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
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Quaternionic analyticity and SU(2) Landau Levels in 3DOutline •Introduction: complex number quaternion. •Quaternionic analytic Landau levels in 3D/4D. 3 Analyticity: a useful rule
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Quaternionic analyticity and SU(2) Landau Levels in 3D
Congjun Wu (UCSD)
Sept 17, 2014, Center of Mathematical Sciences and Applications, Harvard University
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?)
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-