interacting fermionic and bosonic topological insulators, possible connection to standard model and...
TRANSCRIPT
![Page 1: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/1.jpg)
Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies
Cenke Xu
许岑珂
University of California, Santa Barbara
![Page 2: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/2.jpg)
Outline:
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.
![Page 3: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/3.jpg)
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
Collaborators:
Postdoc: Group member:Yi-Zhuang You Yoni BenTov
Very helpful discussions withJoe Polchinski, Mark Srednicki, Robert Sugar, Xiao-Gang Wen, Alexei Kitaev, Tony Zee…….
Wen, arXiv:1305.1045, You, BenTov, Xu, arXiv:1402.4151, Kitaev, unpublished
![Page 4: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/4.jpg)
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.
![Page 5: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/5.jpg)
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
Weyl/chiral fermions:
Weyl fermions can be gapped out by pairing:
![Page 6: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/6.jpg)
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:
![Page 7: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/7.jpg)
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
![Page 8: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/8.jpg)
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
![Page 9: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/9.jpg)
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
gapless gapped
![Page 10: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/10.jpg)
0d boundary of 1d TSC
Consider N copies of 0d Majorana fermions with time-reversal symmetry (in total 2N/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
![Page 11: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/11.jpg)
0d boundary of 1d TSC
Finally, when N = 8,
doublet doublet
GS fully gapped, nondegenerate
Thus, when N = 8, the Majorana fermions can be gapped out by interaction without degeneracy, and
![Page 12: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/12.jpg)
0d boundary of 1d TSC
These 0d fermions are realized at the boundary of 1d TSC:
γ1 γ2
Trivial
TSCE
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 Z8. Fidkowski, Kitaev, 2009
J1 J1
J2 J2
![Page 13: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/13.jpg)
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 Z8, 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:
![Page 14: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/14.jpg)
1d boundary of 2d TSC
Short range interactions reduce the classification of this 2d TSC from Z to Z8, 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:
![Page 15: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/15.jpg)
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/orderϕ disorder, kink condenses
![Page 16: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/16.jpg)
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/orderϕ disorder, kink condenses
![Page 17: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/17.jpg)
3d TSC
Short range interactions reduce the classification of the 3d TSC from Z to Z16, 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
![Page 18: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/18.jpg)
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.
![Page 19: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/19.jpg)
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 Z8, namely the interaction can gap out 16 flavors of 3d left chiral fermions without generating any quadratic fermion mass.
![Page 20: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/20.jpg)
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.
![Page 21: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/21.jpg)
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.
![Page 22: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/22.jpg)
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:
![Page 23: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/23.jpg)
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
gapless gapped
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:
![Page 24: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/24.jpg)
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:
![Page 25: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/25.jpg)
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.
![Page 26: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/26.jpg)
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:
![Page 27: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/27.jpg)
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?
![Page 28: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/28.jpg)
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.
![Page 29: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/29.jpg)
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:
![Page 30: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/30.jpg)
Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk
Knowing this boson state in 4k+2d space (labeled as B4k+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 B4k+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).
![Page 31: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/31.jpg)
Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk
Knowing this boson state in 4k+2d space (labeled as B4k+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 B4k+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.
![Page 32: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/32.jpg)
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?
![Page 33: Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies Cenke Xu 许岑珂 University of](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649cab5503460f9496d169/html5/thumbnails/33.jpg)
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.