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