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Symmetry Protection of Quantum Phases, UCSD Colloquium

Symmetry Protection of Quantum Phases, UCSD Colloquium

Colloquium Talk at Department of Physics, UC San Diego, Oct 6, 2016

Masaki Oshikawa

October 06, 2016
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  1. Symmetry Protection of Quantum Phases Masaki Oshikawa (ISSP, U. Tokyo)

    1 Physics Colloquium@UC San Diego
 Oct 6, 2016 Dan’s bike
  2. “Haldane Conjecture” and Symmetry-Protected Topological Phases Masaki Oshikawa (ISSP, U.

    Tokyo) 4 Physics Colloquium@UC San Diego
 Oct 6, 2016 Dan’s bike
  3. Classification of states of matter 7 = distinction of different

    phases Critical Point (Curie Temp.) ordered phase disordered phase Phase diagram of a ferromagnet Tc T simple model: (classical) Ising model
  4. Are liquid and gas different? 8 Phase transition can be

    “avoided” by going beyond the critical point Liquid/gas are “essentially indistinguishable” Figure from Sonntag R E, Borgnakke C, Van Wylen G J, “Fundamentals of Thermodynamics”
  5. What about solid? 9 Can we avoid the phase transition

    between solid/liquid at, e.g. higher pressures? NO! in solid, translation symmetry is spontaneously broken, while it is not in liquid/gas Spontaneous Symmetry Breakings (SSB)
 clearly distinguish different phases, implying existence of phase transitions
  6. What distinguishes different phases? 10 Different orders (or their absence)

    characterize each phase Ferromagnet: magnetic order Superfluid (3D): off-diagonal long-range order (order of U(1) phase of wavefunctions) etc. “order” ⊇ Spontaneous Symmetry Breaking
  7. However…………. 11 Recently, it has been recognized that there are

    many quantum phases that are beyond understanding in terms of conventional orders/spontaneous symmetry breaking “topological phases” how to define them? how to distinguish different phases?
  8. “Operational” definition 12 If the two gapped ground states are

    connected adiabatically they belong to the same phase Hastings-Wen (2005) Chen-Gu-Wen (2010)
  9. Topological Order 13 If the gapped ground state CANNOT be

    connected adiabatically to a trivial (product) state it belongs to a topologically ordered phase
 (with long-range entanglement) e.g.: Fractional Quantum Hall states Z2 topological phase (RVB spin liquid/Kitaev’s toric code)
  10. Topological Order in 1D 14 Any gapped ground state of

    a local 1D Hamiltonian is connected to a trivial state adiabatically Absence of (genuine) topologically ordered phase in 1D! However, there can be more variety of phases if some symmetries are imposed Verstraete et al. (2005), Hastings (2007), Chen-Gu-Wen (2011)
  11. Imposing Symmetries 15 For a gapped Hamiltonian with a symmetry

    1) the ground state is in a trivial phase, 2) the symmetry is spontaneously broken in the ground state (SSB phase), OR 3) the symmetry is unbroken, but the ground state cannot be connected to a trivial state by any local unitary evolution respecting the symmetry
  12. “SPT phase” 16 The gapped ground state cannot be connected

    to a trivial state adiabatically
 respecting the symmetry The ground state belongs to a Symmetry-Protected Topological (SPT) Phase SPT phases exist in 1 dimension, as well as in higher dimensions
  13. Where did it come from? 17 The concept of SPT

    phases was developed after the discovery of Topological Insulators [Kane-Mele (2005)] e.g. 2D Topological Insulators (Quantum Spin Hall Insulators) can be understood in terms of two quantum Hall states with up- and down- spins, and opposite magnetic field
  14. Topological Insulators 18 The model of two quantum Hall states

    with opposite magnetic fields seems artificial (and indeed is unrealistic) However, presence of the edge states is protected by Kramers degeneracy Topological Insulator is a SPT phase protected by Time-Reversal Symmetry
  15. First Example of SPT phases? Heisenberg antiferromagnetic chain H =

    J j ⇤ Sj · ⇤ Sj+1 S=1/2, 3/2, 5/2........ “massless” = gapless, power-law decay of spin correlations S=1, 2, 3, ..... “massive” = non-zero gap, exponential decay of spin correlations 19 “Haldane conjecture” (1983) based on field-theory mapping with a topological term
  16. AKLT model/state H = J ⇤ j ⇤ Sj ·

    ⇤ Sj+1 + 1 3 (⇤ Sj · ⇤ Sj+1 )2 ⇥ e.g. S=1 Exact groundstate: (Affleck-Kennedy-Lieb-Tasaki 1987) S=1/2 Singlet pair of two S=1/2’s -“valence bonds” Symmetrization (=projection to S=1) ✓non-zero gap, exponential decay of correlations (supporting the Haldane conjecture) 20
  17. Order in AKLT state? Groundstate of the AKLT model: UNIQUE

    (for periodic boundary condition) Correlation function of any local operator decays exponentially There is no local order parameter; no symmetry is broken spontaneously No order, that’s it? 21
  18. “Haldane phase” H = J ⇤ j ⌅ Sj ·

    ⌅ Sj+1 + D(Sz j )2 ⇥ D gap Dc trivial phase (“large-D phase”) “Haldane phase” |D⇥ = | . . . 00000 . . .⇥ D ⇥ : Why quantum phase transition? 22
  19. Edge states “free” S=1/2 appears at each end, interacting with

    each other. Effective coupling: Je e L/ Consider a chain with open boundary condition 2x2=4 groundstates below the Haldane gap (nearly degenerate) 23 Kennedy (1990)
  20. Hidden (string) order + 0 0 - + In Sz

    basis, + and - alternate, with 0’s in between No long-range order w.r.t. local observables, but a hidden (topological) order measurable by the “string order parameter” Ostr lim |j k|⇥⇤ ⇤Sj ei⇥ P k 1 l=j Sl Sk ⌅ Den Nijs & Rommelse (1989) 25
  21. Hidden Z2 x Z2 symmetry Kennedy & Tasaki (1992) H

    = J j ⇤ Sj · ⇤ Sj+1 ˜ H = UHU 1 = J j ⇥ Sx j ei Sx j+1 Sx j+1 + Sy j ei (Sz j +Sx j+1)Sy j+1 + Sz j ei Sz j Sz j+1 ⇤ non-local unitary transformation Global discrete symmetry (π-rotation about x, y, z axes = Z2 x Z2) U = j<k ei Sz j Sx k [simple expression by M.O. (1992)] [well-defined only for open b.c.] 26
  22. Spontaneous breaking of hidden Z2xZ2 symmetry 4-fold groundstate degeneracy for

    ˜ H 4-fold groundstate degeneracy for H only with the open b.c.! = edge states Ferromagnetic order for ˜ H String order for H UHU 1 = ˜ H U Sz j ei Pk 1 l=j Sz l Sz k ⇥ U 1 = Sz j Sz k 28
  23. 29 What was missing? In early 1990s, we already knew

    “Haldane phase” and understood many interesting things on it. However, (as far as I know) nobody asked “when does it work?” A simple analysis reveals that, a certain symmetry is needed for the Haldane phase to be distinct from a trivial phase. Now we expect this from the general argument, which was unknown at that time. Nevertheless, we could have (rather easily) reached the notion of SPT phases back then!
  24. 30 Phys. Rev. B 85 075125 (2012) Phys. Rev. B

    81, 063349 (2010) Frank Pollmann (MPIPKS Dresden) Erez Berg (Weizmann Institute) Ari Turner (Johns Hopkins) In 2009, I came back to the old problem and started a collaboration with
  25. Hidden Z2 × Z2 symmetry 31 The Kennedy-Tasaki transformation is

    nonlocal -- if the transformed Hamiltonian is nonlocal, the argument does not work. Because the transformation is self-dual, for to be local, the original Hamiltonian must have global D2 = Z2 x Z2 symmetry (π-rotation about x, y, z axes) Pollmann, Berg, Turner, M.O. 2009- One of the symmetries needed for the Haldane phase as a SPT phase
  26. Time Reversal 32 AKLT model: edge state with S=1/2 Does

    the edge state survive in more general models? Consider perturbations to AKLT model Generic perturbations will lift the edge degeneracy! However, if the perturbation respect time reversal, it should keep the “Kramers degeneracy” of S=1/2 edge state i.e. time reversal symmetry also protects the Haldane phase cf.) edge state of topological insulator
  27. Yet another symmetry 33 Gu and Wen, 2009 D2 symmetry

    (π-rotation about x,y,z axes): lost string order does not work as an order parameter Time reversal: lost edge state does not characterize the Haldane phase Nevertheless, Haldane phase is still distinct from other phases by QPTs Protected by inversion symmetry! Bx ˠ Dˠ
  28. How inversion works I: bond-centered inversion (parity) valence bond: each

    vb pair is I-even this vb makes AKLT state I-odd! 34 antisymmetric!
  29. S=1 AKLT state is “I-odd”. Now consider any perturbation, keeping

    I-invariance. The adiabatically connected state remains I-odd. On the other hand, a trivial groundstate is I-even. Any adiabatic evolution of the trivial state is also I-even as long as I-invariance is kept. | ⇥ = j | ⇥j There must be a phase transition between the two groundstates (robustness of Haldane phase protected by the inversion symmetry) 35
  30. The concept of hidden Z2xZ2 symmetry by Kennedy- Tasaki was

    generalized to general integer S U = j<k ei Sz j Sx k What did I find back in 1992? The hidden Z2xZ2 symmetry is unbroken in S=2,4,6,8,.... AKLT state while broken in S=1,3,5,7,..... AKLT state! “even-odd effect” 36
  31. What does it mean? The hidden Z2xZ2 symmetry is unbroken

    in the (uniform) S=2 AKLT state. Q (1992): Is it indistinguishable from a trivial state, or are we just unaware of appropriate hidden order/symmetry? 37 remained open until 2009…
  32. Edge state for S=2 S=2 AKLT state: each end has

    S=1 (3-fold deg.) The degeneracy will be lifted by perturbations, and generically no degeneracy remains! (no Kramers degeneracy) 38 This suggests a rather surprising conclusion that S=2 “Haldane phase” is essentially indistinguishable from a trivial state! (although it could be argued back in 1992)
  33. Intrinsic parity for S>1 chains I: lattice inversion The intrinsic

    parity is even, because you flip two valence bonds. In general, intrinsic link parity is even (odd), if the number of valence bonds is even (odd)! 39
  34. 40 S=2 Haldane state The S=2 “Haldane state” could be

    adiabatically connected to a trivial state?! Is this really the case? Yes! There exists a 1-parameter family of Matrix Product State (and corresponding Hamiltonian) interpolating S=2 AKLT state and large-D state Pollmann, Berg, Turner, M.O. 2009
  35. SU(2) symmetry? 42 If we keep the SU(2) symmetry, the

    presence of the S=1 edge state makes the system distinct from trivial states? In general, the answer is NO. The “S=1 edge states” of the “S=2 Haldane phase” are killed by introducing anisotropies
  36. S=1 AF Ladder 43 S. Todo et al. 2001 interchain

    K = 0: 2 x (S=1/2 edge spin) K>0 : rung singlet (trivial), no edge state K<0 : S=1 edge spin (ʙS=2 Haldane phase) NO phase transition at K=0 !! (S=1 Haldane gap at K=0)
  37. Kramers vs. non-Kramers Sb=0 vs. 1 triplet singlet Sb can

    change by level crossing at the edge (w/o bulk transition) Sb=0 vs. 1/2 doublet singlet Kramers theorem requires all the edge levels be doubly degenerate! The degeneracy can be only removed by bulk phase transition. 44 cf.) Todo et al. (2001)
  38. Entanglement Spectrum NA x NB matrix Singular Value Decomposition unitary

    matrices NA x NB diagonal matrix Entanglement Spectrum Entanglement Entropy Entanglement spectrum contains more information than entanglement entropy! Schmidt decomposition 45
  39. Entanglement Spectrum |⇥ = µ µ | A µ A

    | B µ B The entire entanglement spectrum has exact double degeneracy in the Haldane phase! 1 = 2, 3 = 4, 5 = 6, . . . This degeneracy is protected by any one of the three symmetries. A B Minimal entanglement entropy log(2) when 1 = 2 = 1/ ⇥ 2, = 0( 3) 46
  40. “Odd parity” state A B Exact two-fold degeneracy in the

    entire entanglement spectrum | ⇤ ⇥ ⇤ ⇥( ) | , 1⇤A | , 2⇤B | , 2⇤A | , 1⇤B ⇥ 47
  41. 48

  42. Symmetry Protection (I) Spontaneous breaking of hidden Z2xZ2 symmetry, robust

    in the presence of D2(=Z2 x Z2) symmetry [π-rotation about x,y, and z axes] (II) Kramers degeneracy of edge spins, robust in the presence of time-reversal S=1 Haldane phase is “protected” by ANY one of 49 (III) Space Inversion symmetry about a bond center (Gu-Wen/Pollmann-Berg-Turner-M.O.)
  43. Summary Odd # of valence bonds Kramers degeneracy of edge

    spins Hidden Z2xZ2 symmetry breaking (dNR string order) odd intrinsic link parity Exact double degeneracy of entire entanglement spectrum [time-reversal invariance] [inversion symmetry] [D2 symmetry] 50
  44. Summary 51 “Haldane conjecture” was not only - a remarkable

    application of field theory and
 topology to condensed matter physics - a prediction which sounded like crazy at the
 time but eventually is supported by many
 analytical/numerical/experimental evidences
 
 but also
 - an inspiration for many modern concepts
 including Matrix-Product/Tensor-Network States
 and Symmetry-Protected Topological Phases