order and disorder in su(3) and su(4) heisenberg systems

Order and disorder in

SU(3) and SU(4) Heisenberg systems

Correlations and coherence in quantum systemsÉvora, Portugal, 8-12 October 2012

K. PencInstitute for Solid State Physics

and Optics,Wigner Research Centre,

Budapest, Hungary

1Wednesday, October 10, 2012

Supported by: Hungarian OTKA and Swiss National Foundation

In collaboration with:

Miklós Lajkó ISSPO, Wigner RC, Budapest,

Tamás A. Tóth, Laura MessioFrédéric Mila

EPF Lausanne

Bela Bauer, Philippe Corboz,Matthias Troyer

ETH Zürich

Andreas M. Läuchli Uni. Innsbruck


What are the SU(N) symmetric Heisenberg models that we are interested in?

H =�

i,j

Pi,j Pi,j is the transposition operator

i j i jPi,j | 〉→ | 〉 N species on each site

that are treated equally.

simplest example: SU(2) S=1/2 (fundamental representation) [but not the S=1 !]

Pij |βiαj� = |αiβj�


Why do we care about SU(3) or SU(4) Heisenberg models?

(i)They are interesting and intelectually challenging ( funding agencies ?)

(ii) Nostalgia from our younghood, when we all wanted to become particle physicist (SU(3) and quarks)

(iii) Cold atomic gases

(iv) Spin models

(v) Spin-orbital models (SU(4))


SU(2) quantum Heisenberg models

square and honeycomb (bipartite) lattices: happy (mean field) antiferromagnets

triangular lattice: frustrated antiferromagnet

?

happ

y

comp

romi

se

compromise

compromise

3-sublattice 120 degree state5Wednesday, October 10, 2012

SU(2) quantum Heisenberg models

kagome lattice: highly frustrated antiferromagnet, mean field ground state macroscopically degenerate.

Ground state debated, likely a Z2 spin liquid

frustrated square lattice: dimerized state


SU(2) vs. SU(3) - two sites

⊗ = ⊕22 1 3× = +

½ ⊗ ½ = 0 ⊕ 1

Addition of two S=½ SU(2) spins:

using Young diagrams:

↑ or ↓ spin

|↑↓〉−|↓↑〉 singlet, odd (anti-symmetrical)

|↑↑〉, |↑↓〉+|↓↑〉, |↓↓〉 triplet even (symmetrical)

⊗ = ⊕33 3 6× = +

Addition of two SU(3) spins:

|aa〉, |bb〉, |cc〉, |ab〉+|ba〉, |ac〉+|ca〉, and |bc〉+|cb〉.even (symmetrical)

|ab〉−|ba〉, |ac〉−|ca〉, |bc〉−|cb〉: odd (anti-symmetrical).

|a〉, |b〉, and |c〉.

H = P12

P12(|αβ〉 − |βα〉) = −(|αβ〉 − |βα〉)

P12(|αβ〉 + |βα〉) = +(|αβ〉 + |βα〉) E=+1, even wave function

E=−1 , odd wave function1 2


SU(3) singlet

in the SU(3) singlet the spins are fully entangled: we cannot write it in a product form

= |ABC〉 + |CAB〉 + |BCA〉 − |BAC〉 − |ACB〉 − |BCA〉spins fully antisymmetrized

SU(3) irreps on 3 sites

1 2 × 83 = +

Addition of three SU(3) spins (27 states):

⊕⊕ 2×=⊗ ⊗

+ 10× ×3 3


What methods do we use?

(i) Variational - site factorized wave function(ii) Flavor wave calculations (iii) Exact diagonalization of small clusters(iv) iPEPS: infinite project entangled pair states(variational

approach based on tensor ansatz)(v) Variational: Gutzwiller projected fermionic wave

functions


Variational (classical) approacha site-product wave function for SU(3):

minimal, when the di and dj on the bond are orthogonal

two different colors on a bond

Evar =�Ψ|H|Ψ��Ψ|Ψ� = J

�

�i,j�

��di · dj

��2

|ψi� = dA,i|A�i + dB,i|B�i + dC,i|C�i

|Ψ� =�

i

|ψi�


SU(3) flavour-wave theory

States in the fully symmetric multiplet can be represented by 3 Schwinger

bosons aA,aB and aC

1 2 ... M1

N. Papanicolaou, Nucl. Phys. B 305, 367 (1988) A. Joshi et al. PRB 60, 6584 (1999)

We enlarge the fundamental to the fully symmetric representation of M boxes.

The site product wave function is the “classical” solution (no quantum entanglement between sites)

|ψi� = dA,i|A�i + dB,i|B�i + dC,i|C�i

|Ψ� =�

i

|ψi�

|ϑ,ϕ� = cosϑ

2| ↑�+ sin

ϑ

2eiϕ| ↓�

cf. SU(2) spin coherent state for SU(2)

a†A,iaA,i + a†B,iaB,i + a†C,iaC,i = M

Pij =�

µ,ν∈{A,B,C}

a†µ,ia†ν,jaν,iaµ,j


SU(3) flavour-wave theory

a†A,j =�

µ

dµ,ja†µ,j

a†B,j =�

µ

eµ,ja†µ,j

a†C,j =�

µ

fµ,ja†µ,j

Local coordinate system

1/M expansion:

a†A,i, aA,i →�M − a†B,iaB,i − a†C,iaC,i

→√M − 1

2√M

�a†B,iaB,i + a†C,iaC,i

�+ . . .

fd

e

Holstein-Primakoff bosons

|Ψ� =�

i

�a†i

�M|0�

Pij → M�(ei · dj)a

†B,i + (fi · dj)a

†C,i + (di · ej)aB,j + (di · fj)aC,j

�

×�(ei · dj)aB,i + (fi · dj)aC,i + (di · ej)a†B,j + (di · fj)a†C,j

�−M

quadratic in operators: we know how to diagonalize it (spin wave)

H = −MJL+M�

ν

�

k

ων(k)

�α†ν(k)αν(k) +

1

2

�


The models

(i) SU(3) model on the triangular lattice

(ii) SU(3) model on the square lattice

(iii) SU(4) model on the square and cubic lattice

(iv) SU(4) model on the honeycomb lattice


The models






The fate of SU(3) on triangular lattice

“classical” solution?crystal of singlets?

SU(3) classical state is perfectly happy on the triangular lattice - the 3 mutually perpendicular dʼs form a

3 sublattice structure.

SU(2) frustrated!


SU(3) on triangular lattice - exact diagonalization

Signature of SU(3) breaking in the excitation spectrum:

Anderson towers compatible with 3 sublattice order

C2 - Casimir operator, analog of the total spin S^2

02 6 12 20 30 42S(S+1)

16

18

20

22

24

26

28

E

0 3 6 8 12 15 18C2

Kother

E


0 0.1 0.2 0.3 0.4 0.5

0.3

0.4

0.5

0.6

0.7

0.8m

1/D

0 0.05 0.1 0.15 0.2 0.251/L

iPEPS 3x3 unit cellDMRGLFWT

0 0.1 0.2 0.3 0.4 0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

E s [J]

1/D

0 0.05 0.1 0.15 0.2 0.251/L

iPEPS 2x2 unit celliPEPS 3x3 unit cellDMRGLFWTED extrap.

Unbiased calculation: iPEPS, DMRG

B. Bauer, P. Corboz, A. M. Läuchli, L. Messio, K. Penc, M. Troyer, F. Mila:Phys. Rev. B 85, 125116/1-11 (2012)


http://link.aps.org/doi/10.1103/PhysRevB.85.125116


The models






SU(3) classical solutions: macroscopically degenerate

Order by disorder: quantum fluctuations select the

ground state

EZP =M

2

�

ν

�

k

ων(k)

All bonds happy at the mean field level, frustration due to abundance of choices


(b)(a) (b)(a)

SU(3) flavour-wave: dispersions, zero point energy

(a)

−π0

πkx −π

0

π

ky 0

4

ω/J

(b)

−π0

πkx −π

0

π

ky 0

4

ω/J

(c)

−π0

πkx −π

0

π

ky 0

4

ω/J

(a)

−π0

πkx −π

0

π

ky 0

4

ω/J

(b)

−π0

πkx −π

0

π

ky 0

4

ω/J

(c)

−π0

πkx −π

0

π

ky 0

4

ω/J

EZP = 1.27 EZP = 1.68The winner ! T. A. Tóth, A. M. Läuchli, F. Mila, and K. Penc:

PRL 105, 265301 (2010).20Wednesday, October 10, 2012

http://link.aps.org/doi/10.1103/PhysRevLett.105.265301


02 6 12 20 30 42S(S+1)

16

18

20

22

24

26

28

E

0 3 6 8 12 15 18C2

Kother 7

8

9

10

11

12

0 3 6 8

E/J

C2

(a)N=9

16

16.5

17

17.5

18

0 3 6 8C2

(b)N=18

W even A1

M A1 B2 E1

Unbiased calculation: ED, Anderson towerstriangular lattice square lattice

(b)(a)

additional Z2symmetry


Unbiased calculation: iPEPS, DMRG

0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6m

1/D

0 0.05 0.1 0.15 0.2 0.251/L

iPEPS 3x3 unit cellDMRG

0 0.1 0.2 0.3 0.4 0.5

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

E s [J]

1/D

0 0.05 0.1 0.15 0.2 0.251/L

iPEPS 2x2 unit celliPEPS 3x3 unit cellDMRGLFWTED extrap.

B. Bauer, P. Corboz, A. M. Läuchli, L. Messio, K. Penc, M. Troyer, F. Mila:Phys. Rev. B 85, 125116/1-11 (2012)




structure of the flavor wave Hamiltonian

H = (a† + b)(b† + a)

+ (b† + c)(c† + b)

+ (c† + d)(d† + c)

H = (a† + b)(b† + a)

+ (b† + c)(c† + b)

+ (c† + d)(d† + c)

each term separately EZP = 0 the 3-site term gives EZP > 0

a b c d a b c d

nearest neighbor also of different color


The models






SU(4) ladder

!"#!$%&#"'%(")*+*)$$#,

#-.%/)/%("'0

−0.25 0 0.25 0.5 0.75 1 1.25 k

−16.5

−16

−15.5

−15

−14.5

−14

−13.5

−13

−12.5

E(k

)/J

even state odd state

12 1-

3 3 4 45 67

!"#!$%&#"'%(")*+*)$$#,

#-.%/)/%("'0

−0.25 0 0.25 0.5 0.75 1 1.25 k

−16.5

−16

−15.5

−15

−14.5

−14

−13.5

−13

−12.5

E(k

)/J


12 1-

3 3 4 45 67

!"#!$%&#"'%(")*+*)$$#,

#-.%/)/%("'0

−0.25 0 0.25 0.5 0.75 1 1.25 k

−16.5

−16

−15.5

−15

−14.5

−14

−13.5

−13

−12.5

E(k

)/J


12 1-

3 3 4 45 67

M. van den Bossche et al., Phys. Rev. Lett. 86, 4124 (2001).

GS twofold degenerate (translational invariance broken):

gapped excitation spectrum, situation similar to

J1-J2 SU(2) Heisenberg chain:


SU(4) on 2D-square lattice

!"#$%"&'%"()**+,'-"./"0,$%1"2"."!34$"567778

'9$+:"&3$1)%$43;$:3)%-"<="*3:'*

>!?)4&"&'1'%'@$:'"1@)A%&"*:$:'B

CD)!&3E'%*3)%$4"*FA$@'"4$::3+'

'9+3:$:3)%*"5B8G

!!"#

!!$#

!!%#

!!&#

!!'#

()*+,-.( /0,.)1,-.(

2!

2'

23

2$

2'

2&

24

567

89:

Ground state 4-fold degenerate?

Z2 liquid ? Wang & Viswanath (PRB 2009)


SU(4) on 2D-square lattice: breaking SU(4)?

E/N = −1.293 J E/N = −0.727 J

E/N = −1.363 J E/N = −3/2 J

(b)(a)

(c) (d)

Order by disorder:Assuming on-site long

range order, which configuration has the

lowest zero-point energy?

The number of good mean field solution is

highly degenerate.


SU(4) on 2D-square lattice: iPEPS

Tensor network method, with D-dimensional links between sites.

E/N = −1.293 J E/N = −0.727 J

E/N = −1.363 J E/N = −3/2 J

(b)(a)

(c) (d)

D = 2 and a unit cell 4 × 4. The bond- energy pattern of the

ground state is similar to that of the plaquette state selected by

LFWT.



V V V V

A

B

B

A

H

HD = 12 and a unit cell 4 × 2

dimerization and Neel-like state: both spatial and the SU(4)

symmetry is broken

⊗ = ⊕44 6 10× = +

the 6 dimensional irreducible representation is realized on the

dimers, can Neel order



0 0.1 0.2 0.3 0.4 0.5−1.1

−1

−0.9

−0.8

−0.7

−0.6

Es

1/D

2x2 unit cell4x2 unit cell4x4 unit cellVMCED N=16ED N=20

0 0.1 0.2 0.3 0.4 0.5

−0.8

−0.6

−0.4

−0.2

Eb

1/D

H bondsV bondsA/B bonds

0

0.2

0.4

0.6

0.8

CO

P

COP=∑

k|n k − 1/4 |

(c)

(a) (b)

V V V V

A

B

B

A

H

H

bond

ene

rgy

colo

r or

der

para

met

er

ener

gy


0 4 6SU(4) Casimir

-17.5

-17

-16.5

-16

-15.5

-15

-14.5

-14

Tota

l Energ

y [J

]

(0,0) A1

(0,0) B1

(π,0) oe

(π,π/2) e

(π/2,π/2) e

(π,0) ee

(π,π) B2

(π,π/2) o

(π/2,0), e

-π-π/2 0 π/2 π

kx

-π

-π/2

0

π/2

π

ky

(a) (b) (c) (d)

SU(4) on 2D-square lattice: ED

Color structure factor ⟨P(i, j)(a,b)P(k,l)(a,b)⟩bond energy correlations ⟨PijPkl⟩ − ⟨Pij⟩2


SU(4) on 2D-square lattice: our scenario

Dimerization: 6-dimensional irreps are formed,

they can Néel order

P. Corboz, A. M. Läuchli, K. Penc, M. Troyer, F. Mila, PRL 107, 215301 (2011).

SU(4) can be lowered to Sp(4) in cold atoms:H. Hung, Y. Wang, and C. Wu, Phys. Rev. B 84, 054406 (2011).E. Szirmai and M. Lewenstein, EPL 93, 66005 (2011).




The models






(b)(a)(a)

(c) (d)

(b)

Lifting of the degeneracy in flavor wave theory

Basic building blocks:nearest (mean field) and next nearest (fluctuations) neighbor colors different.

The allowed 9-spin configurations, not lifted in harmonic approximation:

iPEPS, D=6

A linear defect


0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

! E

b

1/D

iPEPS 2x2 unit cell

0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

1/D

m

iPEPS 2x2 unit celliPEPS 4x4 unit cell

(c) (d)

(a) (b)

(b)(a)(a)

(c) (d)

(b)(b)(a)(a)

(c) (d)

(b)

iPEPS - local magnetization vanishes

4x4 unit cell, D=6

2x2 unit cell,D=6


0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

! E

b

1/D

iPEPS 2x2 unit cell

0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

1/D

m

iPEPS 2x2 unit celliPEPS 4x4 unit cell

(c) (d)

(a) (b)

(b)(a)(a)

(c) (d)

(b)

iPEPS - dimerization vanishes

2x2 unit cell,D=6


flavor liquid?local magnetization vanishes, no translational symmetry breaking

we use the fermionic representation:

Pij =�

µ,ν∈colors

f†α,ifβ,if

†β,jfα,j

PMFij =

�

α,β∈colors

�fβ,if†β,j�f

†α,ifα,j

= −�

α∈colors

tαijf†α,ifα,j

Using different Ansätze for the hoppings, we evaluate the expectation value of the Hamiltonian

Mean-field decoupling of the fermionic Hamiltonian gives a hopping Hamiltonian and a variational wave function|Ψvari� = PGutzwiller|ΨFS�

Evari =�Ψvari|H|Ψvari��Ψvari|Ψvari�


The fermionic wave function of the pi-flux state

Dirac points

1/4

1/2

3/4

two-fold degenerate bands


96-site cluster - real space correlations from Gutzwiller projected wavefunction

9

26

�13

2

�491

46

�15

9

26

�13

2

9

�11

232

�14

11

�1693

�45

�18

�5

18

�2

Pi-flux

E/N = −0.89466

�1

6

�5

0

�326

5

0

�2

6

�5

0

�1

�14

�188

�18

0

�1597

11

�2

�3

�19

�2

Majorana 0-flux

E/N = −0.822479

�24

�13

�1

1

�324

�73

�1

3

�13

�1

1

�24

�23

�36

�59

�11

�1518

52

�4

2

�6

�15

0-flux

E/N = −0.763304

�4

�12

�5

�1

�266

�10

�1

�2

�12

�5

�1

�4

�33

70

�45

4

�1506

�157

�7

�7

13

�7

Majorana Pi-flux

E/N = −0.754662

marked differences in 3rd neighbor correlations


24-site cluster - real space correlations

Pi-flux

Dimension of the Hilbert space is 24!/(6!)4 = 2 308 743 493 056using symmetries makes it tractable


10-6

10-5

10-4

10-3

10-2

10-1

1

2 4 6 8 10 12 14 16 1820

⟨ P

0δ -

1/4

⟩ (

-1)δ

/2

δ

∼δ−4

∼δ−3

64839220072

60038421696

-0.02

-0.01

0

0.01

0.02

0.03

0 4 8 12 16

×0.2

⟨ P0δ -1/4 ⟩

648chain

algebraic correlations and structure factor


Ground state energy from different methods

0 0.05 0.1 0.15 0.2 0.25

−1.05

−1

−0.95

−0.9

−0.85

−0.8

−0.75

E s

1/D or 1/N

VMC Majorana π−fluxVMC 0−fluxVMC Majorana 0−fluxVMC π−fluxiPEPS 2x2 unit celliPEPS 4x4 unit cellED


SU(N) on honeycomb

SU(4) is most probably an algebraic flavor liquid[P. Corboz, M. Lajkó, A. M. Läuchli, K. Penc, F. Mila, arXiv:1207.6029]

SU(6) is likely a chiral flavor liquid [G. Szirmai E. Szirmai, A. Zamora, and M. Lewenstein, Phys. Rev. A 84, 011611 (2011)],

similarly, SU(N) is also a chiral liquid [extending the results of M. Hermele, V. Gurarie, & A. M. Rey, Phys. Rev. Lett. 103, 135301 (2009)]

honeycomb optical lattices can be realized, see poster of Johannes Hecker-Denschlag

SU(2) is a Néel state

SU(3) is a plaquette state[Y.-W. Lee and M.-F. Yang, Phys. Rev. B 85, 100402 (2012). H.H.Zhao,C.Xu,Q.N.Chen,Z.C.Wei,M.P.Qin,G.M. Zhang, and T. Xiang, Phys. Rev. B 85, 134416 (2012).]

P. Corboz, unpublished


http://arxiv.org/abs/1207.6029

http://arxiv.org/abs/1207.6029

Conclusions

(b)(a)

Phase Diagram

J’/J

1

0

3 sublattice LRO ?

spin-orbital liquid

4 sublattice LRO

QPT

QPT

QPT

spin-orbital liquid?

AB CB

CB

C

A

B C AA

A

A

A

BAB

C

B

D

A

C

B

D C

?resonating

liquid

(b)(a)

SU(2) SU(3) SU(4)

happy

happyfrustrated

fluctuation stabilized dimerization+Neel

squa

retri

angu

lar


Many happy returns of the day to you, Jose!

Many happy workshops in Evora!


the end


order and disorder in su(3) and su(4) heisenberg systems

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