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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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