1
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every story, no matter how much
action-packed it is

2
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begins with introducing the main actors.

3
00:00:06,699 --> 00:00:09,619
our course is certainly going to have a lot

4
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of action and it will touch many
different domains

5
00:00:13,770 --> 00:00:18,119
of physics there are however just three main actors

6
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topology symmetry

7
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and bulk-edge correspondence, so now we'll

8
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introduce them before the main
action begins

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maybe the most frequent question I get
from students who begin their research

10
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in the field is "what is topology?"

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what is so cool or useful about topological systems

12
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the answer is extremely simple

13
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topology is about all the things
that cannot change continuously

14
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so in order to define it we first need
to tell what we call continuous

15
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for the same reason topology in general

16
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is not useful at all especially if we
choose a silly discrete quantity

17
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well for example I could call

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every quantum system with size less than
1 micron

19
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a nano system and study the
topological classifications of every

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experiment into

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nano and macro experiments

22
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so in order to apply to body meaningfully
we need to focus on

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useful physical characteristics and of
course there is no way to make a

24
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complete list of

25
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all the useful topological
properties

26
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so indeed this list keeps growing

27
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however most of the time that you hear
about topological systems

28
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it's all about one extremely simple topological quantity

29
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presence or absence of particles with
zero excitation energy

30
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if you think about it for a second this
property seems

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almost as ridiculous as the division of systems by their size

32
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after all if there's

33
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if there are no zero energy excitations
who cares

34
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maybe they are those with some small
energy

35
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the curious thing is that if we only
consider systems without zero energy

36
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excitations

37
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it may happen that some of these

38
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cannot be transformed into others
without the zero energy excitations

39
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appearing

40
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somewhere on the way. that way one can

41
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group all of the systems with a

42
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gap so with no zero energy excitations
into separate classes

43
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and check what is the difference between
those classes

44
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once again if you remember what I said
before about the usefulness of topology

45
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this difference sometimes won't be
profound at all

46
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sometimes however it will be well worth

47
00:03:04,640 --> 00:03:07,840
decades of research

48
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well for beginning we will focus on the

49
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not profound at all case the

50
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zero dimensional systems or

51
00:03:17,050 --> 00:03:21,750
quantum dots. before we dive into the
topic

52
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I should also say a few words about

53
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the other important player: symmetry

54
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unlike topology it's an extremely well
known concept in physics

55
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and it defines how we think about most
things

56
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so crystals have translation symmetry

57
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energy conservation is a consequence

58
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of the translation symmetry in time,
and finally magnets exist

59
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because they spontaneously break time
reversal

60
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and spin rotation symmetries

61
00:04:00,340 --> 00:04:03,620
to topology symmetry is like an added
flavor

62
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so that when we study the same topological properties

63
00:04:07,890 --> 00:04:11,890
in systems to the different symmetry we
may come to a

64
00:04:11,890 --> 00:04:16,150
completely different answer. but enough
of the introduction

65
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go ahead, study the materials of the
section

66
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since it is the very first one

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I hope it will be simple and it won't
take you much time

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keep track however how topology

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appears and what is the interplay
between topology

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and symmetry