Chiral Topological Insulator on Nambu 3-Algebraic Geometry

Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a $(l,l,l-1)$ Laughlin-Halperin type wavefunction with unique $K$-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge ${1}/({1+2(l-1)^2})$. The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects.


INTRODUCTION
In the past decade, the topological insulars (TIs) with time-reversal symmetry have attracted great attentions. The chiral TI is a new class of TI that has not been experimentally observed. The chiral IT is also known as AIII-class TI which respects the chiral symmetry and lives in arbitrary odd dimension [1]. Since the chiral TI can live in 3D space, the chiral TI is expected to be realized in daily experiments. Indeed, the lattice model of the chiral TI has been discussed in Refs. [2,3], and its possible experimental platform has been proposed in Ref. [4].
Recently, two groups independently applied noncommutative geometry (NCG) techniques to TIs [3,5] and discussed the appearance of quantum Nambu geometry [6][7][8] in the context of TIs. Since quantum Nambu geometry is closely related to the geometry of M-theory [9][10][11][12], the appearance of quantum Nambu geometry in TIs is quite intriguing, however the two groups reached a contradictory conclusion about the Nambu 3-bracket description for TIs; The authors of Ref. [3] insist that 3algebra consistently describes physics of the chiral TI, while the authors of Ref. [5] advocated the 3-algebra is not appropriate because of "pathological" properties of the 3-algebra. Here arises a question: (i) Which statement is correct or is there any compromise between these two? In Ref. [3], the projection density operator method was applied to derive excitation energy within the single mode approximation, however the calculation cannot completely be carried out due to the lack of knowledge of the explicit form of the groundstate. Then arises the second equation: (ii) How can we reasonably construct the explicit groundstate wavefunction of the chiral TI?
In Ref. [13], the author clarified relations between the * On leave from Kagawa National College of Technology, 551 Takuma-cho, Mitoyo, Kagawa 769-1192, Japan. After March 31 2014, email to hasebe@dg.kagawa-nct.ac.jp A-class TIs and quantum Hall effect (QHE) in arbitrary even dimension. A-class and AIII-class TIs share many similar properties: Both A-class and AIII-class are classified by Z topological invariant and regularly appear in even and odd dimensions, and either of them does not respect time-reversal or particle-hole symmetry. However there is one discrepancy: AIII-class respects the chiral symmetry while A-class does not. Since A-class TIs are realized as QHE in even dimensions, the AIII-class TIs may be regarded as odd dimensional analogue of QHE. If so, it is reasonably understood why A and AIII-class TIs are so much like. At the same time, the third question arises: (iii) Why does only AIII-class have the chiral symmetry? There are not many works about QHE in odd dimension except for the pioneering work of Nair and Randjbar-Daemi [14] where they found the Landau level spectrum depends on a "mysterious" extra parameter whose couterpart does not exist in the even dimensional case. Here arises the last question: (iv) What is the physical meaning of the extra parameter found in Nair and Randjbar-Daemi's analysis?
In this paper, we explore 3D chiral TI with emphasis on its relation to quantum Nambu geometry. Through the work, we provide convincing resolutions to all of the controversial issues from (i) to (iv).

THE LANDAU PROBLEM ON S 3
We first revisit the SO(4) Landau model on a threesphere in the SU (2) monopole background [14]: where r denotes the radius of three-sphere. The covariant angular momentum is constructed as Λ µν = −ix µ D ν + ix ν D µ where the covariant derivative is given by D µ = ∂ µ + iA µ with SU (2) monopole gauge field Here, 1 2 σ i denote the SU (2) matrices with spin magnitude I/2. The corresponding field strength 4r 4 I(I + 2) and µ<ν F µνFµν = 0, withF µν = 1 2 ǫ µνρσ F ρσ . The Hamiltonian (1) may respect the SO(4) symmetry since the the SU (2) monopole magnetic field is perpendicular to the surface of S 3 (x µ F µν = F µν x ν = 0). The SO(4) total angular momentum is constructed as The SO(4) Landau Hamiltonian can be decomposed to two SU (2) invariant Hamiltonians: where Notice that though H L/R take a superficially similar form of the SO(3) Landau Hamiltonian on the Haldane's sphere [15], they are SU (2) matrix valued Hamiltonians. The eigenvalues are readily derived as E lL,lR = 1 Mr 2 l L (l L + 1) + l R (l R + 1) − 1 8Mr 2 I(I + 2) where l L and l R denote the SU (2) L ⊗ SU (2) R angular momentum indices. The diagonal SU (2) D operators are constructed as With a given monopole charge I/2, the eigenvalues of the SU (2) L and SU (2) R angular momentum indices are related as [14] l Here n denotes the Landau level index (n = 0, 1, 2, · · · ), and s corresponds to the extra parameter that takes integer of half-integer values [25]. Therefore, l L and l R can respectively be expressed as l L = 1 2 (n + I 2 + s) and l R = 1 2 (n+ I 2 −s). Notice that under the sign change of s, L + and L − are interchanged, and hence s can be identified with the chirality index. The energy eigenvalues are rewritten as and the corresponding nth Landau level degeneracy is given by (6) In the thermodynamic limit I, r → ∞ with fixed B = I/(2r 2 ) and finite s, E (s) n reproduces the ordinary Landau level on 2D-plane, B M (n + 1 2 ). Notice that both of the energy eigenvalue (5) and the degeneracy (6) depend on s, and exhibit the chiral symmetry with respect to s → −s [26]. In the lowest Landau level (LLL) n = 0, the energy is represented as where due to the constraints (4), Therefore, the minimum energy of (7) is achieved at s = 0 for even I, and at s = ±1/2 for odd I. It should be emphasized that for odd I, the LLL has "two fold" degeneracy coming from s = 1/2 and s = −1/2:

THE CHIRAL HOPF MAP AND QUATERNIONS
Let us consider the LLL basis states for s = ±1/2. We derive their functional form instead of the abstract Wigner D-function [14]. For this purpose, we first introduce the chiral Hopf map: The coordinates of S 3 L ⊗ S 3 R are expressed by the twocomponent complex spinors ψ L and ψ R (chiral Hopf spinors) subject to the normalization condition, ψ L † ψ L = ψ R † ψ R = 1 2 , and the chiral Hopf map is explicitly realized as where q µ andq µ denote the quaternions and conjugatequaternions, q µ = (q i , 1) = (−iσ i , 1) andq µ = (−q i , 1) = (iσ i , 1) [27]. It is straightforward to show that x µ (10) x µ are invariant under the simultaneous SU (2) D transformation of ψ L and ψ R : ψ L/R → ψ L/R e αiqi . The explicit form of the chiral Hopf spinors is given by where φ denotes a normalized two-component spinor (S 3fibre) with the normalization φ † φ = 1 and M L/R is the following unitary matrix: with q L µ ≡ q µ and q R µ ≡q µ , and M R is a quaternionic con- Notice that ψ L and ψ R are related by the "parity transformation": , which is equivalent to the quaternionic conjugate in (12). We can derive the SU (2) gauge field , and so ψ L and ψ R respectively transform as SU (2) L ⊗ SU (2) R Weyl spinors, (1/2, 0) and (0, 1/2). The direct product of the two Weyl spinors gives the SU (2) L ⊗ SU (2) R "bi-spin" representation (l L , l R ) that corresponds to the LLL basis state for (l L , l R ) = ( 1 2 ( I 2 +s), 1 2 ( I 2 −s)). The (l L , l R ) representation is explicitly constructed as where with m L = −l L , −l L + 1, · · · , l L − 1, l L . Same for Ψ R lR,mR by replacing L with R. Thus, the LLL basis states are given by the holomorphic function of ψ L and ψ R . The chiral Hopf map is naturally derived by the "dimensional reduction" of the 2nd Hopf map, S 7 S 3 → S 4 . The 2nd Hopf map is realized as a map from a fourcomponent complex spinor ψ subject to ψ † ψ = 1 to x a = ψ † γ a ψ (a = 1, · · · , 5) with the SO(5) gamma matri- 16], and the chiral Hopf map is obtained by imposing an additional con- This implies a geometric embedding of the chiral TI in 4D QHE. Similarities between the chiral TI and 4D QHE can also be found in the SU (2)-bundle topology. With use of the chiral Hopf spinors, Q matrix in Ref. [1] is and the corresponding winding number is evaluated as c 2 = −i 1 24π 2 S 3 tr(−ig † dg) 3 = 1 6 I(I + 1)(I + 2), which is exactly equivalent to the 2nd Chern number of the SU (2) monopole-bundle over S 4 -the set-up of 4D QHE.

QUANTUM NAMBU GEOMETRY
Since the LLL basis states are given by the holomorphic function of the chiral Hopf spinors, the complex conjugation can be regarded as the derivative, ψ L,R * → ∂/∂ψ L,R . From the chiral Hopf map, we obtain the effective operator expression for the S 3 coordinates: where α = 2r/I. From the algebras of quaternions[28], we have where ∂ ∂ψR are two independent SU (2) operators. Eq.(16) realizes the chiral symmetric version of the NC algebra of 4D QHE [16]. We also have [X µ , X In total, the ten operators, X µ , X + i and X − i , amount to form the SO(5) algebra. The parameter s (4) denotes the eigenvalue of the chiral charge operator: Meanwhile in the set-up of 4D QHE, the 5th coordinate of fuzzy four-sphere is given by , and then S = 1 α X 5 . Remember in the set-up of 3D QHE, s was just an internal parameter, but in the "virtual" 4D QHE, s can be interpreted as the latitude of S 4 . The chiral symmetry is realized as the reflection symmetry of the 4D QHE with respect to the equator. This is the resolution for (iv).
In the precedent studies of NCG [17,18], more elegant formulation of the fuzzy three-sphere based on the quantum Nambu bracket has been known [8,9,18]. One may readily confirm that matrix realization of X µ (15) indeed satisfy the quantum Nambu-algebra for the fuzzy three-sphere: where the chiral 3-bracket is defined as Under the ordinary definition of the quantum Nambu 3bracket, [X µ , X ν , X ρ ] ≡ X [µ X ν X ρ] , X µ (15) do not form a closed algebra. Here, several comments are added. Firstly, the chiral 3-bracket can evade the pathological property of the 3-bracket emphasized in Ref. [5] as found [X µ , X ν , 1] χ = 0. Though the ordinary definition of the 3-bracket was adopted in Ref. [3], the whole 3-bracket algebra was not really used in the analysis, and hence the pathological property of the 3-bracket did not apparently appear. This gives the resolution for (i). Secondly, the chiral 3-bracket is neatly fitted in the four-bracket of 4D QHE [13] and concisely given by [X µ , X ν , X ρ ] χ = 1 α [X µ , X ν , X ρ , X 5 ]. This is the algebraic evidence that the chiral TI naturally realizes as a "subspace" embedded in the 4D QHE. Thirdly, the right-hand side of (18) suggests the existence of 3-rank U (1) magnetic field G µνρ = 1 r 4 ǫ µνρσ x σ . The corresponding gauge field is given by a 2-rank antisymmetric tensor field C µν = −C νµ (G µνρ = ∂ µ C νρ + ∂ ν C ρµ + ∂ ρ C µν ) that couples to string-like object. Such 2-rank tensor field is simply obtained by the dimensional reduction of the 3-rank tensor gauge field C abc in the 4D QHE [13] by C µν ≡ C µν5 , and so the string-like object from membrane-like excitation in 4D QHE.
The chiral Nambu 3-algebra gives a crucial implication for the existence of the chiral symmetry. The precedent studies [17,18] tell that the fuzzy three-sphere is realized as a composite of two latitudes s = 1/2 and s = −1/2 of fuzzy four-sphere not just as the equator (s = 0). The reason is simple; If the fuzzy three-sphere was simply the equator, the NC algebra would vanish, [X µ , X ν , X ρ ] χ = [X µ , X ν , X ρ , S] = 0 (s = 0). To incorporate a non-trivial NC structure, the fuzzy three-sphere has to be a composite of two S 3 s with opposite latitudes of same magnitude [18]. This suggests that, in the language of TIs, the chiral TI is given by a superposition of two S 3 s with opposite chiral charges on the virtual 4D QHE [ Fig.1]. In other words, the requirement of NCG necessarily induces the chiral symmetry to the chiral TI. This is the resolution for (iii). s = ±1/2 is achieved as the minimum energy of the Landau Hamiltonian for odd I not even I, and so the system becomes trivial or non-trivial depending on the parity of I. This implies a hidden Z 2 structure of the chiral TI.

THE COLOR HALL EFFECT
While the chiral TI shares similar properties with QHE such as time-reversal breaking and Z classification of topological invariant, Z 2 structure is also incorporated in the chiral TI due to the chiral symmetry, just like the time-reversal symmetry of the QSHE. Therefore, the chiral TI is expected to accommodate a dual property of the QHE and QSHE, and such dual property is manifest in the transport phenomena.
Name the two SU (2) color indices L and R and the three colors of SU (2) gauge field a = 1, 2, 3. Since the color gauge fields are independently coupled to the corresponding color currents, the Hall effect is given by Without loss of generality, we focus on a = 3 in which If there only exist either of L or Rcolor particles, the Hall effect will be given by The L and R-color currents flow in the mutually opposite direction, similar to the spin Hall effect where the flows of up and down spin currents are opposite. However, the color Hall effect does not respect the time reversal symmetry, since the L and R are just labels and are not flipped under the time reversal transformation unlike physical spin of the spin Hall effect. The time reversal transformation just reverses the direction of the L and R-color currents as in the case of the ordinary Hall effect. The quantized version of the color Hall effect can similarly be understood. L and R-color currents respectively contribute to the quantized Hall conductivity as σ L = e 2 2π ν L , σ R = − e 2 2π ν R . The total and different conductances are obtained as For the (l, l, l−1) chiral TI, we have a non-chiral version of the Hall effect: σ = 0, ∆σ = 1 2(l−1) 2 +1 e 2 π , which reduces to the QSH conductance ∆σ QSH = e 2 π [22] for l = 1.

SUMMARY AND DISCUSSIONS
To summarize, we explored one-particle and manybody physics of the chiral TI with Landau levels based on the Nambu 3-algebraic geometry. The chiral TI is a natural 3D generalization of the Haldane's 2D QHE and 3D "reduction" of the Zhang and Hu's 4D QHE. We elucidated the former controversial problems by exploiting the mathematics and physics of the chiral TI. In particular, we clarified that Nambu 3-algebraic geometry is essential for the existence of the chiral symmetry of the chiral TI. Interestingly, the chiral TI exhibits a dual property of QHE and QSHE due to the hidden Z 2 structure of the chiral symmetry.
Recently, 3D AII TI model with Landau levels was constructed by Li and Wu [23] and Dirac-type models in higher dimensions were also explored in Ref. [24]. Though the Li and Wu's model also heavily utilized quaternionic structure, the model respects the time-reversal symmetry and hence describes different physics. like thank Sanjaye Ramgoolam and Congjun Wu for email correspondences. This work was partially supported by a Grant-in-Aid for Young Scientists (B) (Grant No.23740212), Overseas Dispatching Program 2013 of National College of Technology, and The Emma Project for Art and Culture.