SPT order and algebraic topology. Xiao-Gang Wen, MIT/PI, IPAM, Jan. 26, 2015

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1 Xiao-Gan Wen, MIT/PI, IPAM, Jan. 26, 205

2 Two kinds of quantum theories: H-type and L-type H-type: - Hilbert space with tensor structure V = i V i - A local Hamiltonian operator actin on V Space-time path interal well defined on mappin torus Mspace d S. L-type: - Tensors for each cell in space-time lattice defined path interal Space-time path interal well defined on any space-time manifold M d space-time. Lead to different classification of topo. orders and SPT orders H-type models E 8 bosonic quantum Hall state L-type models (E 8 ) 3 bosonic quantum Hall state

3 L-type local bosonic quantum systems A L-type local quantum system in d-dimensional space-time M d is described by (use d = 3 as an exmaple): - a trianlation of M 3 3 with a branchin structure, - a set of indices {v i } on vertecies, e 3 0 e e {e ij } on edes, {φ ijk } on trianles e 0 e 02 - two real and one complex tensors T 3 = 2 {W v0, A e 0 v 0 v (SO ij ), C ± e 0 e 02 e 03 e 2 e 3 e 23 for each 0,, 3-cell (vertex, ede, tetrahedron). v 0 v v 2 v 3 ;φ 23 φ 03 φ 023 φ 02 (SO ij )} v v Unitary condition W v0 > 0, A e 0 v 0 v > 0, e C 0 e 02 e 03 e 2 e 3 e 23 e + v 0 v v 2 v 3 ;φ 23 φ 03 φ 023 φ 02 = (C 0 e 02 e 03 e 2 e 3 e 23 [Kon-Wen 4] v 0 v 3 v v 3 e 2 v v 0 v v 2 v 3 ;φ 23 φ 03 φ 023 φ 02 ), v

4 Partition function and Correlation function Partition function: Z = vertex v 0, ;e 0, ;φ 02, W v0 ede = T 3 A e 0 v 0 v tetra C s023 e 0 e 02 e 03 e 2 e 3 e 23 v 0 v v 2 v 3 ;φ 23 φ 03 φ 023 φ 02 + k j i j k i space time where s 023 = ± depends on the orientation of the tetrahedron. Correlation function on closed space-time manifold physically measurible quantities: Modify tensor on a few simplices ives us a new partition function T 3 T 3 : Z(M d ) Z[ T 3 (x), T 3 (y), ; M d ]: T 3 (x) T 3 (y) = Z[ T 3 (x), T 3 (y); M d ] Z(M d )

5 Short-rane correlated (SRC) system and liquid A infinite-system is not a sinle system but a sequence of systems - with size of space-time M 3 and size of simplices. - each vertex is shared by at most a finite number of simplices. A short-rane correlated (SRC) infinite-system satisfies () T 3 (x) T 3 (y) T 3 (x) T 3 (y) e x y ξ for a fixed ξ in the infinite system-size limit; (2) systems of different sizes in the sequence can deform into each other and keep the SRC property durin the deformation. [Zen-Wen 4] SRC liquid Nk Nk Nk+ LA LU

6 Short-rane correlated (SRC) liquid phases An equivalence relation: Two SRC infinite-systems are equivalent if the two sequences can deform into each other while keepin the SRC property durin the deformation. The resultin equivalent classes are SRC liquid phases or L-type topoloical orders Nk Nk Nk+ [Wen 89] [Chen-Gu-Wen 0] Nk LA Nk LU Nk+ LA LU How to classify SRC liquid phases (L-type topoloical orders) in each dimension?

7 L-type local bosonic systems with symmetry () The indices admit a symmetry action v i v i, e ij e ij, φ ijk φ ijk, where G. (2) T G,3 = {W v0, A e 0 e v 0 v (SO ij ), C 0 e 02 e 03 e 2 e 3 e 23 ± v 0 v v 2 v 3 ;φ 23 φ 03 φ 023 φ 02 (SO ij )} are invariant under the above symmetry action. SRC liquid phases with symmetry: Equivalence relation: Two symmetric SRC infinite-systems can deform into each other while keepin the symmetry property and the SRC property durin the deformation. How to classifies SRC liquid phases with symmetry L-type symmetry enriched topoloical (SET) order

8 Examples of SRC liquids Example : Tensors: non-zero only when all the indices are W = w 0, A = w, C({v i =, e ij =, φ ijk = }) = w 3, All derees of freedom are frozen to. Z(M 3 ) = n=0 (w n) Nn - The above systems with different w n s all belon to the same SRC phase [have a trivial topoloical order (trito)]. Example 2: Tensors: non-zero only when all the ede-indices, face indices, etc are ; the vertex-indices v i G W vi = w 0, A v i v j = w C({v i, e ij =, φ ijk = }) = w 3, Z(M 3 ) = G N 0 n=0 (w n) Nn - The above systems have a symmetry G A conjecture: A system with Z(M d ) = for all closed orientable space-time M d has a trivial topoloical order. Both of the above examples have a trivial topoloical order.

9 A eneral picture for SRC phases. Non trivial TO w/o symm. many phases [Wen 89] Trivial TO w/o symm. one phase (no symm. breakin) Non trivial TO with symm. many phases [Wen 02] Trivial TO with symm. many different phases [Gu-Wen 09] may be called symmetry protected trivial (SPT) phase or symmetry protected topoloical (SPT) phase 2 topoloical order TO TO 2 trito 2 SY TO SY TO 2 topoloical order SY TO 3 SY TO 4 SET phases (PSG,??? ) SY trito SY trito 2 SPT phases SY trito 3 SY trito 4 (???) phase transition preserve symmetry SPT SPT 2 no symmetry SPT phases = equivalent class of symm. smooth deformation Examples: D Haldane phase[haldane 83] 2D/3D TI[Kane-Mele 05; Bernevi-Zhan 06] [Moore-Balents 07; Fu-Kane-Mele 07] D 2D 3D

10 SPT phases the trivial TO with symmetry SPT phases are SRC (apped) quantum phases with a certain symmetry, which can be smoothly connected to the same trivial phase if we remove the symmetry trivial topo order. SPT SPT2 SPT3 with a symmetry G Product state break the symmetry A roup cohomoloy thepry of SPT phases: Usin each element in H d [G, U()] (the d roup cohomoloy class of the roup G with U() as coefficient), we can construct a exactly soluble path interal in d-dimensional space-time, which realize a SPT state with a symmetry G. [Chen-Gu-Liu-Wen ]

11 SPT phases the trivial TO with symmetry SPT phases are SRC (apped) quantum phases with a certain symmetry, which can be smoothly connected to the same trivial phase if we remove the symmetry trivial topo order. SPT SPT2 SPT3 with a symmetry G Product state break the symmetry A roup cohomoloy thepry of SPT phases: Usin each element in H d [G, U()] (the d roup cohomoloy class of the roup G with U() as coefficient), we can construct a exactly soluble path interal in d-dimensional space-time, which realize a SPT state with a symmetry G. [Chen-Gu-Liu-Wen ] How to et the above result? Construct a path interal with symmetry G and Z top (M d ) = on any closed orientable M d.

12 L-type SPT state on +D space-time lattice Path interal on +D space-time lattice described by tensors e T 2 = {W v0, B 0 e 2 e 02 ± v 0 v v 2 (SO ij )} eij = = {W v0 = G, [ν 2 (v 0, v, v 2 )], } e S = ν s ijk 2 ( i, j, k ), Z = G N 0 e S, v i i, i G where ν s ijk (i, j, k ) = e L and s ijk =, The above defines a j k i j G LNσM with taret k i+ j space G on +D ν(,, ) k i j k space-time lattice. i space time The NLσM will have a symmetry G if i G and ν 2 ( i, j, k ) = ν 2 (h i, h j, h k ), h G We will et a SPT state if we choose ν 2 ( i, j, h ) = Z top (M 2 ) = (which is a trivial SPT state).

13 Topoloical path ineral (NLσM) and SPT state ν( i, j, k ) ive rise to a topoloical NLσM if e S fixed = ν s ijk(i, j, k ) = on any sphere, includin a tetrahedron (simplest sphere). ν( i, j, k ) U() On a tetrahedron 2-cocycle condition ν 2 (, 2, 3 )ν 2 ( 0,, 3 )ν 2 ( 0, 2, 3 )ν 2 ( 0,, 2 ) = The solutions of the above equation are called roup cocycle. The 2-cocycle condition has many solutions: ν 2 ( 0,, 2 ) and ν 2 ( 0,, 2 ) = ν 2 ( 0,, 2 ) β (, 2 )β ( 0, ) β ( 0, 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()] = π 0 (space of the solutions). H 2 [G, U()] describes +D SPT phases protected by G

14 Topoloical invariance in topoloical NLσMs As we chane the space-time lattice, the action amplitude e S does not chane: ν 2 ( 0,, 2 )ν 2 (, 2, 3 ) = ν 2 ( 0,, 3 )ν 2 ( 0, 2, 3 ) ν 2 ( 0,, 2 )ν 2 (, 2, 3 )ν 2 ( 0, 2, 3 ) = ν 2 ( 0,, 3 ) as implied by the cocycle condition: ν 2 (, 2, 3 )ν 2 ( 0,, 3 )ν 2 ( 0, 2, 3 )ν 2 ( 0,, 2 ) = The topoloical NLσM is a RG fixed-point. 2

15 Topoloical invariance in topoloical NLσMs As we chane the space-time lattice, the action amplitude e S does not chane: ν 2 ( 0,, 2 )ν 2 (, 2, 3 ) = ν 2 ( 0,, 3 )ν 2 ( 0, 2, 3 ) ν 2 ( 0,, 2 )ν 2 (, 2, 3 )ν 2 ( 0, 2, 3 ) = ν 2 ( 0,, 3 ) as implied by the cocycle condition: ν 2 (, 2, 3 )ν 2 ( 0,, 3 )ν 2 ( 0, 2, 3 )ν 2 ( 0,, 2 ) = The topoloical NLσM is a RG fixed-point. The Z(M d ) = if M d can be obtained by luin spheres. The NLσM describes a SPT state with trivial topo. order.

16 Bosonic SPT phases from H d [G, U()] Symmetry G/ d = U() Z2 T (top. ins.) Z Z 2 (0) Z 2 (Z 2 ) Z 2 2 (Z 2) U() Z2 T trans Z Z Z 2 Z Z 3 2 Z Z 8 2 U() Z2 T (spin sys.) 0 Z2 2 0 Z 3 2 U() 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() Z 0 Z 0 U() 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 topoloical order 2 SY LRE SY LRE 2 SET orders (tensor cateory) (tensor cateory intrinsic topo. order w/ symmetry) LRE LRE 2 SB LRE SB LRE 2 symmetry breakin SRE SB SRE SY SRE SB SRE 2 SY SRE 2 (roup theory) SPT orderes (roup cohomoloy theory) Z2 T : time reversal, trans : translation, 0 only trivial phase. (Z 2 ) free fermion result

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

18 Universal probe for SPT orders How do you know the constructed NLσM round 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:

19 Universal probe for SPT orders How do you know the constructed NLσM round 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: Partitional function as an universal probe, but Ztop SPT (M d ) = does not work. Twist the symmetry by auin the symmetry on M d A G aue field. Ztop SPT (A, M d ). [Levin-Gu 2; Hun-Wen 3]

20 Universal topo. inv.: aued partition function Z(A, M d ) Z(0, M d ) = D e L( ( d i A)) D e L( d) W topinv (A) and W topinv (A) are equivalent if = e i 2π W topinv (A) W topinv(a) W topinv (A) = λ Tr(F 2 ) + The equivalent class of the aue-topoloical term W topinv (A) is the topoloical invariant that probe different SPT state. The topoloical invariant W topinv (A) are Chern-Simons terms or Chern-Simons-like terms. Such Chern-Simons-like terms are classified by H d+ (BG, Z) = H d [G, U()] [Dijkraaf-Witten 92] The topoloical invariant W topinv (A) can probe all the SPT states (constructed so far)

21 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c d = H d [U()] Wtopinv d 0 + Z a Z ac 3 + 0

22 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c In 0 + D, W topinv = k A 2π = ka. d = H d [U()] Wtopinv d 0 + Z a Z ac 3 + 0

23 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c In 0 + D, Wtopinv = k A = ka. 2π Tr(Uθ twist e H ) = e i k S A twist = e i kθ round state carries chare k d = H d [U()] Wtopinv d 0 + Z a Z ac 3 + 0

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

25 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c In 0 + D, Wtopinv = k A = ka. 2π Tr(Uθ twist e H ) = e i k S A twist = e i kθ round state carries chare k In 2 + D, Wtopinv 3 = k A da = kac (2π) 2. d = H d [U()] Wtopinv d 0 + Z a Z ac Hall conductance σ xy = 2k e2 h The ede of U() SPT phase must be apless: left/riht movers + anomalous U() symm.

26 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c In 0 + D, Wtopinv = k A = ka. 2π Tr(Uθ twist e H ) = e i k S A twist = e i kθ round state carries chare k In 2 + D, Wtopinv 3 = k A da = kac (2π) 2. Hall conductance σ xy = 2k e2 h The ede of U() SPT phase d = H d [U()] Wtopinv d 0 + Z a Z ac must be apless: left/riht movers + anomalous U() symm. Probe: 2πm flux in space M 2 induces km unit of chare Hall conductance σ xy = 2ke 2 /h.

27 U() SPT phases and their physical properties Topo. terms for U() SPT state: a A 2π, c da 2π, ac a c In 0 + D, Wtopinv = k A = ka. 2π Tr(Uθ twist e H ) = e i k S A twist = e i kθ round state carries chare k In 2 + D, Wtopinv 3 = k A da = kac (2π) 2. Hall conductance σ xy = 2k e2 h The ede of U() SPT phase d = H d [U()] Wtopinv d 0 + Z a Z ac must be apless: left/riht movers + anomalous U() symm. Probe: 2πm flux in space M 2 induces km unit of chare Hall conductance σ xy = 2ke 2 /h. Mechanism: Start with 2+D bosonic superfluid: proliferate vortices trivial Mott insulator. proliferate vortex+2k-chare U() SPT state labeled by k.

28 A mechanism for 2+D U() Z T 2 SPT state 2+D boson superfluid + as of vortex boson Mott insulator. 2+D boson superfluid + as of S z -vortex boson topoloical insulator (U() Z2 T - The boson superfluid + spin- system S z -vortex = vortex + (S z = +)-spin anti S z -vortex = anti-vortex + (S z = )-spin SPT state) Probin 2+D U() Z T 2 SPT state [Liu-Gu-Wen 4] Let Φ vortex be the creation operator of the vortex. Then T Φ vortex T = Φ vortex, or T Φ Sz -vortext = Φ S z -vortex. U() π-flux in the non-trivial SPT phase is a Kramer doublet

29 H d (G, R/Z) does not produce all the SPT phases with symm. G: Topoloical states and anomalies SPT order from H d (G, R/Z) SPT state theory with with on site aue symmetry (symm.) anomaly 2 topoloical order TO TO 2 2 SY TO SY TO 2 topoloical order SY TO 3 SY TO 4 SET phases (PSG,??? ) SY trito SY trito 2 SPT phases trito SY trito 3 SY trito 4 (???)

30 H d (G, R/Z) does not produce all the SPT phases with symm. G: Topoloical states and anomalies SPT order from H d (G, R/Z) SPT state theory with with on site aue symmetry (symm.) anomaly Topoloical order effective Topoloically theory ordered with ravitational state anomaly 2 topoloical order TO TO 2 2 SY TO SY TO 2 topoloical order SY TO 3 SY TO 4 SET phases (PSG,??? ) SY trito SY trito 2 SPT phases trito SY trito 3 SY trito 4 (???)

31 H d (G, R/Z) does not produce all the SPT phases with symm. G: Topoloical states and anomalies SPT order from H d (G, R/Z) SPT state theory with with on site aue symmetry (symm.) anomaly SPT state theory with with on site mixed symmetry aue rav. anomaly SPT order beyond H d (G, R/Z) Topoloical order effective Topoloically theory ordered with ravitational state anomaly 2 topoloical order TO TO 2 2 SY TO SY TO 2 topoloical order SY TO 3 SY TO 4 SET phases (PSG,??? ) SY trito SY trito 2 SPT phases trito SY trito 3 SY trito 4 (???)

32 CS-like topoloical terms SPT and topo. orders Pure SPT order within H d (G, R/Z): Wtopinv d = ac only depend on A the aue G-connection W d topinv

33 CS-like topoloical terms SPT and topo. orders Pure SPT order within H d (G, R/Z): Wtopinv d = ac only depend on A the aue G-connection W d topinv topoloical order : W d topinv = ω 3, 2 w 2w 3 Wtopinv d only depend on Γ the ravitational SO-connection p is the first Pontryain class, dω 3 = p, and w i is the Stiefel-Whitney classes. [Kapustin 4]

34 CS-like topoloical terms SPT and topo. orders Pure SPT order within H d (G, R/Z): Wtopinv d = ac only depend on A the aue G-connection W d topinv Invertible topoloical order (ito): Wtopinv d = ω 3, w 2 2w 3 Topo. order w/ no topo. excitations Wtopinv d only depend on Γ the ravitational SO-connection p is the first Pontryain class, dω 3 = p, and w i is the Stiefel-Whitney classes. - The Z-class of 2+D ito s are enerated by ω 3, a (E 8 ) 3 state. E 8 state is anomalous as a L-type theory, but not as a H-type - Z 2 -class of 4+D ito s are enerated by w 2 2w 3. [Kapustin 4]

35 CS-like topoloical terms SPT and topo. orders Pure SPT order within H d (G, R/Z): Wtopinv d = ac only depend on A the aue G-connection W d topinv Invertible topoloical order (ito): Wtopinv d = ω 3, w 2 2w 3 Topo. order w/ no topo. excitations Wtopinv d only depend on Γ the ravitational SO-connection p is the first Pontryain class, dω 3 = p, and w i is the Stiefel-Whitney classes. - The Z-class of 2+D ito s are enerated by ω 3, a (E 8 ) 3 state. E 8 state is anomalous as a L-type theory, but not as a H-type - Z 2 -class of 4+D ito s are enerated by w 2 2w 3. [Kapustin 4] Mixed SPT order beyond H d (G, R/Z): depend on both A and Γ W d topinv W d topinv = c ω 3

36 A model to realize all (?) bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders NLσM (roup cohomoloy) approach to pure SPT phases: () NLσM+topo. term: 2λ 2 + 2πiW ( ), G (2) Add symm. twist: ( 2λ ia) 2 + 2πiW [( ia)] (3) Interate out matter field: Z fixed = e 2π i W topinv (A)

37 A model to realize all (?) bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders NLσM (roup cohomoloy) approach to pure SPT phases: () NLσM+topo. term: 2λ 2 + 2πiW ( ), G (2) Add symm. twist: ( 2λ ia) 2 + 2πiW [( ia)] (3) Interate out matter field: Z fixed = e 2π i W topinv (A) G SO NLσM (roup cohomoloy) approach: () NLσM: 2λ 2 + 2πiW ( ), G SO (2) Add twist: ( ia 2λ iγ) 2 + 2πiW [( ia iγ)] (3) Interate out matter field: Z fixed = e 2π i W topinv (A,Γ)

38 A model to realize all (?) bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders NLσM (roup cohomoloy) approach to pure SPT phases: () NLσM+topo. term: 2λ 2 + 2πiW ( ), G (2) Add symm. twist: ( 2λ ia) 2 + 2πiW [( ia)] (3) Interate out matter field: Z fixed = e 2π i W topinv (A) G SO NLσM (roup cohomoloy) approach: () NLσM: 2λ 2 + 2πiW ( ), G SO (2) Add twist: ( ia 2λ iγ) 2 + 2πiW [( ia iγ)] (3) Interate out matter field: Z fixed = e 2π i W topinv (A,Γ) All possible topo. terms are classified by H d (G SO, R/Z) Pure/mixed SPT orders, and itos are classified by H d (G SO, R/Z) = H d (G, R/Z) d k= Hk [G, H d k (SO, R/Z)] H d (SO, R/Z) But not one-to-one. Need to quotient out somethin Γ d (G).

39 Tryin to classify bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders Pure STP: H d (G, R/Z) one-to-one Mixed SPT: d k= Hk [G, H d k (SO, R/Z)] many-to-one ito s: H d (SO, R/Z) many-to-one - Pure STP orders are classified by H d (G, R/Z) CS-like topo. inv. Wtopinv d (A) are classified by H d+ (BG, Z) Pure STP orders are also classified by W d topinv (A) - But itos are not one-to-one classified by Wtopinv d (Γ SO) in H d+ (BSO, Z), because different Wtopinv d (Γ SO) and W topinv d (Γ SO) may satisfy Wtopinv d (Γ SO) = W topinv d (Γ SO) when Γ SO is the SO-connection of the tanent bundle of M d. - Wtopinv d, d W topinv the same ito ito d = H d (SO, R/Z)/Γ d

40 Tryin to classify bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders Pure STP orders: H d (G, R/Z) (the black entries below) ito s: ito d = H d (SO, R/Z)/Γ d (usin Wu class and Sq n ) Mixed SPT order d k= Hk (G, ito d k ) Hk [G,H d k (SO,R/Z)] Γ d (G) 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() Z 0 Z 0 Z Z 0 Z Z Z 2 U() 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() Z2 T 0 2Z 2 0 3Z 2 Z 2 0 4Z 2 3Z 2 2Z 2 Z 2 U() 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

41 Tryin to classify bosonic pure STP orders, mixed SPT orders, and invertible topoloical orders Pure STP orders: H d (G, R/Z) (the black entries below) ito s: ito d = H d (SO, R/Z)/Γ d (usin Wu class and Sq n ) Mixed SPT order d k= Hk (G, ito d k ) Hk [G,H d k (SO,R/Z)] Γ d (G) 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() Z 0 Z 0 Z Z 0 Z Z Z 2 U() 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() Z2 T 0 2Z 2 0 3Z 2 Z 2 0 4Z 2 3Z 2 2Z 2 Z 2 U() 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 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

42 A mechanism for Z T 2 mixed SPT state in 3+D [Vishwanash-Senthil 2, Kapustin 4, Wen 4] The Z2 T mixed SPT states are classified by H (Z2 T, ito 3 ) = Z 2 The topoloical invariant for a Z2 T mixed SPT state (bosonic topoloical super fluid with time reversal symmetry) is Wtopinv 4 = p 2 [W 4] (Wtopinv 4 = p 6 [VS 2, K 4]) Start with a T-symmetry breakin state. Proliferate the symmetry breakin domain walls to restore the T-symmetry. a trivial SPT state. Bind the domain walls to (E 8 ) 3 [W 4] (E 8 [VS 2, K 4]) quantum Hall state, and then proliferate the symmetry breakin domain walls to restore the T-symmetry. a non-trivial Z2 T SPT state.

43 Z 2 SPT phases and their physical properties Topoloical terms: AZ2 = 0, π; a A Z 2 π ; d = H d [Z 2 ] W d topinv 0 + Z 2 2 a Z 2 2 a

44 Z 2 SPT phases and their physical properties Topoloical terms: In 0 + D, W topinv = k A Z 2 2π = ka. AZ2 = 0, π; a A Z 2 π ; d = H d [Z 2 ] W d topinv 0 + Z 2 2 a Z 2 2 a

45 Z 2 SPT phases and their physical properties Topoloical terms: In 0 + D, Wtopinv = k A Z 2 = ka 2π. Tr(Uπ twist e H ) = e 2π i S W topinv = e i kπ S a = e i kπ = ± round state Z 2 -chare: k = 0, AZ2 = 0, π; a A Z 2 π ; d = H d [Z 2 ] W d topinv 0 + Z 2 2 a Z 2 2 a

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

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

48 The ede of Z 2 SPT phase must be apless or symmetry breakin Chen-Liu-Wen ; Levin-Gu 2 Assume the ede of a Z 2 SPT phase is apped with no symmetry breakin. We use Z 2 twist try to create excitations (called Z 2 domain walls) at the ede. 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 sin. So the Z 2 domain walls on the boundary form a non-trivial fusion cateory. the bulk state must carry a non-trivial topoloical order.

49 The boundary of the 2+D Z 2 SPT state has a +D bosonic lobal Z 2 anomaly [Chen-Wen 2] The +D bosonic lobal Z 2 anomaly The ede of Z 2 SPT phase must be apless or symmetry breakin. One realization of the ede is described by +D XY model or U() 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 +D Z 2 symmetry is anomalous: () The XY model has no UV completion in +D such that the Z 2 symmetry is realized as an on-site symm. U = i σx i. (2) If we aue the Z 2, the +D Z 2 aue theory has no UV completion in +D as a bosonic theory. (3) The XY model has a UV completion as boundary of 2+D lattice theory w/ the Z 2 symmetry realized as an on-site symm.