Emergent QCD$_3$ Quantum Phase Transitions of Fractional Chern Insulators

Motivated by the recent work of QED$_3$-Chern-Simons quantum critical points of fractional Chern insulators (Phys. Rev. X \textbf{8}, 031015, (2018)), we study its non-Abelian generalizations, namely QCD$_3$-Chern-Simons quantum phase transitions of fractional Chern insulators. These phase transitions are described by Dirac fermions interacting with non-Abelian Chern-Simons gauge fields ($U(N)$, $SU(N)$, $USp(N)$, etc.). Utilizing the level-rank duality of Chern-Simons gauge theory and non-Abelian parton constructions, we discuss two types of QCD$_3$ quantum phase transitions. The first type happens between two Abelian states in different Jain sequences, as opposed to the QED3 transitions between Abelian states in the same Jain sequence. A good example is the transition between $\sigma^{xy}=1/3$ state and $\sigma^{xy}=-1$ state, which has $N_f=2$ Dirac fermions interacting with a $U(2)$ Chern-Simons gauge field. The second type is naturally involving non-Abelian states. For the sake of experimental feasibility, we focus on transitions of Pfaffian-like states, including the Moore-Read Pfaffian, anti-Pfaffian, particle-hole Pfaffian, etc. These quantum phase transitions could be realized in experimental systems such as fractional Chern insulators in graphene heterostructures.

Understanding phases and phase transitions is the key of condensed matter research.A cornerstone of the modern condensed matter research is the discovery and understanding of integer and fractional quantum Hall phases (IQHs and FQHs) [1][2][3], which inspires lots of interesting concepts, i.e., topological order, fractionalization, emergent gauge fields, in last few decades [4,5].Given the exotic properties of quantum Hall (QH) phases, it is very natural to expect that phases transitions of QH phases will also be strikingly different from conventional phase transitions.
A recent work [14] by one of us has shown that, the phase transitions between any two Abelian QH states in the same Jain sequence [19] are described by the QED 3 -Chern-Simons theory, namely N f flavors of Dirac fermions interact through an emergent U(1) gauge theory at Chern-Simons level K.These critical theories will flow to 3D conformal field theories (CFTs), and their properties depend on the values of N f and K.This family of critical theories have recently generated a surge of interest from both the condensed matter and high energy community due to the duality conjectures (see a review [20]).Some of these theories may have applications to other long-standing condensed matter problem, including spin liquids in frustrated magnetism [21][22][23][24][25][26], high temperature superconductivity [27] as well as deconfined phase transitions [28][29][30][31].Given its interesting properties and wide applications, there has been a large theoretical effort to study the properties of this family of critical theory [32][33][34][35][36][37][38], but its precise properties at small N f and K remain an open issue due to its strongly interacting nature.This makes its experimental exploration exciting, in particular the entire family of the QED 3 -Chern-Simons theory (with any combination of N f and K) is accessible experimentally at the phase transitions of QH/CI phases.
Motivated by the progress on the QED 3 -Chern-Simons universality class of the QH/CI transitions, here we go one step further to think about phase transitions described by the QCD 3 -Chern-Simons theory with emergent non-Abelian gauge fields.Similar to its QED 3 cousin, the QCD 3 -Chern-Simons theory also have interesting properties such as duality [39] and generates considerable interest in past [40][41][42][43][44][45][46][47].In this paper, we mainly focus on two types of such phase transitions that can be understood using non-Abelian parton constructions [48].One type is the phase transitions involving non-Abelian FQH/FCI states [48][49][50][51][52][53][54], at which non-Abelian gauge field naturally emerges.The other type is a bit more surprising as it happens between two Abelian states in different Jain sequences, as opposed to the QED 3 -Chern-Simons critical points between Abelian states in the same Jain sequence.
The rest of the paper is organized as follows.In Section II, we first review some previous works and terminologies that are useful for our discussion.In particular, we will discuss the condensed matter application of level-rank duality [39], and clarify its difference from the usual high-energy literature.Then in Section III, we discuss the QCD 3 -Chern-Simons transitions between Abelian states in different Jain sequences.One example is the transition between σ xy = 1/3 FQH/FCI state and σ xy = −1 IQH/ICI state, which has N f = 2 Dirac fermions coupled to a U (2) Chern-Simons gauge field.In the next two sections we turn to study non-Abelian states, in par-ticular we focus the Pfaffian-like states (e.g.Moore-Read Pfaffian [49], anti-Pfaffian [55,56], etc.) that are close to experimental realizations [57][58][59][60][61].In Section IV we provide parton constructions for these Pfaffianlike states, which are important building blocks for the following analysis on their phase transitions.Some of these constructions are known, but they are scattered in different papers [46,48,51,53,62].In Section V we discuss the QCD 3 -Chern-Simons transitions of these Pfaffian-like states.Finally, we summarize our results and discuss some future directions in Section VI.The appendices contain various technical details regarding non-Abelian Chern-Simons descriptions of several Pfaffian-like states.

A. Chern number changing transitions of free fermion
We start by briefly reviewing transitions between ICIs at fixed electron density and flux, since they are the building block of the later discussion.As will be clear later, the fractionalized transitions can be formulated as the Chern number changing transitions of composite fermions or fermionic partons.Throughout our discussion we use the unit in which the area of unit cell is a 2 = 1 and e 2 = 1, so that the Dirac flux quantum is φ 0 = 2π and the fundamental conductance is e 2 h = 1 2π .However we will express the value of σ xy in units of e 2 h so that the 2π factor can be omitted.The total Chern number of a free fermion insulator, which is equal to its Hall conductance, can be obtained by summing up the Chern numbers of occupied bands, and Chern number changing transitions arise from gap-closings in the underlying band structures.As a parameter such as the strength of the lattice potential changes, the conductance and valence bands may touch and Chern number is transferred between them.The transfer of Chern number ∆C between the two is mediated by the formation of |∆C| Dirac cones.For a transition between two phases with Chern number C 1 and C 2 respectively, the effective theory of the critical point is where Here A is the background electromagnetic field, measured relative to the original flux.AdA is the short hand notation for the Chern-Simons term A ∧ dA (or A is also referred as the probe field, as it can serve as a theoretical device to probe physical properties of the field theory such as Hall response and symmetry properties.The mass m is a phenomenological parameter whose physical meaning is the band gap.Integrating out the fermions when m is finite, we obtain where sgn(m) = m/|m|.We can see that this effective theory produces the desired response on either side of the transition.
It is worth noting that if ∆C is larger than 1, it generically requires to tune more than one parameters (i.e.masses of different Dirac fermions).Such finetuning can be simply avoided in the presence of lattice symmetry [14,15].In particular, it was shown that the magnetic translation symmetry can protect Chern number changing by an arbitrary number [14].Specifically, if the fermion sees a flux p/q per unit cell (with p, q being co-prime) the Chern number changing of fermions is enforced to be |∆C| = q.In the rest of paper, we will always assume the Chern number changing with an arbitrary ∆C is protected by lattice symmetries.

B. Level-rank duality: gauge field versus spin gauge field
Next we will briefly review the level-rank duality [39] as well several terminologies that will be substantially used in the rest of the paper.
When we are using the Chern-Simons theory to describe a topological order, it is very important to distinguish a spin gauge field from a normal gauge field: the former refers to the dynamical gauge field coupled to fermions, while the latter refers to dynamical gauge field coupled to bosons. 1 Throughout this paper we are using a superscript s to denote the spin gauge field (e.g.U (1) s , SU (2) s ).Aslo we denote spin gauge fields by Greek symbols (α, β, • • • ), normal gauge fields are labeled by Latin letters (a, b, • • • ), and probe (background) fields are labeled by capital letters (A, B, • • • ).
Physically the difference between a spin gauge field and a gauge field comes from the subtlety that the fermions (that the spin gauge field couples to) will contribute a nontrivial sign factor to fractional statics of certain anyons.The simplest example to appreciate this is to consider the Kalmeyer-Laughlin chiral spin liquid (i.e.1/2 Laughlin state of bosons) [63].The well known effective theory (also called K−matrix formalism [4]) of this state is The anynon in the above theory is semion, exchanging of which yields a phase factor i. To get the response of the theory one can integrate gauge field b, The first term gives the σ xy = 1/2 electric Hall conductance, while the last term gives the κ xy = 1 thermal Hall conductance.Note that CS g is the gravitational Chern-Simons term [64,65], which physically corresponds to the thermal Hall conductance.The coefficient of gravitational Chern Simons term characterizes thermal Hall conductance in units of (π/6)(k 2 B T / ).It is also known that one can use fermionic parton approach to construct the Kalmeyer-Laughlin chiral spin liquid.We first start with a parton decomposition that fractionalizes a spin−1/2, , and the Kalmeyer-Laughlin chiral spin liquid is realized by putting each parton f ↑,↓ on a C = 1 band.One can notice that there is a U (1) gauge invariance (maximally there is a SU (2) gauge invariance), so we can introduce a dynamical U (1) "gauge field" α.After integrating out gapped fermionic partons we have, Further integrating out α, the above theory yields the same response (σ xy = 1/2, κ xy = 1) as the Kalmeyer-Laughlin chiral spin liquid.However, there seems to have an apparent discrepancy between Eq. (3) and Eq. ( 5) as their Chern-Simons levels are opposite.Naively one may conclude that Eq. ( 5) has anti-semion, exchanging of which yields a phase factor −i.The way to reconcile the inconsistency is to realize that α in Eq. ( 5) is a spin gauge field that couples to fermions, while b in Eq. ( 3) is a gauge field couples to bosons.The coupling with fermions will give an extra sign to the fractional statistics of anyons, which converts the anti-semion to semion.Indeed the equivalence between Eq. (3) and Eq. ( 5) is nothing but a known statement, namely the duality between U (1) 2 and U (1) −2 [39].We remark that in Ref. [39] and many other literature the duality between U (1) 2 and U (1) −2 is achieved by adding a local fermion to both sides of Lagrangian.In the condensed matter context it is more physical to use another version of the duality, namely we have gauge fields on the one side but spin gauge fields on the other side, abstractly written as U (1) 2 ∼ = U (1) s −2 .A formal and systematic approach to understand the relation between spin gauge fields and gauge fields is to use the level-rank (or more generally boson-fermion) duality of TQFTs [39,66].Throughout the paper, we will heavily rely on the level-rank duality to study parton constructions and phase transitions of FCIs/FQHs.Before closing this section, we will discuss an example of a non-Abelian state to show the power of level-rank duality.
Again let us start with the familiar parton construction, Here α is a SU (2) spin gauge field.We emphasize again that the usual terminology of the Chern-Simons description of a topological order is to use the gauge field rather than the spin gauge field.So we can further use the level-rank duality, to convert Eq. ( 6) into a Chern-Simons theory in terms of a U (k) gauge field, a, Now we are ready to read out the topological order of this parton construction.For example, k = 1 reduces to the previously discussed example, the bosonic ν = 1/2 Laughlin described by the U (1) 2 Chern-Simons theory.When k = 2 [48], we have the U (2) 2 topological order.We note that in some literature it was mistaken that the k = 2 parton construction gives the SU (2) 2 topological order (also called bosonic Pfaffian).
C. QED3-Chern-Simons universality and its parton construction The quantum phase transition between Jain sequence state σ xy = C 1 /(kC 1 +1) and σ xy = C 2 /(kC 2 + 1) is described by the QED 3 -Chern-Simons theory [14], Here α µ and b µ are dynamical U (1) gauge fields, A µ is the U (1) background field (e.g.electromagnetic field if we consider electrons).k is an odd integer if one considers Jain sequence state of bosons, while k is an even integer if one considers Jain sequence state of fermions.The dynamical gauge field b µ comes from the Chern-Simons flux attachment for composite fermions.In many literatures, b µ is integrated out, which yields a Chern-Simons term 1/4kπαdα.This new family of QED 3 -Chern-Simons quantum critical points was previously studied using the language of composite fermions, here we will reformulate it using the parton construction that is useful for the rest of this paper.We consider a parton construction that fractionalizes a particle c into c = ϕf .Here f is fermionic, while ϕ is fermionic or bosonic depending on whether c is a fermion or a boson.Jain composite fermion state σ xy = C/(kC + 1) is realized by the mean field ansatz that ϕ is in a 1/k Laughlin state (k is even or odd if ϕ is bosonic or fermionic), and f is in an integer quantum Hall state with Chern number C. One could recognize that the parton ϕ implements the Chern-Simons flux attachment, while fermionic parton f can be viewed as the composite fermion.The phase transition between two Jain sequence states can be formulated as the Chern number changing transition of f parton.

D. Possible experimental realization
The theory of phase transition discussed above (and below) generally applies to many systems.An ideal system for it as well as the QCD 3 -Chern-Simon quantum phase transitions addressed later is the graphene heterostructures with a magnetic field and superlattice potential (e.g. a moire lattice) [11][12][13].This experimental system can be modeled as Landau levels subject to a weak superlattice potential.In specific, we consider the regime that the cyclotron gap between Landau levels is much larger than the superlattice potential strength.In this regime each Landau level splits into sub-bands, and CIs are realized by completely or partially filling some of these sub-bands.The transitions between different FQHs and ICIs/FCIs can be achieved by tuning the lattice potential (e.g.potential strength).Below we discuss a concrete example, namely a phase transition between the σ xy = 1  3 FQH state and σ xy = 1 ICI [14].
A general form of superlattice potential can be written as, where G m are reciprocal vectors of the superlattice.A square lattice potential, for example, has , where a is the lattice constant.We consider flux density φ = 3 2 (in units of Dirac flux quanta) and electron density n = 1 2 per unit cell of the superlattice.Without the superlattice potential, the lowest Landau level (LLL) is 1/3 filled yielding the ν = 1/3 FQH state.Once the superlattice potential is implemented, the LLL splits into three subbands, with a band gap ∆ ∝ U L .In the limit of strong lattice potential (i.e.potential strength much larger than the interaction strength), the lowest sub-band will be completely filled realizing a ICI state.The Chern number will depend on the form of lattice potential.For example, if we take FIG. 1: The band structure of the LLL subject to a weak lattice potential with the form in Eq. ( 10).(a) the lowest band will have C = 1 (see Fig. 1 for detailed band structure).As one tunes the amplitude of the superlattice potential, a phase transition can be realized between σ xy = 1 3 state and σ xy = 1 state.The critical theory is described by Eq. ( 9), in which An interesting question is what will happen if we swap the order the the three subbands such that the lowest one has Chern number C = −1.This can be achieved by reversing the sign of the lattice potential strength used above, as shown in Fig. 1.Now one obtains a transition between ν = 1  3 FQH state and C = −1 ICI state.This transition does not belong to the previously introduced QED 3 -Chern-Simons universality.Instead, an non-Abelian gauge field will emerge at the quantum critical point as we will discuss in the next section.We fractionalize a spinless electron operator into three fermionic partons The σ xy = 1/3 state is realized by a mean-field ansatz that all the fermionic partons are in a C = 1 band (or Landau level), while the σ xy = −1 state is realized by a mean-field ansatz that f is in a C = 1 band and f 1,2 are in a C = −1 band, respectively.The transition between these two states can be described by tuning the Chern number of For each fermionic parton we can use lattice symmetries (eg.magentic translation symmetry) to protect a Chern number transition with ∆C = 2.However, there is no global symmetry that can force f 1 and f 2 to change their Chern number simultaneously.In other words, in a naive setup this phase transition cannot be a direct phase transition.It turns out that the emergent non-Abelian gauge symmetry can help to realize a direct phase transition.The parton construction (Eq.( 11)) has a maximal SU (3) gauge symmetry, (f, f 1 , f 2 ) forms a SU (3) fundamental.This SU (3) gauge structure can be kept in the σ xy = 1/3 state, while it is broken down (the maximally remaining gauge symmetry is fundamental, while the f parton is charged under the diagonal U (1) of the U (2) gauge symmetry.The f 1 and f 2 partons are now related by the U (2) gauge rotation, hence their Chern number has to change simultaneously.The critical theory is described by an effective Lagrangian: where α is a U (2) spin gauge field, and ψ I are U (2) fundamental Dirac fermions.The mass term m 2 I=1 ψI ψ I is the tuning operator for the phase transition.At semi-classical regime |m| 1, the Dirac fermion can be integrated out and we end up with Depending the sign of m, one would either get a σ xy = 1/3 state (m 1) or a σ xy = −1 state (m −1).One may worry that the above U (2) s gauge theory may not describe the Abelian state.To work out its topological order, we need to utilize the level-rank duality.Let us explain it for the case of m 1.We start with the level rank duality, U (N ) s Next we implement S, T transformation S T −3 on the above duality2 , yielding (a background term Adb has been automatically added), Integrating out the U (1) gauge field b in the first Lagrangian, we eventually end up with, which is exactly Eq. ( 13) for m 1.Also the second Lagrangian of the duality in Eq. ( 15) is the effective theory of σ xy = 1/3 state.Therefore, we have shown that m 1 limit gives the σ xy = 1/3 state.Similarly, one can show m −1 gives the σ xy = −1 state.

B. Generalization to U (N ) gauge theories
We now consider a generalization of the previous discussion.We first fractionalize a particle, Here ϕ i , f i are fermionic partons, and k + n is odd (even) if the original particle c is fermionic (bosonic).
We consider the transition between two states: 1) Here α is a U (n) spin gauge field, ψ I is in the U (n) fundamental representation.b is a U (1) gauge field, which comes from the ϕ partons (one can view ϕ 1 Similar as before we can use the level-rank duality to work out the topological order described by the above theory.The U (n) s −1 theory is dual to a topologically trivial theory [39] 1 4π Tr(αdα + 2 3 We then implement a transformation S T −k on both sides of the duality, yielding The right hand side of Eq. ( 21) is exactly the TQFT of σ xy = 1/(n + k) Abelian fractional quantum Hall/Chern insulator state that appears when m 1.The same analysis can also be implemented to the m −1 case, yielding the σ xy = 1/(n − k) Abelian fractional quantum Hall/Chern insulator state.
We remark that the above discussion carries over for k = 0 with a slight modification that the gauge field changes from U (N ) to SU (N ).
Similarly, we use another construction, Here ϕ, ϕ i , f i are all fermionic partons, and k+n is odd (even) if the physical particle c is bosonic (fermionic).We consider the transition between two states, namely 1) The critical theory has again two flavors of Dirac fermions, interacting with a U (n) spin gauge field, Here α is a U (n) spin gauge field, ψ I is in the U (n) fundamental representation.b, β are U (1) gauge field and spin gauge field, respectively.The first three terms in Eq. ( 23) are similar to the critical theory Eq. ( 18).
In the third term we have a dynamical U (1) spin gauge field β, which corresponds to the U (1) gauge symmetry coming from the relative phase rotation between ϕ and Eq. (22).Similar as before one can show that at semiclassical regime |m| 1, the effective theory (after applying the level-rank duality) is which is exactly the effective theory of Abelian composite fermion states with 1) σ xy = p/((k + n)p + 1) state when m 1; and 2) σ xy = p/((k − n)p + 1) state when m −1.

IV. Parton Constructions for Non-Abelian States
In the previous section we showed that a direct phase transition between two FQHs/FCIs in different Jain sequences [68] are necessarily described by a QCD 3 -Chern-Simons theory.In the rest of this paper we study phase transitions involving non-Abelian FQHs/FCIs , i.e., states that have anyons with non-Abelian statistics.We focus on non-Abelian states that are close to the Moore-Read "Pfaffian" state [49], including Pfaffian, anti-Pfaffian, PH-Pfaffian, bosonic Pfaffian states, Ising topological order, etc.Some of them might be realized in the half filled Landau level [57][58][59][60] or frustrated spin systems [69].The transitions involving these states can be naturally captured by QCD 3 -Chern-Simons theories with different gauge groups.For this purpose, we first discuss the parton constructions for these states.Then in Sec.V, the phase transitions will be analyzed.

A. U Sp(4) parton construction
Many of the non-Abelian topological orders we discuss below contain a U Sp(4) s spin gauge field in their TQFT descriptions.In this subsection we will first introduce a parton construction that has emergent U Sp(4) gauge symmetry [51,53].This U Sp(4) parton construction is similar to the familiar SU (2) parton construction ( S = f † σf ), and it is compatible with a global SO(3) spin rotation symmetry.
We consider a spin system (or equivalently a bosonic system), and fractionalize the spin operator into fermionic partons, which maximally has a local U Sp(4) gauge invariance, with Ψ = (ψ 1 , ψ 2 , ψ 4 , ψ 3 ) T being a U Sp(4) vector (fundamental).The local constraint is, on each site.S − can be expressed as which is invariant under a local U Sp(4) transformation Ψ → LΨ.Note that the local Hilbert space has spin S = 1 since (S − ) 2 = 0 but (S − ) 3 = 0.The U Sp(4) gauge invariance and the physical spin rotation symmetry can be easily seen by introducing a 4 × 2 matrix X, which satisfies a reality condition Here µ 2 and σ 2 are Pauli matrices acting from left and right, respectively: The physical spin operators can be expressed as in which spin rotation symmetry acts on X from right It is also easy to see that the spin operator is invariant under a U Sp(4) gauge transformation, acting from left, In general, any elements in U (4) leave spin operators invariant, but the reality condition Eq.( 29) requires L lie in U Sp(4), namely L T µ 2 L = µ 2 .Eq.( 29) on the other hand does not impose further constraints on the global spin rotation symmetry.Note that the global symmetry SU (2) and the gauge group U Sp(4) share a common center −1.Thus the physical global symmetry is actually SU (2)/Z 2 = SO(3).This parton construction can also be understood intuitively by noting that the maximal faithful symmetry of four complex fermions ψ i (i = 1, 2, 3, 4) is O(8).After gauging a U Sp(4) ∼ SO(5) subgroup, an SO(3) subgroup is left as the global spin rotation symmetry.
We also remark that, explicitly the parton construction is, It is straightforward to check that they satisfy the standard commutation relation B. Partons for Pfaffian-like states Using similar results from level-rank duality, we are able to find the parton construction of different non-Abelian states, which are summarized in Table I.Some of these constructions have been discussed in previous papers [46,48,51,53,62].
As discussed above, the bosonic U (2) 2 topological order can be realized in spin/bosonic systems.Without loss of generality, we take the spin-1/2 system as an example.We start with the standard SU (2) parton construction where This parton construction has a local SU (2) gauge structure.The U (2) 2 topological order is realized by putting both f ↑,↓ in C = 2 bands [48,51].After integrating out the partons, one ends up with an SU (2) s −2 Chern-Simons theory 2 4π Tr Here α is a spin SU (2) gauge field.A is the U (1) background field coupled to the global S z spin rotation.Using a level/rank duality SU (2) s −2 ∼ = U (2) 2,2 , the description with normal gauge field can be obtained: Here a is a U (2) gauge field.

Ising topological order
The Ising topological order corresponds to the ν = 1 state in Kitaev's 16-fold-way [69].It can be realized on a model of localized spin-1/2 particles on a honeycomb lattice, with a designed spin interactions.The TQFT description of Ising topological order in terms of spin gauge fields and parton construction were derived in detail in Ref. [46], here we briefly summarize the results.
The Ising topological order is described by , which was shown to be equivalent to the following topological order with Lagrangian where χ is a spin U (2) gauge field and a is a U (1) gauge field.The parton construction is achieved by fractionalizing the physical spin operator as This parton construction has a U(2) gauge invariance: and it is interacting with a U (2) spin gauge field χ; φ carries charge under the U (1) diagonal part of χ.
To get the Ising topological order, we put the bosonic parton φ into a ν = − 1 2 Laughlin state, and put the fermionic partons f i into a topological band with Chern number C = 2.After integrating out the gapped partons, we end up with the TQFT Eq. (43).

Bosonic Pfaffian state
The bosonic Pfaffian state is believed to be realized by bosons in the first Landau level at filling factor ν = a When κxy is concerned in the literature of the ν = 5/2 quantum Hall state, an extra κxy = 2 from the filled first Landau level is usually included.
1, and it can be understood as the p + ip pairing state of composite fermions [50,52].At long wavelength the bosonic Pfaffian state can be described by a SU (2) 2 Chern-Simons gauge theory [50], in which b is a SU (2) gauge field, B is a background probe field.It is known that the bosonic Pfaffian state can also be realized in a spin−1 system preserving the SO(3) spin rotation symmetry, so sometimes it is also refered as a non-Abelian chiral spin liquid [70].To construct the bosonic Pfaffian state, we use Eq.( 25) to fractionalize the spin-1 operator into fermionic partons 3 .The bosonic Pfaffian state is realized by having ψ i in a C = 1 band [51,53], yielding a U Sp(4) s −1 Chern-Simons theory (β is a U Sp(4) spin gauge field), Tr(βdβ + 2 3 which is dual to SU (2) 2 in Eq. ( 45) through the levelrank duality.As a self-consistent check, one can find that after integrating the U Sp(4) gauge field in the above U Sp(4) theory has the same thermal Hall response as of the bosonic Pfaffian (κ xy = 3/2).We can break the U Sp(4) gauge symmetry down to a U (2) gauge symmetry, and the U Sp(4) s −1 Chern-Simons theory will become the U (2) s −2,−2 theory.Interestingly, U ), hence this gives us an alternative way to describe bosonic Pfaffian as well as their transitions.
3 For the bosonic system, one can simply replace spin operators by bosonic annihilation/creation operators.

Pfaffian state
The fermionic Pfaffian state [49] is described by the following TQFT [71] with the Lagrangian [72] Here b is a U (2) gauge field, c is a U (1) gauge field.It is shown in Appendix A that its dual TQFT description in terms of spin gauge fields are in which β is a U Sp(4) spin gauge field and a, b are both U (1) gauge fields.The chiral central charge of the theory is − 5 2 − 1 + 1 + 4 = 3 2 (− 5 2 comes from U Sp(4) −1 part, −1 + 1 comes from the two U (1) parts and 4 comes from 4CS g ), which is identical to that of the Pfaffian state.

Z2
gauge redundancy.Roughly speaking, ψ i (i = 1, 2, 3, 4) furnish a fundamental representation of U Sp(4) subgroup and we will put them on C = 1 Chern bands.We also put ψ on a C = 1 band.Besides the dynamical U Sp(4) spin gauge field β, ψ i 's also couple to a 2 1 4 , where a is a U (1) gauge field which comes from the relative phase rotation between (ψ 1 ψ 4 − ψ 3 ψ 2 ) and ψ.And ψ has charge 1 under A, the external electromagnetic field.Gauge invariance imposes a constraint on the density of the partons: Integrating out the fermions leads to in which the b field describes the ν = 1 integer Quantum Hall state of ψ.We thus recover the TQFT of fermionic Pfaffian state Eq. ( 49).
The fractional U (1) gauge charge of the ψ i partons looks ill-defined.Actually it reflects the fact that the center element of U Sp(4) should be identified with π flux of the U (1) gauge symmetry, and the precise gauge structure of Pfaffian state is × U (1) 1 .Detailed discussion is shown in Appendix A. One more subtlety is that here an ordinary gauge field a, instead of a spin gauge field, is coupled to a fermion ψ.From the parton construction Eq. ( 50), one can see this field couples to both a bosonic part, i.e. ∼ (ψ 1 ψ 4 − ψ 3 ψ 2 ), and a fermionic part, i.e. ψ.These two viewpoints differ by a local fermion, namely the physical electron, which has no influence on the topological order in electronic systems.If we interpret the electromagnetic field A as a spin c connection, the choice we adopt here can be consistently put on a spin c manifold, since (a + A), to which ψ couples, is a spin c connection.

Anti-Pfaffian state
The anti-Pfaffian state [55,56] is described by where b is a U (2) gauge field which is coupled to a bosonic matter, A is the background electromagnetic field.The dual spin gauge field description can be found by the similar method as that described for Pfaffian state and the result is (see Appendix A for details) which is roughly a U (2) s × U (1) theory.The chiral central charge of the TQFT in Eq. ( 54) can then be easily obtained: 5  2 + 1 − 4 = − 1 2 , consistent the Anti-Pfaffian state.
This TQFT motivates a parton construction for Anti-Pfaffian state [62], where we enforce the emergent gauge symmetry is tal coupled to a dynamical U (2) spin gauge field χ, while ψ is charged under the diagonal U (1) of the U (2) gauge field.We put f 1,2 in C = −2 bands and assume ψ forms a ν = 1 3 Laughlin state.Integrating out the fermion fields gives rise to Eq. ( 54).

PH-Pfaffian state
The PH-Pfaffian state [73] can be described by the following Chern-Simons theory in terms of gauge field: with Lagrangian The dual description in terms of spin gauge field is where β is a U Sp(4) spin gauge field, b and a are U (1) gauge fields.The duality is derived in Appendix A. We can do a self-consistent check by examining the gravitational response.Integrating out the gauge fields yields a gravitational Chern Simons term 9 2 CS g .Combined with the last term in Eq. ( 58), the total gravitational response is 1  2 CS g , which is identical to that of the PH-Pfaffian state.The parton construction is straightforward, Similar to the case in Pfaffian state, this parton construction also has a U Sp (4) s ×U (1)

Z2
local gauge redundancy.We put ψ i , i = 1, 2, 3, 4, which are in fundamental representation of U Sp(4) group and carry charge 1/2 under the U (1) gauge field b, in C = −1 bands.The ψ parton, which carries charge 1 under both the U (1) gauge field b and external electromagnetic field A, forms a ν = 1 3 Laughlin state.After integrating out the fermionic matters, one reproduces the TQFT for PH-Pfaffian state, Eq. (58).A detailed analysis on normalization conditions for gauge charge and gauge flux shows that this Lagrangian can be written as (see Appendix.A) 7. Wen's (221)-parton state Wen's (221)-parton state [48,51] was recently proposed as a candidate state of the half filled higher Landau levels of graphene [61].Its parton construction is where f 1 , f 2 and ψ are all fermionic partons.Note that the construction is the same as that for Anti-Pfaffian state, Eq.( 55), which has a U (2) s gauge field χ while ψ is charged under Trχ.We put (f 1 f 2 ) in C = 2 bands and ψ in a C = 1 band.After integrating out the partons, one ends up with an effective theory which is the U (2) s −2,−4 Chern-Simons theory.

V. Phase transitions from the Pfaffian-like states
Following the philosophy used in the discussion about transitions between Abelian FCIs/FQHs in Section III, we can also describe phase transitions involving these Pfaffian-like states.We summarize the key ideas: (1) Phase transitions in general can be understood as Chern number changing transitions of the underlying parton insulators, which are mediated by |∆C| Dirac cones.At the critical point, N f = |∆C| flavors of Dirac fermions interact with non-Abelian gauge fields.
(2) Naively only |∆C| = 1 transitions are generic without fine tuning, however one can use symmetries, in particular the magnetic translation symmetry, to enforce |∆C| > 1 without fine-tuning.Specifically, we can design an external magnetic field and electron density such that the partons see a fractional effective flux.Thus higher Chern number changing transitions can be protected by the magnetic algebra.We focus on those transitions between the Pfaffian-like states and Abelian states, which is more realistic in experimental settings.

A. Phase transitions out of the bosonic Pfaffian
We will first discuss phase transitions out of the bosonic Pfaffian state.From this example it will be clear about how to understand in general the phase transitions involving other similar Pfaffian like states.
As we discussed above the bosonic Pfaffian state can be constructed using the U Sp(4) parton construction, S − = (ψ 1 ψ 4 − ψ 3 ψ 2 ), with each fermionic parton (ψ 1,••• ,4 ) on a C = 1 Chern band.When the fermionic partons have a different Chern number C, a state described by the U Sp(4) s −C TQFT is realized.For example, if C = 0 we have a topologically trivial Mott insulator, while if C = −1 we have a time-reversal reversed bosonic Pfaffian.The phase transitions between the bosonic Pfaffian and these states is described by the Chern number changing transitions of the fermionic partons.Since the fermionic partons are coupled to a U Sp(4) spin gauge field, the critical theory of the phase transitions are Chern-Simons field.It is worth noting that there is a level-rank duality between the U Sp(4) s −1 and U (2) s −2 Chern-Simons theory.Therefore, the bosonic Pfaffian state can be equivalently described by a U (2) s gauge theory.For the U (2) parton construction, we can simply higgs the U Sp(4) s gauge symmetry down to the U (2) s gauge symmetry, for which (ψ 1 , ψ 2 ) and (ψ 4 , ψ 3 ) are the U (2) fundamental and anti-fundamental, respectively.This structure is manifest if one writes the spin operator in matrix form and the U (2) gauge transformation is simply where W is an U (2) matrix in fundamental representation.
By assigning Chern number C 1 and C 2 to (ψ 1 , ψ 2 ) and (ψ 4 , ψ 3 )4 , we will get a state described by the U (2) s −(C1+C2) Chern-Simons theory, whose Lagrangian is Here α is a U (2) spin gauge field and A is a probe field coupled to the S z spin rotation.To get the response (i.e.Hall conductance σ xy and thermal Hall conductance κ xy ) of the state, we can further integrating out TABLE II: Transitions out of the bosonic Pfaffian can be described by the parton construction the U (2) gauge field α, It is worth noting that when C 2 = −C 1 , the effective theory Eq. ( 65) reduces to C1 2π A d Tr(α), which describes a superfluid with charge-2|C 1 | condensation 5 .
The phase transitions between different states are again described by the Chern number changing transitions of partons.However, we shall emphasize that there is no (gauge or global) symmetry to relate C 1 with C 2 , hence a direct phase transition can only have C 1 or C 2 changing, with another one fixed.Table .II summarizes several phases obtained by varying C 1 with C 2 fixed.The phase transitions between the bosonic Pfaffian and these states are described by the critical theory: Chern-Simons gauge field.
One may note that there are two different critical theories for the phase transition between the bosonic Pfaffian and Mott insulator, 1) N f = 1 flavor of Dirac fermions coupled to a U Sp(4) s −1/2 Chern-Simons field; 2) N f = 1 flavor of Dirac fermions coupled to a U (2) s −3/2 Chern-Simons gauge field.Indeed these two critical theories may be dual to each other as discussed in Ref. [39,42].If this duality conjecture is correct, it means that the second theory will have an emergent SO(3) global symmetry rather than the naive U (1) global symmetry. 5Due to the absense of Chern-Simons therm, the U (2) gauge field will confine, and the monopole of the diagonal U (1) field Tr(α)/2 will condense.On the other hand, due to the mutual Chern-Simons term C 1 2π A d Tr(α), the monopole will carry q = 2|C 1 | charge of the physical particle.Therefore, we end up with a charge-2|C 1 | superfluid.

B. Phase transitions of other Pfaffian-like states
Similar to the previous example of the bosonic Pfaffian, phase transitions out of other Pfaffian-like states can be described by changing the Chern number of partons.In Table .IV-VI we have listed the parton states of various parton constructions for the aforementioned Pfaffian-like states: U (2) 2 bosonic state (Table III), Ising topological order (Table IV), anti-Pfaffian state (Table V), Wen's (221)-parton state (Table VI), Pfaffian state (Table VII), and PH-Pfaffian state (Table VIII).Among all the parton states, we would like to say a bit more about the topological superconductors.
There are several ways to use partons to construct topological superconductors, which include the f − if superconductor in Table VI and  (67) Here α is a U (2) spin gauge field, A is the probe field (i.e. the electromagnetic field).To understand why it describes a superconductor, we decompose the U (2) gauge field into a SU (2) and a U (1) gauge field, α = α SU (2) + 1 2 (Trα)1 2 , so that the effective theory is Due to the absence of the Chern-Simons term of the U (1) gauge field-Trα/2, the monopole of Trα/2 will condense which confines the U (1) gauge field.On the other hand, the monopole carries a q = 2 charge of A (hence it is a Cooper pair), the condensation of it will lead to a superconductor.Also we note that the  Ising topological order gauge structure of the theory is

Z2
, the SU (2) gauge fundamental has to carry a charge of the U (1) gauge field (hence it is a U (2) fundamental).This means the confinement of the U (1) gauge field will lead to the confinement of the SU (2) gauge field, even though the latter has a non-trivial Chern-Simons term.Indeed the U (2) fundamental (i.e.(f 1 , f 2 )) is the vortex of the superconductor, and it has a non-Abelian statistics captured by the SU (2) s Chern-Simons term.Moreover, by integrating out α SU (2) , we can find that the superconductor has a thermal Hall conductance κ xy = −3/2, so it is a f − if superconductor.With very similar analysis one can also construct a p − ip superconductor with the parton construction shown in Table VII.
Below we will briefly discuss the phase transitions of all these Pfaffian-like states.

U (2)2 bosonic state
The parton decomposition S − = f 1 f 2 can be used to construct the U (2) 2 bosonic state.This parton construction has an emergent SU (2) gauge symmetry, with (f 1 , f 2 ) T being a SU (2) fundamental.The U (2) 2 state is realized by putting (f 1 , f 2 ) on a C = 2 Chern band, and by changing the Chern number C of (f 1 , f 2 ) we can obtain various states as summarized in Table III.Their transitions are described by, • N f = |C − 2| flavor of Dirac fermions coupled to a SU (2) s −(C+2)/2 Chern-Simons gauge field.

Ising topological order
The parton decomposition S − = f 1 f 2 φ can be used to construct the Ising topological order.This parton construction has an emergent U (2) ∼ = SU (2)×U ( 1) Z2 gauge symmetry, with (f 1 , f 2 ) T being a U (2) fundamental and φ being charged under the diagonal U (1) of the U (2) gauge field.The Ising topological order is realized by putting (f 1 , f 2 ) on a C = 2 Chern band, and putting φ into a ν = −1/2 bosonic Laughlin state.By changing the Chern number C of (f 1 , f 2 ) we can obtain various states as summarized in Table IV.The phase transitions between the Ising topological order and other states are described by a U (2) s gauge theory, Here α is a U (2) spin gauge field, b is U (1) gauge field which describes the ν = −1/2 Laughlin state of φ, A is a U (1) background field.We omit the gravitational Chern-Simons term as well as the mass of Dirac fermions for convenience.We also remark that some phase transitions can be described by changing the state formed by φ.For example, if φ forms a superfluid state, the corresponding parton state is the U (2) 2 bosonic state, hence its transition is a gauged version of the superfluid-ν = −1/2 Laughlin state transition [14, 15]6 .

Anti-Pfaffian state
The parton decomposition c = f 1 f 2 ψ can be used to construct the anti-Pfaffian state.Similar to the parton construction for the Ising topological order, this construction has an emergent U (2) gauge symmetry, with (f 1 , f 2 ) T being a U (2) fundamental and ψ being charged under the diagonal U (1) of the U (2) gauge field.The anti-Pfaffian state is realized by putting (f 1 , f 2 ) on a C = −2 Chern band, meanwhile ψ is forming a ν = 1/3 Laughlin state.By changing the Chern number C of (f 1 , f 2 ) we can obtain various states as summarized in Table V.The phase transitions between the anti-Pfaffian state and other partons states are described by a U (2) s gauge theory, Here α is a U (2) spin gauge field, b is U (1) gauge field which describes the ν = 1/3 Laughlin state of ψ, A is a U (1) probe field (i.e. the electromagnetic field).

Wen's (221)-parton state
The parton decomposition c = f 1 f 2 ψ can also be used to construct Wen's (221)-parton state.The difference from the anti-Pfaffian state is that, here we put (f 1 , f 2 ) on a C = 2 Chern band, and ψ on a C = 1 Chern band.By changing the Chern number C of (f 1 , f 2 ) we can obtain various states as summarized in Table VI.The phase transitions between the Wen's (221)-parton state and other partons states are described by a U (2) s gauge theory, Here α is a U (2) spin gauge field, b is U (1) gauge field which describes the ν = 1 integer quantum Hall state of ψ, A is a U (1) probe field (i.e. the electromagnetic field).
Also one can change the Chern number of ψ instead of (f 1 , f 2 ).For example, if ψ realizes a C = −1 Chern insulator, the parton state will be a f + if superconductor.The phase transition will then be described by a QED 3 -Chern-Simons theory rather than a QCD 3 -Chern-Simons theory.
conductor.The critical theory is, Here α is a U Sp(4) spin gauge field, b is a U (1) gauge field.c is a U (1) gauge field which describes the Chern number 1 insulator of ψ, A is a U (1) probe field (i.e. the electromagnetic field).Note that the ψ i (i = 1, 2, 3, 4) partons couple to an ordinary U (1) gauge field b, since the "fermionic core" of Wilson lines in half-integer spin representations of U Sp(4) has already been taken into account by α.We omit the gravitational Chern-Simons term as well as the mass of Dirac fermions for convenience.
We can also break the U Sp(4)×U (1)

Z2
. For this gauge structure, C 1 and C 2 has to be tuned independently (we consider the case of tuning C 1 ).It can be used to describe different phase transitions from the ones discussed above.The critical theory is, Here χ is a U (2) spin gauge field, b is a U (1) gauge field.c is a U (1) gauge field which describes the ν = 1 Chern insulator of ψ, A is a U (1) probe field (i.e. the electromagnetic field).We omit the gravitational Chern-Simons term as well as the mass of Dirac fermions for convenience.
At last we remark that some phase transitions can be described by changing the state formed by the parton ψ.For example, if ψ forms a ν = −1 integer quantum Hall state the parton construction will then yield the p + ip superconductor.The transition between the p+ip superconductor and the Pfaffian is then described by a QED 3 -Chern-Simons theory, namely N f = 2 Dirac fermions coupled to a U (1) Chern-Simons field (i.e.2b).One can also use the composite fermion approach to describe this phase transition, and it is indeed equivalent to the QED 3 -Chern-Simons theory described above.In constrast, the transition between the p − ip superconductor and Pfaffian state can only be described by the QCD 3 -Chern-Simons theory constructed by the U Sp(4) parton construction.
Here χ is a U (2) spin gauge field, b is a U (1) gauge field.c is a U (1) gauge field which describes the ν = 1 Chern insulator of ψ, A is a U (1) probe field (i.e. the electromagnetic field).

VI. Summary and discussion
We study a new universality class of phase transitions of FCIs/FQHs.These transitions have emergent non-Abelian gauge fields coupled to N f flavors of Dirac fermions, hence are dubbed QCD 3 -Chern-Simons theory.In contrast to the previous study of the QED 3 -Chern-Simons theory [14], the transitions we consider here are either 1) between Abelian FCIs/FQHs in different Jain sequences, or 2) involving non-Abelian FCIs/FQHs.Specifically, we use the non-Abelian parton construction to construct these FCIs/FQHs, and the transitions between them are nothing but the Chern number changing transitions of non-Abelian partons.In order to corretly characterize the topological order from parton constructions, we utilize the level-rank duality [39], and clarify its meaning for the condensed matter application.For the transitions involving non-Abelian states, we focus on the Pfaffian-like states, but the generalization to other more complicated non-Abelian states is straightforward.We shall also mention that the phase transitions discussed here and previously [14] are far from a complete list.For example, a recent numerically observed phase transition [74] falls beyond our study.
We remark that in this paper we only discuss effective theories of these phase transitions without delving into their dynamics (i.e.RG flow, critical exponents, etc.).It is possible that when the Dirac flavor N f is small, the effective theory may not flow into a nontrivial conformal fixed point (in other words it will be weakly first order phase transitions).If this is the case for certain theories, interesting critical phenomenon can still appear at finite temperature due to the proposed scenario of complex fixed point (also dubbed pseudo-criticality ) [31,75,76].We hope this new proposed scenario will also motivate more experimental study on these phase transitions.

A. Duality between different Chern-Simons theories 1. Pfaffian state
The fermionic Pfaffian state is described by the following TQFT [71] Ising × U (1) 8 for the Pfaffian state.From high energy point of view, since this theory Eq. (A2) (without modding out the Z 2 center) is a non-spin TQFT due to the Chern-Simons levels, we need this factor of 2 if we interpret A as a background spin c connection.The procedure of modding out Z 2 center of b and c can be done by gauging a Z 2 one-form global symmetry.As discussed in Ref. [72], one can achieve this by a change of variables to conventionally normalized gauge fields, such that the Z 2 one-form symmetry does not act on them.First we note that if b and c are independent fields, we should have (A9) This is nothing but the particle-hole conjugate of anti-Pfaffian (U (2) −2,4 ).Now we try to find a dual spin gauge field description for the Pfaffian state.From our previous result Eq. (42)
the p − ip superconductor in Table VII.The f − if superconductor is constructed via the parton decomposition c = f 1 f 2 ψ, with (f 1 , f 2 ) on a C = −2 Chern band and ψ on a C = 1 Chern band.The effective theory of this parton construction is a U (2) s Chern-Simons theory, )d(−Trα+A)−3CS g .

6 .
PH-Pfaffian state Similar to the Pfaffian state, the PH-Pfaffian state can also be constructed via the parton decomposition c = (ψ 1 ψ 4 − ψ 3 ψ 2 )ψ/ √ 2. The PH-Pfaffian state is realized by putting (ψ 1 , ψ 2 , ψ 3 , ψ 4 ) in a C = −1 Chern band, and ψ in a ν = 1/3 Laughlin state.The PH-Pfaffian state is well captured by either the maximal U Sp(4)×U (1) Z2 or a smaller U (2)×U (1) Z2 gauge symmetry.Table VIII summarizes various parton states of different Chern number assignment to the fermionic partons.The phase transitions between the PH-Pfaffian state and other states are described by either the U Sp(4)×U (1) Z2 or U (2)×U (1) Z2 gauge depending on whether C 1 and C 2 are changing simultaneously or independently.The U Sp(4)×U (1) Z2 gauge theory is, Here α is a U Sp(4) spin gauge field, b is a U (1) gauge field.c is a U (1) gauge field which describes the ν = 1/3 Laughlin state of ψ, A is a U (1) probe field (i.e. the electromagnetic field).The U (2)×U(1) where b is a U (2) gauge field, and c is a U (1) gauge field.b and c are not independent gauge field as one needs to mod out their Z 2 center.A is the background electromagnetic field, and we need a factor 2 in the last term to reproduce the correct filling fraction ν = 1 2 above two equations means nothing but that only 2π flux of b or c is allowed.Next moding out the Z 2 center of b and c means that: we allow π flux of b and c on a closed surface, but we need to identify them, constraints, we can redefine gauge fields, b = b − c1 2 , (A6) c = 2c.(A7)One can find that the constraints reduced to that only 2π gauge flux of b and c is allowed.In terms of these new variables the Lagrangian can be written as Chern-Simons theory description for Pfaffian state.Indeed we can integrate out the U (1) gauge field c, yielding • • • ϕ k as a 1/k Laughlin state).If k = 1, one can integrate out b yielding a similar form of Eq. (12).For k > 1, one cannot integrate out b as it represents a topological order.The tuning parameter of the transition is again the mass term of the Dirac fermions, m

TABLE I :
Summary of Parton Constructions of Pfaffian-like states.a

TABLE III :
Transition out of U (2) 2 bosonic topological order can be described by the parton construction S − = f 1 f 2 .

TABLE IV :
Transition out of Ising topological order can be described by the parton construction S − = f 1 f 2 φ.

TABLE V :
The transition out of anti-Pfaffian state can be described by the parton construction c = f 1 f 2 ψ.

TABLE VI :
The transition out of Wen's (221)-parton state can be described by the parton construction c = f 1 f 2 ψ.