Quaternionic analyticity and SU(2) Landau Levels in 3D Yi Li (UCSD Princeton), Congjun Wu UCSD) Collaborators: K. Intriligator (UCSD), Yue Yu (ITP, CAS, Beijing), Shou-cheng Zhang (Stanford), Xiangfa Zhou (USTC, China). Sept 17, 2014, Center of Mathematical Sciences and Applications, Harvard University 1 Ref. 1.Yi Li, C. Wu, Phys. Rev. Lett. 110, 216802 (2013) (arXiv:1103.5422). 2.Yi Li, K. Intrilligator, Yue Yu, C. Wu, PRB 085132 (2012) (arXiv:1108.5650). 3.Yi Li, S. C. Zhang, C. Wu, Phys. Rev. Lett. 111, 186803 (2013) (arXiv:1208.1562). 4. Yi Li, X. F. Zhou, C. Wu, Phys. Rev. B. Phys. Rev. B 85, 125122 (2012). Acknowledgements: Jorge Hirsch (UCSD) Xi Dai, Zhong Fang, Liang Fu, Kazuki Hasebe, F. D. M. Haldane, Jiang-ping Hu, Cenke Xu, Kun Yang, Fei Zhou 2 Outline Introduction: complex number quaternion. Quaternionic analytic Landau levels in 3D/4D. Analyticity : a useful rule to select wavefunctions for non-trivial topology. Cauchy-Riemann-Fueter condition. 3D harmonic oscillator + SO coupling. 3D/4D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly?) 3 The birth of i : not from x

2 1 Cardano formula for the cubic equation. x1 c1 c2, x2,3 c1e x 3 px q 0 c1,2 3 q 2 2 i 3 c2e q discriminan 2 t: 2 2 i 3 p 3 3

Start with real coefficients, and end up with three real roots, but no way to avoid i. x 3 3x 0 p 3, q 0 1, x1 0 x2,3 3 4 The beauty of complex Gauss plane: 2D rotation (angular momentum) Euler formula: ei cos i sin QM) (U(1) phase: optics, Complex analyticity: (2D lowest Landau level) f f i 0 x y 1 2i 1 z z0 dz f ( z ) f (z0) Algebra fundamental theorem; Riemann hypothesis distributions of prime numbers, etc. Quan Mech: i appears for the first time in a wave equation. Schroedinger Eq: i H t

5 Further extension: quaternion (Hamilton numbe r) Three imaginary units i, j, k. q x yi zj uk i 2 j 2 k 2 1 Division ab 0 a 0, or, b 0 algebra: 3D rotation: non-commutative. ij ji k ; jk kj i; ki ik j Quaternion-analyticity (Cauchy-Futer integral) f f i x y f f j k 0 z u 1 2 2 1 Dq f (q) 2 | q q0 | ( q q0) f ( q0) 6 Quaternion plaque: Hamilton 10/16/1843

Brougham bridge, Dublin i 2 j 2 k 2 ijk 1 7 3D rotation as 1st Hopf map Rotation axis : rotation R quaternion q: , rotation angle: . imaginary unit:( ) i sin cos j sin sin k cos unit q cos ( ) sin S 3 2 2 r xi yj zk 3D vector r quaternion. imaginary 3D rotation S2.

Hopf map S3 r z k r R( r ) qkq 1 q S3 1st Hopf map qkq 1 S 2 8 Outline Introduction: complex number quaternion. Quaternionic analytic Landau levels in 3D/4D. Analyticity : a useful rule to select wavefunctions for non-trivial topology. Cauchy-Riemann-Fueter condition. 3D harmonic oscillator + SO coupling. 3D/4D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly?) 9 Review: 2D Landau level in the symmetric ga uge H2DLL symmetric gauge: 1 e 2 B (P A) , A z r 2M c

2 Lowest Landau level wavefunction: complex analyticity ( ax ia y ) LLL( z, z ) 0, ( z x iy ) ax LLL( z , z ) f ( z , z )e z f ( z , z ) 0 zz 4l B2 , 1 2 ( x ip x ) ay 1 2 ( y ip y ) f ( z, z ) f ( z ) f ( z ) 1, z, z 2,..., z m ,..., ( m 0) 10 Advantages of Landau levels (2D) Simple, explicit and elegant. Complex analyticity selection of non-trivial WFs. ( x , y ) 1. The 2D ordinary QM WF analysis belongs to real

2. Cauchy-Riemann condition complex analyticity (chirality). 3. Chirality is physically imposed by the B-field. Analytic properties facilitate the construction of Laughlin WF. | z |2 i 2 3 i 4l B sym( z1, z2,.... zn ) ( zi zj ) e i j Pioneering Work: LLs on 4D-sphere ---Zhang and Science 294, 824 Hu (2001). Particles couple to the SU(2) gauge field on the S4 sphere. 2 2 H , ab xa ( ib Ab ) xb ( i a Aa ) ab 2 2MR 1ab5 Second Hopf map. The spin valueI R 2 . S 7 4 S S xa a , ni u i , u 3 R Single particle LLLs xa , ni | m1m2 m3 m4 1m 2m 3m 4m

1 2 3 4 4D integer and fractional TIs with time reversal symmetryD integer and fractional TIs with time reversal symmetry Dimension reduction to 3D and 2D TIs (Qi, Hughes, Zhang). 12 Our recipe 1. 3D harmonic wavefunctions. 2. Selection criterion: quaternionic analyticity (physically imposed by SO coupling). Symmetric-like gauge (3D quantum top): Complex analyticity in a flexible-plane with chirality determined by S along thedirection. Landau-like gauge: spatial separation of 2D Dirac modes with opposite helicites. Generalizable to higher dimensions. S e1 e2 2D LLs in the symmetric gauge H2DLL 1 e 2 B (P A) , A z r 2M c 2 2D LL Hamiltonian = 2D harmonic oscillator (HO)+

orbital Zeeman coupling. H2D LL p2 1 e|B| hc M2r 2 L z , , lB 2M 2 2Mc eB H 2 D LL has the same set of eigenstates as 2D HO. 14 Organization non-trivial topology E2 D , HO /( ) 2nr | m | 1 2 2 m |z| /( 2l B ) LLL z e -3 -1 -2 1 0 -1 3 -1 2 1 0 E2D,LL /( ) 2nr

m 0 0 1 1 2 2 3 m If viewed horizontally, they are topologically trivial. If viewed along the diagonal line, they become LLs. EZeeman /( ) m 15 3D Aharanov-Casher potential !! The SU(2) gauge potential: 1 2 D : A Bz r 2 1 3D : A g r 2 3D LL Hamiltonian = 3D HO + spin-orbit coupling. H LL 3D

P2 1 2 2 M r L 2M 2 1 e 2 M 2 2 ( P A ) r 2M c 2 | eg | c , lg . 2 Mc | eg | The full 3D rotational symm. + time-reversal symm. 16 Constructing 3D Landau Levels E3 D , HO /( ) 2nr l 1 0 3 , 0 E3 D , HO /( ) 1/2- 3/2+ 5/2- 7/2+ SOC : 2 helicity branches 2

1 3 2 l 1 j l . 2 p 1 2 2 m r L 2m 2 2 H LL3 D for j l L (l 1) for j 1/2+ 3/2- 5/2+ 1/2- 3/2+ 1/2+ j E3 D , LL /( ) + 3/2+ 5/2+ 7/2+ + 3/2+ 5/2+ 7/2+ j

17 The coherent state picture for 3D LLL WFs The highest weight state j z j . BothLz andS z are conserved. LLL j ,j z j x iy (r ) 0 l S // e3 r 2 / 4l g2 e Coherent states: spin perpendicular to the orbital plane. e1 l LLL j , high( r ) [( e1 i e2) r ] e3 e2 LLLs in N-dimensions: picking up any two axes and define a complex plane with a spin-orbit coupled helical structure. 18

Comparison of symm. gauge LLs in 2D and 3D 1D harmonic levels: real polynomials. 2D LLs: complex analytic polynomials. sym LLL 2 2 m |z| /( 2l B ) z e , z x iy, m 0. 3D LLs: SU(2) group space quaternionic analytic polynomials. r2 Phase e1 S // e3 e2 2 2l g l jLLL ( r , )

[( e i e ) r ] e 1 2 e3 , high 19 Right-handed triad Quaternionic analyticity Cauchy-Riemann condition and loop integral. g g i 0 x y 1 1 dz g ( z ) g (z 0 ) 2i z z 0 Fueter condition (left analyticity): f (x,y,z,u) quaternionvalued function of 4-real variables. Cauchy-Fueter integrals over closed 3-surface in 4D. f f f f i j k 0

x y z u K(q) 1 2 2 K( q q0 ) Dq f ( q) f (q0 ) 1 x yi zj uk q | q |2 ( x 2 y 2 z 2 u2)2 D(q) dy dz du idx dz du jdx dy du kdx dy dz 20 Mapping 2-component spinor to a single quaternion LLL j ,j z 1,j ,j z ( r , ) e 2,j ,j z r2 4l G2 TR reversal:i y * fj i x 2 e

SU(2) rotation: k f j ,j z( x, y , z ) 1,j ,j z j 2,j ,j z ; U(1) phase ei fe i 2 i y 2 e f;e j 2 i z 2 e f;e Reduced Fueter condition in 3D: i 2 e f f(x,y,z) i

j y z x Fueter condition is invariant under rotation R( , , ) If f satisfies Fueter condition, so does Rf. ( Rf )( x, y , z ) e i j i 2 2 2 e e f ( x , y , z ), 1 r R r 0 . 21 Quaternionic analyticity of 3D LLL f j j z j ( x iy )l The highest state jz=j is obviously analytic. All the coherent states can be obtained from the highest states through rotations, and thus are also analytic. All the LLL states are quaternionic analytic. QED. Completeness: Any quaternionic analytic polynomial corresponds to a LLL wavefunction. f

j j 1 / 2 j z j f c 1 jm j j z m 2 l f j 1 / 2 m0 j j z l (cjm jc j m) f j j z qjm l 0 m0 22 Helical surface states of 3D LLs E / E0 from bulk to surface H 3D bulk

p2 1 2 2 M r L 2M 2 2 l (l 1) 2D H surface l 2 2 MR j 0 l0 j l L R er ( p ) 1 2 H 2D plane v f (l l0 ) / R ver ( p ) py R kf

k f l0 / R px Each LL contributes to one helical Fermi surface. Odd fillings yield odd numbers 23 of Dirac Fermi surfaces. Analyticity condition as Weyl equation (Euclidean) 2D complex analyticity zz LLL( z , z ) f ( z )e 4l 1D chiral edge mode (t , x ) f ( x t ) f f i 0 x y 0 t x 2 B 3D: quaternionic analyticity f f f i j 0

x y z 4D: quaternionic analyticity f f f i j x y z f k 0 u 2D helical Dirac surface mode x y 0 t x y 3D Weyl boundary mode x y z 0 t x y z 24

Outline Introduction: complex number quaternion. Quaternionic analytic Landau levels in 3D/4D. Analyticity : a useful rule to select wavefunctions for non-trivial topology. Cauchy-Riemann-Fueter condition. 3D harmonic oscillator + SO coupling. 3D/4D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly?) 25 Review: 2D LL Hamiltonian of Dirac Fermions H LL 2D e e vF {( p x Ax ) x ( p y Ay ) y }, c c B A z r 2 Rewrite in terms of complex combinations of phonon operators. H 2D LL 0 i (a x ia y ) 2vF , 0

l B i (a x ia y ) LL dispersions:En n Zero energy LL is a branch of halffermion modes due to the chiral symmetry. |z|2 m z 4l B2 LL 0; m e . 0 ai E 1 xi l ( i B pi ), i x, y. 2 lB n 2 n 1 n 0 n 1 n 2 26 3D/4D LL Hamiltonian of Dirac Fermions {1, i } 2D harmonic oscillator {1, i , j , k} (1, i x , i y , i z ) 4D harmonic oscillator {ax , ay } {au , ax , ay , az }

au ia x ja y kaz complex quaternion: 4D Dirac LL Hamiltonian: 4D Dirac HLL 2 l 0 2 0 a ia ja ka x y z u 0 a i a u au ia x ja y kaz 0 au i a 0 27 3D LL Hamiltonian of Dirac Fermions

H LL3 D Dirac 2 0 ( p i r / l0 ) 0 i a l 0 2 2 i a 0 0 2 ( p ir / l0 ) This Lagrangian of non-minimal Pauli coupling. v L {i ( 00 vi i )} 0i F 0i , lg 0i i [ 0, i ], 2 F

0i xi . lg A related Hamiltonian was studied before under the name of Dirac oscillator, but its connection to LL and topological properties was not noticed. Benitez, et al, PRL, 64, 1643 (1990) 28 LL Hamiltonian of Dirac Fermions in Arbitrary Dimensions For odd dimensions (D=2k+1). ii ai 0 2 ii( k ) ai 0 (k ) H D dim LL For even dimensions (D=2k). D dim HLL 0

k 2 ( k 1) a i ai 2k i i 1 a 2k k i i 1 0 a ( k 1) i i 29 A square root problem: 3 D , S ch r o ed in ger 3 D D ir a c H LL H LL

3 D Dirac 3D The square of H LL ( H LL ) gives two copies of opposite helicity eigenstates. with 3 2 L 0 3 D Dirac 2 ( H LL ) p M 2 2 2 r 3 / 2 2M 2 0 ( L ) 2 LL solutions: dispersionless with respect to j. Eigenstates constructed based on non-relativistic LLs. The zeroth LL: ELLn nr , r

LL nr ; j ,l , j z 1 n , j ,l , j 2 i n 1, j ,l 1, j r r z z . LL 0 ; j ,l , j z jLLL,l , j 0 z . 30 Zeroth LLs as half-fermion modes The LL spectra are symmetric with respect to zero

LL contributes energy, thus each state of the zeroth fermion charge depending on the zeroth LL is filled or empty. 2 1e For the 2D case, the vacuum charge density j0 is B 2 h , known as parity anomaly. G. Semenoff, Phys. Rev. Lett., 53, 2449 (1984). For our 3D case, the vacuum E charge density is plus or minus of the half of the particle density 0 of the non-relativistic LLLs. n 0 What kind of anomaly? n 1 n 2 0 n 2 n 1 31 Helical surface mode of 3D Dirac LL The mass of the vacuum outside M H 3 D H LL3 D H 3 D R

p M p M This is the square root problem of the open boundary problem of 3D nonrelativistic LLs. Each surface mode for n>0 of the non-relativistic case splits a pair surface modes for the Dirac case. The surface mode of Dirac zerothLL of is singled out. Whether it is upturn or downturn depends on the sign of the vacuum mass. H 3 D H LL3 D, E n 2 n 1 n 0 j n 1 n 2 32 Conclusions We hope the quaternionic analyticity can facilitate the construction of 3D Laughlin state. The non-relativistic N-dimensional LL problem is a Ndimensional harmonic oscillator + spin-orbit coupling. The relativistic version is a square-root problem corresponding to Dirac equation with non-minimal coupling. Open questions: interaction effects; experimental realizations; characterization of topo-properties with harmonic potentials 33