Contents

Relativistic Landau Models and Generation of Fuzzy Spheres

Kazuki Hasebe

Sendai National College of Technology, Ayashi, Sendai, 989-3128, Japan

khasebe@sendai-nct.ac.jp

August 17, 2019

Non-commutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac-Landau operator eigenstates is obtained based on algebraic methods. We establish “gauge” transformation between the relativistic Landau model and the Pauli-Schrödinger non-relativistic quantum mechanics. After the transformation, the Dirac operator and the angular momentum operastors are found to satisfy the algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac-Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres.

## 1 Introduction

Quantization of the space-time is one of the most fundamental problems in physics. Non-commutative geometry is a promising mathematical framework for the description of quantized space-time [1]. While string theory or matrix theory also suggests appearance of non-commutative geometry [2], the natural energy scale of the non-commutative geometry is considered to be the Planck scale. Interestingly, however, it is well recognized that in low energy physics of some real materials, non-commutative geometry naturally emerges. A well known example is the lowest Landau level physics of the quantum Hall effect, where the electron coordinates effectively satisfy non-commutative algebra due to the presence of strong magnetic field [see [3] and references therein]. More precisely, non-commutative geometry appears in any of the Landau levels as well as the lowest Landau level as a consequence of the level projection. Recently, higher dimensional non-commutative geometry has begun to be applied to studies of topological insulators [4, 5, 6, 7, 8, 9, 10].

Usually, non-commutative geometry is imposed on theories of interest in the beginning, and within the mathematical framework we develop physical theories such as non-commutative quantum field theory. On the other hand, in the set-up of Landau models, non-commutative geometry is not postulated a priori but “generated” as a consequence of level projection. In the work, we proactively utilize the level projection as a tool to derive fuzzy geometries. The merits of this scheme are the following. First, the level projection basically yields a consistent framework of non-commutative geometry. Generally it is far from obvious whether non-commutative geometry can be incorporated in any manifolds, for instance, to curved manifolds, keeping mathematical consistency. However, in the level projection scheme, we have a consistent Hilbert space of the quantum mechanics, and the level projection is just a method to extract a specific subspace of the consistent Hilbert space. Since the whole Hilbert space is well defined, we need not to bother with the mathematical inconsistency in introducing the subspace and the corresponding non-commutative geometry as well. Second, the level projection is rather mechanical, and one can readily introduce fuzzy geometry by following simple instructions to construct effective matrix representation in the subspace. Last, since the level projection scheme is based on physical ideas, mathematics of non-commutative geometry can be understood from a physical point of view, as we shall see in this work.

In the first half of this work, we investigate relativistic Landau models described by Dirac-Landau operator on a sphere. (We shall refer to the Dirac operator in magnetic field as Dirac-Landau operator.) We thus exploit a relativistic counterpart of the Haldane’s sphere [11]. Apart from applications to non-commutative geometry, the relativistic Landau models have increasing importance in recent developments of Dirac matter such as graphene and topological insulator [there are many excellent books and reviews: see [12, 13, 14, 15] for instance]. Theoretical works of Dirac matter with Landau levels on a spherical geometry can be found in Refs.[17, 18] for fullerene, Refs.[19, 20] for the surface of topological insulator, and Refs.[4, 16] for higher dimensional topological insulators. Though the Dirac-Landau equation in flat space has already been intensively investigated in various physical and mathematical contexts [21, 22] and on a sphere as well [23, 24], many studies on a sphere are restricted to zero mode solutions. We present a full analysis of the relativistic Landau model on a sphere including all relativistic Landau level eigenstates. Our method is based on an algebraic method, which provides a concise way to solve the Dirac-Landau operator and highlights a transparent rotational symmetry of the present geometry [Sec.3]111The readers may find an analytic method for solving the Dirac-Landau equation in Ref.[25].. We establish transformation between the relativistic Landau model and the Pauli-Schwinger non-relativistic quantum mechanics obtained by Kazama et al. almost forty years ago [26] [Sec.4]. After the transformation, the transformed Dirac operator and the angular momentum operators are shown to satisfy the algebra, which is the “hidden” symmetry of the system. In the second half, we discuss fuzzy geometries generated by the level projection in the relativistic Landau models. In correspondence to each of the relativistic Landau levels, a relativistic fuzzy sphere is derived. We compare behaviors of the relativistic and non-relativistic fuzzy spheres with respect to magnetic field, where particular properties of the relativistic fuzzy spheres are observed [Sec.5]. We also investigate properties of fuzzy spheres under mass deformation [Sec.6]. Interestingly, th relativistic fuzzy spheres for opposite sign Landau levels balance their sizes keeping the sum of their radii invariant. As the square of the Dirac-Landau operator, a supersymmetric quantum mechanics is constructed, where we demonstrate appearance of super fuzzy spheres [Sec.7]. Finally we apply the results to a realistic Dirac material, graphene, to investigate fuzzy geometries with valley degrees of freedom and behaviors under the change of mass parameter [Sec.8]. Sec.2 is a review about the non-relativistic Landau problem and Sec.9 is devoted to summary and discussions.

## 2 Review of the Non-Relativistic Landau Problem

### 2.1 Monopole harmonics

As a preliminary, we give a rather detail review of non-relativistic quantum mechanics for a charge-monopole system mainly based on Refs.[22, 27, 28]. We use the standard spherical coordinates,

 x=rsinθcosϕ,   y=rsinθsinϕ,   z=rcosθ, (1)

and adopt the Schwinger gauge [27]222We utilize terminology, Schwinger , instead of the Schwinger in Ref.[27]. [see Appendix D for the Dirac gauge] in which the monopole gauge field is given by

 A=gϵij3zr(x2+y2)xjdxi=−gcosθdϕ, (2)

or

 Ax=gzr(x2+y2)y=g1rcotθ⋅sinϕ, Ay=−gzr(x2+y2)x=−g1rcotθ⋅cosϕ, Az=0, (3)

where denotes the monopole charge. In this paper, we consider the case . (It is not difficult to expand similar discussions for .) In the Schwinger gauge the gauge field exhibits an infinite line singularity on the -axis, and the direction of the monopole gauge field on the north hemisphere is opposite to that on the south hemisphere (on the equator, the monopole gauge field vanishes)333In the Dirac gauge [see Appendix D], the singularity of the gauge field is a semi-infinite string either on the positive -axis or on the negative -axis, and the directions of the monopole gauge fields are same on both hemispheres.. The corresponding field strength is given by

 F=dA=gsinθ dθ∧dϕ, (4)

or

 Fi=ϵijk∂jAk=g1r3xi. (5)

The covariant derivative is constructed as

 Di=∂i−iAi, (6)

or

 −iDr=−i∂r,    −iDθ=−i∂θ,     −iDϕ=−i∂ϕ+gcosθ, (7)

and the covariant angular momentum is

 Λ(g)i=−iϵijkxjDk, (8)

or

 Λ(g)x=L(0)x−gz2r(x2+y2)x=L(0)x−gcos2θsinθcosϕ, Λ(g)y=L(0)y−gz2r(x2+y2)y=L(0)y−gcos2θsinθsinϕ, Λ(g)z=L(0)z+g1rz=L(0)z+gcosθ. (9)

Here, represent the free orbital angular momentum operators:

 L(0)x=i(sinϕ∂θ+cotθcosϕ∂ϕ), L(0)y=−i(cosϕ∂θ−cotθsinϕ∂ϕ), L(0)z=−i∂ϕ. (10)

The total angular momentum is constructed as the sum of the covariant and the field angular momenta:

 L(g)=Λ(g)−r2F=Λ(g)−g1rx, (11)

or

 L(g)x=i(sinϕDθ+cosϕcotθDϕ)−gxr, L(g)y=−i(cosϕDθ−sinϕcotθDϕ)−gyr, L(g)z=−iDϕ−gzr. (12)

With use of (10), they are expressed as

 L(g)x=L(0)x−grx2+y2x=L(0)x−gcosϕsinθ, L(g)y=L(0)y−grx2+y2y=L(0)y−gsinϕsinθ, L(g)z=L(0)z. (13)

The square of can be represented as

 L(g)2 =−1sinθ∂∂θ(sinθ∂∂θ)−1sin2θ(∂∂ϕ+igcosθ)2+g2 =L(0)2−2igcosθsin2θ∂∂ϕ+g21sin2θ =L(0)2+2grzx2+y2L(0)z+g2r2x2+y2, (14)

where

 L(0)2=−1sinθ∂θ(sinθ∂θ)−1sin2θ∂ϕ2. (15)

The monopole harmonics are introduced as the simultaneous eigenstates of and :

 L(g)2Ygl,m(θ,ϕ)=l(l+1)Ygl,m(θ,ϕ), L(g)zYgl,m(θ,ϕ)=mYgl,m(θ,ϕ), (16)

where and take the following values [28]:

 l=g+n       (n=0,1,2,⋯), (17a) m=−l,−l+1,⋯,l−1,l. (17b)

The ladder operators are given by

 L(g)+=L(g)x+iL(g)y=eiϕ(∂θ+icotθ∂ϕ−g1sinθ), L(g)−=L(g)x−iL(g)y=e−iϕ(−∂θ+icotθ∂ϕ−g1sinθ), (18)

which act to the monopole harmonics as

 L(g)±Ygl,m=√(l∓m)(l±m+1)Ygl,m±1. (19)

The irreducible representation of the monopole harmonics can be obtained by applying the ladder operators to the lowest or highest weight state. The monopole harmonics are explicitly given by [28, 27]

 Yg l,m(θ,ϕ) =2m√(2l+1)(l−m)!(l+m)!4π(l−g)!(l+g)!(1−x)−m+g2(1+x)−m−g2P(−m−g,−m+g)l+m(x)⋅eimϕ =√(2l+1)(l−m)!(l+m)!4π(l−g)!(l+g)!(sinθ2)−(m+g)(cosθ2)−(m−g)P(−m−g,−m+g)l+m(cosθ)⋅eimϕ, (20)

where denote the Jacobi polynomials [Appendix A]. For uniqueness of the wavefunction, the magnetic quantum number of the azimuthal part of (20) has to take an integer value, . Due to (17b), the monopole charge should be quantized as an integer in the Schwinger gauge [27]. Expressing the Jacobi polynomials by the trigonometric function, (20) can be rewritten as [29]

 Ygl,m(θ,ϕ) =(−1)l+m√(2l+1)4π(l+m)! (l−m)!(l+g)! (l−g)! eimϕ ⋅∑n(−1)n(l−gn)(l+gg−m+n)(sinθ2)2l−2n−g+m(cosθ2)2n+g−m, (21)

or

 Ygl,m(θ,ϕ)= (−1)l+m√(2l+1)4π(l+m)! (l−m)!(l+g)! (l−g)! ⋅∑n(l−gn)(l+gg−m+n) (−1)n ug−m+n vl−n+m u∗n v∗l−g−n, (22)

where and are the components of the Hopf spinor [3]:

 u=cosθ2e−i12ϕ,   v=sinθ2ei12ϕ, (23)

and and are their complex conjugates. For instance, in the case and , we have

 Y12,2=−√5πsin3θ2cosθ2e2iϕ,  Y12,1=12√5π(1+2cosθ)sin2θ2eiϕ, Y12,0=−12√152πsinθcosθ,  Y12,−1=12√5π(−1+2cosθ)cos2θ2e−iϕ, Y12,−2=√5πsinθ2cos3θ2e−2iϕ. (24)

The non-relativistic Landau Hamiltonian in a monopole background is given by [11]

 H =−12M3∑i=1Di2 =−12M(1r2Dr(r2Dr)+1r2sinθDθ(sinθDθ)+1r2sin2θDϕ2) =−12M∂2∂r2−1Mr∂∂r+12Mr2Λ(g)2, (25)

which, on a sphere , reduces to

 H(g)=−12M(1sinθDθ(sinθDθ)+1sin2θDϕ2)=12MΛ(g)2=12M(L(g)2−g2). (26)

In the following, we take . (We sometimes recover to indicate the dimensions of quantities of interest.) Since we have already solved the eigenvalue problem of , the eigenvalues of (26) can readily be obtained as

 E(g)n=12M(n(n+1)+g(2n+1)), (27)

where we used (17a), and the degenerate eigenstates of the th Landau level are the monopole harmonics (20) with degeneracy,

 2l+1=2g+1+2n. (28)

In the lowest Landau level 444For , the monopole harmonics in the lowest Landau level () are given by

(29)
, the monopole harmonics are represented as

 Ygg,m(θ,ϕ) =(−1)m+g√(2g+1)!4π(g+m)!(g−m)!(sinθ2)m+g(cosθ2)−m+geimϕ =(−1)m+g√(2g+1)!4π(g+m)!(g−m)!ug−mvg+m. (30)

The lowest Landau level eigenstates are homogeneous holomorphic polynomials of the Hopf spinor.

### 2.2 Edth operators

The monopole harmonics carry two spin indices, and . (With fixed , both and range from to .555This is the basic observation about the equivalence between the monopole harmonics and spin-weighted spherical harmonics [31, 32].) One may expect that ladder operators for may exist just like the ladder operators, , for . Such operators are known as the edth differential operators [30]666 and respectively correspond to and in Refs.[30, 31, 32]:

 ð(g)+≡(sinθ)g(∂θ+i1sinθ∂ϕ)(sinθ)−g=∂θ−gcotθ+i1sinθ∂ϕ ð(g)−≡(sinθ)−g(∂θ−i1sinθ∂ϕ)(sinθ)g=∂θ+gcotθ−i1sinθ∂ϕ, (31)

or

 ð(g)+=Dθ+i1sinθDϕ, ð(g)−=Dθ−i1sinθDϕ, (32)

where and are the covariant derivatives (7). The edth operators indeed act to the monopole harmonics as [31, 32] 777 In the Cartesian coordinates, the edth operators are represented as

(33)
or, with use of the angular momentum operators (10) and (9),
(34)

 ð(g)+Ygl,m(θ,ϕ)=√(l−g)(l+g+1) Yg+1l,m(θ,ϕ), ð(g)−Ygl,m(θ,ϕ)=−√(l+g)(l−g+1) Yg−1l,m(θ,ϕ). (35)

Notice that, while and respectively increases and decreases the monopole charge by , they are inert with the index (and the magnetic quantum number ). Therefore, in the language of Landau level , the edth operators act as the ladder operators of the Landau levels. In more detail, since / acts as the raising/lowering operator for the monopole charge, as for the Landau levels, / plays the opposite; lowering/raising operator for the Landau level. This implies that the edth operators are the covariant derivatives on a sphere in monopole magnetic field.

From (31), we obtain

 ð(g−1)+ð(g)−−ð(g+1)−ð(g)+=−2g, (36a) and ð(g−1)+ð(g)−+ð(g+1)−ð(g)+=−2(L(g)2−g2). (36b)

These relations are essentially the same as of the ladder operators (in the diagonalized basis) with replacement of with :

 L(g)+L(g)−−L(g)−L(g)+=2m, (37a) and L(g)+L(g)−+L(g)−L(g)+=2(L(g)2−m2). (37b)

From the point of view of three-sphere, the analogies between the edth operators and the angular momentum operators are clearly understood [Appendix B]. The edth and angular momentum operators are mutually commutative:

 L(g+1)ð(g)+−ð(g)+L(g)=0,     L(g−1)ð(g)−−ð(g)−L(g)=0. (38)

In other words, the edth operators are singlet under the angular momentum transformations. Due to the relation (36b), the Landau Hamiltonian (26) can be expressed as888 Alternatively using (32), one may explicitly verify

(39)

 H(g) =−14M(ð(g−1)+ð(g)−+ð(g+1)−ð(g)+) =−12Mð(g+1)−ð(g)++g2M. (40)

Eq.(38) implies that the Hamiltonian (40) is invariant under the rotations:

 [H(g),L(g)]=0. (41)

It is straightforward to confirm that is the eigenstate of the Hamiltonian (40) with the eigenvalues (27) with use of (35). One may find analogies between (40) and the Landau Hamiltonian on a plane, with :

 Hplane =−14M(D¯D+¯DD) =−12M¯DD+B2M, (42)

where and . The covariant derivatives satisfy

 [D,¯D]=−2i[Dx,Dy]=−2B, (43)

which corresponds to (36a). Also from these relations, the edth operators turn out to play the covariant derivatives of the Landau model on the sphere. Furthermore, the center-of-mass coordinates, and , or the magnetic translation operators which commute with the covariant derivatives correspond to the angular momentum operator on the sphere. Then the correspondences between the plane and sphere cases are summarized as

 D,¯D  ⟷  ð+,ð−, X,Y  ⟷  Lx,Ly,Lz. (44)

## 3 Relativistic Landau Problem on a Sphere

### 3.1 Spin connection and the Su(2) angular momentum operator

From the metric on a two-sphere

 ds2=dθ2+sin2θdϕ2, (45)

zweibein can be adopted as [see Appendix C for details]

 e1=dθ,   e2=sinθdϕ. (46)

The torsion free condition, , determines the spin connection:

 ω12(=−ω21)=−cosθdϕ. (47)

We choose the gamma matrices and generator as

 γ1=σx,   γ2=σy, σ12=−σ21=−i14[γa,γb]=12σz, (48)

to have matrix valued spin connection

 ω=12ωabσab=−12σzcosθ dϕ. (49)

Notice that (49) coincides with the monopole gauge field (2) with . This is because that the holonomy of the base-manifold is isomorphic to the gauge group of the monopole. Consequently, the spin connection effectively modifies the monopole charge by depending on up and down-components of the spinor. The components of the Dirac-Landau operator are given by

 −iDμ=−i∂μ+ωμ⊗1−1⊗Aμ=−i∂μ−Aμ, (50)

where denotes a matrix valued gauge field:

 A=−gscosθdϕ, (51)

with

 gs≡1⊗g−12σz⊗1=g−12σz. (52)

(50) is thus obtained as

 −iDθ=−i∂θ,    −iDϕ=−i∂ϕ+gscosθ. (53)

It is straightforward to expand similar discussions to Section 2 with replacement:

 g  →  gs. (54)

The field strength for is derived as

 Fθϕ=−i[Dθ,Dϕ]=∂θAϕ−∂ϕAθ=gssinθ, (55)

or

 Fi=gs1r3xi. (56)

The total angular momentum operator is

 J=L(gs)≡⎛⎝L(g−12)00L(g+12)⎞⎠, (57)

or

 Jx=i(sinϕDθ+cosϕcotθDϕ)+gs1rx, Jz=−iDϕ+gs1rz, (58)

which satisfy the algebra:

 [Ji,Jj]=iϵijkJk. (59)

can be represented as

 Jx=L(0)x−gscosϕsinθ,        Jy=L(0)y−gssinϕsinθ,        Jz=L(0)z, (60)

and the Casimir operator is

 J2 =L(0)2−2igscosθsin2θ∂∂ϕ+gs21sin2θ =L(0)2−2igcosθsin2θ∂ϕ+14sin2θ(1+4g2)+i1sin2θσz(cosθ∂ϕ+ig). (61)

Since commutes with the chiral matrix :

 [σz,Ji]=0, (62)

we can diagonalize in each chiral sector. The eigenvalues of are given by

 j(j+1), (63)

where999Strictly speaking, Eq.(64) holds for non-zero . For , we have   .

 j=g−12+n     (n=0,1,2,⋯). (64)

For , the corresponding eigenstates are

 Υ′gj=g−12,m=⎛⎜⎝Yg−12j=g−12,m(θ,ϕ)0⎞⎟⎠, (65)

with degeneracy , while for , the corresponding eigenstates are

 (66)

with degeneracy .

### 3.2 Dirac-Landau operator and eigenvalue problem

Using (53), we construct the Dirac-Landau operator, , as

 −i\mathrlap⧸DD =−iσx∂θ−i1sinθσy(∂ϕ+igscosθ) =−iσx(∂θ+12cotθ)−iσy1sinθ(∂ϕ+igcosθ) =⎛⎝0−i∂θ−1sinθ(∂ϕ+i(g+12)cosθ)−i∂θ+1sinθ(∂ϕ+i(g−12)cosθ)0⎞⎠. (67)

With the edth operators (31) or101010The edth operators are generally given by . See Appendix D also.

 ð(g)x≡12(ð(g)++ð(g)−)=∂θ=Dθ, ð(g)y≡−i12(ð(g)+−ð(g)−)=1sinθ(∂ϕ+igcosθ)=1sinθDϕ, (68)

the Dirac-Landau operator is concisely expressed as

 −i\mathrlap⧸DD=−iσxð(gs)x−iσyð(gs)y=⎛⎜⎝0−ið(g+12)−−ið(g−12)+0⎞⎟⎠. (69)

Note and . The spin connection term111111The spin connection term yields the non-hermitian term, , in (67). It is well known that on 2D manifolds, the spin connection term vanishes when we modify the Dirac operator to be hermitian [see [33] or [34] for instance]. Though the present Dirac operator contains the non-hermitian term, its eigenvalues are real numbers. induces a difference between monopole charges by 1 in the off-diagonal components, and such “discrepancy” is crucial in the following discussions.

It is not difficult to derive the eigenvalues of the Dirac-Landau operator on a sphere [4, 35]. The square of the Dirac-Landau operator gives the Casimir of the angular momentum :

 (−i\mathrlap⧸DD)2 =J2+14−g2, (70)

where we used (36). Eq.(70) is consistent with the general formula [35, 4]:

 (−i\mathrlap⧸DD)2=J2−g2+R8, (71)

with scalar curvature for two-sphere. Therefore the eigenvalues of are obtained as

 (−i\mathrlap⧸DD)2=(j+12−g)(j+12+g)=n(2g+n), (72)

and those of the Dirac-Landau operator are

 ±λn=±√n(2g+n)      (n=0,1,2,⋯). (73)

The eigenstates of the square of the Dirac-Landau operator are exactly same as of the Casimir . For , the eigenstates of are (65) with degeneracy , and for the eigenstates are and (66) with degeneracy