Ruihua Fan: Floquet conformal field theory

Ruihua Fan: Floquet conformal field theory

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00:00
and hi everyone today I'm going to talk about locate conformal field theory which is based on this archive paper and some other works in progress which is done with my collaborator in feijoada and Ashley okay here's my outline so I start with motivation and then introduce the most general setup and also some conjecture about the general dynamics in the system so I will illustrates the main phenomena and the physics with a very concrete
00:30
example and show you some interesting emergent spatial features in the system and the I will talk a little bit about to the random and also cause the periodic driving and then close with the outlook okay so our motivation or goal is to engineer some interest in phenomena in this many body fluid systems and we still want this we want the system want the dynamics to be exactly solvable and so then we choose the one plus one
01:02
dimensional conformal fuel theory as our platform to do the engineering and because of the powerful conformal symmetry it's it's the model we design can be like exactly solvable and also we want our result to be universal and also robust that gives us some constraint on the driving protocol we should use and the protocol we come up with it turned out turns out to be suitable for starting random and the quasi periodic driving that I'll mention at the end of
01:33
this talk and the one thing I want to address is that symbolization is of course a very interesting phenomena and concept in 911 physics but that's that's not the thing we are going to talk about in this talk okay and here's the general setup so the system we choose is a one plus one dimensional you entry and the rational conformal field theory which covers pretty much all the system that I realized realizable in the rule in our
02:06
lab and the system size is denoted by L and the boundary condition can be either periodic or open it doesn't matter and the driving protocol is implemented by changing the Hamiltonian density and periodically in time so let's say if T naught naught is our Hamiltonian density then we multiply it by such an envelope envelope function f of T and X so it has this spatial dependence in in space and is periodic in time like ft plus capital
02:39
T equals to f of T ok and the reason for this choice is the following so we because we want our result to be universal and this is by construction like universal and applies for any safety because of because the is T not not the stress energy tensor is definable for any safety and all and only involves the V restaura algebra so it's universal and and it's and also it's simple to understand because it has
03:11
a clear physical meaning in terms of the cosy particle picture which is believed to be true for any rational conformal field theory and it's the following so what this f of T does is it modifies the velocity of the quasi particle in our system and let's say we create some excitation which is describable by some quasi particle and for the left mover denoted by this blue dot it will move to the left but now the velocity will depend has a spacetime dependence which
03:43
is proportional to this F of T and X ok the forest for the system and turns out to be that studying the classical motion of this quasiparticle is sufficiently good to understand the general quantum dynamics and the in particular we look for the fixed point of this classical motion meaning that the particle can start from somewhere and come back to the same point after one or several cycles of driving and we have
04:14
the following conjecture is that the existence of such kind of fixed point of the classical motion of quasi particle is the same as saying that the dynamics is heating we don't have a general proof but our illustrators with some concrete example or later okay but for general f of t is too challenging to solve the dynamics for a general of generic fft so we make some further assumption we assume F of T is a
04:46
step function in time and in space it only has a single Fourier component so that it can be written in this form and now and with this simplification and Hamiltonian now only involves the three we are several generators so this a of T will give rise to L naught and the B of T will give rise to L plus and the minus L and this form is a sub algebra of the fukken form whereas our algebra which is isomorphic to this SL to our algebra for
05:17
which we have a very well understanding and now let's do a concrete example to see what really happens so this is the general form and we in the following we choose just n equal to 1 so in general we have say 1 plus 1 minus cosine 2pi all x over L and n equals 1 we can write it as sine 2 times sine squared PI x over L ok and so the system we consider is an open chain with a conformity and
05:48
vary in the boundary condition at x equals 0 and x DX to L and the Hammad in each cycle the Hamiltonian will change between H 1 and H naught H 1 is the I equals is this so-called SSD Hamiltonian and H 1 is our four media Hamiltonian for the general system and the tuning parameters are T naught over L and the T 1 over L which had a time period for which we apply H 1 and H naught
06:19
okay so physically what this h1 does is that the quasiparticle can move to the left or to the right with the diminishing velocity near the boundary okay and with this concrete model you can show that in the high-frequency regime there are no fixed point for the quasi particle motion and if you calculate the energy profile you will see that it just also like keep oscillating in time so means the system is having a non heating dynamics as we lower the frequency you can find that
06:51
there will be two fixed points for the quasi particle one for the left mover and the other for the right mover then the quoted quasi particles can get accumulated at this two fixed point and the form two energy Peaks if you calculate the total energy you'll find that it exponentially grows with time with the coefficient proportional to central charge so it's universal for any CFT so now the system is in heating phase and you can further show that whenever you have two fixed points you'll find that the system is in the
07:23
heating is in a heating phase so there are one-to-one correspondence between having feet having fixed point and having a heating dynamics and further you can show that you can also find some interest in phenomena in the entanglement pattern a namely if we calculate entangled entropy for a subsystem and the entanglement will linearly grows with time as long as the subsystem includes one of the two energy peaks and it doesn't grow at all if if
07:53
the system doesn't include any of the peaks and and here the slope of the linear grill also proportion to central charge which is also universal for any CF t and you can also show that to the total energy of the system and the half system entanglement satisfies this universal relation that doesn't that is true for both chaotic random and the quasi periodic driving and now let's try to understand what happens for general Earth so for general we just replaced it
08:25
that h1 SSD Hamiltonian with each sub and we just put a unhear and now this system actually can be understood as an array of n copies of a equals to one system meaning that if you look at the sub-region of length L over N in the sub region the Hamiltonian looks exactly as we have a h1 like for this sub system okay so with this it's it's quite natural to believe that all the features
08:56
above still holds in this January and indeed by calculation you can find that we will have a two will have two peaks for in each of the sub region so in total we'll have two energy peaks and you can further show that there will be entanglement for every next and nearest neighbor sorry for every nearest name every two nearest neighbor energy Peaks but if you look at the energy Peaks that are next a nearest neighbor with each other they don't share any entanglement and this set up also shows the dynamical
09:28
Casimir in fact more clearly meaning that if you look at the energy expect the energy density away from the critical point it actually gets even smaller than the conventional Casimir effect so without her down turning on the driving it the Casimir energy is this amount but with a driving it would be multiplied by this N squared so it gets smaller and in the real lab noise noises are inevitable it's natural to ask what happens if the driving frequency has some fluctuations denoted
10:00
by this delta T and the answer for that is the non heating phase we immediately disappear meaning that if we start with the parameters that are deep in the heating phase and turn on a little bit fluctuation you will find that energy grows exponentially with time and the entropy grows linearly with time and this absence of non heating phase can be prudery rigorously mathematically which is related to the first in berg theorem and for the heating phase on the other hand for the heating phase you can check
10:32
that all the spatial features like the anterior peaks and the entanglement structure are still robust form all the rate fluctuation okay and now we want to ask what happens if it's between periodic and random namely what happens if we have a quasi-periodic driving what is the phase diagram and the the answer to this question is is the following so we can map the dynamic so we can map the operator evolution to a one dimensional quasicrystal by the
11:03
transfer matrix method meaning the transfer matrix method the transfer matrix for the wave function can be mapped to the operator evolution in the CFT side so on this side it's consecrated crisscross the crystal this side is the quasi periodic driving in C of T so if we have a head if we on this side you if we find the system have bends it means the CFT is in a non heating face and if on this side we don't have any bends meaning the system
11:34
is localized means that the CFT is having a heating dynamics and our experience with the core secrete crystal immediately tells us the existence of a non heating phase actually depends on the realization of the quasi periodicity for example if we use the Fibonacci type of quasi periodicity we don't we won't we won't be able to find any measure nonzero sir we won't be able to find a lot of sorry
12:05
yet we yeah we want to be able to find an that heating phase and now okay now just close with this summary and some outlook so this system provides a tool to to ancient to realize a very robust energy and the Intendant pattern in the feeding face namely we have EPR pairs accumulating at the two energy peaks and it might be useful for quantum
12:35
computation and it's interesting to solve the problem for generic F of T and also for higher dimension and also interesting to consider some holographic interpretation with that I would thank you for your attention

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