=2\sqrt{3}(\theta-\pi/6)$. The Berry
curvature of the two-component spinor system has been well studied:
representing the components of the Dirac hamiltonian
$H_{Dirac}=vk_{x}\gamma^{1}+vk_{y}\gamma^{2}+m\gamma^{0}$ in a unit vector
${\widehat{n}}({\bf
k})=\frac{(vk_{x},vk_{y},m)}{\sqrt{v^{2}(k_{x}^{2}+k_{y}^{2})+m^{2}}}$, the
Berry curvature reads

$\mathcal{F}({\bf k})=\frac{{\widehat{n}}\cdot d_{k_{x}}{\widehat{n}}\times
d_{k_{y}}{\widehat{n}}}{2}.$ (62)

Pictorially the Berry phase accumulated is given by $1/2$ times the solid
angle that ${\widehat{n}}({\bf k})$ sweeps through during an adiabatic
process.

To express the hall conductivity as one varies electron doping $x$, one first
obtains the density of states near the Dirac cone as

$\displaystyle\rho(E)=\frac{dN}{dE}=\frac{2\pi
kdk}{vdk\frac{4\pi^{2}\sin\frac{\pi}{3}}{L_{x}L_{y}}},$ (63)

where we used the area a state occupied in k-space for a triangular system
with size $L_{x},L_{y}$ as $\frac{4\pi^{2}\sin\frac{\pi}{3}}{L_{x}L_{y}}$ and
$k$ as the norm of ${\bf k}$ since the dispersion is uniform around Dirac
cones. At doping level of $x$, the states in a circle around Dirac point with
radius $k_{c}$ in valence band are emptied, $k_{c}$ given by

$\displaystyle\int_{0}^{k_{c}}dE\rho(E)=L_{x}L_{y}\frac{x}{4},$ $\displaystyle
k_{c}^{2}=\pi x\sin\frac{\pi}{3},$ (64)

where the factor $1/4$ comes from the $4$ Dirac cones in $2$ spinon conduction
bands.

The Berry curvature accumulated from states in a circle of radius $k_{c}$ is
given by the solid angle swept by ${\widehat{n}}$ from
$|k|=0,{\widehat{n}}=(0,0,1)$ to $|{\bf k}|=k_{c},{\widehat{n}}({\bf
k})=\frac{(vk_{x},vk_{y},m)}{\sqrt{v^{2}k_{c}^{2}+m^{2}}}$, hence

$\int_{|k|<k_{c}}d^{2}k\mathcal{F}({\bf
k})=\frac{2\pi}{2}(1-\frac{m}{\sqrt{m^{2}+v^{2}k_{c}^{2}}}),$ (65)

substituting $k_{c}$ from eq(D) and $m,v$ from previous analysis, one gets the
hall conductivity as the total Chern number from two spinon valence bands
$C=2$, minus the contributions from states around Dirac cones as

$\sigma_{xy}=\frac{2e^{2}}{h}-\frac{4\pi
e^{2}}{4\pi^{2}\hbar}(1-\frac{2\sqrt{3}\Delta}{\sqrt{12\Delta^{2}+4\sqrt{3}\pi
x}}),$ (66)

where $\Delta=|\pi/6-\theta|$, which is the expression for $\sigma_{xy}$ as in
eq(17).

Figure 8: The relative current strength $I_{bond\langle
ij\rangle}=Im[c^{\dagger}_{i}c_{j}]$ of the chiral metal upon infinitesimal
doping of holes shown as red numbers on the bond ($\theta$ as shown in upper
left of each panel) with arrows indicating current directions. We assume the
holons condense fully at $\Gamma$. The currents for infinitesimal electron
doping are negative of those in hole doped case, and with a shift of the $y$
direction bonds.

##  Appendix E Calculation of spectral function $A(k,\omega=0^{+})$

We define a plane-wave basis for electrons as

$\displaystyle c_{k}^{\dagger}=\frac{1}{L}\sum_{r}e^{ik\cdot
r}c_{i}^{\dagger},$ $\displaystyle c_{k,A}^{\dagger}=\frac{1}{L}\sum_{r\in
A}e^{ik\cdot r}c_{i}^{\dagger},$ $\displaystyle
c_{k,B}^{\dagger}=\frac{1}{L}\sum_{r\in B}e^{ik\cdot r}c_{i}^{\dagger}.$ (67)

The first one is used in defining the spectral function as one measures in
arpes. The latter two correspond to plane waves on sublattices $A,B$,
respectively. Taking the origin at $A$ site, these operators have the
relation,

$\displaystyle c_{k}^{\dagger}=c_{k,A}^{\dagger}+c_{k,B}^{\dagger},$
$\displaystyle c_{k}^{\dagger}=c_{k+{\bf G}_{1},A}^{\dagger}-c_{k+{\bf
G}_{1},B}^{\dagger},$ (68)

where we used $e^{i{\bf G}_{1}\cdot r_{AB}}=-1$, where $r_{AB}$ is the vector
connecting sites on two sublattices.

The spectral function measured in the ground state at frequency $\omega=0^{+}$
is contributed by eigenstates
$u_{k}^{\dagger}=\alpha_{k}c_{k,A}^{\dagger}+\beta_{k}c_{k,B}^{\dagger}$ on
the Fermi pockets. The coefficients are related to those of the spinon
eigenstates by holon condensed values,i.e.

$\displaystyle\alpha_{k}=\sum_{k^{\prime}}u_{b,k^{\prime}}(A)\alpha_{k-k^{\prime},f}$
$\displaystyle\beta_{k}=\sum_{k^{\prime}}u_{b,k^{\prime}}(B)\beta_{k-k^{\prime},f},$
(69)

where $\alpha_{k,f},\beta_{k,f}$ are the coefficients for sublattice $A,B$
wavefunctions for spinon eigenstates and $u_{b,k}(A,B)$ are the holon
condensation discussed in section III eq (11), non-vanishing only for
$k=\Gamma,M_{2}$. Note that spinon states at $k,k+M_{2}$ have the same energy
due to the projective symmetry $T_{x}$ with a momentum boost of $(0,\pi)$, i.e
$M_{2}$ momentum.

This overlap of the plane wave basis for electrons reads

$\displaystyle\langle
u_{k^{\prime}}|c_{k}^{\dagger}|0\rangle=\delta_{k,k^{\prime}}(\alpha_{k}+\beta_{k})+\delta_{k,k^{\prime}\pm{\bf
G}_{1}}(\alpha_{k^{\prime}}-\beta_{k^{\prime}}).$ (70)

The spectral function is given by

$\displaystyle A({\bf k},\omega=0^{+})=2\pi\int_{k^{\prime}\in
FS}dk^{\prime}|\langle u_{k^{\prime}}|c_{k}^{\dagger}|0\rangle|^{2}.$ (71)

Since the original BZ measured in arpes is double the size of the reduced one,
when measured momenta $k$ lie outside of the reduced BZ, one effectively
excites states inside the reduced BZ at its equivalent momentum (i.e.,
obtained by a translation of a reciprocal lattice vector ${\bf G}_{1}$).

Using eq (70) to express the spectral function, one finds a simple expression

$\displaystyle A({\bf
k},\omega=0^{+})=\begin{cases}2\pi|\alpha_{k}+\beta_{k}|^{2}&{\bf
k}\in\textrm{reduced BZ}\\\ 2\pi|\alpha_{ko}-\beta_{ko}|^{2}&{\bf ko}={\bf
k}\pm{\bf G}_{1}\in\textrm{reduced BZ}.\end{cases}$ (72)

where ${\bf ko}$ is the equivalent momenta for ${\bf k}$ in the reduced BZ.

##  Appendix F A concrete mean field theory of the topological superconductor
and deconfined transition

Here we try to construct a concrete mean field theory of the topological
superconductor and writing down a mean field ansatz for it.

To simplify the construction of the bIQHE state, we introduce two slave
bosons, each of which has filling $1/2$ per site. We will hybridize them in
the last stage so the final theory should be smoothly connect to that of one
slave boson. For the discussion of Mott transition at fixed density $n_{c}=1$,
we use $U(1)$ slave boson theory, instead of the $SU(2)$ version. The cost is
that we can not keep track of the $C_{6}$ symmetry. But we can still keep
track of translations $T_{x},T_{y}$. We will show explicitly that our proposed
ansatz is invariant under $T_{x}$ and $T_{y}$.

The modified slave boson theory is:

$\displaystyle c_{i;\uparrow}$ $\displaystyle=b_{i}f_{i;\uparrow}$ (73)
$\displaystyle c_{i;\downarrow}$ $\displaystyle=\tilde{b}_{i}f_{i;\downarrow}$
(74)

This will introduce two independent $U(1)$ gauge field: $a$ and $\tilde{a}$.
But we will lock them together with a Higgs term
$b_{i}^{\dagger}\tilde{b}_{i}+h.c.$. Note the above parton construction does
not satisfy the $SU(2)$ spin rotation symmetry. We can only keep track of the
$S_{z}$ quantum number. As a result, the state constructed in this way may
break the $SU(2)$ spin rotation down to $U(1)$.

Now the constraint is that:

$\displaystyle n_{i;b}$ $\displaystyle=n_{i;f_{\uparrow}}$ $\displaystyle
n_{i;\tilde{b}}$ $\displaystyle=n_{i;f_{\downarrow}}$ (75)

So on average we have $n_{b}=n_{\tilde{b}}=\frac{1}{2}$ per site. To describe
the bIQHE phase, we do a further fractionalization:

$\displaystyle b_{i}$ $\displaystyle=d_{i}\psi_{i;1}$
$\displaystyle\tilde{b}_{i}$ $\displaystyle=\tilde{d}_{i}\psi_{i;2}$

which introduces another two $U(1)$ gauge field $\gamma,\tilde{\gamma}$.

In total, the electron is fractionalized in the following way:

$\displaystyle c_{i;\uparrow}$ $\displaystyle=d_{i}\psi_{i;1}f_{i;\uparrow}$
$\displaystyle c_{i;\downarrow}$
$\displaystyle=\tilde{d}_{i}\psi_{i;2}f_{i;\downarrow}$ (77)

We assign the charge in the following way: $\psi_{1}$ couples to
$-a+A_{c}+\gamma$, $\psi_{2}$ couples to $-\tilde{a}+A_{c}+\tilde{\gamma}$.
$d$ couples to $-\gamma$ and $\tilde{d}$ couples to $-\tilde{\gamma}$.
$f_{\uparrow}$ couples to $\frac{1}{2}A_{s}+a$ and $f_{\downarrow}$ couples to
$-\frac{1}{2}A_{s}+\tilde{a}$.

We consider the follwing mean field ansatz:

$\displaystyle
H=H_{f}+H_{d}+H_{\psi_{1}}+H_{\psi_{2}}-\lambda\sum_{i}(\psi^{\dagger}_{i;1}\psi_{i;2}+h.c.)$
(78)

First, $H_{f}$ is just the mean field ansatz of the fermionic spinons.
$d,\tilde{d}$ hybridize together and have a mean field ansatz with $0$ flux
per unit cell ($T_{x}T_{y}=T_{y}T_{x}$.). Their hybridization locks
$\gamma=\tilde{\gamma}$, so we will only keep $\gamma$ in the following
analysis. We want the PSG of $d$ to be completely trivial. The non-trivial PSG
of $b$ is inherited completely by $\psi$. Because there are two orbitals
formed by $d,\tilde{d}$, there can be a Chern insulator with $C_{d}=1$ even if
the ansatz does not break translation symmetry. We show a generic mean-field
for $d,\tilde{d}$ with such properties:

$\displaystyle H_{d}=\sum_{\langle
ij\rangle}[d_{i}^{\dagger}d_{j}-\tilde{d}_{i}^{\dagger}\tilde{d}_{j}]+\sum_{r}p_{1}d_{r}^{\dagger}(\tilde{d}_{r+{\widehat{x}}}+e^{i\frac{2\pi}{3}}\tilde{d}_{r+{\widehat{y}}}+e^{-i\frac{2\pi}{3}}\tilde{d}_{r-{\widehat{x}}-{\widehat{y}}})$
$\displaystyle+\sum_{r}p_{2}d_{r}^{\dagger}(\tilde{d}_{r-{\widehat{x}}}+e^{i\frac{2\pi}{3}}\tilde{d}_{r-{\widehat{y}}}+e^{-i\frac{2\pi}{3}}\tilde{d}_{r+{\widehat{x}}+{\widehat{y}}})+\textrm{h.c.},$
(79)

which preserves translation and the mixed hopping between $d,\tilde{d}$ has
angular momentum $2$. This mean-field dispersion is generically gapped except
with particular parameters e.g. $p_{1}p_{2}=0$ and may have non-vanishing
Chern number, e.g. when $p_{1}=p_{2}=1+i$, the valence band has chern number
$C_{d}=1$.

$\psi_{1}$ and $\psi_{2}$ have mean field ansatz with
$T_{x}T_{y}=-T_{y}T_{x}$, which should be similar to that of spinon $f$
introduced in the main text. They have Chern number $C_{1}=\pm 1$ and
$C_{2}=1$. The mean-field we use for $\psi$ is those for the CSL1 in eq (II.1)
with $\theta=\pi/6\pm\delta,(\delta>0)$ for Chern number $C=\mp 1$,
respectively.

In the following we analyze the cases with $\lambda=0$ and $\lambda\neq 0$
separately. The topological superconductor discussed in the main text
corresponds to $\lambda\neq 0$.

###  F.1 $\lambda=0$: $p+ip$ superconductor

We deal with the special case of $\lambda=0$ first. In this case, we need to
deal with two $U(1)$ gauge field $a$ and $\tilde{a}$, which couples to
$f_{\uparrow},b$ and $f_{\downarrow},\tilde{b}$ respectively. First,
integration of the spinon $f_{\sigma}$ gives:

$L_{f}=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}(\frac{1}{2}A_{s}+a)d\alpha_{1}+\frac{1}{2\pi}(-\frac{1}{2}A_{s}+\tilde{a})d\alpha_{2}$
(80)

where $\alpha_{1},\alpha_{2}$ are introduced to describe the IQHE of the
fermionic spinons.

Next we consider the ansatz with $C_{d}=-1,C_{1}=C_{2}=1$. The slave boson
part has the action

$L_{b}=\frac{1}{4\pi}\beta d\beta-\frac{1}{2\pi}\gamma
d\beta-\frac{1}{4\pi}\beta_{1}d\beta_{1}-\frac{1}{4\pi}\beta_{2}d\beta_{2}+\frac{1}{2\pi}(A_{c}+\gamma-a)d\beta_{1}+\frac{1}{2\pi}(A_{c}+\gamma-\tilde{a})d\beta_{2}$
(81)

Integration of $\gamma$ locks $\beta=\beta_{1}+\beta_{2}$, then

$L_{b}=\frac{1}{4\pi}\beta_{1}d\beta_{2}+\frac{1}{4\pi}\beta_{2}d\beta_{1}+\frac{1}{2\pi}(A_{c}-a)d\beta_{1}+\frac{1}{2\pi}(A_{c}-\tilde{a})d\beta_{2}$
(82)

The final action is $L_{c}=L_{b}+L_{f}$. By integrating $a$ and $\tilde{a}$,
we can lock $\beta_{1}=\alpha_{1}$ and $\beta_{2}=\alpha_{2}$. We can then
reach the final action:

$L_{c}=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{4\pi}\alpha_{1}d\alpha_{2}+\frac{1}{4\pi}\alpha_{2}d\alpha_{1}+\frac{1}{2\pi}(A_{c}+\frac{1}{2}A_{s})d\alpha_{1}+\frac{1}{2\pi}(A_{c}-\frac{1}{2}A_{s})d\alpha_{2}$
(83)

By relabeling $\alpha_{1}=\alpha_{c}+\alpha_{s}$ and
$\alpha_{2}=\alpha_{c}-\alpha_{s}$, we get

$\displaystyle
L_{c}=\frac{2}{2\pi}A_{c}d\alpha_{c}-\frac{4}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s}$
(84)

The above action is equivalent to the theory of $p+ip$ superconductor, which
is in the $\nu=2$ class of Kitaev’s sixteen fold way classificationKitaev
(2006). $p+ip$ pairing is possible in the spin-triplet channel of
$c_{\uparrow}c_{\downarrow}$. This is also an indication that the full $SU(2)$
spin rotation symmetry is broken in this state. .

###  F.2 $\lambda\neq 0$: $d+id$ superconductor

We will show that the case with $\lambda\neq 0$ gives a superconductor with
the same topological property as the spin singlet $d+id$ superconductor.

With $\lambda\neq 0$, the term $-\lambda\psi^{\dagger}_{i;1}\psi_{i;2}+h.c.$
locks $a=\tilde{a}$ (Note that we already lock $\gamma=\tilde{\gamma}$ by the
hybridization between $d$ and $\tilde{d}$.). After that, the action for the
slave boson part should be modified as

$L_{b}=\frac{1}{4\pi}\beta_{1}d\beta_{2}+\frac{1}{4\pi}\beta_{2}d\beta_{1}+\frac{1}{2\pi}(A_{c}-a)d\beta_{1}+\frac{1}{2\pi}(A_{c}-a)d\beta_{2}$
(85)

which is exactly the action in Eq.(19) of the main text.

$L_{f}$ is also modified to

$L_{f}=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}(\frac{1}{2}A_{s}+a)d\alpha_{1}+\frac{1}{2\pi}(-\frac{1}{2}A_{s}+a)d\alpha_{2}$
(86)

which is the same as Eq.(18) in the main text.

Then the analysis in the main text shows that this state is topologically
equivalent to $d+id$ superconductor.

Finally, let us comment on the spin rotation symmetry. Our formalization only
explicitly preserve $U(1)$ spin rotation corresponding to $S_{z}$ because we
have two different slave bosons $b,\tilde{b}$ for the two spins. When
$\lambda=0$, there are two $U(1)$ gauge fields $a,\tilde{a}$. This strongly
breaks spin rotation symmetry, and thus we get a state consistent with a spin-
triplet $p+ip$ superconductor. When we increase $\lambda$, we couple $b_{i}$
and $\tilde{b}_{i}$ together. In the limit that this hybridization is very
large, we can keep only one
$b^{\prime}_{i}=\frac{1}{\sqrt{2}}(b_{i}+\tilde{b}_{i})$ and then we recover
the usual slave boson theory, which explicitly preserve the $SU(2)$ spin
rotation. Therefore, we believe the phase constructed in this section can
smoothly crossover to the spin-singlet $d+id$ superconductor when we increase
the hybridization $\lambda$.

###  F.3 Model wavefunction

A wavefunction can be written down based on the above parton construction:

$\displaystyle\Psi({\bf x}_{1},\cdots{\bf x}_{N})=P_{G}\Psi_{d,\tilde{d}}({\bf
x}_{1},\cdots{\bf x}_{N})\Psi_{\psi}({\bf x}_{1},\cdots{\bf
x}_{N})\Psi_{f}({\bf x}_{1},\cdots{\bf x}_{N}),$ (87)

where $P_{G}$ is the projection to enforce the constraints in eq (F) and
$\Psi_{d,\tilde{d}},\Psi_{\psi},\Psi_{f}$ are the ground state wave functions
of the mean field discussed in eq (78).

###  F.4 Deconfined CSL to superconductor transition

The critical point is described by changing $C_{1}$ from $-1$ to $1$. This is
captured by mass changing of two Dirac cones $\chi_{1,2}$ with momenta
$(\pi/3,\pi/3),(\pi//3,-2\pi/3)$, obtained by the changing of $\theta$ in
$\psi_{1}$ mean field eq (II.1) from $\pi/6+\delta$ to
$\pi/6-\delta,(\delta>0)$. Because $\psi_{1}$ satisfies the projective
translation symmetry $T_{x}T_{y}=-T_{y}T_{x}$, the mass of the two Dirac cones
$\chi_{1,2}$ at $\theta=\pi/6$ are guaranteed to be the same by the
translation symmetry. The translation acts projectively on two Dirac cones as

$\displaystyle T_{x}:(\chi_{1},\chi_{2})^{T}\rightarrow
e^{-i\frac{\pi}{3}}(\eta^{x}\chi_{2},\eta^{x}\chi_{1})^{T},$ $\displaystyle
T_{y}:(\chi_{1},\chi_{2})^{T}\rightarrow
diag(e^{-i\frac{\pi}{3}},-e^{-i\frac{\pi}{3}})(\chi_{1},\chi_{2})^{T},$ (88)

where $\eta^{x}$ is the Pauli matrix on the Dirac indices of the Dirac
fermions.

Adding up the Dirac fermions and actions for $\psi_{2},d,\tilde{d},f$ etc, one
has for the critical action,(identifying $\gamma=\tilde{\gamma},a=\tilde{a}$)

$\displaystyle\mathcal{L}_{cri}=\sum_{i=1,2}\bar{\chi}_{i}[\eta_{\mu}(i\partial_{\mu}-a_{\mu}+A_{c,\mu}+\gamma_{\mu})\chi_{i}+m_{i}\bar{\chi}_{i}\chi_{i}$
$\displaystyle+\frac{1}{4\pi}(\beta d\beta-2\gamma
d\beta-\beta_{2}d\beta_{2}+2(A_{c}+\gamma-a)d\beta_{2})$
$\displaystyle-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}(\frac{1}{2}A_{s}+a)d\alpha_{1}+\frac{1}{2\pi}(-\frac{1}{2}A_{s}+a)d\alpha_{2},$
(89)

where the first line describes critical theory for $\psi_{1}$ ($m=0$ at
critical point) , the second line describes $d,\tilde{d},\psi_{2}$, and last
line for the spinon action.

With nonzero masses ($m_{1}=m_{2}$ by the projective translation symmetry),
one integrates out $\chi$. To be more precise, we should introduce another
field $\beta_{1}$ for describing the state $C_{1}=\pm 1$ after integrating out
$\chi$: $\frac{\mp
1}{4\pi}\beta_{1}d\beta_{1}+\frac{1}{2\pi}\beta_{1}d(A_{c}+\gamma-a)$.

When $m_{1}=m_{2}>0$, $\psi_{1}$ fills a band with $C_{1}=1$ and the theory
should describe a $d+id$ superconductor.Rewriting
$\gamma=\Gamma+a$,$\alpha_{c}=\frac{\alpha_{1}+\alpha_{2}}{2},\alpha_{s}=\frac{\alpha_{1}-\alpha_{2}}{2}$
and integrating out $a$ gives $\beta=2\alpha_{c}$. The critical action becomes

$\displaystyle\mathcal{L}_{cri}=\sum_{i=1,2}\bar{\chi}_{i}[\eta_{\mu}(i\partial_{\mu}+A_{c,\mu}+\Gamma_{\mu})\chi_{i}+m_{i}\bar{\chi}_{i}\chi_{i}$
$\displaystyle+\frac{1}{4\pi}(2\alpha_{c}d\alpha_{c}-4\Gamma
d\alpha_{c}-\beta_{2}d\beta_{2}+2(A_{c}+\Gamma)d\beta_{2})$
$\displaystyle-\frac{2}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s}.$
(90)

Integrating out $\chi_{i}$, $\Gamma$ gives $2\alpha_{c}=\beta_{1}+\beta_{2}$
and an action

$\displaystyle\mathcal{L}_{d+id}=\frac{1}{4\pi}(-2\alpha_{c}d\alpha_{c}+4\alpha_{c}d\beta_{2}-2\beta_{2}d\beta_{2}+4A_{c}d\alpha_{c})$
$\displaystyle-\frac{2}{4\pi}\alpha_{s}d\alpha_{s}+\frac{1}{2\pi}A_{s}d\alpha_{s}.$
(91)

If one identifies $\tilde{\beta}=\alpha_{c}-\beta_{2}$ and rewriting the
action in $\tilde{\beta},\alpha_{c}$, the action is identical to the $d+id$
superconductor action in eq (25) with gapless $\alpha_{c}$ mode coupled to
$A_{c}$, hence superconducting.

When $m_{1}=m_{2}<0$,$\psi_{1}$ fills a band with $C_{1}=-1$ and the theory
should describe a chiral spin liquid (CSL1). Integrating out $\chi_{i}$ gives
$\frac{1}{4\pi}\beta_{1}d\beta_{1}+\frac{1}{2\pi}\beta_{1}d(A_{c}+\gamma-a)$,
and integrating out $\gamma$ in eq (F.4) gives $\beta=-(\beta_{1}+\beta_{2})$
and an action

$\displaystyle\mathcal{L}_{CSL}=\frac{1}{4\pi}[2\beta_{1}d\beta_{1}+2\beta_{1}d\beta_{2}+2(A_{c}-a)d(\beta_{1}+\beta_{2})]$
$\displaystyle-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}+\frac{1}{2\pi}(a+\frac{1}{2}A_{s})d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}(a-\frac{1}{2}A_{s})d\alpha_{2}.$
(92)

Integrating $\beta_{2}$ locks $\beta_{1}=a-A_{c}$. Substituting it into the
action, we find that the terms from the slave boson part cancel each other and
we are left with the action from the fermionic spinon:

$\displaystyle\mathcal{L}_{CSL}=-\frac{1}{4\pi}\alpha_{1}d\alpha_{1}+\frac{1}{2\pi}(a+\frac{1}{2}A_{s})d\alpha_{1}-\frac{1}{4\pi}\alpha_{2}d\alpha_{2}+\frac{1}{2\pi}(a-\frac{1}{2}A_{s})d\alpha_{2}.$
(93)

Integrating $a$ locks $\alpha_{1}=-\alpha_{2}=\alpha$, and we obtain the final
action for the $U(1)_{2}$ CSL1:

$\displaystyle\mathcal{L}_{CSL}=-\frac{2}{4\pi}\alpha
d\alpha+\frac{1}{2\pi}A_{s}d\alpha.$ (94)

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