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 nontrivial 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.
x  px  q  0
x 1  c 1  c 2 , x 2 ,3  c 1e
3
c 1 ,2 
3

q
2


discriminant:
i
2
3

q
  
2
 c 2e



2
i
2
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
x 2 ,3   3
4
The beauty of “complex”
• Gauss plane: 2D rotation (angular momentum)
i
• Euler formula: e  cos   i sin 
(U(1) phase: optics, QM)
• Complex analyticity: (2D lowest Landau level)
f
x
 i
f
y
 0
1
2 i

1
z  z0
dz f ( z )  f (z 0 )
• 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

t
  H
5
Further extension: quaternion (Hamilton number)
• Three imaginary units i, j, k.
q  x  yi  zj  uk
i  j  k  1
2
2
2
• Division algebra: ab  0  a  0 , or , b  0
• 3D rotation: non-commutative.
ij   ji  k ;
jk   kj  i ; ki   ik  j
• Quaternion-analyticity (Cauchy-Futer integral)
f
x
j
f
z
 i
 k
f
y
f
u

1
2
 0
2
 | q
 f (q 0 )
1
2
 q 0 | (q  q 0 )
Dq f (q )
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
•
ˆ
:
rotation R
ˆ
, rotation angle:  .
imaginary unit:
( ˆ )  i sin  cos   j sin  sin   k cos 
unit quaternion q: q
 cos
• 3D vector r
imaginary quaternion.
• 3D rotation
Hopf map S3 S2.

r  zˆ  k


r   R ( r )  qkq
1
q  S

2
 ( ˆ ) sin

2
 S
3

r  xi  yj  zk
1st Hopf map
3
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 nontrivial 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 gauge
symmetric gauge:
H
e  2

(P 
A) ,
2M
c
1
2 DLL


B
A 

zˆ  r
2
Lowest Landau level wavefunction:
complex analyticity
( a x  ia y ) LLL ( z , z )  0 , ( z  x  iy )
ax 

 LLL ( z , z )  f ( z , z )e
 z f (z , z )  0
1
2
( x  ip x )
ay 
1
2
( y  ip y )
zz
2
4lB
,
f (z , z )  f (z )
2
f ( z )  1, z , z ,...,
z
m
,...,
(m  0)
10
Advantages of Landau levels (2D)
• Simple, explicit and elegant.
• Complex analyticity  selection of non-trivial WFs.
1. The 2D ordinary QM WF  ( x , y ) 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.
 sym (z 1 , z 2 ,.... z n ) 
 (z i
i j

3
 zj )e

i
2
|z i |
2
4lB
Pioneering Work: LLs on 4D-sphere ---Zhang and Hu
Science 294, 824 (2001).
• Particles couple to the SU(2) gauge field on the S4 sphere.

H 
2
  ,   x (  i  A )  x (  i  A )
2
2 MR
2
1 a  b  5
ab
ab
a
b
b
b
a
• Second Hopf map. The spin value I  R 2 .
x a       ,

a
n i  u   i , u 

a
3
S
7
S
 S
4
R
• Single particle LLLs
x a , n i | m1 m 2 m 3 m 4   1  2  3 
m1
m2
m3
m4
4
 4D 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 𝑒1 -𝑒2 plane
with chirality determined by S along
the 𝑒3 direction.
Landau-like gauge: spatial separation of 2D
Dirac modes with opposite helicites.
Generalizable to higher dimensions.

S
eˆ1
eˆ 2
2D LLs in the symmetric gauge
H
e  2

(P 
A) ,
2M
c
1
2 DLL


A 
B
2

zˆ  r
• 2D LL Hamiltonian = 2D harmonic oscillator (HO)+ orbital
Zeeman coupling.
H
2 D LL

p
2
2M

1
2
2
Mω r
2
 ωL z , ω 
e|B|
2 Mc
, lB 
hc
eB
• H 2 D LL has the same set of eigenstates as 2D HO.
14
Organization  non-trivial topology
E 2 D , HO /(   )  2 n r  | m |  1
m
 LLL  z e
-3
-1
-2
1
0
-1
2
2
|z | /( 2 l B )
E 2 D ,LL /(   )  2 n r
3
-1
2
0
1
2
1
0
0
1
m
2
3


m
• If viewed horizontally, they are topologically trivial.
• If viewed along the diagonal line, they become LLs.
E Zeeman /(   )   m
15
3D – Aharanov-Casher potential !!
• The SU(2) gauge potential:
 1

A  B zˆ  r
2
2D :
3D :


1 
A  g    r
2
• 3D LL Hamiltonian = 3D HO + spin-orbit coupling.
H
3D
LL

P
2
2M

1
2

M  r    
2
2

L
 
| eg |
2 Mc
, lg 
c
.
| eg |

e 
M
2
2
2

( P  A  ) 
 r
2M
c
2
1
• The full 3D rotational symm. + time-reversal symm.
16
Constructing 3D Landau Levels
E 3 D , HO /(   )  2 n r  l 
1
0
3
 , 
0
2
1/2- 3/2+ 5/2- 7/2+
SOC : 2 helicity
branches
2
1
E 3 D , HO /(   )
3
l
1
j  l  .
2
p
1
 
2
2

 m r    L
2m 2
2
3D
H LL
for j 
  l
 L  
  ( l  1)  for j 
1/2+ 3/2- 5/2+
1/2- 3/2+
1/2+
j
E 3 D , LL /(   )
½+
3/2+ 5/2+ 7/2+

½+
3/2+ 5/2+ 7/2+

j
17
The coherent state picture for 3D LLL WFs
• The highest weight state j z  j  . Both L z and S z are conserved.

LLL
j  ,j z  j 
 x  iy
(r )  

0

l e
r
2

S // eˆ 3
2
/ 4lg


• Coherent states: spin perpendicular
to the orbital plane.
LLL
j

, high


eˆ1
l
( r )  [( eˆ1  i eˆ2 )  r ]   eˆ3
eˆ 2
• 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
m
 z e
2
2
|z | /( 2 l B )
,
z  x  iy,
m  0.
• 3D LLs: SU(2) group space 
quaternionic analytic polynomials.

LLL
j  , high

eˆ1

l
( r ,  )  [( eˆ1  i eˆ2 )  r ]   eˆ3 e
Phase
r
2

S // eˆ 3
eˆ 2
2
2l g
19
Right-handed triad
Quaternionic analyticity
• Cauchy-Riemann condition and loop integral.
g
x
i
g
y
1
0
2 i

1
z  z0
dz g ( z )  g ( z 0 )
• Fueter condition (left analyticity): f (x,y,z,u) quaternion-valued
function of 4-real variables.
• Cauchy-Fueter integrals over closed 3-surface in 4D.
f
x
i
f
y
 j
f
z
k
f
u
1
0
K (q )
2
1
2
q |q |


2
K (q  q 0 ) Dq f (q )  f (q 0 )
x  yi  zj  uk
(x
2
 y
2
 z
2
2 2
 u )
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 , ˆ )  
e


2
,
j
,
j
  z
• TR reversal: i  y  *
SU(2) rotation:
e

i x
2
r
2
2
4 lG
f j ,j z ( x , y , z )   1,j  ,j z  j  2 ,j  ,j z

; U(1) phase
  fj
k

  e 2f ;e

i y
2
j

  e 2f ;e
• Reduced Fueter condition in 3D:

i z
2
i
e   fe
  e
i
i

2
f
 



 i
 j
y
z
 x

f(x,y,z)

 0
• Fueter condition is invariant under rotation R ( ,  ,  ) .
If f satisfies Fueter condition, so does Rf.
( Rf )( x , y , z )  e
i

2
j
e

2
e
i

2
f ( x , y , z ),

1 
r  R r
21
Quaternionic analyticity of 3D LLL
• The highest state jz=j is obviously f j
analytic.
l
j z
j
 ( x  iy )
• 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
j j z  m 
1 jm
2


l
  f j j (c jm
j  1 / 2 m  0
 z
 jc j  m ) 
l
  f j j q jm
l 0 m 0
 z
22
Helical surface states of 3D LLs
from bulk to surface
E / E0
H bulk 
p
3D
2
2M

1
 
M r    L
2
2
2
 l ( l  1)
2
H
j 0  l0 
2D
surface

2 MR
2
  l
 
 
l     L  R eˆ r  ( p   )
1
2
j
H
2D
plane
 
ˆ
 v f (l  l0 ) / R  v er  ( p   )  
py
R
kf
px
k f  l0  / 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)
2D complex analyticity

 LLL ( z , z )  f ( z )e
f
x
 i
f
y
zz
x
 i
f
y

 0
t
 j
f
z
x
 k
 i
f
u
f
y
 0
 j


x
 0
2D helical Dirac surface mode

 0
4D: quaternionic analyticity
f
 (t , x )  f ( x  t )
2
4lB
3D: quaternionic analyticity
f
1D chiral edge mode
t
 x

x
 y

y
 0
3D Weyl boundary mode
f
z

t
 x

x
 y

y
 z

z
 0
24
Outline
• Introduction: complex number  quaternion.
• Quaternionic analytic Landau levels in 3D/4D.
Analyticity : a useful rule to select wavefunctions for nontrivial 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  v F {( p x 
2D
e
c
A x ) x  ( p y 
e
c

A y ) y },
A 
B
2

zˆ  r
• Rewrite in terms of complex combinations of phonon operators.
H
2D
LL

0
2vF  

  i ( a  ia )
lB
x
y



i ( a x  ia y ) 
,

0

ai 
• LL dispersions: E  n     n
• Zero energy LL is a branch of half-fermion
modes due to the chiral symmetry.
2

LL
0; m
z
 
0
m

e



|z |
2
4lB
.
E
1
(
xi
2 lB
i
lB

p i ),
i  x, y.
n2
n 1
n0
n  1
n  2
26
3D/4D LL Hamiltonian of Dirac Fermions
{1,i }
{1,i , j , k }
(1,i  x ,i  y ,i  z )
2D harmonic oscillator
{a x , a y }
4D harmonic oscillator
{a u , a x , a y , a z }
• “complex quaternion”:
a u  ia x  ja y  ka z
• 4D Dirac LL Hamiltonian:
H
4 D Dirac
LL
  
0

2  a u  ia x  ja y  ka z

0
l 0 


 
2  a u  i   a
au




 ia x  ja y  ka z 

0

 

au  i  a 


0

27
3D LL Hamiltonian of Dirac Fermions
3 D Dirac
H LL

 
0
i   a  l 0 
  
 
 2
2    ( p  i  r / l 0 )
0 
0
 


 
2   i  a


  ( p  i r / l0 ) 
0
2



• This Lagrangian of non-minimal Pauli coupling.
L   {i (  0  0  v  i  i )} 
 0i  
i
2
[  0 ,  i ],
F
0i

x
v
lg
0i
  0i F  ,
i
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).
D  dim
H LL
0
 


2   i i( k )  a i

i i  a i 


0

(k )
• For even dimensions (D=2k).
H
D  dim
LL


0
 

k
2 
( k  1)

a

i


 i ai
2k
i 1

k
 a

2k
 i 

a 




( k  1) 
i
i
i 1
0
29
A square root problem:
3 D , S ch ro ed in ger
3 D D ira c
H LL
 H LL
• The square of H LL3 D Dirac gives two copies of ( H LL3 D )  with opposite
helicity eigenstates.
  3

2
3 D Dirac
2
 L   
( H LL
)
p
M 22
2


 r  
 / 2
2M
2

0




  3
 ( L     ) 
2 
0
• LL solutions: dispersionless with respect to j. Eigen-states
constructed based on non-relativistic LLs.
The zeroth LL:
E n    nr ,
LL
r
  n ; j ,l , j
 n ,j
LL
r
1 
,l , j


2   i  n 1 , j ,l 1 , j
r
z
r

z

z

.


 0 ; j ,l , j
LL
z



0
LLL
j ,l , j z

.


30
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.
1e
2
B ,
• For the 2D case, the vacuum charge density is j 0  
2 h
known as parity anomaly.
G. Semenoff, Phys. Rev. Lett., 53, 2449 (1984).
• 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

0

 0
• What kind of anomaly?
n2
n 1
n0
n  1
n  2
31
Helical surface mode of 3D Dirac LL
3D
H
• The mass of the vacuum outside M  
H   H LL
3D
3D
3D
H
 M
   
 p 
R
 
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 zeroth-LL
of is singled out. Whether it is upturn
or downturn depends on the sign of
the vacuum mass.
3D
H LL ,
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