Real space RG and the emergence of topological order

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

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

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: Symmetry breaking Topological Sz  a

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

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 Tijk, where i,j,k=1,…,D.

12. Definition • Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr …

13. Definition • Define: e-S(i,j,k,…) = Tijk Tilm Tjnp Tkqr … • Partition function: Z = ijk e-S(i,j,k,…) = ijk Tijk Tilm Tjnp …

14. One dimensional case T T T T T T T T T T i j k Z = ijk Tij Tjk …= Tr(TN)

15. One dimensional case T T T T T T T T T T

16. One dimensional case T T T T T T T T T T

17. One dimensional case T T T T T T T T T T T’ T’ T’ T’ T’ T’ik = Tij Tjk

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

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

20. Tensor renormalization group

21. Tensor renormalization group • First step: find a tensor S such that n SlinSjknm Tijm Tklm i l i S l T T  S j j k k

22. Tensor renormalization group

23. Tensor renormalization group • Second step: T’ijk = pqr SkpqSjqr Sirp

24. Tensor renormalization group

25. Tensor renormalization group • 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 ZR

27. Physical picture • Partition function for triangle:

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

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

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

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

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

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

34. Physical picture • First step of TRG: find S such that i S l i l T T  S j j k k = ! 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: 2L 23L … How can truncation to tensor Tijk 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 1D Tijkijk for appropriate basis states {i}. • TRG is iterative construction of Tijk for larger and larger triangles • T* = limL  Tijk i k j

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: T111 = 1, T122 = T212 = T221 = , T112 = T121 = T211 = T222 = 0 where  = e-2K.

44. Example: Triangular lattice Ising model

45. Finding the fixed points • Fixed point tensors S*,T* satisfy: i S* l i l T* T* = S* j j k k i i S* = T* S* S* j k j k

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

47. Physical derivation • Assume no long range order • Recall physical interpretation of T*: i1 i1 i2 k i2 j  T*ijk i j k

48. Physical derivation • Assume no long range order • Recall physical interpretation of T*: i1 k2 k1 i2 j2 j1  T*ijk i j k

49. Physical derivation • Assume no long range order • Recall physical interpretation of T*: i1 k2 k1 i2 j2 j1 T*ijk = i2j1 j2k1 k2i1

50. Physical derivation • Assume no long range order • Recall physical interpretation of T*:  T* =   T*ijk = i2j1 j2k1 k2i1