Quaternionic analyticity and SU2 Landau Levels in 3

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Quaternionic analyticity and SU(2) Landau Levels in 3 D Yi Li (UCSD Princeton), Congjun

Quaternionic analyticity and SU(2) Landau Levels in 3 D 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) (ar. Xiv:

Ref. 1. Yi Li, C. Wu, Phys. Rev. Lett. 110, 216802 (2013) (ar. Xiv: 1103. 5422). 2. Yi Li, K. Intrilligator, Yue Yu, C. Wu, PRB 085132 (2012) (ar. Xiv: 1108. 5650). 3. Yi Li, S. C. Zhang, C. Wu, Phys. Rev. Lett. 111, 186803 (2013) (ar. Xiv: 1208. 1562). 4. Yi Li, X. F. Zhou, C. Wu, 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 3 D/4

Outline • Introduction: complex number quaternion. • Quaternionic analytic Landau levels in 3 D/4 D. Analyticity : a useful rule to select wavefunctions for nontrivial topology. Cauchy-Riemann-Fueter condition. 3 D harmonic oscillator + SO coupling. • 3 D/4 D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly? ) 3

The birth of “i“ : not from • Cardano formula for the cubic equation.

The birth of “i“ : not from • Cardano formula for the cubic equation. discriminant: • Start with real coefficients, and end up with three real roots, but no way to avoid “i”. 4

The beauty of “complex” • Gauss plane: 2 D rotation (angular momentum) • Euler

The beauty of “complex” • Gauss plane: 2 D rotation (angular momentum) • Euler formula: (U(1) phase: optics, QM) • Complex analyticity: (2 D lowest Landau level) • Algebra fundamental theorem; Riemann hypothesis – distributions of prime numbers, etc. • Quan Mech: “i” appears for the first time in a wave equation. Schroedinger Eq: 5

Further extension: quaternion (Hamilton number) • Three imaginary units i, j, k. • Division

Further extension: quaternion (Hamilton number) • Three imaginary units i, j, k. • Division algebra: • 3 D rotation: non-commutative. • Quaternion-analyticity (Cauchy-Futer integral) 6

Quaternion plaque: Hamilton 10/16/1843 Brougham bridge, Dublin 7

Quaternion plaque: Hamilton 10/16/1843 Brougham bridge, Dublin 7

3 D rotation as 1 st Hopf map • Rotation axis • : rotation

3 D rotation as 1 st Hopf map • Rotation axis • : rotation R , rotation angle: . imaginary unit: unit quaternion q: • 3 D vector r imaginary quaternion. • 3 D rotation Hopf map S 3 S 2. 1 st Hopf map 8

Outline • Introduction: complex number quaternion. • Quaternionic analytic Landau levels in 3 D/4

Outline • Introduction: complex number quaternion. • Quaternionic analytic Landau levels in 3 D/4 D. Analyticity : a useful rule to select wavefunctions for nontrivial topology. Cauchy-Riemann-Fueter condition. 3 D harmonic oscillator + SO coupling. • 3 D/4 D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly? ) 9

Review: 2 D Landau level in the symmetric gauge: Lowest Landau level wavefunction: complex

Review: 2 D Landau level in the symmetric gauge: Lowest Landau level wavefunction: complex analyticity 10

Advantages of Landau levels (2 D) • Simple, explicit and elegant. • Complex analyticity

Advantages of Landau levels (2 D) • Simple, explicit and elegant. • Complex analyticity selection of non-trivial WFs. 1. The 2 D ordinary QM WF belongs to real analysis 2. Cauchy-Riemann condition complex analyticity (chirality). 3. Chirality is physically imposed by the B-field. • Analytic properties facilitate the construction of Laughlin WF.

Pioneering Work: LLs on 4 D-sphere ---Zhang and Hu Science 294, 824 (2001). •

Pioneering Work: LLs on 4 D-sphere ---Zhang and Hu Science 294, 824 (2001). • Particles couple to the SU(2) gauge field on the S 4 sphere. • Second Hopf map. The spin value . • Single particle LLLs 4 D integer and fractional TIs with time reversal symmetry Dimension reduction to 3 D and 2 D TIs (Qi, Hughes, Zhang). 12

Our recipe 1. 3 D harmonic wavefunctions. 2. Selection criterion: quaternionic analyticity (physically imposed

Our recipe 1. 3 D harmonic wavefunctions. 2. Selection criterion: quaternionic analyticity (physically imposed by SO coupling). Landau-like gauge: spatial separation of 2 D Dirac modes with opposite helicites. Generalizable to higher dimensions.

2 D LLs in the symmetric gauge • 2 D LL Hamiltonian = 2

2 D LLs in the symmetric gauge • 2 D LL Hamiltonian = 2 D harmonic oscillator (HO)+ orbital Zeeman coupling. • has the same set of eigenstates as 2 D HO. 14

Organization non-trivial topology -3 -1 -2 1 0 -1 2 1 0 3 -1

Organization non-trivial topology -3 -1 -2 1 0 -1 2 1 0 3 -1 0 0 1 1 2 2 3 • If viewed horizontally, they are topologically trivial. • If viewed along the diagonal line, they become LLs. 15

3 D – Aharanov-Casher potential !! • The SU(2) gauge potential: • 3 D

3 D – Aharanov-Casher potential !! • The SU(2) gauge potential: • 3 D LL Hamiltonian = 3 D HO + spin-orbit coupling. • The full 3 D rotational symm. + time-reversal symm. 16

Constructing 3 D Landau Levels 1/2 - 3/2+ 5/2 - 7/2+ 1 0 2

Constructing 3 D Landau Levels 1/2 - 3/2+ 5/2 - 7/2+ 1 0 2 1 0 3 SOC : 2 helicity branches 1/2+ 3/2 - 5/2+ 1/2 - 3/2+ 1/2+ ½+ 3/2+ 5/2+ 7/2+ 17

The coherent state picture for 3 D LLL WFs • The highest weight state

The coherent state picture for 3 D LLL WFs • The highest weight state . Both and are conserved. • Coherent states: spin perpendicular to the orbital plane. • 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 2 D and 3 D • 1 D

Comparison of symm. gauge LLs in 2 D and 3 D • 1 D harmonic levels: real polynomials. • 2 D LLs: complex analytic polynomials. Phase • 3 D LLs: SU(2) group space quaternionic analytic polynomials. 19 Right-handed triad

Quaternionic analyticity • Cauchy-Riemann condition and loop integral. • Fueter condition (left analyticity): f

Quaternionic analyticity • Cauchy-Riemann condition and loop integral. • Fueter condition (left analyticity): f (x, y, z, u) quaternion-valued function of 4 -real variables. • Cauchy-Fueter integrals over closed 3 -surface in 4 D. 20

Mapping 2 -component spinor to a single quaternion • TR reversal: ; U(1) phase

Mapping 2 -component spinor to a single quaternion • TR reversal: ; U(1) phase SU(2) rotation: • Reduced Fueter condition in 3 D: • Fueter condition is invariant under rotation If f satisfies Fueter condition, so does Rf. . 21

Quaternionic analyticity of 3 D LLL • The highest state jz=j is obviously analytic.

Quaternionic analyticity of 3 D LLL • 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. 22

Helical surface states of 3 D LLs from bulk to surface R • Each

Helical surface states of 3 D LLs from bulk to surface R • Each LL contributes to one helical Fermi surface. • Odd fillings yield odd numbers of Dirac Fermi surfaces. 23

Analyticity condition as Weyl equation (Euclidean) 2 D complex analyticity 1 D chiral edge

Analyticity condition as Weyl equation (Euclidean) 2 D complex analyticity 1 D chiral edge mode 3 D: quaternionic analyticity 2 D helical Dirac surface mode 4 D: quaternionic analyticity 3 D Weyl boundary mode 24

Outline • Introduction: complex number quaternion. • Quaternionic analytic Landau levels in 3 D/4

Outline • Introduction: complex number quaternion. • Quaternionic analytic Landau levels in 3 D/4 D. Analyticity : a useful rule to select wavefunctions for nontrivial topology. Cauchy-Riemann-Fueter condition. 3 D harmonic oscillator + SO coupling. • 3 D/4 D Landau levels of Dirac fermions: complex quaternions. An entire flat-band of half-fermion zero modes (anomaly? ) 25

Review: 2 D LL Hamiltonian of Dirac Fermions • Rewrite in terms of complex

Review: 2 D LL Hamiltonian of Dirac Fermions • Rewrite in terms of complex combinations of phonon operators. • LL dispersions: E • Zero energy LL is a branch of half-fermion modes due to the chiral symmetry. 26

3 D/4 D LL Hamiltonian of Dirac Fermions 2 D harmonic oscillator 4 D

3 D/4 D LL Hamiltonian of Dirac Fermions 2 D harmonic oscillator 4 D harmonic oscillator • “complex quaternion”: • 4 D Dirac LL Hamiltonian: 27

3 D LL Hamiltonian of Dirac Fermions • This Lagrangian of non-minimal Pauli coupling.

3 D LL Hamiltonian of Dirac Fermions • This Lagrangian of non-minimal Pauli coupling. • A related Hamiltonian was studied before under the name of Dirac oscillator, but its connection to LL and topological properties was noticed. Benitez, et al, PRL, 64, 1643 (1990) 28

LL Hamiltonian of Dirac Fermions in Arbitrary Dimensions • For odd dimensions (D=2 k+1).

LL Hamiltonian of Dirac Fermions in Arbitrary Dimensions • For odd dimensions (D=2 k+1). • For even dimensions (D=2 k). 29

A square root problem: • The square of helicity eigenstates. gives two copies of

A square root problem: • The square of helicity eigenstates. gives two copies of with opposite • LL solutions: dispersionless with respect to j. Eigen-states constructed based on non-relativistic LLs. The zeroth LL: 30

Zeroth LLs as half-fermion modes • The LL spectra are symmetric with respect to

Zeroth LLs as half-fermion modes • The LL spectra are symmetric with respect to zero energy, thus each state of the zeroth LL contributes ½- fermion charge depending on the zeroth LL is filled or empty. • For the 2 D case, the vacuum charge density is known as parity anomaly. , G. Semenoff, Phys. Rev. Lett. , 53, 2449 (1984). • For our 3 D case, the vacuum charge density is plus or minus of the half of the particle density of the nonrelativistic LLLs. E • What kind of anomaly? 31

Helical surface mode of 3 D Dirac LL • The mass of the vacuum

Helical surface mode of 3 D Dirac LL • The mass of the vacuum outside • This is the square root problem of the open boundary problem of 3 D nonrelativistic LLs. R E • 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 zeroth-LL of is singled out. Whether it is upturn or downturn depends on the sign of the vacuum mass. 32

Conclusions • We hope the quaternionic analyticity can facilitate the construction of 3 D

Conclusions • We hope the quaternionic analyticity can facilitate the construction of 3 D 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