SPT order - new state of quantum matter. Xiao-Gang Wen, MIT/Perimeter Taiwan, Jan., 2015

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1 Xiao-Gang Wen, MIT/Perimeter Taiwan, Jan., 2015

2 Symm. breaking phases and topo. ordered phases We used to believe that all phases and phase transitions are described by symmetry breaking Counter examples: 2 - Quantum Hall states σxy = mn eh - Spin liquid states, Organic κ-(et)2 X and herbertsmithite FQH states and spin-liquid states have different phases with no symmetry breaking, no crystal order, no spin order,... so they must have a new order topological order [Wen 89]

3 Topological order and chiral spin state The discovery of chiral spin state: - Lee and Joannopoulos, unpub.: RVB bosonic ν = 1/2 state - Equivalence of the RVB and FQH states Kalmeyer and Laughlin, Phys. Rev. Lett. 59, 2095 (1987) - Chiral spin states and superconductivity Wen, Wilczek, and Zee, Phys. Rev. B 39, (1989) (1) S 1 (S 2 S 3 ) 0 (2) U k (1) CS π-flux and π/2-flux chiral spin states break the same time-reversal symm.: the notion of topological order The dependence of vaccum degeneracy on the topology of space relfects some sort of topological ordering...

4 Probe/define topological order via topo. invariants Robust topology-dependent ground state degeneracy D g is a topological invariant = a property of H that is invariant under any perturbations: Prop(H) = Prop(H + δh), δh 1 - Topological invariants a new order Topological order g=1 g=0 g=2 [Wen 89, Wen & Niu 90] Deg.=1 Deg.=D 1 Deg.=D 2

5 Probe/define topological order via topo. invariants Robust topology-dependent ground state degeneracy D g is a topological invariant = a property of H that is invariant under any perturbations: Prop(H) = Prop(H + δh), δh 1 - Topological invariants a new order Topological order g=1 g=0 g=2 [Wen 89, Wen & Niu 90] Deg.=1 Deg.=D 1 Deg.=D 2 Robust non-abelian geometric s phases of the degenerate ground states by deforming the shape of torus through a deformation loop: Ψ α Ψ α = U αβ Ψ β [Wen 90]

6 Probe/define topological order via topo. invariants Robust topology-dependent ground state degeneracy D g is a topological invariant = a property of H that is invariant under any perturbations: Prop(H) = Prop(H + δh), δh 1 - Topological invariants a new order Topological order g=1 g=0 g=2 [Wen 89, Wen & Niu 90] Deg.=1 Deg.=D 1 Deg.=D 2 Robust non-abelian geometric s phases of the degenerate ground states by deforming the shape of torus through a deformation loop: Ψ α Ψ α = U αβ Ψ β [Wen 90] Robust gapless edge state in QH and chiral spin liquid [Wen 91] the gapless excitations are shown to form a representation of the U(1) or SU(2) Kac-Moody algebras

7 Vector bundle over the moduli space [Wen 90,12] T S ST The shape of a torus is described by the moduli space: upper-half plane τ M = = A sphere with three punctures. PSL(2,Z)=MCG(T 2 ) Ground states a U(D 1 ) vector bundle over M = the robust non-abelian geometric s phases. - Non-Abelian geometric phase of Dehn twist for torus: Ψ α Ψ α = e i φ T (N k ) T αβ Ψ β - 90 rotation Ŝ for τ = i torus: Ψ α e i an ks αβ Ψ β - 60 rotation Ŝ ˆT for τ = 1+ 3 i : Ψ 2 α e i bn k (ST )αβ Ψ β Find a N 1 and N k = kn 1, so that Ŝ, Ŝ ˆT do not depend on N k.

8 Non-Abelian geom. phase and topo. partition func. Let the space-time = a fiber bundle over S 1 t = mapping torus. Such a fiber bundle is described an element in Ŵ MCG(M) (such as Ŝ, ˆT ). So we denote space-time = M Ŵ S 1 t Topo. (fixed-point) partition function[kong-wen 14] Z(M Ŵ S 1 t ) = Z top (M Ŵ S 1 t )e # Vol Z top (M Ŵ S 1 t ) = Tr(W ) Z top (M S 1 t ) = the ground state degeneracy on space M. d+1 M W twist d d M 1 M 2 Z top (S d S 1 t ) = 1 Z top (S d 1 S 1 S 1 t ) = number of topological particle types. Topological partition function and the non-abelian geometric phases are the same type of topological invariants for topologically ordered states

9 Gapped/Short-range-correlated phases & topology Z(M Ŵ S 1 t ) = Z top (M Ŵ S 1 t )e # Vol, Z top (M Ŵ S 1 t ) = Tr(W ) Vol length of t A conjecture : On closed space-time manifold M d+1 in large system-size limit Z(M d+1 ) = Z top (M d+1 )e # Vol M d+1 where Z top (M d+1 ) (a map M d+1 C ) is a topological invariants of M d+1. For trivial phase Z top (M d+1 ) = 1 Another conjecture : The topological invariants of M d+1 classify non-trivial gapped/src phases and non-trivial gapped/src phases classify the topological invariants of M d+1. [Wen 89] Non-trivial gapped/src phases = topologically ordered phases

10 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical)

11 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum)

12 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) entangled

13 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled

14 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled =... unentangled

15 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled =... unentangled = ( ) ( )... short-range entangled (SRE) state

16 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled =... unentangled = ( ) ( )... short-range entangled (SRE) state Crystal order: Φ crystal = = 0 x1 1 x2 0 x3... = direct-product state unentangled state (classical)

17 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled =... unentangled = ( ) ( )... short-range entangled (SRE) state Crystal order: Φ crystal = = 0 x1 1 x2 0 x3... = direct-product state unentangled state (classical) Particle condensation (superfluid) Φ SF = all conf.

18 Microscopic picture of topological order Topological order comes from quantum entanglement = direct-product state unentangled (classical) + entangled (quantum) = ( + ) ( + ) = x x unentangled =... unentangled = ( ) ( )... short-range entangled (SRE) state Crystal order: Φ crystal = = 0 x1 1 x2 0 x3... = direct-product state unentangled state (classical) Particle condensation (superfluid) Φ SF = all conf. = ( 0 x1 + 1 x1 +..) ( 0 x2 + 1 x2 +..)... = direct-product state unentangled state (classical)

19 Long-range entanglement by scrambling the phase Φ SF ({z 1,..., z N }) = 1 unentangled product state Local dancing rules of a FQH liquid: (1) every electron dances around clock-wise (Φ FQH only depends on z = x + iy) (2) takes exactly three steps to go around any others (Φ FQH s phase change 6π) Global dancing pattern Φ FQH ({z 1,..., z N }) = (z i z j ) 3 A theory of multi-layer Abelian FQH state: I ;i<j (z I i z I j ) K II I <J;i,j (z I i z J j ) K IJ e 1 4 i,i zi i 2 Low energy effective theory is K-matrix Chern-Simons theory L = K IJ a 4π I da J Number of edge modes c = dim(k) = number of layers. Even K-matrix classifies all 2+1D Abelian topological order (but not one-to-one)

20 What is long-range entanglement? For gapped systems with no symmetry: According to Landau theory, no symm. to break all systems belong to one trivial phase

21 What is long-range entanglement? For gapped systems with no symmetry: According to Landau theory, no symm. to break all systems belong to one trivial phase Thinking about entanglement: there are [Chen-Gu-Wen 10] - long range entangled (LRE) states - short range entangled (SRE) states LRE product state = SRE local unitary transformation LRE state SRE product state

22 What is long-range entanglement? For gapped systems with no symmetry: According to Landau theory, no symm. to break all systems belong to one trivial phase Thinking about entanglement: there are [Chen-Gu-Wen 10] - long range entangled (LRE) states many phases - short range entangled (SRE) states one phase g LRE product state = SRE 2 topological order local unitary transformation LRE state SRE product state local unitary transformation SRE product state SRE product state local unitary transformation LRE 1 LRE 2 LRE 1 LRE 2 All SRE states belong to the same trivial phase LRE states can belong to many different phases: different patterns of long-range entanglements [defined by LU trans.] = different topological orders [Wen 1989] SRE phase transition g 1

23 How to classify topological order

24 How to classify topological order of a simple type Monoid and group structures of topological orders Let C d+1 = {a, b, c, } be a set of topologically ordered phases in d dimensions. Stacking a-to state and b-to state a c-to state: a b = c, a, b, c C d+1 c TO a TO b TO make C d+1 a monoid (a group without inverse). Consider topological order a and topological order a Ztop(M a d+1 ) = [Ztop(M a d+1 )], then Ztop a a (M d+1 ) = Ztop(M a d+1 )Ztop(M a d+1 ) In general, Ztop(M a d+1 )Ztop(M a d+1 ) 1 a a is a non trivial topological order, and a-to has no inverse. A topological order is invertible iff its Z top (M d+1 ) = e i θ [Freed 14; Kong-Wen 14]

25 Invertible bosonic topological orders ito d+1 Invertible topological orders are simplest topological orders In 2+1D: ito 3 = Z Z top (M d+1 ) = e i 2πk M 3 ω 3 (g µν) where ω 3 is the gravitational Chern-Simons term: dω 3 = p 1 and p 1 is the Pontryagin class. The quantization of the topological term: k = 1 int. ω 3 M 3(g µν ) = ω M 3(gµν W ) mod 3, since N closed p 1 = 0 mod 3. But actually (?), k = int. Z-class: ito 3 = Z Z top (M 3 ) = e i π M 3 w 3 Z 2 -class? (where w i H i (M, Z 2 ) is the i th Stiefel-Whitney class w M 3 3 = 0, 1 mod 2) No, since w 3 = 0 for tangent bundle of any orientable M 3 In 3+1D: ito 4 = 0 Z top (M 4 ) = e i π M 4 w 2 w 2 Z 2 -class? But w 2 w 2 = p 1 mod 2 and Z top (M 4 ) = e i π M 4 p 1 e i θ M 4 p 1.

26 Invertible bosonic topological orders ito d+1 In 4+1D: ito 5 = Z 2 Z top (M 5 ) = e i π M 5 w 2 w 3 Z 2 -class. We find w M d+1 2 w 3 = 1 (or Z top (M 5 ) = 1) when M 5 = CP 2 W S 1 and W : CP 2 (CP 2 ) In 6+1D: ito 7 = 2Z Two independent grav. Chern-Simons terms: Z top (M 7 ) = e 2πi M 7 (k 1 ω 7 +k 2 ω 7 ) where dω 7 = p 2, d ω 7 = p 1 p 1 Z Z-class (k 1, k 2 ). [Kapustin 14,Kong-Wen 14] 1 + 1D 2 + 1D 3 + 1D 4 + 1D 5 + 1D 6 + 1D Boson: 0 Z (E8 ) 3 0 Z 2 0 Z Z Fermion: Z 2 Z (p+ip) 2????

27 Short-range entanglement + symm. SPT phase For gapped systems with a symmetry H = U g HU g, g G LRE symmetric states many different phases SRE symmetric states one phase (no symm. breaking)

28 Short-range entanglement + symm. SPT phase For gapped systems with a symmetry H = U g HU g, g G LRE symmetric states many different phases SRE symmetric states many different phases We may call them symm. protected trivial (SPT) phase g 2 topological order LRE 1 SRE LRE 2 g 1 g 2 SY LRE 1 SY LRE 2 topological order SB LRE 1 SB SRE 1 SY SRE 1 SB LRE 2 SB SRE 2 SY SRE 2 g 1 topological orders (??? ) symmetry breaking (group theory) SPT phases (??? ) phase transition preserve symmetry SPT 1 SPT 2 no symmetry SPT phases = equivalent class of symmetric LU trans.

29 Short-range entanglement + symm. SPT phase For gapped systems with a symmetry H = U g HU g, g G LRE symmetric states many different phases SRE symmetric states many different phases We may call them symm. protected trivial (SPT) phase or symm. protected topological (SPT) phase g 2 topological order LRE 1 SRE LRE 2 g 1 g 2 SY LRE 1 SY LRE 2 topological order SB LRE 1 SB SRE 1 SY SRE 1 SB LRE 2 SB SRE 2 SY SRE 2 g 1 topological orders (??? ) symmetry breaking (group theory) SPT phases (??? ) phase transition preserve symmetry SPT 1 SPT 2 no symmetry SPT phases = equivalent class of symmetric LU trans. Examples: 1D Haldane phase[haldane 83] 2D/3D TI[Kane-Mele 05; Bernevig-Zhang 06] [Moore-Balents 07; Fu-Kane-Mele 07] 1D 2D 3D

30 Symm. breaking phases group theory SPT phases group cohomology theory SPT phases gapped quantum phases with a certain symmetry, which can be smoothly connected to the same trivial phase if we remove the symmetry. SPT1 SPT2 SPT3 with a symmetry G Product state break the symmetry

31 Symm. breaking phases group theory SPT phases group cohomology theory SPT phases gapped quantum phases with a certain symmetry, which can be smoothly connected to the same trivial phase if we remove the symmetry. SPT1 SPT2 SPT3 with a symmetry G Product state break the symmetry [Chen-Gu-Liu-Wen 11] A systematic construction of many SPT phases: Using each element in H d+1 (G, U 1 ) (the d + 1 group cohomology class of the group G with U 1 U(1) as coefficient), we can construct a d + 1D exactly soluble Hamiltonian, whose ground state is a SPT state with an on-site symmetry G.

32 Non-linear σ-model (NLσM) and SPT states Our construction of the SPT states is based on NLσM. Consider an d + 1D system: S = d d xdt 1 g 1 g 2, 2λ h, g are unitary matrices in G. Symmetry g(x) hg(x). λ is small the ground state is ordered g(x, t) = g 0. λ is large the ground state is disordered g(x, t) = 0. The fixed point action of the disordered phases: Under RG, λ the disordered symmetric phase is described by a fixed point theory S fixed = 0 or e S fixed = 1. All correlations are short ranged. All excitations are gapped. We thought that the disordered ground states are trivial, no excitations at low energies. But the disordered ground states can be non-trivial: they can belong to different phases, even without symmetry breaking. the notions of topological orders and SPT orders.

33 NLσM with topological terms Different disordered phases can arise from NLσM with different topological terms.

34 NLσM with topological terms Different disordered phases can arise from NLσM with different topological terms. Another G symmetric system S = ( ) d d 1 x dt g 1 g 2 + 2πiW 2λ G-top where W G-top [g(x i, t)] is a topological term. The topo. term W G-top [g(x i, t)] can appear if π d+1 (G) 0. Example 1: 0 + 1D with G = U 1 = S 1 (g = e i θ ) W G-top = k i g 1 2π t g = k θ, k Z. 2π dt WG-top = k winding number. Example 2: 2 + 1D with G = SU 2 = S 3 (g = n 0 + n σ) W G-top = k 1 ɛ µνλ (ig 1 24π 2 µ g)(ig 1 ν g)(ig 1 λ g), k Z. dt d 2 x W = k winding number. The topological term has no dynamical effect e S fixed = 1, but can give rise to different symmetric phases classified by k

35 Fixed-point eff. Lagrangian for symmetric phases If λ is large disordered state. Under RG, λ disordered symmetric ground state is described by a low energy fixed-point theory S fixed = 2πi W G-top topological non-linear σ-model with pure topological term.

36 Fixed-point eff. Lagrangian for symmetric phases If λ is large disordered state. Under RG, λ disordered symmetric ground state is described by a low energy fixed-point theory S fixed = 2πi W G-top topological non-linear σ-model with pure topological term. Fixed point theories (2π-quantized topological terms) symmetric phases: The symmetric phases are classified by different 2π-quantized topological terms (Hom(π d+1 (G), Z))

37 Fixed-point eff. Lagrangian for symmetric phases If λ is large disordered state. Under RG, λ disordered symmetric ground state is described by a low energy fixed-point theory S fixed = 2πi W G-top topological non-linear σ-model with pure topological term. Fixed point theories (2π-quantized topological terms) symmetric phases: The symmetric phases are classified by different 2π-quantized topological terms (Hom(π d+1 (G), Z)) But in the λ limit, g(x i, t) is not a continuous function. The mapping classes π d+1 (G) does not make sense. The above result is not valid. However, the idea is OK.

38 Fixed-point eff. Lagrangian for symmetric phases If λ is large disordered state. Under RG, λ disordered symmetric ground state is described by a low energy fixed-point theory S fixed = 2πi W G-top topological non-linear σ-model with pure topological term. Fixed point theories (2π-quantized topological terms) symmetric phases: The symmetric phases are classified by different 2π-quantized topological terms (Hom(π d+1 (G), Z)) But in the λ limit, g(x i, t) is not a continuous function. The mapping classes π d+1 (G) does not make sense. The above result is not valid. However, the idea is OK. Can we define topological terms and topological non-linear σ-models when space-time is a discrete lattice?

39 Path integral on space-time lattice 0+1D space-time lattice Z = Tre βh = ν 1 (g i, g i+1 ), ν 1 (g 0, g 1 ) = (e βh ) g0,g 1 {g i } i For example, on 0+1D space-time lattice with branching structure Z = t ν 1 (g 0, g 2 )ν 1 (g 2, g 1 )ν 1 (g 1, g 3 )ν 1 (g 3, g 4 ) g 0,,g 4 =g 0 = ν 1 (g 0, g 2 )ν (g 1, g 2 )ν 1 (g 1, g 3 )ν 1 (g 3, g 4 ) g 0,,g 4 =g 0 = {g i } ν s ij (g i, g j ) ij where s ij = 1 if the edge ij has a +-orientations, and s ij = if the edge ij has a -orientations.

40 NLσM on 1+1D space-time lattice Path integral on 1+1D space-time lattice with branching structure: e S = ν s ijk 2 (g i, g j, g k ), where ν s ijk (gi, g j, g k ) = e L and s ijk = 1, The above defines a j k i LNσM with target k i j space G on 1+1D gk space-time lattice. + g g g space time The NLσM will have a symmetry G if g i G and ν 2 (g i, g j, g k ) = ν 2 (hg i, hg j, hg k ), h G The above is the lattice version of NLσM field theory: L = 1 λ g 1 g 2 gj ν(,, ) g i i j k G

41 Topo. term and topo. NLσM on space-time lattice ν(g i, g j, g k ) give rise to a topological NLσM if g3 g e S 0 fixed = ν s ijk(gi, g j, g k ) = 1 on any sphere, including a tetrahedron (simplest sphere). ν(g i, g j, g k ) U 1 g1 g On a tetrahedron 2-cocycle condition ν 2 (g 1, g 2, g 3 )ν 2 (g 0, g 1, g 3 )ν 1 2 (g 0, g 2, g 3 )ν 1 2 (g 0, g 1, g 2 ) = 1 The solutions of the above equation are called group cocycle. The 2-cocycle condition has many solutions: ν 2 (g 0, g 1, g 2 ) and ν 2 (g 0, g 1, g 2 ) = ν 2 (g 0, g 1, g 2 ) β 1(g 1,g 2 )β 1 (g 0,g 1 ) β 1 (g 0,g 2 ) are both cocycles. We say ν 2 ν 2 ( equivalent). The set of the equivalent classes of ν 2 is denoted as H 2 (G, U 1 ) = π 0 (space of the solutions). H 2 (G, U 1 ) describes 1+1D SPT phases protected by G. 2

42 Group cohomology H d [G, U 1 ] in any dimensions d-cochain: U 1 valued function of d + 1 variables ν d (g 0,..., g d ) = ν d (gg 0,..., gg d ) U 1, on-site G -symmetry δ-map: ν d with d + 1 variables (δν d ) with d + 2 variables (δν d )(g 0,..., g d+1 ) = ν ( )i d (g 0,..., ĝ i,..., g d+1 ) i Cocycles = cochains that satisfy (δν d )(g 0,..., g d+1 ) = 1. Equivalence relation generated by any d 1-cochain: ν d (g 0,..., g d ) ν d (g 0,..., g d )(δβ d 1 )(g 0,..., g d ) H d+1 (G, U 1 ) is the equivalence class of cocycles ν d. d + 1D lattice topological NLσMs with symmetry G in are classified by H d+1 (G, U 1 ): e S = ν s(i,j,...) d+1 (g i, g j,...), ν d+1 (g 0, g 1,..., g d+1 ) H d+1 (G, U 1 )

43 Topological invariance in topological NLσMs g g 0 2 g 0 g 2 g g g 0 g 2 g g g 1 3 g 1 As we change the space-time lattice, the action amplitude e S does not change: g 3 ν 2 (g 0, g 1, g 2 )ν 1 2 (g 1, g 2, g 3 ) = ν 2 (g 0, g 1, g 3 )ν 1 2 (g 0, g 2, g 3 ) g 1 g 1 ν 2 (g 0, g 1, g 2 )ν 1 2 (g 1, g 2, g 3 )ν 2 (g 0, g 2, g 3 ) = ν 2 (g 0, g 1, g 3 ) as implied by the cocycle condition: ν 2 (g 1, g 2, g 3 )ν 2 (g 0, g 1, g 3 )ν 1 2 (g 0, g 2, g 3 )ν 1 2 (g 0, g 1, g 2 ) = 1 The topological NLσM is a RG fixed-point.

44 The NLσM ground state is short-range entangled g 1 g 3 g* g 5 g 2 g 4 gg 1 3 gg* = = gg 5 gg The ground state wave function Ψ({g i }) = i ν 2(g i, g i+1, g ) 2 gg gg 4 gg gg 1 5 g* gg 2 gg gg 4 3

45 The NLσM ground state is short-range entangled g 1 g 3 g* g 5 g 2 g 4 gg 1 3 gg* = = gg 5 The ground state wave function Ψ({g i }) = i ν 2(g i, g i+1, g ) It is symmetric under the G-transformation gg 2 gg gg 4 gg gg 1 5 g* gg 2 gg gg 4 3

46 The NLσM ground state is short-range entangled g 1 g 3 g* g 5 g 2 g 4 gg 1 3 gg* = = gg 5 The ground state wave function Ψ({g i }) = i ν 2(g i, g i+1, g ) It is symmetric under the G-transformation It is equivalent to a product state Ψ 0 = i g i g i under a LU transformation (note that Ψ 0 ({g i }) = 1) Ψ({g i }) = ν 2 (g i, g i+1, g ) ν 2 (g i, g i+1, g )Ψ 0 ({g i }) i=even gg 2 gg gg 4 i=odd = Ψ 0 ({g i }) Short-range entangled The ground state is symmetric with a trivial topo. order gg gg 1 5 g* gg 2 gg gg 4 3

47 Bosonic SPT phases from H d+1 (G, U 1 ) Symmetry G d = 0 d = 1 d = 2 d = 3 U 1 Z2 T (top. ins.) Z Z 2 (0) Z 2 (Z 2 ) Z 2 2 (Z 2) U 1 Z2 T trans Z Z Z 2 Z Z 3 2 Z Z 8 2 U 1 Z2 T (spin sys.) 0 Z2 2 0 Z 3 2 U 1 Z2 T trans 0 Z2 2 Z 4 2 Z 9 2 Z2 T (top. SC) 0 Z 2 (Z) 0 (0) Z 2 (0) Z2 T trans 0 Z 2 Z 2 2 Z 4 2 U 1 Z 0 Z 0 U 1 trans Z Z Z 2 Z 4 Z n Z n 0 Z n 0 Z n trans Z n Z n Z 2 n Z 4 n D 2h = Z 2 Z 2 Z2 T Z 2 2 Z 4 2 Z 6 2 Z 9 2 SO(3) 0 Z 2 Z 0 SO(3) Z2 T 0 Z 2 2 Z 2 Z 3 2 g g 2 topological order 2 SY LRE 1 SY LRE 2 SET orders (tensor category) (tensor category intrinsic topo. order w/ symmetry) LRE 1 LRE 2 SB LRE 1 SB LRE 2 symmetry breaking SRE SB SRE 1 SY SRE 1 SB SRE 2 SY SRE 2 (group theory) SPT orderes (group cohomology g 1 g SPT order - new state of quantum 1 theory) matter Z2 T : time reversal, trans : translation, 0 only trivial phase. (Z 2 ) free fermion result

48 Universal probe for SPT orders How do you know the constructed NLσM ground states carry non-trivial SPT order? How do you probe/measure SPT order? Universal probe = one probe to detect all possible orders.

49 Universal probe for SPT orders How do you know the constructed NLσM ground states carry non-trivial SPT order? How do you probe/measure SPT order? Universal probe = one probe to detect all possible orders. Universal probe for crystal order = X-ray diffraction:

50 Universal probe for SPT orders How do you know the constructed NLσM ground states carry non-trivial SPT order? How do you probe/measure SPT order? Universal probe = one probe to detect all possible orders. Universal probe for crystal order = X-ray diffraction: Z SPT top (M d+1 ) = 1 G Nv Universal probe for SPT order = gauging the symmetry - Under an uniform symm. trans.: g(x) hg(x), [Levin-Gu 12] L(g 1 dg) L(g 1 dg) - Under a non-uniform symm. trans.: g(x) h(x)g(x), L(g 1 dg) L[g 1 (d + h 1 dh)g] - Replace h 1 dh by ia to get the gauged Lagrangian: L[g 1 (d + h 1 dh)g] L[g 1 (d ia)g]

51 Universal topo. inv.: gauged partition function Gauging or symmetry twist is our way to probe SPT order. To measure the SPT order, we measure the change of the partition function in response to the symmetry twist : Z[A] Dg e L(g 1 ( d i A)g) Z[0] = Dg e = Z L(g 1 dg) top [A] = e i 2π W A-top (A)

52 Universal topo. inv.: gauged partition function Gauging or symmetry twist is our way to probe SPT order. To measure the SPT order, we measure the change of the partition function in response to the symmetry twist : Z[A] Dg e L(g 1 ( d i A)g) Z[0] = Dg e = Z L(g 1 dg) top [A] = e i 2π W A-top (A) For gapped symmetric states, the above change/responce is universal independent of any perturbations that do not break the symmetry:

53 Universal topo. inv.: gauged partition function Gauging or symmetry twist is our way to probe SPT order. To measure the SPT order, we measure the change of the partition function in response to the symmetry twist : Z[A] Dg e L(g 1 ( d i A)g) Z[0] = Dg e = Z L(g 1 dg) top [A] = e i 2π W A-top (A) For gapped symmetric states, the above change/responce is universal independent of any perturbations that do not break the symmetry: - If globally we have A = ih 1 dh, then Z[A]/Z[0] = 1. So W A-top (A) is non-trivial only when A 0 for some loops. loop - If we choose A such that A 0 only for large loops, then loop the corresponding W A-top (A) represents a global topological response, and is universal. In other words, we choose (1) da 1/(system size) 2 if G is continuous; (2) da = 0, but loop A 0 for non-contractible loops if G is discrete.

54 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A)

55 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A) An example: Topo. term for SU 2 SPT state: In 2 + 1D π 3 (SU 2 ) = Z: W 3 G-top = k Tr( i g 1 dg) 3 24π 2.

56 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A) An example: Topo. term for SU 2 SPT state: SU 2 connection A ig 1 dg In 2 + 1D π 3 (SU 2 ) = Z: A = 2x2 matrix; WG-top 3 = k Tr( i g 1 dg) 3. 24π 2 WA-top 3 = ktr A3. 24π 2

57 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A) An example: Topo. term for SU 2 SPT state: SU 2 connection A ig 1 dg In 2 + 1D π 3 (SU 2 ) = Z: A = 2x2 matrix; F da + [A, A] W 3 G-top = k Tr( i g 1 dg) 3 24π 2. W 3 A-top = ktr A3 +3AF 24π 2.

58 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A) An example: Topo. term for SU 2 SPT state: SU 2 connection A ig 1 dg In 2 + 1D π 3 (SU 2 ) = Z: A = 2x2 matrix; F da + [A, A] W 3 G-top = k Tr( i g 1 dg) 3 24π 2. W 3 A-top = ktr A3 +3AF 24π 2. d = H d (SU 2 ) WA-top d Z Tr A3 +3AF 24π

59 Calculate the topo. responce for SPT phases Topological NLσM, as a fixed-point theory, contain only the pure topological term, and it is easy to calculate W A-top (A): cocycle ν d ({g i }) = G-topo. term W G-top (g 1 dg) = A-topo. term W A-top (A) ν d ({g i }) W G-top (g 1 dg) = g 1 dg AW A-top (A) An example: Topo. term for SU 2 SPT state: SU 2 connection A ig 1 dg In 2 + 1D π 3 (SU 2 ) = Z: A = 2x2 matrix; F da + [A, A] W 3 G-top = k Tr( i g 1 dg) 3 24π 2. WA-top 3 = ktr A3 +3AF. 24π 2 Spin quantum Hall conductance σxy spin Gapless state if the = k 4π d = H d (SU 2 ) WA-top d Z Tr A3 +3AF 24π SU 2 symm. is not broken. (No topo. order, need symm. protection) [Liu-Wen 12]

60 The edge of the SU(2) SPT state must be gapless Bulk fixed-point action: [Liu & Wen 12] k S bulk = i Tr(g 1 dg) 3, k Z, g SU(2) The 12π M 3 SU(2) symmetry g(x) hg(x), h, g(x) SU(2)

61 The edge of the SU(2) SPT state must be gapless Bulk fixed-point action: [Liu & Wen 12] k S bulk = i Tr(g 1 dg) 3, k Z, g SU(2) The 12π M 3 SU(2) symmetry g(x) hg(x), h, g(x) SU(2) The edge excitations on M 3 described by fixed-point WZW: S edge = k Tr( g 1 g) i k Tr(g 1 dg) 3, M 3 8π M 3 12π At the fixed point, we have a equation of motion z [( z g)g 1 ] = 0, z [( z g 1 )g] = 0, z = x + it. Right movers [( z g)g 1 ](z) SU(2)-charges Left movers [( z g 1 )g]( z) SU L (2)-charges, g(x) g(x)h L Level-k Kac-Moody algebra [Witten 84] The SU(2) symmetry is anomalous at the edge. In general, G SPT state has anomalous G-symmetry at the boundary a defining property of SPT phases.

62 H d (G, R/Z) does not produce all the SPT phases with symm. G: Topological states and anomalies SPT order from H d (G, R/Z) SPT state theory with with on site gauge symmetry (symm.) anomaly Topological order effective Topologically theory ordered with gravitational state anomaly g g 2 topological order 2 SY LRE 1 SY LRE 2 SET orders (tensor category) (tensor category intrinsic topo. order w/ symmetry) LRE 1 LRE 2 SB LRE 1 SB LRE 2 symmetry breaking SB SRE 1 SB SRE 2 (group theory) SRE SY SRE 1 SY SRE 2 SPT orderes (group cohomology g 1 g 1 theory)

63 H d (G, R/Z) does not produce all the SPT phases with symm. G: Topological states and anomalies SPT order from H d (G, R/Z) SPT state theory with with on site gauge symmetry (symm.) anomaly SPT state theory with with on site mixed symmetry gauge grav. anomaly SPT order beyond H d (G, R/Z) Topological order effective Topologically theory ordered with gravitational state anomaly g g 2 topological order 2 SY LRE 1 SY LRE 2 SET orders (tensor category) (tensor category intrinsic topo. order w/ symmetry) LRE 1 LRE 2 SB LRE 1 SB LRE 2 symmetry breaking SB SRE 1 SB SRE 2 (group theory) SRE SY SRE 1 SY SRE 2 SPT orderes (group cohomology g 1 g 1 theory)

64 SPT order beyond H d (G, R/Z): mixed SPT order Pure SPT order within H d (G, R/Z): only depend on A W d A-top W d A-top = Tr A3 +3AF 24π 2

65 SPT order beyond H d (G, R/Z): mixed SPT order Pure SPT order within H d (G, R/Z): only depend on A W d A-top W d A-top = Tr A3 +3AF 24π 2 Invertible topological order: WA-top d = ω 3, 1w 2 2w 3 WA-top d only depend on Γ p 1 is the first Pontryagin class, dω 3 = p 1, and w i is the Stiefel-Whitney classes.

66 SPT order beyond H d (G, R/Z): mixed SPT order Pure SPT order within H d (G, R/Z): only depend on A W d A-top W d A-top = Tr A3 +3AF 24π 2 Invertible topological order: WA-top d = ω 3, 1w 2 2w 3 WA-top d only depend on Γ p 1 is the first Pontryagin class, dω 3 = p 1, and w i is the Stiefel-Whitney classes. mixed SPT order beyond H d (G, R/Z): depend on both A and Γ W d A-top W d A-top = p 1Tr A3 +3AF 24π 2

67 A general theory for bosonic pure STP orders, mixed SPT orders, and invertible topological orders NLσM (group cohomology) approach to pure SPT phases: 1 (1) NLσM+topo. term: 2λ g 2 + 2πiW (g 1 g), g G 1 (2) Add symm. twist: ( 2λ ia)g 2 + 2πiW [( ia)g] (3) Integrate out matter field: Z fixed = e 2π i W A-top (A) G SO NLσM (group cohomology) approach: 1 (1) NLσM: 2λ g 2 + 2πiW (g 1 g), g G SO 1 (2) Add twist: ( ia 2λ iγ)g 2 + 2πiW [( ia iγ)g] (3) Integrate out matter field: Z fixed = e 2π i W A-top (A,Γ) Pure STP orders, mixed SPT orders, and invertible topological orders are classified by H d (G SO, R/Z) = H d (G, R/Z) d 1 k=1 Hk [G, H d k (SO, R/Z)] H d (SO, R/Z) after quotient out something Γ d (G).

68 Trying to classify bosonic pure STP orders, mixed SPT orders, and invertible topological orders Pure STP orders: H d (G, R/Z) mixed SPT orders: d 1 k=1 Hk [G, H d k (SO, R/Z)] ito s: H d (SO, R/Z) The above are one-to-one description of pure SPT orders, but only many-to-one description of mixed SPT orders and ito s - G-symmetry twists A WA-top d (A) can fully detect/distinguish all elements of H d (G, R/Z). - SO-symmetry twists Γ SO WA-top d (Γ SO) can fully detect/distinguish all elements of H d (SO, R/Z). - SO-symmetry twists Γ from the tangent bundle of M d are only special SO-symmetry twists (which are arbitrary SO-bundles on M d ) WA-top d (Γ) cannot fully detect/distinguish all elements of H d (SO, R/Z). ito d = H d (SO, R/Z)/Γ d

69 Trying to classify bosonic pure STP orders, mixed SPT orders, and invertible topological orders Pure STP orders: H d (G, R/Z) (the black entries below) mixed SPT order d 1 k=1 Hk (G, ito d k ) = Hk [G,H d k (SO,R/Z)] Γ d (G) ito s: ito d = H d (SO, R/Z)/Γ d Probe mixed SPT order described by H k [G, H d k (SO, R/Z)]: put the state on M d = M k M d k and add a G-symmetry twist on M k Induce a state on M d k described by H d k (SO, R/Z) a ito state in ito d k G \ d = ito d 0 0 Z 0 Z Z n Z n 0 Z n 0 Z n Z n Z n,2 Z n Z n Z n,2 Z2 T 0 Z. 2 0 Z 2 Z 2 0 Z 2 2Z 2 Z 2 U(1) Z 0 Z 0 Z Z 0 Z Z Z 2 U(1) Z 2 Z 2 Z 2 Z Z 2 Z 2 2Z 2 Z 2 2Z 2 2Z 2 Z 2Z 2 Z 2Z 2 U(1) Z2 T 0 2Z 2 0 3Z 2 Z 2 0 4Z 2 3Z 2 2Z 2 Z 2 U(1) Z2 T Z Z 2 Z 2 2Z 2 Z 2 Z Z 2 Z 2Z 2 2Z 2 2Z 2 3Z 2 Z 2

70 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac

71 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: In 0 + 1D, WA-top 1 = k A = ka. 2π a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac

72 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: In 0 + 1D, WA-top 1 = k A = ka. 2π Tr(Uθ twist e H ) = e i k S 1 A twist = e i kθ ground state carries charge k a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac

73 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: In 0 + 1D, WA-top 1 = k A = ka. 2π Tr(Uθ twist e H ) = e i k S 1 A twist = e i kθ ground state carries charge k In 2 + 1D, WA-top 3 = k A da = kac (2π) 2 1. a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac

74 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: In 0 + 1D, WA-top 1 = k A = ka. 2π Tr(Uθ twist e H ) = e i k S 1 A twist = e i kθ ground state carries charge k In 2 + 1D, WA-top 3 = k A da = kac (2π) 2 1. Hall conductance σ xy = 2k e2 h The edge of U 1 SPT phase must a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac be gapless with left/right movers and anomalous U(1) symm.

75 U(1) SPT phases and their physical properties Topo. terms for U 1 SPT state: In 0 + 1D, WA-top 1 = k A = ka. 2π Tr(Uθ twist e H ) = e i k S 1 A twist = e i kθ ground state carries charge k In 2 + 1D, WA-top 3 = k A da = kac (2π) 2 1. Hall conductance σ xy = 2k e2 h The edge of U 1 SPT phase must a A 2π, c 1 da 2π, ac 1 a c 1 d = H d [U 1 ] WA-top d Z a Z ac be gapless with left/right movers and anomalous U(1) symm. Dimension reduction: choose space-time M 3 = S 1 M 2 and put 2πm flux through M 2. L 2+1D = kada/2π L 0+1D = k M 2 AdA/2π = 2kmA. - The 2+1D U 1 SPT state labeled by k reduces to a 0+1D U 1 SPT state labeled by km (with charge 2km in ground state). - 2πm flux in space M 2 induces km unit of charge Hall conductance σ xy = 2ke 2 /h.

76 From probe to mechanism of SPT states 2π flux inducing 2k charge probes U(1) SPT state. Attaching 2k charges to a U(1) vortex makes U(1) SPT state.

77 From probe to mechanism of SPT states 2π flux inducing 2k charge probes U(1) SPT state. Attaching 2k charges to a U(1) vortex makes U(1) SPT state. Start with 2+1D bosonic superfluid: proliferate vortices trivial Mott insulator. proliferate vortex+2k-charge U(1) SPT state labeled by k. - Why? U(1) flux is the U(1) symmetry twist. A vortex in U(1) superfluid is a 2π U(1) symmetry twist = 2π flux. U(1) twist θ flux U θ The vortex condensed state (or the vortex proliferated state) remembers the binding of vortex and 2k-charge: a 2π U(1) symmetry twist (2π flux) carries 2k charges.

78 From probe to mechanism of SPT states 2π flux inducing 2k charge probes U(1) SPT state. Attaching 2k charges to a U(1) vortex makes U(1) SPT state. Start with 2+1D bosonic superfluid: proliferate vortices trivial Mott insulator. proliferate vortex+2k-charge U(1) SPT state labeled by k. - Why? U(1) flux is the U(1) symmetry twist. A vortex in U(1) superfluid is a 2π U(1) symmetry twist = 2π flux. U(1) twist θ flux U θ The vortex condensed state (or the vortex proliferated state) remembers the binding of vortex and 2k-charge: a 2π U(1) symmetry twist (2π flux) carries 2k charges. Bulk effective theory of the U(1) SPT state labeled by k L bulk = 1 2π a 1 da π Ada 1 + k 2π Ada 2 Boson current J b = da 1 /2π; Vortex current J v = da 2 /2π.

79 Why bind even charge to vortex? Can we bind charge-1 to a vortex, to make a new U(1) SPT state beyond group cohomology, which has σ xy = e2 h?

80 Why bind even charge to vortex? Can we bind charge-1 to a vortex, to make a new U(1) SPT state beyond group cohomology, which has σ xy = e2? No! h (1) Inserting 2π flux will always create a quasiparticle. In the new U(1) SPT state, such a quasiparticle will carry a unit U(1) charge. The bound state of charge-1+2π-flux is a fermion, which implies that the new U(1) SPT state must carry a non-trivial topological order and is not a SPT state.

81 Why bind even charge to vortex? Can we bind charge-1 to a vortex, to make a new U(1) SPT state beyond group cohomology, which has σ xy = e2? No! h (1) Inserting 2π flux will always create a quasiparticle. In the new U(1) SPT state, such a quasiparticle will carry a unit U(1) charge. The bound state of charge-1+2π-flux is a fermion, which implies that the new U(1) SPT state must carry a non-trivial topological order and is not a SPT state. (2) The bound state of charge-1+vortex is a fermion. They cannot condense to make the superfluid into an insulator. But a (charge-1+vortex)-pair is a boson. Proliferate/condensing such (charge-1+vortex)-pairs can make the superfluid into an insulator with non-trivial Z 2 topological order described by Z 2 gauge theory.

82 Why bind even charge to vortex? Can we bind charge-1 to a vortex, to make a new U(1) SPT state beyond group cohomology, which has σ xy = e2? No! h (1) Inserting 2π flux will always create a quasiparticle. In the new U(1) SPT state, such a quasiparticle will carry a unit U(1) charge. The bound state of charge-1+2π-flux is a fermion, which implies that the new U(1) SPT state must carry a non-trivial topological order and is not a SPT state. (2) The bound state of charge-1+vortex is a fermion. They cannot condense to make the superfluid into an insulator. But a (charge-1+vortex)-pair is a boson. Proliferate/condensing such (charge-1+vortex)-pairs can make the superfluid into an insulator with non-trivial Z 2 topological order described by Z 2 gauge theory. Boson system with no topological order: 2k AdA 4π

83 Why bind even charge to vortex? Can we bind charge-1 to a vortex, to make a new U(1) SPT state beyond group cohomology, which has σ xy = e2? No! h (1) Inserting 2π flux will always create a quasiparticle. In the new U(1) SPT state, such a quasiparticle will carry a unit U(1) charge. The bound state of charge-1+2π-flux is a fermion, which implies that the new U(1) SPT state must carry a non-trivial topological order and is not a SPT state. (2) The bound state of charge-1+vortex is a fermion. They cannot condense to make the superfluid into an insulator. But a (charge-1+vortex)-pair is a boson. Proliferate/condensing such (charge-1+vortex)-pairs can make the superfluid into an insulator with non-trivial Z 2 topological order described by Z 2 gauge theory. Boson system with no topological order: 2k AdA 4π k Fermion system with no topo. order: AdA (spin struc.) 4π

84 Z 2 SPT phases and their physical properties Topological terms: AZ2 = 0, π; a 1 A Z 2 π ; d = H d [Z 2 ] W d A-top Z a Z a

85 Z 2 SPT phases and their physical properties Topological terms: In 0 + 1D, W 1 A-top = k A Z 2 2π = ka 1. AZ2 = 0, π; a 1 A Z 2 π ; d = H d [Z 2 ] W d A-top Z a Z a

86 Z 2 SPT phases and their physical properties Topological terms: In 0 + 1D, WA-top 1 = k A Z 2 = ka 2π 1. Tr(Uπ twist e H ) = e 2π i S 1 W A-top = e i kπ S 1 a 1 = e i kπ = ±1 ground state Z 2 -charge: k = 0, 1 AZ2 = 0, π; a 1 A Z 2 π ; d = H d [Z 2 ] W d A-top Z a Z a

87 Z 2 SPT phases and their physical properties Topological terms: In 0 + 1D, WA-top 1 = k A Z 2 = ka 2π 1. Tr(Uπ twist e H ) = e 2π i S 1 W A-top = e i kπ S 1 a 1 = e i kπ = ±1 ground state Z 2 -charge: k = 0, 1 In 2 + 1D, W 3 M 3 A-top = 1 M 3 2 a3 1. Here we do not view a 1 as 1-form AZ2 = 0, π; a 1 A Z 2 π ; d = H d [Z 2 ] W d A-top Z a Z a but as 1-cocycle a 1 H 1 (M 3, Z 2 ), and a 3 1 a 1 a 1 a 1 : M 3 a 1 a 1 a 1 = 0 or 1 e 2π i M 3 W A-top = e π i M 3 a 3 1 = ±1

88 Z 2 SPT phases and their physical properties Topological terms: In 0 + 1D, WA-top 1 = k A Z 2 = ka 2π 1. Tr(Uπ twist e H ) = e 2π i S 1 W A-top = e i kπ S 1 a 1 = e i kπ = ±1 ground state Z 2 -charge: k = 0, 1 In 2 + 1D, W 3 M 3 A-top = 1 M 3 2 a3 1. Here we do not view a 1 as 1-form AZ2 = 0, π; a 1 A Z 2 π ; d = H d [Z 2 ] W d A-top Z a Z a but as 1-cocycle a 1 H 1 (M 3, Z 2 ), and a1 3 a 1 a 1 a 1 : a M 3 1 a 1 a 1 = 0 or 1 e 2π i M 3 W A-top = e π i M 3 a1 3 = ±1 Poincaré duality: 1-cocycle a 1 2-cycle N 2 (2D submanifold) N 2 is the surface across which we do the Z 2 symmetry twist. Choose M 3 = S 1 M 2. As we go around S 1 : (a) (b) (c) a 3 M 3 1 = # of loop creation/annihilation + # of line reconnection.

89 The edge of Z 2 SPT phase must be gapless or symmetry breaking Chen-Liu-Wen 11; Levin-Gu 12 Assume the edge of a Z 2 SPT phase is gapped with no symmetry breaking. We use Z 2 twist try to create excitations (called Z 2 domain walls) at the edge. We may naively expect those Z 2 -domain walls are trivial, but they are not. They have a non-trivial fusion property: different fusion order can differ by a sign. So the Z 2 domain walls on the boundary form a non-trivial fusion category. the bulk state must carry a non-trivial topological order.

90 The boundary of the 2+1D Z 2 SPT state has a 1+1D bosonic global Z 2 anomaly [Chen-Wen 12] The 1+1D bosonic global Z 2 anomaly The edge of Z 2 SPT phase must be gapless or symmetry breaking. One realization of the edge is described by 1+1D XY model or U(1) CFT. The primary field (vertex operator) V l,m has dimensions (h R, h L ) = ( (l+2m)2, (l 2m)2 ). 8 8 The Z 2 symmetry action V l,m ( ) l+m V l,m Such a 1+1D Z 2 symmetry is anomalous: (1) The XY model has no UV completion in 1+1D such that the Z 2 symmetry is realized as an on-site symm. U = i σx i. (2) If we gauge the Z 2, the 1+1D Z 2 gauge theory has no UV completion in 1+1D as a bosonic theory. (3) The XY model has a UV completion as boundary of 2+1D lattice theory w/ the Z 2 symmetry realized as an on-site symm.

91 xxx

92 A mechanism for Z T 2 mixed SPT state The topological term for a Z2 T mixed SPT state (bosonic topological super fluid with time reversal symmetry): WA-top 4 = 1p 2 1 Start with a T-symmetry breaking state. Proliferate the symmetry breaking domain walls to restore the T-symmetry. a trivial SPT state. Bind the domain walls to (E 8 ) 3 quantum Hall state, and then proliferate the symmetry breaking domain walls to restore the T-symmetry. a non-trivial Z2 T SPT state.