Dirichlet Type Problem for 2D Quaternionic Time-Harmonic ...
Quaternionic states of matter from synthetic gauge...
Transcript of Quaternionic states of matter from synthetic gauge...
Congjun Wu University of California, San Diego
Quaternionic states of matter from synthetic gauge fields
ISCAP Taiyuan,
Jun 16, 2014
Collaborators: Yi Li (UCSD Princeton), X. F. Zhou (USTC, Hefei)
2e
1e
S
Q-BEC with Hopf invariant 3D Q-analytic Landau level
2
AcknowledgementCollaborations on related topics: Vincent Liu, Zi Cai, Luming Duan, Yue Yu, K. Intrilligator, S. C. Zhang.
Helpful discussions with Jorge Hirsch, Kazuki Hasebe, T. L. Ho, Jiang-ping Hu, Nai Phuan Ong, Cenke Xu, Kun Yang, Yue Yu, S. C. Zhang, Fei Zhou
Support from NSF-DMR, AFOSR.
Ref. 1) Y. Li, X. F. Zhou, C. Wu, arxiv1205.2162.2) Y. Li, C. Wu, PRL 110, 216802 (2013), arxiv:1103.5422.3) Y. Li, Xiangfa Zhou, C. Wu, Phys. Rev. B 85, 125122 (2012).4) Review on synthetic SO coupling: X. F. Zhou, Y. Li, Z. Cai, C. Wu, J
Phy. B 46, 134001 (2013).
Other related work1) C. Wu, I. Mondragon-Shem, X. F. Zhou, Chin. Phys. Lett. 28, 097102, 2011(arXiv:0809.3532).
2) Y. Li, S. C. Zhang, C. Wu, PRL 111, 186803(2013).3) Y. Li, K. Intrilligator, Y. Yu, C. Wu, PRB 85, 85132(2012).
Outline
• Real numbers complex numbers quaternions (Q).
• Bosons: real (positive) BEC complex BEC Q-BEC
unconventional symm. beyond the “no-node” theorem.
p-wave BEC, and topological spin textures
• Fermions: Q-analytic Landau levels in 3D.
3
harmonic potential+ spin-orbit couplingCauchy-Riemann-Fueter condition.
• Complex quaternions: 3D Landau levels of Dirac fermions.
033 xx
History: how did people accept “i”?
• Not because of the “ridiculous”Eq. , but solving the cubic Eq (Cardano formula).
32
32
pq
3
2
23
2
13,2211 ,ii
ececxccx
32,1
2
qc
03 qpxx
• Start up with real coefficients, and end up with three real roots, but no way to avoid “i”.
4
12 x
!!!1,1
discriminant:
3,0 3,21 xx
The beauty of complex and elegancy
• 2D rotation: Gauss plane.
• Euler formula:
• Complex analysis based on complex analyticity:
• Applications: algebra fundamental theorem; Riemann hypothesis – distributions of prime numbers.
1ie
Quantum mechanics: the most important quantity in SchroedingerEq is not hbar but “i”.
Ht
i
5
0
y
gi
x
g)()(
1
2
10
0
zgzgdzzzi
Quaternion: a further extension
• Three imaginary units i, j, k.
ukzjyixq 1222 kji
• Hamilton: Non-commutative division algebra (3D rotation is non-commutative).
• Division algebra:
jikkiikjjkkjiij ;;
• Q-analyticity (Cauchy- Futer integral)
0
u
fk
z
fj
y
fi
x
f)()(
)(||
1
2
10
02
02
qfqfDqqqqq
0,00 baab
Quaternion plaque: Hamilton 10/16/1843
1222 ijkkjiBrougham bridge, Dublin
7
3D rotation as 1st Hopf map
• : imaginary quaternion unit:
3D rotation: unit quaternion q:2D rotation: unit complex phase
cossinsincossin)ˆ( kji
3
2sin)ˆ(
2cos Sq
• 3D rotation and Hopf map.
8
• 3D vector r imaginary quaternion. zkyjxir
• Rotation axis , rotation angle: .
21 Sqkq 3Sq
1st Hopf map
1
ˆ
qkqr
kzr
Outline
• Real numbers complex numbers quaternions (Q).
• Bosons: real (positive) BEC complex BEC Q-BEC
unconventional symm. beyond the “no-node” theorem.
p-wave BEC, and topological spin textures
• Fermions: Q-analytic Landau levels in 3D.
9
harmonic potential+ spin-orbit couplingCauchy-Riemann-Fueter condition.
• Complex quaternions: 3D Landau levels of Dirac fermions.
• “No-node” theorem: many-body ground state wavefunctions of bosons in the coordinate representation are positive-definite .
10
0),...,(21
n
rrr
N
jiji
N
iiex
iN
i
rrVrVM
H )()(2
int1
22
1
• A ground state property valid under general conditions (no rotation).
R. P. Feynman
Conventional BECs based on positive numbers
• No-go: Conventional BEC CANNOT break time-reversal symmspontaneously.
An intuitive proof
0),...,(21
n
rrr),...,( 21 nrrr |),...,(| 21 nrrr
)(|),..(|
)(|),..(||),...(|2
...||
int
2
1
1
2
1
2
11
2
1
jiji
n
n
iiexnni
n
in
rrVrr
rUrrrrm
drdrH
11
• More formally and rigorously, c.f. Perron-Frobenius theorem.
Conventional BEC (invariant under rotations, s-wave-like)
1212
• “No-node” theorem forbids unconventional symmetries, say, p, d, etc.
• Cf. conventional superconductivity: the pair WF belongs to the trivial (s-wave) representation of the rotation group.
)()()(
21
21 kekdrrrrik
k
k
• High Tc d-wave superconductivity tested by phase sensitive Josephson junction experiment by Van Harlingen et al.
Unconventional BEC: beyond “no-node”
• The condensate wavefunction belongs to a non-trivial (non-s-wave) representation of the lattice point group.
)(r
• No-node theorem does NOT apply to excited states.
• Example: the p-orbital bands can have degenerate band minima.
Interaction selects complex p+ip UBEC:
W. V. Liu, and C. Wu, PRA, 2006; Zi Cai, C. Wu, PRA 84, 033635(2011).Review: C. Wu, Mod. Phys. Lett. 23, 1(2009). Wirth, Olschlager, Hemmerich, Nat. Phys. 2011, N. Y. Kim, et al, Nat. Phy. Physics 2010.
13
)()()(21
rirrKK
K1 K2
-K1-K2
• Solid state experiments: d-wave BEC in exiton-polarition lattices (Kim, Yamamoto, Wu)
Outline
• Real numbers complex numbers quaternions (Q).
• Bosons: real (positive) BEC complex BEC Q-BEC
unconventional symm. beyond the “no-node” theorem.
p-wave BEC, and topological spin textures
• Fermions: Q-analytic Landau levels in 3D.
14
harmonic potential+ spin-orbit couplingCauchy-Riemann-Fueter condition.
• Complex quaternions: 3D Landau levels of Dirac fermions.
Two-component spinor quaternion
)ˆ,(r
• A quaternion can also be understood as a pair of complex numbers just like complex number is a pair of real numbers.
15kj
ir
ImRe
ImRe)ˆ,(
• Spinor wavefunction quaternion wavefunction;Spin density distribution 1st Hopf-mapping.
)ˆ,(2
)ˆ,()ˆ,( rrrS
kkSjSiS zyx2
1
The 2D version: BEC with Rashba SO coupling
16
• Free space: Spin spiral stripe for density-only interactions. (“order from disorder” mechanism beyond G-P level.)
• Spin-textures in SO coupled excitoncondensates observed from photoluminescence by L. Butov.
C. Wu , I. Mondragon Shem, and X.F. Zhou, Chin. Phys. Lett. 28, 097102 (2011)
(arXiv:0809.3532). – We are grateful to Chinese Physics Letters for the
acceleration of publication.
Exciton: A.A. High et al., Nature 483, 584 (2012). A.A. High et al., arXiv:1103.0321.
Cold atoms: Y. J. Lin et al, Nature 2011.
• Solid state SO coupled boson system: excitons in semiconductors.
Trap: Prediction of spontaneous generation of “baby” skyrmion spin-texture and half-quantum vortex.
3D SO coupling in hyperfine states of 40K atoms
)(2
1
2
2222
3
irmm
H D
Artificial SO coupling from light-atom interaction
).8
177,0,0(),0,
2
3,
2
1(
),0,1,0(),0,2
3,
2
1(
43
21
kk
kk
.166.0
).ˆ)(ˆˆ(166.0
m
eIeeA zzyyxx
Yi Li, Xiangfa Zhou, C. Wu, PRB
85, 125122 (2012).
Anderson, Juzeliunas, Spielman
and Galitskil,.Phys. Rev. Lett.
108, 235301 (2012).
Spherical rotator subjected to a fundamental monople
• Low energy SO sphere with radius kso. Dimensionless SO coupling strength
,)()(22
2
122
2
1 kAiMrMrVkTTex
22
22
2
1)(
2rmi
mH
T
sok
22
kAkd
• Momentum space:
TsoTso lkm
lmk
,,
• Angular momentum quantization change to half-integer values.18
19
• LLL wavefunctions through SO coupled harmonics:
Yi Li, Xiangfa Zhou, C. Wu, PRB 85, 125122 (2012); Gosh et al, PRA 84,
053629 (2011).
E
j0
rn
1r
n
• Angular dispersion suppressed by strong SO coupling forming nearly flat Landau levels.
TrTjn njj
Ezr
)2
1(
2
)1(2,
TR invariant parity breaking 3D Landau-levels
The single-particle ground state (mixed s and p-partial wave)
)sin
cos)(
0
1)(()ˆ,( 10
2
2
1
2
2
isoso
l
r
jj erkijrkjer T
z
• Quaternion representation:
20
)sin)ˆ((cos|)(|
|)(|
ImReImRe)ˆ,(
)()ˆ(
r
er
kjir
r
argument imaginary unit
• Imaginary unit solid angle direction
cossinsincossin)ˆ( kji
• Along any , winds in (1, )-plane as r increases. at zero points of g(r).
BEC WF as a quaternionic defect
• Mapping from 3D real space to quaternionic phase space – 3D skrymion.
2|)(| rf
2|)(| rgfgr /)(tan
)sin)ˆ((cos|)(|)ˆ,( rr
3S
• Condensation WF based on single-particle GS.
nrn)(
)ˆ()()ˆ( re
cossinsincossin)ˆ( kji
Quaternionic defect with non-zero Hopf invariant
22
• 3D spin distribution: the horizontal cross-section z=0 is a baby (2D) skrymion.
xy-plane
)2cossin)(cos()(
cos
cossin
cos,sin
sin,cossinsin)(
)(
)(
222
2
rrS
rrS
rS
z
y
x
5.0/ Tlz
See the linking number
• The trajectory of spin with the same orientation form a closed loop (in our case, the loop may be cut by the boundary).
• For any two loops, they link each other.
23
)2cossin)(cos()( 222 rrSz
• For ; reduced to a straight line of the z-axis
For ; a circle in the equatorial plane
,0,// zS
2,
2ˆ//
zS
3D quaternion analogy to 2D Abrikosov vortex lattice
24
2.0/ Tlz
• As SO coupling goes strong, the condensate WF mixes different single particle states within the same LL driven by interaction.
• Rotational symmetry is broken forming texture lattices.
0/ Tlz
1,8.0,15 c
Outline
• Real numbers complex numbers quaternions.
• Positive BEC complex BEC quaternionic BEC
go beyond the “no-node” theorem
p-wave BEC, and topological spin textures
• Quaternionic analytic Landau levels in 3D.
25
harmonic potential+ spin-orbit couplingCauchy-Riemann-Fueter condition.
• Complex quaternions: 3D Landau levels of Dirac fermions.
26
3D SU(2) LLs
2D LLs (continuum, flat spectra and complex analyticity); IQHE & FQHE.
2D QAH, Chern Insulator, TI; Frac-TI (lattice)
(symm-like gauge) degenerate helical modes;math: quaternionic analyticity
Dirac fermion: flat band of zero modes from non-minimal coupling; novel anomaly?
(Landau-like gauge) 4DQHE: spatially separated (3+1)d chiral anomaly
Exp. realization: cold atom; semiconductors?
3D TI (lattice)Frac-TI (3D
Laughlin WF)?
3ˆ// eS
2e
1e
QM: (Final exam for grad students. )harmonic oscillator + spin-orbit coupling
0
xk
yk
u
zk
1ˆ
1ˆ
BEh
enju
2
0
0
2
3
ai
aiH LLDiracD
symm
3D quaternionic (SU(2)) Landau levels
.0,
,)2/(|| 22
miyxz
ez BlzmsymLLL
Return to Landau levels (a review of 2D)
• Simple, explicit and elegant.
xiklkyyLandau
LLL
xBx ee)2/())(( 22
0
Landau gauge: 1D harmonic oscillators with center-shift (kx dep.)spatial separation of chiral modes.
xBkly 2
0
.0,,)2/(|| 22
miyxzez Blzmsym
LLL
Symmetric gauge: analytic functions of complex variables (chiral).
y
x
0
0xk
0xk
Comparison of symm gauge LLs in 2D and 3D
28
• 3D LLs: SU(2) group space
quaternionic analytic polynomials.
2
2
3
2
ˆ21,])ˆˆ[(),( gl
r
e
l
high
LLL
jereier
• 2D LLs: complex analytic polynomials.
.0,,)2/(|| 22
miyxzez Blzmsym
LLL Phase
Right-handed triad
• 1D harmonic levels: real polynomials.
3ˆ// eS
2e
1e
Conclusions
29
• Quaternion is a beautiful and useful concept.
• The 3D spin-orbit coupled BEC exhibit the 3D structure of quaternionic defects with non-zero Hopf invariants.
• Landau levels are generalized to 3D with the full rotational symmetry and TR symmetry.
• Quanternionic analyticity is a useful criterion to select good 3D wavefunctions for non-trivial topology.
Proposal: Quantum Mechanics class instruction (Final or qualexams).
k
RashbaE
sok
Landau-level (LL) type single particle quantization (2D)
• Ring rotor in momentum space subjected to a flux half integer quantized angular momentum.
,)()(22
2
122
2
1 kAiMrMrVkTTex
• The Rashba ring in momentum space with the radius kso. Dimensionless SO coupling strength .
1,/ TsoTTlkml
C. Wu et al arXiv:0809.3532; C. Wu et al,
Chin. Phys. Lett. 28, 097102 (2011). Hui Hu,
et al PRL 2012. 30
• Angular dispersion is suppressed by strong SO coupling forming nearly flat Landau levels.
const 2 2
2
,
T
z
Trjn
jnE
zr
dkkA )(
SO coupled bosons in trap: spin textures and half-quantum vortex
31
• Weak SO coupling. Condensate WFs:
,)(
)(2/1
ierg
rf
• Friedel-like oscillation of (up), and (down) at the pitch value of kso.
• The relative phase shift between f(r) and g(r).
2|)(| rf2|)(| rg
2|)(| rf
2|)(| rg
2
• 2D (baby) skyrmion type spin texture formation and half-quantum vortex.
sin,cos,|||| 22 fgSfgSgfSyxz
31
2/1 SLjz
)(rS
Spin lines with non-zero Hopf invariant
32
• 3D spin configuration with non-trivial Hopf-invariant; twisted baby (2D) skrymion as moving along z-axis.