# Symmetry protected fractional Chern insulators and fractional topological insulators

###### Abstract

In this paper we construct fully symmetric wavefunctions for the spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant fractional topological insulators (FTI) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction (and the same quantized Hall conductance in FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number one. The low-energy gauge groups of a generic FCI wavefunctions can be either or the discrete group , and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground state degeneracy. Examples of FCI/FTI wavefunctions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number two are discussed. Generic FTI wavefunctions in the absence of spin conservation are also presented whose low-energy gauge groups can be either or . The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations.

###### pacs:

71.27.+a, 71.70.Ej, 73.43.-f## I Introduction

Phases of matters in condensed matter systems can almost always be characterized by the Landau-Ginzburg symmetry breaking theoryLandau (1937a, b). Experimental discovery of integer and fractional quantum Hall states in 2-D electron gas under a strong external magnetic fieldKlitzing et al. (1980); Tsui et al. (1982) has provided striking counter examples of this paradigm. The fractional quantum Hall liquids are particularly fascinating in the sense that their low energy excitations are quasi-particles carrying fractional electric chargeLaughlin (1983) and obeying anyonic statisticsArovas et al. (1984). Although these liquid phases do not break physical symmetries, they are still different quantum phases. One measurable difference is their edge states: despite the fact that these liquids are all insulators in the bulk, they all possess certain edge metallic modesWen (1995). In general different bulk phases host different edge states which can be detected by various experimental probes such as electric transportWen (1991a).

A few years after the experimental discovery of integer quantum Hall effect(IQHE), Haldane showed that the essence of it is *not* the external magnetic fieldHaldane (1988), by explicitly writing down a lattice model Hamiltonian of IQHE with zero net magnetic field. However, it takes more than two decades for people to show that similar statement is true even for FQHE. Recently results from a series of model studiesTang et al. (2011); Sun et al. (2011); Neupert
et al. (2011a); Sheng et al. (2011); Wang et al. (2011); Regnault and Bernevig (2011); Neupert
et al. (2011b); Xiao et al. (2011); Hu et al. (2011), including convincing evidences from exact diagonalizationsNeupert
et al. (2011a); Sheng et al. (2011); Wang et al. (2011); Regnault and Bernevig (2011); Neupert
et al. (2011b); Xiao et al. (2011), indicate that fractional quantum hall states exist in the ground states of interacting lattice models, in the absence of an external magnetic field. It is found that the ground state is likely to respect the full lattice symmetry. Here we call these fractional ground states spin-polarized “fractional Chern insulators” (FCI) to distinguish from the traditional fractional quantum Hall states in an external magnetic field. These proposed lattice models share a common feature: a partially filled nearly flat two-dimensional band with non-trivial band topology.

The concept of band topology originates from the well-known TKNN index (or Chern number) of an IQH insulatorThouless et al. (1982). In the past few years, this concept has been generalized to time-reversal symmetric systems, and triggers the theoretical and experimental discoveries of topological insulators in spin-orbital coupled compounds in both two and three spatial dimensionsQi and Zhang (2010); Hasan and Kane (2010); Moore (2010). In two dimension (2D), a time-reversal symmetric band insulator is characterized by a topological index. Experimentally, HgTe quantum heterostructure has been shown to be a 2D topological insulatorKonig et al. (2007). In the simplest limit, 2D topological insulator can be viewed as a direct product of the up-spin and down-spin wavefunctions hosting opposite TKNN index.

It is then quite natural to ask whether similar time-reversal-invariant (TRI) versions of two-dimensional fractional topological insulators (FTI) exist or not, and there has been a lot of interest in this issueBernevig and Zhang (2006); Levin and Stern (2009); Maciejko et al. (2010); Vaezi (2011); Swingle et al. (2011). In the simplest limit when spin along the -direction is conserved, it can be understood as the direct product of wavefunctions of the up spin and the down spin with opposite FQHE. Clearly this direct product is a fully gapped stable phase. In addition it must have non-trivial ground state degeneracy on a torus even in the presence of a small conservation breaking perturbation, because the ground state degeneracy cannot be lifted by an arbitrary local perturbation. So there is no question that in principle this fractionalized phase could exist. One important issue is whether this phase hosts stable gapless edge excitations. This problem has been studied in by Levin and SternLevin and Stern (2009). Another important open question is that whether TRI FTI can exist in a reasonable Hamiltonian.

In order to realize the FCI or FTI phases in experiments, one should find a compound with a nearly flat topological non-trivial band so that correlation effect is strong. Naively this is unnatural because usually a flat band is realized by spatially localized orbitals which do not support topological non-trivial hopping terms. However, a very recent theoretical investigationXiao et al. (2011) on transition metal oxide heterostructures indicates that a nearly flat topological non-trivial band can be naturally realized in the orbital double-layer perovskite grown along the [111] direction. Exact diagonalization in the same work shows that fractional quantum hall state can be realized in principle when the nearly flat band is partially filled. Because the temperature scale of the FCI/FTI physics in this system is controlled by short-range Coulomb interaction, it can be a high-temperature effect.

Fractional quantum Hall states, especially the non-Abelian ones, have been shown to be very useful building blocks of quantum computers. If high temperature FCI/FTI physics can be realized experimentally, it will certainly have deep impact in condensed matter physics, including the efforts on topological quantum computationNayak et al. (2008).

Motivated by the recent progresses on FCI/FTI physics, in this paper we try to address several important issues: *what are the many-particle wavefunctions of 2-D FCIs/FTIs? Can there be more than one FCI/FTI phases with the same filling fraction? If the answer is positive, can we classify these quantum phases (or ground state wavefunctions)?*

Historically, Laughlin’s wavefunctions of FQH states in a magnetic fieldLaughlin (1983) have been shown to be one of the most important theoretical progresses in many-particle physics. It allows people to understand a lot of properties of FQH liquids in a compact fashion, including the fractionalized quasi-particle excitationsArovas et al. (1984), topological ground state degeneraciesHaldane and Rezayi (1985a); Wen and Niu (1990), as well as constructing the low energy effective theoriesGirvin and MacDonald (1987); Zhang et al. (1989); Read (1989). Here in the case of FCI/FTI systems, analytical understanding of the ground state wavefunctions will help us extract various measurable information in a similar way.

Recently there was an interesting work to construct FCI wavefunctions by proposing a one-to-one mapping between the lattice problem and the magnetic field problemQi (2011). We would like to emphasize that the wavefunction problem for FCI is related to that for the magnetic field case, yet they are very different from each other. This is because the lattice symmetry of FCI is fundamentally different from the continuum case of the 2-D electron gas. In fact, the recently discovered FCI states preserve all the lattice point group symmetry as well as translational symmetry. ^{1}^{1}1Because the wavefunction constructed in Ref.Qi (2011) is based on one-dimensional Wannier function which explicitly select a special direction of the lattice, the constructed wavefunctions do not obviously respect the lattice point group symmetry. Here in this paper, we point out that as a consequence of the lattice symmetry, there exist many different quantum FCI phases, all respecting the full lattice symmetry, even at the same filling fraction with the same quantum Hall conductance. These different FCI phases are distinct in the bulk in a more subtle way. One hand-waving statement is that the bulk quasi-particle excitations of these phases carry different lattice quantum numbers. These distinct FCI phases cannot be adiabatically connected with each other without a phase transition while the lattice symmetry is respected. Similar phenomena of distinct topologically ordered phases protected by symmetry is known in the context of quantum spin liquidsWen (2002) and other low dimensional topological phasesChen et al. (2011).

Now we outline the content of this paper. We start with the spin-polarized FCI at filling ( is an odd number). In section II the parton construction of the fractional quantum Hall states (or spin-polarized FCI states) is introduced on a lattice, which is a natural generalization of the continuum caseLaughlin (1983); Jain (1989a). We argue that a general FCI wavefunction could break the gauge group down to , and consequently the low-energy dynamics is described by Chern-Simons-Higgs theories. We explicitly write down the form of the electronic FCI wavefunctions which will be useful for future variational Monte Carlo study. We construct quasiparticle excitations of such FCI states. To demonstrate how lattice symmetry restricts the structure of the wavefunctions, we introduce the concept of projective symmetry group (PSG)Wen (2002) which serves as the mathematical language to classify different symmetry protected FCI phases.

With these theoretical preparations, in section III we discuss one particular example, *i.e. *the checkerboard lattice modelSun et al. (2011); Neupert
et al. (2011a) and write down two FCI wavefunctions and two FCI wavefunctions in distinct universality classes for . These wavefunctions support the same 2011a); Sheng et al. (2011); Regnault and Bernevig (2011). Which state is realized in the simulated modelNeupert
et al. (2011a); Sheng et al. (2011); Regnault and Bernevig (2011) would be determined by energetics. Because our proposed wavefunctions has the form of a Slater determinant and can be effectively implemented by variational Monte Carlo approach, the energetics of the proposed states can be studied by future numerical investigation. In Appendix G we present another four examples of distinct FCI phases in the honeycomb lattice modelHaldane (1988): two are states and the other two are states.
We also propose spin-polarized FCI states with non-Abelian quasiparticles, which might be realized in nearly flat bands with Chern number . Such non-Abelian FCIs might be used to build a universal quantum computerFreedman et al. (2002); Nayak et al. (2008). quantized Hall conductance and similar topological properties. They are characterized by different PSGs in the bulk. These states can all serve as candidate states for the FCI state found in numerical simulationsNeupert
et al. (

In section IV we demonstrate that our parton construction can be used to compute the topological ground state degeneracy. This is particularly important for the states, which belong to a new class of FQH wavefunctions.

In section V we generalize our efforts to construct ground state wavefunctions of TRI FTIs. When the mixing between the up and down spins is weak in the electronic hamiltonian, it is natural to generalize our spin polarized results to this case. For filling fraction (on average for each spin), we present classes of and wavefunctions and discuss their properties including quasi-particle statistics and ground state degeneracies. We also propose a new parton construction formalism which allows one to write down generic electron wavefunctions for TRI FTI states in the absence of spin conservation. We can deform such a generic TRI FTI wavefunction in the absence of spin conservation into a -conserved TRI FTI wavefunction (where spin- and spin- decouple) by continuously tuning a parameter. Stability of such a state against perturbations are briefly discussed.

## Ii parton construction of spin-polarized fractional Chern insulator states

### ii.1 A brief review of Laughlin’s FQH state from parton construction

Soon after the experimental discovery of fractional quantum Hall (FQH) effectsTsui et al. (1982), Laughlin proposed a series of variational wavefunctionsLaughlin (1983) which were shownHaldane and Rezayi (1985b) numerically to be a very good description of FQH states at odd-denominator filling fraction . Later this idea of constructing trial wavefunctions was generalized to other filling fractionsHaldane (1983); Halperin (1984); Jain (1989b). An important lesson we can learn from Laughlin’s wavefunction is as follows. With a fixed filling fraction (or a fixed number of flux quanta through the sample), the many-body wavefunction tends to vanish as fast as possible when two electrons approach each other so that the repulsive Coulomb energy between electrons could be minimized. As an example, Laughlin’s state at is nothing but the cube of the wavefunction for a filled lowest Landau level. We can construct this wavefunction by splitting an electron into three fermionic *partons*:

(1) |

Naturally from (1) we can see each parton carries electric charge where stands for the electron charge. The electron wavefunction is obtained through the following projection

(2) |

where represents the parton vacuum and can be any mean-field state of the three partons . When each of the three partons occupy the lowest Landau level (LLL) one immediately obtains the Laughlin’s state . We have chosen the disc geometry and the symmetric gauge. are complex coordinates, is the electron magnetic length and is the parton magnetic length.

Since each kind of parton occupies a LLL, the electro-magnetic response of the FQH state is characterized by Hall conductivity

(3) |

where is the effective mass of each parton. This mean-field Hamiltonian preserves the gauge symmetry and partons will also couple to a internal gauge field. Its effective theory is the Chern-Simons gauge theory, which explains the 3-fold topological ground state degeneracy on a torusWen and Zee (1998). These partons are nothing but charge quasiparticle excitationsLaughlin (1983) of Laughlin state. Indeed after projection (2) the three species of partons becomes indistinguishable thanks to the internal symmetry: each parton creates a charge quasihole upon acting on the ground state . It is straightforward to verify that the following wavefunction

(4) | |||

reproduces the Laughlin wavefunction with three quasiholes at up to a constant factor. Hence these partons are indeed charge anyons obeying fractional statistics with statistical angle .

### ii.2 FCI state and its quasiparticles from parton construction

Since the three seemingly-different partons are essentially the same quasihole excitations with the same quantum numbers, physically it is attempting to include the tunneling terms in the mean-field Hamiltonian. By mixing different partons, these terms will break the internal gauge symmetry down to a a subgroup of , which is , the center of the group, in the most generic case where terms are present. In general the projected wavefunction (2) is different from its parent projected wavefunction. For a 2-D electron gas in a magnetic field, however, people usually focus on the LLL within which the many-body wavefunction is an analytic function (*e.g. *in the symmetric gauge on a disc). It is straightforward to show that as long as the mixing terms act inside the Hilbert space of LLL, the corresponding electron wavefunction (2) for a state remains the same as that of its parent state. This is because the parton wavefunction describes a state with LLL fully filled. Mixing between different partons within the LLL Hilbert space only gives a unitary transformation of basis and does not modify the parton wavefunction. For a lattice model, it is natural to consider mixing terms acting between all bands (rather than within the filled bands), and the corresponding electron wavefunction of a state will be a different wavefunction from that of its parent state. For a filling fraction , our discussion straightforwardly generalizes to the corresponding (the center of the group) state and its parent state.

To our knowledge, the parton states of FQHE have not been proposed before. For this new class of wavefunctions, several natural questions need to be answered. What are the quasi-particles in the state? What is the low-energy effective theory of the state? Will it preserves the topological properties, such as ground state degeneracy? We answer these questions in this paper and find the topological properties of the states are identical to the states. Their difference lies in the projective symmetry group, which is protected by lattice symmetry. In general, states and states both serve as candidate ground states for the FCI states of a filled band with Chern number one.

To begin with, let us consider the quasiparticle excitations in a state. The physical quasiparticle excitations in a state are constructed by inserting fluxes in the mean-field ansatz of terms and simultaneously creating vortices (or defects) in the Higgs condensates . In 2-D, because , these defects are the point-like vortices carrying gauge fluxes. Because the flux can be considered to be localized in a single plaquette, one can effectively interpret it as a overall gauge flux of all the -partons. Namely when a -parton winds around a fundamental vortex, it experiences a flux. Due to the Chern numbers of the filled parton bands, this vortex also binds with a single -parton gauge charge and thus carries electric charge . Because this object carries both flux and gauge charge, the fractional statistical angle results. These are exactly the same electric charge and statistics that a quasiparticle carries in the state. We conclude that the topological properties of quasiparticles are the same in both the and its parent state.

Following the above discussions, we can write down the wavefunctions with low-energy anyonic excitations in a state. At filling fraction , in order to create one quasiparticle at and its antiparticle at , we need to insert flux in a plaquette at position and flux in a plaquette at position with . carries flux and charge while carries flux and charge. Both and have statistical angle and their mutual statistical angle is . They are realized by creating phase shift for all mean-field amplitudes on the string connecting two plaquettes and , on top of the mean-field ansatz for the ground state. An example of such a pair of quasiparticle and its antiparticle in a state on honeycomb lattice is schematically shown in FIG. 1. The corresponding electron wavefunction is obtained by the projection of this new mean-field ansatz to the electronic degrees of freedom. When , this projection is given by Eq.(2).

The ground state degeneracy of a FCI state at on a torus can also be understood once we know its quasiparticle statisticsWen and Niu (1990); Oshikawa and Senthil (2006). Consider the following tunneling process : a pair of quasiparticle (with flux and charge ) and its antiparticle (with flux and charge ) are created and the quasiparticle is dragged around the non-contractible loop along direction on the torus before it is finally annihilated with it anti-particle. This tunneling process will leave a string of phase shift (as shown in FIG. 1) along this loop , therefore has the same physical effects as adiabatically inserting a flux in the non-contractible loop along direction on the torus. Note that when the quasiparticle-anti-quasiparticle pair carries flux and charge the corresponding tunneling process is realized by . Similarly we can define a tunneling process by dragging the fundamental quasiparticle around non-contractible loop once, which is physically equivalent to inserting flux in non-contractible loop . In the thermodynamic limit, the Hilbert space of degenerate ground states should be expanded by these tunneling processesOshikawa and Senthil (2006). The two tunneling operators satisfy the following “magnetic algebra”:

(5) |

This is straightforward to understand from the point of view of Aharonov-Bohm effect. Another way to understand it is because the tunneling process can de continuously deformed into two linked loopsWen and Niu (1990) and corresponds to a phase of , where is the statistical angle of the fundamental quasiparticle. All degenerate ground states can be labeled by *e.g. *eigenvalues of unitary operators and (since they commute with each other). In this basis acts like a ladder operator and changes the eigenvalue of by a phase . In this way one can see the ground state degeneracy of a state on torus is -fold. We can easily generalize this discussion to a genius- Riemann surface with pairs of non-contractible loops and the corresponding ground state degeneracy is -fold. This is consistent with the ground state degeneracy calculated from the low-energy effective theory as will be shown in section IV in a formal way.

Because the discussion on low-energy effective theory of the state involves more technical details, we postpone it to Section IV, where we compute its ground state degeneracy. We’ll show that the ground state degeneracy of a state is the same as that of a state: -fold on a genus- Riemann surface.

### ii.3 Regarding lattice symmetries

In the numerical simulations of FCI phasesSheng et al. (2011); Neupert et al. (2011a), only the -fold topological degeneracy of FCI is observed on torus. This indicates that the FCI wavefunctions respect the full lattice symmetry, since otherwise there should be extra degeneracies due to lattice symmetry breaking. This motivates us to write down the fully symmetric FCI wavefunctions.

In the following we outline the general strategy to construct fully symmetric FCI states on a lattice in the parton approach. Here we focus on spin-polarized FCI states with filling fraction through the parton construction. The electron operator is given by

(6) |

where is the coordinate of a lattice site. For simplicity we assume there is only one orbital per lattice site. As mentioned earlier, this parton construction has a local symmetry since the electron operator is invariant under any local transformation where . A generic parton mean-field ansatz is written as

(7) |

where is a matrix assuming there are one electron orbital per site. Under a local gauge transformation it transforms as . Again once we obtain a mean-field state with the right filling number from (7), the corresponding electron wavefunction is obtained through

(8) |

whose explicit form is a Slater determinant as given later in (10).

Not all parton mean-field ansatzs correspond to FCI states.
Let’s start from a mean-field state with : where each flavor of the parton has the same filling number as the electron. For FCI states in topological flat bands, the filling fraction is such that on average there is one electron (hence one parton with each flavor) per unit cells. If the mean-field ansatz (7) has explicit lattice translation symmetry, however, the corresponding state with filling would most likely be a gapless metallic state^{2}^{2}2When there are parton mixing terms which breaks the gauge symmetry from down to , one can construct a gapped state with filling by just filling the lowest parton band, since there is on average one parton (including all flavors) in each unit cell. However, the Hall conductance of such a state is is the Chern number of lowest parton band. Unless this lowest parton band has Chern number (which is unlikely), this gapped state at filling will have a Hall conductance different from and is not a good candidate for the FCI states realized in recent numerical studies. where since only a fraction () of the lowest band is filled. How to construct a gapped mean-field ansatz of FCI with filling fraction ?

The answer lies in the gauge structure of the parton construction (6). This gauge structure allows the physical (lattice) symmetry to be realized projectively in the parton mean-field ansatz, which gives rise to a symmetric electron wavefunction after projection. Briefly speaking, the mean-field state itself can explicitly break lattice symmetries (such as lattice translations) while the electron wavefunction after projection (8) remains fully symmetric.

By inserting *e.g. * flux in each original unit cell, one can enlarge the unit cell by times. Therefore the corresponding mean-field state with filling is a state filling the lowest bands of mean-field ansatz (7). If each of the lowest bands have a Chern number , the mean-field state filling these bands would have total Chern number , and the corresponding Hall conductivity is

(9) |

because each parton carries electric charge . This gives the correct electromagnetic response of a spin-polarized FCI state.

Here the mean-field Hamiltonian (7) explicitly breaks lattice translation symmetry due to the unit cell enlargement, but as long as the translated mean-field ansatz can be transformed to the original ansatz by a local gauge rotation, the corresponding electron wavefunction (8) still respect the translation symmetry. This is because any two mean-field ansatzs differ by a local gauge rotation give exactly the same electron wavefunction after projection. Similarly, even the mean-field Hamiltonian breaks other lattice symmetries such as the point group symmetry, the electron wavefunction after projection still can be fully symmetric. The mean-field ansatz simply forms a projective representation of the symmetry group. The mathematical framework of constructing fully symmetric electron wavefunctions based on parton mean-field ansatzs is the projective symmetry group (PSG), which will be introduced shortly. The technique of enlarging the unit cell by times without physically breaking any lattice symmetry will be generalized to the case of time-reversal-invariant FTI states with filling fraction in section V.

Following this strategy, we always require that the parton mean-field ansatz of FCI state breaks lattice translation symmetry explicitly and enlarges the unit cell by times, so that the resultant mean-field state is an insulator. The partons will fill the lowest bands of the mean-field spectrum and the corresponding electron state after projection would still be gapped. Now that the number of momentum points of each band in the (reduced) 1st Brillouin zone equals the electron number , we can see the electron wavefunction (8) is nothing but a Slater determinant

(10) | |||

where represents the eigenvector of mean-field Hamiltonian (7). To be specific, corresponds to the parton component in momentum- single-particle eigenvector of the bottom-up -th band. Here and where is the total electron number at filling fraction . Note that for a mean-field ansatz (25) in the absence of mixing terms, the lowest bands are all degenerate and we have . The corresponding electron wavefunction (10) reduces to the product of copies of a Slater determinant:

(11) |

where is the momentum- single-particle eigenvector of parton mean-field Hamiltonian with . This is a lattice version of Laughlin’s state in free spaceLaughlin (1983). However once we add lattice-symmetry-preserving parton mixing terms which breaks gauge symmetry from to , the electron wavefunction (10) of a FCI state, as well as its projective symmetry group which will be introduced shortly, will immediately become different from its parent state (11). We emphasize again that only when the unit cell is enlarged by times, we will have the same number of momentum points as the electron number . The mean-field amplitudes can be determined by variational Monte Carlo study of the energetics of electronic wavefunctions (10).

Considering flux insertion in order to enlarge the unit cell in the mean-field ansatz (7), another question arises: can there be more than one way of inserting fluxes into plaquettes without breaking physical lattice symmetries? If yes, how to classify different mean-field ansatzs (7)? The answer of the 1st question is positive and to answer the 2nd question, we need to introduce a mathematical structure: projective symmetry group (PSG)Wen (2002) in order to characterize different “universality classes” of symmetric FCI states. PSG classifies different mean-field ansatz which forms a projective representation of the physical symmetry group. In the following we give a brief introduction of PSG.

Note that there is a many-to-one correspondence between parton mean-field states and physical electron states due to the above projection operation: any two parton mean-field states related to each other by a gauge transformation correspond to the same electron state. As a result, although the physical electron state preserves all lattice symmetry, its parton mean-field ansatz may or may not explicitly preserve these lattice symmetries. The physical lattice symmetries are realized projectively in the mean-field ansatz. More precisely, in a generic case the parton mean-field ansatz (7) should be invariant under a combination of lattice symmetry operation and a corresponding gauge transformation :

(12) |

Different universality classes of parton mean-field ansatzs are characterized by different PSGsWen (2002), *i.e. *different gauge transformations associated with symmetry operations :

(13) |

The low-energy gauge fluctuation of a mean-field ansatz is controlled by its invariant gauge groupWen (2002) (IGG)

where represents the identity operator of the (lattice) symmetry group (SG). In other words, IGG is a subgroup of the internal gauge group (which is here) that keeps the mean-field ansatz (7) invariant. Hereafter we would call a parton mean-field state with *e.g. * state a state. We can see that the IGG of mean-field ansatz (7) always contains the following group as a subgroup:

(14) |

where is the identity matrix. This group is the center of the group. A mean-field ansatz with is called a state^{3}^{3}3In principle the IGG of a mean-field ansatz can be any subgroup of the internal gauge group, *i.e. * here. For example when by adding terms to a mean-field ansatz, the corresponding IGG becomes . Here we focus on the cases with or .. The low-energy theory of this state will be described by fermionic partons interacting with gauge fields.

The classification of PSGs with (which we call PSGs in this paper) are easy to carry out. The only gauge invariant quantities of a ansatz is the gauge-invariant flux through each plaquette, which must belong to the center of the gauge group, namely the group in Eq.14, because otherwise the gauge group would be broken and cannot be . Two states have the same PSG if and only if they have the same gauge flux in each given plaquette. Therefore distinct PSGs have different gauge flux pattern and vice versa.

The classification of PSGs with (which we call PSGs in this paper) involves more technical details and we leave this analysis for the honeycomb lattice modelHaldane (1988) and the checkerboard lattice modelSun et al. (2011); Neupert et al. (2011a) in Appendix B and D.

## Iii Examples

The first part of this section shows four concrete examples of mean-field ansatzs corresponding to spin-polarized FCI states with filling fraction in the checkerboard lattice modelSun et al. (2011); Neupert et al. (2011a). These include two ansatzs and their two parent states. As discussed in the previous section and proved in Appendix A-D, although the mean-field ansatz explicitly breaks lattice translation symmetry by tripling the unit cell, the physical electron state after projection (8) preserves all lattice symmetry. Four examples for the honeycomb lattice model with filling fraction are displayed in Appendix G. We show the Hall conductance of all these states are .

In the second part of this section, we present a scenario which might realize non-Abelian FCI with by partially filling nearly flat bands with Chern number , and two examples of such states in the parton construction are presented in Appendix H.

### iii.1 Four examples of spin-polarized FCI states with in the checkerboard lattice model

The checkerboard lattice modelSun et al. (2011); Neupert
et al. (2011a) has been shown to support nearly flat bands with non-zero Chern numbers. Its symmetry group is shown in Appendix C and FIG. 2. Each lattice site is labeled by coordinate as shown in Appendix C. Recently there are numerical evidenceNeupert
et al. (2011a); Sheng et al. (2011); Regnault and Bernevig (2011) of FCI states with filling fraction and in this model. By parton construction, we use PSG to classify different mean-field ansatz ( is an *odd* integer) as shown in Appendix D. In the following we show two mean-field states that belong to different universality classes. They correspond to two gauge-inequivalent solutions of (72) when . We show mean-field amplitudes up to the next next nearest neighbor. Their two parent states also have different PSGs because they have different patterns of the gauge invariant fluxes. All four states are candidates of the FCI state found in the exact diagonalizations of checkerboard lattice modelNeupert
et al. (2011a); Sheng et al. (2011); Regnault and Bernevig (2011).

#### iii.1.1 The FCI state CB1 with and its parent state

In the FCI state CB1 the gauge transformations associated with lattice symmetries are listed below:

(15) | |||

where . are lattice translation operations and are reflections. They are schematically shown in Fig.2(b) and defined mathematically in Appendix C. As shown in Appendix D the symmetry allowed mean-field amplitudes are:

(\@slowromancap[email protected]) For nearest neighbor (NN) amplitude

(16) |

*i.e. * can be any complex symmetric matrix. All other NN amplitudes can be generated from by symmetry operations through (12).

There are two independent NNN amplitudes:

(\@slowromancap[email protected]) For next nearest neighbor (NNN) amplitude

(17) |

*i.e. * can be any real symmetric matrix. Half of NNN mean-field amplitudes can be generated from by symmetry operations through (12).

(\@slowromancap[email protected]) For next nearest neighbor (NNN) amplitude

(18) |

where can be any real symmetric matrix. Half of NNN mean-field amplitudes can be generated from by symmetry operations through (12).

(\@slowromancap[email protected]) For next next nearest neighbor (NNNN) amplitude

(19) |

*i.e. * can be any Hermitian matrix. All other NNNN mean-field amplitudes can be generated from by symmetry operations through (12).

These mean-field amplitudes of , , and should be treated as variational parameters. Their optimal values which minimize the variational energy of electron wavefunction (10) can be determined in variational Monte Carlo simulations.

The corresponding parent state has . Choosing parameters , and we have the Chern numbers of the 6 bands for each parton species as and the lowest band is well separated from other bands, as shown in FIG. 5. This qualitative band structure persists for a large parameter range. Each band of the parton ansatz is 3-fold degenerate, corresponding to the 3 parton flavors . By adding small parton mixing terms to the mean-field state, each 3-fold degenerate band splits into 3 non-degenerate parton bands in a mean-field state, as shown in FIG. 6. By filling the resulting 3 lowest bands (all with Chern number ) we obtain a FCI state whose Hall conductivity is in the unit of . and

#### iii.1.2 The FCI state CB2 with and its parent state

In the FCI state CB2 the gauge transformations associated with lattice symmetry are listed below:

(20) | |||

where . As shown in Appendix D the symmetry allowed mean-field amplitudes are:

(\@slowromancap[email protected]) For nearest neighbor (NN) amplitude

(21) |

*i.e. * can be any complex symmetric matrix. All other NN amplitudes can be generated from by symmetry operations through (12).

There are two independent NNN amplitudes:

(\@slowromancap[email protected]) For next nearest neighbor (NNN) amplitude

(22) |

where can be any real symmetric matrix. Half of NNN mean-field amplitudes can be generated from by symmetry operations through (12).

(\@slowromancap[email protected]) For next nearest neighbor (NNN) amplitude

(23) |

*i.e. * can be any real symmetric matrix. Half of NNN mean-field amplitudes can be generated from by symmetry operations through (12).

(\@slowromancap[email protected]) For next next nearest neighbor (NNNN) amplitude

(24) |

*i.e. * can be any Hermitian matrix. All other NNNN mean-field amplitudes can be generated from by symmetry operations through (12).

The corresponding parent state has . From FIG. 7 we can see that the pattern of fluxes in CB2 state is different from that in CB1 state. For example, in CB1 state flux (shown by triple arrows) are inserted in one half of the NN square plaquettes (those enclosed by two horizontal blue sites and two vertical red sites) while in CB2 state flux are inserted in the other half of the NN square plaquettes (those enclosed by two horizontal red sites and two vertical blue sites). Notice that these two types of NN square plaquettes are inequivalent, not related to each other by any lattice symmetry (such as translation or mirror reflection) in the checkerboard lattice modelSun et al. (2011). Therefore CB1 and CB2 are different mean-field FCI states belonging to distinct universality classes. Choosing parameters , and we have the Chern numbers of the 6 bands for each parton species as , and the lowest band is well separated from other bands, as illustrated in FIG. 8. This qualitative band structure persists for a large parameter range. Each band of a parton ansatz is 3-fold degenerate, corresponding to the 3 parton flavors . By adding small parton mixing terms to the mean-field state, each 3-fold degenerate band splits into 3 non-degenerate parton bands in a mean-field state. By filling the resultant 3 lowest bands (all with Chern number ) we obtain a FCI state whose Hall conductivity is again in unit of . and

### iii.2 Possible non-Abelian states by partially filling a nearly flat band with Chern number

It has been shownWen (1991b); Wen and Zee (1998) that in parton construction, when each parton species fills Landau levels, the effective theory of the corresponding electron state is the Chern-Simons theory and the system has non-Abelian quasiparticle excitations when . Moreover, the non-Abelian quasiparticles of this state can be used as topologically-protected qubits in a universal quantum computerNayak et al. (2008) as long as . This motivates us to propose possible realization of non-Abelian FCI states realized in a partially-filled nearly flat band with Chern number . When this band is partially filled with a filling fraction *e.g. *, the Hall conductance of the corresponding FCI state would be parton bands all have Chern number instead of , the FCI state obtained by filling lowest parton bands indeed has Hall conductance . If the lowest

We discuss the case as an example. In Appendix H using the parton construction, we show two examples of FCIs, one on the honeycomb lattice and the other on the checkerboard lattice, both have 3 lowest parton bands with Chern number . These two FCIs with and states is Chern-Simons theoryWen and Zee (1998), featured by -fold ground state degeneracy on the torus and non-Abelian quasiparticle excitationsWitten (1989). These results indicate that once a nearly flat band with Chern number is found, by partially filling it one may realize non-Abelian FCIs, which have the potential to build a universal quantum computerFreedman et al. (2002); Nayak et al. (2008). are non-Abelian FCIs. Their low-energy effective theory of these

## Iv Effective theory and ground state degeneracy of spin-polarized FCI states

As mentioned earlier, the partons in our construction not only couples to the external electromagnetic gauge field, but also couples to a dynamical internal gauge field, which is a () gauge field for a () state. The low energy effective theories of the partons coupled with the internal gauge fields control all the topological properties of the systems. The topological properties of a state has been studied beforeWen and Zee (1998); Wen (1999). In this section we will analyze the low-energy effective theory of spin-polarized FCI state (7) from the parton construction. To our knowledge, the FQH states have not been proposed and studied before. We’ll try to answer the following questions: what is the ground state degeneracy of the FCI state? Is it the same as or different from that of a FCI state?

We start from a mean-field state which has been shownWen and Zee (1998) to describe the Laughlin state in the continuum limit. Its mean-field ansatz is

(25) |

In other words, there is no hopping between partons of different species and the species of partons have exactly the same band structure. As shown in (11) its electron wavefunction is a lattice version of Laughlin stateLaughlin (1983). Apparently this mean-field state doesn’t break the gauge symmetry which leaves the electron operator (6) invariant. Since here the partons couple to both the electromagnetic field and the gauge field, the Lagrangian writes

(26) | |||

where means path-ordered integral. and are the and the gauge fields respectively. Strictly speaking they are both defined on the link of a lattice here. To linear order the above action can be written as

(27) | |||

where stands for electromagnetic gauge field while represents the internal gauge field. is the electric charge of each parton. Here and are conserved and parton currents respectively. To be precise, and . In the long-wavelength limit the spatial components of the parton currents in momentum space (with momentum q) writes:

Since the partons form a band insulator, the band gap allows us to safely integrate out the partons and obtain an effective action for the gauge fields. Let’s assume all the filled lowest parton bands have Chern number . Upon integrating out partons , the effective Lagrangian density writes

(28) | |||

the first term corresponds to the quantized Hall conductance , while the second term, *i.e. *a Chern-Simons term describes the low-energy gauge fluctuations. As shown in Appendix E, the Chern-Simons theory of gauge field can be reduced to Chern-Simons theory of gauge fields . The gauge field configuration is given by where are matrices defined in (76). In the gauge, and are conjugate variables since the Lagrangian density for internal gauge fields writes