every story, no matter how much action-packed it is
begins with introducing the main actors.
our course is certainly going to have a lot
of action and it will touch many different domains
of physics there are however just three main actors
topology symmetry
and bulk-edge correspondence, so now we'll
introduce them before the main action begins
maybe the most frequent question I get from students who begin their research
in the field is "what is topology?"
what is so cool or useful about topological systems
the answer is extremely simple
topology is about all the things that cannot change continuously
so in order to define it we first need to tell what we call continuous
for the same reason topology in general
is not useful at all especially if we choose a silly discrete quantity
well for example I could call
every quantum system with size less than 1 micron
a nano system and study the topological classifications of every
experiment into
nano and macro experiments
so in order to apply to body meaningfully we need to focus on
useful physical characteristics and of course there is no way to make a
complete list of
all the useful topological properties
so indeed this list keeps growing
however most of the time that you hear about topological systems
it's all about one extremely simple topological quantity
presence or absence of particles with zero excitation energy
if you think about it for a second this property seems
almost as ridiculous as the division of systems by their size
after all if there's
if there are no zero energy excitations who cares
maybe they are those with some small energy
the curious thing is that if we only consider systems without zero energy
excitations
it may happen that some of these
cannot be transformed into others without the zero energy excitations
appearing
somewhere on the way. that way one can
group all of the systems with a
gap so with no zero energy excitations into separate classes
and check what is the difference between those classes
once again if you remember what I said before about the usefulness of topology
this difference sometimes won't be profound at all
sometimes however it will be well worth
decades of research
well for beginning we will focus on the
not profound at all case the
zero dimensional systems or
quantum dots. before we dive into the topic
I should also say a few words about
the other important player: symmetry
unlike topology it's an extremely well known concept in physics
and it defines how we think about most things
so crystals have translation symmetry
energy conservation is a consequence
of the translation symmetry in time, and finally magnets exist
because they spontaneously break time reversal
and spin rotation symmetries
to topology symmetry is like an added flavor
so that when we study the same topological properties
in systems to the different symmetry we may come to a
completely different answer. but enough of the introduction
go ahead, study the materials of the section
since it is the very first one
I hope it will be simple and it won't take you much time
keep track however how topology
appears and what is the interplay between topology
and symmetry