Quaternionic analyticity and SU(2) Landau Levels in 3D Yi Li (UCSD Princeton), Congjun Wu ( UCSD)...
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Transcript of Quaternionic analyticity and SU(2) Landau Levels in 3D Yi Li (UCSD Princeton), Congjun Wu ( UCSD)...
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