# Paired parton quantum Hall states: a coupled wire construction

Alexander Sirota  Department of Physics, University of Virginia,
Charlottesville, VA22904, USA  Sharmistha Sahoo  Department of Physics,
University of Virginia, Charlottesville, VA22904, USA  Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, British Columbia,
Canada V6T 1Z4  Gil Young Cho  Department of Physics, Korea Advanced Institute
of Science and Technology, Daejeon 305-701, Korea  Department of Physics,
Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic
of Korea  Jeffrey C. Y. Teo<EMAIL_ADDRESS>Department of Physics,
University of Virginia, Charlottesville, VA22904, USA

###### Abstract

The Pfaffian fractional quantum Hall (FQH) states are incompressible non-
Abelian topological fluids present in a half-filled electron Landau level,
where there is a balanced population of electrons and holes. They give rise to
half-integral quantum Hall plateaus that divide critical transitions between
integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH
states, such as the Laughlin state, that can be understood using partons,
which are fermionic divisions of the electron. In this paper, we propose a new
family of incompressible paired parton FQH states at filling $\nu=1/6$ (modulo
1) that emerge from critical transitions between IQH states and Abelian FQH
states at filling $\nu=1/3$ (modulo 1). These paired parton states are
originated from a half-filled parton Landau level, where there is an equal
amount of partons and holes. They generically support Ising-like anyonic
quasiparticle excitations and carry non-Abelian Pfaffian topological orders
(TO) for partons. We prove the principle existence of these paired parton
states using exactly solvable interacting arrays of electronic wires under a
magnetic field. Moreover, we establish a new notion of particle-hole (PH)
symmetry for partons and relate the PH symmetric parton Pfaffian TO with the
gapped symmetric surface TO of a fractional topological insulator in three
dimension.

##  I Introduction

Topological phases of matter Wen (2017) have been playing a pivotal role in
the development of modern condensed matter physics. Several fundamental
concepts such as fractionalization Laughlin (1983), topological order Wen
(1990), and anyon Arovas  _et al._ (1984); Wilczek (1990) have emerged from
the studies of topological states and made striking experimental success. For
example, the celebrated Laughlin state Laughlin (1983) and more general
fractional quantum Hall (FQH) states Cage  _et al._ (2012) are inarguably the
most well-understood strongly-correlated electronic phases beyond one
dimensional systems, and have been a ‘motif’ for many interesting quantum
phases. On the other hand, another experimentally-observed ‘topological’
state, namely topological band insulator Hasan and Kane (2010); Qi and Zhang
(2011); Chiu  _et al._ (2016), embodies a remarkable interplay between
symmetries and topologies of electronic wavefunctions. The topological
insulator has made a broad impact on a number of disciplines of physics, from
experimental and theoretical condensed matter physics to theoretical high-
energy physics, and is still being actively investigated. In particular,
physical properties of fractional topological insulators Maciejko  _et al._
(2010); Swingle  _et al._ (2011); Levin  _et al._ (2011); Maciejko  _et al._
(2012); Ye  _et al._ (2016); Maciejko and Fiete (2015); Stern (2016); Ye  _et
al._ (2017) (FTI), which are fractional analogues of topological insulators,
are largely unexplored.

Conventional topological insulators (TI) in three dimensions are intricately
related to the quantum Hall effect in two dimensions. The massless surface
Dirac fermions on the boundary of a TI inspired a new dual low-energy
description Son (2015); Wang and Senthil (2015); Metlitski and Vishwanath
(2016) of the compressible composite Fermi liquid at a half-filled Landau
level Kalmeyer and Zhang (1992); Halperin  _et al._ (1993) at the critical
transition between adjacent integer quantum Hall plateaus. Under the duality,
the role of time-reversal (TR) symmetry of the former is replaced by particle-
hole symmetry Girvin (1984); Son (2015); Barkeshli  _et al._ (2015); Wang and
Senthil (2016); Balram and Jain (2017); Nguyen  _et al._ (2018) of the latter.
As a consequence, a thin TI slab with finite thickness and TR breaking massive
top and bottom surfaces can be regarded as a quasi-2D system that is
topologically equivalent to an incompressible quantum anomalous Hall state Liu
_et al._ (2016) which violates the particle-hole symmetry. This is because the
two systems share identical 2D Chern invariant Thouless  _et al._ (1982) in
the bulk as well as chiral charge and energy transport along their edges.
Moreover, the symmetry-preserving many-body interacting gapped surface states
Wang  _et al._ (2013); Metlitski  _et al._ (2015); Chen  _et al._ (2014);
Bonderson  _et al._ (2013) of a TI exhibit similar topological orders to the
incompressible Pfaffian quantum Hall state Read and Moore (1991) at half-
filling. For instance, the T-Pfaffian TI surface state Chen  _et al._ (2014)
has an identical anyon structure to the particle-hole symmetric Pfaffian
quantum Hall state Son (2015).

Similar correspondences hold between 3D FTI and 2D FQH states. The FTI
considered in this article are electronic states where the electrons are
divided into emergent charge $e/3$ fermionic components, referred to as
partons. More precisely, we use the parton ansatz Jain (1989); WEN (1992);
Maciejko  _et al._ (2010)

$\displaystyle\Psi_{\mathrm{el}}=\pi^{1}\pi^{2}\pi^{3},$ (1)

where $\pi^{a}$, for $a=1,2,3$, are the fermionic partons. The partons are
coupled to a dynamical $\mathbb{Z}_{3}$ gauge field Maciejko  _et al._ (2012).
Each $\pi^{a}$ carries a unit $\mathbb{Z}_{3}$ gauge charge so that the
electronic quasiparticle $\Psi_{\mathrm{el}}$ is $\mathbb{Z}_{3}$ neutral.
Moreover, each parton species fills a topological band insulator spectrum.
From the construction, the FTI surface must have the “fractional parity
anomaly” $\sigma_{xy}=\frac{e^{2}}{6h}$, which is the same Hall conductivity
as of a FQH state with filling $\nu=1/6$. In previous works Sahoo  _et al._
(2017); Cho  _et al._ (2017), we showed that the FTI can host two types of
charge-conserving and $\mathbb{Z}_{3}$-symmetric surface states with a finite
excitation energy gap. The first is a ferromagnetic surface state that breaks
TR symmetry. The second is an anomalous symmetry-preserving surface state,
referred to as the fractionalized T-Pfaffian state (denoted by
$\mathcal{T}$-$\mathrm{Pf}^{\ast}$ in our previous work). This state is
topologically-ordered and supports further fractionalized surface
quasiparticle excitations, such as the charge $e/12$ Ising anyon.

Figure 1: Correspondence between thin FTI slabs with gapped surfaces and
parton fractional quantum Hall states.

The three-dimensional FTI in a slab geometry with a finite thickness can be
topologically regarded as a quasi-two-dimensional system (see figure 1). When
the two horizontal dimensions are infinite or much longer than the vertical
one, the third axis reduces to a finite set of local variables in long length
scale. If the top and bottom surfaces are gapped, then the FTI slab is
topologically equivalent to a 2D FQH state. While there is currently no
confirmation of 3D FTI materials, the thin slab correspondence allows
theoretical proposals and predictions of new FQH states, which may arise in
existing 2D quantum Hall materials or heterostructures given the right
conditions.

Partons FQH | $\mathfrak{L}_{q}$ | $\mathfrak{P}_{p,q}$
---|---|---
states
filling fraction | $\nu=1/3$ | $\nu=1/6$
central charge | $c=1+2q$ | $c=1+q+p/2$
known examples | Laughlin state $\mathfrak{L}_{0}$ | Q-Pfaffian state $\mathfrak{P}_{1,0}$
FTI-inspired | Dirac parton $\mathfrak{L}_{1}$ | Particle-hole symmetric
examples | parton Pfaffian $\mathfrak{P}_{-1,1}$
Table 1: Summary of the series of Abelian parton FQH states $\mathfrak{L}_{q}$
and paired parton FQH states $\mathfrak{P}_{p,q}$.

In this article, we propose two new classes of parton FQH states inspired by
FTI slabs (see table 1 for a summary). Each FQH state is characterized by its
filling fraction $\nu$ and chiral central charge $c$ that respectively specify
the electric and thermal Hall conductance Kane and Fisher (1997); Cappelli
_et al._ (2002); Kitaev (2006)

$\displaystyle\sigma_{xy}=\frac{dI_{\mathrm{charge}}}{dV}=\nu\frac{e^{2}}{h},\quad\kappa_{xy}=\frac{dI_{\mathrm{energy}}}{dT}=c\frac{\pi^{2}k_{B}^{2}}{3h}T,$
(2)

where $dV$ and $dT$ are the transverse potential and temperature differences
across sample. The first class is a series of Abelian FQH states
$\mathfrak{L}_{q}$ with filling fraction $\nu=1/3$ and chiral central charge
$c=1+2q$, where $q$ is an integer. For example, $\mathfrak{L}_{q=0}$ is the
Laughlin FQH state $U(1)_{3}$ where partons are confined. $\mathfrak{L}_{q=1}$
represents a deconfined parton FQH phase
$U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\otimes SU(3)_{1}$, where each of the three
partons $\pi^{a}$ completely fills a Landau level and an electrically neutral
$SU(3)_{1}$ sector emerges. This $(2+1)$D phase is topologically equivalent to
a thin FTI slab with TR breaking top and bottom surfaces Sahoo  _et al._
(2017). The second class is a series of paired parton FQH states
$\mathfrak{P}_{p,q}$ with filling $\nu=1/6$ and chiral central charge
$c=1+q+p/2$, where $p$ and $q$ are integers. They generically carry non-
Abelian topological orders and support Ising anyons with charge $e/12$. For
example, $\mathfrak{P}_{p=1,q=0}$ is identical to one of the $Q$-Pfaffian
states in ref. Read and Moore, 1991 where partons are confined. On the other
hand, $\mathfrak{P}_{p=-1,q=1}$ represents a FQH phase that extend the
Pfaffian topological order by deconfined partons. We refer to this phase as
the parton Pfaffian state and denote it by $\mathrm{Pf}^{\ast}$. It is
topologically equivalent to a thin FTI slab where one of the two surfaces are
TR preserving and the other is TR breaking Sahoo  _et al._ (2017).

One of the most important issues in condensed matter theory is to design and
find microscopic electronic Hamiltonians for the strongly-interacting
topological states. The parton ansatz (1) that artificially divides an
electron into fractional components is an adhoc theoretical construction for
the purpose of providing an interpretation of a topological state. It is only
valid when the emergence of partons is supported by a microscopic Hamiltonian
of many-body interacting electrons. Instead of relying on the parton ansatz as
a premise, in this article, we will use the exactly solvable model approach
where Hamiltonians of interacting electrons are first designed and
subsequently shown to carry gapped parton excitations in the $(2+1)$D bulk and
gapless parton $(1+1)$D conformal field theories (CFT) along the boundary.
These model Hamiltonians of parton FQH states will be constructed by the
coupled wire method. It involves a highly anisotropic description where the
low-energy electronic degrees of freedom are confined along an array of
continuous one-dimensional wires and many-body interactions take the form of
inter- and intra-wire electron backscatterings. The technique was derived from
sliding Lutthinger liquids O’Hern  _et al._ (1999); Emery  _et al._ (2000);
Vishwanath and Carpentier (2001); Sondhi and Yang (2001); Mukhopadhyay  _et
al._ (2001) and was first employed by Kane, Mukhopadhyay and Lubensky Kane
_et al._ (2002) in the study of Laughlin Laughlin (1983) and Haldane-Halperin
hierarchy Haldane (1983); Halperin (1984) FQH states. The method was later
applied to more general FQH states Teo and Kane (2014); Klinovaja and Loss
(2014); Meng  _et al._ (2014); Sagi  _et al._ (2015); Kane  _et al._ (2017)
such as the Moore-Read Pfaffian Read and Moore (1991, 1992) and Read-Rezayi
Read and Rezayi (1999) FQH states. In particular, our article can be regarded
as a parton generalization of Kane, Stern and Halperin’s work Kane  _et al._
(2017), which provided a coupled-wire description of a sequence of
“conventional” Pfaffian FQH states at filling $\nu=1/2$.

Apart from being a tool to build exactly solvable models, the coupled wire
construction has two more valuable features. First, the topological order of
the $(2+1)$D bulk can be inferred from the low-energy CFT that describes the
system’s $(1+1)$D boundary. The coupled wire Hamiltonian introduces an
excitation energy gap in the bulk but leaves behind gapless degrees of freedom
along the edge. The gapless boundary modes of the Abelian parton states
$\mathfrak{L}_{q}$ and the paired parton states $\mathfrak{P}_{p,q}$ are
effectively described by the $(1+1)$D CFTs

$\displaystyle\begin{split}\mathfrak{L}_{q}&=U(1)_{3}\otimes[SU(3)_{1}]^{q},\\\
\mathfrak{P}_{p,q}&=U(1)_{24}\otimes_{\mathrm{el}}U(1)^{q}_{12}\otimes_{\mathrm{el}}\mathrm{Ising}^{p}.\end{split}$
(3)

Second, the series of paired parton states $\mathfrak{P}_{p,q}$ exhibits a new
notion of parton particle-hole (PH) symmetry, which applies to the half-filled
parton Landau levels. These paired parton FQH states can be related to one
another under the PH conjugation (to be defined in details in the main text)

$\displaystyle\mathrm{PH}_{\pi}\left(\mathfrak{P}_{p,q}\right)\equiv\mathfrak{L}_{1}\boxtimes\overline{\mathfrak{P}_{p,q}}=\mathfrak{P}_{-p-2,2-q}$
(4)

that “subtracts” $\mathfrak{P}_{p,q}$ from the parton Landau level
$\mathfrak{L}_{1}=U(3)_{1}/\mathbb{Z}_{3}$. In particular, the parton Pfaffian
state $\mathfrak{P}_{-1,1}=\mathrm{Pf}^{\ast}$ is PH symmetric. It is not a
coincidence that the parton Pfaffian state contains the fractional
$\mathcal{T}$-$\mathrm{Pf}^{\ast}$ topological order that describes the
symmetric gapped surface state of the FTI. A parallel analogy can be drawn at
the half-filled electronic Landau level, where the PH symmetric Pfaffian state
Son (2015) has identical topological order to the T-Pfaffian state Chen  _et
al._ (2014) of the symmetric gapped surface of a conventional TI. The exact
correspondence between the symmetric FTI surface and the PH symmetric half-
filled parton Landau levels is out of the scope of this article and we leave
this implication for future studies.

This article will be presented in the following order. In section II, we
describe (1+1)D chiral CFTs, which will serve as the building blocks for the
(2+1)D parton FQH states and will emerge at the system edge. They also
describe gapless modes at the domain walls between gapped regions on the
surface of FTIs. We will introduce the Dirac parton CFT in section II.1, the
parton Pfaffian CFT in section II.2, and their relationship through “gluing
and splitting” in section II.3. In section III.1 and III.2, we present the
models for the series of Abelian parton FQH states $\mathfrak{L}_{q}$ at
filling one-third and the non-Abelian paired parton FQH state
$\mathfrak{P}_{p,q}$ at filling one-sixth by coupling electron wires in the
presence of magnetic field. In section III.3, we present a model for the
surface topological order of the FTI that is closely-related to parton
Pfaffian quantum Hall state at filling one-sixth. In section IV, we introduce
the emergent notion of particle-hole conjugation and symmetry in the context
of partons and demonstrate the PH action on the paired parton states
$\mathfrak{P}_{p,q}$ with respect to each of the Abelian states
$\mathfrak{L}_{q_{0}}$ that we found in the section III. Finally, in section
V, we summarize our results and discuss open implications. For completeness,
we provide in appendix A a review of the relevant Kac-Moody algebras.

##  II Parton conformal field theories

In this section, we present the relevant conformal field theories Di Francesco
_et al._ (1999) (CFTs) that describe the low-energy degrees of freedom along
the $(1+1)$D edges of the parton fractional quantum Hall (FQH) states. In
particular, in section II.1 and II.2, we focus on the Dirac parton triplet and
the parton Pfaffian CFTs that live on the edge of the parton Landau levels
$\mathfrak{L}_{1}$ and the particle-hole symmetric parton Pfaffian FQH state
$\mathfrak{P}_{-1,1}$ respectively. In addition to the local electron, these
CFTs carry charge $e/3$ fermionic partons along with other fractional
quasiparticles as primary fields. We will characterize the charge and energy
transport in these CFTs, and show that the parton Pfaffian carries half the
degrees of freedom in the Dirac parton triplet (see figure 2). This means a
pair of parton Pfaffians can be glued into a Dirac parton triplet by a
condensation Hamiltonian consists of many-body electron backscattering (see
(53) in section II.3). On the other hand, a Dirac parton triplet can be split
into a pair of parton Pfaffians by a fractional basis transformation (see
figure 3 and eq.(72) in section II.3). The $(2+1)$D parton FQH states that
support these fractional boundary modes will be constructed in the next
section, and their topological orders can be inferred from the edge CFTs
through the bulk-boundary correspondence Read and Moore (1991).

###  II.1 The Dirac parton triplet

We begin with the triplet of chiral Dirac parton channels. It appears along
the 1D interface separating two time-reversal symmetry breaking domains with
opposite orientations on the surface of the fractional topological insulator
(FTI). It also appears along the 1D boundary of the 2D parton FQH state
$\mathfrak{L}_{1}$ where each of the three species of deconfined partons
completely occupies a Landau level.

In low energy, the triplet is described by the orbifold Di Francesco  _et al._
(1999) CFT $U(3)_{1}/\mathbb{Z}_{3}$ (to be defined below). The $U(3)_{1}$
theory consists of three copies of chiral Dirac parton channels, where the
fermionic partons can be represented by normal-ordered bosonized vertex
operators $\pi^{a}\sim e^{i\tilde{\phi}_{a}}$, for $a=1,2,3$. In low-energy,
the bosonized variables are described by the $U(3)_{1}$ Lagrangian density

$\displaystyle\mathcal{L}_{U(3)_{1}}=\frac{1}{2\pi}\sum_{a=1}^{3}\partial_{\mathsf{t}}\tilde{\phi}_{a}\partial_{\mathsf{x}}\tilde{\phi}_{a}$
(5)

up to non-universal kinetic velocity terms. They follow the equal-time
commutation relation (ETCR)

$\displaystyle\left[\partial_{\mathsf{x}}\tilde{\phi}_{a}(\mathsf{x}),\tilde{\phi}_{b}(\mathsf{x}^{\prime})\right]=2\pi
i\delta_{ab}\delta(\mathsf{x}-\mathsf{x}^{\prime}),$ (6)

which is equivalent to the time-ordered correlation

$\displaystyle\langle\tilde{\phi}_{a}(\mathsf{z})\tilde{\phi}_{b}(\mathsf{w})\rangle=-\delta_{ab}\log(\mathsf{z}-\mathsf{w})+\frac{i\pi}{2}S_{ab},$
(7)

where $\mathsf{z},\mathsf{w}\sim\tau+i\mathsf{x}$ are complex space-time
parameters (or $\mathsf{z},\mathsf{w}\sim e^{\tau+i\mathsf{x}}$ in a radially-
ordered complex space-time geometry where $\mathsf{x}$ is periodic and the
Euclidean time $\tau$ parametrizes the radial direction). The non-singular
second term containing

$\displaystyle S=(S_{ab})_{3\times 3}=\begin{pmatrix}0&1&-1\\\ -1&0&1\\\
1&-1&0\end{pmatrix}$ (8)

is put in (7) to ensure anticommuting correlations between distinct fermions,
$\langle e^{i\tilde{\phi}_{a}}e^{i\tilde{\phi}_{b}}\rangle=-\langle
e^{i\tilde{\phi}_{b}}e^{i\tilde{\phi}_{a}}\rangle$. The particular form of
$S_{ab}$ is chosen to respect the cyclic threefold rotation of parton labels
$a,b=1,2,3$.

Each parton carries the electric charge $e^{\ast}=e/3$, where $e$ is the
charge of the electron. The diagonal combination
$\Psi_{\mathrm{el}}\sim\pi^{1}\pi^{2}\pi^{3}\sim
e^{i(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3})}$ is a local
electronic quasiparticle with charge $e$. In addition to $\Psi_{\mathrm{el}}$,
there are local spin 1 bosons generated by the electrically neutral and
$\mathbb{Z}_{3}$ neutral combinations
$E^{ab}=e^{i(\tilde{\phi}_{a}-\tilde{\phi}_{b})}\sim\pi^{a}(\pi^{b})^{\dagger}$,
for $a\neq b$. They form the roots of a $SU(3)$ affine Kac-Moody current Lie
algebra at level 1. It will become clear in section III.1 that these operators
are integral combinations of electrons.

The $\mathbb{Z}_{3}$ symmetry rotates the phases of the partons $\pi^{a}\to
e^{2\pi i/3}\pi^{a}$, and leaves the electronic quasiparticle
$\Psi_{\mathrm{el}}$ invariant. The orbifold construction allows additional
twist fields that corresponds to twisted boundary conditions according to the
$\mathbb{Z}_{3}$ symmetry when the $(1+1)$D system is compactified on a torus
where both the spatial and temporal directions are periodic. Physically, the
orbifold construction addresses the issue of electron locality. If $\pi^{a}$
were local fermions, the CFT would not support any additional twist fields
that carry non-trivial monodromy with $\pi^{a}$. However, since the partons
$\pi^{a}$ are fractional quasiparticles that carry a non-trivial
$\mathbb{Z}_{3}$ gauge charge, they are non-local with respect to the
$\mathbb{Z}_{3}$ gauge fluxes.

The orbifold theory includes twist field that carries a $\mathbb{Z}_{3}$ flux
component, such as the Laughlin quasiparticle $\lambda\sim
e^{i(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3})/3}$. It carries the
electric charge $e^{\ast}=e/3$ and spin $h_{\lambda}=1/6$, and generate the
$U(1)_{3}$ charge sector. The bosonic twist fields
$\zeta^{ab}=e^{-i(\tilde{\phi}_{a}+\tilde{\phi}_{b})}\sim(\pi^{a})^{\dagger}(\pi^{b})^{\dagger}$,
for $a\neq b$, serve as representatives for the pure $\mathbb{Z}_{3}$ charge.
This is because they obey the fractional mutual statistics with the Laughlin
quasiparticle

$\displaystyle\langle\lambda(\mathsf{z})\zeta^{ab}(\mathsf{w})\rangle=\frac{1}{(\mathsf{z}-\mathsf{w})^{2/3}}$
(9)

The fractional exponent corresponds to a branch cut in the correlation
function and gives rise to the non-trivial $e^{2\pi i/3}$ monodromy after a
braiding cycle, $\mathsf{z}(s)=\mathsf{w}+|r|e^{is}$, where $s$ runs from 0 to
$2\pi$. Electrically neutral combinations of the Laughlin quasiparticle and
$\mathbb{Z}_{3}$ charges, such as $\lambda^{2}\zeta^{23}\sim
e^{i(2\tilde{\phi}_{1}-\tilde{\phi}_{2}-\tilde{\phi}_{3})/3}$, generate the
primary fields for an emergent $SU(3)_{1}$ sector (see (17) below).

The charged $U(1)_{3}$ and the electrically neutral $SU(3)_{1}$ sectors fully
decouple from each other, and their mutual operator product expansions (OPEs)
are non-singular. The decomposition

$\displaystyle\mathfrak{L}_{1}$ $\displaystyle=(\mbox{Dirac parton})^{\otimes
3}$ $\displaystyle=U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\otimes SU(3)_{1}.$ (10)

can be summarized by the following fractional basis transformation of
bosonized variables.

$\displaystyle\begin{pmatrix}\phi^{0}_{\pi}\\\ \phi^{1}_{\pi}\\\
\phi^{2}_{\pi}\end{pmatrix}=\frac{1}{3}\begin{pmatrix}1&1&1\\\ 2&-1&-1\\\
1&1&-2\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{1}\\\ \tilde{\phi}_{2}\\\
\tilde{\phi}_{3}\end{pmatrix},$ (11)

where the above $3\times 3$ transformation matrix is the inverse of

$\displaystyle A=\begin{pmatrix}1&1&0\\\ 1&-1&1\\\ 1&0&-1\end{pmatrix},$ (12)

whose columns are the simple roots of $U(1)\times SU(3)$. In (11),
$\tilde{\phi}_{a}$ ($\phi^{I}$) are referred to as bosonized variables in the
Cartan-Weyl (resp. Chevalley) basis. The Lagrangian density (5) turns into

$\displaystyle\mathcal{L}_{U(3)_{1}/\mathbb{Z}_{3}}=\frac{1}{2\pi}K_{IJ}\partial_{\mathsf{x}}\phi^{I}_{\pi}\partial_{\mathsf{t}}\phi^{J}_{\pi}$
(13)

under the basis transformation. The $K$-matrix in the Chevalley basis is given
by

$\displaystyle K=A^{T}A=\begin{pmatrix}3&0&0\\\ 0&2&-1\\\
0&-1&2\end{pmatrix},$ (14)

and it governs the ETCR of the Chevalley bosonized variables

$\displaystyle\left[\partial_{\mathsf{x}}\phi^{I}_{\pi}(\mathsf{x}),\phi^{J}_{\pi}(\mathsf{x}^{\prime})\right]=2\pi
i(K^{-1})^{IJ}\delta(\mathsf{x}-\mathsf{x}^{\prime}).$ (15)

The (1,1)-entry of (14) recovers the $K$-matrix of the Laughlin $\nu=1/3$ FQH
state. The lower $2\times 2$ block of (14) is identical to the Cartan matrix
of $SU(3)$. Primary fields are in general represented by vertex operators
$e^{im_{I}\phi^{I}_{\pi}}=e^{im_{I}(A^{-1})^{Ia}\tilde{\phi}_{a}}$, where
${\bf m}=(m_{0},m_{1},m_{2})$ are vectors with integral entries. In
particular, vectors that fall inside the image of $K$, i.e. ${\bf m}=K{\bf n}$
for some integral vector ${\bf n}$, correspond to local fields that have
trivial monodromy with all primary fields and account for all integral
electronic combinations. For example, $\Psi_{\mathrm{el}}\sim
e^{i(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3})}=e^{i3\phi^{0}_{\pi}}$
is the smallest local electronic quasiparticle in the Laughlin charge sector.
The neutral combinations $\pi^{a}(\pi^{b})^{\dagger}\sim
E^{ab}=e^{i(\tilde{\phi}_{a}-\tilde{\phi}_{b})}=e^{\pm
i(2\phi^{1}_{\pi}-\phi^{2}_{\pi})}$, $e^{\pm
i(-\phi^{1}_{\pi}+2\phi^{2}_{\pi})}$ or $e^{\pm
i(\phi^{1}_{\pi}+\phi^{2}_{\pi})}$, for $a\neq b$, are local fields that form
the 6 roots of the $SU(3)$ affine Kac-Moody current Lie algebra Di Francesco
_et al._ (1999) at level 1. The $U(1)_{3}\times SU(3)_{1}$ currents obey the
singular OPE

$\displaystyle
i\partial\tilde{\phi}_{a}(\mathsf{z})i\partial\tilde{\phi}_{b}(\mathsf{w})$
$\displaystyle=\frac{\delta_{ab}}{(\mathsf{z}-\mathsf{w})^{2}}+\ldots,$
$\displaystyle i\partial\tilde{\phi}_{a}(\mathsf{z})E^{bc}(\mathsf{w})$
$\displaystyle=\frac{\delta_{a}^{b}-\delta_{a}^{c}}{\mathsf{z}-\mathsf{w}}E^{bc}(\mathsf{w})+\ldots,$
(16) $\displaystyle E^{ab}(\mathsf{z})E^{cd}(\mathsf{w})$
$\displaystyle=\frac{\delta^{ad}\delta^{bc}}{(\mathsf{z}-\mathsf{w})^{2}}+\frac{if^{(ab)(cd)(ef)}}{\mathsf{z}-\mathsf{w}}E^{ef}(\mathsf{w})+\ldots$

where
$f^{(ab)(cd)(ef)}=\varepsilon^{abd}\delta^{bc}\delta^{ae}\delta^{df}-\varepsilon^{abc}\delta^{ad}\delta^{bf}\delta^{ce}$
is the structure factor of $SU(3)$ where $a\neq b$, $c\neq d$ and $e\neq f$.
The derivations of the current algebra (16) as well as OPEs between vertex
operators in general are available in appendix A.1.

The Laughlin quasiparticle $\lambda\sim
e^{i\phi^{0}_{\pi}}=e^{i(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3})/3}$
is the smallest non-trivial primary field in the $U(1)_{3}$ charge sector. It
decouples from the $SU(3)_{1}$ neutral sector and the OPE between $\lambda$
and any of the $SU(3)_{1}$ roots $E^{ab}$ is non-singular. The non-trivial
primary fields of $SU(3)_{1}$ are given by the two fundamental representations
of the Lie algebra. Linear combinations of primary fields in the super-
selection sectors

$\displaystyle\mathcal{E}=\mathrm{span}\left\\{\mathcal{E}^{1},\mathcal{E}^{2},\mathcal{E}^{3}\right\\},\quad\overline{\mathcal{E}}=\mathrm{span}\left\\{(\mathcal{E}^{1})^{\dagger},(\mathcal{E}^{2})^{\dagger},(\mathcal{E}^{3})^{\dagger}\right\\},$
$\displaystyle\mathcal{E}^{a}=\lambda^{\dagger}\pi^{a}\sim
e^{i[\tilde{\phi}_{a}-(\tilde{\phi}_{1}+\tilde{\phi}_{2}+\tilde{\phi}_{3})/3]}$
(17)

rotate according to the OPEs

$\displaystyle
E^{ab}(\mathsf{z})\mathcal{E}^{c}(\mathsf{w})=-iS_{ab}\frac{\delta^{bc}\delta^{a}_{d}}{\mathsf{z}-\mathsf{w}}\mathcal{E}^{d}(\mathsf{w})+\ldots$
(18) $\displaystyle
E^{ab}(\mathsf{z})\mathcal{E}^{c}(\mathsf{w})^{\dagger}=iS_{ab}\frac{\delta^{ac}\delta^{b}_{d}}{\mathsf{z}-\mathsf{w}}\mathcal{E}^{d}(\mathsf{w})^{\dagger}+\ldots$

and form a conjugate pair of three dimensional irreducible representations of
$SU(3)$. For instance, the matrices $L^{ab}=(L^{abc}_{d})_{3\times 3}$ and
$\overline{L}^{ab}=(\overline{L}^{abc}_{d})_{3\times 3}$, for
$L^{abc}_{d}=\delta^{bc}\delta^{a}_{d}$ and
$\overline{L}^{abc}_{d}=\delta^{ac}\delta^{b}_{d}$, are the raising and
lowering matrices for the off-diagonal Gell-Mann matrices
$L^{ab}+\overline{L}^{ab}$ and $i(L^{ab}-\overline{L}^{ab})$, for $a\neq b$.
All primary fields in $\mathcal{E}$ and $\overline{\mathcal{E}}$ are
electrically neutral and carry spin $h=1/3$. Since the root operators $E^{ab}$
are local electronic, the OPEs in (18) show that any two primary fields within
the same sector $\mathcal{E}$ or $\overline{\mathcal{E}}$ are equivalent up to
local electrons. They obey the fusion rules

$\displaystyle\mathcal{E}\times\mathcal{E}=\overline{\mathcal{E}},\quad\overline{\mathcal{E}}\times\overline{\mathcal{E}}=\mathcal{E},\quad\mathcal{E}\times\overline{\mathcal{E}}=1,$
(19)

which abbreviate the OPEs

$\displaystyle\mathcal{E}^{a}(\mathsf{z})\mathcal{E}^{b}(\mathsf{w})=\frac{-i\varepsilon^{abc}}{(\mathsf{z}-\mathsf{w})^{1/3}}\mathcal{E}^{c}(\mathsf{w})^{\dagger}+\ldots,$
$\displaystyle\mathcal{E}^{a}(\mathsf{z})^{\dagger}\mathcal{E}^{b}(\mathsf{w})^{\dagger}=\frac{-i\varepsilon^{abc}}{(\mathsf{z}-\mathsf{w})^{1/3}}\mathcal{E}^{c}(\mathsf{w})+\ldots,$
$\displaystyle\mathcal{E}^{a}(\mathsf{z})\mathcal{E}^{b}(\mathsf{w})^{\dagger}=\frac{\delta^{ab}}{(\mathsf{z}-\mathsf{w})^{2/3}}+\ldots.$
(20)

The Dirac parton triplet carries electric and energy transport. The external
electromagntic gauge field is coupled to the theory through the $U(1)_{EM}$
transformation $\tilde{\phi}_{a}\to\tilde{\phi}_{a}+\Lambda/3$ [or
equivalently $\phi^{I}_{\pi}\to\phi^{I}_{\pi}+\Lambda(K^{-1})^{IJ}t_{J}$,
where the charge vector is ${\bf t}^{T}=(t_{0},t_{1},t_{2})=(1,0,0)$], when
the electronic quasiparticle transforms by
$\Psi_{\mathrm{el}}=\pi^{1}\pi^{2}\pi^{3}\to e^{i\Lambda}\Psi_{\mathrm{el}}$.
In particular, the differential conductance of the parton channel is given by

$\displaystyle\sigma=\frac{dI}{dV}=\frac{e^{2}}{h}{\bf t}^{T}K^{-1}{\bf
t}=\frac{1}{3}\frac{e^{2}}{h},$ (21)

which associates to the filling fraction $\nu=1/3$ of a FQH state that
supports a boundary Dirac parton triplet CFT. As all three bosons propagate in
the same direction, the channel carries a chiral central charge
$c=c_{R}-c_{L}=3$, which dictates the differential thermal conductance Kane
and Fisher (1997); Cappelli  _et al._ (2002); Kitaev (2006)

$\displaystyle\kappa=\frac{dI_{E}}{dT}=c\left(\frac{\pi^{2}k_{B}^{2}}{3h}\right)T$
(22)

in low temperature.

###  II.2 The parton Pfaffian

The parton Pfaffian CFT, denoted by $\mathrm{Pf}^{\ast}$, carries exactly half
of the degrees of freedom of the Dirac parton triplet. It consists of an
electrically charged $U(1)_{24}$ sector and a neutral
$U(1)_{12}\times\overline{\mathrm{Ising}}$ sector.

$\displaystyle\mathfrak{P}_{-1,1}=\mathrm{Pf}^{\ast}=U(1)_{24}\otimes_{\mathrm{el}}U(1)_{12}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}.$
(23)

Each sector is generated by a central “almost local” primary field that has
non-trivial monodromy only with Ising anyons or $\pi$-fluxes.

$\displaystyle\begin{split}&U(1)_{24}\ni\mbox{charge $e$ spin 3 boson
}e^{i\Phi_{\rho}}\\\ &U(1)_{12}\ni\mbox{neutral spin $3/2$ Dirac fermion
}e^{i\Phi_{d}}\\\ &\overline{\mathrm{Ising}}\ni\mbox{Majorana fermion
}\psi.\end{split}$ (24)

The electrically charged $U(1)_{24}$ and neutral $U(1)_{12}$ sectors are
Abelian and can be described by a two-component bosonized theory

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2\pi}\left(K_{\rho}\partial_{\mathsf{t}}\phi_{\rho}\partial_{\mathsf{x}}\phi_{\rho}+K_{d}\partial_{\mathsf{t}}\phi_{d}\partial_{\mathsf{x}}\phi_{d}\right)$
(25)

up to non-universal velocity terms, where the $K$-matrices

$\displaystyle K_{\rho}=24,\quad K_{d}=12$ (26)

define the levels of the two $U(1)$ sectors. The $\overline{\mathrm{Ising}}$
sector is generated by a Majorana fermions $\psi$, which is described by the
Lagrangian density

$\displaystyle\mathcal{L}=i\psi\left(\partial_{t}+v\partial_{x}\right)\psi.$
(27)

The Majorana fermion propagates in the opposite direction to the two $U(1)$
sectors.

The primary field content of the Abelian part of (23) can be represented by
vertex operators $e^{i(a\phi_{\rho}+b\phi_{d})}$, where $a,b$ are integers.
The charged $U(1)_{24}$ sector supports primary fields $e^{ia\phi_{\rho}}$,
denoted by $[a]_{\rho}$, that carry charge $q_{a}=a/12$ (in units of the
electric charge $e$) and spin $h_{a}=a^{2}/48$. In particular, $[12]_{\rho}$
represents the “almost local” charge $e$ boson $e^{i\Phi_{\rho}}$, where
$\Phi_{\rho}=12\phi_{\rho}$. The external $U(1)_{\mathrm{EM}}$ that transforms
$e^{i\Phi_{\rho}}\to e^{i\Phi_{\rho}}e^{i\Lambda}$ shifts
$\phi_{\rho}\to\phi_{\rho}+\Lambda/12=\phi_{\rho}+\Lambda
K_{\rho}^{-1}t_{\rho}$, where the charge vector is $t_{\rho}=2$. This set the
filling fraction

$\displaystyle\nu=t_{\rho}K_{\rho}^{-1}t_{\rho}=\frac{1}{6}$ (28)

that determines the differential conductance $\sigma=\nu(e^{2}/h)$ of the
charge sector. The primary fields satisfy the fusion rule

$\displaystyle[a]_{\rho}\times[a^{\prime}]_{\rho}=[a+a^{\prime}]_{\rho},$ (29)

which is an abbreviation of the operator product expansion
$e^{ia\phi_{\rho}(\mathsf{z})}e^{ia^{\prime}\phi_{\rho}(\mathsf{w})}=e^{i(a+a^{\prime})\phi_{\rho}(\mathsf{w})}(\mathsf{z}-\mathsf{w})^{aa^{\prime}/24}+\ldots$.

The neutral $U(1)_{12}$ sector supports similar primary fields. The vertex
operators $e^{ib\phi_{d}}$ are denoted by $[b]_{d}$. They carry spin
$h_{b}=b^{2}/24$ and follow similar fusion rules in (29). $[6]_{d}$ represents
the “almost local” neutral spin-$3/2$ Dirac fermion $e^{i\Phi_{d}}$, where
$\Phi_{d}=6\phi_{d}$. Both the charged $U(1)_{24}$ and neutral $U(1)_{12}$
sectors are $\mathbb{Z}_{2}$ graded with respect to their central elements
$[12]_{\rho}$ and $[6]_{d}$ respectively. Each primary field is assigned a
parity according its monodromy with the central element. The monodromy can be
determined by the time-ordered correlations

$\displaystyle\begin{split}&\left\langle
e^{ia\phi_{\rho}(\mathsf{z})}e^{i\Phi_{\rho}(\mathsf{w})}\right\rangle\sim(\mathsf{z}-\mathsf{w})^{a/2},\\\
&\left\langle
e^{ib\phi_{d}(\mathsf{z})}e^{i\Phi_{d}(\mathsf{w})}\right\rangle\sim(\mathsf{z}-\mathsf{w})^{b/2},\end{split}$
(30)

which carry branch cuts (i.e. odd monodromy) when $a,b$ are odd. Odd primary
fields are also referred to as $\pi$-fluxes.

The $\overline{\mathrm{Ising}}$ CFT supports two non-trivial primary fields.
The first is the Majorana fermion $\psi$. It carries spin $h_{\psi}=-1/2$ and
follows the fusion rule $[\psi]\times[\psi]=1$, which is an abbreviation for
the operator product expansion
$\psi(\overline{\mathsf{z}})\psi(\overline{\mathsf{w}})=1/(\overline{\mathsf{z}}-\overline{\mathsf{w}})+\ldots$.
The second is the Ising twist field $\sigma$. It carries spin $h=-1/16$ and
following the non-Abelian fusion rules

$\displaystyle[\sigma]\times[\psi]=[\sigma],\quad[\sigma]\times[\sigma]=1+[\psi].$
(31)

They corresponds to the OPEs
$\sigma(\overline{\mathsf{z}})\psi(\overline{\mathsf{w}})=\sigma(\overline{\mathsf{w}})/(\overline{\mathsf{z}}-\overline{\mathsf{w}})^{1/2}+\ldots$
and
$\sigma(\overline{\mathsf{z}})\sigma(\overline{\mathsf{w}})=1/(\overline{\mathsf{z}}-\overline{\mathsf{w}})^{1/8}+\psi(\overline{\mathsf{w}})(\overline{\mathsf{z}}-\overline{\mathsf{w}})^{3}/8+\ldots$.
Like the other two Abelian sectors, the $\overline{\mathrm{Ising}}$ CFT is
also $\mathbb{Z}_{2}$-graded with respect to the fermion $\psi$. The 1 and
$[\psi]$ primary fields are even, and the Ising twist field $[\sigma]$ is odd
as it has odd monodromy with $\psi$.

We associate the three primary field combinations

$\displaystyle\begin{split}&\Psi_{\mathrm{el}_{1}}=[12]_{\rho}\otimes[\psi]\sim
e^{i\Phi_{\rho}}\psi\\\ &\Psi_{\mathrm{el}_{2}}=[12]_{\rho}\otimes[6]_{d}\sim
e^{i\Phi_{\rho}}e^{i\Phi_{d}}\\\
&\Psi_{\mathrm{el}_{3}}=[12]_{\rho}\otimes[-6]_{d}\sim
e^{i\Phi_{\rho}}e^{-i\Phi_{d}}\end{split}$ (32)

to be electronic. This means that they are treated as combinations of integral
products of electronic operators. All of them are charge $e$ Dirac fermions.
They carry spin $h_{1}=5/2$ and $h_{2}=h_{3}=9/2$ respectively. In general, a
primary field in the parton Pfaffian CFT is local electronic if it can be
expressed as combinations of $\Psi_{\mathrm{el}_{1}}$,
$\Psi_{\mathrm{el}_{2}}$ and $\Psi_{\mathrm{el}_{3}}$. For example,
$\Psi_{\mathrm{el}_{1}}^{\dagger}\Psi_{\mathrm{el}_{2}}=[6]_{d}\otimes[\psi]$
is a spin 1 neutral boson, and is an integral combination of electrons. The
locality of the electron forbids the presence of any twist field that exhibit
non-trivial monodromy with any of the $\Psi_{\mathrm{el}}$’s. This includes
all the $\pi$ fluxes to the electron, such as $[1]_{\rho}$, $[1]_{d}$ and
$[\sigma]$. The “confinement” of $\pi$ fluxes is indicated by the “electronic”
tensor product $\otimes_{\mathrm{el}}$ in (23). A general physical primary
field of the parton Pfaffian theory takes a tensor product form, and belong in
one of the following types

$\displaystyle\openone_{a,b}$ $\displaystyle=[a]_{\rho}\otimes[b]_{d}$
$\displaystyle\sim e^{i(a\phi_{\rho}+b\phi_{d})},\quad\mbox{for $a$ and $b$
even},$ $\displaystyle\Psi_{a,b}$
$\displaystyle=[a]_{\rho}\otimes[b]_{d}\otimes[\psi]$ $\displaystyle\sim
e^{i(a\phi_{\rho}+b\phi_{d})}\psi,\quad\mbox{for $a$ and $b$ even},$ (33)
$\displaystyle\Sigma_{a,b}$
$\displaystyle=[a]_{\rho}\otimes[b]_{d}\otimes[\sigma]$ $\displaystyle\sim
e^{i(a\phi_{\rho}+b\phi_{d})}\sigma,\quad\mbox{for $a$ and $b$ odd}.$

In other words, the three sector components must be either all even or all odd
according to the $\mathbb{Z}_{2}$ grading. All other combinations are not
allowed by electron locality.

The electric charges, spins and fusion rules of these primary fields can be
deduced from that of the three components of $\mathrm{Pf}^{\ast}$ in (23).
$X_{a,b}$ carries the electric charge $q_{X_{a,b}}=a/12$ (in unit of the
electric charge $e$), for $X=\openone,\Psi,\Sigma$. Their spins are given by

$\displaystyle h_{\openone_{a,b}}$
$\displaystyle=\frac{a^{2}}{48}+\frac{b^{2}}{24},$ $\displaystyle
h_{\Psi_{a,b}}$ $\displaystyle=\frac{a^{2}}{48}+\frac{b^{2}}{24}-\frac{1}{2},$
$\displaystyle h_{\Sigma_{a,b}}$
$\displaystyle=\frac{a^{2}}{48}+\frac{b^{2}}{24}-\frac{1}{16}.$ (34)

They follow the fusion rule

$\displaystyle\openone_{a,b}\times\openone_{a^{\prime},b^{\prime}}=\Psi_{a,b}\times\Psi_{a^{\prime},b^{\prime}}=\openone_{a+a^{\prime},b+b^{\prime}}$
$\displaystyle\openone_{a,b}\times\Psi_{a^{\prime},b^{\prime}}=\Psi_{a+a^{\prime},b+b^{\prime}}$
$\displaystyle\openone_{a,b}\times\Sigma_{a^{\prime},b^{\prime}}=\Psi_{a,b}\times\Sigma_{a^{\prime},b^{\prime}}=\Sigma_{a^{\prime},b^{\prime}}$
$\displaystyle\Sigma_{a,b}\times\Sigma_{a^{\prime},b^{\prime}}=\openone_{a+a^{\prime},b+b^{\prime}}+\Psi_{a+a^{\prime},b+b^{\prime}}.$
(35)

Within (33), $\openone_{4,\pm 2}$ and $\Psi_{4,\pm 4}$ are charge $e/3$ spin
$1/2$ fermions and serve as the deconfined partons that decompose the
electronic quasiparticles

$\displaystyle\Psi_{\mathrm{el}_{1}}$ $\displaystyle=\Psi_{4,\pm
4}\times\openone_{4,\mp 2}\times\openone_{4,\mp 2}=\Psi_{4,\pm
4}\times\Psi_{4,\pm 4}\times\Psi_{4,\pm 4},$
$\displaystyle\Psi_{\mathrm{el}_{2}}$
$\displaystyle=\Psi_{4,4}\times\Psi_{4,4}\times\openone_{4,-2}=\openone_{4,2}\times\openone_{4,2}\times\openone_{4,2},$
(36) $\displaystyle\Psi_{\mathrm{el}_{3}}$
$\displaystyle=\Psi_{4,-4}\times\Psi_{4,-4}\times\openone_{4,2}=\openone_{4,-2}\times\openone_{4,-2}\times\openone_{4,-2}.$

By summing the contributions from the three components, the overall electric
and thermal conductance $\sigma=\nu e^{2}/h$ and
$\kappa=c(\pi^{2}k_{B}^{2}/3h)T$ are specified by

$\displaystyle\nu=\frac{1}{6}+0+0=\frac{1}{6},\quad
c=1+1-\frac{1}{2}=\frac{3}{2},$ (37)

each of which is half of that of the parton Dirac triplet
$U(3)_{1}/\mathbb{Z}_{3}$ described in the previous subsection. This suggests
the decomposition or conformal embedding

$\displaystyle
U(3)_{1}/\mathbb{Z}_{3}\sim\mathrm{Pf}^{\ast}\otimes\mathrm{Pf}^{\ast},$ (38)

which will be discussed in the following subsection.

Before doing so, it is worth noticing that within the parton Pfaffian theory,
there is a subset of primary fields

$\displaystyle I_{a}$ $\displaystyle=[a]_{\rho}\otimes[a]_{d}\sim
e^{ia(\phi_{\rho}+\phi_{d})},\quad\mbox{for $a$ even},$
$\displaystyle\psi_{a}$
$\displaystyle=[a]_{\rho}\otimes[a]_{d}\otimes[\psi]\sim
e^{ia(\phi_{\rho}+\phi_{d})}\psi,\quad\mbox{for $a$ even},$ (39)
$\displaystyle\sigma_{a}$
$\displaystyle=[a]_{\rho}\otimes[a]_{d}\otimes[\sigma]\sim
e^{ia(\phi_{\rho}+\phi_{d})}\sigma,\quad\mbox{for $a$ odd},$

that are bosonic, fermionic, or semionic combinations with spins

$\displaystyle h_{I_{a}}$ $\displaystyle=\frac{(a/2)^{2}}{4},\quad\mbox{for
$a$ even},$ $\displaystyle h_{\psi_{a}}$
$\displaystyle=\frac{(a/2)^{2}}{4}-\frac{1}{2},\quad\mbox{for $a$ even},$
$\displaystyle h_{\sigma_{a}}$
$\displaystyle=\frac{a^{2}-1}{16},\quad\mbox{for $a$ odd},$ (40)

and are closed under fusion

$\displaystyle I_{a}\times
I_{a^{\prime}}=\psi_{a}\times\psi_{a^{\prime}}=I_{a+a^{\prime}},$
$\displaystyle
I_{a}\times\psi_{a^{\prime}}=\psi_{a+a^{\prime}},\quad\psi_{a}\times\sigma_{a^{\prime}}=\sigma_{a+a^{\prime}}$
$\displaystyle\sigma_{a}\times\sigma_{a^{\prime}}=I_{a+a^{\prime}}+\sigma_{a+a^{\prime}}.$
(41)

This sub-collection generates a theory, referred to as the parton $T$-Pfaffian
state and denoted by $\mathcal{T}$-$\mathrm{Pf}^{\ast}$, that describes the
gapped time-reversal symmetric surface state of a fractional topological
insulator Sahoo  _et al._ (2017); Cho  _et al._ (2017). The
$\mathcal{T}$-$\mathrm{Pf}^{\ast}$ state has a similar topological anyon
content as the conventional T-Pfaffian Chen  _et al._ (2014) surface state of
a single-body topological insulator. Except the charge assignment of each
anyon is $1/3$ of the conventional one. Also, there is a threefold increase of
periodicity. In particular, the spin-$1/2$ quasiparticle $\psi_{4}$ in the
$\mathcal{T}$-$\mathrm{Pf}^{\ast}$ takes the role of the fermionic parton and
carries electric charge $e/3$. $\psi_{12}$ is the electronic quasiparticle and
$1_{24}$ is the charge $2e$ bosonic Cooper pair.

###  II.3 Gluing and splitting

Figure 2: Gluing and splitting between a pair of parton Pfaffian CFTs and a
Dirac parton triplet. The Dirac parton and parton Pfaffian CFTs are defined in
(10) and (23). The chiral central charge $c$ and “filling fraction” $\nu$
indicate the differential thermal and electric conductance
$\kappa=c(\pi^{2}k_{B}^{2}/3h)T$ and $\sigma=\nu(e^{2}/h)$ respectively.

A pair of parton Pfaffian channels can be glued into a parton Dirac channel
through an anyon condensation Bais and Slingerland (2009) process

$\displaystyle
U(3)_{1}/\mathbb{Z}_{3}=\mathrm{Pf}^{\ast}\boxtimes\mathrm{Pf}^{\ast},$ (42)

which can be carried out explicitly by a sine-Gordon Hamiltonian. We begin
with a pair of parton Pfaffian theories
$\mathrm{Pf}^{\ast}_{A}\otimes\mathrm{Pf}^{\ast}_{B}$, each described by (23).
First, we focus on the two neutral Ising sectors
$\overline{\mathrm{Ising}}_{A}\otimes\overline{\mathrm{Ising}}_{B}$. By
condensing the bosonic fermion pair $[\psi]_{A}\otimes[\psi]_{B}$, they
becomes

$\displaystyle\overline{SO(2)_{1}}=\overline{\mathrm{Ising}}_{A}\boxtimes\overline{\mathrm{Ising}}_{B}$
(43)

where the product notation $\boxtimes$ here stands for a tensor product and
the condensation of the fermion pair. We define the electrically neutral spin
$h=-1/2$ Dirac fermion

$\displaystyle d_{0}=\frac{1}{\sqrt{2}}\left(\psi_{A}+i\psi_{B}\right)\sim
e^{i2\phi_{0}}$ (44)

The neutral fermion $d_{0}$ is fractional, however it can be made local
electronic by combining with the charge $e$ boson from either one of the
charge sectors $U(1)^{A}_{24}$ or $U(1)^{B}_{24}$. The bosonized variable
$\phi_{0}$ is described by the Lagrangian density

$\displaystyle\mathcal{L}_{0}=\frac{1}{2\pi}K_{0}\partial_{\mathsf{t}}\phi_{0}\partial_{\mathsf{x}}\phi_{0}$
(45)

up to non-universal velocity terms, where $K_{0}=-4$. The vertex operators
$s_{\pm}=e^{\pm i\phi_{0}}$ are spin $h=-1/8$ spinor fields that originated
from the pair of Ising twist fields

$\displaystyle[\sigma]_{A}\otimes[\sigma]_{B}=s_{+}+s_{-}.$ (46)

Similar to the Ising twist fields, these spinor fields $s_{\pm}$ have non-
trivial monodromy with the electronic quasiparticles
$\Psi_{\mathrm{el}_{j}}^{A,B}$ in (32) and are confined. However, they can be
combined with the $\pi$-fluxes in the Abelian sectors $U(1)_{24}^{A,B}$ and
$U(1)_{12}^{A,B}$ to form deconfined excitations. The spin $-2$ boson
$d_{0}^{2}\sim e^{i4\phi_{0}}$ condenses because $\psi\times\psi=1$. This
fixes the fusion rules

$\displaystyle d_{0}\times d_{0}=1,\quad d_{0}\times s_{\pm}=s_{\mp},$
$\displaystyle s_{\pm}\times s_{\pm}=d_{0},\quad s_{\pm}\times s_{\mp}=1.$
(47)

Next, we take the rest of the Abelian components $\left[U(1)_{24}^{A}\times
U(1)_{12}^{A}\right]\times\left[U(1)_{24}^{B}\times U(1)_{12}^{B}\right]$ into
account. The parton Pfaffian pair is now described by the five-component
bosonized Lagrangian density

$\displaystyle\mathcal{L}_{\mathrm{Pf}^{\ast}\otimes\mathrm{Pf}^{\ast}}$
$\displaystyle=\mathcal{L}_{0}+\mathcal{L}_{d}^{A}+\mathcal{L}_{d}^{B}+\mathcal{L}_{\rho}^{A}+\mathcal{L}_{\rho}^{B},$
(48) $\displaystyle\mathcal{L}_{d}^{A,B}$
$\displaystyle=\frac{1}{2\pi}K_{d}\partial_{\mathsf{t}}\phi_{d}^{A,B}\partial_{\mathsf{x}}\phi_{d}^{A,B},$
$\displaystyle\mathcal{L}_{\rho}^{A,B}$
$\displaystyle=\frac{1}{2\pi}K_{\rho}\partial_{\mathsf{t}}\phi_{\rho}^{A,B}\partial_{\mathsf{x}}\phi_{\rho}^{A,B}$

up to non-universal velocity terms, where $K_{\rho}=24$ and $K_{d}=12$ for
both the $A$ and $B$ sectors. The first three bosonzied variables represent
the neutral degrees of freedom and are invariant under $U(1)_{\mathrm{EM}}$.
The final two represent the charged sectors and transforms under
$\phi_{\rho}^{A,B}\to\phi_{\rho}^{A,B}+\Lambda/12$ when electron operators
change by $\Psi_{\mathrm{el}}\to e^{i\Lambda}\Psi_{\mathrm{el}}$.

We define a new basis within the neutral sectors by the transformation

$\displaystyle\begin{pmatrix}\phi^{n0}_{L}\\\ \phi^{n1}_{R}\\\
\phi^{n2}_{R}\end{pmatrix}=2\begin{pmatrix}1&-1&1\\\ -1&2&-1\\\
-1&1&-2\end{pmatrix}\begin{pmatrix}\phi_{0}\\\ \phi_{d}^{A}\\\
\phi_{d}^{B}\end{pmatrix},$ (49)

where the $3\times 3$ matrix is unimodular and has the inverse

$\displaystyle B=\begin{pmatrix}1&-1&1\\\ -1&2&-1\\\
-1&1&-2\end{pmatrix}^{-1}=\begin{pmatrix}3&1&1\\\ 1&1&0\\\
-1&0&-1\end{pmatrix}.$ (50)

The new bosonized variables are described by the Lagrangian density

$\displaystyle\mathcal{L}_{n}$
$\displaystyle=\mathcal{L}_{0}+\mathcal{L}_{d}^{A}+\mathcal{L}_{d}^{B}=\frac{1}{2\pi}(K_{n})_{IJ}\partial_{\mathsf{t}}\phi^{nI}\partial_{\mathsf{x}}\phi^{nJ},$
(51)

where the $K$-matrix (suppressing all zeros) is

$\displaystyle K_{n}=\begin{pmatrix}-3&&\\\ &2&-1\\\
&-1&2\end{pmatrix}=B^{T}\begin{pmatrix}-1&&\\\ &3&\\\ &&3\end{pmatrix}B.$ (52)

Lastly, we introduce the gluing sine-Gordon potential

$\displaystyle\mathcal{H}_{\mathrm{gluing}}$
$\displaystyle=u\cos\left(12\phi^{n0}_{L}-24\phi_{\rho}^{A}+24\phi_{\rho}^{B}\right)$
$\displaystyle=u\cos\left[24\left(\phi_{0}-\phi_{d}^{A}+\phi_{d}^{B}-\phi_{\rho}^{A}+\phi_{\rho}^{B}\right)\right].$
(53)

It is straightforward to check that the angle variable
$\Theta=\phi_{0}-\phi_{d}^{A}+\phi_{d}^{B}-\phi_{\rho}^{A}+\phi_{\rho}^{B}$
satisfies the “Haldane’s nullity condition” Haldane (1995)
$[\Theta(\mathsf{x}),\Theta(\mathsf{x}^{\prime})]=0$. The factor of 24 makes
sure $\mathcal{H}_{\mathrm{gluing}}$ is constructed by integral combinations
of local electronic operators. The potential therefore introduces a finite
excitation energy gap to a counter-propagating pair of boson modes and removes
them from low-energy. It is charge preserving as $\Theta$ is invariant under
$U(1)_{\mathrm{EM}}$. The gapping potential pins a finite ground state
expectation value for $\langle\Theta(\mathsf{x})\rangle$ and condenses the
bosonic combination

$\displaystyle 1\propto e^{i\langle\Theta\rangle}$
$\displaystyle\propto\left\langle
e^{i\phi_{0}}e^{-i(\phi^{A}_{\rho}+\phi^{A}_{d})}e^{i(\phi^{B}_{\rho}+\phi^{B}_{d})}\right\rangle$
$\displaystyle\sim\left\langle
s_{+}{I^{A}_{1}}^{\dagger}I^{B}_{1}\right\rangle=\left\langle{\sigma^{A}_{1}}^{\dagger}\sigma^{B}_{1}\right\rangle.$
(54)

This is because the Ising anyon $\sigma_{1}$ (defined in (39)) is the product
$\sigma I_{1}$ for both the $A$ and $B$ sectors, and from (46), the even
spinor $s_{+}$ is originated from the product $\sigma^{A}\sigma^{B}$. Together
with the bosonic fermion pair $\psi_{0}^{A}\psi_{0}^{B}$ that was condensed
previously in (43), they generate all the electrically neutral bosonic pairs
in $\mathrm{Pf}^{\ast}_{A}\otimes\mathrm{Pf}^{\ast}_{B}$

$\displaystyle\mathcal{B}=\left\\{\begin{array}[]{*{20}c}{I_{4m}^{A}}^{\dagger}I_{4m}^{B},{\psi_{4m}^{A}}^{\dagger}\psi_{4m}^{B},{\psi_{4m+2}^{A}}^{\dagger}I_{4m+2}^{B},\\\
{I_{4m+2}^{A}}^{\dagger}\psi_{4m+2}^{B},{\sigma^{A}_{2m+1}}^{\dagger}\sigma^{B}_{2m+1}\end{array}\right\\}_{m\in\mathbb{Z}}.$
(57)

Bosons in $\mathcal{B}$ have trivial mutual monodromy, and the sine-Gordon
potential (53) condenses all bosons in $\mathcal{B}$. For example, the Ising
pairs take non-vanishing ground state expectation values thanks to
$\mathcal{H}_{\mathrm{gluing}}$

$\displaystyle 1$ $\displaystyle\propto e^{i(2m+1)\langle\Theta\rangle}$
$\displaystyle\propto\left\langle
e^{i(2m+1)\phi_{0}}e^{-i(2m+1)(\phi^{A}_{\rho}+\phi^{A}_{d})}e^{i(2m+1)(\phi^{B}_{\rho}+\phi^{B}_{d})}\right\rangle$
$\displaystyle\sim\left\langle
s_{\pm}{I^{A}_{2m+1}}^{\dagger}I^{B}_{2m+1}\right\rangle=\left\langle{\sigma^{A}_{2m+1}}^{\dagger}\sigma^{B}_{2m+1}\right\rangle,$
(58)

where the parity of the spinor field is fixed by the fusion rule (47),
$e^{i(2m+1)\phi_{0}}=s_{+}$ (or $s_{-}$) if $m$ is even (resp. odd). We denote
the relative tensor product
$\mathrm{Pf}^{\ast}_{A}\boxtimes_{\mathcal{B}}\mathrm{Pf}^{\ast}_{B}$ to be
the remaining low-energy CFT that is unaffected by the condensation.

From the $K$-matrix (52), it is clear that the two electrically neutral modes
$\phi^{n1}$ and $\phi^{n2}$ are completely decoupled from the rest and
therefore are unaffected by the sine-Gordon potential
$\mathcal{H}_{\mathrm{gluing}}$. It is also straightforward to check that the
sum $\phi_{\rho}=2(\phi_{\rho}^{A}+\phi_{\rho}^{B})$ commutes with $\Theta$
and thus decouples from $\mathcal{H}_{\mathrm{gluing}}$ as well. Here we
normalizes $\phi_{\rho}$ with the factor of 2 because the odd vertices, such
as $e^{i(\phi_{\rho}^{A}+\phi_{\rho}^{B})}$, have non-trivial monodromy with
the electronic quasiparticles $\psi_{12}^{A}$ and $\psi_{12}^{B}$ and are not
allowed by electron locality. Grouping together, the bosonized variables
$(\phi^{0}_{\pi},\phi^{1}_{\pi},\phi^{2}_{\pi})=(\phi_{\rho},\phi^{n1},\phi^{n2})$
generate the relative tensor product
$\mathrm{Pf}^{\ast}_{A}\boxtimes_{\mathcal{B}}\mathrm{Pf}^{\ast}_{B}$. They
obey the same equal-time commutation relation in (15), and therefore are
described by the same $U(1)_{3}\times SU(3)_{1}$ parton Lagrangian density
(13).

Moreover, the bosonized variables $(\phi_{\rho},\phi^{n1},\phi^{n2})$ respect
the same charge assignment and electron locality as the Dirac parton triplet.
$\phi_{\rho}$ is the only bosonized variable that transform under
$U(1)_{\mathrm{EM}}$. The quasiparticle
$e^{i\phi_{\rho}}=e^{i2(\phi_{\rho}^{A}+\phi_{\rho}^{B})}$ carries charge
$e/3$ and represents the Laughlin quasiparticle. We now show the triple
$e^{i3\phi_{\rho}}=e^{i6(\phi_{\rho}^{A}+\phi_{\rho}^{B})}$ is an integral
electronic combination. First, it can be changed into
$I_{-6}^{A}\psi_{6}^{B}\sim
e^{i6(-\phi_{\rho}^{A}-\phi_{d}^{A}+\phi_{\rho}^{B}+\phi_{d}^{B})}\psi^{B}$ by
absorbing the electronic combination
${\Psi_{\mathrm{el}_{2}}^{A}}^{\dagger}\Psi_{\mathrm{el}_{1}}^{B}{\Psi_{\mathrm{el}_{3}}^{B}}^{\dagger}=e^{-i(12\phi_{\rho}^{A}+6\phi_{d}^{A})}e^{i6\phi_{d}}\psi^{B}_{0}$.
Second, we observe that
$I_{-6}^{A}\psi_{6}^{B}={I_{6}^{A}}^{\dagger}\psi_{6}^{B}$ is one of the boson
in (57) that condensed to the ground state. This proves $e^{i3\phi_{\rho}}$
represents an electronic quasiparticle after the condensation. Furthermore, in
the neutral sector, the simple $SU(3)$ roots $e^{i(2\phi^{n1}-\phi^{n2})}$ and
$e^{i(-\phi^{n1}+2\phi^{n2})}$ are also local electronic. Using the basis
transformation (49), the roots are $e^{-i(2\phi_{0}-6\phi_{d}^{A})}$ and
$e^{-i(2\phi_{0}+6\phi_{d}^{B})}$, which are respectively identical to the
electronic operators
$\psi^{A}_{0}e^{i6\phi_{d}^{A}}=\Psi_{\mathrm{el}_{1}}^{A}{\Psi_{\mathrm{el}_{3}}^{A}}^{\dagger}$
and
$\psi^{B}_{0}e^{-i6\phi_{d}^{B}}={\Psi_{\mathrm{el}_{1}}^{B}}^{\dagger}\Psi_{\mathrm{el}_{3}}^{B}$
up to the condensate $\psi^{A}_{0}\psi^{B}_{0}$.

This concludes the gluing procedure

$\displaystyle\mathrm{Pf}^{\ast}\otimes\mathrm{Pf}^{\ast}\xrightarrow{\mathcal{H}_{\mathrm{gluing}}}$
$\displaystyle\mathrm{Pf}^{\ast}\boxtimes_{\mathcal{B}}\mathrm{Pf}^{\ast}=U(3)_{1}/\mathbb{Z}_{3}.$
(59)

Similar gluing procedure was also presented by us in an earlier work Sahoo
_et al._ (2017).

Figure 3: Splitting of a chiral Dirac triplet channel into a pair of parton
Pfaffian channels. The basis transformations (I), (II), (III) and (IV) are
defined in (60), (63), (68) and (70) respectively. The backscattering
Hamiltonian is defined in (62).

Next, we present the reversal – the splitting of the chiral Dirac parton
triplet $U(3)_{1}/\mathbb{Z}_{3}$ into a pair of decoupled parton Pfaffian
CFTs. The fractionalization is facilitated by an extension of the Dirac
partons that includes an additional counter-propagating pair of neutral
spin-$3/2$ Dirac fermions, $U(1)_{3}\times\overline{U(1)_{3}}$. We first show
that the extension can be supported in a purely 1D setting, such as an edge
reconstruction, and does not require additional 2D topological order. In other
words, the non-chiral $U(1)_{3}\times\overline{U(1)_{3}}$ CFT can be realized
in a 1D electronic wire without being holographically supported on the
boundary of a 2D topological state. To achieve this, we start with two
electronic wires that contains two counter-propagating pairs of Dirac
electrons $c^{\sigma}_{a}\sim e^{i\tilde{\Phi}^{\sigma}_{a}}$ at Fermi level,
where $a=0,1$ is the wire index and $\sigma=R,L=+,-$ labels the propagating
direction. They are represented by the solid black lines in figure 3. We will
introduce a many-body interacting potential that leaves behind a pair of
neutral Dirac modes in low-energy.

The interaction is based on the following fractional basis transformation of
the bosonized variables

$\displaystyle\begin{pmatrix}\tilde{\Phi}^{R}_{\rho}\\\
\tilde{\Phi}^{R}_{n}\\\ \tilde{\Phi}^{L}_{\rho}\\\
\tilde{\Phi}^{L}_{n}\end{pmatrix}=\begin{pmatrix}3/2&1&-1/2&-1\\\
-1/2&1&-1/2&0\\\ -1/2&-1&3/2&1\\\
-1/2&0&-1/2&1\end{pmatrix}\begin{pmatrix}\tilde{\Phi}^{R}_{0}\\\
\tilde{\Phi}^{R}_{1}\\\ \tilde{\Phi}^{L}_{0}\\\
\tilde{\Phi}^{L}_{1}\end{pmatrix}.$ (60)

This transformation will also be useful in the coupled wire construction in
section III.2 (c.f. eq.(143)). We notice that the new bosonized variables
contain half-integral combinations of the original ones. Consequently, the
vertex operators
$e^{i\tilde{\Phi}^{\sigma}_{\rho}},e^{i\tilde{\Phi}^{\sigma}_{n}}$ are non-
local and cannot be expressed as integral combination of electronic ones. On
the other hand, the sum and differences
$\tilde{\Phi}^{R}_{\rho}\pm\tilde{\Phi}^{L}_{\rho}$,
$\tilde{\Phi}^{R}_{n}\pm\tilde{\Phi}^{L}_{n}$ and
$\tilde{\Phi}^{\sigma}_{\rho}\pm\tilde{\Phi}^{\sigma^{\prime}}_{n}$ are
integral. The variables $\tilde{\Phi}^{R,L}_{\rho}$ represent electrically
charged sector and shift by
$\tilde{\Phi}^{\sigma}_{\rho}\to\tilde{\Phi}^{\sigma}_{\rho}+\Lambda$ under a
$U(1)_{\mathrm{EM}}$ transformation that change the phase of an electron $c\to
e^{i\Lambda}c$. The other two variables $\tilde{\Phi}^{R,L}_{n}$ represent
electrically neutral modes and are invariant under $U(1)_{\mathrm{EM}}$. The
bosonized variables are described by the Lagrangian density

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2\pi}\sum_{\sigma=+,-}\sum_{a=0,1}\sigma\partial_{\mathsf{t}}\tilde{\Phi}^{\sigma}_{a}\partial_{\mathsf{x}}\tilde{\Phi}^{\sigma}_{a}$
$\displaystyle=\frac{1}{2\pi}\sum_{\sigma=+,-}\sigma\left(\frac{1}{2}\partial_{\mathsf{t}}\tilde{\Phi}^{\sigma}_{\rho}\partial_{\mathsf{x}}\tilde{\Phi}^{\sigma}_{\rho}+\partial_{\mathsf{t}}\tilde{\Phi}^{\sigma}_{n}\partial_{\mathsf{x}}\tilde{\Phi}^{\sigma}_{n}\right)$
(61)

up to non-universal velocity terms.

We introduce the sine-Gordon potential

$\displaystyle\mathcal{H}_{\rho}$
$\displaystyle=u\cos\left(\tilde{\Phi}^{R}_{\rho}-\tilde{\Phi}^{L}_{\rho}\right)$
$\displaystyle=u\cos\left(2\tilde{\Phi}^{R}_{0}+2\tilde{\Phi}^{R}_{1}-2\tilde{\Phi}^{L}_{0}-2\tilde{\Phi}^{L}_{1}\right),$
(62)

which can be constructed from electron backscattering and preserves charge
$U(1)_{\mathrm{EM}}$. This gives a finite excitation energy gap to the charged
sector $\rho$ and removes it from low-energy, and leaves behind a counter-
propagating pair of spin-$1/2$ neutral Dirac fermions
$c^{i\tilde{\Phi}^{R,L}_{n}}$. Next, we perform another fractional basis
transformation

$\displaystyle\begin{pmatrix}\phi^{R}_{D}\\\
\phi^{L}_{D}\end{pmatrix}=\frac{1}{3}\begin{pmatrix}2&-1\\\
-1&2\end{pmatrix}\begin{pmatrix}\tilde{\Phi}^{R}_{n}\\\
\tilde{\Phi}^{L}_{n}\end{pmatrix}.$ (63)

The new variables are described by the Lagrangian density

$\displaystyle\mathcal{L}_{D}$
$\displaystyle=\frac{1}{2\pi}\sum_{\sigma=+,-}\sigma\partial_{\mathsf{t}}\tilde{\Phi}^{\sigma}_{n}\partial_{\mathsf{x}}\tilde{\Phi}^{\sigma}_{n}$
$\displaystyle=\frac{1}{2\pi}\sum_{\sigma=+,-}\sigma
K_{D}\partial_{\mathsf{t}}\phi^{\sigma}_{D}\partial_{\mathsf{x}}\phi^{\sigma}_{D},$
(64)

where $K_{D}=3$. The composite vertices
$e^{i3\phi^{R}_{D}}=e^{i(2\tilde{\Phi}^{R}_{n}-\tilde{\Phi}^{L}_{n})}$ and
$e^{i3\phi^{L}_{D}}=e^{i(-\tilde{\Phi}^{R}_{n}+2\tilde{\Phi}^{L}_{n})}$ are
two decoupled counter-propagating spin $\pm 3/2$ neutral Dirac fermions, and
generate the non-chiral $U(1)_{3}\times\overline{U(1)_{3}}$ CFT. They are
represented by the dashed blue lines in figure 3.

Before proceeding, we notice that these neutral fermions are not integral
combinations of the original electrons. Therefore, to be precise, the bosonic
doubles $e^{i6\phi^{\sigma}_{D}}$ should be regarded as the local fundamental
constituents instead. This changes the compactification radius and the level
of the $U(1)$ CFT. In additional to the original primary fields
$e^{ia_{\sigma}\phi^{\sigma}_{D}}$, this also allows $\pi$-fluxes to the
neutral Dirac fermions, which are represented by half-vertex twist fields
$e^{ia_{\sigma}\phi^{\sigma}_{D}/2}$. We rescale
$\phi^{\sigma}_{d}=\phi^{\sigma}_{D}/2$, which turns (64) into

$\displaystyle\mathcal{L}_{D}=\frac{1}{2\pi}\sum_{\sigma=+,-}\sigma
K_{d}\partial_{\mathsf{t}}\phi^{\sigma}_{d}\partial_{\mathsf{x}}\phi^{\sigma}_{d},$
(65)

where $K_{d}=12$ and matches the $\mathcal{L}_{d}$ in (25) and (48). The
rescaled bosonized variables $\phi^{R,L}_{d}$ generate the non-chiral
$U(1)_{12}\times\overline{U(1)_{12}}$ CFT. The inclusion of the $\pi$-fluxes
allows additional characters that correspond to anti-periodic boundary
conditions of the Dirac fermion in a closed $(1+1)$D system. This process is
known as $\mathbb{Z}_{2}$-orbifolding Di Francesco  _et al._ (1999) in the CFT
context, and in this case, it extends $U(1)_{3}$ to
$U(1)_{3}/\mathbb{Z}_{2}=U(1)_{12}$. Nevertheless, to simplify the notations,
we suppress the rescaling/orbifolding and return back to the previous
normalization $\mathcal{L}_{D}$ in (64) until the subtlety of electron
locality arises again.

We now combine this counter-propagating pair of neutral Dirac fermions with
the Dirac parton triplet. To avoid confusion, we differentiate the auxiliary
and the parton sectors by $D$ and $\pi$. We denote the auxiliary neutral Dirac
fermion sectors by $U(1)^{D}_{3}\times\overline{U(1)^{D}_{3}}$ and its
bosonized variables by $\phi^{R,L}_{D}$. We denote the the Dirac partons by
$U(1)^{\pi}_{3}\times SU(3)_{1}^{\pi}$, which we recall is identical to
$U(3)_{1}^{\pi}/\mathbb{Z}_{3}$ through a basis transformation (11), and its
(right-moving) bosonzied variables by
$\phi^{0}_{\pi},\phi^{1}_{\pi},\phi^{2}_{\pi}$. The total Lagrangian density
is the combination of (13) and (64). We first perform a basis transformation
between $U(1)^{\pi}_{3}\times U(1)^{D}_{3}$

$\displaystyle\begin{array}[]{*{20}l}4\phi^{A}_{\rho}=\phi^{0}_{\pi}+\phi^{R}_{D}\\\
4\phi^{B}_{\rho}=\phi^{0}_{\pi}-\phi^{R}_{D}\end{array}.$ (68)

Recall $\phi^{0}_{\pi}$ generates the charged Laughlin sector, and
$\lambda\sim e^{i\phi^{0}_{\pi}}$ is the Laughlin $q=e/3$ quasiparticle. The
$A$ and $B$ bosons here correspond to two decoupled charged sectors
$U(1)^{A}_{24}\times U(1)^{B}_{24}$

$\displaystyle\mathcal{L}_{\pi}^{0}+\mathcal{L}_{D}^{R}$
$\displaystyle=\frac{1}{2\pi}\left(3\partial_{\mathsf{t}}\phi^{0}_{\pi}\partial_{\mathsf{x}}\phi^{0}_{\pi}+3\partial_{\mathsf{t}}\phi^{R}_{D}\partial_{\mathsf{x}}\phi^{R}_{D}\right)=\mathcal{L}^{A}_{\rho}+\mathcal{L}^{B}_{\rho},$
$\displaystyle\mathcal{L}_{\rho}^{A,B}$
$\displaystyle=\frac{1}{2\pi}K_{\rho}\partial_{\mathsf{t}}\phi^{A,B}_{\rho}\partial_{\mathsf{x}}\phi^{A,B}_{\rho},$
(69)

where $K_{\rho}=24$ which is identical to that of (25) and (48). The bosons
are shifted by $\phi^{A,B}_{\rho}\to\phi^{A,B}_{\rho}+\Lambda/12$ under a
$U(1)_{\mathrm{EM}}$ transformation that adds the phase $c\to e^{i\Lambda}c$
to electrons. Each of the charged sectors carries the differential electric
conductance $\sigma^{A,B}=(1/6)e^{2}/h$, and the two add up to the conductance
of the Laughlin channel.

There are three modes $\phi^{L}_{D},\phi^{1}_{\pi},\phi^{2}_{\pi}$ remaining
and they correspond to the electrically neutral sectors
$\overline{U(1)^{D}_{3}}\times SU(3)_{1}^{\pi}$. We rotate to a new basis

$\displaystyle\begin{pmatrix}\phi_{0}\\\ \phi^{A}_{d}\\\
\phi^{B}_{d}\end{pmatrix}=\frac{1}{2}B\begin{pmatrix}\phi^{L}_{D}\\\
\phi^{1}_{\pi}\\\ \phi^{2}_{\pi}\end{pmatrix}$ (70)

using the unimodular $B$ transformation defined in (50). This turns the
neutral sectors into $\overline{SO(2)_{1}}\times U(1)_{12}^{A}\times
U(1)_{12}^{B}$

$\displaystyle\mathcal{L}^{L}_{D}+\mathcal{L}_{SU(3)_{1}^{\pi}}$
$\displaystyle=-\frac{1}{2\pi}3\partial_{\mathsf{t}}\phi^{L}_{D}\partial_{\mathsf{x}}\phi^{L}_{D}$
$\displaystyle\;\;\;+\frac{1}{2\pi}\sum_{i,j=1,2}K_{ij}^{SU(3)}\partial_{\mathsf{t}}\phi^{i}_{\pi}\partial_{\mathsf{x}}\phi^{j}_{\pi}$
$\displaystyle=\mathcal{L}_{\overline{SO(2)_{1}}}+\mathcal{L}^{A}_{d}+\mathcal{L}^{B}_{d},$
(71) $\displaystyle\mathcal{L}_{\overline{SO(2)_{1}}}$
$\displaystyle=\frac{1}{2\pi}K_{0}\partial_{\mathsf{t}}\phi_{0}\partial_{\mathsf{x}}\phi_{0},$
$\displaystyle\mathcal{L}^{A,B}_{d}$
$\displaystyle=\frac{1}{2\pi}K_{d}\partial_{\mathsf{t}}\phi_{d}^{A,B}\partial_{\mathsf{x}}\phi_{d}^{A,B},$

where $K_{0}=-4$ and $K_{d}=12$, both of which match that of the previous
Lagrangian densities (45) and (25) respectively, and $K^{SU(3)}$ is the
$2\times 2$ Cartan matrix of $SU(3)$ defined in the lower $2\times 2$ block of
$\mathcal{L}_{n}$ in (52).

Lastly, we decompose the spin $1/2$ Dirac fermion $d_{0}\sim
e^{i2\phi_{0}}=(\psi_{A}+i\psi_{B})/\sqrt{2}$ into Majorana components (see
eq.(44)). They generate two decoupled Ising sectors
$\overline{\mathrm{Ising}^{A}}\times\overline{\mathrm{Ising}^{B}}$. This
completes the splitting process, which can be summarized by the following flow
chart.

$\displaystyle\begin{diagram}$ (72)

We conclude this subsection by addressing electron locality in the splitting
process. First, we show that

$\displaystyle\openone_{12,\pm 6}^{A,B}=e^{i12\phi_{\rho}^{A,B}}e^{\pm
i6\phi_{d}^{A,B}},\quad\Psi_{12,0}^{A,B}=e^{i12\phi_{\rho}^{A,B}}\psi_{A,B}$
(73)

are electronic quasiparticles in both sectors (c.f. eq.(32)). Without loss of
generality, we illustrate this on the $A$ sector. Written in terms of the
basis of the original Dirac partons and the auxiliary electrons,

$\displaystyle\openone_{12,\pm 6}^{A}$
$\displaystyle=e^{i3\phi^{0}_{\pi}+i3\phi^{R}_{D}}e^{\pm i3\phi^{L}_{D}\pm
i3\phi^{1}_{\pi}}$
$\displaystyle=e^{i3\phi^{0}_{\pi}}e^{i3(\phi^{R}_{D}\pm\phi^{L}_{D})}e^{\pm
i3\phi^{1}_{\pi}}$ (74) $\displaystyle\Psi_{12,0}^{A}$
$\displaystyle=e^{i3\phi^{0}_{\pi}+i3\phi^{R}_{D}}\cos\left(3\phi^{L}_{D}+\phi^{1}_{\pi}+\phi^{2}_{\pi}\right)$
$\displaystyle\sim
e^{i3\phi^{0}_{\pi}}e^{i3(\phi^{R}_{D}+\phi^{L}_{D})}e^{i(\phi^{1}_{\pi}+\phi^{2}_{\pi})}$
$\displaystyle\;\;\;+e^{i3\phi^{0}_{\pi}}e^{i3(\phi^{R}_{D}-\phi^{L}_{D})}e^{-i(\phi^{1}_{\pi}+\phi^{2}_{\pi})}.$
(75)

The first piece $e^{i3\phi^{0}_{\pi}}$ is the combination of three Laughlin
quasiparticles, and is identical to the electronic spin $3/2$ Dirac fermion
$\lambda^{3}\sim\pi^{1}\pi^{2}\pi^{3}$. From the basis transformations (60)
and (63), it is straightforward to check that the second pieces
$e^{i3(\phi^{R}_{D}+\phi^{L}_{D})}=e^{i(\tilde{\Phi}^{R}_{0}-\tilde{\Phi}^{R}_{1}-\tilde{\Phi}^{L}_{0}+\tilde{\Phi}^{L}_{1})}$
and
$e^{i3(\phi^{R}_{D}-\phi^{L}_{D})}=e^{i(3\tilde{\Phi}^{R}_{0}-3\tilde{\Phi}^{L}_{1})}$
are integral combination of the auxiliary electrons. Using the basis
transformation (11), the third and last pieces are
$e^{i3\phi^{1}_{\pi}}=e^{i(2\tilde{\phi}_{1}-\tilde{\phi}_{2}-\tilde{\phi}_{3})}$
and
$e^{i(\phi^{1}_{\pi}+\phi^{2}_{\pi})}=e^{i(\tilde{\phi}_{1}-\tilde{\phi}_{3})}$,
which are identical to the neutral local electronic combinations
$E^{12}E^{13}=(\pi^{1})^{2}(\pi^{2})^{\dagger}(\pi^{3})^{\dagger}$ and
$E^{13}=\pi^{1}(\pi^{3})^{\dagger}$ respectively. The electron locality of
$\pi^{1}\pi^{2}\pi^{3}$ and $\pi^{a}(\pi^{b})^{\dagger}$ will be shown later
in section III.1 when the Dirac parton triplet is constructed explicitly from
electronic wires. In particular, $\pi^{1}\pi^{2}\pi^{3}\sim
e^{i\Phi^{R}_{\rho}}$, $\pi^{1}(\pi^{2})^{\dagger}\sim e^{i\Phi^{R}_{n1}}$ and
$\pi^{2}(\pi^{3})^{\dagger}\sim e^{i\Phi^{R}_{n2}}$ can be expressed
explicitly using (92) and (98) (also see figure 4) in terms of electronic
operators. It is important to acknowledge that the parton Pfaffian channels
descend from electronic degrees of freedom. Electron locality forbids
excitations that have non-trivial monodromy to any electronic quasiparticle.
This restriction was addressed in (33) and will not be repeated here.

##  III Coupled wire models of fractional quantum Hall states

In this section, we present the coupled wire models that represent the parton
fractional quantum Hall (FQH) states $\mathfrak{L}_{q}$ and
$\mathfrak{P}_{p,q}$ (see table 1 and (3)). In section III.1, we construct a
sequence of Abelian FQH states $\mathfrak{L}_{q}$ at filling $\nu=1/3$. Each
of them carries a Laughlin $U(1)_{3}$ charge sector and $q$ copies of neutral
$SU(3)_{1}$ sectors that support the deconfinement of partons (except for
$q=0$, which corresponds to the Laughlin state where partons are confined). In
particular, $\mathcal{L}_{1}$ represents the FQH state where each of the three
deconfined partons fills a Landau level. Its $1+1$D boundary hosts the Dirac
parton triplet conformal field theory (CFT)
$U(3)_{1}/\mathbb{Z}_{2}=U(1)_{3}\times SU(3)_{1}$, which was discussed in
detail in section II.1. This new parton FQH state is topologically equivalent
to and inspired by a 3D fractional topological insulator (FTI) slab Sahoo  _et
al._ (2017) with finite thickness and time-reversal (TR) symmetry breaking
surfaces (see figure 1).

In section III.2, we construct a sequence of incompressible FQH states
$\mathfrak{P}_{p,q}$ at filling $\nu=1/6$. We speculate that they may
originate from the transition between an integral quantum Hall plateau and a
$\nu=n+1/3$ plateau occupied by one of the $\mathfrak{L}_{q}$ states. The
compressible parton liquid at the transition then gains a many-body excitation
energy gap by a parton pairing mechanism. It is out of the scope of this paper
to address the compressible parton liquid theory that may describe an integral
to fractional quantum Hall plateau transition and relate to the surface state
of a FTI. Instead, we focus on incompressible FQH states that exhibit parton
pairing. The $\mathfrak{P}_{p,q}$ state carries a $U(1)_{24}$ charge sector,
which has three times the periodicity of the $U(1)_{8}$ charge sector of the
Moore-Read Pfaffian state Read and Moore (1991) due to the charge $e/3$
partons. Its neutral sector consists of $q$ copies of $U(1)_{12}$, each
contains a neutral spin $3/2$ Dirac fermion, and $p$ Ising copies, each
generated by a neutral spin $1/2$ Majorana fermion. In particular, the
boundary of $\mathfrak{P}_{-1,1}$ supports the parton Pfaffian CFT
$\mathrm{Pf}^{\ast}=U(1)_{24}\otimes_{\mathrm{el}}U(1)_{12}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}$,
which was presented in section II.2. Similar to $\mathfrak{L}_{1}$,
$\mathfrak{P}_{-1,1}$ is also equivalent to a 3D FTI slab with a TR symmetric
and a TR breaking surface (see figure 1). Moreover, as shown in section II.3,
the parton Pfaffian CFT is exactly half of the Dirac parton triplet. This
infers that the parton Landau levels $\mathcal{L}_{1}$ can be split into a
pair of parton Pfaffian FQH states $\mathfrak{P}_{-1,1}$. In other words,
$\mathfrak{P}_{-1,1}$ is symmetric under a parton “particle-hole” conjugation,
because the “particle” state identical to the “hole” state, which is obtained
from subtracting the parton Pfaffian $\mathfrak{P}_{-1,1}$ from the parton
Landau levels $\mathcal{L}_{1}$. This new notion of parton “particle-hole”
conjugation will be discussed in the next section.

The construction of these exactly solvable FQH models relies on the coupled
wire method. It was pioneered by Kane, Mukhopadhyay and Lubensky Kane  _et
al._ (2002) in modelling the Laughlin Laughlin (1983) and Haldane-Halperin
hierarchy Haldane (1983); Halperin (1984) FQH states. The construction begins
with a 2D array of parallel metallic electron wires under a magnetic field.
Under the Lorentz gauge $A_{\mathsf{x}}=-B\mathsf{y}$, the Fermi momenta of
the wires are displaced $k_{f}\to k_{f}+(e/\hbar c)B\mathsf{y}$. Many-body
interactions are restricted only to momentum preserving combinations of inter-
wire and intra-wire electron backscatterings. A combination with unbalanced
momentum contains an oscillating factor $e^{ik_{\mathrm{total}}\mathsf{x}}$
and vanishes upon the integration of $\mathsf{x}$. These backscattering
combinations may involve multiple electrons, and take the form of
${c^{\sigma_{1}}_{a_{1}}}^{\dagger}\ldots{c^{\sigma_{n}}_{a_{n}}}^{\dagger}c^{\sigma_{n+1}}_{b_{1}}\ldots
c^{\sigma_{2n}}_{b_{n}}$, where $\sigma=R,L$ labels the propagating direction
of the plane wave electron modes $c^{\sigma}_{a}$ at Fermi level, and $a$ is
some wire index. They can be generated by higher order corrections to the
interacting action in the partition function
$Z=\int\mathcal{D}c\mathcal{D}c^{\dagger}\exp\left[i\left(S_{\mathrm{kinetic}}+S_{\mathrm{interaction}}\right)\right]$,
for some bare local interaction such as
$S_{\mathrm{interaction}}=\sum_{\mathsf{y}}\int\prod_{r=1,2,3,4}d\mathsf{x}_{r}u^{abcd}_{\mathsf{x}_{1}\mathsf{x}_{2}\mathsf{x}_{3}\mathsf{x}_{4}}c^{\dagger}_{a}(\mathsf{x}_{1})c^{\dagger}_{b}(\mathsf{x}_{2})c_{c}(\mathsf{x}_{3})c_{d}(\mathsf{x}_{4})$,
where $c_{a}(\mathsf{x})$ is the electron (annihilation) operator at position
$\mathsf{x}$. The relevance of these backscattering combinations in the
renormalization group sense depends on forward scattering interactions. In
this paper, we do not address the origins and energetics of these
backscattering terms. Instead, we take the strong coupling limit and study the
gapped topological state associated to a given backscattering Hamiltonian.
These models are exactly solvable in the sense that they consist of mutually
commuting and non-competing terms. They freeze all low-energy degrees of
freedom in the 2D bulk but leave behind chiral 1D boundary modes.

###  III.1 Filling one-third

Figure 4: Bundle arrangement and basis transformation for the $\nu=1/3$ parton
FQH state. The momentum displacement between channels (directed lines) and
bundles (green cylinders) are shown in $nk_{f}$, where $2k_{f}$ is the
displacement between $R$ and $L$-movers on same wire at Fermi level. Dashed
boxes represent products of electronic operators that correspond to (92) and
(98). $\bullet$ and $\circ$ represents electron creation and annihilation
operators respectively.

We begin with an array of electronic bundles. Each bundle consists of five
wires, and each wire carries a right ($R$) and a left ($L$) moving electronic
Dirac fermion channel that are separated in momentum space by $2k_{f}$ at
Fermi level. We label each bundle by an integer $y$ which represents its
vertical position. The wires within a bundle are labeled by $a=0,\ldots,4$. We
bosonize the Dirac fermions so that the electron (annihilation) operator at
$(y,a,\sigma)$, for $\sigma=R,L=+,-$, is represented by the vertex operator

$\displaystyle
c_{ya}^{\sigma}(\mathsf{x})\sim\exp\left[i\left(\tilde{\Phi}^{\sigma}_{ya}(\mathsf{x})+k^{\sigma}_{ya}\mathsf{x}\right)\right],$
(76)

where $k^{\sigma}_{ya}$ is the $\mathsf{x}$-momentum of the Dirac fermion. The
bosonized variables obey the equal-time commutation relation (ETCR)

$\displaystyle\left[\partial_{\mathsf{x}}\tilde{\Phi}_{ya}^{\sigma}(\mathsf{x}),\tilde{\Phi}_{y^{\prime}a^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}\delta_{aa^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime}).$
(77)

The complete commutation relation that includes the zero modes is presented in
(253) in appendix A.2.

The filling fraction of the system under a perpendicular magnetic field ${\bf
B}=B\hat{\bf z}$ is

$\displaystyle\nu=\frac{N_{e}}{N_{\Phi_{B}}}=\frac{5(2k_{f})/(2\pi)}{Bd/(hc/e)}=\frac{5(2k_{f})}{\left(\frac{eB}{\hbar
c}\right)d},$ (78)

where $N_{e}=5(2k_{f})/(2\pi)$ is the number electrons per unit length in a
bundle of 5 wires, $N_{\Phi_{B}}=Bd/\phi_{0}$ is the number of fluxes (in
units of the flux quantum $\phi_{0}=hc/e$) between adjacent bundles per
length, and $d$ is the bundle displacement in $\mathsf{y}$. Under the Lorentz
gauge ${\bf A}=-B\mathsf{y}\hat{\bf x}$, the field shifts the
$\mathsf{x}$-momentum

$\displaystyle k^{\sigma}_{ya}=\frac{e}{\hbar c}B\mathsf{y}+\sigma
k_{f}=\frac{5(2k_{f})}{\nu}\frac{\mathsf{y}}{d}+\sigma k_{f}$ (79)

of each electronic channel according to its vertical position
$\mathsf{y}=\mathsf{y}(y,a)$. In the coupled wire model, we arrange the wires
so that the channel momentums are given by

$\displaystyle\begin{array}[]{*{20}l}k^{\sigma}_{y0}=(30y+\sigma)k_{f}\\\
k^{\sigma}_{ya}=(30y+4a+4+\sigma)k_{f},\quad\mbox{for $a=1,2,3$}\\\
k^{\sigma}_{y4}=(30y+24+\sigma)k_{f}\end{array}.$ (83)

This pattern of momentum shifting is shown in figure 4. In particular, the
shift of $\mathsf{x}$-momentum between adjacent bundles in this configuration
is

$\displaystyle\delta k=k^{\sigma}_{y+1,a}-k^{\sigma}_{ya}=30k_{f}.$ (84)

This fixes the filling fraction to be $\nu=1/3$ by comparing with (79) and
$d=\mathsf{y}(y+1,a)-\mathsf{y}(y,a)$.

Next, within each bundle, we perform a basis transformation and group the
electronic channels into three counter-propagating pairs of electrically
charged $U(1)_{3}$ sectors

$\displaystyle\begin{array}[]{*{20}l}\Phi^{R}_{y,\rho}=2\tilde{\Phi}^{R}_{y4}-\tilde{\Phi}^{L}_{y4}\\\
\Phi^{R}_{y,c1}=2\tilde{\Phi}^{R}_{y0}-\tilde{\Phi}^{L}_{y0}\\\
\Phi^{R}_{y,c2}=-\tilde{\Phi}^{R}_{y1}+\tilde{\Phi}^{R}_{y2}+3\tilde{\Phi}^{R}_{y3}+2\tilde{\Phi}^{L}_{y1}-2\tilde{\Phi}^{L}_{y3}\end{array},$
(88)
$\displaystyle\begin{array}[]{*{20}l}\Phi^{L}_{y,\rho}=2\tilde{\Phi}^{L}_{y0}-\tilde{\Phi}^{R}_{y0}\\\
\Phi^{L}_{y,c1}=2\tilde{\Phi}^{L}_{y4}-\tilde{\Phi}^{R}_{y4}\\\
\Phi^{L}_{y,c2}=3\tilde{\Phi}^{L}_{y1}+\tilde{\Phi}^{L}_{y2}-\tilde{\Phi}^{L}_{y3}-2\tilde{\Phi}^{R}_{y1}+2\tilde{\Phi}^{R}_{y3}\end{array},$
(92)

and a pair of electrically neutral $SU(3)_{1}$ ones

$\displaystyle\begin{array}[]{*{20}l}\Phi^{R}_{y,n1}=2\tilde{\Phi}^{R}_{y1}-\tilde{\Phi}^{L}_{y1}-\tilde{\Phi}^{L}_{y2}\\\
\Phi^{R}_{y,n2}=2\tilde{\Phi}^{R}_{y2}-\tilde{\Phi}^{L}_{y2}-\tilde{\Phi}^{L}_{y3}\end{array},$
(95)
$\displaystyle\begin{array}[]{*{20}l}\Phi^{L}_{y,n1}=2\tilde{\Phi}^{L}_{y2}-\tilde{\Phi}^{R}_{y2}-\tilde{\Phi}^{R}_{y1}\\\
\Phi^{L}_{y,n2}=2\tilde{\Phi}^{L}_{y3}-\tilde{\Phi}^{R}_{y3}-\tilde{\Phi}^{R}_{y2}\end{array}.$
(98)

These new local combinations are diagrammatically represented in figure 4.
They obey the equal-time commutation relation

$\displaystyle\left[\partial_{\mathsf{x}}\Phi_{y,\mu}^{\sigma}(\mathsf{x}),\Phi_{y^{\prime},\mu^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}K_{\mu\mu^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime}),$
(99)

where the $K$-matrix (suppressing all 0’s) is

$\displaystyle K=\begin{pmatrix}3&&&&\\\ &3&&&\\\ &&3&&\\\ &&&2&-1\\\
&&&-1&2\end{pmatrix}$ (100)

and the bozonized variables $\Phi^{\sigma}_{y,\mu}$ are ordered in
$(\Phi^{\sigma}_{y,\rho},\Phi^{\sigma}_{y,c1},\Phi^{\sigma}_{y,c2},\Phi^{\sigma}_{y,n1},\Phi^{\sigma}_{y,n2})$.
The external electromagnetic gauge transformation, that rotates the phases of
electronic operators by $e^{i\tilde{\Phi}_{ya}}\to
e^{i\Lambda}e^{i\tilde{\Phi}_{ya}}$, shifts the new local bosonized variables
by

$\displaystyle\Phi^{\sigma}_{y,\mu}\to\Phi^{\sigma}_{y,\mu}+t_{\mu}\Lambda,$
(101)

where the charge vector is ${\bf
t}=(t_{\rho},t_{c1},t_{c2},t_{n1},t_{n2})=(1,1,3,0,0)$. Each combination
$\Phi^{\sigma}_{y,\mu}$ in (92) and (98) corresponds to a product of
electronic operators
$e^{i\Phi^{\sigma}_{y,\mu}(\mathsf{x})+ik^{\sigma}_{y,\mu}\mathsf{x}}$ that
carries the net $\mathsf{x}$-momentum

$\displaystyle\begin{array}[]{*{20}l}k^{\sigma}_{y,\rho}=3(4+10y+5\sigma)k_{f}\\\
k^{\sigma}_{y,c1}=3(4+10y-3\sigma)k_{f}\\\
k^{\sigma}_{y,c2}=3(12+30y+\sigma)k_{f}\\\
k^{\sigma}_{y,n1}=k^{\sigma}_{y,n2}=0\end{array}.$ (106)

The basis transformations (92) and (98) were designed to facilitate the Dirac
partons (10). To see this, we first introduce the intra-bundle interactions
for each $y$,

$\displaystyle\mathcal{H}^{\mathrm{intra}}_{y}=u_{I}\cos\Theta^{\mathrm{intra}}_{y,I}+u_{II}\cos\Theta^{\mathrm{intra}}_{y,II},$
(107)
$\displaystyle\begin{array}[]{*{20}l}\Theta^{\mathrm{intra}}_{y,I}=2\Phi^{R}_{y,c1}+\Phi^{L}_{y,c1}+\Phi^{R}_{y,c2}-2\Phi^{L}_{y,c2}\\\
\Theta^{\mathrm{intra}}_{y,II}=\Phi^{R}_{y,c1}+2\Phi^{L}_{y,c1}-2\Phi^{R}_{y,c2}+\Phi^{L}_{y,c2}\end{array}.$
(110)

The two linearly independent angle variables $\Theta^{\mathrm{intra}}_{y,I}$,
$\Theta^{\mathrm{intra}}_{y,II}$ satisfy the “Haldane’s nullity” gapping
condition Haldane (1995)
$\left[\Theta^{\mathrm{intra}}_{y,l}(\mathsf{x}),\Theta^{\mathrm{intra}}_{y^{\prime},l^{\prime}}(\mathsf{x}^{\prime})\right]=0$.
(See (269) in appendix A.2.) Thus, along each bundle $y$,
$\mathcal{H}^{\mathrm{intra}}_{y}$ turns two (out of five) pairs of modes,
$\Phi^{\sigma}_{y,c1},\Phi^{\sigma}_{y,c2}$, massive and removes them from
low-energy. We also see that the interaction preserves charge $U(1)$ and
$\mathsf{x}$-momentum conservation. This is because the angle variables are
invariant under the $U(1)_{EM}$ gauge transformation (101) and the two sine-
Gordon terms are products of electronic operators that have trivial net
$\mathsf{x}$-momenta,
$2k^{R}_{y,c1}+k^{L}_{y,c1}+k^{R}_{y,c2}-2k^{L}_{y,c2}=k^{R}_{y,c1}+2k^{L}_{y,c1}-2k^{R}_{y,c2}+k^{L}_{y,c2}=0$.

The intra-bundle interaction (107) leaves behind, along each bundle $y$, three
pairs of low-energy modes

$\displaystyle\boldsymbol{\Phi}^{\sigma}_{y}=(\Phi^{\sigma}_{y,0},\Phi^{\sigma}_{y,1},\Phi^{\sigma}_{y,2})=(\Phi^{\sigma}_{y,\rho},\Phi^{\sigma}_{y,n1},\Phi^{\sigma}_{y,n2}).$
(111)

These modes are not affected by $\mathcal{H}^{\mathrm{intra}}_{y}$ because the
three bosonized variables commute with the sine-Gordon angle variables
$\Theta^{\mathrm{intra}}_{y,M}$. (See (269) in appendix A.2.) It is important
to acknowledge – from (92) and (98) – that the three $\Phi^{\sigma}_{y,J}$
associate to integral combinations of electronic operators which are local.
Locality only allows excitations that have non-fractional mutual monodromy
with these local electronic combinations. Let $e^{-i\phi}$ be the vertex
operator that creates such an excitation, where $\phi$ is some non-integral
combination $b^{J}_{\sigma}\Phi^{\sigma}_{y,J}$. Locality requires the equal-
time commutator

$\displaystyle\left[N^{\sigma^{\prime}}_{y^{\prime},J^{\prime}},\phi(\mathsf{x})\right]=i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}K_{JJ^{\prime}}b^{J}_{\sigma}$
(112)

to have integral value, where the $K$-matrix was defined in (14) and

$\displaystyle
N^{\sigma}_{y,J}=\frac{1}{2\pi}\int\partial_{\mathsf{x}}\Phi^{\sigma}_{y,J}(\mathsf{x})d\mathsf{x}$
(113)

is the number operator of the local electronic combination
$e^{i\Phi^{\sigma}_{y,J}}$. The integrality of (112) comes from the fact that
$N^{\sigma}_{y,J}$ is a physical observable and can only take integral
eigenvalues in the electronic many-body Hilbert space. Locality therefore only
allows vertex excitations $e^{ib^{J}_{\sigma}\Phi^{\sigma}_{y,J}}$, where
$K_{JJ^{\prime}}b^{J}$ are integers.

Integral combinations $\Phi_{\alpha}=\alpha^{J}_{\sigma}\Phi^{\sigma}_{y,J}$
of the local bosonized variables span the root lattice

$\displaystyle\Lambda_{U(1)_{3}\times SU(3)_{1}}^{\sigma}$
$\displaystyle=\mathrm{span}_{\mathbb{Z}}\left\\{\Phi^{\sigma}_{y,\rho},\Phi^{\sigma}_{y,n1},\Phi^{\sigma}_{y,n2}\right\\}$
(114)

of the affine Kac-Moody algebra $U(1)_{3}\times SU(3)_{1}$. We define the dual
bosonized variables (in the Chevalley basis), which are fractional
combinations of the local ones

$\displaystyle\phi_{y}^{\sigma,J}=(K^{-1})^{JJ^{\prime}}\Phi^{\sigma}_{y,J},$
(115)

where $J=(0,1,2)=(\rho,n1,n2)$. Integral combinations
$a_{J}^{\sigma}\phi_{y}^{\sigma,J}$ of these dual variables span a dual
lattice, known as the weight lattice

$\displaystyle{\Lambda_{U(1)_{3}\times SU(3)_{1}}^{\sigma}}^{\ast}$
$\displaystyle=\mathrm{span}_{\mathbb{Z}}\left\\{\phi_{y}^{\sigma,0},\phi_{y}^{\sigma,1},\phi_{y}^{\sigma,2}\right\\}$
$\displaystyle=K^{-1}\Lambda_{U(1)_{3}\times SU(3)_{1}}^{\sigma}$ (116)

whose elements correspond to vertex excitations
$e^{ia_{J}^{\sigma}\phi_{y}^{\sigma,J}}$ allowed by electron locality. The
dual variables obey the equal-time commutation relation (15). They can be
transformed into the Cartan-Weyl basis
$\tilde{\phi}^{\sigma}_{y,a}=A_{a}^{J}\phi^{\sigma,J}_{y}$ by (11) that gives
the Dirac fermionic partons $\pi_{y}^{\sigma,a}\sim
e^{i\tilde{\phi}^{\sigma}_{y,a}}$.

Figure 5: Emergence of Dirac partons from five electronic wires. The basis
transformation is defined in (92) and (98). The intra-bundle interaction is
defined in (107).

The basis transformation (92), (98) and the intra-bundle gapping interaction
(107) now turn each bundle of five electronic wires into a counter-propagating
pair of Dirac parton triplets $U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\times
SU(3)_{1}$. This process is summarized in figure 5. As promised in section
II.1, it is now evident that the charge $e$ fermion
$\Psi_{\mathrm{el}}=e^{i3\phi^{\sigma}_{\rho}}=e^{i\Phi^{\sigma}_{\rho}}$ as
well as the six neutral bosonic $SU(3)$ roots
$E^{ab}=e^{i(\tilde{\phi}^{\sigma}_{a}-\tilde{\phi}^{\sigma}_{b})}=e^{\pm
i\Phi^{\sigma}_{n1}}$, $e^{\pm i\Phi^{\sigma}_{n2}}$ or $e^{\pm
i(\Phi^{\sigma}_{n1}+\Phi^{\sigma}_{n2})}$ are all integral combinations of
local electrons. We notice in passing that this is not the simplest way in
realizing the Dirac partons. For instance, similar procedure can be applied to
a bundle of four electronic wires instead of five. The current method is
presented so that the neutral $SU(3)_{1}$ sector carries zero
$\mathsf{x}$-momentum (see eq.(106)). This will become useful in the next
subsection in the discussion of the parton Pfaffian FQH state.

Figure 6: Coupled wire models and gapless edge modes of (a) the parton state
$\mathfrak{L}_{1}=U(1)_{3}\times SU(3)_{1}$ with $c=3/2$, (b) the Laughlin
state $\mathfrak{L}_{0}=U(1)_{3}$ with $c=1$, and (c) the
$\mathfrak{L}_{2}=U(1)_{3}\times(SU(3)_{1})^{2}$ state with $c=5$.
$\mathcal{H}_{\rho}$ and $\mathcal{H}_{n}^{(q)}$ are defined (117) and (118)
respectively.

We now describe the coupled wire construction of the $\mathfrak{L}_{q}$ series
of FQH state with filling $\nu=1/3$. We introduce the interwire gapping
interactions

$\displaystyle\mathcal{H}_{\rho}$
$\displaystyle=u_{\rho}\sum_{y}\cos\left(\Phi^{R}_{y,\rho}-\Phi^{L}_{y+1,\rho}\right),$
(117) $\displaystyle\mathcal{H}_{n}^{(q)}$
$\displaystyle=u_{n}\sum_{y}\left[\sum_{j=1,2}\cos\left(\Phi^{R}_{y,nj}-\Phi^{L}_{y+q,nj}\right)\right.$
(118)
$\displaystyle\;\;\;\;+\cos\left(\Phi^{R}_{y,n1}+\Phi^{R}_{y,n2}-\Phi^{L}_{y+q,n1}-\Phi^{L}_{y+q,n2}\right)\Big{]},$

where $q$ is a fixed integer. The coupled wire model Hamiltonian density
contains all three interactions from (107), (117) and (118)

$\displaystyle\mathcal{H}^{(q)}=\mathcal{H}_{\rho}+\mathcal{H}_{n}^{(q)}+\sum_{y}\mathcal{H}^{\mathrm{intra}}_{y}.$
(119)

All the sine-Gordon potential preserves charge $U(1)_{\mathrm{EM}}$ and
$\mathsf{x}$-momentum conservation. This can be verified using the charge
assignment (101) and momenta (106). The interactions introduce a finite
excitation energy gap in the bulk but leaves behind gapless edge modes. See
figure 6 for illustration when $q=0,1,2$. In low energy, the gapless edge
modes are described by the Kac-Moody CFT

$\displaystyle\mathfrak{L}_{q}=U(1)_{3}\times\left[SU(3)_{1}\right]^{q}$ (120)

and carry the chiral central charge $c=1+2q$. We take the notation so that the
neutral sector $\left(SU(3)_{1}\right)^{q}$ is right-moving if $q>0$ and
$\left(SU(3)_{1}\right)^{q}=\left(\overline{SU(3)_{1}}\right)^{-q}$ is left-
moving if $q<0$. For example, $q=0$ corresponds to the Laughlin $\nu=1/3$ FQH
state where the topological state has no non-trivial neutral sectors and all
partons are confined. We focus on the case when $q=1$ that corresponds to the
parton FQH state $U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\times SU(3)_{1}$ at filling
$\nu=1/3$ and central charge $c=3$. This has the identical topological order
to a slab of fractional topological insulator with time reversal breaking and
conjugate top and bottom surfaces Sahoo  _et al._ (2017) (see figure 1).

We notice that the first line in (118) of $\mathcal{H}^{(q)}_{n}$ alone can
already introduce an energy gap for the neutral $SU(3)_{1}$ sectors in the
bulk. The last term is included so that the interaction takes the form of
$SU(3)_{1}$ Kac-Moody current backscattering

$\displaystyle\mathcal{H}^{(q)}_{n}=\frac{u_{n}}{2}\sum_{y}\sum_{\boldsymbol{\alpha}}E^{\boldsymbol{\alpha}}_{y,R}E^{-\boldsymbol{\alpha}}_{y+q,L},$
(121)

where $\boldsymbol{\alpha}=\pm(1,0),\pm(0,1),\pm(1,1)$ label the six roots of
$SU(3)_{1}$

$\displaystyle
E^{\boldsymbol{\alpha}}_{y,\sigma}=e^{i\alpha^{j}\Phi^{\sigma}_{y,nj}},$ (122)

which are identical to the six current operators $E^{ab}$ in the parton basis
in (16) of section II.1 (or (243) in appendix A.1). In this case, the $SU(3)$
symmetric interaction (121) introduces a finite excitation energy gap if
$u_{n}$ is negative, so that the additional sine-Gordon term on the second
line of (118) does not compete with the two terms on the first line.

Lastly, we observe that the $SU(3)_{1}$ current operators have zero
$\mathsf{x}$-momentum. This allows the backscattering interaction (121) of an
arbitrary hopping range $q$ to preserve momentum conservation. The coupled
wire model does not provide a preference towards a particular
$\mathfrak{L}_{q}$ phase. While the charge sector is frozen by
$\mathcal{H}_{\rho}$, $SU(3)$ current backscattering Hamiltonians
$\mathcal{H}^{(q)}_{n}$ with different ranges compete. A generalized coupled
wire model that simultaneously includes multiple $\mathcal{H}^{(q)}_{n}$’s
could potentially describe a $SU(3)$ spin liquid with a complex phase diagram
connected by a web of topological phase transitions.

###  III.2 Filling one-sixth

Figure 7: Bundle arrangement and basis transformation for the $\nu=1/6$
$\mathrm{Pf}^{\ast}$ FQH state. The momentum displacement between channels
(directed lines) and bundles (green cylinders) are shown in $nk_{f}$, where
$2k_{f}$ is the displacement between $R$ and $L$-movers on same wire at Fermi
level. Dashed boxes represent products of electronic operators. $\bullet$ and
$\circ$ represents electron creation and annihilation operators respectively.

The model for filling one-sixth consists of half as many wires as in the one-
third case. This is shown in figure 7 where half the bundles from figure 4 are
taken out. The bundles now has a staggered and dimerized configuration where
translation symmetry is broken and two bundles, labeled by $A=0,1$, now form a
super-unit cell. The electron (annihilation) operators are

$\displaystyle
c^{\sigma}_{yAa}(\mathsf{x})\sim\exp\left[i\left(\tilde{\Phi}^{\sigma}_{yAa}(\mathsf{x})+k^{\sigma}_{yAa}\mathsf{x}\right)\right],$
(123)

where $y$ is an integer that labels the vertical position of the 2-bundle
super-unit cell, $a=0,\ldots,4$ labels the five electronic wires in each
bundle, and $\sigma=R,L=+,-$ corresponds to the two counter-propagating
electronic channels in each wire. The bosonized variables obey the equal-time
commutation relation (ETCR)

$\displaystyle\left[\partial_{\mathsf{x}}\tilde{\Phi}_{yAa}^{\sigma}(\mathsf{x}),\tilde{\Phi}_{y^{\prime}A^{\prime}a^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}\delta_{AA^{\prime}}\delta_{aa^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime}).$
(124)

The model is under a perpendicular magnetic field so that the
$\mathsf{x}$-momenta of the electronic channels $c^{\sigma}_{yAa}$ are

$\displaystyle\begin{array}[]{*{20}l}k^{\sigma}_{yA0}=(120y+30A+\sigma)k_{f}\\\
k^{\sigma}_{yAa}=(120y+30A+4a+4+\sigma)k_{f}\\\
k^{\sigma}_{yA4}=(120y+30A+24+\sigma)k_{f}\end{array},\quad\mbox{for
$a=1,2,3$}.$ (128)

Similar to the $\nu=1/3$ case, we first perform the basis transformation (92)
and (98) for each bundle $(y,A)$. Then, we introduce the charge
$U(1)_{\mathrm{EM}}$ and momentum conserving intra-bundle gapping potential
$\mathcal{H}^{\mathrm{intra}}_{y,A}$ defined in (107) to turn each bundle into
the Dirac parton triplet $U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\times SU(3)_{1}$.
(Also see figure 5.) We here repeat the transformed bosonized variables that
are unaffected by $\mathcal{H}^{\mathrm{intra}}_{y,A}$.

$\displaystyle\begin{array}[]{*{20}l}\Phi^{R}_{yA\rho}=2\tilde{\Phi}^{R}_{yA4}-\tilde{\Phi}^{L}_{yA4}\\\
\Phi^{L}_{yA\rho}=2\tilde{\Phi}^{L}_{yA0}-\tilde{\Phi}^{R}_{yA0}\end{array},$
(131)
$\displaystyle\begin{array}[]{*{20}l}\Phi^{R}_{yA,n1}=2\tilde{\Phi}^{R}_{yA1}-\tilde{\Phi}^{L}_{yA1}-\tilde{\Phi}^{L}_{yA2}\\\
\Phi^{R}_{yA,n2}=2\tilde{\Phi}^{R}_{yA2}-\tilde{\Phi}^{L}_{yA2}-\tilde{\Phi}^{L}_{y3}\\\
\Phi^{L}_{yA,n1}=2\tilde{\Phi}^{L}_{yA2}-\tilde{\Phi}^{R}_{yA2}-\tilde{\Phi}^{R}_{yA1}\\\
\Phi^{L}_{yA,n2}=2\tilde{\Phi}^{L}_{yA3}-\tilde{\Phi}^{R}_{yA3}-\tilde{\Phi}^{R}_{yA2}\end{array}.$
(136)

The bosonzied variabes obey the ETCR

$\displaystyle\left[\partial_{\mathsf{x}}\Phi_{yAJ}^{\sigma}(\mathsf{x}),\Phi_{y^{\prime}A^{\prime}J^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})\right]$
$\displaystyle=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}\delta_{AA^{\prime}}K_{JJ^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime}),$
(137)

where the $K$-matrix is

$\displaystyle K=\begin{pmatrix}3&&\\\ &2&-1\\\ &-1&2\end{pmatrix}$ (138)

and is ordered according to $J=\rho,n1,n2=0,1,2$. The first bosonzied variable
shifts by $\Phi^{\sigma}_{yA\rho}\to\Phi^{\sigma}_{yA\rho}+\Lambda$ under the
external $U(1)_{\mathrm{EM}}$ transformation of an electron operator $c\to
ce^{i\Lambda}$. The other two bosonized variables
$\Phi^{\sigma}_{yA,n1},\Phi^{\sigma}_{yA,n2}$ corresponds to neutral sectors
and are invariant under $U(1)_{EM}$. The vertices
$e^{i\Phi^{\sigma}_{yAJ}(\mathsf{x})+ik^{\sigma}_{yAJ}\mathsf{x}}$ are
integral combinations of electronic operators, which are diagramatically
represented as the yellow boxes in figure 7. They carry the
$\mathsf{x}$-momenta

$\displaystyle\begin{array}[]{*{20}l}k^{\sigma}_{yA\rho}=3(4+40y+10A+5\sigma)k_{f},\\\
k^{\sigma}_{yA,n1}=k^{\sigma}_{yA,n2}=0.\end{array}$ (141)

Figure 8: Emergence of parton Pfaffian from ten electronic wires. Each of the
pair of basis transformations in the yellow boxes is defined in (92) and (98).
The second basis transformation in the blue box is the product of (143) and
(147). $\mathcal{H}^{\mathrm{intra}}$, $\mathcal{H}^{\mathrm{dimer}}_{SU(3)}$,
$\mathcal{H}^{\mathrm{dim}}_{DF}$ and $\mathcal{H}^{\mathrm{dim}}_{MF}$ are
defined in (107), (142), (153) and (154) respectively.

At this point, there are two counter-propagating pairs of Dirac parton
triplets in each unit cell $y$ in low-energy. We introduce the intra-cell
$SU(3)$ current backscattering

$\displaystyle\mathcal{H}^{\mathrm{dimer}}_{y,SU(3)}=\frac{u_{III}}{2}\sum_{\boldsymbol{\alpha}}E^{\boldsymbol{\alpha}}_{y,A=0,R}E^{-\boldsymbol{\alpha}}_{y,A=1,L}$
(142)
$\displaystyle=u_{III}\left[\sum_{j=1,2}\cos\left(\Phi^{R}_{y,A=0,nj}-\Phi^{L}_{y,A=1,nj}\right)\right.$
$\displaystyle\;\;\;\;+\cos\left(\Phi^{R}_{y,A=0,n1}+\Phi^{R}_{y,A=0,n2}-\Phi^{L}_{y,A=1,n1}-\Phi^{L}_{y,A=1,n2}\right)\Big{]},$

where $u_{III}<0$. It mirrors the neutral inter-bundle gapping potential (118)
and (121) at $q=1$ for the $\nu=1/3$ case, except now it acts only within the
dimerized super-unit cell. This leaves two counter-propagating pairs of
charged modes $\Phi^{\sigma}_{yA\rho}$ and the two counter-propagating pairs
of neutral modes $\Phi^{R}_{y,nj}=\Phi^{R}_{y,A=1,nj}$ and
$\Phi^{L}_{y,nj}=\Phi^{L}_{y,A=0,nj}$, for $j=1,2$, unaffected.

Next, we perform a fractional basis transformation to the charged modes

$\displaystyle\begin{pmatrix}\Phi^{R}_{y\rho}\\\ \Phi^{R}_{y,n0}\\\
\Phi^{L}_{y\rho}\\\
\Phi^{L}_{y,n0}\end{pmatrix}=\begin{pmatrix}3/2&1&-1/2&-1\\\ -1/2&1&-1/2&0\\\
-1/2&-1&3/2&1\\\
-1/2&0&-1/2&1\end{pmatrix}\begin{pmatrix}\Phi^{R}_{y,A=0,\rho}\\\
\Phi^{R}_{y,A=1,\rho}\\\ \Phi^{L}_{y,A=0,\rho}\\\
\Phi^{L}_{y,A=1,\rho}\end{pmatrix}.$ (143)

The $\rho$ ($n$) sectors are electrically charged (resp. neutral). They obey
the ETCR

$\displaystyle\left[\partial_{\mathsf{x}}\Phi^{\sigma}_{y\rho}(\mathsf{x}),\Phi^{\sigma^{\prime}}_{y^{\prime}\rho}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}K_{\rho}\delta(\mathsf{x}-\mathsf{x}^{\prime})$
$\displaystyle\left[\partial_{\mathsf{x}}\Phi^{\sigma}_{y,nJ}(\mathsf{x}),\Phi^{\sigma^{\prime}}_{y^{\prime},nJ^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}K_{JJ^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime})$
$\displaystyle\left[\partial_{\mathsf{x}}\Phi^{\sigma}_{y\rho}(\mathsf{x}),\Phi^{\sigma^{\prime}}_{y^{\prime},nJ}(\mathsf{x}^{\prime})\right]=0,$
(144)

where $K$ is the same as that in (138) and

$\displaystyle K_{\rho}$ $\displaystyle=6.$ (145)

They carry the $\mathsf{x}$-momenta

$\displaystyle k^{\sigma}_{y\rho}=3\left(9+40y+20\sigma\right)k_{f},\quad
k^{\sigma}_{y,nJ}=0.$ (146)

It should be noticed that $\Phi^{\sigma}_{y\rho}$ and $\Phi^{\sigma}_{y,n0}$
are half-integral combinations of electronic bosonized variables – previously
referred to as “almost local” variables in section II.2 – and therefore they
do not associate to local electronic vertex operators. On the other hand,
their sums and differences $\Phi^{R}_{y\rho}\pm\Phi^{L}_{y\rho}$,
$\Phi^{R}_{y,n0}\pm\Phi^{L}_{y,n0}$ and
$\Phi^{\sigma}_{y\rho}\pm\Phi^{\sigma^{\prime}}_{y,n0}$ correspond to integral
electronic combinations.

It will be convenient later to perform a second unimodular basis
transformation within the neutral sectors

$\displaystyle\begin{pmatrix}\Phi^{R}_{y,d0}\\\ \Phi^{R}_{y,d1}\\\
\Phi^{R}_{y,d2}\\\ \Phi^{L}_{y,d0}\\\ \Phi^{L}_{y,d1}\\\
\Phi^{L}_{y,d2}\end{pmatrix}=\begin{pmatrix}1&0&0&0&1&0\\\ 0&1&-1&1&0&0\\\
0&2&1&1&0&0\\\ 0&1&0&1&0&0\\\ 1&0&0&0&1&-1\\\
1&0&0&0&2&1\end{pmatrix}\begin{pmatrix}\Phi^{R}_{y,n0}\\\ \Phi^{R}_{y,n1}\\\
\Phi^{R}_{y,n2}\\\ \Phi^{L}_{y,n0}\\\ \Phi^{L}_{y,n1}\\\
\Phi^{L}_{y,n2}\end{pmatrix}.$ (147)

This changes the ETCR to

$\displaystyle\left[\partial_{\mathsf{x}}\Phi^{\sigma}_{y,dJ}(\mathsf{x}),\Phi^{\sigma^{\prime}}_{y^{\prime},dJ^{\prime}}(\mathsf{x}^{\prime})\right]$
$\displaystyle=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{yy^{\prime}}(K_{d})_{JJ^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime})$
(148)

where the $K_{d}$-matrix is now diagonal

$\displaystyle K_{d}=\begin{pmatrix}1&&\\\ &3&\\\ &&3\end{pmatrix}.$ (149)

The complete ETCR between the bosonized variables
$\Phi^{\sigma}_{y\rho},\Phi^{\sigma}_{y,dJ}$ can be found in (285) in appendix
A.2.2.

We introduce the neutral Dirac fermions

$\displaystyle
d_{y,J}^{\sigma}=e^{i\Phi^{\sigma}_{y,dJ}}\prod_{y^{\prime}<y}(-1)^{N^{d}_{y^{\prime}}}.$
(150)

The operator $N^{d}_{y^{\prime}}$ is a combination of local electronic number
operators and is defined explicitly in (297) in appendix A.2.2. It is chosen
in a way to ensure mutual commutativity between the sine-Gordon angle
variables in the upcoming interactions (153), (155), (156),(157) as well as
(LABEL:PfrhogammaHpq) and (LABEL:PfdrhoHpq). The fermion parity operator
$(-1)^{N^{d}_{y}}$ anticommutes with the vertex operators
$e^{i\Phi^{\sigma}_{y,dJ}}$ in the same unit-cell $y$. Similar to the Jordan-
Wigner Jordan (1927); Jordan and Wigner (1928) fermionization of the Ising
model, the string of fermion parity operators in (150) ensures the mutual
anticommutation relations between the neutral Dirac fermions in (150)

$\displaystyle\left\\{d_{y,J}^{\sigma}(\mathsf{x}),d_{y^{\prime},J^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})\right\\}=\left\\{d_{y,J}^{\sigma}(\mathsf{x}),d_{y^{\prime},J^{\prime}}^{\sigma^{\prime}}(\mathsf{x}^{\prime})^{\dagger}\right\\}=0$
(151)

for $(y,J,\mathsf{x})\neq(y^{\prime},J^{\prime},\mathsf{x}^{\prime})$.

From (149), we see that $d_{y,0}^{\sigma}$ is a spin $h=\pm 1/2$ fermion and
$d_{y,1}^{\sigma},d_{y,2}^{\sigma}$ have spin $h=\pm 3/2$. The former can be
decomposed into a pair of Majorana fermions

$\displaystyle
d^{\sigma}_{y,0}=\frac{1}{\sqrt{2}}\left(\gamma^{\sigma}_{y}+i\delta^{\sigma}_{y}\right),$
$\displaystyle\begin{split}\gamma^{\sigma}_{y}=\sqrt{2}\cos\Phi^{\sigma}_{y,d0}\prod_{y^{\prime}<y}(-1)^{N^{d}_{y^{\prime}}},\\\
\delta^{\sigma}_{y}=\sqrt{2}\sin\Phi^{\sigma}_{y,d0}\prod_{y^{\prime}<y}(-1)^{N^{d}_{y^{\prime}}}.\end{split}$
(152)

We introduce the Dirac fermion backscattering dimerization

$\displaystyle\mathcal{H}^{\mathrm{dimer}}_{y,DF}$
$\displaystyle=-i\frac{u_{IV}}{2}{d^{R}_{y,2}}^{\dagger}d^{L}_{y,2}+h.c.$
(153) $\displaystyle=u_{IV}\cos\left(\Phi^{R}_{y,d2}-\Phi^{L}_{y,d2}\right),$

and the Majorana fermion backscattering dimerization

$\displaystyle\mathcal{H}^{\mathrm{dimer}}_{y,MF}$
$\displaystyle=-iu_{V}\delta^{R}_{y}\delta^{L}_{y}$ (154)
$\displaystyle=u_{V}\left[\cos\left(\Phi^{R}_{y,d0}-\Phi^{L}_{y,d0}\right)+\cos\left(\Phi^{R}_{y,d0}+\Phi^{L}_{y,d0}\right)\right].$

(The factors of $i$ appearing in (153) and (154) are consequences of the
Baker-Campbell-Hausdorff formula $e^{A}e^{B}=e^{A+B+[A,B]/2}$ and the constant
terms in the complete ETCR (285) presented in appendix A.2.2.) These fermion
bilinear potentials can be expressed as integral products of the original
electron operators $c^{\sigma}_{yAa}$. These intra-unit cell interactions
leave behind the following degrees of freedom unaffected: (a) the electrically
charged spin $h=\pm 3$ bosons $e^{i\Phi^{\sigma}_{y,\rho}}$ that generate a
$U(1)_{6}$ sector for each propagating direction $\sigma=R,L$, (b) the
electrically neutral spin $h=\pm 3/2$ Dirac fermions $d^{\sigma}_{y,d1}$ that
generate the counter-propagating pair of $U(1)_{3}$ sectors, and (c) the
electrically neutral spin $h=\pm 1/2$ counter-propagating pair of Majorana
fermions $\gamma^{\sigma}_{y}$. These combine into the non-chiral parton
Pfaffian CFT. (See figure 8 for a summary.)

The remaining interactions in the model couple degrees of freedom between unit
cells. They are

$\displaystyle\mathcal{H}_{\rho\gamma}$
$\displaystyle=-u_{\rho\gamma}\sum_{y}\cos\left(\Phi^{R}_{y\rho}-\Phi^{L}_{y+1,\rho}-\pi
N^{d}_{y}\right)i\gamma^{L}_{y}\gamma^{R}_{y+1}$ (155)
$\displaystyle=\frac{u_{\rho\gamma}}{2}\sum_{y}\sum_{s_{1},s_{2}=\pm}s_{1}s_{2}\cos\left(\Phi^{R}_{y\rho}-\Phi^{L}_{y+1,\rho}+s_{1}\Phi^{L}_{y,d0}+s_{2}\Phi^{R}_{y+1,d0}\right),$
$\displaystyle\mathcal{H}_{d\rho}$
$\displaystyle=-u_{d\rho}\sum_{y}\left(i{d^{R}_{y,1}}^{\dagger}d^{L}_{y+1,1}+h.c.\right)\cos\left(\Phi^{R}_{y\rho}-\Phi^{L}_{y+1,\rho}-\pi
N^{d}_{y}\right)$ (156)
$\displaystyle=-u_{d\rho}\sum_{y}\sum_{s=\pm}\cos\left(\Phi^{R}_{y\rho}-\Phi^{L}_{y+1,\rho}+s\Phi^{R}_{y,d1}-s\Phi^{L}_{y+1,d1}\right),$
$\displaystyle\mathcal{H}_{\gamma d}$ $\displaystyle=-u_{\gamma
d}\sum_{y}\left(i{d^{R}_{y,1}}^{\dagger}d^{L}_{y+1,1}+h.c.\right)i\gamma^{L}_{y}\gamma^{R}_{y+1}$
(157) $\displaystyle=u_{\gamma
d}\sum_{y}\sum_{s_{1},s_{2}=\pm}s_{1}s_{2}\cos\left(\Phi^{R}_{y,d1}-\Phi^{L}_{y+1,d1}+s_{1}\Phi^{L}_{y,d0}+s_{2}\Phi^{R}_{y+1,d0}\right).$

The interwire potentials preserves the external $U(1)_{\mathrm{EM}}$ as well
as $\mathsf{x}$-momentum. The factors of $i$ and signs $s_{1}$, $s_{2}$ in the
interactions (155), (156) and (157) are consequences of the Baker-Campbell-
Hausdorff formula $e^{A}e^{B}=e^{A+B+[A,B]/2}$ and the constant terms in the
complete ETCRs (285) and (298) in appendix A.2.2. Any two out of the three
sets of interactions are enough to introduce a finite excitation energy gap in
the bulk. When all three terms are present, they are not competing when the
product $u_{\rho\gamma}u_{d\rho}u_{\gamma d}$ is positive. They collectively
pin the ground state expectation values of the order parameters

$\displaystyle\begin{split}&e^{i\left\langle\Theta^{\rho}_{y+1/2}(\mathsf{x})\right\rangle}=e^{i\left\langle\Phi^{R}_{y,\rho}(\mathsf{x})-\Phi^{L}_{y+1,\rho}(\mathsf{x})-\pi
N^{d}_{y}\right\rangle},\\\ &i\left\langle
d^{R}_{y,1}(\mathsf{x})^{\dagger}d^{L}_{y+1,1}(\mathsf{x})\right\rangle,\quad
i\left\langle\gamma^{L}_{y}(\mathsf{x})\gamma^{R}_{y+1}(\mathsf{x})\right\rangle\end{split}$
(158)

to be either all positive or all negative. Figure 9 summarizes the three sets
of interactions.

Figure 9: Coupled wire model and its gapless edge modes of the $\nu=1/6$
parton Pfaffian state. $\mathcal{H}_{\rho\gamma}$, $\mathcal{H}_{d\rho}$ and
$\mathcal{H}_{\gamma d}$ are defined in (155), (156) and (157).

To summarize, the coupled wire model involves the following gapping potentials

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{\rho\gamma}+\mathcal{H}_{d\rho}+\mathcal{H}_{\gamma
d}$
$\displaystyle\;\;\;+\sum_{y}\mathcal{H}^{\mathrm{dimer}}_{y,MF}+\mathcal{H}^{\mathrm{dimer}}_{y,DF}+\mathcal{H}^{\mathrm{dimer}}_{y,SU(3)}$
$\displaystyle\;\;\;+\sum_{y}\sum_{A=0,1}\mathcal{H}^{\mathrm{intra}}_{y,A}.$
(159)

The first line consists of the inter-bundle terms (155), (156) and (157),
which are also shown in figure 9. The second line contains the terms (142),
(153) and (154) that act within a super-unit cell. The last line includes
intra-bundle terms that were defined previously in (107). The intra-cell and
intra-bundle terms are summarized in figure 8.

It is important to notice that these interactions are consistent with electron
locality. Paradoxically, the potentials involve backscatterings of fractional
fields. For example, $\mathcal{H}^{\mathrm{dimer}}_{y,MF}$,
$\mathcal{H}^{\mathrm{dimer}}_{y,DF}$ and $\mathcal{H}_{\gamma d}$ backscatter
Majorana and neutral Dirac fermions. $\mathcal{H}_{\rho\gamma}$ and
$\mathcal{H}_{d\rho}$ couples the fractional charge $e$ boson
$e^{i\Phi_{\rho}}$ between wires. However, these potentials are specifically
designed so that under a basis transformation of the bosonized variables, each
of them can be expressed as integral combinations of local electrons. (See
(307), (314) and (314) in appendix A.2.2.) In particular, each of the inter-
bundle terms $\mathcal{H}_{\gamma d}$, $\mathcal{H}_{\rho\gamma}$ and
$\mathcal{H}_{d\rho}$ simultaneously backscatters two of the three fractional
fields $e^{i\Phi_{\rho}}$, $d_{1}\sim e^{i\Phi_{d1}}$ and
$\gamma\sim\cos\Phi_{d0}$. The bosonized variables $\Phi_{\rho}$, $\Phi_{dJ}$
are half-integral in the sense that any sum or difference between any pair is
an integral combination of local electron variables. As a result, the inter-
bundle potentials are local. On the other hand, potentials that backscatter a
single fractional species cannot be constructed from integral combinations of
electrons, and are not allowed by electron locality.

The gapping potential (159) leave behind the gapless edge mode

$\displaystyle\mathrm{Pf}^{\ast}=U(1)_{6}\times
U(1)_{3}\times\overline{\mbox{Majorana fermion}}.$ (160)

Along the top edge in figure 9, the three sectors are generated by

$\displaystyle 2\phi^{R}_{\rho}=\Phi^{R}_{\rho}/6,\quad
2\phi^{R}_{d}=\Phi^{R}_{d1}/3,\quad\psi^{L}=\gamma^{L}$ (161)

and are described by the Lagrangian densities (25) and (27). The CFT supports
the charge $(a/12)e$ Abelian primary fields

$\displaystyle\begin{split}\openone_{a,b}&=e^{ia\phi^{R}_{\rho}}e^{ib\phi^{R}_{d}}\\\
\Psi_{a,b}&=e^{ia\phi^{R}_{\rho}}e^{ib\phi^{R}_{d}}\gamma^{L}\end{split}$
(162)

for even integers $a,b$ (c.f. the notation from (33)). In the bulk, these
operators $\openone_{a,b}(\mathsf{x}_{0},y),\Psi_{a,b}(\mathsf{x}_{0},y)$
create (or annihilate) gapped excitations in the form of vortices/kinks of the
order parameters in (158) at $\mathsf{x}_{0}$. As a consequence of the pair
backscattering structures of the inter-bundle interactions
$\mathcal{H}_{\gamma d}$, $\mathcal{H}_{\rho\gamma}$ and
$\mathcal{H}_{d\rho}$, they allow deconfined half-excitations that correspond
to half-vortices/$\pi$-kinks of all three sectors. They associate to the
charge $(a/12)e$ non-Abelian bosonic products

$\displaystyle\Sigma_{a,b}=e^{ia\phi^{R}_{\rho}}e^{ib\phi^{R}_{d1}}\sigma^{L}_{0}$
(163)

for odd integers $a,b$, where $\sigma^{L}_{0}$ is the Ising twist field of the
Majorana fermion $\gamma^{L}$. Each $\Sigma_{a,b}$ has $\pi$ monodromy with
all three generators $e^{i\Phi_{\rho}^{R}}$, $d_{1}^{R}\sim
e^{i\Phi^{R}_{d1}}$ and $\gamma^{L}$ of $U(1)_{6}$, $U(1)_{3}$ and
$\overline{\mathrm{Ising}}$. The parton Pfaffian state therefore has the
extended product structure

$\displaystyle\mathrm{Pf}^{\ast}=U(1)_{24}\otimes_{\mathrm{el}}U(1)_{12}\otimes_{\mathrm{el}}\overline{\mbox{Ising}}$
(164)

described in section II.2. Eq.(162) and (163) accounts for all primary fields.
All other vortices – such as $\openone_{a,b},\Psi_{a,b}$, for $a\not\equiv b$
modulo 2, and $\Sigma_{a,b}$, for $a\equiv b$ modulo 2 – are non-local with
respect to the electron and are confined. For example, the pair
$\openone_{1,0}(\mathsf{x}_{0},y),\openone_{-1,0}(\mathsf{x}_{1},y)$ creates a
$\pi$-kink dipole for $\Theta^{\rho}_{y+1/2}$ alone, and associates to a
linearly diverging excitation energy
$2(u_{\rho\gamma}+u_{d\rho})|\mathsf{x}_{0}-\mathsf{x}_{1}|$. The confinement
of these vortices is remembered by the “electronic tensor product”
$\otimes_{\mathrm{el}}$.

####  III.2.1 Variations

The coupled wire model (159) described the “particle-hole” symmetric parton
Pfaffian state. By rearranging the inter-bundle terms
$\mathcal{H}_{\rho\gamma}$, $\mathcal{H}_{d\rho}$ and $\mathcal{H}_{\gamma
d}$, one can construct a sequence of paired parton states at the same filling
fraction $\nu=1/6$ but with different edge modes and topological order. In
section III.1, the coupled wire models (119) (see also figure 6) at filling
$\nu=1/3$ generate a series of Abelian parton states
$\mathfrak{L}_{q}=U(1)_{3}\times[SU(3)_{1}]^{q}$ by allowing $q^{\mathrm{th}}$
nearest neighbor bundle backscatterings. A similar construction can be applied
for the paired parton sequence

$\displaystyle\mathfrak{P}_{p,q}=U(1)_{24}\otimes_{\mathrm{el}}U(1)_{12}^{q}\otimes_{\mathrm{el}}\mathrm{Ising}^{p},$
(165)

where $p$, $q$ are integers. Negative powers associate to counter propagating
sectors. For example, $U(1)_{6}^{-q}=\overline{U(1)_{6}^{q}}$ and
$\mathrm{Ising}^{-p}=\overline{\mathrm{Ising}^{p}}$. For instance, the
particular parton Pfaffian state described previously in (164) is
$\mathrm{Pf}^{\ast}=\mathfrak{P}_{p=1,q=-1}$. The chiral central charge of
(165) depends on its neutral sectors, and is given by

$\displaystyle c_{p,q}=1+q+p/2.$ (166)

The case when both powers $p,q$ are trivial is special. It corresponds to an
Abelian state with strong electronic quasiparticle pairing. It has a strongly-
paired topological order and edge CFT

$\displaystyle\mathrm{SP}^{\ast}=\mathfrak{P}_{p=0,q=0}=U(1)_{24}.$ (167)

There is no gapless charge $e$ electronic quasiparticles on the boundary.
Instead the smallest local electronic primary field is a charge $2e$ Cooper
pair.

Figure 10: Coupled wire model and its gapless edge modes of the $\nu=1/6$
parton Pfaffian state $\mathfrak{P}_{2,-1}$ in (165).
$\mathcal{H}_{\rho\gamma}^{(2)}$ and $\mathcal{H}_{d\rho}^{-1}$ are defined in
(LABEL:PfrhogammaHpq) and (LABEL:PfdrhoHpq).

We first consider the variations when the powers $p$ and $q$ are not both
trivial. The general coupled wire model is

$\displaystyle\mathcal{H}^{(p,q)}$
$\displaystyle=\mathcal{H}^{(p)}_{\rho\gamma}+\mathcal{H}^{(q)}_{d\rho}$
$\displaystyle\;\;\;+\sum_{y}\mathcal{H}^{\mathrm{dimer}}_{y,MF}+\mathcal{H}^{\mathrm{dimer}}_{y,DF}+\mathcal{H}^{\mathrm{dimer}}_{y,SU(3)}$
$\displaystyle\;\;\;+\sum_{y}\sum_{A=0,1}\mathcal{H}^{\mathrm{intra}}_{y,A}.$
(168)

The intra-cell and intra-bundle terms in the second and third line are
identical to that of (159), and were defined in (107), (142), (153) and (154)
(see also figure 8). The first two terms $\mathcal{H}^{(p)}_{\rho\gamma}$ and
$\mathcal{H}^{(q)}_{d\rho}$ involve $p^{\mathrm{th}}$ and $q^{\mathrm{th}}$
nearest neighbor backscattering. We ignore terms like $\mathcal{H}_{\gamma d}$
in (157) that couple only neutral modes because they are redundant for the
bulk energy gap.

The generalized inter-bundle terms are defined by

These interactions conserve charge $U(1)_{\mathrm{EM}}$ and
$\mathsf{x}$-momentum. They can be re-expressed as integral combinations of
electron operators (see (307), (314) and (314) in appendix A.2.2), and are
therefore consistent with electron locality. They introduce a finite
excitation energy in the bulk and leave behind the edge CFT (165) in low-
energy. An example for $p=2$ and $q=-1$ is illustrated in figure 10.

Special care is needed for the trivial case when $p=q=0$. This is because
$\mathcal{H}^{(0)}_{\rho\gamma}$ and $\mathcal{H}^{(0)}_{d\rho}$ in
(LABEL:PfrhogammaHpq) and (LABEL:PfdrhoHpq) only backscatter the Majorana and
Dirac fermions $\gamma$ and $d_{1}$ within a super-cell. They do not involve
the charge boson $e^{i\Phi_{\rho}}$, which still remains massless in the bulk.
The charge sector can be turned massive by the boson backscattering

$\displaystyle\mathcal{H}_{\rho}^{(0,0)}=-u_{\rho}\sum_{y}\cos\left(2\Phi^{R}_{y\rho}-2\Phi^{L}_{y+1,\rho}\right),$
(169)

where the factor of 2 ensures that $\mathcal{H}_{\rho}^{(0,0)}$ is an integral
combination of electron operators. By including (169) in the coupled wire
model (168) for $p=q=0$, $\mathcal{H}^{(0,0)}_{\rho}+\mathcal{H}^{(0,0)}$
removes all low-energy degrees of freedom in the bulk and leave behind the
strongly-paired edge CFT (167).

###  III.3 The parton T-Pfaffian surface state of a fractional topological
insulator

In previous works Sahoo  _et al._ (2017); Cho  _et al._ (2017), we proposed
the symmetry-preserving $\mathcal{T}-\mathrm{Pf}^{\ast}$ surface topological
order of a fractional topological insulator (FTI) Maciejko  _et al._ (2010);
Swingle  _et al._ (2011); Levin  _et al._ (2011); Maciejko  _et al._ (2012);
Ye  _et al._ (2016); Maciejko and Fiete (2015); Stern (2016); Ye  _et al._
(2017). The FTI consists of deconfined partons coupled to a discrete
$\mathbb{Z}_{3}$ gauge theory. Each of the three parton species occupies a
time-reversal (TR) symmetric fermionic topological band and hosts a parton
Dirac surface state. The surface can be turned massive without breaking TR
symmetry or charge $U(1)$ conservation, and similar to the T-Pfaffian surface
state Chen  _et al._ (2014), the parton version
$\mathcal{T}-\mathrm{Pf}^{\ast}$ admits a fractional anyonic excitation
structure and hosts an Ising-like topological order. The surface quasiparticle
structure was introduced in ref. Sahoo  _et al._ , 2017; Cho  _et al._ , 2017
and was reviewed in (39) in section II.2. In this section, we propose a
coupled wire description to this surface topological order. The construction
parallels the coupled wire description by Mross, Essin and Alicea Mross  _et
al._ (2015) of the T-Pfaffian surface state of a conventional topological
insulator.

Figure 11: Coupled wire model of the $\mathcal{T}-\mathrm{Pf}^{\ast}$ surface
state of a fractional topological insulator. The energy gap is introduced by
the many-body electron backscattering $\mathcal{U}$ defined in (177).

The massless parton Dirac fermions on the surface of the FTI can be mimicked
by a 2D array of chiral parton Dirac channels, labeled by their vertical
position $y$, with alternating propagating directions
$\sigma=R,L=+,-=(-1)^{y}$. These channels are represented by the red wires in
figure 11. Each channel carries a parton Dirac CFT
$U(3)_{1}/\mathbb{Z}_{3}=U(1)_{3}\times SU(3)_{1}$ described in section II.1.
It consists of three parton Dirac fermions that propagate in a single-
direction. The emergence of these parton channels can be facilitated by an
antiferromagnetic stripe order on the FTI surface. It introduces a time-
reversal symmetry-breaking parton energy gap on each stripe, where the Dirac
mass $m(y)=m_{0}\mathrm{sgn}[\sin(\pi y)]$ flips sign across adjacent stripes.
This removes the 2D parton Dirac surface fermions and leaves behind a chiral
Dirac channel $U(3)_{1}/\mathbb{Z}_{3}$ along each 1D interface. The
unbalanced chirality is allowed by TR symmetry-breaking. On the other hand,
the 2D array collectively recovers an antiferromagnetic time-reversal (AFTR)
symmetry, which combines the local TR conjugation with the half-translation
$y\to y+1$.

The chiral parton Dirac fermion at wire $y$ can be bosonized into
$\pi^{a}_{y}(\mathsf{x})\sim e^{i\tilde{\phi}_{y,a}(\mathsf{x})}$, for
$a=1,2,3$. The bosonized variables $\tilde{\phi}^{a}_{y}$ fields satisfy the
equal-time commutation relation (ETCR)

$\displaystyle\left[\partial_{\mathsf{x}}\tilde{\phi}_{y,a}(\mathsf{x}),\tilde{\phi}_{y^{\prime},a^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i(-1)^{y}\delta_{y,y^{\prime}}\delta_{a,a^{\prime}}\delta(\mathsf{x}-\mathsf{x}^{\prime}).$
(170)

The symmetry-preserving many-body gapping interaction relies on the
bipartition of each parton Dirac channel into a pair of parton Pfaffians,
$U(3)_{1}/\mathbb{Z}_{3}=\mathrm{Pf}^{\ast}\boxtimes\mathrm{Pf}^{\ast}$. The
splitting allows the parton Pfaffian channels to be backscattered
independently in opposite directions (see figure 11). This splitting basis
transformation was introduced in section II.3 (see figure 3 for a summary). It
requires a channel reconstruction that extends the parton Dirac CFT by two
counter-propagating pairs of electron modes, $c^{\sigma}_{y,b}(\mathsf{x})\sim
e^{i\tilde{\Phi}^{\sigma}_{y,b}(\mathsf{x})}$, for $\sigma=R,L=+,-$ and
$b=0,1$. The electronic bosonized variables obey the ETCR

$\displaystyle\left[\partial_{\mathsf{x}}\tilde{\Phi}^{\sigma}_{y,b}(\mathsf{x}),\tilde{\Phi}^{\sigma^{\prime}}_{y^{\prime},b^{\prime}}(\mathsf{x}^{\prime})\right]=2\pi
i\sigma\delta_{y,y^{\prime}}\delta_{b,b^{\prime}}\delta^{\sigma,\sigma^{\prime}}\delta(\mathsf{x}^{\prime}-\mathsf{x}).$
(171)

The AFTR symmetry, represented by the anti-unitary operator $\mathcal{T}$,
sends the partons and the auxiliary electrons from the $y^{\mathrm{th}}$ wire
to the $(y+1)^{\mathrm{th}}$ one

$\displaystyle\begin{split}\mathcal{T}\tilde{\phi}_{y,a}\mathcal{T}^{-1}&=-\tilde{\phi}_{y+1,a}+\frac{(-1)^{y}}{2}\pi,\\\
\mathcal{T}\tilde{\Phi}^{\sigma}_{y,b}\mathcal{T}^{-1}&=-\tilde{\Phi}^{\sigma}_{y+1,b}+\frac{(-1)^{y}}{2}\pi.\end{split}$
(172)

The AFTR operator squares to

$\displaystyle\mathcal{T}^{2}=(-1)^{F}\mathrm{translation}_{y\to y+2},$ (173)

where $(-1)^{F}$ is the total fermion parity operator. The additional phases
to the AFTR transformation are included so that
$\mathcal{T}^{2}\tilde{\phi}_{y,a}\mathcal{T}^{-2}=(-1)^{F}\tilde{\phi}_{y+2,a}(-1)^{F}=\tilde{\phi}_{y+2,a}+(-1)^{y}\pi$
and similarly for the electronic bosonized variables
$\tilde{\Phi}^{\sigma}_{y,b}$.

The basis transformations (60), (63), (68) and (70) together with the
backscattering (62) split the chiral parton Dirac CFT
$U(3)_{1}/\mathbb{Z}_{3}$ along each interface into a pair of parton Pfaffians
$\mathrm{Pf}^{\ast}_{A/B}=U(1)_{24}^{A/B}\otimes_{\mathrm{el}}U(1)_{12}^{A/B}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}^{A/B}$.
The bosonized variables $\phi^{A/B}_{\rho}$, $\phi^{A/B}_{d}$ and $\phi_{0}$
in $U(1)_{24}^{A/B}$, $U(1)_{12}^{A/B}$ and
$\overline{SO(2)_{1}}=\overline{\mathrm{Ising}}^{A}\times\overline{\mathrm{Ising}}^{B}$
are described by the Lagrangian densities (up to non-universal velocity terms)

$\displaystyle\mathcal{L}_{\mathrm{tot}}=\sum_{y}\left[\left(\sum_{C=A,B}\mathcal{L}^{C}_{y,\rho}+\mathcal{L}^{C}_{y,d}\right)+\mathcal{L}_{y,0}\right],$
(174)

The neutral Dirac fermion
$e^{i\phi_{y,0}}\sim\cos(2\phi_{y,0})+i\sin(2\phi_{y,0})$ can be split into
Majorana components

$\displaystyle\begin{split}&\psi^{A}_{y}=\sqrt{2}\cos(2\phi_{y,0})\prod_{y^{\prime}<y}(-1)^{N_{y^{\prime}}},\\\
&\psi^{B}_{y}=\sqrt{2}\sin(2\phi_{y,0})\prod_{y^{\prime}<y}(-1)^{N_{y^{\prime}}},\end{split}$
(175)

where the Jordon-Wigner string is a product of some electronic number
operators $N_{y}$ similar to that in (152) and it ensures the mutual anti-
commutation relations between Majorana fermions in different wires. The
bosonized variables and Majorana fermions transform according to the
antiferromagnetic time-reversal symmetry (172)

$\displaystyle\mathcal{T}\phi^{A/B}_{y,\rho}\mathcal{T}^{-1}=-\phi^{A/B}_{y+1,\rho}+\frac{(-1)^{y}}{8}\pi,$
$\displaystyle\mathcal{T}\phi^{A/B}_{y,d}\mathcal{T}^{-1}=-\phi^{A/B}_{y+1,d},\quad\mathcal{T}\phi_{y,0}\mathcal{T}^{-1}=-\phi_{y+1,0},$
$\displaystyle\mathcal{T}\psi^{A}_{y}\mathcal{T}^{-1}=\psi^{A}_{y+1},\quad\mathcal{T}\psi^{B}_{y}\mathcal{T}^{-1}=-\psi^{B}_{y+1}.$
(176)

There are three primitive electronic quasiparticles $\openone_{12,\pm
6}=e^{i12\phi_{\rho}}e^{\pm i6\phi_{d}}$, $\Psi_{12,0}=e^{i12\phi_{\rho}}\psi$
in each parton Pfaffian channel. They all carries electric charge $e$ and were
defined in (32) in section II.2 and (73) in section II.3. The inter-wire
gapping interactions in figure 11 consists of the backscatterings of these
electronic quasiparticles.

$\displaystyle\mathcal{U}=\sum_{y}\mathcal{U}_{y+1/2}=\sum_{y}\left(\mathcal{U}^{1}_{y+1/2}+\mathcal{U}^{2}_{y+1/2}+\mathcal{U}^{3}_{y+1/2}\right)$
(177)

where the interactions between each pair of wires are

$\displaystyle\mathcal{U}^{1}_{y+1/2}$
$\displaystyle=-u_{1}(-1)^{y}{\openone_{12,6,y+1}^{A}}^{\dagger}\openone_{12,6,y}^{B}+h.c.$
$\displaystyle=-\tilde{u}_{1}(-1)^{y}\cos\left(\Theta^{\rho}_{y+1/2}+\Theta^{d}_{y+1/2}\right),$
$\displaystyle\mathcal{U}^{2}_{y+1/2}$
$\displaystyle=-u_{2}(-1)^{y}{\openone_{12,-6,y+1}^{A}}^{\dagger}\openone_{12,-6,y}^{B}+h.c.$
$\displaystyle=-\tilde{u}_{2}(-1)^{y}\cos\left(\Theta^{\rho}_{y+1/2}-\Theta^{d}_{y+1/2}\right),$
$\displaystyle\mathcal{U}^{3}_{y+1/2}$
$\displaystyle=-u_{3}(-1)^{y}i{\Psi_{12,0,y+1}^{A}}^{\dagger}\Psi_{12,0,y}^{B}+h.c.$
$\displaystyle=-\tilde{u}_{3}(-1)^{y}\cos(\Theta^{\rho}_{y+1/2})i\psi^{A}_{y+1}\psi^{B}_{y}.$
(178)

The sine-Gordon angle variables are
$\Theta^{\rho}_{y+1/2}=12\phi_{y,\rho}-12\phi_{y+1,\rho}+\pi N_{y}$ and
$\Theta^{d}_{y+1/2}=6\phi_{y,d}-6\phi_{y+1,d}+\pi N_{y}$. They transform
according to the AFTR symmetry
$\mathcal{T}\Theta^{\rho}_{y+1/2}\mathcal{T}^{-1}=-\Theta^{\rho}_{y+3/2}+3\pi(-1)^{y}$
and $\mathcal{T}\Theta^{d}_{y+1/2}\mathcal{T}^{-1}=-\Theta^{d}_{y+3/2}$.
Together with the AFTR transformation (176) that sends
$i\psi^{A}_{y+1}\psi^{B}_{y}\to i\psi^{A}_{y+2}\psi^{B}_{y+1}$, this shows the
collection of inter-wire backscatterings (177) preserves the antiferromagnetic
time-reversal symmetry. Moreover, they all preserve charge $U(1)$ conservation
because each term simply brings a charge $e$ electronic quasiparticle from one
wire to the next.

The coupled wire model (177) resembles the one that defines the paired parton
Pfaffian state in (155), (156) and (157). The model is exactly solvable and
the three gapping interactions are non-competing when
$\tilde{u}_{1}\tilde{u}_{2}>0$ and $\tilde{u}_{3}\neq 0$ so that they pin the
ground state expectation values $\langle
i\psi^{A}_{y+1}\psi^{B}_{y}\rangle=(-1)^{m_{\psi}}$ and
$\langle\Theta^{\rho}_{y+1/2}\rangle=m_{\rho}\pi$,
$\langle\Theta^{\rho}_{y+1/2}\rangle=m_{d}\pi$, where
$(-1)^{m_{\psi}+m_{\rho}}=\mathrm{sgn}(\tilde{u}_{3})$ and
$(-1)^{m_{\rho}+m_{d}}=\mathrm{sgn}(\tilde{u}_{1})$. Therefore they freeze all
low-energy degrees of freedom on the surface and introduce a finite excitation
energy gap in strong coupling.

Figure 12: A parton Pfaffian CFT (highligted blue line) at the domain wall
separating two gapped regions with symmetry-preserving interaction
$\mathcal{U}$ and AFTR symmetry breaking interaction $\mathcal{U}_{f}$.

The array of parton Dirac channels $U(1)_{3}\times SU(3)_{1}$ can also model a
gapped FTI surface that violates the AFTR symmetry. The “ferrimagnetic”
interaction is given by

$\displaystyle\mathcal{U}_{f}=$ $\displaystyle-
u_{f}\sum_{y}\Big{[}\cos\left(3\phi^{0}_{2y-1,\pi}-3\phi^{0}_{2y,\pi}\right)$
$\displaystyle\;\;\;+\sum_{a\neq b}E^{ab}_{2y-1}E^{ab}_{2y}\Big{]}$
$\displaystyle-
m_{e}\sum_{y}\sum_{b=0,1}{c^{R}_{y,b}}^{\dagger}c^{L}_{y,b}+h.c.$ (179)

where
$3\phi^{0}_{y,\pi}=\tilde{\phi}_{y,1}+\tilde{\phi}_{y,2}+\tilde{\phi}_{y,3}$
is the electronic bosonized variable of the charge sector $U(1)_{3}$ and
$E^{ab}_{y}=e^{i(\tilde{\phi}_{y,a}-\tilde{\phi}_{y,b})}$ are the roots of the
neutral $SU(3)_{1}$ sector (see section II.1). The last line consists of
intra-wire electronic backscatterings that removes the two pairs of counter-
propagating auxiliary electrons from low-energy. (179) dimerizes the array of
parton Dirac channels and introduces a finite excitation energy gap on the
surface.

When the AFTR symmetry-breaking surface is juxtaposed with the parton
$\mathcal{T}-\mathrm{Pf}^{\ast}$ surface enabled with interaction
$\mathcal{U}$ in (177), they leave behind a parton Pfaffian CFT at the domain
wall interface (see figure 12). This verifies one of the propositions in our
previous work Sahoo  _et al._ (2017) that a FTI slab with a symmetry-
preserving top surface and a time-reversal breaking bottom surface hosts a
parton Pfaffian CFT along the boundary (see figure 1). The bulk-boundary
correspondence implies the topological equivalence between the FTI thin film
(treated as a quasi-2D topological phase) and the particle-hole symmetric
paired parton FQH state $\mathfrak{P}_{1,-1}=\mathrm{Pf}^{\ast}$.

##  IV Particle-hole symmetry for partons

Particle-hole (PH) conjugation Girvin (1984); Son (2015); Barkeshli  _et al._
(2015); Wang and Senthil (2016); Balram and Jain (2017); Nguyen  _et al._
(2018) inverts a FQH state of electrons to a FQH state of holes in the lowest
Landau level (LLL). The PH conjugate of a FQH state $\mathfrak{F}$ is a new
FQH state obtained by subtracting $\mathfrak{F}$ from the LLL. The
“subtraction” is topologically captured by a product

$\displaystyle\mathrm{PH}_{\mathrm{LLL}}(\mathfrak{F})=U(1)_{1}\otimes\overline{\mathfrak{F}}$
(180)

between the lowest Landau level $U(1)_{1}$ and the time-reversal conjugate
$\overline{\mathfrak{F}}$ of $\mathfrak{F}$. This product describes the bulk
topological order as well as the gapless edge conformal field theory (CFT).
For instance, the time-reversal conjugate $\overline{\mathfrak{F}}$ refers to
the same edge CFT as $\mathfrak{F}$ except with the opposite propagating
direction. The LLL’s electric and thermal Hall transport, which are specified
by the filling fraction $\nu_{\mathrm{LLL}}=1$ and edge chiral central charge
$c_{\mathrm{LLL}}=c_{R}-c_{L}=1$ in (2), are reduced by the subtraction

$\displaystyle\begin{split}\nu\left(\mathrm{PH}_{\mathrm{LLL}}(\mathfrak{F})\right)=1-\nu(\mathfrak{F}),\\\
c\left(\mathrm{PH}_{\mathrm{LLL}}(\mathfrak{F})\right)=1-c(\mathfrak{F}).\end{split}$
(181)

For example, this relates FQH states in the Jain’s sequence Jain (1990)
$\mathfrak{J}^{+}_{p}$ with filling $\nu=p/(2p+1)$ and central charge $c=p$ to
another sequence $\mathfrak{J}^{-}_{p+1}$ with filling $\nu=(p+1)/(2(p+1)-1)$
and central charge $c=1-p$. The former (latter) has an effective Chern-Simons
description $\mathcal{L}=\frac{1}{2\pi}K_{IJ}a_{I}\wedge da_{J}$ with the
$p\times p$ $K$-matrix $(K_{\mathfrak{J}^{+}_{p}})_{IJ}=\delta_{IJ}+2$ [resp.
$(p+1)\times(p+1)$ $K$-matrix
$(K_{\mathfrak{J}^{-}_{p+1}})_{IJ}=-\delta_{IJ}+2$]. The two are related by
(180) which identifies

$\displaystyle
K_{\mathfrak{J}^{-}_{p+1}}=G^{T}\left[1\oplus(-K_{\mathfrak{J}^{+}_{p}})\right]G=G^{T}\begin{pmatrix}1&\\\
&-K_{\mathfrak{J}^{+}_{p}}\end{pmatrix}G,$ (182)

where 1 is the $K$-matrix of the lowest Landau level $U(1)_{1}$ and $G$ can be
chosen to be the $GL(p+1,\mathbb{Z})$ transformation

$\displaystyle G=\left(\begin{smallmatrix}1&2&2&\ldots&2\\\ &-1&&&\\\
&&-1&&\\\ &&&\ddots&\\\ &&&&-1\end{smallmatrix}\right)_{(p+1)\times(p+1)}.$
(183)

Particle-hole symmetry acts within the half-filled Landau level. It flips
between FQH states that shares the same filling at $\nu=1/2$. For example,
(180) conjugates the Read-Moore Pfaffian state Read and Moore (1991)
$\mathrm{Pf}_{1}=U(1)_{8}\otimes_{\mathrm{el}}\mathrm{Ising}$ to the anti-
Pfaffian state Levin  _et al._ (2007); Lee  _et al._ (2007)
$\mathrm{PH}_{\mathrm{LLL}}(\mathrm{Pf})=U(1)_{1}\otimes\overline{U(1)_{8}}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}$,
which is equivalent to
$\mathrm{Pf}_{-3}=U(1)_{8}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}^{3}$
up to anyon condensation Bais and Slingerland (2009). The equivalence can be
explicitly demonstrated by the basis transformation

$\displaystyle\begin{pmatrix}2\phi^{R}\\\
\Phi^{L}_{n}\end{pmatrix}=\begin{pmatrix}1&-1\\\
-1&2\end{pmatrix}\begin{pmatrix}\tilde{\Phi}^{R}_{\mathrm{LLL}}\\\
2\phi^{L}\end{pmatrix},$ (184)

where $e^{i\tilde{\Phi}^{R}_{\mathrm{LLL}}}$ is the edge chiral $U(1)_{1}$
Dirac electron of the LLL, $e^{i4\phi^{R}}$ and $e^{i4\phi^{L}}$ are the
charge $e$ spin 1 bosons in $U(1)_{8}$ and $\overline{U(1)_{8}}$, and
$d^{L}\sim e^{i\Phi^{L}_{n}}\sim\gamma^{L}+i\delta^{L}$ is a spin $1/2$
neutral Dirac fermion, which can be split into a pair of Majorana fermions and
correspond to a pair of $\overline{\mathrm{Ising}}$ CFTs. More recently, a
particle-hole symmetric Pfaffian state
$\mathrm{Pf}_{-1}=U(1)_{8}\otimes_{\mathrm{el}}\overline{\mathrm{Ising}}$ was
proposed by Son Son (2015) that is invariant under the particle-hole
conjugation (180), and has the identical topological structure of anyon
excitations as the time-reversal symmetric T-Pfaffian surface state Chen  _et
al._ (2014) of a 3D topological insulator. The equivalence between
$\mathrm{Pf}_{-1}$ and its conjugate
$\mathrm{PH}_{\mathrm{LLL}}(\mathrm{Pf}_{-1})=U(1)_{1}\otimes\overline{U(1)_{8}}\otimes_{\mathrm{el}}\mathrm{Ising}$
can be shown by a basis transformation similar to (184), which leaves behind a
$L$-moving neutral Dirac fermion $d^{L}\sim\gamma^{L}+i\delta^{L}$ that
reduces to a single Majorana fermion $\gamma^{L}$ upon removing $\delta^{L}$
from low-energy by backscattering to the $R$-moving Ising sector. Later, Kane,
Stern and Halperin Kane  _et al._ (2017) generalized a sequence of Pfaffian