Interacting topological superconductors and possible origin

Report
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:
Yi-Zhuang You
Group member:
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Kitaev, unpublished
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
Motivation:
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.
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.
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
Weyl/chiral fermions:
Weyl fermions can be gapped out by pairing:
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:
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
Popular alternative: Realize chiral fermions on the 3d boundary of a
4d topological insulator/superconductor
3d boundary,
16 chiral fermions
Mirror sector
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
3d boundary,
16 chiral fermions
Mirror sector
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.
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)
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
3d boundary,
16 chiral fermions
Mirror sector,
gapped by interaction
A different question: Can we gap out the mirror sector with short
range interaction, while
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
gapless
0
gapped
+infty
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
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
0d boundary of 1d TSC
These 0d fermions are realized at the boundary of 1d TSC:
γ1
E
γ2
J2
Trivial
TSC
J1
J2
J1
E
In the bulk:
With N flavors, at the boundary
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
1d boundary of 2d TSC
1d boundary of 2d p±ip 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 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:
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.
ϕ condense/order
ϕ disorder,
kink condenses
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.
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.
ϕ condense/order
ϕ disorder,
kink condenses
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….
2d boundary of 3d TSC
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
Consider a modified boundary Hamiltonian (Wang, Senthil 2014):
Consider an enlarged O(2) symmetry.
When ϕ condenses/orders, it breaks T, breaks O(2), but keeps
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.
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.
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.
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:
Interacting TSC and Possible Origin of 16n chiral fermions in Standard Model
Further thoughts:
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)
gapless
0
gapped
+infty
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
Conclusion for part 1:
3d boundary,
16 chiral fermions
Mirror sector,
gapped by interaction
When and only when there are 16 chiral fermions, we can gap out
the mirror sector by interaction with
Then only the 16 left fermions survive at low energy.
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.
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:
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?
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: (AlvarezGauze, 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.
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:
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).
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.
……
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?
Gravitational Anomalies and Bosonic phases with Gapless boundary and Trivial bulk
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.

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