realizing symmetry-protected topological phases with su(n) ultra

Realizing symmetry-protected topological phases with SU(N) ultra-cold

fermions

Keisuke Totsuka Condensed-Matter Theory Group, Yukawa Institute for Theoretical Physics, Kyoto University

November 14, 2014 talk @ NQS2014, YITP

Ref: Nonne et al. Phys.Rev.B 82, 155134 (2010) Euro.Phys.Lett. 102, 37008 (2013), Bois et al., arXiv:14102974 (2014), Tanimoto-KT (2014)

Collaborators

Sylvain Capponi (IRSAMC,

Univ. Toulouse)

Marion Moliner (Univ. Strasbourg)

Philippe Lecheminant (LPTM,

Univ. Cergy-Pontoise)

Héroïse Nonne (Dept. Phys. Technion)

Kazuhiko Tanimoto (YITP, Kyoto Univ.)

Adrien Bolens (YITP, Kyoto Univ.↔ EPFL)

Valentin Bois (LPTM,

Univ. Cergy-Pontoise)

Outline

• Introduction ... 1D topological phases

• Alkaline-earth cold atom & SU(N) Hubbard model

• Phase diagram and Mott phases

• SU(N) (symmetry-protected) topological phase

• Summary

Topological Order in 1D

• ONLY Symmetry-protected topological (SPT) phases possible

✓ protecting symmetry “G” necessary (otherwise trivial)

✓ only short-range entanglement, no topological degeneracy

✓ topological insulator HgTe, superfluid 3He (B-phase), ....

• Key points:

1. featureless / no fractionalization in the bulk...

2. existence of (non-propagating) edge states

3. classification of interacting phases

4. ex) Haldane phase of integer-spin chains

trivial phase

protecting sym.

trivial phase topological

Miyashita-Yamamoto ’93

Verstraete et al. ’05, Chen et al ’10

Chen et al. ’11

S=1 Heisenberg chain (QMC)


• Matrix-Product States (MPS):

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

• existence of “edge states” (α,β)✓ featureless in the bulk (no local order, exp-decaying correlation, etc.)✓ 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

spin-1 chain


• edge states & “symmetry fractionalization”:

✓ symmetry operation on MPS :

• classification of SPT phases:✓ classifying “topological phases” = classifying possible “edge states”

✓ robust criteria (against small parameter change/perturbations) ??

✓ “V” = “projective representation”:

✓ group cohomology as a tool to label “topological phases”

Chen-Gu-Wen ’11, Schuch et al. ’11

symmetry fractionalizes

Perez-Garcia et al. ’08

Pollmann et al. ’10, ’12

mathematical representation of

“physical” edge states

V (u1)V (u2) = !(u1, u2)V (u1u2)

Topological Order in 1D ... Haldane phase

• “Haldane phase” .... a prototypical example

✓ massive short-range phases in S=integer spin chains

✓ G.S. of spin-1 parent Hamiltonian:

• “emergent” Sedge= S/2 edge spins (discrete edge states)

✓ exist for any integer spins (ex. Zn-doped NENP)

✓ fine structure: Pollmann et al. ’10, ’12, Gu et al. ’11,..

Haldane ’83

HVBS =X⇢

Sj ·Sj+1 +1

3(Sj ·Sj+1)

3

�

‣ Sedge=1/2, 3/2,.. “topological”

‣ Sedge=0,1,2,.. “trivial”

S=1 VBS

cf) group cohomology, NLσ w/ Θ-term

Z2 classification


• Zoo of “SPT” phases from group cohomology Chen et al. ’11, ’12, ’13

But where can we find these SPT phases ???

alkaline-earth ultracold fermions and SU(N) Hubbard model

Alkaline-earth cold atoms... an introduction

• alkaline-earth atoms: 8A1A

2A

3B 4B 5B 6B 7B 8B 11B 12B

3A 4A 5A 6A 7A

element names in blue are liquids at room temperatureelement names in red are gases at room temperatureelement names in black are solids at room temperature

Periodic Table of the Elements

Los Alamos National Laboratory Chemistry Division

11

1

3 4

12

19 20 21 22 23 24 25 26 27 28 29 30

37 38 39 40 41 42 43 44 45 46 47 48

55 56 57

58 59 60

72 73 74 75 76 77 78 79 80

87 88 89

90 91 92 93 94 95 96

104 105 106 107 108 109 110 111 112

61 62 63 64 65 66 67

97 98 99

68 69 70 71

100 101 102 103

114 116 118

31

13 14 15 16 17 18

32 33 34 35 36

49 50 51 52 53 54

81 82 83 84 85 86

5 6 7 8 9 10

2H

Li

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc Ti V Cr Mn Fe Co Ni Cu Zn

Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

La* Hf Ta W Re Os Ir Pt Au Hg

Ac~ Rf Db Sg Bh Hs Mt Ds Uuu Uub Uuq Uuh Uuo

B C N O F

Al Si P S Cl

Ga Ge As Se Br

In Sn Sb Te I

Tl Pb Bi Po At

He

Ne

Ar

Kr

Xe

Rn

39.10

85.47

132.9

(223)

9.012

24.31

40.08

87.62

137.3

(226)

44.96

88.91

138.9

(227)

47.88

91.22

178.5

(257) (260) (263) (262) (265) (266) (271) (272) (277) (296) (298) (?)

50.94

92.91

180.9

52.00

95.94

183.9

54.94

(98)

186.2

55.85

101.1

190.2 190.2

102.9

58.93 58.69

106.4

195.1 197.0

107.9

63.55 65.39

112.4

200.5

10.81

26.98

12.01

28.09

14.01

69.72 72.58

114.8 118.7

204.4 207.2

30.97

74.92

121.8

208.9 (209) (210) (222)

16.00 19.00 20.18

4.003

32.07 35.45 39.95

78.96 79.90 83.80

127.6 126.9 131.3

140.1 140.9 144.2 (147) (150.4) 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0

232.0 (231) (238) (237) (242) (243) (247) (247) (249) (254) (253) (256) (254) (257)

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

hydrogen

barium

francium radium

strontium

sodium

vanadium

berylliumlithium

magnesium

potassium calcium

rubidium

cesium

helium

boron carbon nitrogen oxygen fluorine neon

aluminum silicon phosphorus sulfur chlorine argon

scandium titanium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton

yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon

lanthanum hafnium

cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium

tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon

actinium

thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium

rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium

1.008

6.941

22.99

Lanthanide Series*

Actinide Series~

1s1

[Ar]4s23d104p3[Ar]4s23d3[Ar]4s13d10

[Ne]3s23p6[Ne]3s23p4

[Ar]4s1[Ar]4s23d10

1s2

[He]2s1 [He]2s2

[Ar]4s23d7

[Ne]3s23p5

[He]2s22p1[He]2s22p2 [He]2s22p3

[Ar]4s23d5

[He]2s22p4 [He]2s22p5 [He]2s22p6

[Ar]4s23d104p5

[Ne]3s1 [Ne]3s23p1 [Ne]3s23p3[Ne]3s23p2

[Rn]7s25f146d2

[Ne]3s2

[Ar]4s2 [Ar]4s23d1 [Ar]4s23d2 [Ar]4s13d5[Ar]4s23d6

[Ar]4s23d8 [Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p4 [Ar]4s23d104p6

[Kr]5s1 [Kr]5s2 [Kr]5s24d1 [Kr]5s24d2 [Kr]5s14d4 [Kr]5s14d5 [Kr]5s24d5 [Kr]5s14d7 [Kr]5s14d8 [Kr]4d10 [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 [Kr]5s24d105p2 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6

[Xe]6s1 [Xe]6s2 [Xe]6s25d1

[Xe]6s24f15d1 [Xe]6s24f3 [Xe]6s24f4 [Xe]6s24f5 [Xe]6s24f6 [Xe]6s24f7 [Xe]6s24f75d1 [Xe]6s24f9 [Xe]6s24f10 [Xe]6s24f11 [Xe]6s24f12 [Xe]6s24f13 [Xe]6s24f14 [Xe]6s24f145d1

[Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9[Xe]6s14f145d10

[Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5[Xe]6s24f145d106p6

[Rn]7s1 [Rn]7s2 [Rn]7s26d1

[Rn]7s26d2 [Rn]7s25f26d1 [Rn]7s25f36d1 [Rn]7s25f46d1 [Rn]7s25f6 [Rn]7s25f7 [Rn]7s25f76d1 [Rn]7s25f9 [Rn]7s25f10 [Rn]7s25f11 [Rn]7s25f12 [Rn]7s25f13 [Rn]7s25f14 [Rn]7s25f146d1

[Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9

2 electrons in outer-most shell

Alkaline-earth cold atoms... an introduction

• alkaline-earth atoms: ... what’s so nice ??

✓ atomic ground state: 1S0 ➡ J=0 electronic w.f.

✓ decoupling of “nuclear spin I” from electronic states ( )➡ Iz-independent (s-wave) scattering length

✓ SU(N) (N≦2I+1) symmetry (almost perfect...)

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

✓ tunability: interaction strength, lattice structure, ....

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

Gorshkov et al ’10

ex. 173Yb: Kitagawa et al., ’08, Scazza et al. ’14, 87Sr: Pagano et al. ’14

I·J = 0

Alkaline-earth cold atoms... orbitals

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

• use ground state (1S0) and excited state (3P0)

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

• SU(N) (N=2I+1) symmetry

• two-orbital SU(N) system (crucial for “topological phase”)...

1S0 3P0(G.S.) forbidden(exc.)

Alkaline-earth cold atoms... orbitals

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

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

Kobayashi et al. ’12, ’14

1S0 3P0

z // chain

✓ 1D optical lattice✓ confinement in (xy)

not too strong✓ fill s-band only✓ px , py

instead of “g” and “e”

(G.S.)

px py

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

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

✓ optical lattice (3D), large-U (U∼102nK, U/t∼10-102): Mott phase (w/ n=1, i.e. 1/N-filling)

✓ charge gap, doublon, compressibility ➡ “Mott core”

✓ But.... still in high-T region (i.e. U≫T≫t2/U, SU(N) paramagnet)

Taie et al. ’12

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

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

✓ optical lattice (3D), large-U (U∼102nK, U/t∼10-102): Mott phase (w/ n=1, i.e. 1/N-filling)

✓ charge gap, doublon, compressibility ➡ “Mott core”

✓ But.... still in high-T region (i.e. U≫T≫t2/U, SU(N) paramagnet)

Taie et al. ’12

lower-T

higher-T

The Model

2-orbital SU(N) fermion model ... minimal models ?

• “Ingredients”

✓ charge

✓ nuclear-spin multiplet (Iz=−I,...,+I, N=2I+1, α=1,...,N)✓ orbital: 2 atomic states “g” and “e” (m=1,2)

or 2-orbitals px, py in p-band model

✓ strong correlation (Mott physics)

• Control parameters:

✓ # of flavors: N (N≦2I+1)

✓ (short-range) interactions: U, VG, Vex (t: fixed)✓ filling (# of fermions/site ≦2N) ➡ fix n=N (half-filled)

Kobayashi et al. ’12, ’14

competition of the two determines

SU(N) magnetism

frozen@low-T (≪U)

c†m↵,i

2-orbital SU(N) Hubbard model ... two minimal models

• Hubbard-like Hamiltonian-(A):

Gorshkov et al ’10, Nonne-Moliner-Capponi-Lecheminant-KT ’13

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”

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

U(1)c

⇥SU(N)s

⇥U(1)o

generic symmetry:

“g-e Hamiltonian”

T aj ⌘ 1

2c†m↵,j(�

a)mncn↵,j

g.s. “g” + metastable exc. “e”

2-leg ladder rep.(single chain, in fact)

2-orbital SU(N) Hubbard model ... two minimal models

• Hubbard-like Hamiltonian-(B):

Kobayashi et al ’12, Bois et al. ’14

c†m↵,i : creation op. for “orbital” px= , py= ), nuclear-spin=αgeneric symmetry:

Hp-band =� t

NX

↵=1

2X

a=p

x

,p

y

X

j

⇣c†m↵, j

cm↵, j+1 + h.c.

⌘� µ

X

j

nj

+1

4(U1 + U2)

X

j

n2j

+X

j

�2U2(T

x

j

)2 + (U1 � U2)(Tz

j

)2

.

“p-band model”

T aj ⌘ 1

2c†m↵,j(�

a)mncn↵,jU(1)c

⇥SU(N)s

⇥Z2,o

✓ much less control parameters✓ But with orbital anisotropy ...

(∵ pair hopping)

pair hopping

orbital anisotropy U1 = 3U2

* for axially-symmetric trap:

single-band SU(N) Hubbard model ... (unsuccessful) hunt for gapped topological phases

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

• insulating phases:

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

✓ 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

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

Uc=0 Uc>0(BKT)

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

N=2 N >2

Szirmai et al. ’05, ’08, ’13Kobayashi et al. 25aBA-7No featureless fully gapped phase

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

• Necessary to consider multi-orbital systems... ✓ 2-orbital SU(N) Hubbard (N: even)

✓ half-filling (i.e. N fermions/site)

✓ forget about “harmonic trap” (for the moment...)

• Plan of attack:✓ strong-coupling expansions ... large-|U| mapping

✓ weak-coupling RG ... fermionization + duality + RG

✓ numerics (DMRG)

✓ map out the phase diagram

Look for insulating phases without symmetry breaking(≈ topological phases) in realistic systems!!

crash course in SU(N)

• irreducible representations:

✓ SU(2): “spin” S

✓ SU(N): “Young tableau”

✓ “singlet” ... dimer vs. N-mer

✓ two S=1/2 ➡ singlet “dimer”

✓ N boxes (vertically aligned) ➡ SU(N) singlet (N-mer)

S=1/2 S=1 S=3/2 S=2

➡N

SU(N)-singlet

Phase diagram

2-orbital SU(N) Hubbard ... “SU(N)” vs “orbital”

• Strong-coupling: t=0 “atomic limit” (U>0, N particles/site)

• inter-orbital exchange (Vex) = Hund coupling ➡ “dual nature” of orbital & SU(N)-spin

SU(N)-“spin”

orbital pseudo-spin

HG = (· · · )� V g-eex

X

i

0

@N2�1X

A=1

SAg,iS

Ae,i

1

A

| {z }Hund

= (· · · ) + V g-eex

X

i

(Ti)2

| {z }Hund

,

Nonne et al. ’13

HH =� tX

i

2X

m=1

NX

↵=1

⇣c†m↵, i

cm↵, i+1 + h.c.

⌘

� µX

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

Vex>0 Vex<0

orbital quenched maximized

SU(N) maximized quenched

• Strong-coupling: t=0 “atomic limit” (U>0, N particles/site)

Nonne et al. ’13

T = N/2 T = N/2 T = N/2

T = 0 T = 0 T = 0

HH =� tX

i

2X

m=1

NX

↵=1

⇣c†m↵, i

cm↵, i+1 + h.c.

⌘

� µX

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 <0 (or Vex<0):SU(N)-“spin” quenchedorbital-spin T=N/2

J >0 (or Vex>0):large SU(N)-“spin”orbital-spin quenched

dual nature of “orbital” & SU(N) spin

2-orbital SU(N) Hubbard ... “SU(N)” vs “orbital”

2-orbital SU(N) Hubbard ... global phase structure

• Strong-coupling: “effective Hamiltonian” for U>0

T = N/2 T = N/2 T = N/2

T = 0 T = 0 T = 0

Bois et al. ’14

T=N/2 XXZ + D-termJ <0 (or Vex<0):

J >0 (or Vex>0):

Bois et al. ’14

He↵ = JN2�1X

A=1

SAi SA

i+1

J ⌘ 1

2

((t(g))2

U + Udi↵ + J + Jz2

+(t(e))2

U � Udi↵ + J + Jz2

)

SU(N) Heisenberg model

Horbital

=X

i

�Jxy

�T x

i

T x

i+1

+ T y

i

T y

i+1

�+ J

z

T z

i

T z

i+1

�(J � Jz

)(T z

i

)2 �H

e↵

X

i

T z

i

2-orbital SU(N) Hubbard ... global phase structure

• Strong-coupling: “effective Hamiltonian” for U<0

T = 0 T = 0 T = 0

Moliner et al. ’14

s=1/2 AF Ising

T = 0 T = 0 T = 0

HCDW =

⇢2(t(g))2

E�+

2(t(e))2

E+

�X

i

�szi s

zi+1 � 1/4

�.

E±(J, Jz, Udi↵) ⌘ J + Jz/2 + (2N � 1)|U |± (N � 1)Udi↵

sz = +1/2 sz = �1/2 sz = +1/2

sz = �1/2 sz = �1/2sz = +1/2

cf) Zhao et al. ’07 for single-band cases

➡ different story from U>0 cases...

➡ leads to 2kF-CDW

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

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

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 diagram

(RG)

Nonne et al. ’10

degenerate non-degenerate, Mott-like cf. Dalla Torre et al. ’06

Berg et al. ’08VGVG

orbital -charge exchange

orbital-Haldane

charge-Haldane

spin-Haldane

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

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

Vex

= �0.03t Vex

= 0.01t

DMRG

Nonne et al. ’10

weak-couplingphase diagram

Vex

= �t Vex

= t

cf) Zhao et al. ’07orbital-charge exch.

VG

! �VG

+ V g-eex

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

• 4 Mott phases without G.S. degeneracy: spin-HaldaneSspin=1

rung-singlet(orbital large-D)

charge-Haldane

orbital-HaldaneSorb=1

“charge” “orbital” “spin”

(a) spin-Haldane gapped local singlet collective singlet

(b) orbital-Haldane gapped collective singlet local singlet

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

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

orbital-”e”

orbital-”g”

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

Kobayashi et al. ’12

Nonne et al. ‘10,11

trivial state ➡

SPT-I

SPT-II

SPT-III

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

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

Nonne et al. ’13, Bois et al. ’14

RG-flow pattern different for N=2 and N>2 ...

✓ spin-Peierls✓ 2kF-CDW, 2kF- ODW✓ orbital-Haldane (T=N/2)✓ orbital-liquid (gapless, c=1) ⬅ regardless of “N”

g1 =N

4⇡g21 +

N

8⇡g22 +

N

16⇡g23 +

N + 2

4⇡g27 +

N � 2

4⇡

�2g28 + g29

�

g2 =N

2⇡g1g2 +

N2 � 4

4⇡Ng2g3 +

1

2⇡(g2g5 + g3g4) +

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)

⇡Ng8g9

g5 =N2 � 1

2⇡N2g22 +

1

2⇡g24 +

2(N � 1)

⇡Ng28

g6 =N + 1

4⇡Ng27 +

N � 1

2⇡Ng28 +

N � 1

4⇡Ng29

g7 =(N + 2)(N � 1)

2⇡Ng1g7 +

2

⇡g6g7 +

N � 1

4⇡(2g2g8 + g3g9)

g8 =N + 1

4⇡g2g7 +

2

⇡g6g8 +

1

2⇡(g4g9 + g5g8) +

(N � 2)(N + 1)

4⇡N(2g1g8 + g2g9 + g3g8)

g9 =N + 1

4⇡g3g7 +

1

⇡(g4g8 + 2g6g9) +

(N � 2)(N + 1)

2⇡N(g1g9 + g2g8) ,

4N Majoranas + interactions

RG+duality

2-orbital SU(N) (N>2) Hubbard ... global structure

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

•

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J/t

U/t

✓ spin-Peierls (SP)✓ 2kF CDW (U<0)✓ orbital T=N/2 SU(2) HAF (J<0)✓ SU(N) HAF (J>0) (not by RG!)

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

cf) Zhao et al. ’07

pseudo-spin=N/2Heisenberg (∀N)

SU(N) Heisenberg (N=even)

Spin-PeierlsCDW

2-orbital SU(N) (N>2) Hubbard... DMRG phase diagram for SU(4)

• N=4 (SU(4)) g-e model, N fermions/site (half-filling):

• Phase diagram (numerical & weak-coupling RG): Moliner et al. ’14

H0H =� t

X

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

SU(N): quenchedorbital: T=N/2

orbital-Haldane

Jz=J (orbital-SU(2)) Jz=0

our SU(4) topological phaseDMRG, L=32,36

Moliner et al. ’14

CDW

SP

CDWTz=0 phase(large-D)

SU(N): maximizedorbital: quenched

2-orbital SU(N) (N>2) Hubbard... DMRG phase diagram for SU(4)

• SU(N) p-band model, N fermions/site (half-filling):

• Phase diagram (N=2,4, numerical):

Bois et al. ’14

DMRG, L=32,36

Hp-band =� t

NX

↵=1

2X

a=p

x

,p

y

X

j

⇣c†m↵, j

cm↵, j+1 + h.c.

⌘� µ

X

j

nj

+1

4(U1 + U2)

X

j

n2j

+X

j

�2U2(T

x

j

)2 + (U1 � U2)(Tz

j

)2

.

-8 -6 -4 -2 0 2 4 6 8U

1

-5

0

5

U2

CDW

ODW

SP

singlet

SU(4) Haldane

SU(4)

our SU(N) topological phase

Bois et al. ’14

CDW

Tz=0 phase(large-D)

-10 -5 0 5 10U

1

-10

-5

0

5

10

U2

SPsinglet

Haldane (spin)

Haldane (charge)

U2=3U

1

SU(2)

(U1,U2) ⇒ (－U1,－U2)

harmonic trap

charge-Haldane(SPT)

c=2 QPT

Tx=0 phase(large-Dx)

Tz=0 phase(large-Dz)

➡ U1=3U2 for axially-sym. trap

ODWSP

SU(4) top(SPT).

spin-Haldane(SPT)

SU(N) topological phases

2-orbital SU(N) (N>2) Hubbard... SU(4) topological phase

• Nature of SU(4) topological phase ??

• strong-coupling (large-U) effective Hamiltonian (N=even):

• cook up Hamiltonian sharing the same topological properties...

• SU(4) VBS Hamiltonian

N/2 rows, 2 columns

Nonne et al. ’13, Bois et al., ’14

N=2N=4

N=6Moliner et al. ’14

He↵ = JN2�1X

A=1

SAi SA

i+1

J ⌘ 1

2

((t(g))2

U + Udi↵ + J + Jz2

+(t(e))2

U � Udi↵ + J + Jz2

)

HVBS = JsX

i

⇢SAi S

Ai+1 +

13

108(SA

i SAi+1)

2 +1

216(SA

i SAi+1)

3

�parent Hamiltonian:

similar strategy applies to general N(=even)additional terms

orbital-Haldane

CDW

SP

2-orbital SU(N) (N>2) Hubbard... SU(4) topological phase

• SU(4) VBS state:

✓ given by 6×6 MPS, 6-fold degenerate edge states (➡ DMRG)

✓ exponentially-decaying cor. fn., featureless in the bulk

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

✓ QPT between topological and (non-top.) SP phases

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. ’13N=4 (DMRG, L=36)

Duivenvoorden-Quella ’12, ’13

edge states (VBS)(nα)

construction

QPT SU(N)2

Fingerprint of Topological Phase... “entanglement spectrum”

• group-cohomology “classification”:

• SU(4) topological phase: (protected by PSU(4)=SU(4)/Z4)

• 4 classes (Z4 classifiaction): trivial, class-1, class-2, class-3

• related to edge statesDuivenvoorden-Quella ‘12,13

# of boxes mod 4

emergent SU(4)“spin”@edge

Gu et al. ’11, Schuch et al. ’11

✓ look at emergent “edge” spins: Sedge

✓ with SO(3)=PSU(2)=SU(2)/Z2 ... 2 classes (Sedge=0,1,.. or Sedge=1/2,3/2,...)

emergent SU(2) “spin”@edgetrivial non-trivial

our state

Pollmann et al. ’10, ’12

Fingerprint of Topological Phase... “entanglement spectrum”

• SU(4) topological phase: (protected by PSU(4)=SU(4)/Z4)

• entanglement spectrum (E.S.) ?1. topological phase ➡ E.S. consistent with irreps. allowed...2. non-topological phase ➡ no special structure

Duivenvoorden-Quella ‘12,13

H(↵) =X

j

SAj S

Aj+1 + ↵

X

j

⇢13

108(SA

j SAj+1)

2 +1

216(SASA

j+1)3

�

# of boxes mod 4

Tanimoto-KT ’14

emergent SU(4)“spin”@edge

↵ = 0

Heisenberg

iTEBD

Entanglement spectrum and non-local order parameters

• (A) entanglement spectrum as a probe of SPTO:

✓ special level structure = “topological”

✓ otherwise... non-topological

• (B) string order parameters (our old friend...):

• Q) Relation between the two methods ??

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.

den Nijs-Rommelse ’89, Kennedy-Tasaki ’92

still tempting to have “order parameters”...

modern

⬅ hidden AF order

cf. Pollmann’s talk

Entanglement spectrum and non-local order parameters ... Haldane phase

• “string order parameter” :

• relation to group cohomology:

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.

den Nijs-Rommelse ’89, Kennedy-Tasaki ’92

proj.rep.

Pollmann-Turner ’12, Hasebe-KT ’13

Both Ostr(x,z)≠0 ➡ UxUz =－UzUx (Ux, Uz: proj. rep.)

➡ topological Haldane phase...sufficient condition

for SPT

boundary term

Entanglement spectrum and non-local order parameters ... SU(4) phase

• “string order parameters” O(m,n):

• Non-local unitary connecting different topological classes:

Tanimoto-KT ’14

UKT = exp

8<

:i2⇡

N

X

k<j

H⇢(k)G⇢(j)

9=

; ˆQ ⌘ exp

✓i2⇡

NˆH⇢

◆, ˆP ⌘ exp

✓i2⇡

NˆG⇢

◆

Heisenberg

VBS

2 commuting ZN for PSU(N)

O1,2(1, 1) = O1,2(1, 3) = 0

cf. Kennedy-Tasaki ’92, Oshikawa ’92 for Haldane

O1(m,n) ⌘ lim|i�j|%1

*

n

XP (i)om

8

<

:

Y

ik<j

Q(k)n

9

=

;

n

X†P (j)

om+

O2(m,n) ⌘ lim|i�j|%1

*

n

XQ(i)om

8

<

:

Y

i<kj

P (k)n

9

=

;

n

X†Q(j)

om+

ZN-OP ZN-OPZN-string

class-1: O(1,3), class-2: O(2,1), class-3: O(3,3)

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)

N=3

S=3/2 Heisenberg(gapless)

3 phases (2 gapped, 1 gapless)

cf. Moliner et al. ’14

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2kF-CDW

SP

N=4

2kF-CDWSU(4) topological

SP

orbital-Haldane

QPT: SU(4)-top⇔SP

spin-orbital coupled

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,..)

• Outlook:

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

✓ Quantum-optical detection of topological order ??

✓ Mapping out N=odd phases (SU(N)-orbital)

✓ other fillings ??, Higher-D ???, topological terms ???Kobayashi et al. ’12

“Mott core”

Endres et al. ’11

realizing symmetry-protected topological phases with su(n) ultra

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