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PHYSICAL REVIEW LETTERS

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Topological Phases in the Zeroth Landau Level of Bilayer Graphene 1

Z. Papić1,2,3 and D. A. Abanin2,3

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3 Institute for Quantum Computing, Waterloo, Ontario N2L 3G1, Canada (Received 15 July 2013; published 28 January 2014)

We analyze the phase diagram of the zeroth Landau level of bilayer graphene, taking into account the realistic effects of screening of the Coulomb interaction and strong mixing between two degenerate sublevels. We identify robust quantum Hall states at filling factors ν ¼ −1, − 43, − 53, − 85, − 12 and discuss the nature of their ground states, collective excitations, and relation to the more familiar states in GaAs using a tractable model. In particular, we present evidence that the ν ¼ − 12 state is non-Abelian and described by either the Moore-Read wave function or its particle-hole conjugate, while ruling out other candidates such as the 331 state. DOI: 10.1103/PhysRevLett.112.046602

PACS numbers: 72.80.Vp, 63.20.Pw, 63.22.-m, 87.10.-e

Introduction.—Following rapid progress in graphene sample quality, the fractional quantum Hall effect (FQHE) was discovered in this material [1–6]. A novel feature of graphene [7] compared to GaAs-based two-dimensional electron gas (2DEG) is the fourfold degeneracy of Landau levels (LLs) due to spin and valley degrees of freedom. The interplay of long-range SU(4)-symmetric Coulomb interactions and smaller symmetry-breaking terms gives rise to an unusual sequence of FQHEs in the zeroth LL, in which certain states are absent or weak, while others exhibit phase transitions as a function of the magnetic field [4,6,8]. It was suggested [9–12] that SU(4) symmetry may give rise to new states not found in GaAs 2DEG and other semiconducting systems (see Ref. [13] for a review). Very recently, an observation of robust FQHE in the zeroth LL of bilayer graphene (BG) was also reported [14]. Remarkably, both odd-denominator (filling factor ν ¼ − 43) and even-denominator (ν ¼ − 12) fractions were observed. In contrast, monolayer graphene (MG) has so far exhibited only odd-denominator FQHE. What is the nature of the quantum Hall states expected in BG? In particular, is the observed half-integer state an Abelian or a non-Abelian state? In this Letter, partly motivated by experiment [14], we provide an insight into the nature of various topological phases arising in the zeroth Landau level of BG. In contrast to previous work [15–19], our study fully takes into account the realistic effects of screening of the Coulomb interaction in BG and LL mixing between the degenerate sublevels. Using exact diagonalization, we identify robust quantum Hall states at filling factors ν ¼ −1, − 43, − 53, − 85, − 12. Furthermore, we introduce a simple model obtained by truncating the Coulomb interaction and show that it is a helpful guide in understanding the nature of the incompressible ground states and their collective excitations. In particular, we find that odd-denominator states are related to

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the Laughlin state [20] and the unprojected composite fermion states [21,22]. Furthermore, we present evidence that ν ¼ − 12 is a non-Abelian state, described by the MooreRead wave function [23] or its particle-hole conjugate [24], and rule out the 331 state [25] from possible candidates. Model.—The effective low-energy Hamiltonian of BG in a magnetic field near K point is given by [26] HK ¼ ð1=4mÞðσ þ p~ 2þ þ σ − p~ 2− Þ, in terms of Pauli matrices σ  ¼ σ x  iσ y and canonical momentum p~  ¼ p~ x  ip~ y , p~ ¼ p þ eA. Effective mass derives from the interlayer tunneling amplitude and the velocity of Dirac excitations in MG [26]. Near the K 0 point we have HK0 ¼ HK . The LL spectrum of HK contains two zero-energy levels (0, j0; mi), (0, j1; mi), where jn; mi denotes the wave function in the nth nonrelativistic LL with angular momentum m. These orbitals will be referred to as 0,1 below. We mostly focus on the vicinity of the filling factor ν ¼ 0, where two pairs of 0,1 orbitals with the same valley and spin index are filled [27,28]. In this regime, spin and valley SU(4) symmetry of Coulomb interactions is broken down by the short-range valley-anisotropic terms and Zeeman interactions [8]. Recent experimental work [29] implies that the symmetry breaking is quite strong. Thus, fractional states in the interval jνj < 2 are likely spin and valley polarized, and it is sufficient to restrict the Hamiltonian to the 0,1 LL orbitals with fixed spin and valley index. Interactions.—The key challenge for the theoretical analysis of FQHE in the zeroth LL of BG lies in the mentioned “orbital” degeneracy [26,30]. This leads to strong LL mixing effects beyond perturbation theory [31] and makes direct numerical approaches difficult. Because of this challenge, previous work [15–19] was limited to FQHE in the nonzero LLs of BG. Another feature of BG that distinguishes it from both MG and the nonrelativistic 2DEG is the screening of the Coulomb interactions due to virtual excitations of

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PHYSICAL REVIEW LETTERS

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electron-hole pairs between filled and empty n ≠ 0 LLs [32,33]. In momentum space, the screened interaction VðkÞ is well described by [33] p VðkÞ ¼ V 0 ðkÞ= ffiffiffiffiffiffiffiffiffiffiffi ½1 þ a tanhðbk2 l2B Þ=ðklB Þ, where lB ¼ ℏ=eB is the magnetic length, V 0 ðkÞ ¼ ð2πe2 Þ=ðκkÞ is the bare Coulomb interaction, κ is the screening constant due to the substrate, and parameter a is related to the ratio of Coulomb and cyclotron energy [33]. Cyclotron energy is small in BG (∼2.16 ½meVB½T); thus, a becomes large for magnetic fields B ∼ 10 T, and screening strongly alters the effective Coulomb interactions, as we show below. Without loss of generality, we fix b ¼ 12 and vary a. Note that there are also higher-order contributions to screening (3-body terms, etc.), which we neglect here. To quantify the effect of screening, we project the Coulomb interaction into the space of 0,1 orbitals with a given spin or valley polarization. Within this subspace, due to the extra degree of freedom, the Haldane pseudopotentials [34,35] V m for pairs of particles with a relative angular momentum m must be generalized into 16 different types, Z

fn g

pffiffiffi d2 k −k2 e VðkÞFnn31 ðkÞFnn42 ð−kÞFm m0 ð 2kÞ: (1) 2 ð2πÞ

Vm i ¼

Here ni ¼ 0, 1 denote indices of LL orbitals involved in the scattering process defined by the momentum conser0 vation m0 ¼ m þ ðn3 þ n4 Þ − ðn1 þ n2 Þ. Form factors Fnn can be expressed in terms of the generalized Laguerre polynomials [36]. fn g The first few pseudopotentials V m i are evaluated in fn g Fig. 1. Although there are linear relations [37] between V m i , the case of BG is clearly much more difficult than that of GaAs where retaining only V nnnn is an excellent approxim mation in the nth LL. Furthermore, for unscreened Coulomb 1

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FIG. 1 (color online). Generalized Haldane pseudopotentials for the zeroth LL of BG, for unscreened Coulomb interaction (left) and for screening a ¼ 6 (right). Values of the pseudopotentials are quoted relative to V 0000 . Four types of “diagonal” 0 pseudopotentials (red, blue, and green) define the truncated model (2), which becomes accurate for large a. Off-diagonal pseudopotentials are shown in gray.

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interactions (Fig. 1, left), different types of V m i are of comparable magnitude, leading to strong LL mixing effects whose influence on the many-body states is generally not understood. Interestingly, in the case of BG where Coulomb interaction is strongly screened, we find an emergent hierarchy of different types of interaction terms (Fig. 1, right). For large a, the diagonal pseudopotentials (i.e., those with n1 ¼ n3 , n2 ¼ n4 ) become dominant, while the off-diagonal ones are strongly suppressed. This can be understood from the real-space structure of the potential VðrÞ ∝ lnða=rÞ, which is quite flat in the important interval of distances 1 ≪ r=lB ≪ a. Moreover, for constant VðrÞ, the off-diagonal pseudopotentials identically vanish. Thus, it is natural that flattening of the potential reduces the importance of mixing, as Fig. 1 illustrates. Motivated by these considerations, we introduce the following decomposition of the many-body Hamiltonian: Hfull ¼ H0000 þ H1111 þ H1010 þ H0101 þ λH rest ;

(2)

where λ ∈ ½0; 1 and Hfni g denotes part of the full fni g Hamiltonian containing the terms V m¼0;1;… . Point λ ¼ 0 defines what we refer to as the “truncated model,” which is shown below to be useful for developing an intuitive understanding of the physics in the large-screening limit. The truncated model would be SU(2) invariant if 0 and 1 LL orbitals were identical. Because the form factors of 0 and 1 LL are different, SU(2) symmetry is slightly broken. In the case of gapped phases, we generally find (see the Supplemental Material [38]) that the ground states adiabatically evolve as λ is reduced from unity down to zero. We can, thus, use the truncated model to obtain a simple picture of the ground state when quantum fluctuations arising from the mixing terms, present in Hrest , are frozen out. Results.—In the following, we use exact diagonalization to study the many-body problem defined by Eq. (2), including all types of mixing terms between 0 and 1 LLs. We consider N electrons in an area enclosing N Φ flux quanta, assuming fully periodic boundary conditions (torus) [39,40]. In this case, the BG filling factor is simply defined as ν ¼ −2 þ N=N Φ . Because of orbital degeneracy in BG and asymmetry between 0 and 1 LL, the electronhole symmetry is broken and substituted by ν → ν þ 2 symmetry. In addition to the torus, we also consider spherical geometry [34], to verify the insensitivity of our conclusions to the particular type of boundary condition. Similar to Ref. [39], in Fig. 2 we first compute the ground-state energy per particle E0 =N, as a function of ν. Cusps in E0 =N indicate discontinuities in the chemical potential leading to gaps for creating charged excitations. This represents unbiased evidence for incompressibility that does not depend, e.g., on the choice of a particular trial wave function. Our analysis shows that incompressibility

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FIG. 2 (color online). Ground-state energy per particle as a function of the filling factor in the range −2 < ν < −1, for the full Coulomb interaction (left) and the truncated model (right). Vertical lines denote the robust filling factors ν ¼ −1, − 43, − 85, − 53, while weaker features are also observed at ν ¼ − 32, − 75. Data shown are for N ¼ 8–12 particles and screening strength a ¼ 6.

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arises at ν ¼ −1, which are expected to develop into stable fractions in experiment. Some features in the data are also observed at ν ¼ − 32, − 75, but they are sensitive to system size and less likely to lead to develop into robust states. As advertised above, the truncated model (Fig. 2, right) displays the same cusps as the full Coulomb Hamiltonian (Fig. 2, left), but easier to discern in our finite systems. To avoid clutter, in Fig. 2 we have omitted the onebody Madelung correction [39], which vertically displaces the curves corresponding to different values of N but does not affect the existence of the cusps. The strongest state in Fig. 2 is clearly ν ¼ −1. Previous work [27] has suggested that this state might have unique features that distinguish it from the states found in spinful and bilayer 2DEGs, such as the gapless neutral mode dispersing as k3=2 . Quite surprisingly, our calculations show that this state is fully gapped in both charge and neutral sectors. A strong cusp in Fig. 2 is an indication of a robust charge gap, which is confirmed by finite-size scaling (see the Supplemental Material [38]) that yields the gap of Ec ≈ 0.06e2 =ϵlB . Similarly, the neutral mode shown in Fig. 3(a) remains gapped in the thermodynamic limit; see Fig. 3(c). The reason for discrepancy with the mean-field theory [27] might lie in the lack of complete polarization of the ground state in n ¼ 0 LL. As shown in Fig. 3(b), roughly 1 out of 10 particles is promoted from 0 into 1 LL in the thermodynamic limit. Once again, the truncated model in Eq. (2) is particularly useful for understanding the physics at ν ¼ −1; see Fig. 3, inset. In the model defined by Eq. (2), the ground state is indeed a single Slater determinant corresponding to the fully filled n ¼ 0 LL. The branch of low-lying excitations in Fig. 3 (inset) is also given by single Slater determinants formed by promoting a single particle from n ¼ 0 into n ¼ 1 and boosting it with the momentum k. When quantum fluctuations are turned on,

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FIG. 3 (color online). Gapped neutral mode of the ν ¼ −1 state. Energy spectrum as a function of momentum for the full Coulomb interaction with a ¼ 6 (a) and the truncated model (inset). Finite scaling of the average population of n ¼ 0 LL (b) suggests that the ground state has a finite component (∼10%) in n ¼ 1 LL. Scaling of the neutral gap is shown in (c). Data are for system sizes N ¼ 12−16 particles.

these states become dressed with further excitations into n ¼ 1 LL; e.g., the ground state begins to acquire configurations with two particles in n ¼ 1 LL, the first excitation in the k ¼ 0 sector will contain also three particles in n ¼ 1 LL, etc. The weight of such configurations can be computed in perturbation theory and will be presented elsewhere [37]. In the remainder of this Letter, we focus on the strongest fractional states ν ¼ − 53, − 43 and discuss in detail the intriguing case of half-integer fractions, ν ¼ − 32, − 12. To understand the nature of these fractions in an unbiased way, it is instructive to first determine P the average population of each of the levels, N n¼0;1 ≡ h m c†m;n cm;n i, in these manybody ground states. In Fig. 4 we plot N n per particle as a function of system size for ν ¼ − 32, − 12 as well as ν ¼ − 43, − 53 (inset). Whereas the available system sizes are not sufficient to draw definite

FIG. 4 (color online). Scaling of average populations N νn of the levels n ¼ 0, 1 (per particle) as a function of system size, for filling factors ν ¼ − 32, − 12 and ν ¼ − 43, − 43 (inset). Data are consistent with ν ¼ − 32, − 43, − 53 being fully polarized in n ¼ 0 LL (N 0 =N → 1, N 1 =N → 0). In contrast, at ν ¼ − 12 the populations converge to n ¼ 0 LL being completely filled and n ¼ 1 LL being half filled, i.e., N 0 =N → 2=3, N 1 =N → 1=3 (dashed lines).

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conclusions, we observe strong indications that at ν ¼ − 32, − 43, − 53 nearly all particles reside in n ¼ 0 LL in the thermodynamic limit, i.e., N 0 =N → 1, N 1 =N → 0. This is further corroborated in the truncated model, where the ground state to be fully polarized in n ¼ 0 LL for any finite system (N 0 =N ¼ 1). Since the ground state at ν ¼ − 43, − 53 has almost all the weight in 0 LL, one might expect it to be related to the usual Laughlin state. Indeed, the projection of these ground states to 0 LL has a large overlap with the Laughlin wave function (∼98%) and weakly depends on the screening. Additional evidence that these are Laughlin-like fractions can be found in the truncated model, where the gap does not close upon interpolating from λ ¼ 1 to 0 (Supplemental Material [38]). For the latter value of λ, the ground state experiences no fluctuations (N 0 =N ¼ 1), and it is essentially identical to the Laughlin wave function. Given that ν ¼ − 43 and ν ¼ − 53 ground states are described by the Laughlin state with smaller corrections due to mixing with 1 LL, are the quasiparticles of these two states also similar? In the truncated limit λ ¼ 0, the first excited state in these two cases is also fully polarized in 0 LL; hence, ν ¼ − 43 and ν ¼ − 53 should have identical gaps because full 0 LL polarization restores particle-hole symmetry. In contrast, we find that the gaps for the full Coulomb interaction appear to be 2–3 times greater at ν ¼ − 43 compared to those of ν ¼ − 53, suggesting that LL mixing terms do produce a significant difference in the relative stability of these two fractions in experiment. Finally, we come to the case of half filling. At ν ¼ − 32, 0 LL is nearly completely filled (Fig. 4), and we expect a compressible state related to the composite Fermi liquid [41]. On the other hand, the populations of levels at ν ¼ − 12 (Fig. 4) approach N 0 =N → 2=3 and N 1 =N → 1=3, which suggests that in this case 0 LL is likely completely filled, while 1 LL is half filled and thus could be expected to be described by the Moore-Read Pfaffian state [23]. Assuming that polarizations of 0,1 LLs have saturated (i.e., N 0 =N ¼ 1 for ν ¼ − 32 and N 0 =N ¼ 2=3 for ν ¼ − 12), in Fig. 5 we study the effect of screening on the competition between the Fermi liquid and the Pfaffian state in a single isolated (0 or 1) LL. To this end, we use the spherical geometry [34], where the Pfaffian and Fermi liquid states are distinguished by a special quantum number called the shift [42]. In Fig. 5(a) we show the scaling of the ground-state energy per particle with system size, for Pfaffian or Fermi liquid shifts in both 0 and 1 LL and for the fixed screening a ¼ 4. The extrapolated ground-state energy in 0 LL is lower for the Fermi liquid shift, suggesting that the ground state is compressible. We cannot rule out the possibility that larger screening might drive a compressibleincompressible phase transition, even without involving 1 LL; however, the resulting state is expected to have a small gap in that case.

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FIG. 5 (color online). Screening favors the paired state over the Fermi liquid at ν ¼ − 12. (a) Extrapolated ground-state energies per particle at ν ¼ − 12 and ν ¼ − 32 for the Fermi liquid and Pfaffian shifts the sphere and screening a ¼ 4. At ν ¼ − 12, the Pfaffian shift gives lower energy, while the Fermi liquid wins at ν ¼ − 32. (b) Screening improves the overlap of the ν ¼ − 12 ground state with the Pfaffian wave function.

On the other hand, in 1 LL the situation is reverse, and the ground-state energy is minimized by choosing the Pfaffian shift. For reference, we note that the extrapolated energy difference between the two shifts is comparable to the value in the first-excited LL of GaAs (Supplemental Material [38]). Additionally, the overlap of the ground state with the Moore-Read wave function is significantly enhanced by the screening; Fig. 5(b). Combined with the polarization data in Fig. 4, this evidence altogether points to the non-Abelian physics at ν ¼ − 12. Because of the peculiarity of the half filling, in addition to the Pfaffian, its particle-hole conjugate—the “antiPfaffian” [24]—is also a possible candidate state. These two states are degenerate in the absence of mixing and yield the same overlap with the Coulomb ground state shown in Fig. 5(b). To reliably determine which state is favored, one must include all mixing terms but also higher order (3-body) corrections to the screening, which lies outside the scope of the present Letter. Finally, we note that 331-based wave functions [25] appear to be unlikely candidates for ν ¼ − 12 in BG. The correlations in the 331 wave function enforce N 0 and N 1 to be equal, and a small imbalance between them quickly destroys the state [43,44]. Since N 0 is nearly twice as large as N 1 at ν ¼ − 12 (Fig. 4), the 331 state is ruled out as a viable candidate (see the Supplemental Material [38]). In conclusion, we have presented a method to analyze the effect of screening and strong LL mixing in the zeroth LL of BG. We have identified several robust fractions with Abelian (ν ¼ −1, − 43, − 53, − 85) and non-Abelian (ν ¼ − 12) topological order, some of which have been observed in recent experiments [14]. Future work will address the microscopic characterization of the excitations in these states and ways to further increase their gaps or perhaps stabilize new states using the potential tunability of the interactions in BG [16,17,45].

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We thank B. Feldman, B. Halperin, A. Kou, D.-K. Ki, A. Morpurgo, and A. Yacoby for useful discussions. This work was supported by DOE Grant No. DE-SC0002140.

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Topological phases in the zeroth Landau level of bilayer graphene.

We analyze the phase diagram of the zeroth Landau level of bilayer graphene, taking into account the realistic effects of screening of the Coulomb int...
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