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
