quaternionic analyticity and su(2) landau levels in 3doutline •introduction: complex number...

Quaternionic analyticity and SU(2) Landau Levels in 3D

Congjun Wu （UCSD)

Sept 17, 2014, Center of Mathematical Sciences and Applications, Harvard University

Collaborators:Yi Li (UCSD Princeton)

K. Intriligator (UCSD), Yue Yu (ITP, CAS, Beijing),Shou-cheng Zhang (Stanford),Xiangfa Zhou (USTC, China).

1

2

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

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).

Outline

• Introduction: complex number quaternion.

• Quaternionic analytic Landau levels in 3D/4D.

3

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?)

The birth of “i“ : not from

• Cardano formula for the cubic equation.

4

12 x

0

33

xx

,1

0,3

qp

3

0

3,2

1

x

x

32

32

pq

3

2

23

2

13,2211 ,ii

ececxccx

32,1

2

qc

03 qpxx

discriminant:

• Start with real coefficients, and end up with three real roots, but no way to avoid “i”.

The beauty of “complex”

• Gauss plane: 2D rotation (angular momentum)

• Euler formula: (U(1) phase: optics, QM)

• Complex analyticity: (2D lowest Landau level)

• Algebra fundamental theorem; Riemann hypothesis – distributions of prime numbers, etc.

sincos iei

• Quan Mech: “i” appears for the first time in a wave equation.

Ht

i

5

0

y

fi

x

f)()(

1

2

10

0

zfzfdzzzi

Schroedinger Eq:

• Three imaginary units i, j, k.

ukzjyixq 1222 kji

• Division algebra:

jikkiikjjkkjiij ;;

• Quaternion-analyticity (Cauchy-Futer integral)

0

u

fk

z

fj

y

fi

x

f

)(

)()(||

1

2

1

0

02

02

qf

qfDqqqqq

0,or,00 baab

Further extension: quaternion (Hamilton number)

• 3D rotation: non-commutative.

6

Quaternion plaque: Hamilton 10/16/1843

1222 ijkkjiBrougham bridge, Dublin

7

3D rotation as 1st Hopf map

• : imaginary unit:

rotation R unit quaternion q:

cossinsincossin)ˆ( kji

3

2sin)ˆ(

2cos Sq

• 3D rotation Hopf map S3 S2.

8

• 3D vector r imaginary quaternion. zkyjxir

• Rotation axis , rotation angle: .

21 Sqkq 3Sq

1st Hopf map

1)(

ˆ

qkqrRr

kzr

Outline

• Introduction: complex number quaternion.

• Quaternionic analytic Landau levels in 3D/4D.

9

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?)

Review: 2D Landau level in the symmetric gauge

10

rzB

AAc

eP

MH DLL

ˆ

2,)(

2

1 22

)(2

1)(

2

1yyxx ipyaipxa

symmetric gauge:

Lowest Landau level wavefunction:complex analyticity

0),(

,),(),(24

zzf

ezzfzz

z

l

zz

LLLB

)0(,...,,...,,,1)(

)(),(2

mzzzzf

zfzzfm

)(,0),()( iyxzzziaa LLLyx

Advantages of Landau levels (2D)

• Simple, explicit and elegant.

• Analytic properties facilitate the construction of Laughlin WF.

i B

i

l

z

jijinsym ezzzzz

2

2

4

||

321 )(),....,(

),( yx

• Complex analyticity selection of non-trivial WFs.

1. The 2D ordinary QM WF belongs to real analysis

2. Cauchy-Riemann condition complex analyticity (chirality).

3. Chirality is physically imposed by the B-field.

1212

• Particles couple to the SU(2) gauge field on the S4 sphere.

12

)()(,2 51

2

2

2

aabbbaabba

abAixAix

MRH

• Second Hopf map. The spin value .

R4

3

7 SS

S

uunxii

a

a ,,

• Single particle LLLs

4D integer and fractional TIs with time reversal symmetry

Dimension reduction to 3D and 2D TIs (Qi, Hughes, Zhang).

4321

43214321|,

mmmm

iammmmnx

2RI

Science 294, 824 (2001).

Pioneering Work: LLs on 4D-sphere ---Zhang and Hu

Symmetric-like gauge (3D quantum top):Complex analyticity in a flexible 𝑒1-𝑒2 plane

with chirality determined by S along

the 𝑒3 direction.

Our recipe

1. 3D harmonic wavefunctions.

2. Selection criterion: quaternionic analyticity (physically imposed by SO coupling).

2e

1e

S

Landau-like gauge: spatial separation of 2D Dirac modes with opposite helicites.

Generalizable to higher dimensions.

2D LLs in the symmetric gauge

• 2D LL Hamiltonian = 2D harmonic oscillator (HO)+ orbital Zeeman coupling.

14

eB

hcl,

Mc

e|B|ω,ωLrMω

M

pH BzLLD

22

1

222

2

2

rzB

AAc

eP

MH DLL

ˆ

2,)(

2

1 22

• has the same set of eigenstates as 2D HO.LLDH

2

1515

Organization non-trivial topology

m0

-3

-1 1

2-2

-1 1 3

0

)2/(|| 22Blzm

LLL ez

• If viewed horizontally, they are topologically trivial.

• If viewed along the diagonal line, they become LLs.

mEZeeman

)/(

1||2)/(,2

mnErHOD

rLLD nE 2)/(,2

m0 1 2 3

-1 0 1 2

1616

222

22

2

3

2)(

2

1

2

1

2

rM

Ac

eP

M

LrMM

PH D

LL

rgAD

2

1:3rzBAD

ˆ

2

1:2

• The SU(2) gauge potential:

• 3D LL Hamiltonian = 3D HO + spin-orbit coupling.

3D – Aharanov-Casher potential !!

.||

,2

||

eg

cl

Mc

egg

16

• The full 3D rotational symm. + time-reversal symm.

17

Constructing 3D Landau Levels

)/(,3

HOD

E

j

3/2+

3/2+

3/2-

5/2-

5/2+

7/2+

1/2+

1/2+

1/2-

1/2-

SOC : 2 helicity

branches

.2

1 ljl

0

1

2

1 3

0

2

32)/(

,3 lnE

rHOD

jl

jlL

for)1(

for

)/(,3

LLD

E

j½+ 3/2+ 5/2+ 7/2+

½+ 3/2+ 5/2+ 7/2+

Lrmm

pH D

LL

22

2

3

2

1

2

,

1818

The coherent state picture for 3D LLL WFs

321, ])ˆˆ[()( el

high

LLLj reier

• Coherent states: spin perpendicular

to the orbital plane.

• The highest weight state . Both and are conserved. jj

z

22 4/

,

0)( g

z

lrl

LLLjjj e

iyxr

zL

zS

• LLLs in N-dimensions: picking up any two axes and define a complex plane with a spin-orbit coupled helical structure.

3ˆ// eS

2e

1e

Comparison of symm. gauge LLs in 2D and 3D

19

• 3D LLs: SU(2) group space

quaternionic analytic polynomials.

2

2

3

2

ˆ21, ])ˆˆ[(),( gl

r

el

high

LLLj ereier

• 2D LLs: complex analytic polynomials.

.0,)2/(|| 22

miy,xzez Blzmsym

LLL Phase

Right-handed triad

• 1D harmonic levels: real polynomials.

3ˆ// eS

2e

1e

Quaternionic analyticity

20

• Cauchy-Riemann condition and loop integral.

0

y

gi

x

g )()(1

2

10

0

zgzgdzzzi

• Fueter condition (left analyticity): f (x,y,z,u) quaternion-valued function of 4-real variables.

0

u

fk

z

fj

y

fi

x

f )()()(2

1002qfqfDqqqK

222222 )(||

1)(

uzyx

ukzjyix

qqqK

dzdykdxdudyjdxdudzidxdudzdyqD )(

• Cauchy-Fueter integrals over closed 3-surface in 4D.

Mapping 2-component spinor to a single quaternion

21

zzz

G

z

z

z jjjjjj

l

r

jj

jjLLLjj jzyxfer ,,2,,1,

4

,,2

,,1

, ),,()ˆ,(2

2

0

f(x,y,z)

zj

yi

x• Reduced Fueter condition in 3D:

• Fueter condition is invariant under rotation .If f satisfies Fueter condition, so does Rf.

• TR reversal: ; U(1) phase

SU(2) rotation:

fji y * ii fee

feefeefeeiijiki zyx222222 ;;

rRrzyxfeeezyxRfiji 1222 ),,,(),,)((

),,( R

Quaternionic analyticity of 3D 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.

ljjj iyxf

z)(

• Completeness: Any quaternionic analytic polynomial corresponds to a LLL wavefunction.

0 02/1 02/1 2

1 )(l

l

mjmjj

j

l

mmjjmjj

j

j

jjjm

mjjqfjccfcffzz

zz

• All the LLL states are quaternionic analytic. QED.

22

Helical surface states of 3D LLs

from bulk to surface

2

100 lj

0/ EE

j

• Each LL contributes to one helical Fermi surface.

• Odd fillings yield odd numbers of Dirac Fermi surfaces.

lMR

llH D

surface

2

22

2

)1(

LrMM

pH D

bulk

22

2

3

2

1

2

)(ˆ/)( 0

2 pevRllvH rf

D

plane

)(ˆ

peRLl r

yp

xpfk

Rlkf

/0

R

23

Analyticity condition as Weyl equation (Euclidean)

24

2D complex analyticity

0

y

fi

x

f

0

yxtyx

0

xt

24)(),( Bl

zz

LLL ezfzz

)(),( txfxt

1D chiral edge mode

0

z

fj

y

fi

x

f

3D: quaternionic analyticity 2D helical Dirac surface mode

0

u

fk

z

fj

y

fi

x

f

3D Weyl boundary mode

0

zyxtzyx

4D: quaternionic analyticity

Outline

• Introduction: complex number quaternion.

• Quaternionic analytic Landau levels in 3D/4D.

25

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?)

Review: 2D LL Hamiltonian of Dirac Fermions

rzB

A

ˆ2

.,),(2

1yxip

li

l

xa i

B

B

ii

},)(){(2

yyyxxxF

D

LLA

c

epA

c

epvH

• Rewrite in terms of complex combinations of phonon operators.

,0)(

)(022

yx

yx

B

FD

LLiaai

iaai

l

vH

• LL dispersions: nEn

E

0n

1n

2n

1n

2n

26.0

2

2

4

||

;0Bl

zm

LLm e

z

• Zero energy LL is a branch of half-fermion modes due to the chiral symmetry.

0

0

2

0

0

2

0

4

aia

aial

kajaiaa

kajaiaaH

u

u

zyxu

zyxuDiracDLL

27

3D/4D LL Hamiltonian of Dirac Fermions

2D harmonic oscillator },{ yx aa

},,,{ zyxu aaaa

},1{ i

),,,1(

},,,1{

zyx iii

kji

• 4D Dirac LL Hamiltonian:

4D harmonic oscillator

• “complex quaternion”: zyxu kajaiaa

3D LL Hamiltonian of Dirac Fermions

0)/(

)/(0

20

0

2 2

0

2

003

lrip

lripl

ai

aiH DiracD

LL

.],,[2

,)}({

000

0000

g

ii

ii

ii

g

ii

l

xF

i

Fl

vviL

• 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 not noticed.

Benitez, et al, PRL, 64, 1643 (1990)

28

29

0

0

2

1

)1(2

1

)1(2

dim

k

ii

kik

k

ii

kik

DLL

aia

aiaH

LL Hamiltonian of Dirac Fermions in Arbitrary Dimensions

0

0

2 )(

)(

dim

i

k

i

i

k

iD

LL

ai

aiH

• For odd dimensions (D=2k+1).

• For even dimensions (D=2k).

30

• The square of gives two copies of with opposite helicity eigenstates.

D ira cD

LL

erS ch ro ed in gD

LL HH 3,3

)2

3(0

02

3

222/

)( 22

223

L

Lr

M

M

pH DiracD

LL

DiracD

LLH 3

)( 3 D

LLH

• LL solutions: dispersionless with respect to j. Eigen-states constructed based on non-relativistic LLs.

.2

1

,

,1,,1

,,,

,,;

zr

zr

zr

r

jljn

jljnLL

jljn

r

LL

n

i

nE

A square root problem:

The zeroth LL:

.0

,,

,,;0

LLL

jljLL

jlj

z

z

• For the 2D case, the vacuum charge density is , known as parity anomaly.

3131

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.

G. Semenoff, Phys. Rev. Lett., 53, 2449 (1984).

Bh

ej

2

02

1

• For our 3D case, the vacuum charge density is plus or minus of the half of the particle density of the non-relativistic LLLs.

E

0n

1n

2n

1n

2n

0

0

• What kind of anomaly?

Helical surface mode of 3D Dirac LL

D

LLH 3

,

R

DH 3

• The mass of the vacuum outside M

Mp

pMH D

3D

LL

D HH 33

• This is the square root problem of the

open boundary problem of 3D non-

relativistic 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 zeroth-LL

of is singled out. Whether it is upturn

or downturn depends on the sign of

the vacuum mass.

E

j0n

1n

2n

1n

2n32

33

Conclusions

• We hope the quaternionic analyticity can facilitate the

construction of 3D Laughlin state.

• The non-relativistic N-dimensional LL problem is a N-

dimensional 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