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Formulation of the Relativistic Quantum Hall Effect And Formulation of the Relativistic Quantum Hall Effect and “Parity Anomaly” Kouki Yonaga1, Kazuki Hasebe2, and Naokazu Shibata1 1Department of Physics, Tohoku University, Sendai, 980-8578, Japan 2Sendai National College of Technology, Ayashi, Sendai, 989-3128, Japan (Dated: March 8, 2018) We present a relativistic formulation of the quantum Hall effect on Haldane sphere. An explicit form of the pseudopotential is derived for the relativistic quantum Hall effect with/without mass term. We clarify particular features of the relativistic quantum Hall states with the use of the exact diagonalization study of the pseudopotential Hamiltonian. Physical effects of the mass term to the relativistic quantum Hall states are investigated in detail. The mass term acts as an interpolating parameter between the relativistic and non-relativistic quantum Hall effects. It is pointed out that the mass term unevenly affects the many-body physics of the positive and negative Landau levels as a manifestation of the “parity anomaly”. In particular, we explicitly demonstrate the instability of the Laughlin state of the positive first relativistic Landau level with the reduction of the charge gap. I. INTRODUCTION (a) Massless Dirac matter has attracted considerable attention in condensed matter physics for its novel properties and recent experimental realizations in solid materials. In contrast to normal single-particle excitations in solids, Dirac particles exhibit linear dispersion in a low energy region and continuously vanishing density of states at the charge-neutral point [1]. These features are actually re- alized in graphene [2, 3] and on topological insulator sur- face [4]. Besides, in the presence of a magnetic field, the relativistic quantum Hall effect was observed in graphene (b) Massive [5–7] and also on topological insulator surface recently [8, 9]. One of the most intriguing features of the relativistic quantum Hall effect is the effect of mass term; in the non-relativistic quantum Hall effect, the mass parameter just tunes the Landau level spacing, while in the rela- tivistic quantum Hall effect the mass term is concerned with interesting physics such as the semi-metal to insula- tor transition and the time reversal symmetry breaking of the topological insulators [10]. In experiments, disor- FIG. 1. Schematic of the energy spectrum and Landau level. der and interaction with a substrate in the atomic layer of (a) and (b) show the massless and the massive cases (M > 0), graphene cause the asymmetry in the two sublattices of a respectively. (g represents a monopole charge.) The right honeycomb structure to induce a mass term [11, 12], and figure of (b) shows asymmetry between the positive and neg- magnetic doping in topological insulators yields a mas- ative energy levels due to the absence of −M. (In general, there exists the energy level E = +sgn(g · M) |M| while not sive gap of the surface Dirac cone [13]. Interestingly, in E = −sgn(g · M) |M|.) The original reflection symmetry of the presence of an external magnetic field, the mass term the energy levels with respect to the zero-energy is broken due arXiv:1602.02820v4 [cond-mat.str-el] 29 Jul 2016 brings the physics associated with the “parity anomaly” to the mass term. [14]; in the absence of a magnetic field, the mass term does not change the equivalence between the positive and negative energy levels [15], while in the presence ot a tential of the infinite disk geometry in the previous study magnetic field, asymmetry occurs between the positive [18], we construct an exact form of the pseudopotential and negative energy levels depending on the sign of the based on the relativistic Landau model recently analyzed mass parameter [16] (see Fig.1). by one of the authors [19]. Previous numerical studies on In this paper, we establish a relativistic formulation of fractional quantum Hall states in graphene show the ex- the quantum Hall effect on a two-sphere and perform a istence of a Laughlin state at ν =1/3 even in the n =1 first investigation of the “parity anomaly” in the context Landau level [20, 21] where the charge excitation gap is of the relativistic many-body physics. For concrete calcu- larger than that of the n = 0 Landau level. This sta- lations, the spherical geometry called Haldane sphere [17] bility of the Laughlin state in the n = 1 Landau level is adopted. Instead of using the approximate pseudopo- is a unique feature of the linear dispersion of the Dirac 2 equation. We study how this stability of the Laughlin The degenerate eigenstates of the Landau level are states changes with increase of mass. Based on the ex- act diagonalization, we numerically obtain a many-body n =1, 2, : ··· 1 ground state of the relativistic pseudopotential Hamilto- g− 2 1 Y 1 (θ, φ) nian, and analyze the mass effect to the Laughlin state g j=(g− 2 )+n,m ψ (θ, φ)= 1 , ±λn,m + at ν =1/3 in the n = 1 relativistic Landau level. √2 g 2 iY =( + 1 )+( 1) (θ, φ) ∓ j g 2 n− ,m (2.7a) 1 II. RELATIVISTIC LANDAU PROBLEM ON A g− 2 g Y 1 (θ, φ) j=g− 2 ,m SPHERE n = 0 : ψλ0=0,m(θ, φ)= , (2.7b) 0 ! In this section, we give a brief review of the relativistic g with m = g + 1 + n, g + 3 + n, ,g 1 + n. Y Landau problem on the Haldane’s sphere [19] and discuss − 2 − 2 ··· − 2 j,m its mass deformation. The monopole gauge field is given denote the monopole harmonics [22]: by [22, 23] (2l + 1)(l m)!(l + m)! Y g (θ, φ)=2m − A = g cos θ dφ, (2.1) l,m 4π(l g)!(l + g)! − s − m+g m−g (−m−g,−m+g) where g denotes the monopole charge. (In the following, (1 x)− 2 (1 + x)− 2 P (x) eimφ, we assume that g is positive for simplicity.) The Dirac × − l+m · (2.8) operator on the Haldane sphere can be represented as µ m µ iD = iem γ (∂µ + iωµ iAµ) where em (m =1, 2, (α,β) µ− =6 θ, φ−) denote the zweibein− of two-sphere whose non- where x = cos θ and Pn (x) stand for the Jacobi poly- nomials. zero components are e θ =1/R, e φ =1/(R sin θ) (R is 1 2 The magnetic field does not affect the spectrum sym- the radius of the Haldane sphere) and ω stands for the µ metry between the positive and negative Landau levels spin connection. When we adopt the 2D gamma matrices (2.5), but acts unevenly on the upper and lower com- as (γ1,γ2) = (σ , σ ), the spin connection is expressed as x y ponents of the eigenstates, which can most apparently ω =0,ω = 1 σ , and then the Dirac operator takes the θ φ 2 z be seen from the absence of the lower component of the form of zero-mode (2.7b). Also notice that the components of 1 1 1 ψg (2.7a) consist of the monopole harmonics in dif- iD = i σ ∂ i σ (∂ + i(g σ )cos θ) ±λn,m − 6 − R x θ − R sin θ y φ − 2 z ferent non-relativistic Landau levels, n and n 1, and 1 1 1 carry the same SU(2) index − = i σ (∂ + cot θ) iσ (∂ + ig cos θ), − R x θ 2 − y R sin θ φ (2.2) 1 j = g + n, (2.9) or − 2 g (g+ 1 ) which implies that ψ itself transforms as the SU(2) ð 2 ±λn,m 1 0 i − iD = (g− 1 ) − . (2.3) irreducible representation. Such SU(2) angular momen- − 6 R ið 2 0 ! − + tum operators are given by (g+ 1 ) Here ð 2 are the edth operators [24]: 1 xi ± J = iǫ x (∂ i ) (g σ ) , (2.10) i − ijk j k − Ak − − 2 z r 1 ð(g) = ∂ ig cot θ i ∂ . (2.4) ± θ ∓ ± sin θ φ with For graphene, two components of the Dirac spinor indi- 1 = dx = (g σ )cos θdφ. (2.11) cate sublattice degrees of freedom, while for a topological A Ai i − − 2 z insulator the real spin degrees of freedom of surface elec- tron. The eigenvalues of the Dirac operator (2.2) are Ji (2.10) is formally equivalent to the total angular mo- derived as mentum of the non-relativistic charge-monopole system with the replacement of the monopole charge g to a ma- 1 1 λ = n(n +2g), (n =0, 1, 2, ) (2.5) trix value, g σz. The Dirac operator is a singlet under ± n ±R ··· − 2 the SU(2) transformation generated by Ji, where n correspondsp to the relativistic Landau level in- [J , iD]=0, (2.12) dex. Notice that the spectrum (2.5) exhibits the reflec- i − 6 tion symmetry with respect to the zero energy. Each and so there exist the simultaneous eigenstates (2.7) of Landau level λn accommodates the following degener- acy ± the Dirac operator and the SU(2) Casimir. Each rela- tivistic Landau level thus accommodates the SU(2) de- dλn = d−λn = 2(g + n). (2.6) generacy (2.6), 2j +1=2(g + n). 3 The Dirac operator also respects the chiral symmetry, +3 +2 iD, σz =0, (2.13) +1 {− 6 } nNR = 0 +3 and the spectrum of the Dirac operator is symmetric with +2 respect to the zero eigenvalue (2.5). The non-zero Lan- +1 dau level eigenstates of the same eigenvalue magnitude n = 0 R −1 (2.7a) are related by the chiral transformation: −2 −3 g g ψ = σzψ .
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