interacting fermionic and bosonic topological insulators, possible connection to standard model and...

Interacting Fermionic and Bosonic Topological Insulators, possible Connection to Standard Model and Gravitational Anomalies

Cenke Xu

许岑珂

University of California, Santa Barbara

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.

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Wen, arXiv:1305.1045, You, BenTov, Xu, arXiv:1402.4151, Kitaev, unpublished


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.


Weyl/chiral fermions:

Weyl fermions can be gapped out by pairing:


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:


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

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)



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.



Our conclusion: this is only possible with 16 chiral fermions, i.e. classification of 4d TSC is reduced by interaction

0+infty

gapless gapped

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


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


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


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:


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:


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


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

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


Consider an enlarged O(2) symmetry.When ϕ condenses/orders, it breaks T, breaks O(2), but keeps


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.

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.



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.



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.


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:

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


Further thoughts:

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.



Conclusion for part 1:

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.


2d E8 state (Kitaev): does not need any symmetry; chiral bosons with chiral c=8 at the 1d boundary.

Effective field theory:


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?


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.


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:


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).


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.


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?

Conclusion for part 2:


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.

interacting fermionic and bosonic topological insulators, possible connection to standard model and...

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