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.