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

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

Yi Li (UCSD Princeton), Congjun Wu （ UCSD)

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

Collaborators: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

32

232

13,2211 ,ii

ececxccx

32,1 2

q

c

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

yf

ixf )()(

121

00

zfzfdzzzi

Schroedinger Eq:

• Three imaginary units i, j, k.

ukzjyixq 1222 kji

• Division algebra:

jikkiikjjkkjiij ;;

• Quaternion-analyticity (Cauchy-Futer integral)

0

uf

kzf

j

yf

ixf

)(

)()(||

121

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

AAce

PM

H DLL

ˆ

2,)(

21 2

2

)(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

ab AixAixMR

H

• Second Hopf map. The spin value .

R4

3

7 SS

S uunx ii

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

ia mmmmnx

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-plane with chirality determined by S along thedirection.

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

eBhc

l,Mc

e|B|ω,ωLrMω

Mp

H BzLLD 22

12

222

2

rzB

AAce

PM

H DLL

ˆ

2,)(

21 2

2

• 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 mnE rHOD

rLLD nE 2)/(,2

m0 1 2 3

-1 0 1 2

1616

222

222

3

2)(

2

12

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 HODE

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 LLDE

j½+ 3/2+ 5/2+ 7/2+

½+ 3/2+ 5/2+ 7/2+

Lrmm

pH D

LL

222

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.

jjz

22 4/,

0)( g

z

lrl

LLLjjj eiyxr

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

2e1e

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

19

• 3D LLs: SU(2) group space quaternionic analytic polynomials.

2

2

3

2ˆ21, ])ˆˆ[(),( gl

r

el

high

LLLj ereier

• 2D LLs: complex analytic polynomials.

.0,)2/(|| 22

miy,xzez BlzmsymLLL Phase

Right-handed triad

• 1D harmonic levels: real polynomials.

3ˆ// eS

2e1e

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 )()()(21

002qfqfDqqqK

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 jjjjjjl

r

jj

jjLLLjj jzyxfer ,,2,,1,

4

,,2

,,1

, ),,()ˆ,(2

2

0

f(x,y,z)z

jy

ix

• 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 zyx 222222 ;;

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

mjjqfjccfcff

zz

zz

• All the LLL states are quaternionic analytic. QED.

22

Helical surface states of 3D LLs

from bulk to surface

21

00 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

222

3

2

1

2

)(ˆ/)( 02

pevRllvH rfD

plane

)(ˆ peRLl r

yp

xpfk

Rlk f /0

R

23

Analyticity condition as Weyl equation (Euclidean)

24

2D complex analyticity

0

yf

ixf

0

yxt yx

0

xt

24)(),( Bl

zz

LLL ezfzz

)(),( txfxt

1D chiral edge mode

0

zf

jyf

ixf

3D: quaternionic analyticity

2D helical Dirac surface mode

0

uf

k

zf

jyf

ixf

3D Weyl boundary mode

0

zyxt zyx

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

LL Ac

epA

c

epvH

• Rewrite in terms of complex combinations of phonon operators.

,0)(

)(022

yx

yx

B

FDLL iaai

iaai

l

vH

• LL dispersions: nE n

E

0n

1n2n

1n2n

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

02

0

02

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

gii

lx

Fi

Flv

viL

• 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

dimk

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

LLai

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 ir a cDLL

erS ch r o ed in gDLL HH 3,3

)2

3(0

02

3

222/

)( 22223

L

Lr

M

M

pH DiracD

LL

DiracDLLH 3

)( 3DLLH

• 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

0 2

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

1n2n

1n2n

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

1n2n

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