Hello, so today we have learned the tenfold
classification of topological insulators and
superconductors.
So this table here is pretty much the summary
of the day.
In closing, let me make a few comments.
First, this is the periodic table of topological
insulators and superconductors.
This is the periodic table of chemical elements.
When people first made the chemical periodic
table, some elements were not yet discovered.
Nevertheless, they conjectured the existence
of undiscovered chemical elements together
with possible chemical properties.
And in fact, later they were experimentally
discovered.
One may say we are in a somewhat similar situation
here.
This table unifies many known existing topological
states of matter.
However, there are also undiscovered topological
states of matter predicted by the periodic
table.
For example symmetry class CI in three dimensions:
the table says there is a non-trivial topological
superconductor here, and symmetry class CI
is a symmetry class of singlet superconductors
with time-reversal symmetry.
So the topological singlet superconductor
in class CI has not yet been discovered.
Anyway, as in the chemical periodic table
it is an interesting challenge to find a physical
realization for all entries in the periodic
table.
The second comment is about the other symmetries.
In the opening video I emphasized that symmetries
in the tenfold classification are carefully
chosen, and that is true.
But however, it is still possible to include
more symmetry conditions than time-reversal,
particle-hole and chiral symmetry.
From a solid-state physics point of view,
natural symmetries to think about are symmetries
of a crystal, and in fact topological phases
can exist in the presence of, or because of,
a crystalline symmetry.
Such topological states of matter are widely
called topological crystalline materials.
For example a topological crystalline insulator
has been experimentally discovered recently.
So I think that it is reasonable to expect
that there will be more discoveries of this
kind in the future.
But I'd however emphasize that what we have
learned today, the techniques and the concept
developed in the original tenfold classification,
are basics even in the presence of other symmetries.
They can be suitably generalized and extended
to discuss various kinds of topological crystalline
materials.
The final issue I want to mention is the effect
of interactions.
Of course, the interaction effects in topological
phases of matter in general are quite non-trivial
problems, but however in certain cases it
was shown that interaction effects invalidate
or modify the tenfold classification presented
here in this table.
So for example, class BDI in one dimension,
the table says the classification is an integer.
But however once you include inter-particle
interactions, it was shown that this classification
is actually given by Z_8 - not integer, when
you have inter-particle interactions.
And there are more known examples of this
kind.
Although, I would say, we do not yet have
complete theory of interacting topological
phases with symmetries, there has been a lot
of recent activity concerning the effect of
interactions.
We just started to learn the interaction effect
in classifying topological phases of matter
with interactions.
So, that's all I want to say, so thank you
for your attention.
