1. Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of California, Santa Barbara

2. Outline: Part 1: Interacting Topological Superconductor and Possible Origin of 16n chiral fermions in Standard Model Part 2: Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk without assuming any symmetry.

3. Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Collaborators: Postdoc: Group member: Yi-Zhuang You Yoni BenTov Very helpful discussions with Joe Polchinski, Mark Srednicki, Robert Sugar, Xiao-Gang Wen, Alexei Kitaev, Tony Zee……. Wen, arXiv:1305.1045, You, BenTov, Xu, arXiv:1402.4151, Kitaev, unpublished

4. Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Motivation: Current understanding of interacting TSC: Interaction may not lead to any new topological superconductor, but it can definitely “reduce” the classification of topological superconductor, i.e. interaction can drive some noninteracting TSC trivial, in other words, interaction can gap out the boundary of some noninteracting TSC, without breaking any symmetry. 1. Finding an application for interacting topological superconductors, especially a non-industry application; 2. Many high energy physicists are studying CMT using high energy techniques, we need to return the favor.

5. Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Weyl/chiral fermions: Weyl fermions can be gapped out by pairing:

6. Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Very high energy In Standard Model (higher than EW unification energy), every generation has (effectively) 16 massless Left chiral fermions coupled with gauge field (spinor rep of SO(10) in GUT): This theory is difficult to regularize on a 3d lattice. Because on a 3d lattice, if we want to realize left fermions, we also get right fermions coupled to the same gauge theory For example: Weyl semimetal has both left, and right Weyl fermions in the 3d BZ:

7. Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Very high energy In Standard Model (higher than EW unification energy), every generation has (effectively) 16 massless Left chiral fermions coupled with gauge field (spinor rep of SO(10) in GUT): Popular alternative: Realize chiral fermions on the 3d boundary of a 4d topological insulator/superconductor 3d boundary, 16 chiral fermions Mirror sector This theory is difficult to regularize on a 3d lattice. Because on a 3d lattice, if we want to realize left fermions, we also get right fermions coupled to the same gauge theory

8. However, this approach requires a subtle adjustment of the fourth dimension. If the fourth dimension is too large, there will be gapless photons in the bulk; if the fourth dimension is too small, the mirror sector on the other boundary will interfere. Mirror sector Key question: Can we gap out the mirror sector (chiral fermions on the other boundary) without affecting the SM at all? This cannot be done in the standard way (spontaneous symmetry breaking, condense a boson that couples to the mirror fermion mass) 3d boundary, 16 chiral fermions Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model

9. A different question: Can we gap out the mirror sector with short range interaction, while Mirror sector, gapped by interaction If this is possible, then only16 left fermions survive at low energy. 3d boundary, 16 chiral fermions Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Our conclusion: this is only possible with 16 chiral fermions, i.e. classification of 4d TSC is reduced by interaction 0 +infty gaplessgapped

10. 0d boundary of 1d TSC Consider N copies of 0d Majorana fermions with time-reversal symmetry (in total 2 N/2 states): Breaks time-reversal For N = 2, the only possible Hamiltonian is But it breaks time-reversal symmetry, thus with time-reversal symmetry, H = 0, the state is 2-fold degenerate. For N = 4, the only T invariant Hamiltonian is

11. 0d boundary of 1d TSC Finally, when N = 8, doublet GS fully gapped, nondegenerate Thus, when N = 8, the Majorana fermions can be gapped out by interaction without degeneracy, and

12. 0d boundary of 1d TSC These 0d fermions are realized at the boundary of 1d TSC: γ1γ1 γ2γ2 Trivial TSC E E With N flavors, at the boundary In the bulk: This implies that, with interaction, 8 copies of such 1d TSC is trivial, i.e. interaction reduces the classification from Z to Z 8. Fidkowski, Kitaev, 2009 J1J1 J1J1 J2J2 J2J2

13. 1d boundary of 2d TSC The system has time-reversal symmetry, which forbids any quadratic mass for odd flavors, but does not forbid mass for even flavors. Define another Z2 symmetry: The T and Z2 together guarantee that the 1d boundary of arbitrary copies remain gapless, without interaction, i.e. Z classification. Short range interactions reduce the classification of this 2d TSC from Z to Z 8, namely its edge (8 copies of 1d Majorana fermions) can be gapped out by interaction, with Qi, 2012, Yao, Ryu 2012, Ryu, Zhang 2012, Gu, Levin 2013 1d boundary of 2d p±ip TSC:

14. 1d boundary of 2d TSC Short range interactions reduce the classification of this 2d TSC from Z to Z 8, namely its edge (8 copies of 1d Majorana fermions) can be gapped out by interaction, with Qi, 2012, Yao, Ryu 2012, Ryu, Zhang 2012, Gu, Levin 2013 This can be shown with accurate bosonization calculation (Fidkowski, Kitaev 2009) One can also demonstrate this result with an argument, which can be generalized to higher dimensions. Consider Hamiltonian:

15. 1d boundary of 2d TSC If ϕ orders/condenses, fermions are gapped, breaks T and Z2, but preserves T’ If ϕ disorders, all symmetries are preserved, integrating out ϕ will lead to a local four fermion interaction. The symmetries can be restored by condensing the kinks of ϕ (transverse field Ising). A fully gapped and nondegenerate symmetric 1d phase is only possible when kink is gapped and nondegenerate. ϕ condense/order ϕ disorder, kink condenses

16. 1d boundary of 2d TSC If ϕ orders/condenses, fermions are gapped, breaks T and Z2, but preserves T’ If ϕ disorders, all symmetries are preserved, integrating out ϕ will lead to a local four fermion interaction. A kink of ϕ has N flavors of 0d Majorana fermion modes, with We know that with N = 8, interaction can gap out kink with no deg, so…. ϕ condense/order ϕ disorder, kink condenses

17. 3d TSC Short range interactions reduce the classification of the 3d TSC from Z to Z 16, namely its edge (16 copies of 2d Majorana fermions) can be gapped out by interaction, with Kitaev (unpublished) Fidkowski, et.al. 2013, Wang, Senthil 2014, Metlitski, et.al. 2014, You, Xu, arXiv:1409.0168 2d boundary of 3d TSC

18. Consider an enlarged O(2) symmetry. When ϕ condenses/orders, it breaks T, breaks O(2), but keeps 2d boundary of 3d TSC Consider a modified boundary Hamiltonian (Wang, Senthil 2014): All the symmetries can be restored by condensing the vortices of the ϕ order parameter. A fully gapped, nondegenerate, symmetric state is only possible if the vortex is gapped, nondegenerate. A vortex core has one Majorana mode, and With N = 16, interaction can gap out the 2d boundary with no deg.

19. 3d boundary of 4d TSC (sketch) The 3d boundary of a 4d TSC with U(1) x T x Z2 symmetry: These symmetries guarantee that no quadratic mass terms are allowed at the 3d boundary. So without interaction the classification of this 4d TSC is Z. We want to argue that, with interaction, the classification is reduced to Z 8, namely the interaction can gap out 16 flavors of 3d left chiral fermions without generating any quadratic fermion mass.

20. 3d boundary of 4d TSC (sketch) The 3d boundary of a 4d TSC with U(1) x T x Z2 symmetry: Now consider U(1) order parameter: The U(1) symmetry can be restored by condensing the vortex loops of the order parameter. For N=1 copy, the vortex line is a gapless 1+1d Majorana fermion with T and Z2 symmetry (same as 1d boundary of 2d TSC) Then when N=8 (16 chiral fermions at the 3d boundary), interaction can gap out vortex loop.

21. 3d boundary of 4d TSC (sketch) The 3d boundary of a 4d TSC with U(1) x T x Z2 symmetry: Now consider three component order parameter: All the symmetries can be restored by condensing the hedgehog monopole of the order parameter. For N=1 copy, the monopole is a 0d Majorana fermion with T symmetry Then when N=8 (16 chiral fermions at the 3d boundary), interaction can gap out monopole.

22. 3d boundary of 4d TSC (sketch) Dual theory for hedgehog monopole: Hedgehog monopole can be viewed as a domain wall of two flavors of vortex loops. Dual theory for SF Goldstone mode: Dual theory for one flavor of vortex loop: Dual theory for two flavors of vortex loops plus monopole:

23. Question 1: what is the maximal symmetry of the interaction term? Question 2: Is this phase transition continuous? If so, what is the field theory for this phase transition? (Numerical data suggests this is indeed a continuous phase transition. To appear) 0 +infty gaplessgapped Question 3: properties of the strongly coupled “trivial” state? The fermion Green’s function has an analytic zero, G(ω) ~ ω arXiv:1403.4938 Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Further thoughts:

24. When and only when there are 16 chiral fermions, we can gap out the mirror sector by interaction with Mirror sector, gapped by interaction Then only the 16 left fermions survive at low energy. 3d boundary, 16 chiral fermions Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model Conclusion for part 1:

25. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk Introduction for part 2: Fermionic TI and TSC: systems with trivial bulk spectrum, but gapless boundary; 2d IQH and p+ip TSC: does not need any symmetry; 2d QSH: U(1) and time-reversal 3d TI: U(1) and time-reversal 3d He3B: time-reversal Bosonic analogue: 2d E8 state (Kitaev): does not need any symmetry; chiral bosons with chiral central charge c=8 at the 1d boundary Bosonic “topological insulators”, or bosonic symmetry protected topological states: Chen, Gu, Liu, Wen, 2011.

26. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk 2d E8 state (Kitaev): does not need any symmetry; chiral bosons with chiral c=8 at the 1d boundary. Effective field theory:

27. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk 2d E8 state (Kitaev): does not need any symmetry; chiral bosons with chiral c=8 at the 1d boundary. Chiral boson will lead to gravitational anomaly at the 1+1d boundary (namely general coordinate transformation is no longer a symmetry). Goal: Can we find higher dimensional analogues of this state? Key: can we find higher dimensional (boundary) bosonic theories which are gapless without assuming any symmetry? Or: can we find higher dimensional (boundary) bosonic theories with gravitational anomalies?

28. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk In (4k+2)d space-time (4k+1d space), the following “self-dual” rank-2k tensor boson field Θ has gravitational anomalies: (Alvarez- Gauze, Witten 1983) When k=0 (1+1d space-time), the self-dual condition becomes: The 4k+3d bulk field theory for this self-dual boson field is C is a (2k+1)-form antisymmetric gauge field. Recall: 2+1d CS field has 1+1d chiral boson at its boundary.

29. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk The K matrix has to satisfy the following conditions to construct the desired bosonic phase: 1, Det[K] = 1, otherwise the bulk will have topological degeneracy; 2, local excitations of this system are all bosonic; The same K for E8 state in 2d satisfies both conditions:

30. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk Knowing this boson state in 4k+2d space (labeled as B 4k+2 state), we can construct other bosonic state in other dimensions. In every 4k+3d space, there is a bosonic state with time-reversal symmetry, which can be viewed as proliferating T-breaking domain walls with B 4k+2 sandwiched in each T domain wall. Its 4k+4d bulk space-time action is: This state has Z2 classification, namely it is only a nontrivial BSPT with θ = π mod 2π (analogue of 3d TI).

31. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk Knowing this boson state in 4k+2d space (labeled as B 4k+2 state), we can construct other bosonic state in other dimensions. In every 4k+4d space, there is a bosonic state with U(1) symmetry, which can be viewed as proliferating U(1) vortex with B 4k+2 stuffed in each vortex. After “gauging” this U(1) global symmetry, its 4k+5d bulk space-time action is: …… This state has Z classification. At the 4k+4d boundary, there is a mixed U(1) and gravitational anomaly.

32. Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk Further thoughts: We used the perturbative gravitational anomalies to construct higher dimensional bosonic TI without any symmetry; What about global gravitational anomalies? In 8k and 8k+1d space-time, single Majorana fermions have global gravitational anomalies (Witten 1983), namely partition function changes sign under a “large” general coordinate transformation. Global gravitational anomaly (Z2 classified) corresponds to the Z2 classification of 1d, 8d and 9d fermionic TI without any symmetry. By contrast, perturbative gravitational anomaly (Z classified) corresponds to the Z classification at 2d, 6d, 10d… But is there a bosonic theory with global gravitational anomalies?

33. Conclusion for part 2: Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk In every 4k+2d space, there is a bosonic state with trivial bulk spectrum, but gapless boundary states and boundary gravitational anomalies, without assuming any symmetry. Descendant bosonic SPT states in other dimensions can be constructed. All these states are beyond the group cohomology classification of bosonic SPT states.