June 14, 2013 talk @ EQPCM (ISSP) Symmetry-protected topological phases of alkaline-earth ultra-cold fermionic atoms in one dimension Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University 1. Introduction 2. Symmetry-protected topological order in 1D 3. Alkaline-earth cold atom & SU(N) Hubbard model 4. SU(N) symmetry-protected topological phase 5. 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) Marion Moliner (LPTM, Univ. Cergy-Pontoise) Philippe Lecheminant (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) 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.. 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
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) 1983-- 1987-- a new paradigm?? topological order (Wen 89) the new holy bible
(1+1)D 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) Hastings 07 Verstraete et al. 05 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) order Gu-Wen 09, Chen et al. 11 protecting sym. trivial phase trivial phase topological
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) Haldane 83, Affleck et al. 88 Miyashita -Yamamoto 93 Non-local (string) order parameters: Z2 Z2 symmetry relation to SPT order Kennedy-Tasaki 92 Pollmann-Turner 12, Hasebe-KT 13 O z string O x string " D Xj 1 lim Si z exp i i j %1 k=i " D jx lim Si x exp i i j %1 S z k k=i+1 # S x k S z j # E S x j E.
Topological Order in 1D Matrix-Product States (MPS): p 0ii 2 1ii p2 1ii 0i i : S=1 VBS convenient rep. for generic gapped states in 1D (w/ SRE) physical state existence of edge states (α,β) featureless in the bulk special structure ( edge states ) localized at boundaries cf. FQHE, topological insulators bulk Kintaro Ame
Topological Order in 1D edge states & symmetry fractionalization : symmetry operation on MPS : Perez-Garcia et al. 08 edge states = obey projective representation classifying topological phases = classifying possible edge states V does not necessarily follow group-multiplication rule: V (g 1 )V (g 2 )=!(g 1,g 2 )V (g 1 g 2 ) catalogue of gapped symmetry-protected topological phases in 1D (in terms of proj. rep.) symmetry fractionalizes Pollmann et al. 10, 12 Chen-Gu-Wen 11, Schuch et al. 11
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 1 S0 (G.S.) 3 P0 forbidden J=0 for both G.S. (1 S0) and (metastable) excited state ( 3 P0) decoupling of nuclear spin I from electronic states I-independent scattering length SU(N) (N=2I+1) symmetry (to a very good approx.) Gorshkov et al 10 171 Yb (I=1/2, SU(2)), 173 Yb (I=5/2, SU(6)), 87 Sr (I=9/2, SU(10)),... Takahashi group 10-12 DeSalvo et al. 10
Alkaline-earth cold atoms... a realistic system bearing SPT phases alkaline-earth atoms: two electrons in the outer shell 1 S0 (G.S.) 3 P0 p-band model... another way of introducing 2-orbitals 1D optical lattice confinement in (xy) not too strong fill s-band only z // chain Kobayashi et al. 12
Alkaline-earth cold atoms... Mott insulating phase (expt.) alkaline-earth atoms ( 173 Yb, 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 ingredients charge... a minimal model nuclear-spin multiplet (I z = 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 10 Nonne-Moliner-Capponi-Lecheminant-KT 13 H G = t X i,m + V G X c m,i c m, i c m, i+1 +h.c. n g, i n e, i + V e-g ex i Coulomb between g - e X i, Machida et al. 12 X µ G n i + X i i c g, i c e, i c g, ic e, i e - g exchange X m=e,g U G 2 n m, i(n m, i 1) : creation op. for orbital =m, nuclear-spin=α Hubbard-like int. within e & g generic symmetry: U(1) c SU(N) s U(1) o
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 Lieb-Wu 68, Assaraf et al. 99 Manmana et al. 11 charge: gap, spin : critical TLL charge: gap, spin : critical Uc=0 N=2 Uc>0 N >2 (BKT) deep inside Mott phase: SU(N)-Heisenberg (typically, U /t >11) Sutherland 75, Affleck 88 Manmana et al. 11 for other commensurate fillings... f=p/q?? Sutherland 75 Affleck 88 Assaraf et al. 99
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 Szirmai et al. 05, 08 q < N: full gap (C0S0), spatially non-uniform insulating phase No featureless fully-gapped phase possible to have fully-gapped uniform states?? SU(5) Hubbard@comm.fillings Szirmai et al. 05, 08 in realistic interacting models...
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-haldane orbital- g orbital- e Sspin=1 rung-singlet orbital large-d charge-haldane orbital-haldane Sorb=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 Kobayashi et al. 12 Della Torre et al. 06 Nonne et al. 10
Phases of 2-orbital SU(N) Hubbard... N=2, half-filling (weak-coupling) 4 Mott phases + 3 phases w/ degenerate G.S.: CDW (2kF) spin-haldane Sspin=1 ( RT ) rung-singlet orbital large-d ( RS ) degenerate ODW (2kF) non-degenerate, Mott-like charge-haldane ( HC ) orbital-haldane Sorb=1 ( HO ) weak-coupling phase diag. Nonne et al. 10 V ex = 0.03t V ex =0.01t
Phases of 2-orbital SU(N) Hubbard... N=2, half-filling (numerical) 4 Mott phases + 3 insulating phases w/ degenerate G.S.: weak-coupling phase diag. V ex = 0.03t V ex =0.01t DMRG Nonne et al. 10 V ex = t V ex = t
2-orbital SU(N) (N>2) Hubbard... half-filling weak-coupling: (very) complicated, but still doable... g 1 = N 4 g2 1 + N 8 g2 2 + N 16 g2 3 + N +2 4 g2 7 + N 2 2 g2 8 + N 2 4 g2 9 g 2 = N 2 g 1g 2 + N 2 4 4N g 2g 3 + 1 2 g 3g 4 + 1 2 g 2g 5 + N g 7g 8 + N 2 g 3 = N 2 g 1g 3 + N 2 4 4 N g2 2 + 1 g 2g 4 + N g 7g 9 + N 2 g8 2 g 4 = 1 2 g 4g 5 + N 2 1 2 N 2 g 2(N 1) 2g 3 + g 8 g 9 N g 5 = N 2 1 2 N 2 g2 2 + 1 2 g2 4 + 2(N 1) g8 2 N g 6 = N +1 4 N g2 7 + N 1 2 N g2 8 + N 1 4N g2 9 g 7 = N 2 + N 2 2N g 8 = N +1 g 9 = N +1 g 1 g 7 + 2 g 6g 7 + N 1 4 g 3g 9 + N 1 2 g 2g 8 4 g 2g 7 + N 2 N 2 4N 4 g 3g 7 + 2 g 6g 9 + N 2 N 2 2N g 8 g 9 g 2 g 9 + 2 g 6g 8 + N 2 N 2 g 1 g 8 + 1 2N 2 g 5g 8 + 1 g 1 g 9 + 1 g 4g 8 + N 2 N 2 g 2 g 8 2N RG-flow pattern different from N=2 SP 2kF-CDW orbital-heisenberg (S=N/2) regardless of N 2 g 4g 9 + N 2 N 2 g 3 g 8 4N Nonne et al. 13
2-orbital SU(N) (N>2) Hubbard... half-filling (very) schematic phase diagram for N=even: 2X NX H H = c m, i c m, i+1 +h.c. n i + U X 2 t X i + J X i m=1 =1 X (Ti x ) 2 +(T y i )2 + J z (Ti z ) 2, i µ X i i n 2 i J = V e-g ex, J z = U G V G, U = U G + V G 2, µ = U G + Vex e-g 2 spin-peierls (SP) CDW orbital T=N/2 SU(2) HAF SU(N) HAF SU(N) Heisenberg (N=even) + µ G Nonne et al. 13 magic line J/t U/t pseudo-spin=n/2 Heisenberg
2-orbital SU(N) (N>2) Hubbard Magic line : N=even... SU(4) SPT phase (new!) J z = J, J= 2NU N +2 strong-coupling effective Hamiltonian: H e = NX 2 1 A=1 S A i S A i+1 generalize AKLT-idea: ex) SU(4) VBS parent Hamiltonian: H VBS = J s X i N/2 rows, 2 columns Nonne et al. 13 N=2 N=4 S A i S A i+1 + 13 108 (SA i S A i+1) 2 + 1 216 (SA i S A i+1) 3 N=6 N=4 hs A (j)s A (j + n)i = ( 12 5 1 5 n n 6= 0 4 5 n =0 NB) consistent w/ DMRG sim. N=6 Nonne et al. 13
SU(4) VBS 2-orbital SU(N) (N>2) Hubbard... SU(4) SPT phase (new!) H VBS = J s X 6-fold degenerate edge states one of N(=4) SPT phases (protected by SU(N)) i S A i S A i+1 + 13 108 (SA i S A i+1) 2 + 1 216 (SA i S A i+1) 3 Duivenvoorden-Quella 12, 13 When U, QPT out to ( spin -orbital entangled) SP phase (univ. class SU(N)2) N=4 (DMRG, L=36) N=4 QPT: top SP N=6 edge states (nα) Nonne et al. 13
Comparison: N=3 and N=4 cases... numerics (DMRG) 20 10 drastic difference between N=odd / even (preliminary) DMRG results: L=36 sites order parameter & edge states N=3 phase diag. based on L=36 DMRG (Nf=3*L) J CDW SP critical J=6U/5 cf. Nonne et al. 13 15 10 2kF-CDW SU(4) topological 5 J 0-10 U J/t 0-5 CDW SP HO topo J=4U/3 S=3/2 Heisenberg (gapless) -20-6 -4-2 0 2 4 6 U -10-6 -4-2 0 2 4 6 U/t N=3 N=4 3 phases (2 gapped, 1 gapless) SU(4)-topSP QPT orbital-haldane
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 ( 171 Yb), 5/2 ( 173 Yb), 9/2 ( 87 Sr),... Mott core Outlook: Effects of trap (SPT phase & Mott core)? quantum-optical detection of topological order?? Kobayashi et al. 12 higher-d???