Non-Abelian Fermionization and the Landscape of Quantum Hall Phases

In this section, we first briefly review the dualities relevant to describing the $\nu=1/2$ bosonic Laughlin state. The recently proposed non-Abelian Chern-Simons-matter dualities relate theories of Wilson-Fisher bosons coupled to a Chern-Simons gauge field to theories of Dirac fermions also coupled to a Chern-Simons gauge field, with the matter content in the fundamental representation of the gauge group. We can write these dualities schematically as

	$N_{f}$ scalars + $U(N)_{k,k}$	$\displaystyle\longleftrightarrow$	$\displaystyle\text{$N_{f}$ fermions + $SU(k)_{-N+N_{f}/2}$}\,,$		(2.1)
	$N_{f}$ scalars + $SU(N)_{k}$	$\displaystyle\longleftrightarrow$	$\displaystyle\text{$N_{f}$ fermions + $U(k)_{-N+N_{f}/2,-N+N_{f}/2}$}\,,$		(2.2)
	$N_{f}$ scalars + $U(N)_{k,k+N}$	$\displaystyle\longleftrightarrow$	$\displaystyle\text{$N_{f}$ fermions + $U(k)_{-N+N_{f}/2,-N-k+N_{f}/2}$}\,.$		(2.3)

Our conventions and notation concerning non-Abelian Chern-Simons theories are presented in Appendix A.

The dualities, Eqs. (2.1)-(2.3), can be shown to imply the quadrality described in the Introduction [22],

	$\displaystyle\text{a scalar}+U(1)_{2}$	$\displaystyle\longleftrightarrow\text{a fermion}+SU(2)_{-1/2}$
	$\displaystyle\updownarrow\hskip 34.14322pt$			(2.4)
	$\displaystyle\text{a fermion}+U(1)_{-3/2}$	$\displaystyle\longleftrightarrow\text{a scalar}+SU(2)_{1}\,.$

The first theory (top left) is the relativistic version of the usual Landau-Ginzburg theory for the $\nu=1/2$ bosonic Laughlin state. Explicitly, it is described by the Lagrangian,

$$\mathcal{L}_{\Phi}=|D_{a}\Phi|^{2}-|\Phi|^{4}+\frac{2}{4\pi}ada+\frac{1}{2\pi}Ada\,.$$

(2.5)

Here $a$ is an emergent $U(1)$ gauge field; $A$ the background electromagnetic (EM) field; we use the notation $D_{a}^{\mu}=\partial^{\mu}-ia^{\mu}$ and $adb=\varepsilon^{\mu\nu\lambda}a_{\mu}\partial_{\nu}b_{\lambda}$; and the term $-|\Phi|^{4}$ denotes tuning to the Wilson-Fisher fixed point. The dual non-Abelian bosonic theory (bottom right) is given by

$$\mathcal{L}_{\phi}=|D_{u-A\mathbf{1}/2}\phi|^{2}-|\phi|^{4}+\frac{1}{4\pi}\operatorname{Tr}\left[udu-\frac{2i}{3}u^{3}\right]-\frac{1}{2}\frac{1}{4\pi}AdA\,,$$

(2.6)

where $u$ is an $SU(2)$ gauge field, $\mathbf{1}$ is the $2\times 2$ identity matrix in color space, and $-|\phi|^{4}$ again denotes tuning to the Wilson-Fisher fixed point. By turning on mass operators, both theories can be shown to describe the transition between the $\nu=1/2$ bosonic Laughlin state and the trivial insulator, which correspond to their gapped ($\langle\Phi\rangle=0$, $\langle\phi\rangle=0$) and condensed ($\langle\Phi\rangle\neq 0$, $\langle\phi\rangle\neq 0$) phases, respectively. Essential to this conclusion is the fact that $SU(2)_{1}$, the TQFT obtained when $\Phi$ is gapped, is Abelian at the level of the braid group: it is equivalent to $U(1)_{2}$ by level-rank duality.

The $SU(2)$ theory, $\mathcal{L}_{\phi}$, served as the main building block in our construction of the Read-Rezayi states in Ref. [15], in which we considered multiple layers of the $\nu=1/2$ bosonic Laughlin state and introduced an interlayer pairing interaction for the $\Phi$ particles. The paired phase of this theory yielded the Read-Rezayi states. However, in that work we did not consider the landscape of non-Abelian phases accessible by dual theories of Dirac fermions, which we now turn to.

Before describing the composite fermion theories of interest, we mention here a subtlety that arises when considering topological orders of composite fermions. When assessing the anyon content of the corresponding gauge theory, it is necessary to account for the the fact that the degrees of freedom charged under the gauge field are fermions, which affects the statistics of certain anyons by a minus sign. More technically, this is a result of the fact that gauge fields which couple to fermions are spin (actually spin_c, see Appendix A) connections, as opposed to the $U(1)$ connections that couple to bosons. We will refer to such gauge fields throughout this paper as spin gauge fields, and we will denote their associated TQFTs with the superscript ‘spin.’ In general, level-rank duality can be thought of as relating a TQFT with a spin gauge field (spin TQFT) to one with a $U(1)$ gauge field.²²2Level-rank duality can be equivalently formulated to relate two spin-TQFTs by adding an invisible spin-1/2 line (also known in the condensed matter literature as a local spin-1/2 particle) to each side of the duality [22]. This formulation is less physical if we wish to view the fundamental charges at short distances as bosons, so we will refrain from using it. Therefore, since composite fermion theories give rise to spin TQFTs, we will frequently invoke level-rank duality below and refer to a state’s topological order via its corresponding (non-spin) TQFT. For example, if a composite fermion theory yields a $SU(2)_{-k}^{\mathrm{spin}}$ TQFT, we will refer to the associated topological phase by its level-rank dual, $U(k)_{2}$.

This formal discussion has physical implications. For example, consider the topological order of the $\nu=1/2$ Laughlin state. The anyons of this state are semions, with $\pi/2$ statistics. This state can be equally well described by a $U(1)_{2}$ TQFT or a $U(1)_{-2}$ TQFT with a spin gauge field, which we will denote $U(1)_{-2}^{\mathrm{spin}}$. As we will see below, this theory arises on integrating out a Landau level of composite fermions. While it appears that the anyons in this theory are antisemions (statistics $-\pi/2$), the $\pi$ statistics of the composite fermions converts them into semions. This is a type of level-rank duality, relating $U(1)_{2}$ to $U(1)_{-2}^{\mathrm{spin}}$. In this sense, level-rank dualities can generally be viewed as boson-fermion dualities, with some interesting exceptions, e.g. in the $SU(2)_{1}\leftrightarrow U(1)_{2}$ duality mentioned above, neither theory is spin.

The bosonic theories described above are dual to a theory of Dirac fermions coupled to a $U(1)_{-3/2}$ Chern-Simons gauge field. For clarity, we will refer to this theory as Theory A,

$$\mathcal{L}_{A}=i\bar{\psi}\not{D}_{a}\psi-\frac{3}{2}\frac{1}{4\pi}ada-\frac{1}{2\pi}adA-\frac{1}{4\pi}AdA+\cdots\,,$$

(3.1)

where $a$ is a $U(1)$ gauge field and we use the notation $\not{D}=D_{\mu}\gamma^{\mu}$, where $\gamma^{\mu}$ are the Dirac gamma matrices. This theory is also dual to a theory of Dirac fermions coupled to a non-Abelian, $SU(2)_{-1/2}$ gauge field, which we will refer to as Theory B,

$$\mathcal{L}_{B}=i\bar{\chi}\not{D}_{b-A\mathbf{1}/2}\chi-\frac{1}{2}\frac{1}{4\pi}\mathrm{Tr}\left[bdb-\frac{2i}{3}b^{3}\right]-\frac{1}{4}\frac{1}{4\pi}AdA+\cdots\,,$$

(3.2)

where $b$ is an $SU(2)$ gauge field and $\mathbf{1}$ is the $2\times 2$ identity matrix.³³3Throughout this work, we approximate the Atiyah-Patodi-Singer $\eta$-invariant as a level-$\frac{1}{2}$ Chern-Simons term and explicitly include it in the Lagrangian. Here the $\chi$ fields transform as a doublet under $SU(2)$, and they have charge $-1/2$ under the global EM symmetry, $U(1)_{\mathrm{EM}}$. The fundamental (unit) charges are therefore the baryons, $\varepsilon_{\alpha\beta}\chi^{\alpha}\chi^{\beta}$, where $\alpha,\beta=1,2$ are $SU(2)$ color indices. Finally, the ellipses refer to irrelevant operators, such as Maxwell or Yang-Mills terms for the gauge fields. These operators are normally dropped since the duality is only valid in the IR limit, in which these operators are taken to zero, and their usual purpose is to provide UV regularization. However, we will see in the sections below that these operators can play important roles in determining low energy physics when background fields are turned on.

Being dual to the bosonic theories discussed in Section 2, Theory A and Theory B each describe a transition from the $\nu=1/2$ bosonic Laughlin state, which has $U(1)_{2}$ topological order, to a trivial insulator. This can be seen by introducing mass terms, $-m_{\psi}\bar{\psi}\psi$ and $-m_{\chi}\bar{\chi}\chi$, to their respective theories. For $m_{\psi}>0,m_{\chi}>0$, integrating out $\psi$ and $\chi$ can be seen to immediately yield an insulating state with vanishing Hall conductivity, $\sigma_{xy}=0$. On the other hand, when $m_{\psi}<0,m_{\chi}<0$, integrating out the composite fermions yields a state with $\sigma_{xy}=-\frac{1}{2}\frac{1}{2\pi}$ ($m_{\psi}<0,m_{\chi}<0$), with Theory A yielding a $U(1)_{-2}^{\mathrm{spin}}$ gauge theory and Theory B yielding $SU(2)_{-1}^{\mathrm{spin}}$. These constitute the same topological order as the $U(1)_{2}$ state by level-rank duality.

By differentiating this pair of Lagriangians with respect to the background EM gauge field, $A_{\mu}$, to obtain the global EM charge current, $J^{\mu}$, one observes that, under the duality, the monopole current of Theory A is related to the baryon number current of Theory B,

$$J_{e}^{\mu}=\frac{1}{2\pi}\,\varepsilon^{\mu\nu\lambda}\partial_{\nu}(a_{\lambda}-A_{\lambda})\leftrightarrow-\frac{1}{2}\,j^{\mu}_{\chi}-\frac{1}{2}\frac{1}{2\pi}\varepsilon^{\mu\nu\lambda}\partial_{\nu}A_{\lambda}\,,$$

(3.3)

where $j^{\mu}_{\chi}=\bar{\chi}\gamma^{\mu}\chi$ is the $\chi$ charge current of the Theory B. The interpretation of this dictionary is analogous to charge-vortex duality [42, 43]: flux of the gauge field $a$ in Theory A maps to charge of the $\chi$ fermions in Theory B. The same interpretation applies to the pair of bosonic theories discussed in the previous Section.

In contrast to their bosonic counterparts, these composite fermions each satisfy the Pauli exclusion principle. As a result, it is natural to consider the gapped phases accessible by filling up Landau levels and forming IQH states, in analogy to the construction of the Jain sequences, in which FQH phases are obtained as IQH states of composite fermions [3]. In particular, integer quantum Hall states of the $SU(2)$ doublet composite fermions, $\chi$, can be expected to yield non-Abelian topological orders. This method of forming non-Abelian quantum Hall phases appears to be quite natural, but we will quickly learn that the non-Abelian dualities imply that things are not so simple, and the ultimate choice of ground state will be sensitive to the order in which the lowest Landau level and IR limits are taken.

To this end, we begin by relating the electronic filling fraction, $\nu$, to the filling fractions of the $\psi$ and $\chi$ fermions using the dictionary, Eq. (3.3). Focusing first on the Abelian Theory A, the physical electric charge density is given in terms of the magnetic flux felt by the composite fermions,

$\displaystyle\rho_{e}=\left\langle J^{0}_{e}\right\rangle=-\frac{1}{2\pi}\langle\varepsilon^{ij}\partial_{i}a_{j}\rangle-\frac{1}{2\pi}B,$

(3.4)

where $B=\varepsilon^{ij}\partial_{i}A_{j}$ is the background magnetic field. We use brackets here to emphasize that that we define $\rho_{e}$ to be the expectation value of the charge density operator. We can relate $\rho_{e}$ to the composite fermion charge density through the equation of motion for $a_{0}$,

$\displaystyle 0=\langle\psi^{\dagger}\psi\rangle-\frac{3}{4\pi}\langle\varepsilon^{ij}\partial_{i}a_{j}\rangle-\frac{1}{2\pi}B\,,$

(3.5)

If we define the composite fermion filling fraction of Theory A to be

$\displaystyle\nu_{\psi}=2\pi\,\frac{\langle\psi^{\dagger}\psi\rangle}{\langle\varepsilon^{ij}\partial_{i}a_{j}\rangle},$

(3.6)

we obtain a relation between the composite fermion and electronic filling fractions,

$\displaystyle\nu=-2\pi\,\frac{\rho_{e}}{B}=\frac{\nu_{\psi}-1/2}{\nu_{\psi}-3/2}\,.$

(3.7)

Note that we have absorbed a minus sign into the definition of $\nu$ for notational convenience. From this formula, we see that IQH states of the $\psi$ fermions, which occur at fillings $\nu_{\psi}=p-1/2$, correspond to the known (descendent) bosonic Jain sequence states,

$$\nu_{p}=\frac{p-1}{p-2}\,,\,p\in\mathbb{Z}\,.$$

(3.8)

Indeed, integrating out the composite fermions yields the Lagrangian,

$$\mathcal{L}_{A,\mathrm{eff}}=\frac{p-2}{4\pi}ada-\frac{1}{2\pi}adA-\frac{1}{4\pi}AdA.$$

(3.9)

Each of these states (with the exception of the states at $p=1,2$, which are respectively a trivial insulator and a superfluid) is an Abelian FQH phase of the physical charges, which here are bosons.

The same type of analysis can be carried out for Theory B, leading to a non-Abelian version of the bosonic Jain sequence. Recalling Eq. (3.3), the electric charge density is directly related to the density of $\chi$ fermions, $\chi^{\dagger}\chi$, via

$\displaystyle\rho_{e}=-\frac{1}{2}\langle\chi^{\dagger}\chi\rangle-\frac{1}{4}\frac{1}{2\pi}B.$

(3.10)

Because the $\chi$ fermions are coupled to a Chern-Simons gauge field, they do not confine, meaning that a non-zero magnetic field, $B$, will cause them to form Landau levels with degeneracy,

$\displaystyle d_{LL}=\frac{BA}{2\pi}\times|q_{\chi}|\times(\text{color degeneracy})=\frac{BA}{2\pi}\,,$

(3.11)

where $A$ in this expression is the area of the system and $q_{\chi}=-1/2$ is the EM charge of the $\chi$ fermions. Therefore, the filling fraction of the $\chi$ fermions is

$\displaystyle\nu_{\chi}=2\pi\,\frac{\langle\chi^{\dagger}\chi\rangle}{B}\,.$

(3.12)

Plugging this into Eq. (3.10) yields a relation between $\nu$ and $\nu_{\chi}$,

$\displaystyle\nu=\frac{1}{2}\nu_{\chi}+\frac{1}{4}.$

(3.13)

When the $\chi$ fermions fill an integer number of Landau levels, $\nu_{\chi}=s-1/2$, and the filling of the physical charges is

$$\nu_{s}=\frac{s}{2}\,,s\in\mathbb{Z}\,.$$

(3.14)

On integrating out the composite fermions, one obtains a $SU(2)^{\mathrm{spin}}_{-s}$ theory with Lagrangian,

$\displaystyle\mathcal{L}_{B,\,\mathrm{eff}}$

$\displaystyle=-\frac{s}{4\pi}\mathrm{Tr}\left[bdb-\frac{2i}{3}b^{3}\right]-\frac{s}{2}\frac{1}{4\pi}AdA\,,$

(3.15)

which describes a non-Abelian, $U(s)_{2}$, topological order when $|s|>1$. This approach recalls earlier approaches to non-Abelian FQH states using parton constructions [7, 30, 31]. However, unlike in those cases, in which the elctron operator is fractionalized by hand and it is necessary to require that the partons do not confine by fiat, here the duality provides a clear basis for the presence of a (deconfining) non-Abelian gauge field.

Having obtained the sequences of incompressible filling fractions associated with Theory A and Theory B, one can immediately observe several special (non-trivial) filling fractions, $\nu_{*}$, at which they coincide, indicating the presence of competing ground states. Comparing Eqs. (3.8) and (3.14), these fillings occur when $\nu_{p}=\nu_{s}=\nu_{*}$, i.e.

$\displaystyle s=2\left(1+\frac{1}{p-2}\right)\,,\,p,s\in\mathbb{Z}.$

(3.16)

Here we recall that $p$ and $s$ are the number of Landau levels filled by the $\psi$ and $\chi$ fermions, respectively. This equation has several solutions, which are organized in Table 1. Two of these solutions, $(p=1,s=0)$ and $(p=0,s=1)$, respectively correspond to the $\nu=1/2$ bosonic Laughlin state, which has $U(1)_{2}\leftrightarrow SU(2)_{-1}^{\mathrm{spin}}$ topological order. This is consistent with the fact that at criticality these theories describe a plateau transition between these states.

It is the presence of other solutions, at $(p=3,s=4)$ and $(p=4,s=3)$, corresponding to $\nu_{*}=2$ and $\nu_{*}=3/2$ respectively, that reveals new physics. On the one hand, Theory A predicts the $\nu_{*}=3/2$ and $\nu_{*}=2$ states to have $U(1)_{-2}$ and $U(1)_{1}$ (i.e. trivial) topological orders. On the other hand, Theory B predicts the same states to have non-Abelian $SU(2)_{-3}^{\mathrm{spin}}\leftrightarrow U(3)_{2}$ and $SU(2)_{-4}^{\mathrm{spin}}\leftrightarrow U(4)_{2}$ topological orders. While it is common in quantum Hall physics for different competing states to be proposed for the same filling fraction, the duality of Theory A and Theory B identifies the two theories’ ground states. Therefore, the consistency of the duality implies that the conditions under which the $\psi$ fermions form an IQH state are not the same as those of the $\chi$ fermions, and the two possible states must be separated by a phase transition. A theory of this phase transition requires short-distance dynamical information not specified by, but consistent with, the duality. We now present a possible scenario for a transition of this kind.

We now provide a possible explanation of the physics occurring at the special filling fractions $\nu_{*}=3/2,2$. For now, we will work from the point of view of the non-Abelian Theory B, and we will begin by considering what happens as we fill Landau Levels. From Table 1, we see that filling the zeroth and first Landau Levels of the $\chi$ fermions corresponds to the expected trivial insulator ($\nu=0$) and bosonic Laughlin $(\nu=1/2)$ states. What occurs when the non-Abelian composite fermions fill two Landau levels is also quite non-trivial, but we will table that discussion until the next subsection. For now, our concern will be what happens when we fill the third Landau level, corresponding to $\nu_{*}=3/2$. Our proposal for $\nu_{*}=2$ will prove to be essentially identical. At $\nu_{*}=3/2$, Theory B predicts an incompressible state with $SU(2)_{-3}^{\mathrm{spin}}\leftrightarrow U(3)_{2}$ topological order, while Theory A predicts $U(1)_{2}$. This suggests that it should be possible to trigger an instability in the non-Abelian theory as we fill this Landau level, which preempts the $U(3)_{2}$ topological order and yields the same Abelian phase predicted by Theory A (and vice versa). Consequently, both the $U(1)_{2}$ and $U(3)_{2}$ phases must exist in the $\nu=3/2$ phase diagram. This is the only state of affairs consistent with the duality.

How might such an instability occur? It is here that the ellipses in Eqs. (3.1) and (3.2) become crucial.⁴⁴4We thank Chong Wang for enlightening discussions on this point. These ellipses include operators that are irrelevant at tree level but may nevertheless play an important role in determining the low energy physics when a magnetic field and chemical potential are introduced. Indeed, there is no sense in which such fields are ever perturbative, as they reorganize the spectrum of a theory in dramatic ways. To make this discussion more precise, consider the Yang-Mills term in Theory B,

$$\mathcal{L}_{YM}=-\frac{1}{2g_{YM}^{2}}\operatorname{Tr}[f_{\mu\nu}f^{\mu\nu}]\,,$$

(3.17)

where $f_{\mu\nu}=\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu}-i[b_{\mu},b_{\nu}]$ is the field strength of the $SU(2)$ gauge field, $b$. At tree level, the mass dimension of the operator $\operatorname{Tr}[f^{2}]$ is 4, meaning that $[g_{YM}^{2}]=1$: it is an energy scale. Commonly, the IR limit in which the duality of Eqs. (3.1) and (3.2) holds is phrased as the limit $g_{YM}^{2}\sim\Lambda\rightarrow\infty$, where $\Lambda$ is a UV cutoff, but strictly speaking this is only true in the absence of a background magnetic field, which provides its own energy scale in the form of the cyclotron frequency, $\omega_{c}\sim\sqrt{B}$ (for massless Dirac fermions).

Consequently, one can form a dimensionless coupling,

$$\bar{\lambda}=\frac{g_{YM}^{2}}{\omega_{c}}\,,$$

(3.18)

that can have non-trivial running as a result of strong interaction effects. This would mean that the ground state ultimately chosen by Theory A can depend on the order of the limits, $g_{YM}^{2}\rightarrow\infty$ and $\omega_{c}\rightarrow\infty$. Importantly, one order of limits in Theory B may not correspond to an analogous order of limits in Theory A. In other words, Theory B may be in a strongly coupled regime ($0\leq\bar{\lambda}<\infty$) while Theory A may behave as if all of the irrelevant operators have been taken to zero prior to taking $\omega_{c}\rightarrow\infty$.

This is the essence of our proposal⁵⁵5We emphasize that while we focus on the example of the Yang-Mills term, there are many other operators that might be responsible for the behavior we propose, and it may be more correct to consider a linear combination of these operators as being responsible. Another likely family of examples is four-fermion operators, which have the same tree level dimension as the Yang-Mills operator. for the phase diagram of Theory B, shown schematically in Fig. 1. Whether the theory is in the $U(3)_{2}$ or $U(1)_{2}$ phase is determined by the value of $\bar{\lambda}$, with the two phases being separated by a phase transition at a value $\bar{\lambda}=\bar{\lambda}_{*}\sim\mathcal{O}(1)$. For $\bar{\lambda}>\bar{\lambda}_{*}$, $\bar{\lambda}$ runs large, corresponding to the limit $g_{YM}^{2}\rightarrow\infty$ followed by $\omega_{c}\rightarrow\infty$. In this phase, the Yang-Mills term disappears, leaving the Chern-Simons term, which ensures that the composite fermions, $\chi$, are deconfined. It is in this regime that we expect the picture described in the previous subsection of deconfined $\chi$ fermions filling Landau levels to hold, making this the phase with $U(3)_{2}$ topological order. On the other hand, for $\bar{\lambda}<\bar{\lambda}_{*}$, we propose that $\bar{\lambda}$ runs small. Here the Yang-Mills term becomes important, and the assumption of deconfined composite fermions breaks down: the Landau levels mix. As a result, the composite fermions will tend to form bound states that are neutral under $SU(2)$: the baryons, $\varepsilon_{\alpha\beta}\chi^{\alpha}\chi^{\beta}$. These are bosons of charge $-1$ since the fermions are doublets under $SU(2)$. Being the physical electric charges in this theory, these bosonic baryons will find themselves at filling $\nu=3/2$, and the resulting topological order will ultimately be $U(1)_{2}$, as predicted by Theory A. Note that this final conclusion is conjecture based on the consistency of the duality – the physics in such a phase is very strongly coupled, and we cannot show explicitly that this is the true ground state. We also point out that $\bar{\lambda}$ may not run to zero in this phase, instead running to a small but finite value. This represents a very interesting but theoretically daunting possibility.

We now comment on what occurs in the dual description of Theory A, in which we can define a similar running coupling⁶⁶6Note that unlike the fermions of Theory B, the magnetic field felt by the $\psi$ fermions depends both on the background chemical potential and magnetic field, so $\omega_{c}$ is not precisely the Landau level gap, which is determined by $\langle\varepsilon^{ij}\partial_{i}a_{j}\rangle$., $\bar{\lambda}_{A}=g_{\mathrm{Maxwell}}^{2}/\omega_{c}$, where $g_{\mathrm{Maxwell}}$ is the coupling associated with the Maxwell term for the $U(1)$ gauge field, $a$. In this theory, the limit $\bar{\lambda}_{A}\rightarrow\infty$ corresponds to the $U(1)_{2}$ (IQH) state at filling $\nu_{*}=3/2$ ($\nu_{*}=2$), as the Maxwell term vanishes. However, since the Maxwell term is not dual to the Yang-Mills term, it is not clear whether the phase transitions described for Theory B correspond to transitions tuned by $\bar{\lambda}_{A}$ or another coupling, such as one associated with a linear combination of four-fermion operators. Nevertheless, unless $\bar{\lambda}_{*}=\infty$ in Theory B (in which case the non-Abelian phase is nowhere stable, having no basin of attraction), the duality indicates that the non-Abelian states should be accessible to Theory A as well, and that these transitions should also be present in that theory.

The nature of the transition between these phases depends on microscopic details, and it is not immediately clear how to study the strongly coupled physics when $\bar{\lambda}\sim\mathcal{O}(1)$. One exciting possibility is that the phase transition is continuous, which would exist beyond the Landau paradigm since it separates two distinct topological orders. Furthermore, this would imply the existence of an unstable conformal field theory (CFT) fixed point, which we expect would be quite exotic and perhaps involve emergent symmetries. In the spirit of universality, we attempt here to write down the simplest possible theory of such a transition. The theory we find consists of $N_{f}=4$ flavors of electrically neutral Dirac fermions, $\xi_{i}$, coupled to a $SU(2)_{-1}^{\mathrm{spin}}$ Chern-Simons gauge field, $c$,

$$\mathcal{L}=\sum_{i=1}^{N_{f}=4}\bar{\xi}_{i}(i\not{D}_{c}-m_{\xi})\xi_{i}-\frac{1}{4\pi}\operatorname{Tr}\left[cdc-\frac{2i}{3}c^{3}\right]-\frac{3}{2}\frac{1}{4\pi}AdA\,.$$

(3.19)

We emphasize that the fields $\xi$ and $c$ need not have any local relationship with the fields in Theory A nor Theory B. This theory has a $U(4)$ global symmetry rotating the fermion flavors. For $m_{\xi}\gg 0$, the theory is in the Abelian, $SU(2)_{+1}^{\mathrm{spin}}\leftrightarrow U(1)_{-2}$ phase, and for $m_{\xi}\ll 0$ the theory is in the non-Abelian, $SU(2)_{-3}^{\mathrm{spin}}\leftrightarrow U(3)_{2}$ phase. While the flavor index can be interpreted as a kind of Landau level index, there is no a priori reason for this symmetry to be enforced, rendering this theory at best a multicritical point. What’s more, this theory has $N_{f}>2|k|=2$, meaning that it falls outside of the Chern-Simons-matter dualities of Eqs. (2.1)-(2.3) and may spontaneously break the flavor symmetry at small values of $m_{\xi}$ [37]. We therefore find the presence of a direct second-order transition unlikely, and we conjecture that any other possible CFT with these two phases also has an enlarged set of global symmetries compared to the underlying UV problem.⁷⁷7While one may also wish to consider theories with bosonic matter, we note that it is not possible to condense bosonic operators to transition between $U(3)_{2}$ and $U(1)_{-2}$ due to the difference in the signs of the level. This cannot be repaired with level-rank duality because the $U(1)_{-2}$ gauge field is not spin. We also conjecture that this is the case for the other transitions discussed in this work.

Therefore, it is perhaps more natural to expect the mundane scenario in which the two phases are separated by a first order transition. Starting in, say, the $U(1)_{2}$ phase, as $\bar{\lambda}$ is increased, phase separation will set in, yielding bubbles and stripes of the $U(3)_{2}$ phase, which eventually fill the system. It is also possible that the transition is not direct, and that several different phases arise when $\bar{\lambda}\sim\mathcal{O}(1)$. We cannot exclude this possibility. That we cannot present a thorough description of the transition is not surprising, given the complexity of the phase diagrams for electrons in magnetic fields at fractional filling.

In the above subsection, we conspicuously left out a discussion of the case when the $\chi$ composite fermions fill two Landau levels, corresponding to filling $\nu_{*}=1$. At this filling, Eqs. (3.4) and (3.5) indicate that the $\psi$ composite fermions of Theory A feel a vanishing magnetic field and therefore form a metallic state with density set by the background magnetic field, $\langle\psi^{\dagger}\psi\rangle=B/2\pi$. This is the well known metallic, composite Fermi liquid state of bosons with unit filling [44]. On the other hand, the analysis of Section 3.2 finds Theory B in an IQH state, yielding $U(2)_{2}$ topological order. Although one of the predicted phases is gapless, we nevertheless propose that the same scenario presented above holds in this theory: the ultimate choice of ground state is determined by the order in which the IR and $\omega_{c}\rightarrow\infty$ limits are taken. These states are again separated by a phase transition, which is tuned by the dimensionless coupling of the kind defined in Eq. (3.18).

In contrast to the cases in which both phases were gapped, here we can clearly understand this transition in terms of the $\psi$ composite fermions of Theory A. Indeed, the $U(2)_{2}$ state can be shown to be one of a range of non-Abelian phases of bosons at $\nu=1$ that can be obtained as paired states of the composite Fermi liquid [45], the most famous of which being the $SU(2)_{2}$ bosonic Pfaffian state [13]. The major difference between the $U(2)_{2}$ state and the $SU(2)_{2}$ state lies in the topological spin of the non-Abelian half-vortex, which is set by the pairing momentum channel. As we will explain in more detail in the following subsection, the appropriate channel to obtain the $U(2)_{2}$ order is $l=2$. Consequently, this transition can be understood in terms of the flow of the pairing interaction, which is a four-fermion operator in the Abelian Theory A. Interestingly, it is expected from numerical simulations and recent analytic calculations that the composite Fermi liquid state of bosons at $\nu=1$ is unstable to pairing in the lowest Landau level limit [46, 47], perhaps suggesting that the metallic state seen in Theory A has no basin of attraction unless the theory is modified in some way.

From the point of view of Theory B, it is natural to expect that $\bar{\lambda}$ is again the correct running coupling, with the choice of ground state again being viewed as a question of the order in which the lowest Landau level and IR limits are taken. As above, for $\bar{\lambda}>\bar{\lambda}_{*}$, the $\chi$ fermions find themselves in an IQH state, yielding the $U(2)_{2}$ topological order. For $\bar{\lambda}<\bar{\lambda}_{*}$, this picture breaks down, leading to a theory involving bosonic baryons, which we conjecture form the $\nu=1$ metallic state.

Remarkably, at filling $\nu_{*}=1$, we have observed a duality between composite fermion pairing in Theory A and the IQH effect in Theory B. This is surprising, as there is no known dictionary of local operators that makes this connection explicit. Indeed, while the $U(2)_{2}$ state has previously been obtined both in the Read-Green pairing picture [12, 13] and as an IQH state of non-Abelian partons [7, 30, 31, 48], it has never before been suggested that these two constructions may be dual to one another. We will see below that this duality is not limited to the particular case of bosons at $\nu=1$: in Section 3.6, we will encounter a parallel story involving the anti-Pfaffian state of fermions at $\nu=1/2$.

Similar competing ground states are found as the external magnetic field is turned off, i.e. $\nu_{*}\rightarrow\infty$ (see Table 2). In this case, the analysis of Section 3.2 indicates that Theory A predicts a superfluid state, the usual fate of bosons at finite density and $B=0$. On the other hand, the composite fermions of Theory B form a metal with density equal to the background charge density, a far more exotic non-Fermi liquid state which surely requires that the fundamental bosonic charges be very strongly interacting. Nevertheless, the interpretation of the transition between these states is natural from the point of view of Theory B: again adopting the notation of Section 3.3, for $\bar{\lambda}>\bar{\lambda}_{*}$, the theory remains metallic, while for $\bar{\lambda}<\bar{\lambda}_{*}$, the $\chi$ fermions confine to form bosonic baryons. Since these bosons feel no magnetic field, they would then condense and spontaneously break $U(1)_{\mathrm{EM}}$, forming the superfluid state seen in Theory A. Given how natural the state predicted in Theory A is, one might wonder if the metallic state predicted by Theory B is ever stable to baryon condensation. We leave this question for future work.

Thus far, we have focused our analysis on the dual fermionic theories appearing in the $SU(2)$ quadrality. However, the above considerations are broadly applicable to any pair of fermionic theories related by a Chern-Simons matter duality. To illustrate this point, we consider a more general composite fermion duality describing a transition between the $\nu=1/k$ Laughlin state and an insulator. This duality relates a theory of Dirac composite fermions with $k-1$ (Abelian) fluxes attached (Theory A^′) to a theory of composite fermions coupled to a $U(N)$ gauge field (Theory B^′) [49],

	$\displaystyle\mathcal{L}_{A^{\prime}}(k)=i\bar{\psi}\not{D}_{a}\psi-\frac{1}{2}\frac{1}{4\pi}ada+\frac{1}{2\pi}adv$	$\displaystyle+\frac{k-1}{4\pi}vdv+\frac{1}{2\pi}vdA+\dots\,,$		(3.20)
		$\displaystyle\updownarrow$
	$\displaystyle\mathcal{L}_{B^{\prime}}(k,N)=i\bar{\eta}\not{D}_{u}\eta-\frac{1}{2}\frac{1}{4\pi}\operatorname{Tr}\left[udu-\frac{2i}{3}u^{3}\right]$	$\displaystyle-\frac{N-k}{4\pi}bdb-\frac{1}{2\pi}\operatorname{Tr}[u]db+\frac{1}{2\pi}bdA+\dots$		(3.21)

In Theory A^′, $\psi$ is a Dirac fermion charged under an emergent $U(1)$ gauge field $a$, and $v$ is another $U(1)$ gauge field. In Theory B^′, $\eta$ is a Dirac fermion in the fundamental representation of $U(N)$, $u$ is a $U(N)$ gauge field, and $b$ is a $U(1)$ gauge field. These dualities are derived in Appendix B.1, where it is also shown that these theories are dual composite fermion descriptions of the bosonic Landau-Ginzburg theory for the $\nu=1/k$ Laughlin state: when $k$ is even, the fundamental charges are bosons, while when $k$ is odd they are fermions (note that when $N=k=2$, we recover Theory A and Theory B, which are associated with the $\nu=1/2$ bosonic Laughlin state). We emphasize that the above duality holds regardless of the rank $N$ of the gauge group $U(N)$ in Theory B^′, and hence amounts to the statement that $\mathcal{L}_{A^{\prime}}(k)$ is dual to an infinite number of theories $\mathcal{L}_{B^{\prime}}(k,N)$ parameterized by the integer $N$.

As in the examples encountered thus far, we find special filling fractions, $\nu_{*}$, at which the dual theories predict differing ground states. The details of this analysis are essentially identical to that of the preceding $SU(2)$ examples and are presented in detail in Appendix B.2. Our results are summarized in Table 3. For example, at the filling $\nu_{*}=2/(2k-3)$, Theory A^′ predicts the usual Abelian Jain state, while Theory B^′ predicts the more exotic, non-Abelian $U(4)_{2,2(2k-3)}$ state. Our dynamical proposal for understanding the transitions between these states, as well as the others featured in Table 3, is essentially identical to that of Section 3.3, and so we will not comment on it further.

One state of particular note is the $U(3)_{2,-1}$ topological order, which is level-rank dual to $U(2)_{-3,-1}^{\mathrm{spin}}$, predicted by Theory B^′ with $k=1$ at $\nu_{*}=-3$. Now, the non-spin $U(2)_{3,1}$ Chern-Simons theory is dual to the (also non-spin) $(G_{2})_{1}$ Chern-Simons theory [50]. Remarkably, $(G_{2})_{1}$ describes precisely the Fibonacci topological order, which supports a single non-trivial anyon, $\tau$, obeying the fusion rule $\tau\times\tau=1+\tau$, and is of particular import from the perspective of topological quantum computation [34]. It is rather interesting that Theory B^′ predicts the (spin) Fibonacci topological order to appear in competition with the $\nu_{*}=-3$ IQH state. In future work, we will explore how other mechanisms can lead to the emergence of this exotic order, using bosonic theories, in the spirit of our earlier construction [15].

Additionally, we wish to highlight the cases at fillings $\nu_{*}=1/(k-1)$, in which the non-Abelian $U(2)_{2,2(k-1)}$ state predicted by Theory B^′ can again be understood as a pairing instability of the Abelian composite Fermi liquid state in Theory A^′. The argument that $U(2)_{2,2(k-1)}$ can be obtained from pairing in Theory A^′ parallels that for the $U(2)_{2}$ state discussed in the previous subsection, which corresponds to the special case $k=2$. Indeed, the non-Abelian part of $U(2)_{2,2(k-1)}$ is again $SU(2)_{2}$, and the major difference from this state lies in the topological spins of the non-Abelian half vortices, which are modified by the level of the Abelian sector. Moreover, one can check explicitly that pairing of the Theory A^′ composite fermions in the $l=2$ angular momentum channel yields the expected $U(2)_{2,2(k-1)}$ topological order for all $k$ by considering the edge spectrum: this topological order supports three chiral, charge-neutral Majorana fermions and one anti-chiral $U(1)_{k-1}$ bosonic charge mode. From the point of view of Theory A^′, $l=2$ pairing leads to the three chiral Majorana fermions, in addition to a $U(1)_{k-1}$ charge mode coming from the left over Abelian sector. That such a large class of non-Abelian topological orders which can be understood via pairing in a composite fermion theory, in this case Theory A^′, has a dual description as IQH states of composite fermions in a dual non-Abelian theory, Theory B^′ is quite remarkable.

We close this section by commenting on some particular examples of physical interest. First, for $k=1$ and $\nu_{*}=\infty$, Theory A^′ is a free Dirac fermion at finite chemical potential but vanishing magnetic field. In this case, in Theory B^′, it is possible to integrate out the auxiliary gauge field $b$ without violating flux quantization: its Chern-Simons level is $-(N-k)=-1$. This cancels the Chern-Simons term for the Abelian part of the $U(2)$ gauge field, $\operatorname{Tr}[u]$, Higgsing the background EM field, $A_{\mu}$, and leading to a chiral superconductor with topological order is $U(2)_{2,0}=[SU(2)_{2}\times U(1)_{0}]/\mathbb{Z}_{2}\cong SO(3)_{1}$, which is Abelian. This state contains only a single anyon, a Majorana fermion⁸⁸8This can be viewed as a consequence of the condensation of the Majorana fermion of the $SU(2)_{2}$ topological order, which confines the non-Abelian “half-vortex” of the $SU(2)_{2}$ factor. [26, 51]. Such a state can be accessed via a pairing instability in Theory A^′. It is natural to expect this instability to arise due to the effects of local four-fermion operators that destabilize Theory A^′ in the order of limits in which the $\eta$ fermions of Theory B^′ form an IQH state.

For $k=-1$ and $\nu_{*}=-1/2$, Theory A^′ is Son’s Dirac composite Fermi liquid theory of the half-filled Landau level [52]. In this case, the topological order predicted by Theory B^′ is $U(2)_{2,-4}=[SU(2)_{2}\times U(1)_{-8}]/\mathbb{Z}_{2}$, which is the (time-reversed) topological order of the famous anti-Pfaffian state, another paired state of composite fermions [35, 36]! Additionally, for $k=3$ and $\nu_{*}=1/2$, Theory B^′ predicts the $[SU(2)_{2}\times U(1)_{8}]/\mathbb{Z}_{2}=U(2)_{2,4}$ order, which is that of Wen’s $(221)$ parton state [7, 30], another proposed ground state of fermions at half filling that can be understood in terms of pairing in Theory A^′, which now has additional attached fluxes. Now, although these states are all accessible within parton constructions (see, for example, Ref. [48]), we must again emphasize that the projective framework hinges on the dynamical assumption that the electron fractionalizes into partons which do not confine. Finally, another non-trivial bosonic example is $k=0$ and $\nu_{*}=1$, in which Theory B^′ predicts a $[SU(2)_{2}\times U(1)_{-4}]/\mathbb{Z}_{2}=U(2)_{2,-2}$ ground state, which describes the Ising topological order [53]. While these results are reminiscent of the parton constructions giving these states, the use of duality provides a connection between partonic intuition and the dynamics of pairing. Additionally, it is straightforward to check in these examples that $l=2$ pairing in Theory A^′ yields the expected $U(2)_{2,2(k-1)}$ order by comparing the edge theories. It would be interesting to determine if other dual descriptions exist which yield the other proposed Pfaffian-like states, which arise from pairing the composite fermions of Theory A^′ in other angular momentum channels, as IQH phases of dual composite fermions, and we leave this to future work.