Introduction—

Anyons carry fractional charge and fractional statistics [1]–[3]. Soon after their introduction, Laughlin and Chern-Simons approaches suggested that a finite-density gas of charged anyons could superconduct [4]–[9]. Fractional Chern insulators (FCIs) sharpen this question because their anyons live on a lattice, carry crystal quantum numbers, and can acquire dispersive bands [10]–[14]. This problem is now concrete in moiré Chern bands. Fractional anomalous Hall and FCI regimes have been observed in twisted transition-metal dichalcogenides [15]–[17] and rhombohedral graphene moiré superlattices [18], [19]. Superconductivity has recently been reported nearby in both twisted MoTe\(_2\) and rhombohedral graphene platforms [20]–[22].

A key numerical motivation is the chiral \(f\)-wave superconducting phase found near a spin-polarized FCI at both \(\nu=2/3\) and finite doping [23]. After particle-hole transformation, this phase is a charge-\(2e\) topological superconductor with \(c_-=3/2\) at \(\nu = 1/3\). Related flat-Chern-band calculations also find chiral \(f\)-wave superconductivity near fractionalized Chern-band regimes [24], [25]. The same experimental and numerical phenomenology also contains reentrant insulating or Hall features. In the orientation used below, the relevant nearby normal descendant is a \(\sigma_{xy}=0\) charge density wave (CDW) state. We ask whether an Abelian Laughlin-type FCI can organize the FCI, this normal descendant, and the \(c_-=3/2\) superconductor within one framework.

Several Abelian FCI descendants are known. Doping or bandwidth tuning can produce topologically trivial anyon superconductors [23], [26]–[31]. Topological anyon superconductors with an odd number of Majorana edge modes are more constrained. Existing controlled QCD\(_3\)–Chern-Simons constructions produce topological superconducting descendants of Abelian FCIs, but do not directly realize the charge-\(2e\), \(c_-=3/2\) topological superconductor indicated above [26], [32].

We give such a route. The guiding analogy is superconductivity from a fractionalized spin liquid, where chargons condense and spinons pair [33]–[38]. Here the spinons are replaced by \(U(3)\) composite fermions which represent charge-\(e/3\) Laughlin anyons. The formalism keeps track of charge, flux quantization, and pair Higgsing: one electron fractionalizes into three Abelian anyons, and superconductivity arises when three anyon pairs combine into an ordinary charge-\(2e\) Cooper pair.

Starting from a \(\nu=1/3\) FCI, we obtain charge-\(2e\) anyon superconductors with a common charge sector. In the color-isotropic branch, the neutral descendants are an SC\(^\ast\) with residual Laughlin \(U(1)_3\) topological order, a strong-pairing superconductor with \(c_-=3\), and an \(SO(3)^s_1\) topological superconductor with chiral \(f\)-wave electron pairing, three chiral Majorana edge modes, and \(c_-=3/2\). At finite density, doped anyons form a chargon metal [39] whose weak pairing in the target chirality reaches the same \(SO(3)^s_1\) infrared superconducting response. Zero doping and finite doping are therefore unified in the same determinant-locked \(U(3)\) Higgs theory.

We also find a \(\sigma_{xy}=0\), \(c_-=0\) Mott descendant. In a microscopic Hofstadter realization, the \(2\pi/3\) magnetic-translation algebra forces it into a period-three charge density wave (CDW), matching after particle-hole transformation the RIQH features seen in numerics and experiment [20], [23]. The commensurate \(SO(3)^s_1\) superconductor is tied to the same period-three background and coexists with density order, even though its gauge-invariant Cooper pair has zero momentum. Experiments cannot yet resolve whether the superconducting phase breaks translations [20], [21], but numerics show suggestive hints [23].

\(U(3)\) partons and determinant locking—

Before entering the formalism, we outline the intuition behind the construction. In a Laughlin \(\nu=1/3\) state, an electron fractionalizes into three charge-\(e/3\) constituents, \[\psi_{\rm el}\sim f_1f_2f_3 . \label{eq:schematicparton}\tag{1}\] The \(f_a\) are composite fermions rather than physical anyon-creation operators. In the Laughlin branch a dressed \(f\) line carries charge \(e/3\) and Laughlin statistics. Unlike constructions that put the three constituents in different states [23], [26]–[31], we keep them equivalent by promoting \(a\) to \(SU(3)\) gauge color. Then \(C_f=1\) gives the \(\nu=1/3\) FCI [40], \(C_f=0\) gives the confined \(\sigma_{xy}=0\) Mott/CDW descendant, and most interestingly pairing the \(f\) partons gives the superconducting descendants.

We now implement Eq. 1 as a parton construction: \[\psi_{\rm el}=b\,f_1f_2f_3 = \frac{b}{3!}\epsilon_{abc}f_af_bf_c . \label{eq:parton}\tag{2}\] The second form displays the determinant representation. The field \(f=(f_1,f_2,f_3)^T\) is a fundamental of \[U(3)=\frac{SU(3)\times U(1)}{\mathbb{Z}_3}, \label{eq:U3quotient}\tag{3}\] and carries no microscopic electromagnetic charge. The determinant Higgs field \(b\) transforms as \(\det^{-1}\) under \(U(3)\) and has physical charge \(+1\). Writing \[c=\alpha+a\mathbf{1}_3, \qquad a=\frac{1}{3}\operatorname{Tr}c, \qquad \operatorname{Tr}\alpha=0 , \label{eq:decompose}\tag{4}\] the covariant derivatives are \[\begin{align} D_\mu f&=(\partial_\mu-i c_\mu)f,\\ D_\mu b&=(\partial_\mu-iA_\mu+i\operatorname{Tr}c_\mu)b . \end{align} \label{eq:covder}\tag{5}\] Condensing \(b\) is invariantly described by \[\mathcal{L}_b=\frac{1}{2\pi}\,\beta_b\,\mathrm d(A-\operatorname{Tr}c), \label{eq:detlock}\tag{6}\] with compact dual gauge field \(\beta_b\). Locally this gives \(a\simeq A/3\), or equivalently \(f\) carries \(e/3\) charge ¹. At this level it is determinant locking, not superconductivity. After determinant locking, a dressed fundamental \(f\) line sees \(A/3\) and carries charge \(e/3\), so \(f_1f_2f_3\) is the electron fractionalized into three Laughlin anyons.

If the \(U(3)\) partons are taken as microscopic degrees of freedom, we must keep track of how magnetic translations act on them. Since a fundamental parton carries charge \(e/3\), they experience background \(2\pi/3\) flux per unit cell. Equivalently, the parton translations obey the projective algebra \[T_xT_yT_x^{-1}T_y^{-1}=\omega,\qquad \omega=e^{2\pi i/3}.\] Thus a translation-preserving microscopic Hofstadter regularization requires a three-valley magnetic multiplet for each color. In such a strictly translation-preserving construction, normal Chern numbers and BdG Chern numbers can change only in multiples of three, and the pairing problem must distinguish intra-valley from inter-valley pairing. The single-valley \(C_f\) and BdG indices used below should therefore be read as infrared continuum indices, or equivalently as indices defined in an enlarged period-three unit cell, whenever the relevant Chern-Simons or BdG level changes by an amount not divisible by three.

Normal descendants—

Our convention for a \(U(3)\) Chern-Simons response is \[\mathcal{L}_{U(3)_{k,r}}[c] =\frac{k}{4\pi}\operatorname{Tr}\left(c\mathrm dc+\frac{2}{3}c^3\right) +\frac{r-k}{12\pi}(\operatorname{Tr}c)\mathrm d(\operatorname{Tr}c), \label{eq:U3kl}\tag{7}\] so the trace field \(a=(1/3)\operatorname{Tr}c\) has level \(3r\). A \(C_f=1\) band of the \(U(3)\) fundamental gives the spin theory \(U(3)^s_{1,1}\). After determinant locking, it gives the standard \(SU(3)^s_1\) description of the \(\nu = 1/3\) FCI with \(\sigma_{xy}=(1/3)e^2/h\) and \(c_-=1\) from the level-rank duality [32], [40], [41], \[3\Omega_g-SU(3)^s_1\simeq U(1)_3, \qquad c_-=1, \label{eq:levelrank}\tag{8}\] where \(\Omega_g\) is the gravitational Chern-Simons term. The counter term supplies three complex chiral modes, and gauging color removes the \(SU(3)_1\) current algebra of central charge two, leaving \(c_-=1\). The superscript \(s\) denotes a spin gauge field.

A \(C_f=0\) parton band gives \(U(3)^s_{0,0}\). The neutral \(SU(3)^s_0\) gauge theory is expected to confine, so the normal response has \(\sigma_{xy}=0\) and \(c_-=0\). Since the parton Chern number \(C_f\) only changes by one, this breaks microscopic translation symmetry and gives rise to a charge density wave (CDW). Equivalently, the filling-\(1/3\) LSM anomaly forbids a featureless translation-preserving insulator, so the confined descendant must be a CDW rather than a trivial Mott insulator [42]–[44].

Pair Higgs and charge sector—

Having treated the three charge-\(e/3\) constituents as equivalent color components, we take the simplest pair Higgs to be color symmetric, \[P_{ab}=P_{ba}\in {\rm Sym}^2{\boldsymbol{3}}, \qquad P\mapsto gPg^T . \label{eq:Prep}\tag{9}\] A local Hubbard-Stratonovich coupling is \[\mathcal{L}_Y= \eta P^\dagger_{ab}f_aD_- f_b+{\rm h.c.}, \qquad D_-=D_x-iD_y . \label{eq:Yukawa}\tag{10}\] Since the pair is symmetric in color, Fermi statistics requires the orbital part to be antisymmetric, so the angular momentum \(\ell\) must be odd. We focus on the lowest angular momentum channel with chirality opposite to the \(C_f=1\) anyon band, implemented by \(D_-\). For a weakly paired single \(C_f=1\) color band, \[C_{\rm BdG}=2C_f+\ell .\] Thus the \(p_x-ip_y\) channel has \(\ell=-1\), giving \(C_{\rm BdG}=1\) and one chiral Majorana mode per color. In the color-symmetric triplet, this is the minimal weak-pairing branch that becomes the \(SO(3)^s_1\) infrared response with physical \(c_-=3/2\). The opposite chirality has \(\ell=+1\) and \(C_{\rm BdG}=3\) per color, and represents a distinct higher-chirality descendant with a different residual topological order.

The gauge-invariant physical Cooper pair is \[\Delta_{2e}=b^2\det P, \qquad \det P=\frac{1}{6}\epsilon_{abc}\epsilon_{def}P_{ad}P_{be}P_{cf} . \label{eq:Cooper}\tag{11}\] Since \(\det P\mapsto(\det g)^2\det P\), the operator \(b^2\det P\) is gauge invariant and carries physical charge \(2e\). In a color-diagonal gauge, suppressing the odd-parity form factor, \[\Delta_{2e}\sim b^2(f_1D_-f_1)(f_2D_-f_2)(f_3D_-f_3), \label{eq:sixanyonpair}\tag{12}\] namely a Cooper pair made from six charge-\(e/3\) anyons.

Only the trace charges of the two condensates determine the physical charge response. The determinant Higgs couples to \(A-3a\), while the common phase of a full-rank pair Higgs couples to \(2a\). Dualizing these phases gives \[\mathcal{L}_{\rm a} =\frac{3r}{4\pi}a\,\mathrm da +\frac{1}{2\pi}\beta_b\mathrm d(A-3a) +\frac{1}{2\pi}\beta_P\mathrm d(2a). \label{eq:BFstartMain}\tag{13}\] The dynamical \(K\) matrix has a null vector for any integer \(r\) as shown in the End Matter. After an integral change of variables, the massive block has determinant \(-1\), and the remaining \(U(1)\) gauge field \(\tilde{a}\) has no Chern-Simons term. The universal charge-sector response is \[\mathcal{L}_{2e}=\frac{\kappa}{2}(\mathrm d\tilde{a})^2+ \frac{2}{2\pi}\tilde{a}\mathrm dA+\cdots . \label{eq:chargeSC}\tag{14}\] This is the dual response of a charge-\(2e\) superconductor: a \(2\pi\) monopole of \(\tilde{a}\) creates \(b^2\det P\). The quotient in Eq. 3 prevents the lifted \(SU(3)\times U(1)\) normalization from being misread as a charge-\(6e\) condensate.

Color-isotropic branch—

We first take \[\langle P_{ab}\rangle=\Delta\delta_{ab} . \label{eq:Pdelta}\tag{15}\] We name pairing chirality by the gauge-invariant electron-pair amplitude \(\langle\psi_{{\rm el},\mathbf{k}}\psi_{{\rm el},-\mathbf{k}}\rangle\propto b^2\det P(\mathbf{k})\), following Ref. [23], [45]. The target branch uses \[P_{ab}(\mathbf{k})=\Delta\,(k_x-i k_y)\delta_{ab} , \label{eq:PwaveIso}\tag{16}\] which gives \[\det P(\mathbf{k})=\Delta^3(k_x-i k_y)^3, \label{eq:fif}\tag{17}\] so the physical Cooper pair has \(f-if\) chirality.

The stabilizer of this symmetric tensor is \(O(3)\) inside \(U(3)\). After determinant locking, the internal neutral stabilizer is \[\{g\in U(3):\det g=1, \;gg^T=1\}=SO(3). \label{eq:SO3higgs}\tag{18}\] Thus every region in this branch is the charge-\(2e\) superconductor of Eq. 14 , stacked with an \(SO(3)\) neutral sector. The parent \(\mathbb{Z}_3\) quotient affects the flux lattice but produces no further quotient of \(SO(3)\).

The neutral level follows from the paired BdG invariant of the infrared partons. In the \(P\)-unitary gauge, \[f_a=\frac{1}{\sqrt2}(\chi_{+,a}+i\chi_{-,a}), \label{eq:fMajorana}\tag{19}\] where each \(\chi_\pm\) is a real \(SO(3)\) vector. A minimal single-valley continuum regularization is \[h_\pm(\mathbf{k})=k_1\sigma_x+k_2\sigma_y+(m\pm\Delta-Bk^2)\sigma_z, \qquad B>0 . \label{eq:BdG}\tag{20}\] We define \[\nu_+=\Theta(m+|\Delta|), \qquad \nu_-=\Theta(m-|\Delta|), \qquad n=\nu_++\nu_- . \label{eq:nudef}\tag{21}\] Each topological vector-Majorana block contributes one unit to the spin \(SO(3)\) response, \[\mathcal{L}_{\rm neutral}^{(n)}=3\Omega_g-SO(3)^s_n . \label{eq:SO3n}\tag{22}\] With this convention, the middle branch of Fig. 1 has \(n=1\) and is the desired \(c_-=3/2\) superconductor.

As explained above, at commensurate filling the single-valley BdG index is an infrared or enlarged-cell index whenever its change is not a multiple of three. Thus the \(SO(3)^s_0\) and \(SO(3)^s_1\) commensurate branches, whose neutral levels differ from the parent \(SO(3)^s_2\) branch, are naturally read in a period-three background. The gauge-invariant Cooper pair \(b^2\det P\) still carries zero momentum, so this is not a pair-density wave. The \(SO(3)^s_2\) branch is instead the pure-Higgs descendant of the FCI and can preserve microscopic translations.

For \(SO(3)^s_n\), equivalently the integer-spin sector of \(SU(2)_{2n}\), \(c[SO(3)^s_n]=3n/(n+1)\) for \(n=0,1,2\) [46], [47]. Hence \(c_-=3-3n/(n+1)\):

All entries share the charge-\(2e\) response in Eq. 14 . Only the neutral color sector changes.

The \(SO(3)^s_2\) superconductor is the pure-Higgs descendant of the \(C_f=1\) FCI. Condensing \(P=\Delta I\) in the determinant-fixed \(U(3)^s_{1,1}\) state leaves the charge sector in Eq. 14 and restricts the neutral gauge field through the standard embedding \(SO(3)\subset SU(3)\). The fundamental \({\boldsymbol{3}}_{SU(3)}\) becomes the real vector of \(SO(3)\), and the embedding has Dynkin index two [48], [49], \[SO(3)^s_2 \subset SU(3)^s_1. \label{eq:SU3SO3embedding}\tag{23}\] Equivalently, this is the \(N=3\) case of the small-level relation \(SO(N)_2\leftrightarrow SU(N)_1\) used in Ref. [50]. BdG-wise, the complex \(C_f=1\) color band becomes two topological real vector-Majorana bands, so \((\nu_+,\nu_-)=(1,1)\).

This branch is an SC\(^\ast\): its superconducting sector is the ordinary charge-\(2e\) sector of Eq. 14 , while its neutral complement carries the same electronic Laughlin order as the parent FCI. The End Matter uses the spin level-rank duality to obtain \[3\Omega_g-SO(3)^s_2\simeq U(1)_3 . \label{eq:SO3LaughlinMain}\tag{24}\] Thus the pure-Higgs branch inherits both the chiral \(\mathbb{Z}_3\) topological order and the chiral central charge \(c_-=1\) of the parent \(\nu=1/3\) FCI. No period-three density order is required for this branch.

The \(SO(3)^s_1\) superconductor is the new branch. Exactly one real color-Majorana block remains topological. The neutral response contains \(SO(3)^s_1\), giving three chiral Majorana modes and \[c_-=3-\frac{3}{2}=\frac{3}{2} . \label{eq:c32}\tag{25}\] This phase has the pairing symmetry and topology found numerically [23], [25]. It appears because the color-isotropic pair Higgs resolves the complex color band into two real vector-Majorana blocks.

The \(SO(3)^s_0\) branch is the strong-pairing superconductor with \(c_-=3\). Both real color-Majorana blocks are inverted, no neutral Chern-Simons sector remains, and the response is only the fixed background \(3\Omega_g\). Equivalently, \((\nu_+,\nu_-)=(0,0)\) is adiabatically connected to the BEC limit of the paired anyon fluid. The remaining low-energy charged objects are \(P\sim ff\), each carrying charge \(2e/3\) in the determinant-Higgs background [23], [26], [51], [52].

Following Ref. [53], \(SO(3)^s_0\) denotes an IR phase with a magnetic \(\mathbb{Z}_2\) symmetry, which in the anyon superconductor language is the parity of neutral \(SO(3)\) color vortices. Depending on the UV realization, this magnetic \(\mathbb{Z}_2\) can be either spontaneously broken or explicitly broken by monopole perturbations.

Finite density and criticality—

The same determinant-locked charge sector also describes finite density. Doping the electron density by \(\delta\) creates total \(f\)-density \(n_f=3|\delta|\) in the determinant-Higgs background. A color-singlet saddle has \[\langle f^\dagger_a f_b\rangle=\frac{n_f}{3}\delta_{ab}, \qquad \langle f^\dagger T^a f\rangle=0, \label{eq:colorsinglet}\tag{26}\] for every traceless \(SU(3)\) generator \(T^a\). The non-Abelian Gauss law then enforces zero average color flux, \[\frac{k}{2\pi}\langle F^a_{\alpha,12}\rangle +\langle f^\dagger T^a f\rangle=0 \quad\Rightarrow\quad \langle F^a_{\alpha,12}\rangle=0 . \label{eq:Gauss}\tag{27}\] The doped state is therefore a chargon metal: fractionalized charge-\(e/3\) carriers form small Fermi pockets with no average Cartan color flux, giving a compressible metal with finite clean-limit Drude weight [39]. This differs from a composite Fermi liquid of flux-attached particles at special effective filling. When the chemical potential is tuned into a parton band, the BdG phase boundaries deform to \[m=\pm R_\mu, \qquad R_\mu=\sqrt{|\Delta|^2+\mu^2}. \label{eq:Rmu}\tag{28}\] At finite density, \(|m|<|\mu|\), and hence weak pairing lies on the middle BdG branch. The color-symmetric Fermi surface then pairs into the \(SO(3)^s_1\) phase with \(c_-=3/2\), giving the same charge-\(2e\) topological superconductor.

The zero-density limit is more subtle. The direct FCI-to-CDW route involves a parton band inversion and hence a Dirac gap closing. Near this critical point, gauge fluctuations can make the pair-Higgs channel relevant and deflect the flow into the superconducting phase before the system reaches the CDW [52]. Thus the zero-doping FCI-to-SC boundary should be viewed as emanating from the FCI-to-CDW critical point. At that point the gauge-invariant electron operator has gapless spectral weight, since all three parton constituents are gapless.

Relation to three-pocket work—

While completing this manuscript, we became aware of a contemporaneous three-pocket construction [52]. The two approaches share the same Abelian low-energy charge sector, but realize microscopic translations and pairing differently. To see the common charge sector, restrict the \(U(3)\) connection to diagonal entries \(c_1,c_2,c_3\) with \(c_1+c_2+c_3=A\). Choosing \(a_1=c_1\) and \(a_2=c_2\), the three diagonal components carry Cartan charges \[q_1=(1,0),\qquad q_2=(0,1),\qquad q_3=(-1,-1),\] and the resulting Abelian Chern-Simons theory is \[K=\begin{pmatrix}2&1\\1&2\end{pmatrix}.\] Thus the diagonal \(U(1)^2\) sector of the \(U(3)\) construction explains the intervalley flux attachment scheme used in Ref. [52].

The difference is the ultraviolet symmetry embedding. In the three-pocket construction, the three components are physical valleys carrying microscopic crystal-momentum quantum numbers. In the present \(U(3)\) construction, they are gauge colors of a parton field, so their apparent transformation under translation is gauge-dependent rather than a physical valley quantum number. Correspondingly, Ref. [52] uses inter-valley pairing of physical valleys, whereas the color-isotropic branch constructed here uses intra-color pairing in gauge space.

Discussion—

We have presented a determinant-locked \(U(3)\) parton route from a \(\nu=1/3\) FCI to charge-\(2e\) superconductivity. The main result is that the FCI, the nearby \(\sigma_{xy}=0\) CDW/Mott descendant, the finite-density chargon metal, and the \(c_-=3/2\) chiral topological superconductor are organized by the same infrared theory. In the color-isotropic pair-Higgs branch, all superconducting phases share the ordinary charge-\(2e\) response, while their neutral color sectors distinguish an SC\(^\ast\) descendant with \(c_-=1\), the target \(SO(3)^s_1\) topological superconductor with \(c_-=3/2\), and a strong-pairing anyon superconductor with \(c_-=3\).

Our construction gives several numerical predictions. At zero doping, the superconducting instability should emerge from the FCI–CDW critical region, where the parton band inversion closes the gaps of the three constituents forming the physical electron. The electron gap should therefore soften, accompanied by enhanced low-energy spectral weight near the critical point. Numerics can also search for the nearby \(c_-=1\) SC\(^\ast\) branch and the \(c_-=3\) strong-pairing branch using entanglement spectrum, edge central charge, and modular data [54]–[56]. Translation symmetry provides another sharp diagnostic. At commensurate filling, the \(c_-=3/2\) branch is naturally tied to the same period-three density background as the \(\sigma_{xy}=0\) CDW descendant, while the Cooper pair remains a zero-momentum charge-\(2e\) order parameter rather than a pair-density wave. This directly distinguishes our construction from the translation-symmetric setting of Ref. [52].

Experimentally, the chiral \(f\)-wave character of the paired state can be constrained by phase-sensitive Josephson or SQUID measurements [57], [58]. The chiral Majorana boundary sector is branch dependent: it is present for some superconducting descendants but absent for others, making thermal Hall transport and edge tunneling useful diagnostics rather than universal signatures [45], [47], [59]. A more characteristic local consequence of our mechanism is the tendency of the superconducting state to coexist with the period-three density background. Scanning tunneling microscopy can image this modulation directly and also probe the softening of the electron gap near the FCI–CDW boundary.

Acknowledgments. We especially thank Ashvin Vishwanath and Chong Wang for discussions on magnetic translations. We also thank Da-Chuan Lu, Zhaoyu Han, Zhi-Qiang Gao, and Zhengyan Darius Shi for helpful discussions. T.W. is grateful for support from the Harvard Quantum Initiative Fellowship and the Simons Collaboration on Ultra-Quantum Matter, a grant from the Simons Foundation (Grant No. 651440).

Magnetic translations and PSG—

Here we justify the translation convention used in the main text. If the \(U(3)\) partons are taken as microscopic Hofstadter degrees of freedom, the magnetic-translation algebra must be kept explicitly. After determinant locking, a fundamental parton carries charge \(e/3\), so one electron flux quantum per microscopic unit cell gives a \(2\pi/3\) flux for \(f\). A convenient PSG representative uses the clock and shift matrices \[U=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}, \qquad \Omega=\begin{pmatrix}1&0&0\\0&\omega&0\\0&0&\omega^2\end{pmatrix}, \qquad \omega=e^{2\pi i/3}, \label{eq:clockshiftEnd}\tag{29}\] which obey \[U\Omega=\omega\Omega U . \label{eq:clockshiftalgEnd}\tag{30}\] The parton translations act projectively, \[T_x:f(\mathbf{r})\mapsto U f(\mathbf{r}+\hat{x}), \qquad T_y:f(\mathbf{r})\mapsto \Omega f(\mathbf{r}+\hat{y}), \label{eq:PSGEnd}\tag{31}\] so that \[T_xT_yT_x^{-1}T_y^{-1}=\omega \label{eq:magtransEnd}\tag{32}\] on a fundamental parton. The center phase is invisible on the gauge-invariant electron \(b\,\epsilon_{abc}f_af_bf_c\).

A representative hopping ansatz is \[H_f=-t_x\sum_{\mathbf{r}}f^\dagger_{\mathbf{r}}U f_{\mathbf{r}+\hat{x}} -t_y\sum_{\mathbf{r}}f^\dagger_{\mathbf{r}}\Omega f_{\mathbf{r}+\hat{y}}+{\rm h.c.}+\cdots . \label{eq:MFEnd}\tag{33}\] This form uses an unfolded projective Brillouin zone and should not be read as removing the Hofstadter valleys in a microscopic translation-preserving regularization. Indeed, the spatially dependent gauge transformation \[W_{x,y}=\Omega^y U^x\] brings Eq. 33 to a Landau-gauge representative \[T_x'=1, \qquad T_y'(x)=\omega^{x}\mathbf{1}_3 . \label{eq:LandauGaugeEnd}\tag{34}\] Thus the same PSG can be written with a three-site magnetic unit cell. In this gauge the color index is a spectator and each color carries the three magnetic sublattices required by \(2\pi/3\) flux. Therefore, in a strictly translation-preserving microscopic Hofstadter construction, normal Chern numbers and BdG Chern numbers change only in multiples of three. The commensurate \(C_f=0\) route in the main text is instead interpreted in a tripled unit cell, or in a period-three density background. In that enlarged cell each color can be described by a single Dirac Hamiltonian, so \(C_f\) and the color-isotropic BdG index can change by one. This enlarged-cell statement is not applied to the \(SO(3)^s_2\) SC\(^\ast\) branch, which is the pure-Higgs descendant of the translation-preserving FCI.

Charge-sector derivation—

The charge response in Eq. 14 follows only from the trace charges of the two Higgs fields. In a local \(SU(3)\times U(1)\) lift, \(b\) has physical charge \(+1\) and trace gauge charge \(-3\), while the common phase of a full-rank \(P\) has trace gauge charge \(+2\). Dualizing the two condensed phases gives \[\mathcal{L}_{\rm tr} = \frac{3r}{4\pi}a\,\mathrm da +\frac{1}{2\pi}\beta_b\mathrm d(A-3a) +\frac{1}{2\pi}\beta_P\mathrm d(2a). \label{eq:BFstartEnd}\tag{35}\] With \(A\) suppressed, the dynamical matrix in the basis \((a,\beta_b,\beta_P)\) is \[K_{\rm full}= \begin{pmatrix} 3r & -3 & 2\\ -3&0&0\\ 2&0&0 \end{pmatrix}, \qquad \det K_{\rm full}=0 . \label{eq:KfullEnd}\tag{36}\] The null direction is the superconducting dual photon. An integral change of variables \[\lambda=-3\beta_b+2\beta_P, \qquad \tilde{a}=-\beta_b+\beta_P, \label{eq:unimodEnd}\tag{37}\] or equivalently \(\beta_b=2\tilde{a}-\lambda\) and \(\beta_P=3\tilde{a}-\lambda\), gives \[\mathcal{L}_{\rm tr} = \frac{3r}{4\pi}a\,\mathrm da +\frac{1}{2\pi}\lambda\mathrm d(a-A) +\frac{2}{2\pi}\tilde{a}\mathrm dA . \label{eq:BFendEnd}\tag{38}\] The \((a,\lambda)\) block has \[K_{\rm massive}= \begin{pmatrix} 3r&1\\ 1&0 \end{pmatrix}, \qquad \det K_{\rm massive}=-1, \label{eq:KmassiveEnd}\tag{39}\] and leaves no Abelian topological order. The remaining compact field \(\tilde{a}\) has no Chern-Simons term and couples to electromagnetism as \[\mathcal{L}_{\rm charge} = \frac{\kappa}{2}(\mathrm d\tilde{a})^2 +\frac{2}{2\pi}\tilde{a}\mathrm dA+\cdots . \label{eq:chargeSCend}\tag{40}\] Thus a \(2\pi\) monopole of \(\tilde{a}\) creates the gauge-invariant Cooper pair \(b^2\det P\). The condensate is therefore an ordinary charge-\(2e\) superconductor. The quotient \(U(3)=(SU(3)\times U(1))/\mathbb{Z}_3\) fixes the flux normalization and removes the apparent charge-\(6e\) ambiguity of the lifted description.

Level-rank duality and line dictionary—

We use the superscript \(s\) to denote spin Chern-Simons gauge fields. The spin level-rank duality for orthogonal theories is \[SO(N)^s_k\times SO(0)^s_1 \simeq SO(k)_{-N}\times SO(Nk)^s_1 . \label{eq:SOspinLevelRankEnd}\tag{41}\] Here \(SO(0)^s_1\) keeps the transparent local fermion explicit, while \(SO(Nk)^s_1\) is an invertible spin TQFT with framing anomaly \(c=Nk/2\) [50]. For \(N=3\) and \(k=2\), \[SO(3)^s_2\times SO(0)^s_1 \simeq SO(2)_{-3}\times SO(6)^s_1 \simeq U(1)_{-3}\times SO(6)^s_1 . \label{eq:SO3LRend}\tag{42}\] Since \(SO(6)^s_1\) has \(c_-=3\), this gives \[3\Omega_g-SO(3)^s_2\simeq U(1)_3 . \label{eq:SO3toU13EndClean}\tag{43}\] The same Laughlin spin theory is obtained from the \(SU(3)\) presentation, \[3\Omega_g-SU(3)^s_1\simeq U(1)_3 . \label{eq:SU3toU13EndClean}\tag{44}\] Equivalently, if the common invertible \(3\Omega_g\) sector is absorbed into the convention, Eq. 44 is often written as \(SU(3)^s_{-1}\simeq U(1)_3\) [32], [41].

The line dictionary can be stated directly in the electronic spin theory. The \(SO(3)^s_2\) theory is the integer-spin sector of \({\rm Spin}(3)_2=SU(2)_4\), with lines \(j=0,1,2\). In the present Higgs completion the bosonic simple current \(j=2\) is screened by the condensed pair-Higgs sector, and the \(j=1\) line splits into two Abelian lines in the fractional complement \(3\Omega_g-SO(3)^s_2\), \[j=1\longrightarrow x\oplus x^2,\qquad x^3=1,\qquad \theta_x=\theta_{x^2}=e^{-2\pi i/3}. \label{eq:xsplitEnd}\tag{45}\] Equivalently, these are the fundamental and antifundamental lines of the \(SU(3)^s_{-1}\) complement, \[x\simeq {\boldsymbol{3}},\qquad x^2\simeq \bar{\boldsymbol{3}}. \label{eq:xSU3End}\tag{46}\] In the Laughlin \(U(1)_3\) convention, let \(q\) be the generator with \(q^3=\psi_{\rm el}\) and \(\theta_q=e^{\pi i/3}\). The electronic identification is \[x\longleftrightarrow q\psi_{\rm el},\qquad x^2\longleftrightarrow q^2. \label{eq:anyonDictionaryEndClean}\tag{47}\] Indeed, \(\theta_{q\psi_{\rm el}}=-e^{\pi i/3}=e^{-2\pi i/3}\). Thus \(3\Omega_g-SO(3)^s_2\simeq U(1)_3\) as electronic spin topological orders, not as an equality of bosonic modular tensor categories.

Anisotropic \(P=\Delta(E-I)\) branch—

For comparison, one may also consider an anisotropic full-rank pair-Higgs slice \(P=\Delta(E-I)\). A symmetric inter-component pairing pattern with zero diagonal and equal off-diagonal entries is represented in the \(U(3)\) infrared basis by \[E-I= \begin{pmatrix} 0&1&1\\ 1&0&1\\ 1&1&0 \end{pmatrix}. \label{eq:EI}\tag{48}\] This slice is not used in the commensurate color-isotropic construction of the main text. Since \(E-I\) is full rank, the trace-sector reduction is unchanged and all regions in this slice have the same charge-\(2e\) response in Eq. 14 . Only the neutral path changes.

The matrix \(E-I\) has a one-dimensional real eigenspace spanned by \((1,1,1)\) and a degenerate two-dimensional plane orthogonal to it. Therefore the subgroup preserving the pair-Higgs form is \[S[O(1)\times O(2)]\simeq O(2). \label{eq:O2stab}\tag{49}\] Under the connected component \(SO(2)\simeq U(1)\), the real \(SO(3)\) vector decomposes as \[{\boldsymbol{3}}\to {\boldsymbol{1}}\oplus{\boldsymbol{2}}. \label{eq:SO3toO2}\tag{50}\] The corresponding pairing amplitudes are obtained by diagonalizing \(E-I\) in this symmetry-adapted basis. Let \(f_s=(f_1+f_2+f_3)/\sqrt3\), and let \(f_{d,1},f_{d,2}\) span the plane orthogonal to \((1,1,1)\). Then \[\Delta(E-I)\longrightarrow \Delta\,{\rm diag}(2,-1,-1). \label{eq:EIdiag}\tag{51}\] Thus the singlet channel has pairing eigenvalue \(2\Delta\), while the doublet channel has pairing eigenvalue \(-\Delta\). The sign of the doublet eigenvalue only exchanges the two Majorana blocks, so the gap closings are controlled by the Takagi singular values \(2|\Delta|\) and \(|\Delta|\). A channel with pairing singular value \(\tilde{\Delta}\) has Majorana masses \(m\pm\tilde{\Delta}\), and therefore closes at \(m=\pm\tilde{\Delta}\). Hence the singlet gap closes at \(m=\pm2|\Delta|\), while the doublet gap closes at \(m=\pm|\Delta|\). We record this sequence by \[\begin{gather} n_s=\Theta(m+2|\Delta|)+\Theta(m-2|\Delta|),\\n_d=\Theta(m+|\Delta|)+\Theta(m-|\Delta|). \label{eq:O2levels} \end{gather}\tag{52}\] The singlet is neutral under the connected \(SO(2)\), so \(n_s\) counts the gauge-neutral Majorana stack. The doublet carries the connected \(SO(2)\) gauge response, so \(n_d\) is the connected \(SO(2)\) Chern-Simons level. The connected neutral response is \[\mathcal{L}^{E-I}_{\rm neutral}=\mathcal{L}_{O(2)^s_{n_d}}+n_s\mathcal{L}_\gamma, \label{eq:O2neutral}\tag{53}\] where \(\mathcal{L}_\gamma\) denotes one gauge-neutral chiral Majorana.

Eq. 53 should be read as a bookkeeping of the gap closings and the connected Chern-Simons response, not as a complete classification of the \(O(2)\) topological order. The full line content depends on the treatment of the reflection component in \(O(2)=U(1)\rtimes\mathbb{Z}_2\), possible discrete theta terms, and the ultraviolet Higgs completion that determines which flux lines are screened or can end on monopoles. We therefore do not assign an independent residual topological order to each subregion of Fig. 2. The purpose of this slice is only to illustrate that different full-rank pair structures can share the same determinant-locked charge-\(2e\) response while following different neutral Higgs paths. The central \(O(2)^s_1+\gamma\) region is an anisotropic representative of the odd-Majorana topological superconducting branch when it is connected to the color-isotropic \(SO(3)^s_1\) region.

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