# Composite Fermion Insulator: Neutral Fermi Surface with Quantum Oscillations Satisfying the Onsager Rule

###### Abstract

We propose a time reversal invariant Composite Fermion Insulator(CFI) state with neutral Fermi surface and gauge field. We consider a system consisting of two quantum Hall layers with opposite magnetic fields (or two nearly flat bands with opposite Chern numbers ) at filling . We add inter-layer repulsive interaction starting from two decoupled Composite Fermion Liquids (CFL) with opposite chiralities. In this case physical exciton is frustrated from condensation, unlike the conventional quantum Hall bilayers. We argue that more natural phases are the exciton condensates between composite fermions or between slave bosons. The resulting states are insulators with neutral Fermi surfaces coupled to an emergent gauge field without Chern-Simons term. We consider this CFI phase for the following two motivations. (1) In this state, the internal magnetic field is perfectly locked to an external magnetic field : . As a result, this phase can show quantum oscillations satisfying the Onsager condition. (2) The proposed CFI state is one example of a family of states which are generalization of the well-studied Composite Fermion Liquid (CFL) state. The CFL can be constructed as a neutral Fermi surface coupled to an gauge field , which further couples to a special bosonic topological order state: or Laughlin state. The CFI state in this paper can be viewed as replacing the bosonic sector with another time reversal invariant bosonic topological order: double semion state. Finally we also comment on other possible states in this kind of system with flat bands. One interesting possibility is a topological superconductor in the class.

###### pacs:

Valid PACS appear here## I Introduction

After the great discovery of the Fractional Quantum Hall statesStormer (1999), exotic gapped and gapless states with fractionalization have been quickly generalized to the time reversal invariant systemsWen (2004, 2002); Kitaev (2006). Most of the new states are proposed in frustrated spin models, which can be realized in the Mott insulating phase of some Hubbard models at integer fillings. One may wonder whether exotic topological orders or gapless states can also be found in fractional filling of a strongly correlated band. The answer is certainly yes given the observations of the pseudogap phase and the strange metal phase in the under-doped and optimal doped cupratesLee et al. (2006). However, the interpretations of these phases are quite hard. First, it is not clear whether these phases can exist as a ground state or necessarily need a thermal ensemble. Second, the unconventional behaviors are probably coming from both spin and charge parts, which entangle with each other and complicate the theoretical understanding. To simplify the analysis, it is probably helpful to start from pure spinless fermionic models to build some intuitions. As far as we are aware, the currently known exotic states for spinless fermions are limited to “quantum Hall like” states at fractional filling of a Chern bandBergholtz and Liu (2013), which needs either external or spontaneous time reversal symmetry breaking. The purpose of this paper is to propose one example of gapless insulator in a time reversal invariant spinless fermion system. We hope the current paper can motivate future studies to search for other examples in this flavor, some of which may be even beyond our current imagination.

Similar to the generalization of the bosonic Laughlin state to the double semion stateLevin and Wen (2005), we generalize the well-studied composite fermion liquid (CFL) in the half-filled Landau level to a time reversal invariant composite fermion insulator (CFI) phase. The CFL phase can be constructed in a parton construction of electron operator: . is a neutral fermion which forms a Fermi surface, while the charged boson is in the Laughlin state(or in a topological ordered state). The proposed CFI state is simply making the boson part in a double semion state. Because there is no Chern-Simons term for the emergent gauge field , the resulting phase is an insulator. This CFI phase is different from the spinon Fermi surface stateMotrunich (2005); Lee and Lee (2005) in a spin model. The neutral fermion in the CFI phase does not need to carry any quantum number, unlike the spinon in the more familiar spin liquid phases. The neutral fermion here should be better viewed as a composite fermion which bind one electron and two quasi-holes with charge , similar to the composite fermion in the half-filled Landau levelHalperin et al. (1993). Also, because of the non-trivial topological order of the bosonic sector, this CFI phase show quite different responses to an applied magnetic field.

We show that this CFI phase can show quantum oscillations satisfying the Onsager rule under an external magnetic field. Quantum oscillations of a neutral Fermi surface was proposed before in the weak Mott insulator close to a metal-insulator transitionMotrunich (2006); Chowdhury et al. (2018); Sodemann et al. (2018). However, in these proposals, the internal magnetic field is generically only partially locked to an applied magnetic field : with . Therefore the frequency of the quantum oscillations violates the Onsager rule and may also have temperature dependence from the factor. We will show that the CFI phase has . Therefore, the CFI phase constructed here can provide the first “proof of existence” example of an insulator with quantum oscillations satisfying the Onsager rule, which may also shed light to the recent experimental observations of quantum oscillations in three dimensional mixed valence insulatorsLi et al. (2014); Tan et al. (2015); Xiang et al. (2018).

For simplicity, we construct the CFI phase in a quantum Hall bilayers with opposite magnetic field. However, the state can also emerge in systems with two nearly flat bands with opposite Chern numbers. This kind of systems have been shown to be realizable in the moire superlattices from graphene layers or Transition metal dichalcogenide (TMD) layersZhang et al. (2018); Chittari et al. (2018); Wu et al. (2018); Zhang and Senthil (2018). Therefore this exotic state may show up in future experiments, though our main focus of this paper is purely theoretical.

## Ii Set up of the system and Summary of the result

The proposed state can be realized in a quantum Hall bilayers with opposite magnetic fields, or equivalently in a system consisting of two narrow bands with opposite Chern numbers . We consider this model to simplify the construction of the phase. However, the state should also emerge in more general models which are far from decoupled quantum Hall systems.

At the filling of the quantum Hall bilayers with opposite magnetic fields, the decoupled limit have two CFLs with opposite chiralities. Each CFL can be described by the standard Halperin-Lee-Read (HLR) theory from flux attachmentHalperin et al. (1993). In this paper we use a modified version of the HLR theory for two reasons: (1) The coefficient of the Chern-Simons term should be an integer to be mathematically well-defined; (2) The composite fermions are neutral and should not couple to the physical gauge field . For each layer , we do parton construction . Then we let the neutral fermion form a Fermi surface and let the charged boson go to the Laughlin state. The low energy theory is:

(1) |

where is the layer index. is an gauge field from the parton . describes a Fermi surface formed by the neutral composite fermion coupled to the gauge field . is the theory for the bosonic Laughlin state coupled to gauge field :

(2) |

where corresponds to layer and separately. is another gauge field to describe the bosonic sector. The above action can be also understood as the double-semion state for the two component boson . Integrating recovers the HLR theory for two decoupled CFLs. But to be more rigorous we will use the above theory as the starting point instead of the original version of the HLR theory.

The above action is invariant to the time reversal symmetry which interchanges the two layers:

(3) |

(4) |

(5) |

and

(6) |

Note that the time reversal transformation for the gauge field looks like a symmetry.

We can redefine and . couples to the physical charge, while couples to the layer charge, which we view as of a pseudospin. We also define and . From the Laughlin argumentLaughlin (1981), a flux carries charge of , while flux of carries charge of . Because of this non-trivial response of the bosonic double-semion sector, carries a charge, while carries a physical charge.

### ii.1 Summary of several CFI Phases

Starting from the decoupled two CFLs at the infinite distance separating the two layers (), we can reduce the distance and discuss the possible phases. Because of the non-trivial response of the bosonic state, magnetic flux and carry either or physical charge . In the tradtional quantum Hall bilayers, the adding intra-layer repulsion is represented by . Therefore is suppressed while is enhanced. can mediates an inter-layer pairing through the “Amperean Pairing” mechanism, which then leads to the exciton condensation phaseSodemann et al. (2017) at infinitesimal finite distance . In the quantum Hall bilayers with opposite magnetic field, the intra-layer repulsion is represented by . In this case is actually suppressed while is enhanced. generically suppresses the zero-momentum pairingMetlitski et al. (2015). Therefore we expect that the state with two decoupled CFLs is stable to infinitesimal finite distance for this new set up.

In the intermediate distance it is natural to expect that the decoupled CFLs are unstable to another phase. Inter-layer repulsion naturally bound an electron in one layer with a hole in the other layer to a bosonic exciton. In the traditional case this bosonic exciton can then condense and the system is in a symmetry breaking state with an additional integer Hall conductivity. In our new model, the excitons still fell a doubly enlarged magnetic field and can not condense. Therefore we can avoid the exciton condensation phase in this new set up.

Instead of the physical exciton condensation, more natural possibilities are the exciton condensation between the partons or between . There are three possibilities, which are summarized in Table. 1. All of these phases are insulators which can show quantum oscillations under an external magnetic field . We dub these phases as composite fermion insulator (CFI). They are distinguished by their different responses to . CFI3 is a superfluid under . Both CFI1 and CFI2 are metals under . However, the gapless charge of is carried by the internal flux in CFI1 and is carried by the composite fermions in CFI2. We are going to discuss these three CFI phases separately in the following sections.

Phase | Mechanism | Low Energy Theory | Under | Under | Under | under |

CFI1 | Insulator | Metal | changes | |||

CFI2 | Insulator | Metal | Quantum Oscillations | |||

CFI3 | Insulator | Superfluid | Meissner Effect | |||

## Iii CFI1: Exciton Condensation of Fermions

In the intermediate distance, composite fermions should also fell a residual short ranged inter-layer repulsion: . We can see that the exciton condensation between the composite fermions are favored by this inter-layer repulsion. The exciton condensation between composite fermions have been proposed in the traditional quantum Hall bilayers with the same magnetic fieldAlicea et al. (2009); You (2017). However in the numerical simulationZhu et al. (2017) exciton condensation is favored. The quantum Hall bilayers with opposite magnetic field is probably a better platform for composite fermion exciton condensation because the physical exciton condensation can be avoided. We propose the phase with in the inter-mediate region and discuss its properties. Such a bosonic exciton condensation apparently higgses the gauge field. We can then identify . The final theory is:

(7) |

where

(8) |

contain two Fermi surfaces coupled to the gauge field . Because of the exciton condensation , the areas of the two Fermi surfaces are not equal and depend on , though the sum is fixed.

We can also integrate now. Equivalently the phase can be described as:

(9) |

Because of the term, and are mixed and we use to denote them. Both and monopole of are neutral. Therefore this phase is an insulator under . The monopole of flux for carries charge under . Therefore this phase is metallic under , similar to the behavior of the CFL under physical gauge field. The fermion can still be understood as a composite fermion. The bosonic double-semion state contains quasi-electron and quasi-hole excitations with charge . The composite fermion is a neutral fermion formed by one electron and two quasi-holes. The composite fermion in the CFL state is known to have a dipole structure. It is hard to figure out the internal structure of the composite fermion in the CFI state from the low energy theory in Eq. 9. We will only consider the low energy properties of the CFI state in this paper, which is not influenced by the internal structure of the composite fermion.

## Iv CFI2: Exciton Condensation of Bosons

In the previous section we construct a CFI phase from the exciton condensation of the neutral fermions. In this section we further show that the exciton condensation of bosons in Eq. 1 can give a CFI phase with similar properties. Considering the exciton condensation that while in Eq. 1. From Eq. 1 and Eq. 2, such an exciton condensation higgses . After identifying , the final theory of the second CFI phase is:

(10) |

where,

(11) |

describes a double-semion state for the two component boson which breaks the “” rotation symmetry. After integrating and , we find that there is no Chern-Simons term for and . The second CFI phase can be described by:

(12) |

Here and form two Fermi surfaces with equal areas. and carries different charges under . The phase is still an insulator under and a metal under . The difference from the CFI1 is that the gapless charge under is carried by the composite fermions, instead of the monopole. In this CFI2 phase, the composite fermions can also be viewed as spinon, which is from the fractionalization of the physical exciton, similar to Ref. Barkeshli et al., 2018; Zaletel et al., 2018. But here we also have an internal gauge field and as a result the physical properties are essentially different from that in the Ref. Barkeshli et al., 2018; Zaletel et al., 2018.

If we further add , the physical symmetry corresponding to is broken. The resulting phase is still an insulator. It has additional goldstone mode corresponding to . We will not go to details of this third CFI phase.

## V Quantum Oscillation of the CFI Phase

In this section we show the most amazing property of the CFI phase: it can show quantum oscillation satisfying the Onsager rule under an external magnetic field ^{1}^{1}1The external magnetic field here is in addition to the original opposite magnetic fields to define the Landau levels for the two layers. One can consider two flat Chern bands to avoid the introduction of the background opposite magnetic fields.. This is, as far as we know, the first known example of insulator to show such a property.

First let us understand why the CFL phase in the half-filled Landau level show quantum oscillation, which has been well established experimentally. This property of the CFL is actually quite non trivial because the Fermi surface is neutral and only couples to the internal gauge field^{2}^{2}2In the original HLR theory, the Fermi surface also couples to the external gauge field and therefore show quantum oscillation. However, this understanding is not precise and can not be easily generalized to the CFI phase. In this paper, we insist that the composite fermions should be neutral in the CFL phase.. To show quantum oscillation response to , the internal magnetic field must locks to the external : . We do parton . forms a Fermi surface and couples to . forms the state and couples to . One can easily get the free energy under and as:

(13) |

Here and are diamagnetism constant discussed in Ref. Sodemann et al., 2018; Chowdhury et al., 2018. For the state, we has an additional term. Under , the state excites quasi-electron or quasi-hole, which has an energy cost proportional to . Under the condition that , the free energy is minimized by . Therefore the state can have a huge diamagnetism like “Meissner effect” although it is an insulator. This is the reason why the CFL phase show quantum oscillation.

The above analysis can be generalized to any bosonic quantum Hall insulators with . If is in such a state, it must excite charged excitation after , which costs an energy . Therefore this family of CFL states show quantum oscillation under . Amazingly the double-semion state in Eq. 2 also have to excite quasi electron and quasi-hole pair under . This excitation also cost an energy linear to as long as the bosonic sector is still gapped. Therefore is also true for the CFI phases considered in this papers. We expect that CFI phase has de Haas-van Alphen quantum oscillation in the measurement of the layer polarization. At finite temperature, it should also show Shubnikov-de Haas Oscillations in the resistivity measurements.

Recently there are experimental evidences of quantum oscillations in the mixed-valence insulatorLi et al. (2014); Tan et al. (2015); Xiang et al. (2018). If such a state can also be understood as a neutral Fermi surface state from the parton construction . We still needs . This again requires the bosonic insulator to have very large diamagnetism. Ref. Chowdhury et al., 2018; Sodemann et al., 2018 suggest that the boson is close to a metal-insulator transition and therefore shows a large response. In this case internal magnetic field is only partially locked to the external one: with . We expect that , where is the charge gap. Therefore this description needs to be fine tuned to small charge gap region. If we want to have an insulator with even deep inside the insulating phase, we need the boson sector to have huge diamagnetism response, like the double semion state for the CFI phases proposed here. It is not clear which insulating phase in three dimension can have such a property.

Last we have a short discussion of the responses to the layer magnetic field . CFI1 and CFI2 respond differently to the . In the CFI1, because of the term , the composite fermion can be identified as a vortex under . Therefore adding changes the fermion density: . The internal magnetic field is still zero and there is no quantum oscillation.

For CFI2, the composite fermions and carry opposite charges under . Therefore under , and form Jain squences with and . This is a sequence of fractional topological insulator states.

## Vi Other Possible States in Flat Chern Bands

The main focus of this paper is on the CFI phase. We illustrate the construction of the phase in a quantum Hall bilayers with opposite magnetic fields, or in a system with two flat Chern bands. In this section we provide some other examples of possible interesting states in this kind of system to motivate future studies of this system.

We start from the decoupled CFLs in Eq. 1 and consider the pairing between composite fermions.

### vi.1 Non-Abelian Fractional Topological Insulator

One interesting case is when forms a pairing while forms a pairing. The final state is a time reversal invariant Pfaffian anti-Pfaffian state. This state can be realized in the second Landau levels. It is stable to finite layer distance . It is interesting to numerically check the stability at limit.

### vi.2 Integer and Fractional Quantum Hall Insulator

Starting from Eq. 1 and Eq. 2, we consider the inter-layer triplet pairing of the composite fermions with the angular momentum , or a singlet pairing with angular momentum . The composite fermions are then in the class of topological superconductorKitaev (2006). The effective Chern-Simons theory is:

(14) |

where is from the particle-hole duality of the composite fermion pairing. is introduced to capture the property of this topological pairing.

Combining with Eq. 2 and integrating and , we identify and . Finally we have

(15) |

The response to and are described by:

(16) |

We can see that this state has a charge Hall conductivity from spontaneous time reversal breaking. For the charge Hall conductivity is an integer. It is a little surprising to get an integer Hall conductivity from fractionally filled Landau levels. We emphasize this Integer quantum Hall insulator has excitation with fractional statistics and is therefore different from the state which is simply polarized to one layer.

### vi.3 Topological Superconductor

A more interesting possibility is the realization of a physical topological superconductor from charge frustration induced by magnetic fields. In the system of quantum Hall bilayers with opposite magnetic fields (or flat bands), the physical exciton is frustrated. In contrast, the interlayer electron-electron pair does not feel any magnetic field and can therefore has a well defined momentum. Such a bosonic Cooper pair can condense, leading to a superconductor with inter-layer pairing. Because of the large repulsion, we expect a pairing with higher angular momentum.

Such a topological superconductor is easy to describe in a composite boson theory. We make a particle hole transformation for the second layer . and can be viewed as in the traditional quantum Hall bilayers. Then we use a composite boson theory from flux attachment:

(17) |

where is the action for the boson coupled to the gauge field . For the traditional quantum Hall bilayers, condensations and lead to the physical exciton condensation. In our case, because of the opposite charge carried by , the final state is physically different. Doing particle vortex dualities for both and , we get:

(18) |

Integrating and defining and , we have

(19) |

The above action describes a topological superconductor in the class. It either has a triplet pairing with symmetry or a singlet pairing.

## Vii Conclusion

In summary, we propose a composite fermion insulator phase with neutral Fermi surface coupled to an internal gauge field. Such a phase may be realized in a system consisting of nearly Flat Chern bands with opposite numbers, which can be engineered in the graphene or TMD moiré superlatticesZhang et al. (2018); Zhang and Senthil (2018); Wu et al. (2018). It is interesting to search for the CFI phase and the topological superconductor phase both numerically and experimentally in this kind of system. Theoretically the CFI phase can be viewed as a simple generalization of the celebrated CFL phase in the half-filled landau level to the time reversal invariant case. As a matter of principle, a family of CFI phases with other bosonic sectors (for example, toric code) can emerge in strongly correlated spinless fermionic models.

## Viii Conclusion

We thank Debanjan Chowdhury, Max Metlitski and T. Senthil for useful discussions. The work is supported by NSF grant DMR-1608505 to Senthil Todadri.

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- (29) The external magnetic field here is in addition to the original opposite magnetic fields to define the Landau levels for the two layers. One can consider two flat Chern bands to avoid the introduction of the background opposite magnetic fields.
- (30) In the original HLR theory, the Fermi surface also couples to the external gauge field and therefore show quantum oscillation. However, this understanding is not precise and can not be easily generalized to the CFI phase. In this paper, we insist that the composite fermions should be neutral in the CFL phase.