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00:00

let's get started welcome everyone to
detent uqm virtual seminar today we are
very happy to have Shweta when who is
gonna tell us about periodic quasi
periodic and random Drive in conformal
field theories and opponent exponent
Shweta please go ahead yeah okay yeah
yeah yeah
first I hope everyone is doing good at
home so today yeah I will introduce some

00:33

very simple story on some nightly broom
driving system this work is in
collaboration with real high envy and
Ashwin and hover okay
so first let me tell our motivation the
motivation is very simple we want to
understand some very basic questions
like what kind of non equilibrium phases
converge when we drive a many-body
system and are there any other

01:06

parameters to help categorize as a
possible Chris diagram and how does the
phase diagram depend on the type of
driving for example you can do the
periodic quasi periodic and random
driving surely we can have different
emergent phase with different kind of
driving okay we add the initial effort
we want to fund some exact exactly sobel
setup to understand such questions in

01:37

particular we hope the result should be
universal which means the result should
be independent of the model we study
okay the outline of this talk is as
follows first I introduce to you the
setup and then I introduce what is the
operator evolution with driving okay and
I will introduce what is Lyapunov
exponent and how to use the output

02:07

Lyapunov exponent to determine the phase
diagram and how the different kind
of driving resulting in different kind
of face I will also give some examples a
periodic quasi periodic and the random
driving in particular there is a nice
example on the exact meeting between
some Kazakh here local driving CFG and
the some Kazakh crystal insulator

02:39

physics and the finally I would give the
conclusion okay let's start from the
setup as I mentioned we hope the our
study in the universal so which should
fill the theory I which is a conformal
field theory here we know that it can
form of your theory can be considered at
the effective field theory the low
energy you felt you feel Syria was some
interesting gaseous system system like

03:12

the critical icing that you know you
later prefer me and so on okay we use
CRT in one plus one be okay second how
do we try with the system now suppose we
start from some initial state cuz I zero
the driving is here we use is discrete
driving what do I mean by discredit
driving at each step at a you just step
we drive with a system with a fixed
Ahamed honing for some time interval T

03:44

then in the second step which we can
drive it with another Hamiltonian with
another time interval t2 okay we can
drive we can repeat this procedure so on
it so long so this procedure I think is
a very very universal one thing I want
to commentate that the initial state is
not limited to the pure state our setup
can be applied to mix this data like the

04:16

thermal ensemble okay so the clip
hardness set have is exist how do we
true the Hamiltonian
Vasudha hamiltonian here we emit honey
at is driving at his driving step in the
following form first we should see of T
Hammond honey which which have uniform
Hamiltonian density now we want to
deform we want to be formed the

04:48

Hamiltonian density with some function f
FX okay so so in short we we can choose
many different Hamiltonian for each
Hamiltonian we can be formatted in
different way the way you did
hamiltonians to drive the CFG
and si or whatever you happen okay here
the t 0 0 is nothing but the Hamiltonian
density which is the sum of the chiral
part and the unequal part okay you early

05:21

for very general deformation I mean if
you choose the FX arbitrarily then if we
look at the Fourier components if we
look at the Fourier components of this
Hamiltonian we can fun oh there are many
Fourier components which we call it
Ellen Ellen is in the for a Fourier
component or the chiral the
stress-energy tensor T and similarly we
can define the the antequera part of the

05:54

Fourier component of the stress energy
tensor T bar okay now the question is
this if we choose the deformation FX
arbitrarily this question will be very
challenging
we're very challenging because why this
is the challenging because the Fourier
components Ellen we satisfy these
satisfy some algebra which is called a
virus or algebra so yeah the the the

06:25

relation Amala generate heard of is this
this is a well-known of our storage
where you see of T this algebra is
infinite dimensional this is why it's
hard to study
because a dimensional if you use this
generator to drive the system I don't
know how to calculate it
exactly
so our fig our trick here is no will you
not have consider a general deformation

06:57

we will assimilate we will simply focus
all very simple case in the following
sense
first we know that there the sub algebra
of this virus or algebra which is called
SL to our algebra which means if you
should m and M in this way mm equals
minus n you can find all this algebra is
closed it is also finite dimensional
because what happening by closed a this

07:29

Esper is only generated by three
generators
l0l plus and minus n this algebra the
fun has dimers no good so then you may
ask
Oh what kind of a deformation of the
Hamiltonian corresponding to this sub
algebra the other is this if we deform
the Hamiltonian density with such kind
of function FX you can find it there the
constant there's a cosine term or the

08:00

sine term with arbitrary coefficient
okay this is if we deform the
Hamiltonian in this way so the only
algebra involved in this problem is SL
to our algebra and then we can handle
this question by hand
okay I think now the theta will be clear
we simply deformed come Tony with some
simple function so only a financial
algebra is involved that we have solved

08:32

this problem by hand for arbitrary kind
of driving so why this kind of
deformation is a simple but the reason
is very very the derivative as follows
if we deform the Hamiltonian with this
SL to our editor
and in the Heisenberg picture if we
study how to operate her oh it will
under the Hamiltonian under the unitary
evolution you have fun oh the operator

09:04

Oh seedy bar is awarded to operator Oh
tinu the new bar there is a simple
relation between T new and pay the the
relation the way it similarly a Mobius
transformation this is simply called a
Mobius transformation okay now you have
you have I think you are more clear why
I choose this kind of deformation
because under this deformation to
operate her evolve in a very simple way

09:38

only as modulus transformation it is
involved now let's go of one more step
further and we drive the system with a
serie with a sequence of Hamiltonians
whatever you happen and this operator of
you we will simply evolved from the bar
to some provision TN and at the end bar
so DN is related with the original
traditions e through a measure product

10:13

of Si or two metrics you see the product
of Si O 2 metrics right
this means oh when you try with the
operator for arbitrary long sequence
finally the provision of the operator it
is similarly related with a real
original well it was through a product
of metrics so what what is this now I
want to remind you of what we learn you
saw later physics okay
before I mention what happened in your

10:44

solid physics let's make one more
comment this is SL two magics here has
the general form here so each matrix
element can be some complex number so
this matrix is SL to Z matrix
more precisely it is su 1 comma 1
it's metrics is isomorphic to SL to are
yeah this is similarly some mathematical

11:17

effects now let me remind you what's it
well what kind of similar things we see
in the solid physics in solely the
physics when we studied a tight banding
model we use the transfer matrix
transfer matrix clear the scien is a
amplitude in a wave function for the
electron on the site
I'm and we in here it is some outside

11:49

chemical potential and when we solve
this Hamiltonian we saw this ordinary
equation we simply do the transfer
matrix so what is the transfer matrix
suppose we know that the wave function
at the side 1 and the 2 then we can
obtain the we function at his side N and
minus 1 through this relation ok we
define capital cyton we function on the

12:21

side n minus 1 then we have this
relation here the product of metrics
from t1 to TN is simulated for dr. the
transfer matrix here
here we n is the potential e in the
possible energy its Apogee so this
matrix is si or to our ok - now I think
I simply want to make an analogy in our

12:52

solely the physics when we study the the
wave function your lettuce we use the
transfer matrix which is a product of
some SAR to our matrix and here when we
study the CFT the time-dependent driving
CRT we also consider a product we also
consider a product of MAGIX which is as
u 1 comma 1 you can say the structure is
very similar
right actually although the concrete
form of the metrics to look quite

13:24

different we will see many similar
properties between the two different
models okay good now let me mention some
course bonus first before before we
study the concrete concrete model okay
so once we early once we see the product
of metrics we can define the Lyapunov
exponent in the following way now we
multiply in metrics then we take the log

13:56

all the metrics product here is the norm
you're ready for the norm of the metrics
you can choose you can use different
definitions well here different
definitions is not essential here
okay the Lyapunov exponents is defined
by PI over n log of this measures
product it can norm here and then we
take n go to infinity okay
you're asked oh how why this Lyapunov
exponent is useful okay why it's useful

14:30

surely if if you see a positive positive
Lyapunov exponent it means the transfer
measures the norm can we grow
exponentially large which means you see
a localized the state you will see a
localized or still in the lattice on the
CFT side I will show you if the Lyapunov
exponent is positive we will have a
heating phase which means the system
will absorb energy on and on on the

15:02

other hand if the Lyapunov exponent is
zero it means the wave function can be
extended or critical here what what what
what do I mean by the create critical
wave function critical function means
the wave function is neither extended
nor localized for example it can be
power-law decay this is a tactical
cannibal or a function which I call
critical will function linear we will
say it in the lattice we

15:33

we see the extended will function it
means the in the CFDs I will see some
non-teaching phase which means when you
drive the system the system will not
absorb energy and at the phase
transition the energy that the system
will absorb energy but you know very
slowly they say what I mean by christian
vision in the CRT side yeah actually
here I is the behavioral energy grows in

16:04

the three possible emergent phase which
are we introduced in the following slide
okay so besides the energy growth so we
can also use the entanglement entropy to
measure which face are we in we have fun
in total there will be three kind of
behavior for the entanglement growth
which is a linear growth oscillating and
the log log log grows okay this is this
till now it is simply some very general

16:36

cause the correspondence I will show you
example now okay we will study different
types of driving from easy to difficult
we will introduce the result according
to the following order
periodic random and causing periodic so
the periodical driving is very simple
because the driving sequence we we have
we will repeat really repeated the
driving sequence with some period so

17:07

essentially we simply need to focus
focus all our attention in a unit cell
you know you need cell if we understand
the property of the the metrics product
within one unit cell we can understood
it what heaven after in unit cells it
turns out there is a very simple way to
determine the possible face the possible
emergent phase in this case what we

17:37

simply need to do is we simply need to
check the product of the metrics from m1
to MP within within one unit cell
the only quantity we need to check in
the trees in the trees of this matrix
product if the absolute value of these
trees we call the product as PI P if the
absolute value of these trees PI P is
bigger than 2 we have a positive with

18:11

the ethanol exponent the CFG will be in
a sitting face if this quantity is
smaller than 2 we will have some
nourishing face the system will not
absorb energy if this value equal to it
means the CFG will be at some transition
how do you understand this result now
let me remind you what happened in
transfer matrix when we studied transfer
matrix in lattice model if the trace of

18:42

this PI P is the absolute value it is
smaller than 2 it means the energy E is
in the energy spectrum and the wave
function is extended if you choose some
other energy e so the trace
so this quantity is bigger than 2 it
means e the energy is not in the it's
not in the spectrum but in the gap and
in this case the wave function will be
localized ok there is some in intuition

19:17

in the lattice system now I will give
you a example of the minimal minimal set
half of periodical driving CFG we simply
drive the system with two different
Hamiltonian recall H 1 and H 0 so we we
drive the system with each one for time
interval T 1 and drive it with type
drive with a hamiltonian h0 for time T 0
okay the Hamiltonian information H 1 we

19:50

simply choose it for example with this
form okay there the parameter theta when
we chew on see how you can have a
different a deformation
okay now if we try with the CFG with two
different Hamiltonians if we use look
retiring here we have fun we can
determine the phase diagram here I
simply choose a different to count out
deformation we can have we can have some

20:23

rich crazy diagram the blue region is
enough teaching phase and the red region
is a heating phase so if we look at the
entanglement entropy for example of the
half system the entanglement obey of the
half system will find o in this blue
region the in terms of entropy simply
oscillate at the function with time in
this red region which is the heating
phase the entanglement of a will grow
the linearly in time at the boundary

20:55

between the red region and the blue
region the entanglement of a will grows
like a log P knot he behavior okay this
is how we we can use the entanglement
way to to characterize the different
kind of face you know drag a
time-dependent periodical driving CFG
okay
later we found oh there are some very
interesting fine structure within the
fist within each field so the phase

21:27

diagram what we found is followed and
followed it with the C of T is in the
heating phase in the heating phase oh we
thought that the CFG we observe this
system will absorb energy exponentially
in time but then the question is this it
with us if the energy is uniformly
distribute is tributed in the system or
not Oh what's amazing in that we found
the energy density here in the this is

21:59

the real space the y axis is energy
density we what we found is oh there are
two energy Peaks within the within the
heating phase what's more interesting
one one big one
the chiral the other people's aunty
Carol what I mean so suppose we stop
driving the system then YP this peak
will move westward and at this this peak
will move left we have we have one

22:30

chiral pink and one aunt in Cairo peak
what's more interesting is that the
total the the the total the entanglement
entropy in the system is mainly
contributed by the entanglement between
the two peaks which means as I mention
in the heating phase the entanglement
entropy here will grow the linearly in
time all the contribution of this
entanglement over here is contributed by
the two peaks one chiral and the other

23:01

is an hi Carol this is quite amazing
because I think at least if we have
studied the ABS the RT uality
what is the possible structure in the in
the gravity do it is possible we have
some long hole connect connecting the
two the two peaks in the real space we
don't know yeah so what I want to say
that in the heating phase of the driving
CFG there are some there are many

23:34

interesting structures in the in the
energy density distribution okay Q now
this is the minimal set half of the pure
pure periodic driving CFG next I will go
to the random random driving random
driving CFG for random driving sake so
what happened we want to fund some
analogy in latias right the first thing
will come come in comes into our money's
under symbolization

24:06

right I think all we all of us know the
internalization it assembly means in the
title body model for the own soil
chemical potential region if we is
randomly chosen if there are some
randomness you know potential V then we
have fun Oh for arbitrary energy e
when we saw the transfer matrix we can
find that we function is localized this
is the understand localization so

24:37

actually there is a mathematical cerium
which can prove understand localization
which is called Furstenberg Syrian first
Firstenberg serum serum is very powerful
it it is simple it tells us in the
following thing suppose we have s
suppose we choose some random matrix
which is SL and are another period SL
and now but not si R to R so as they are

25:08

to our matrix is a symmetrically
understand organization which is
similarly a a specific case in
Firstenberg of the cerium so first of
burg cerium is more powerful because we
can study arbitrary and here if we
choose arbitrarily random matrix in SL
and R and we multiply them together so
Firstenberg cerium tells us and there
are what kind of conditions the lyapunov

25:39

the Lyapunov exponent is always positive
so first a more fun there if we can
satisfy the following two conditions the
then the product of the random matrix
must have positively happened of
exponent so what two conditions are
these one is
non-compact and the other is strongly
irreducible let me explain what's this
suppose we choose a random metrics and

26:10

suppose the genie team you jamie-lynn
the smallest a subgroup which can which
contains the random metrics in you
truths so this this smallest subgroup
must have been an compact this is a
condition one and a second we should the
hell for arbitrary subgroup oh this
Jameel for each subgroup it should be
irreducible it means strongly
irreducible supporter we satisfy the two

26:42

conditions then first event the first
murder serum tells us the product or
this message the random matrix must have
a positive Lyapunov exponent so for
understand localization we can easily
check for understand localization both
conditions are satisfied this is why
there is always localize the face but no
extended face this is how to use the
fürstenberg theorem to prove understand
localization and now we want to use

27:13

fürstenberg serum in random driving CFT
in the random driving CFG we also have a
product of two by two matrix but this
matrix is not as well - our PR SAR to
see how how do we use furstenberg
theorem we simply need to embed a cell
to the SL to see into SL for R we
thought we have always embed our assay
OTC matches in in sa our 2d our matrix

27:44

then we can use the first marker cerium
okay
before I introduce before I move move oh
you may guess in the random CFG maybe
there's only one phase which correspond
to understand correlation right yeah I
will always we also believe so but it
turns out this is not the truth let me
tell you why
okay to study the random driving CRT we
need to study the product of many su 1

28:16

comma 1 matrix which is a sub subgroup
of Sao Tuesday okay yeah now usually
when we do the when we do this
calculation we really tried people in
the kind of randomness we found Oh the
theory is always in the heating phase
and later when we check some
mathematical mathematical serum which is
called the Simon theorem so Barry salmon
in 2005 he proved that for a random

28:49

product of as u 1 comma 1 metrics
you so when the furstenberg condition I
mentioned two condition compact a non
compact and a stronger irreducible in in
which cases the two the two conditions
are satisfied and in which condition in
what case the two conditions are not
satisfied he simply exhaust so so berry
salmon he simply exhaust all all the
kisses that the Furstenberg serum is not

29:21

satisfied yeah only fork different case
I mean all goes through one by one I
will simply tell what I simply want to
tell if that when we go through the four
different case we found oh well our
random driving CFG when we through the
Hamiltonian H 0 and H 1 in some specific
case the Furstenberg serum does not
satisfy which means there's no under

29:53

centralization we cannot nanu Lee
consider there is always localization in
the random driving CFG there are some
except exceptional pawns where the
random driving sake we will not be
heated up let me tell you what is this
point okay
yeah first I want to tell that if we
choose dry with a CFG with two random
I'm tuning if we choose the two
Hamiltonian randomly if you study the

30:24

internal entropy the integumentary will
always grow the linearly in time we've
always grow linearly but there is one
single point which we call it
exceptional point when you track when
you travel the system travel the safety
randomly the system we never absorb
energy which means and you know if we
look at the entanglement abate and this
exceptional point the entanglement
repair we simply oscillate all the way

30:56

so what are these what is this
exceptional point okay
let me introduce this point as follows
suppose we choose to Hamiltonian H a and
H B with simply this deformation we have
a caseta parameter within this
deformation now we try with the to the
safety wheels I mean Tony is Ray and H B
but a you know random away okay which
means with the probability 1/2 and 1/2

31:28

for example so if we travel if we travel
the system with the Hamiltonian H a with
some time interval T a and we try with
the Hamiltonian H B for some time in
here would t be F ETA and the TB is
chosen in some at some specific value
here then the this is exceptional point
we will we found ok

32:01

what's the meaning of T a star in the TV
star here so with the form the
Hamiltonian so it's each Hamiltonian we
will have an effective lens because
because we you deform the Hamiltonian
some if you put a coded particle in the
system the quasi particle will move
faster in some location and slower in
some other place
so we can be fine effective a lens of
the CRT so when the time interval key

32:34

and the PB are chosen and 1/2 half of
the effectual lens over the system okay
then this is the exceptional point okay
let me give a shorter were very short
summary here so what I want to say that
if you random randomly dry with SLT with
with probability 1 the system will be
heated up whether there is some
exceptional point which we follow from
some mathematical theorem and and some

33:08

exceptional point the system will not be
heated up what's more interesting like
this suppose as the exception
exceptional point if we drive the system
not randomly but periodically with the
finer the system is in the heating phase
the in the heating phase so so so this
is this is the amazing thing if you
drive the system randomly it's not
fitted but if you drive it periodically

33:40

it is hid more interestingly later I
will show you if we try with the at the
exceptional point and cosy periodically
it is the in the a first idiot at the
phase transition so this accept
exception opponent very interesting
because when we try rated in different
way we can say hating hating on the face
foundation okay okay
I will give a short summary to now just

34:12

not simply imagine is a periodical
driving CFG under the random driving CH
e in the periodical driving sake
in general we have heating phase not
heating phase and the first transition
in the random driving CFG we in general
we only have our heating fields but
there is some exceptional point in the
net in a feeding face it is only a
single point so the system will at this
exceptional pollen in the system will

34:44

not be heated up yeah
so the - now this is a periodical
driving and of random driving CRT if
there is any question you can ask her
now I have a question yeah so do you see
these the hot spots also in random oh
yeah feeding face yeah I didn't measure
some did you no matter in random driving
or periodical driving or the CAHSEE
periodical driving I will introduce the

35:16

later
suppose we are in the heating phase we
will always observe two energy Peaks two
energy peaks in the real space this is a
common feature
not my this is a straw basket
stroboscopic measurement right yes yo
how do you define a stroboscopic
measurement for the random I thought you
appear there's no period oh good yeah
yeah this is a random driving in general

35:48

and you will already see that the two
peaks - energy density Peaks but in
random driving in general the tradition
of the two peaks will move can move in
the in the in the system they are not
fixed in the periodical driving the the
two energy peaks we will not move they
are they are always fixed when you
observe the system in the integer time
or driving
so that is yes okay good so and the

36:22

other other question
okay now just now I simply met admit to
you that what happened you know
periodical driving and the random
driving C of T next I want to tell what
happened for the Kazi period driving
okay what cause if you're also driving
there are many interesting models the
most a beautiful one in my opinion is a
Fibonacci Fibonacci called crystal you
won't be so what is this model again we

36:53

have this tied banning model but the
potential we potentially is not a
periodical and it's also not a random
but the potential has some sequence like
a be a a be some sequence here you may
ask
oh why this sequence of a be a a B is
called quasi periodic so the reason how
to generate this sequence of outside a
chemical potential the way to generate

37:25

is use original rotation which simile
means okay we would hate the sum
function we need the frequency Omega and
the time is in his written an x-ray
beside a number of the site and so when
you would hit you rotated out the
variable if the variable in recently we
will choose other potential at v8 if the
variable is in region B we will choose

37:56

the potential at the VB this is the way
we define some coffee or some Fibonacci
sequence we simply use some original
original rotation suppose we should
actually hear this frequency Omega is
the inverse of the good and region
this is why it's called people not
Fibonacci sequence if you choose this
Amida at some regional number we always
get a periodic periodic sequence here

38:29

there are different way though without
generating this Fibonacci sequence but
this one is original rotation is the
most intuitive what intuitive way so I
mean there are some other method I will
not introduce then you will ask oh how
do people study the property or this
Fibonacci cause a crystal okay
in the early river of this area people
cannot study this Fibonacci called

38:59

crystal strictly but people can study
the Fibonacci crisscrossed crystal by
using the by using the periodical one to
approaching to approaching the Fibonacci
limit what do I mean now as I mentioned
the frequency Omega is a inverse of the
Gooden couldn't ritual now if we choose
Omega as FN minus 1 here F n minus 1

39:30

over F 1 F 1 is the Fibonacci number if
n goes to infinity we will approach the
Fibonacci limit for finite n we will
always get a some original personal
number Omega and the crystal is not our
causing crystal but a crystal then we
can we have people similarly take take n
from smaller number to the larger and a
larger number and look at how how the

40:02

energy spectrum or the Casa Cristo
evolve as we increase our number and
what people find is very interesting and
we approach the Fibonacci limit we can
find it there are more and more energy
bands um we finally will have infinite
energy band but the band the band width
of each energy depend upon the will of
the in the energy spectrum
the smaller and the smaller so this is

40:34

how people studied the Fibonacci Casa
Cristo in practice then mathematically
it was accrued that for arbitrary we eat
does not equal equal weeby weeby
in the potential the potential sequence
which should in the Fibonacci quality
crystal well arbitrary choice of a
potential the spectrum is a cantor that
have zero mirror what's more interesting
in the spectrum the wavefunction is

41:05

polynomially bounded which simply means
in the spectrum of the fibonacci because
the crystal the wave function is
power-law decay but not localized and
not extended then we will ask her what
happened for a cosy Bureau who could
driving CFG okay before any mass medical
study that's let's simply check they use
it using the less simply use the same
logic in the quasi crystal we use some

41:39

regional number to approach the
Fibonacci number and see what happens in
the phase diagram of this driving CFG
what we simply observe is that as we
approach the Fibonacci limit from left
to right okay for some small smaller a
Fibonacci number which is which means
the system we are dragging the system
periodically
we have fun oh there there are posts

42:10

heating phase and an antiquing phase so
the reasoning in yellow is the heating
phase the region in blue is an oxygen
phase now as we approach the Fibonacci
limit which should the omega which is f
minus 1 over F 1 increase in oh and all
what we observe in that oh the knocking
fears will disappear finally
for example in this plot you will say Oh

42:41

almost all the reason are in a hidden
place that the yellow the yellow region
okay here I didn't show some some white
structure in this in the hidden face
actually if you plot Burnley you can
define the Lyapunov exponent in this
hitting face you will see many
interesting self similar structure in
the Lyapunov exponents which I added in
the show here
okay you need in this in this body I

43:12

simply choose some specific Hamiltonian
H a and H B if I simply change the
Hamiltonian H a and H B we will again
some other different Chris diagram yeah
we are here from left to right we are
doing the same thing as before but we
change the the Hamiltonian is a and H B
in some other form what I want to say
that the feature it is robust so what we
try to do bust and we approach the

43:42

Fibonacci limit but we always say oh the
heating phase is dominated actually we
find about not heating phase we always
disappear then we will ask the following
question in the Fibonacci limit can we
prove that the non heating phase is a
Cantor set of Mayer 0 what is the
feature of the entanglement and energy
evolution in the international peace I
mean although if you simply look at the

44:17

phase diagram through your eyes
oh there's finally you funny in the
Fibonacci limit there's no knocking face
but the author is there it and then the
probability is 0 I want to emphasize
that probability 0 doesn't mean
impossible there is positive but the
possibility 0 probability is 0 so 2 2
because ok we have enough people now

44:49

a crystal there are many there are
thousands of people are in mathematics
which prove some which proves strictly
the spectrum property and the wave
function property of the CAHSEE crystal
so now we simply think if we can map it
with a map our phase diagram in the CFG
to the CAHSEE crystal so we can use
every mathematical result in literature
right this is our strategy actually with

45:19

fun indeed we can make exact mapping
between the phase diagram or the
fibonacci driving Z of T and the energy
spectrum in the Kazi crystal the menu
here is a little bit technical but let
me sketch the main procedure of this
meeting so in the in the code in the
fibonacci Casa Cristo when we consider
the product of the transfer matrix we
can define mm2 de Evelyn at the product

45:54

of the metrics so the number of the
metrics a the Fibonacci number from m1
to mm so what's amazing so the beautiful
property in this Fibonacci sequence is
that this m2 de satisfies some very nice
property here there are some nice
property of the metrics in this
Fibonacci sequence so if we can define
the trees of the top of the trees of the

46:25

metrics we can find there is some
recursion relation between the trees
here we call the equation of motion if
the equation of motion satisfy some here
in some constant of motion in this in
this Fibonacci sequence which relates
the trees of different metrics this is
called a constant of motion which
related the trees of other X this is the

46:54

X F n xn minus 1 xn minus 2 so the three
variables
these men avoid this 2d manifold so
depending on the style of this in wiring
the I if I in the pontiff will have this
kind of manifold if I equal zero so the
there is a touching touching point here
if the you aren t is negative this this
party in the middle is decoupled in the

47:25

other part so so in the Fibonacci
sequence
the problem is reduced it to study that
how a point evolve on this many hood how
upon a wall on the manifold so this
point means the trees the trees or the
matrix because yes your time is yeah so

47:57

yeah this is too technical
I will simply show you the result
because because the nice property of the
people not a sequence we can study we
can study the trees though the metrics
but not the concrete form of the matrix
finally we can we can find an exact
meaning between the phase diagram here
in the in the in the bottom in the phase
diagram of the oh the people not
striving sequence and in the top is the

48:28

energy spectrum of crystal and we
approach the other people not limit we
can find there is an exact exact meeting
between the two oh yeah okay
actually I have more to say but I I me
not have enough time I will give to give
them in this slide and the next one I
simply want to say that enough people
not driving CFG in the NASA team faced
the the entanglement repay will growth

49:00

log at the logo time and the energy grew
the power power law in time this is
similar to the behavior of the we
function in the Bonacci quasi crystal
yeah one more comment actually we
starting a different kind of cozy period
riving we also study something called
abre andrey cozy here with the driving
here is some this diagram I don't have
time to show sorry okay

49:34

I think I'll kill now we have bought
some partial answer to our motivation at
the very beginning that what kind of
non-equilibrium faces converging the
time-dependent driving many body system
we have hitting face not hitting face
and a phase transition but how the other
parameters we use it we can use a
Lyapunov exponent we can use in Tanguma
entropy evolution we can use energy
evolution and how does the field diagram
depending on the type of driving yeah

50:06

how we have show some different examples
right and at the summary in the summary
in the periodical driving we have a
heating phase now heating phase and the
phase transition in the random driving
wheel we have a heating apiece but we do
some exceptional exceptional points in
the nakajima phase for the people
naughty driving we have a heating phase
and a mirror zero critical critical

50:38

phase for more general cause appear on
the driving we can have heating phase
not getting filled and critical which I
didn't have time to to tell and in all
the three cases as long as the system in
the heating phase the enhancer entropy
of the Hoth system entanglement repiy
always grow linearly proportional to the
diagonal exponent and other energy total
energy grows exponentially with read the

51:09

original rated with the ik naught
exponent in the future it is interesting
to generalize the SL to our edge bruh to
the virus or algebra it is also
interesting to add the nine integrable
terms in the driving
yeah sorry I assumed I I prepared two
too many materials and they cannot have
tell the details I'm did you yeah but if

51:40

you have a question we can we can tell
you later yeah okay thank you so much
for sharing your interesting results
with us so now we have time for some
questions
very quick questions
and maybe III president I tell the story
- yeah very quick away
I have a question yeah so you have Cece
make made some general statements about

52:28

phases and then you showed us winding
one sort of example using these one
quasi periodic example yeah so what I
think you said some more along the lines
something along the lines of this being
these results being general in some way
but what indication is there for that
good so what do I mean by general as
long as we try the CFT use sl2 algebra
and saliva you travel your system with
SL to algebra all my concluding up here

52:59

I okay I see so this was just one C
already worked out the general case this
was just to illustrate that I say oh
yeah I thought you are suppose this you
know the very interesting future
question is you C of T from the very
basic algebra the virus or algebra so I
sell to our it's the unique is the only
sub algebra a virus or algebra which is
simple here the next step we can I mean
there's no other sub sub algebra in

53:32

where else or algebra if we if you want
to move one step further you need to go
to the where's or algebra that will be
more interesting I believe yeah okay
thank you
yeah you're welcome I have also a quick
comment about what you said yeah because
I'm also kind of working at this I think
I feel like there is some notion of
rigidity here like even if you deform
your so2 are a little bit all these

54:03

things are all these claims are valid if
I think oh yeah okay it doesn't have to
be litter exactly maybe as well I don't
know I feel like if you even like add
more
Fourier components like I don't know
maybe let's say you have L plus 1 minus
1 and L zero point then you ask
I don't know L 2 and L 3 but the
coefficients are small enough then so
that you're not
kind of moving too far it's like I don't

54:34

know some attractive fixed points yeah
yeah yeah rigidity to some extent I feel
like yeah yeah I don't know how to prove
this strictly but hey you know you know
physically if you add some perturbation
yeah it's totally possible the feature
if you don't take very long very long
time I mean if you use some finite time
maybe the heating on in there hitting a
structure is always there but if you

55:05

added a perturbation by hitting some
much longer time maybe it will give toy
some property here yeah for me because I
might even use it and another generous
origin air is her last minute in class -
Emma we need to go to the web sorry
Edward if if I want to solve this
problem problem exactly you see we will
be very very challenging well it's very
challenging

55:36

yeah maybe we can make some
approximation some good approximation I
wish I don't know maybe we have this
castle this bitter yeah all right so if
there is no other question let's thank
the speaker again and thank you everyone
for you know joining us today and we'll
have our next meeting in two weeks so
stay safe everyone and have a good day

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