Interacting - PowerPoint Presentation

Slide1

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

Cenke

Xu

许岑珂

University of California, Santa BarbaraSlide2

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.Slide3

Interacting TSC and Possible Origin of 16n chiral fermions in Standard ModelCollaborators:

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

, unpublishedSlide4

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.Slide5

Interacting

TSC

and Possible Origin of 16n chiral fermions in Standard Model

Weyl

/chiral fermions:

Weyl

fermions can be gapped out by pairing:Slide6

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:Slide7

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 theorySlide8

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 ModelSlide9

A different question:

C

an 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

gappedSlide10

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 isSlide11

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, andSlide12

0d boundary of 1d TSC

These 0d fermions are realized at the boundary of 1d TSC:

γ

1

γ

2

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

J

1

J

1

J

2

J

2Slide13

1

d

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:Slide14

1

d

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:Slide15

1

d

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 condensesSlide16

1

d

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 condensesSlide17

3

d 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

2

d

boundary of 3

d

TSCSlide18

Consider an enlarged O(2) symmetry.

When

ϕ

condenses/orders, it breaks T, breaks O(2), but keeps

2

d

boundary of 3

d

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.Slide19

3

d

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. Slide20

3

d

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.Slide21

3

d

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.Slide22

3

d

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:Slide23

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:Slide24

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:Slide25

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

3

d TI: U(1) and time-reversal

3

d 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

.Slide26

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:Slide27

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?Slide28

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.Slide29

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:Slide30

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 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 2

π

(analogue of 3d TI).Slide31

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 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. Slide32

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?Slide33

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.