Symmetry-protected topological phases of alkaline-earth ultra-cold fermionic atoms

in one dimension Keisuke Totsuka

Yukawa Institute for Theoretical Physics, Kyoto University

June 14, 2013 talk @ EQPCM (ISSP)

1. Introduction2. Symmetry-protected topological order in 1D3. Alkaline-earth cold atom & SU(N) Hubbard model4. SU(N) symmetry-protected topological phase5. Summary

Ref: Nonne et al. Phys.Rev.B 82, 155134 (2010) Euro.Phys.Lett. 102, 37008 (2013), and work in progress

Collaborators• Sylvain Capponi

(IRSAMC, Univ. Toulouse)

• Philippe Lecheminant (LPTM, Univ. Cergy-Pontoise)

• Marion Moliner (LPTM, Univ. Cergy-Pontoise)

• Héroïse Nonne (U-Cergy ⇒ Dept. Phys. Technion)

Symmetry and Phase Transitions• Landau picture of “phase transitions” (from L.L. vol. 5)

✓ phase transitions: spontaneous breaking of symmetry G (original) ⇒ H (subgroup) ex) G=SO(3), H=SO(2) for (collinear) magnetic order

✓ “Order Parameters”✓ classification in terms of SSB

patterns of G

1. “... the transition between phases of different symmetry .... cannot occur in a continuous manner”

2. “... in a phase transition of the second kind .... the symmetry of one phase is higher than that of the other..”

Challenges from quantum systems• fractional quantum-Hall systems

✓ Laughlin wave function ...“ψm” (ν=1/m)

✓ describes featureless uniform ground states,no symmetry breaking (locally)

✓ Nevertheless, different “phases” for different “m”

• quantum spin liquids (Z2/chiral-SL, quantum dimer,...)

✓ no magnetic LRO, no translation SSB, ...

✓ global (topological) properties (e.g. γtop)(≠boring paramagnet)

the new “holy bible”

1983--

1987--

a new paradigm?? ➡ topological order (Wen ’89)

“Topological Order” in 1D

• Q-information tells us:

✓ any gapped states can be approximated by MPS

✓ only short-range entanglement in 1D (i.e. γtop=0)

✓ can be smoothly deformed to trivial product state ➡ No (genuine) topological order in 1D !!

• but... still possible to define featureless “topological phases” protected by some symmetry

symmetry-protected topological (SPT) orderGu-Wen ’09, Chen et al. ’11

trivial phase trivial phase topological

Verstraete et al. ’05

Hastings ’07

protecting sym.

(1+1)D

“Topological Order” in 1D

• Phases-I ... long-range entanglement (LRE):

✓ genuine topological phases in (2+1)D, (3+1)D

✓ w/ symmetries, possible to have topological order + SSB (“symmetry-enriched” etc.)

• Phases-II ... short-range entanglement (SRE)✓ wo/ symmetries, only a single trivial product state ✓ w/ “protecting symmetries”, various topological phases

“Topological Order” in 1D

• Paradigmatic example: “Haldane phase”

✓ 1D integer-S spin chain

✓ featureless non-magnetic state w/ exponentially-decaying cor.

✓ existence of edge states (cf. ESR)

• Non-local (string) order parameters:✓ Z2×Z2 symmetry

✓ relation to SPT order

Miyashita-Yamamoto ’93

Oz

string ⌘ lim

|i�j|%1

DSz

i

exp

"i⇡

j�1X

k=i

Sz

k

#Sz

j

E

Ox

string ⌘ lim

|i�j|%1

DSx

i

exp

"i⇡

jX

k=i+1

Sx

k

#Sx

j

E.

Kennedy-Tasaki ’92

Pollmann-Turner ’12, Hasebe-KT ’13

Haldane ’83, Affleck et al. ’88

“Topological Order” in 1D

• Matrix-Product States (MPS):

✓ convenient rep. for generic gapped states in 1D (w/ SRE)

• existence of “edge states” (α,β)✓ featureless in the bulk ✓ special structure (“edge states”) localized at boundaries

bulk

cf. FQHE, topological insulators

✓|0ii

p2|�1ii

�p2|1ii �|0ii

◆

Kintaro Ame

: S=1 VBS

physical state

“Topological Order” in 1D

• edge states & “symmetry fractionalization”:

✓ symmetry operation on MPS :

• edge states = obey “projective representation”✓ classifying “topological phases” = classifying possible “edge states”

✓ “V” does not necessarily follow group-multiplication rule:

✓ catalogue of gapped symmetry-protected topological phases in 1D (in terms of proj. rep.)

Chen-Gu-Wen ’11, Schuch et al. ’11

symmetry fractionalizes

Perez-Garcia et al. ’08

Pollmann et al. ’10, ’12

V (g1)V (g2) = !(g1, g2)V (g1g2)

“Topological Order” in 1D

• Motivation....

• Given a “protecting symmetry”

✓ zoo of symmetry-protected topological (SPT) phases

• Realization of topological phases ✓ existence of (many) symmetry-breaking perturbations in real

systems (fine-tuning necessary to have high sym.)

✓ SPT phases protected by high symmetry unlikely ??

How to realize SPT phases in realistic systems ??

Alkaline-earth cold atoms... a realistic system bearing SPT phases

• alkaline-earth atoms: two electrons in the outer shell

• J=0 for both G.S. (1S0) and (metastable) excited state (3P0)

• decoupling of “nuclear spin I” from electronic states➡ I-independent scattering length

• SU(N) (N=2I+1) symmetry (to a very good approx.)

• 171Yb (I=1/2, SU(2)), 173Yb (I=5/2, SU(6)), 87Sr (I=9/2, SU(10)) ,....

Gorshkov et al ’10

Takahashi group ’10-’12 DeSalvo et al. ’10

1S0 3P0(G.S.) forbidden

Alkaline-earth cold atoms... a realistic system bearing SPT phases

• alkaline-earth atoms: two electrons in the outer shell

• p-band model ... another way of introducing 2-orbitals

Kobayashi et al. ’12

1S0 3P0

z // chain

✓1D optical lattice✓confinement in (xy) not

too strong✓fill s-band only

(G.S.)

Alkaline-earth cold atoms... Mott insulating phase (expt.)

• alkaline-earth atoms (173Yb, I=5/2): SU(6) Mott phase

✓ optical lattice, large-U: Mott phase (w/ n=1)

✓ charge gap, doublon, compressibility

✓ But.... still in high-T region (i.e. SU(N) paramagnet)

Taie et al. ’12

2-orbital SU(N) Hubbard model ... a minimal model

• “ingredients”

✓ charge

✓ nuclear-spin multiplet (Iz=−I,...,+I, N=2I+1, α=1,...,N)

✓ orbitals: ground state “g”, excited state “e” (m=1,2)cf. 2-orbitals px, py in p-band model

• Hubbard-like Hamiltonian: Gorshkov et al ’10Nonne-Moliner-Capponi-Lecheminant-KT ’13

c†m↵,i : creation op. for “orbital”=m, nuclear-spin=α

Machida et al. ’12

HG

=� tX

i,m↵

⇣c†m↵, icm↵, i+1

+ h.c.⌘� µ

G

X

i

ni +X

i

X

m=e,g

UG

2nm, i(nm, i � 1)

+ VG

X

i

ng, ine, i + V e-gex

X

i,↵�

c†g↵, ic†e�, icg�, ice↵, i

Hubbard-like int. within “e” & “g”

“e”-“g” exchangeCoulomb between “g”-”e”

U(1)c

⇥SU(N)s

⇥U(1)o

generic symmetry:

SU(N) Hubbard model ... simpler case

• quench “e”-orbital ➡ (single-band) SU(N) Hubbard

• insulating phases:

✓ 1/N-filling: MIT@U=Uc ➡ charge: gapped, “SU(N)-spin”: gapless

✓ deep inside Mott phase: SU(N)-Heisenberg

✓ for other commensurate fillings... f=p/q ??

Lieb-Wu ’68, Assaraf et al. ’99Manmana et al. ’11

Sutherland ’75Affleck ’88

Uc=0 Uc>0(BKT)

TLL charge: gap, “spin”: criticalcharge: gap, “spin”: critical

N=2 N >2

Sutherland ’75, Affleck ’88Manmana et al. ’11

Assaraf et al. ’99

(typically, U /t >11)

SU(N) Hubbard model ... simpler case

• single-band SU(N) Hubbard

• For other commensurate fillings ... f=p/q

✓ q > N: gapless charge/”spin”, C1S(N-1) (c=n) ➡ critical phase

✓ q < N: full gap (C0S0), spatially non-uniform insulating phase

Szirmai et al. ’05, ’08

possible to have “fully-gapped uniform states” ??

SU(5) Hubbard@comm.fillingsin realistic interacting models....

Szirmai et al. ’05, ’08

➡ No featureless fully-gapped phase

Phases of 2-orbital SU(N) Hubbard

• plan of attack:

✓ 2-orbital (“g” & “e”), SU(N)-fermion, @half-filling

✓ Maximal filling: 2N fermions/site (2N boxes)ex) half-filling = N fermions/site (N boxes out of 2N)

✓ Mathods:

1. weak-coupling (RG + duality)➡ low-energy field theory: 4N Majorana fermions (+ int.)

2. strong-coupling (approach from Mott sides)➡ effective spin/orbital models

3. numerical ➡ DMRG

Phases of 2-orbital SU(N) Hubbard... N=2, half-filling

• 4 Mott phases wo/ G.S. degeneracy:

spin-HaldaneSspin=1

rung-singletorbital large-D

charge-Haldane

orbital-HaldaneSorb=1

“charge” “orbital” “spin”

(a) spin-Haldane gapped local singlet spin-1 Haldane

(b) rung-singlet gapped Tz=0 (large-D) local singlet

(c) charge-Haldane gapped singlet/triplet local singlet

(d) orbital-Haldane gapped spin-1 Haldane local singlet

orbital-”e”

orbital-”g”

Della Torre et al. ’06Nonne et al. ’10

Kobayashi et al. ’12

Phases of 2-orbital SU(N) Hubbard... N=2, half-filling (weak-coupling)

• 4 Mott phases + 3 phases w/ degenerate G.S.:

Vex

= �0.03t Vex

= 0.01t

spin-HaldaneSspin=1 (“RT”)

rung-singletorbital large-D (“RS”)

charge-Haldane (“HC”)

orbital-HaldaneSorb=1 (“HO”)

CDW(2kF)

ODW(2kF)

weak-couplingphase diag.

Nonne et al. ’10

degenerate non-degenerate, Mott-like

Phases of 2-orbital SU(N) Hubbard... N=2, half-filling (numerical)

• 4 Mott phases + 3 insulating phases w/ degenerate G.S.:

Vex

= �0.03t Vex

= 0.01t

DMRG

Nonne et al. ’10

weak-couplingphase diag.

Vex

= �t Vex

= t

2-orbital SU(N) (N>2) Hubbard ... half-filling

• weak-coupling: (very) complicated, but still doable...

Nonne et al. ’13

g1 =N

4⇡g21 +

N

8⇡g22 +

N

16⇡g23 +

N + 2

4⇡g27 +

N � 2

2⇡g28 +

N � 2

4⇡g29

g2 =N

2⇡g1g2 +

N2 � 4

4N⇡g2g3 +

1

2⇡g3g4 +

1

2⇡g2g5 +

N

⇡g7g8 +

N � 2

⇡g8g9

g3 =N

2⇡g1g3 +

N2 � 4

4⇡Ng22 +

1

⇡g2g4 +

N

⇡g7g9 +

N � 2

⇡g28

g4 =1

2⇡g4g5 +

N2 � 1

2⇡N2g2g3 +

2(N � 1)

N⇡g8g9

g5 =N2 � 1

2⇡N2g22 +

1

2⇡g24 +

2(N � 1)

N⇡g28

g6 =N + 1

4⇡Ng27 +

N � 1

2⇡Ng28 +

N � 1

4N⇡g29

g7 =N2 +N � 2

2N⇡g1g7 +

2

⇡g6g7 +

N � 1

4⇡g3g9 +

N � 1

2⇡g2g8

g8 =N + 1

4⇡g2g7 +

N2 �N � 2

4N⇡g2g9 +

2

⇡g6g8 +

N2 �N � 2

2N⇡g1g8 +

1

2⇡g5g8 +

1

2⇡g4g9 +

N2 �N � 2

4N⇡g3g8

g9 =N + 1

4⇡g3g7 +

2

⇡g6g9 +

N2 �N � 2

2N⇡g1g9 +

1

⇡g4g8 +

N2 �N � 2

2N⇡g2g8

RG-flow patterndifferent from N=2

✓ SP✓ 2kF-CDW✓ orbital-Heisenberg (S=N/2) ⬅ regardless of “N”

2-orbital SU(N) (N>2) Hubbard ... half-filling

• (very) schematic phase diagram for N=even:

•

pseudo-spin=N/2Heisenberg

SU(N) Heisenberg (N=even) J/t

U/t“magic line”✓ spin-Peierls (SP)

✓CDW✓ orbital T=N/2 SU(2) HAF✓ SU(N) HAF

Nonne et al. ’13

HH =� tX

i

2X

m=1

NX

↵=1

⇣c†m↵, i

cm↵, i+1 + h.c.

⌘� µ

X

i

ni

+U

2

X

i

n2i

+ JX

i

�(T x

i

)2 + (T y

i

)2 + J

z

X

i

(T z

i

)2,

J = V e-gex

, Jz = UG

� VG

,

U =UG

+ VG

2, µ =

UG

+ V e-gex

2+ µ

G

2-orbital SU(N) (N>2) Hubbard... SU(4) SPT phase (new!)

• “Magic line”: N=even

• strong-coupling effective Hamiltonian:

• generalize AKLT-idea: ex) SU(4) VBS

He↵ =N2�1X

A=1

SAi S

Ai+1

Jz = J , J =2NU

N + 2

HVBS = JsX

i

⇢SAi S

Ai+1 +

13

108(SA

i SAi+1)

2 +1

216(SA

i SAi+1)

3

�

N/2 rows, 2 columns

parent Hamiltonian:

hSA(j)SA(j + n)i =(

125

�� 1

5

�nn 6= 0

45 n = 0N=4

N=6 Nonne et al. ’13

N=2N=4

N=6

NB) consistent w/ DMRG sim.

Nonne et al. ’13

2-orbital SU(N) (N>2) Hubbard... SU(4) SPT phase (new!)

• SU(4) VBS

✓ 6-fold degenerate edge states

✓ one of N(=4) SPT phases (protected by SU(N))

✓ When U↘, QPT out to (“spin”-orbital entangled) “SP” phase (univ. class SU(N)2)

HVBS = JsX

i

⇢SAi S

Ai+1 +

13

108(SA

i SAi+1)

2 +1

216(SA

i SAi+1)

3

�

N=4

N=6

Nonne et al. ’13

N=4 (DMRG, L=36)

Duivenvoorden-Quella ’12, ’13

QPT: top ⇒ SP

edge states(nα)

Comparison: N=3 and N=4 cases... numerics (DMRG)

• drastic difference between N=odd / even

• (preliminary) DMRG results:

✓ L=36 sites

✓ order parameter & edge states

-6 -4 -2 0 2 4 6U

-20

-10

0

10

20

J

CDWSPcriticalJ=6U/5

N=3 phase diag. based on L=36 DMRG (Nf=3*L)

-6 -4 -2 0 2 4 6U/t

-10

-5

0

5

10

15

J/t

CDWSPHOtopo

J=4U/3

N=3 N=4

S=3/2 Heisenberg(gapless)

SU(4) topological

orbital-Haldane

3 phases (2 gapped, 1 gapless)

cf. Nonne et al. ’13

2kF-CDW

SU(4)-top⇔SP QPT

J

U

Summary• Symmetry-protected topological order in 1D

• SU(N) Hubbard model with 2-orbitals (N=2I+1)

✓ alkalline-earth Fermi gas in optical lattice (@half-filling)

✓ phase diagram depends on N=even / odd

✓ symmetry-protected topological phases for N=even (I=1/2,3/2,..)

✓ stable topological phases for N/2=odd, i.e. I=1/2 (171Yb), 5/2 (173Yb), 9/2 (87Sr), ...

• Outlook:

✓ Effects of trap (SPT phase & Mott core) ?

✓ quantum-optical detection of topological order ??

✓ higher-D ???

Kobayashi et al. ’12

“Mott core”