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)
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 (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
Topological Order in 1D
• 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
Topological Order in 1D
• 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:
•
�����
����
��� � �������
����
���
�����
����
������������
���
����
���
��� �
����� !���������"""�����
�����
���
�����
�����!
����
����
#$��
%�!�
�����&������
����&������&� �����&������&� �
�'�
�(�)
&*
����(
�&*
*
�
+++
�
,
*
�
+++
�
,
-����
-���
��$�
+++
*
�.�
����� +++
*
�#*�
+++
*
�&*�
/
�����!���
�&.#�*����!���
0�1 !� !�! 2�)(3
�$�����
���
�����
���
�
�45�!�.�6�������6���� !�! �
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
�����
����
��� �
�������
����
���
�����
����
������������
���
����
���
��� �
����� !���������"""�����
�����
���
�����
�����!
����
����
#$��
%�!�
�����&������
����&������&� �����&������&� �
�'�
�(�)
&*
����(
�&*
*
�
+++
�
,
*
�
+++
�
,
-����
-���
��$�
+++
*
�.�
����� +++
*
�#*�
+++
*
�&*�
/
�����!���
�&.#�*����!���
0�1 !� !�! 2�)(3
�$�����
���
�����
���
�
�45�!�.�6�������6���� !�! �
J/t
U/t
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
Top Related