Xueda Wen: Periodic,Quasi-periodic,and Random driving conformal field theories and Lyapunov exponent

Xueda Wen: Periodic,Quasi-periodic,and Random driving conformal field theories and Lyapunov exponent

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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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