quaternionic states of matter from synthetic gauge...
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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.