1. Real space RG and the emergence of topological order Michael Levin Harvard University Cody Nave MIT

2. Basic issue Fractional statistics Ground state deg. Topological order Lattice scaleLong distances Consider quantum spin system in topological phase:

3. Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order:

4. Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order: SzSz  a Symmetry breakingTopological

5. Topological order is an emergent phenomena No signature at lattice scale Contrast with symmetry breaking order: SzSz  a Symmetry breakingTopological

6. Problem Hard to probe topological order - e.g. numerical simulations Even harder to predict topological order - Very limited analytic methods - Only understand exactly soluble string-net (e.g. Turaev-Viro) models where  = a

7. One approach: Real space renormalization group Generic models flow to special fixed points: Expect fixed points are string-net (e.g. Turaev-Viro) models

8. Outline I. RG method for (1+1)D models A. Describe basic method B. Explain physical picture (and relation to DMRG) C. Classify fixed points II. Suggest a generalization to (2+1)D A. Fixed points  exactly soluble string-net models (e.g. Turaev-Viro)

9. Hamiltonian vs. path integral approach Want to do RG on (1+1)D quantum lattice models Could do RG on (H,  ) (DMRG) Instead, RG on 2D “classical” lattice models (e.g. Ising model) with potentially complex weights

10. Tensor network models Very general class of lattice models Examples: - Ising model - Potts model - Six vertex model

11. Definition Need: Tensor T ijk, where i,j,k=1,…,D.

12. Definition Define: e -S(i,j,k,…) = T ijk T ilm T jnp T kqr …

13. Definition Define: e -S(i,j,k,…) = T ijk T ilm T jnp T kqr … Partition function: Z =  ijk e -S(i,j,k,…) =  ijk T ijk T ilm T jnp …

14. One dimensional case TTTTTTTTTT ij Z =  ijk T ij T jk …= Tr(T N ) k

15. One dimensional case TTTTTTTTTT

16. TTTTTTTTTT

17. TTTTTTTTTT T’ T’ ik = T ij T jk

18. Higher dimensions TT T TT T T’ Naively:

19. Higher dimensions TT T TT T T’ Naively: But tensors grow with each step

20. Tensor renormalization group

21. il j k i j k l TT S S  First step: find a tensor S such that  n S lin S jkn   m T ijm T klm

22. Tensor renormalization group

23. Second step : T’ ijk =  pqr S kpq S jqr S irp

24. Tensor renormalization group

25. Iterate: T  T’  T’’  … Efficiently compute partition function Z Fixed point T * captures universal physics

26. Physical picture Consider generic lattice model: Want: partition function Z R

27. Physical picture Partition function for triangle:

28. Physical picture Think of  (  a,  b,  c ) as a tensor   Then: Z R =      …

29. Physical picture Think of  (  a,  b,  c ) as a tensor   Then: Z R =      … Tensor network model!

30. Physical picture First step of TRG: find S such that j k i j k l TT S S  il

31. Physical picture First step of TRG: find S such that j k i j k l TT S S  il

32. Physical picture First step of TRG: find S such that j k i j k l TT S S  il ?? 

33. Physical picture First step of TRG: find S such that j k i j k l TT S S  il =

34. Physical picture First step of TRG: find S such that j k i j k l TT S S  il = S is partition function for !

35. Physical picture Second step:

36. Physical picture Second step:

37. Physical picture TRG combines small triangles into larger triangles

38. Physical picture But the indices of tensor   have larger and larger ranges: 2 L  2 3L  … How can truncation to tensor T ijk possibly be accurate?

39. Physical interpretation of   is a quantum wave function

40. Non-critical case System non-critical   is a ground state of gapped Hamiltonian   is weakly entangled: as L  , entanglement entropy S  const.

41. Non-critical case (continued)  Can factor  accurately as    1 D T ijk  i  j  k for appropriate basis states {  i }. TRG is iterative construction of T ijk for larger and larger triangles T * = lim L   T ijk ii jj kk

42. Critical case  is a gapless ground state  as L  , S ~ log L Method breaks down at criticality Analogous to breakdown of DMRG

43. Example: Triangular lattice Ising model Z =   exp(K   i  j ) Realized by a tensor network with D=2: T 111 = 1, T 122 = T 212 = T 221 = , T 112 = T 121 = T 211 = T 222 = 0 where  = e -2K.

44. Example: Triangular lattice Ising model

45. Finding the fixed points Fixed point tensors S *,T * satisfy: j k i j k l T*T* T*T* S*S* S*S*  il  S*S* S*S* S*S* T*T* i jk k j i

46. Physical derivation Assume no long range order Recall physical interpretation of T * : ii jj kk    T * ijk  i  j  k

47. Physical derivation Assume no long range order Recall physical interpretation of T * : jj kk    T * ijk  i  j  k i1i1 i2i2  i 1 i 2

48. Physical derivation Assume no long range order Recall physical interpretation of T * :    T * ijk  i  j  k i1i1 i2i2 k1k1 k2k2 j2j2 j1j1

49. Physical derivation Assume no long range order Recall physical interpretation of T * : i1i1 i2i2 k1k1 k2k2 j2j2 j1j1 T * ijk = i 2 j 1 j 2 k 1 k 2 i 1

50. Physical derivation Assume no long range order Recall physical interpretation of T * : T * ijk = i 2 j 1 j 2 k 1 k 2 i 1 T*T* =

51. Fixed point solutions Are these actually solutions? Yes.

52. Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on?

53. Fixed point solutions Are these actually solutions? Yes. But we have too many solutions! What’s going on? Coarse graining is incomplete! Fixed point still contains some lattice scale physics

54. Fixed points

55. Fixed surfaces

56. The points on each surface differ in short distance physics

57. Classification of fixed surfaces Two cases: 1. No symmetry: - Can continuously change any T * ijk = i 2 j 1 j 2 k 1 k 2 i 1  T * ijk = 1 Only one (trivial) universality class

58. Classification of fixed surfaces 2. Impose some symmetry (invariance under |  i >  O i j |  j >): - Can classify possibilities for each group G - Fixed surfaces  {Proj. rep.  of G such that    is a rep. of G} - e.g., G = SO(3),  = spin-1/2: Haldane spin-1 chain! Only nontrivial possibilities are generalizations of spin-1 chain

59. Generalization to (2+1)D? (1+1)D(2+1)D

60. Generalization to (2+1)D? T ijk Regular triangular lattice (1+1)D(2+1)D i j k

61. Generalization to (2+1)D? T ijk T ijkl Regular triangular lattice Regular triangulation of R 3 (1+1)D(2+1)D i j k

62. Generalization to (2+1)D? (1+1)D(2+1)D

63. Generalization to (2+1)D? (1+1)D(2+1)D

64. Fixed point ansatz in (2+1)D? Expect that faces can be labeled by indices corresponding to boundaries: i

65. Fixed point ansatz in (2+1)D? Expect that faces can be labeled by indices corresponding to boundaries: i1i1 i2i2 i3i3 b c a

66. Fixed point ansatz in (2+1)D? Expect that faces can be labeled by indices corresponding to boundaries: i1i1 i2i2 i3i3 b c a d e f

67. Fixed point ansatz in (2+1)D? Expect that faces can be labeled by indices corresponding to boundaries: i1i1 i2i2 i3i3 b c a T * ijkl = F abc def  i 1 j 1 k 1 i 2 j 2 l 2 … d e f

68. Fixed point solutions in (2+1)D? Substituting into RG transformation gives fixed point constraints of form  n F mlq kpn F jip mns F jsn lkr = F jip qkr F riq mls etc. (but no constraint on )

69. Fixed point solutions in (2+1)D? Substituting into RG transformation gives fixed point constraints of form  n F mlq kpn F jip mns F jsn lkr = F jip qkr F riq mls etc. (but no constraint on ) Exactly constraints for Turaev-Viro (or string-net) models!

70. Conclusion TRG approach gives: 1. Understanding of emergence of topological order. 2. Classification of fixed points 3. Powerful numerical method in (1+1)D Does it work in (2+1)D?