Fractional Quantum Hall Effect in Graphene: Quantitative Comparison between Theory and Experiment

A longstanding issue in the field of the fractional quantum Hall effect (FQHE)Tsui et al. (1982) has been that the quantitative agreement between theory and experiment is less precise than what one would expect. It is possible to obtain very accurate numbers for many quantities of interest from the composite fermion (CF) theory Jain (1989, 2007, 2015). Detailed comparisons have been carried out for activations gaps Du et al. (1993), collective mode dispersionsKukushkin et al. (2009), and spin-polarization phase transitions Jain (2007). In all cases, the measured numbers are generally consistent with those predicted by theory, but the agreement is worse than that suggested by the accuracy of the theory as determined from comparisons with exact diagonalization resultsDev and Jain (1992a); Wu et al. (1993); Jain (2007); Balram et al. (2013). It is believed that the deviation arises from corrections due to effects extraneous to the FQHE physics, such as finite quantum well width, Landau level (LL) mixing and disorder, which provide sizable corrections that are hard to deal with in a quantitative manner.

The observation of FQHE in graphene Du et al. (2009); Bolotin et al. (2009); Dean et al. (2011); Feldman et al. (2012, 2013); Amet et al. (2015); Lin et al. (2014) provides a unique opportunity in this context, because finite width corrections are essentially absent in graphene. Furthermore, as noted by Peterson and NayakPeterson and Nayak (2014), unlike in GaAs quantum wells, in the $n=0$ LL of graphene, LL mixing does not produce any effective three body interaction (which incorporates the breaking of particle-hole symmetry), but only corrections to the pairwise interaction. One might therefore expect smaller corrections due to LL mixing in graphene than in GaAs. The FQHE in graphene may thus provide an opportunity for better quantitative comparisons between theory and experiments, and a better appreciation of our understanding of the role of LL mixing.

The spin polarization transitions provide some of the most precise tests of the quantitative accuracy of the FQHE theory, for several reasons. First, it is a thermodynamic measurement (as opposed to the excitation energies of charged or neutral modes), and therefore is likely to be less susceptible to the presence of disorder. Second, the critical Zeeman energy where a transition between two differently spin polarized FQHE states occurs provides a direct measure of the rather small Coulomb energy differences between the two states, and thus enables a detailed and exceedingly sensitive test of our quantitative understanding of the FQHE. Finally, there is an extensive amount of experimental phenomenology associated with the spin physics. Phase transitions as a function of the Zeeman energy have been measured in various semiconductor based two-dimensional systems Eisenstein et al. (1989); Du et al. (1995); Yeh et al. (1999); Kukushkin et al. (1999, 2000); Melinte et al. (2000); Tiemann et al. (2012); Liu et al. (2014a, 2015); Bishop et al. (2007); Padmanabhan et al. (2009); Shabani et al. (2010); Padmanabhan et al. (2010); Gokmen et al. (2010); Liu et al. (2014b); Betthausen et al. (2014), and recently also in graphene by Feldman et al.Feldman et al. (2013).

Hoping to obtain a better comparison between theory and experiment, we have determined the spin-polarization phase diagram as accurately as possible for an ideal two-dimensional system in the $n=0$ LL. (These results apply to graphene as well as to narrow GaAs quantum wells, because in the limit of zero thickness, the $n=0$ LL wave functions are identical for the two.) We perform large scale exact diagonalization studies of various FQHE states for this purpose. This also allows us to determine the quantitative accuracy of the CF theoryJain (1989, 2007); Lopez and Fradkin (1991); Halperin et al. (1993). We also estimate the effect of LL mixing by modifying the pseudopotentials according to Ref. Peterson and Nayak (2014). Our conclusions, briefly, are as follows.

From exact diagonalization calculations, we have obtained essentially the exact theoretical phase diagram of the spin polarization of FQHE states without allowing for LL mixing (see blue crosses in Fig. 1). We find that it is in excellent agreement with the experimental phase diagram in graphene (see dots in Fig. 1). The agreement between theory and experiment becomes significantly worse when LL mixing is included to linear order in the LL mixing parameter $\lambda$ using the pseudopotentials of Ref. Peterson and Nayak, 2014. This suggests that the current experimental systems are outside the linear regime, which is not surprising given that $\lambda$ is typically of order 1 and sometimes much larger (e.g. 2.2 for suspended graphene). This is relevant to the issue of whether the Pfaffian or the anti-Pfaffian stateMoore and Read (1991); Levin et al. (2007); Lee et al. (2007) at $\nu=1/2$ in the second LL is selected by LL mixing, which has received much attention recentlyWójs et al. (2010); Rezayi and Simon (2011); Peterson and Nayak (2014); Zaletel et al. (2015); Pakrouski et al. (2015); Tylan-Tyler and Lyanda-Geller (2015).

The allowed spin polarizations for various FQHE states at $n/(2n\pm 1)$ and their energy ordering are correctly predicted by the CF theory. The mean-field model in which composite fermions are treated as free particles at an effective magnetic fieldHalperin et al. (1993); Park and Jain (1998); Jain (2007) remains satisfactory, and we obtain a precise estimation for the CF mass. We also carry out quantitative tests of the CF theory, which are extremely precise because the critical Zeeman energies depend on very small energy differences between different states, and even a slight error in the energy can lead to large corrections in the critical Zeeman energies. For the fractions $\nu=n/(2n+1)$, calculations based on Jain’s wave functionsJain (1989) predict the critical Zeeman energies with 15% accuracy, which can be further improved by the method of CF diagonalization that incorporates $\Lambda$ level ($\Lambda$L) mixing, where a $\Lambda$L refers to a CF Landau level in an effective magnetic field. For the states at $\nu=n/(2n-1)$, the results depend sensitively on the lowest Landau level (LLL) projection method. In particular, the treatment with Jain-Kamilla (JK) projectionJain and Kamilla (1997a, b); Jain (2007) underestimates the critical Zeeman energies by a factor of 2-3 (i.e., relatively overestimates the energies of the non-fully spin polarized states). This is not an intrinsic deficiency of the CF theory but rather a technical issue, as can be seen from the fact that the “hard-core projection” introduced previously in Ref. Wu et al., 1993 produces very accurate states for $\nu=n/(2n-1)$. (This projection is presently not amenable to calculations for large systems.)

For completeness, we have also considered the states of composite fermions with four vortices attached. A previous predictionPark and Jain (1999) of the absence of spin transitions for the $n/(4n+1)$ states is confirmed by more rigorous calculations. Here the states remain spontaneously spin polarized even in the absence of a Zeeman energy. Spin transitions are, however, possible for the ⁴CF FQHE state at $\nu=n/(4n-1)$, and we estimate the critical Zeeman energies for the prominent transition.

The physics of graphene differs from that of GaAs in two important aspects: the dispersion of electrons is linear and there are four Dirac cones. The linear dispersion of Dirac fermions leads to LLs which have a cyclotron energy of sgn$(n)\sqrt{2|n|}\hbar v_{F}/\ell$, where $v_{F}$ is the Fermi velocity and $\ell$ is the magnetic length and $n$ is any integerCastro Neto et al. (2009); Goerbig (2011). (This is to be contrasted from GaAs or other conventional semiconductors wherein electrons have a parabolic dispersion and LLs which have a cyclotron energy given by $\hbar\omega_{c}(n+1/2)$, where $n$ is a non-negative integer and $\omega_{c}$ is the cyclotron energy. ) Also, the LL wave functions are two component wave functions, corresponding to two sublattices of graphene. In general this leads to different interaction pseudopotentialsApalkov and Chakraborty (2006); Nomura and MacDonald (2006); Tőke et al. (2006) than in systems with parabolic dispersion. However, it turns out that the electron wave functions as well as the interaction pseudopotentials in the $n=0$ LL of graphene are identical to those in the LLL of GaAs.

The second difference is that each LL of graphene has four bands, which arise from the valley and spin degrees of freedom, while GaAs has two bands from the spin degree of freedom. Within each band, the physics of FQHE in the $n=0$ LL is identical in the two systems, apart from corrections due to finite width and LL mixing. Much work has been done toward understanding the origin of the lifting of various degeneraciesMorpurgo and Guinea (2006); Aleiner et al. (2007); Semenoff (2012); Roy et al. (2014), but we will assume below that all bands are well separated. This assumption is justified for the experiments with which we compare our results.

We refer the reader interested in further information to an extensive literature on the physics of Dirac fermions in a magnetic fieldCastro Neto et al. (2009); Goerbig (2011).

A good qualitative and semiquantitative theoretical understanding of these transitions has been obtained in terms of integer or fractional quantum Hall effect of spinful composite fermions Wu et al. (1993); Park and Jain (1998); Park et al. (1998); Park and Jain (1999, 2001); Chang et al. (2003); Murthy and Shankar (2003); Wójs et al. (2007); Davenport and Simon (2012); Mukherjee et al. (2014); Balram et al. (2015a, b), which successfully predicts the allowed spin polarizations at all of these filling factors and also provides an estimate of the the critical Zeeman energy where transitions between them occur. While these quantitative estimates are a good zeroth order approximation, their accuracy has not been carefully evaluated in the past.

The FQHE state at $\nu=n/(2pn\pm 1)$ maps into integer quantum Hall effect (IQHE) state of composite fermions with $n$ filled $\Lambda$ levels, where a composite fermion is defined as a bound state of an electron and $2p$ vortices. For spinful composite fermions, the CF filling is written as $n=n_{\uparrow}+n_{\downarrow}$, where $n_{\uparrow}$ and $n_{\downarrow}$ are the number of filled spin-up and spin-down $\Lambda$Ls. The different states will be denoted as $(n_{\uparrow},n_{\downarrow})$, and we will use the convention $n_{\uparrow}\geq n_{\downarrow}$ without loss of generality. One can list all possible states and their spin polarization $\gamma=(n_{\uparrow}-n_{\downarrow})/(n_{\uparrow}+n_{\downarrow})$. For example, for 2/5 and 2/3, which both map into $n=2$ of composite fermions, we have a fully spin polarized state $(2,0)$ (with $\gamma=1$) and a spin singlet state $(1,1)$ (with $\gamma=0$). To take another example, 6/13 and 6/11 map into $n=6$, where we have four possible states $(6,0)$, $(5,1)$, $(4,2)$ and $(3,3)$, with $\gamma=1$, 2/3, 1/3, and 0, respectively. One expects one transition at 2/5 and 2/3 and three at 6/13 and 6/11. The possible states and spin polarizations of other fractions can be similarly enumerated.

The CF theory also identifies the flux values where these states occur in the spherical geometry Jain (2007). These are the flux values at which our calculations below are carried out. All our calculations are performed in the spherical geometry, where the ground states have total orbital angular momentum $L=0$ and a total spin $S$ that can be ascertained by the CF theory. We will assume the ideal limit of zero thickness and neglect disorder. In quoting the energies below, we include the electron-background and background-background interaction. The density for a finite system (in the spherical geometry) depends on the number of particles $N$ and is slightly different from its thermodynamic limit. To eliminate this effect we use the density corrected energy Morf and Halperin (1986) $E_{N}^{{}^{\prime}}=(2Q\nu/N)^{1/2}E_{N}$ for extrapolation to the thermodynamic limit $N^{-1}\rightarrow 0$, where the integer $2Q$ is the magnetic flux (in units of $\phi_{0}=hc/e$) to which the electrons are subjected. All energies quoted below are the thermodynamic limits of the per particle density corrected energies $\text{lim}_{N\rightarrow\infty}E_{N}^{{}^{\prime}}/N$.

To avoid clutter, we give only give the spin-polarization phase diagram in the main text. All of the individual numbers as well as extrapolations are given in various tables and figures in Appendix A.

Fig. 3 shows a comparison between the critical values of $\kappa$ obtained from exact diagonalization (blue crosses) with the measured ones. The results for graphene are taken from Feldman et al.Feldman et al. (2013). The value of $\kappa$ depends on the product $\epsilon g$ where $\epsilon$ is the dielectric function and $g$ is the Landé g factor, for which we have taken the values $\epsilon=3.0$ and $g=2.0$ for graphene in the figure. An excellent agreement between the measured and the theoretical values is evident. We stress, however, that we have not yet included effects of LL mixing, which is discussed below.

For completeness, Fig. 3 also shows critical Zeeman energies measured in other systems, taken from Padmanabhan et al. (AlAs quantum well) Padmanabhan et al. (2009) and Du et. al Du et al. (1995) (GaAs-AlGaAs heterojunction). In all cases, the critical values of $\kappa$ as well as their filling factor dependences are roughly consistent with theory. The difference between the critical values of $\kappa$ in graphene and in heterojunction samples is somewhat surprising, because the heterojunction samples also correspond to a very small thicknesses. We believe that part of the difference might result from the fact that Feldman et al. varied $\kappa$ by changing the density, whereas Du et al. Du et al. (1995) accomplished that by tilting the magnetic field. We believe that high parallel fields in the latter experiments may cause additional corrections which have not been considered.

It was predicted in Ref. Park and Jain, 1999 that the spin physics of the FQHE states at $n/(4n+1)$, described in terms of composite fermions carrying four vortices (⁴CFs), is qualitatively different from that at $n/(2n\pm 1)$. Calculations based on the JK wave functions indicated that the fully spin polarized state $(n,0)$ at $n/(4n+1)$ is the ground state even at zero Zeeman energy, and consequently there are no spin-polarization phase transitions. The failure of the free-CF model was interpreted in Ref. Park and Jain, 1999 in terms of a Bloch ferromagnetism for composite fermions, caused by a large exchange interaction that dominates their CF-cyclotron energy and favors the fully polarized state even in the absence of the Zeeman energy.

We have seen above that the JK projection overestimates the energies of the non-fully spin polarized state by a larger amount than of the fully spin polarized states. One may therefore ask if the result in Ref. Park and Jain, 1999 is an artifact of the JK projection scheme. Furthermore, the spin physics at $n/(4n-1)$ has not been investigated so far. Are these states also always fully spin polarized?

We have investigated these questions both by exact and CF diagonalizations for some of these fractions. For 2/9 FQHE state, exact diagonalization results are inconlusive but both the JK wave function and CF diagonalization (see Fig. 5) find that the fully spin polarized state has lower energy, thus confirming absence of any spin-polarization phase transition. The same is true for the $3/13$ and $4/17$ FQHE states wherein both the JK wave function and CF diagonalization find the state with larger spin polarization to be lower in energy than the state with a smaller spin polarization (see Fig. 5). We have also studied the first meaningful member of the Jain sequence $n/(4n-1)$, namely 2/7. Here, the calculations based on the JK wave functions as well as exact diagonalization show that the spin singlet state has a slightly lower energy (see Fig. 5), giving $\kappa_{2/7}=0.0013(3)$ (exact).

For the ^2pCF states with $p>1$, one can also perform the JK projection slightly differently as:

Because these wave functions apparently build better short distance correlations, one may expect them to have lower energies than their counterparts in Eqs. (8) and (9). Contrary to this expectation we find that at $\nu=n/(2pn+1)$ the wave functions in Eq. 13 in general have slightly higher energies for finite systems than those in Eqs. 8, as seen in Fig. 6. (We note that for states restricted to two $\Lambda$ levels, the wave functions of Eq. 13 and Eq. 8 are identical.). In contrast, for states at $\nu=n/(2pn-1)$ we find that the ground state energies obtained from the wave functions in Eq. 14 are lower than those obtained from Eq. 9 (see Fig. 7). Therefore, for states at $\nu=n/(2pn+1)$ with $p>1$ we quote energies obtained from Eqs. (8) while for states at $\nu=n/(2pn-1)$ we quote energies obtained from Eq. 14.

The critical Zeeman energies where transitions between differently spin polarized states occur are a direct measure of the energy difference between the states, and thus serve as a very sensitive test of the quantitative accuracy of the theory of the FQHE. These critical Zeeman energies are in general subject to corrections due to finite thickness of the quantum well and also Landau level mixing. In GaAs both effects are present, and it is not straightforward to disentangle their contributions, although progress has been made in experiments that study the effect systematically as a function of the quantum well width Liu et al. (2014a). Because graphene has negligible finite thickness corrections, this gives an opportunity to obtain an accurate test of the CF theory, and also to gain insight into our understanding of the role of LL mixing Peterson and Nayak (2014).

We have evaluated accurate spin-polarization phase diagram for the FQHE for an ideal two-dimensional system confined to the LLL with no LL mixing and no disorder. We have also evaluated corrections due to LL mixing, assuming that these are linear in the parameter $\lambda$. We find that the experimental results of Feldman et al. Feldman et al. (2013) are in excellent agreement with theory that neglects LL mixing. Somewhat unexpectedly, if we include LL mixing in a linear approximation, the agreement becomes significantly worse, indicating that the amount of LL mixing in experiments is too large to be captured by a first order perturbative treatment. These results underscore our lack of a quantitative understanding of the effect of LL mixing on various quantities.

We have shown that the critical Zeeman energies are well captured in terms of an effective mass model of composite fermions. We have shown that the CF states with Jain-Kamilla projection produce the correct energy ordering for these states, and are fairly accurate for the “parallel flux attached” states at $\nu=n/(2n+1)$, producing the critical Zeeman energies with better than $\sim$15% accuracy. In contrast, for the “reverse flux attached” states at $\nu=n/(2n-1)$ the JK projection obtains the correct energy ordering of the states but obtains the critical Zeeman energies that are off by a factor of two to three. For these states, the hard-core projection method is very accurate but difficult to implement for large systems.

The work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0005042 (ACB, JKJ), Hungarian Scientific Research Funds No. K105149 (CT), the Polish NCN grant 2014/14/A/ST3/00654 and the EU Marie Curie Grant PCIG09-GA-2011-294186 (AW). We thank Research Computing and Cyberinfrastructure at Pennsylvania State University (supported in part through instrumentation funded by the National Science Foundation through grant OCI-0821527), the HPC facility at the Budapest University of Technology and Economics and Wrocław Centre for Networking and Supercomputing and Academic Computer Centre CYFRONET, both parts of PL-Grid Infrastructure. Csaba Tőke was supported by the Hungarian Academy of Sciences.