Fractional quantum Hall effect at $\nu=2+4/9$

The fractional quantum Hall effect (FQHE) Tsui et al. (1982); Laughlin (1983) forms a paradigm in our understanding of strongly correlated quantum phases of matter. Of particular interest among the panoply of FQHE phases are the ones observed in the second Landau level (SLL) of ordinary semiconductors such as GaAs. These have attracted widespread attention because of the possibility that the excitations of these phases obey non-Abelian braiding statistics, which could potentially be utilized in carrying out fault-tolerant topological quantum computation Kitaev (2003); Nayak et al. (2008).

FQHE was first observed at filling factor $\nu=1/3$ Tsui et al. (1982) in the lowest LL (LLL) and was explained by Laughlin using his eponymous wave function Laughlin (1983). Soon a whole zoo of fractions were observed, primarily along the sequence $n/(2pn\pm 1)$ ($n$ and $p$ are positive integers) and its particle-hole conjugate Eisenstein et al. (1990). These FQHE states can be understood as arising from the integer quantum Hall effect (IQHE) of composite fermions (CFs) Jain (1989a, 2007), which are bounds states of electron and an even number ($2p$) of quantized vortices. The theory of weakly interacting CFs captures almost all of the observed FQHE phenomenology in the LLL.

In comparison to the FQHE in the LLL, the FQH states in the SLL are fewer in number and are more fragile Willett et al. (1987). Moreover, the nature of many FQH states in the SLL is dramatically different from their LLL counterparts. In particular, one of the strongest FQH states in the SLL occurs at $\nu=2+1/2$ Willett et al. (1987), whereas at the corresponding half-filled LLL a compressible state is observed. A key breakthrough in the field of FQHE came about by a proposal of Moore and Read Moore and Read (1991), who posited a “Pfaffian” wave function to describe the state at $\nu=5/2$. Subsequently, it was understood that the Pfaffian wave function can be interpreted as a $p$-wave paired state of composite fermions Read and Green (2000). The excitations of the Pfaffian state are Majorana fermions which, owing to their non-Abelian braiding properties, could form building blocks of a topological quantum computer Kitaev (2003); Nayak et al. (2008).

The nature of the state at $1/3$ state in the SLL, although believed to be Laughlin-like, has been under intense debate d’Ambrumenil and Reynolds (1988); Balram et al. (2013); Johri et al. (2014); Peterson et al. (2015); Kleinbaum et al. (2015); Jeong et al. (2017); Balram et al. (2020). FQHE has been observed at $\nu=2+2/5$ Xia et al. (2004); Choi et al. (2008); Pan et al. (2008); Kumar et al. (2010) but is widely believed to be of a parafermionic nature, unlike the Abelian LLL Jain CF state at $\nu=2/5$ Read and Rezayi (1999); Rezayi and Read (2009); Wójs (2009); Bonderson et al. (2012); Sreejith et al. (2013); Zhu et al. (2015); Mong et al. (2017); Pakrouski et al. (2016). Furthermore, as yet there is no conclusive experimental evidence of FQHE at the next three members of the $n/(2n+1)$ Jain sequence, namely $\nu=3/7$, $4/9$, and $5/11$, in the SLL Shingla et al. (2018), though some features of FQHE have been reported in the literature at some of these fillings Pan et al. (2008); Choi et al. (2008); Reichl et al. (2014) (A non-Abelian parton state at $3/7$ was constructed in Ref. Faugno et al. (2020) and shown to be feasible in the SLL.). However, FQHE has been observed at $\nu=2+6/13$ Kumar et al. (2010). These observations collectively point to the fact that the nature of FQHE states in the LLL and SLL are different from each other, and a description of FQHE states in the SLL likely entails going beyond the framework of noninteracting composite fermions.

Recently, candidate “parton” states have been constructed to describe the FQHE at many of the experimentally observed fillings in the SLL Balram et al. (2018a, b, 2019). The parton theory Jain (1989b) produces model incompressible states beyond the Laughlin Laughlin (1983) and the CF theory Jain (1989a). The “$\bar{n}\bar{2}111$” parton states for $n=1,2,3$ give a good description of the SLL FQHE states observed at $\nu=2+2/3,2+1/2$ and $2+6/13$ Balram et al. (2018a, b). In particular, the $\bar{n}\bar{2}111$ sequence underscores the unusual stability of $6/13$ in the SLL.

In this work, we consider the next member of the $\bar{n}\bar{2}111$ parton sequence, namely $\bar{4}\bar{2}111$ and consider its feasibility at $\nu=2+4/9$. While FQHE at $2+4/9$ has not been established conclusively, indications for it in the form of a minimum in the longitudinal resistance have already been seen in experiments Choi et al. (2008); Reichl et al. (2014). Future experiments on high-quality samples could likely establish FQHE at this filling by an observation of a well-quantized plateau in the Hall resistance. We show that the $\bar{4}\bar{2}111$ parton state gives a good description of the exact SLL Coulomb ground state seen in numerics. Furthermore, the parton state is energetically favorable compared to the $411$ Jain CF state at $2+4/9$ in the thermodynamic limit. Therefore, if FQHE is established at $\nu=2+4/9$, it is highly likely to be distinct from its LLL counterpart at $\nu=4/9$, which is well described by the $411$ Jain CF state. The parton state is topologically distinct from the Jain CF state and we make predictions for experimentally measurable quantities that can unambiguously distinguish the two. In particular, the parton state supports counter-propagating edge modes that do not occur in the Jain CF state.

The parton theory put forth by Jain Jain (1989b) constructs FQHE states as a product of IQH states. The essential idea is to break each electron into fictitious objects called partons, place the partons into incompressible IQH states, and recover the final state by fusing the partons back into the physical electrons. The $N$-electron wave function for the $l$-parton Jain state, denoted as “$n_{1}n_{2}\cdots n_{l}$”, is given by

Here $z_{j}=x_{j}-iy_{j}$ is the two-dimensional coordinate of electron $j$ which is parametrized as a complex number, $\mu$ denotes the parton species and $\mathcal{P}_{\rm LLL}$ implements projection into the LLL. Each parton species is exposed to the external magnetic field and occupies the same area, which fixes the charge of the $\mu$ parton species to $q_{\mu}=-e\nu/n_{\mu}$, with $\sum_{\mu}q_{\mu}=-e$, where $-e$ is the charge of the electron. The state $\Phi_{n_{\mu}}$ is the Slater determinant IQH wave function of $N$ electrons filling the lowest $|n_{\mu}|$ LLs. We allow for $n_{\mu}<0$ and negative values are denoted by $\bar{n}$, with $\Phi_{\bar{n}}=\Phi_{-|n|}=[\Phi_{n}]^{*}$. The parton state of Eq. (1) occurs at a filling factor of $\nu=(\sum_{\mu}n_{\mu}^{-1})^{-1}$ and has a shift Wen and Zee (1992) of $\mathcal{S}=\sum_{\mu}n_{\mu}$ in the spherical geometry.

The Laughlin state Laughlin (1983) is a “$11\cdots$” parton state. The parton state $n11$ ($\bar{n}11$) with wave function $\Psi^{\rm CF}_{n/(2n+1)}=\mathcal{P}_{\rm LLL}\Phi_{n}\Phi^{2}_{1}$ ($\Psi^{\rm CF}_{n/(2n-1)}=\mathcal{P}_{\rm LLL}[\Phi_{n}]^{*}\Phi^{2}_{1}$) correspond to the Jain CF states Jain (1989a). Recently, it has been shown that parton states of the form “$221\cdots,$” which are not composite fermion states, could be viable candidates to describe certain FQH states observed in the LLL in wide quantum wells Faugno et al. (2019) and in LLs of graphene Wu et al. (2017); Kim et al. (2019).

The motivation for considering the $\bar{4}\bar{2}111$ parton state stems from the recent application of parton theory to capture states in the SLL Balram et al. (2018a, b, 2019). Consider the family of parton states described by the wave function

The $\sim$ sign in Eq. (2) indicates that the states written at both sides of the sign differ slightly in the details of the projection. We expect that such details do not change the topological nature of the states Balram and Jain (2016). Throughout this text, for the $\bar{n}\bar{2}111$ parton states we use the form given on the rightmost side of Eq. (2). A nice feature of the parton wave functions stated in Eq. (2) is that they can be evaluated for large system sizes which allow a reliable extrapolation of their thermodynamic energies. One can construct the above parton states for large system sizes because the constituent Jain CF states can be evaluated for hundreds of electrons using the Jain-Kamilla method of projection Jain and Kamilla (1997); Möller and Simon (2005); Jain (2007); Davenport and Simon (2012); Balram et al. (2015a). The Jain CF states in this work are evaluated using the Jain-Kamilla method.

The $n=1$ member, namely $\bar{1}\bar{2}111$, is likely topologically equivalent to the $\nu=2/3$ $\bar{2}11$ Jain CF state Balram and Jain (2016). The $n=2$ member $\bar{2}\bar{2}111$ has a good overlap with the exact SLL Coulomb ground state at $\nu=5/2$ Balram et al. (2018a). The $n=3$ state, $\bar{3}\bar{2}111$, gives a good description of the Coulomb ground state at $\nu=2+6/13$ Balram et al. (2018b). In the Supplemental Material (SM) SM we provide further evidence in favor of the feasibility of the $\bar{3}\bar{2}111$ parton state to describe the $\nu=2+6/13$ FQHE. We shall consider the $n=4$ member of this sequence which occurs at filling factor $4/9$.

Although there is no definitive observation of FQHE at $4/9$ in the SLL, signatures of incompressibility have been seen at $\nu=2+4/9$ and its particle-hole conjugate at $\nu=2+5/9$ Choi et al. (2008); Reichl et al. (2014). FQHE at $\nu=2+4/9$ is likely swamped by a bubble phase Shingla et al. (2018); however, it is likely that with improvements in the sample quality or for some interaction parameters close to that of the SLL Coulomb one, FQHE will ultimately be observed at $2+4/9$.

For all our calculations, we deploy Haldane’s spherical geometry Haldane (1983), in which $N$ electrons reside on the surface of a sphere in the presence of a radial magnetic field generated by a monopole of strength $2Q(hc/e)$ located at the center of the sphere. FQHE ground states occur at flux values $2Q=\nu^{-1}N-\mathcal{S}$, where $\mathcal{S}$ is a rational number called the shift, which is useful in characterizing the topological nature of the FQHE state Wen and Zee (1992). All FQHE ground states are uniform on the sphere and thus have total orbital angular momentum $L=0$. The parton states $\Psi^{\bar{n}\bar{2}111}_{\nu=2n/(5n-2)}$ of Eq. (2) satisfy the flux-particle relationship $2Q=[(5n-2)/(2n)]N-(1-n)$; i.e., their filling factors are $\nu=2n/(5n-2)$ and their shifts are $\mathcal{S}=1-n$. Of particular interest to us in this work is the $\bar{4}\bar{2}111$ parton state which has a shift of $\mathcal{S}=-3$. This parton state is topologically distinct from the $411$ Jain CF state which also occurs at $\nu=4/9$ but has a shift of $\mathcal{S}=6$. We assume a single-component system and neglect the effects of LL mixing and disorder. Under these assumptions, states related by particle-hole conjugation are considered on the same footing.

Throughout this work, we shall write wave functions in the LLL, which is where they are easily evaluated, even though they might apply to states occurring in the SLL. Haldane Haldane (1983) showed that the physics of the SLL can be simulated in the LLL by using an effective interaction that has the same set of Haldane pseudopotentials in the LLL as the Coulomb interaction has in the SLL. In this work, we have used the form of the effective interaction described in Ref. Shi et al. (2008) to simulate the physics of the SLL in the LLL.

Let us begin by testing the viability of the $\bar{4}\bar{2}111$ parton state for $\nu=2+4/9$ FQHE. In Fig. 1 we compare the energies of the $\bar{4}\bar{2}111$ parton and the $411$ Jain CF states at $\nu=4/9$ in the LLL and the SLL. In the LLL, as expected, we find that the Jain CF state has lower energy than the parton state. However, in the SLL we find that the $\bar{4}\bar{2}111$ parton state is energetically more favorable compared to the Jain CF state. For the sake of completeness, we have also investigated the competition between the parton and Jain CF states in the $n=1$ LL of monolayer graphene. The effective interaction we use to simulate the physics of the $n=1$ LL of monolayer graphene in the LLL is described in Ref. Balram et al. (2015b). We find the $411$ Jain CF state has lower energy here, consistent with the fact that experimentally observed FQHE states in the $n=1$ LL of monolayer graphene are well described by the CF paradigm Amet et al. (2015); Balram et al. (2015b); Zeng et al. (2019). Results for $n=0$ LL of graphene are identical to those in the LLL of GaAs under our working assumptions of neglecting effects of finite width and LL mixing.

Next, we turn to comparisons of the parton state with the exact SLL Coulomb ground state. The smallest system accessible to exact diagonalization (ED) is that of $N=16$ electrons at a flux of $2Q=39$ which has a Hilbert space dimension of $7\times 10^{8}$. We have evaluated the ground state for this system with the truncated pseudopotentials from the disk geometry, which differ slightly from the spherical pseudopotentials but are known to provide a more reliable extrapolation to the thermodynamic limit Peterson et al. (2008a, b). The exact SLL Coulomb ground obtained by using the truncated disk pseudopotentials has $L=0$. In Fig. 2 we compare the pair-correlation function foo of this exact SLL Coulomb ground state with that of the $\bar{4}\bar{2}111$ parton state. Both these pair-correlation functions show oscillations that decay at long distances, which is a typical characteristic of incompressible states Kamilla et al. (1997); Balram et al. (2015c). Moreover, the two pair-correlation functions are in reasonable agreement with each other. For completeness, we have also evaluated the exact LLL Coulomb ground state for the same system. The overlaps of the LLL and SLL Coulomb ground states obtained using the disk pseudopotentials is $0.3663$ and their pair-correlation functions are also very different from each other SM which indicate that the nature of the ground state in the two LLs are very different.

Currently, we do not have a reliable estimate of the thermodynamic values of the gaps predicted by our parton ansatz. However, we can extract the charge and neutral gaps for $N=16$ particles from exact diagonalization of the SLL Coulomb interaction at the parton shift. The charge gap here is defined as $[\mathcal{E}(2Q=40)+\mathcal{E}(2Q=38)-2\mathcal{E}(2Q=39)]/4$, where $\mathcal{E}(2Q)$ is the exact ground state energy at flux $2Q$ and the factor of $4$ in the denominator accounts for the fact that the addition of a single flux quantum in the parton state produces four fundamental quasiholes. The neutral gap is defined as the difference between the two lowest exact energies at the flux of $2Q=39$. The charge and neutral gaps for $N=16$, evaluated using exact diagonalization with the disk pseudopotentials, are $0.009~{}e^{2}/\epsilon\ell$ and $0.005~{}e^{2}/\epsilon\ell$ respectively, where $\ell=\sqrt{\hbar c/(eB)}$ is the magnetic length and $\epsilon$ is the dielectric constant of the background host material.

We next consider the effect of finite width on the system, which we model by taking the transverse wave function to be the ground state for an infinite square quantum well of width $w$ (see SM SM for details). For the disk pseudopotentials, we find that the ground state for $16$ electrons has $L=0$ for (at least) $w\leq 5\ell$ SM . Moreover, the pair-correlation function of the exact ground state agrees well with that of the parton state for the entire range of widths considered in this work (see Fig. 2 and Ref. SM ). Furthermore, we find that the system has robust charge and neutral gaps for all the widths considered. We note that the $L=0$ ground state is delicate. In particular, the exact SLL Coulomb ground state obtained using the spherical pseudopotentials has $L=2$. However, the overlap between the lowest energy $L=0$ state obtained using the spherical pseudopotentials and the ground state obtained using the disk pseudopotentials is $0.9692$, which indicates that these two states are close to each other. Encouragingly, with the spherical pseudopotentials, as the quantum well width is increased the ground state turns uniform in the range $w\in(0.5,1]\ell$ and stays uniform for $w\in[1,10]\ell$ SM . These results indicate that finite thickness enhances the stability of the parton state.

Now that we have made a case for the plausibility of the $\bar{4}\bar{2}111$ parton state to occur in the SLL, we shall turn to deduce the experimental consequences of this parton ansatz. An additional particle in the factor $\Phi_{\bar{4}}$ has charge $e/9$, whereas that in the factors $\Phi_{\bar{2}}$ and $\Phi_{1}$ has a charge $2e/9$ and $-4e/9$ respectively. All the quasiparticles of the $\bar{4}\bar{2}111$ parton state obey Abelian braid statistics Wen (1991). The $4/9$ Jain CF state is also an Abelian state and hosts quasiparticles of charge $-e/9$ and $-4e/9$.

Next, to infer other topological consequences of the $\bar{4}\bar{2}111$ ansatz, we consider the low-energy effective theory of its edge, which is described by the Lagrangian density WEN (1991); Wen (1992, 1995); Moore and Wen (1998):

Here $\epsilon^{\alpha\beta\gamma}$ is the fully anti-symmetric Levi-Civita tensor, $A$ is the vector potential corresponding to the external electromagnetic field, $a$ is the internal gauge field, and we have used the Einstein’s convention of summing repeated indices. The integer-valued symmetric $K$ matrix and the charge vector $t$ of Eq. (3) for the parton state are given by (see SM SM for a derivation)

The above $K$ matrix has four negative and one positive eigenvalues and thus the $\bar{4}\bar{2}111$ state hosts four upstream and one downstream edge modes. A naive counting suggests that there are a total of nine edge states for the $\bar{4}\bar{2}111$ ansatz: four from the factor $\Phi_{\bar{4}}$, two from $\Phi_{\bar{2}}$, and one from each factor of $\Phi_{1}$. However, these edges states are not all independent since the density variations of the five partons must be identified. This results in four constraints and leads to five edge states consistent with that ascertained from the above $K$ matrix.

Assuming full equilibration of the edge modes, the thermal Hall conductance $\kappa_{xy}$ at temperatures much smaller than the gap takes a quantized value proportional to the chiral central charge $c_{-}$, which is defined as the difference in the number of downstream and upstream modes: $\kappa_{xy}=c_{-}[\pi^{2}k_{\rm B}^{2}/(3h)]T$ Kane and Fisher (1997). For $\bar{4}\bar{2}111$ ansatz, we thus predict a thermal Hall conductance of $\kappa_{xy}=-3[\pi^{2}k_{\rm B}^{2}/(3h)]T$. The Hall viscosity of the $\bar{4}\bar{2}111$ state is also expected to be quantized Read (2009): $\eta_{\rm H}=\hbar\rho_{0}\mathcal{S}/4$, where $\rho_{0}=(4/9)/(2\pi\ell^{2})$ is the electron density and $\mathcal{S}=-3$ is the shift of the parton state. The ground-state degeneracy of the parton state on a topologically nontrivial manifold with genus $g$ is $|{\rm Det}(K)|^{g}=18^{g}$. Besides the $n\bar{n}111$ parton states Balram et al. (2020), the $\bar{4}\bar{2}111$ ansatz provides another example of a fully spin polarized Abelian FQH state at $\nu=a/b$ (with $a,b$ coprime), which has a ground state degeneracy on the torus that is greater than $b$.

The $4/9$ Jain CF state is described by the $4\times 4$ $K$ matrix $K=2C_{4}+\mathbb{I}_{4}$, where $C_{k}$ is the $k\times k$ matrix of all ones and $\mathbb{I}_{k}$ is the $k\times k$ identity matrix, and charge vector $t=(1,1,1,1)^{T}$. In contrast to the $\bar{4}\bar{2}111$ state, assuming the absence of edge reconstruction, the $4/9$ Jain CF state has four downstream edge states and no upstream modes. The $4/9$ Jain CF state thus has a thermal Hall conductance of $\kappa_{xy}=4[\pi^{2}k_{\rm B}^{2}/(3h)]T$. Moreover, the Hall viscosity of the $4/9$ Jain CF state is given by $\eta_{\rm H}=(3/2)\hbar\rho_{0}$, corresponding to shift ${\cal S}=6$. On a manifold of genus $g$, the $4/9$ Jain CF state has a degeneracy of $9^{g}$.

The presence of upstream neutral modes can be detected in shot noise experiments Bid et al. (2010); Dolev et al. (2011); Gross et al. (2012); Inoue et al. (2014). Recently, thermal Hall measurements have been carried out at several filling factors in the lowest as well as the second LL Banerjee et al. (2017, 2018); Srivastav et al. (2019). These experiments can be used to test the predictions of the parton theory and therefore can unambiguously distinguish between the topological nature of the 4/9 states in the SLL and the LLL. In particular, including the contributions of the filled LLLs of spin up and spin down, the thermal Hall conductance of the $\bar{4}\bar{2}111$ state in the SLL is $-[\pi^{2}k_{\rm B}^{2}/(3h)]T$ which is different from what one would expect from the $4/9$ Jain CF state in the SLL, which has $\kappa_{xy}=6[\pi^{2}k_{\rm B}^{2}/(3h)]T$.

In summary, we have considered the viability of the “$\bar{4}\bar{2}111$” parton state for FQHE at $\nu=2+4/9$ where the first signs of incompressibility in the form of minimum in longitudinal resistance have already been observed experimentally Choi et al. (2008); Reichl et al. (2014). Interestingly, if FQHE eventually stabilizes at this filling factor, then it is likely to be topologically different from its LLL counterpart at $\nu=4/9$, which is described by a Jain CF state. We also proposed experimental measurements that can reveal the underlying topological structure of the parton state and decisively distinguish it from the $4/9$ state occurring in the lowest Landau level.

Supplemental Material for “Fractional Quantum Hall Effect at $\nu=2+4/9$”

In this Supplemental Material (SM) we provide

• •

  a derivation of the low-energy effective edge theory of the $\bar{4}\bar{2}111$ parton state (Sec. I),

• •

  a detailed study of the effect of finite width at $\nu=2+4/9$ (Sec. II), and

• •

  a comparison of the Laughlin, Pfaffian and the $\bar{3}\bar{2}111$ parton states with the exact Coulomb ground state at filling factors $\nu=7/3,~{}5/2$ and $2+6/13$ respectively (Sec. III).