Thanks Martin for
the nice introduction Hello everyone
today I'm going to talk about
local temperature on thoughts.
Before going into details.
Let me set the stage and
motivate the problem first and
then all walk you through the details
of this problem step by step.
What is the main problem the main
problem is quantum mechanics basically
in quantum mechanics we war with
states which are victorious and
operators which are matrices and
the dimension of those matrices and.
States or victors depends on the number of
particles that we have in the system for
example let's consider the highs and
Berg model of five spins so
we have Heisenberg interaction between
of these spins and any operator in this
space can be represented using
chronic curtains of product.
As you see of Sigma Z.
A point two for
example can be shown in this way
Sigma why a point five can be
shown in this way and so on.
And because at each point the Hilbert
space is to buy to power the matrices
are two dimensional matrices then for
five spins we have a Hilbert space
which is two to the five dimensional.
It's very easy today are going to lies the
Hamiltonian associated with this system
and we can do it.
We can consider more spins and
spins in general and
the dimension of Hilbert space for N.
spins would be true to the N.
and if N.
is below thirty six you can do it on
your laptop of course you need to use
some symmetries But
if you use also metry than
thirty four five is still
doable on your laptop but
beyond thirty six is not doable
with the best available computing
clusters you can go up to like forty
something but not beyond that so
we have problems we cannot solve exactly
quantum systems which are slightly big.
But what can we do what are the possible
solutions of such problems
there are a few possibilities
one is we can use Integra
billet in the system meaning
that instead of solving this big
space to to the end we can solve some
much smaller of matrices matrices
was the mentions or put in only of the big
IN LIKE AND OR and square or in Q.
and by combining eigenvalues and
eigenvectors of those.
Matrices we can get the results for
the actual problem for
the two to the end dimensional says.
Three formulas and
a few bows ons are examples of that.
We can also use perturbation
theory of an interaction is we and
if we ignore interaction we can solve
the rest of the system for example
we can start with three fermions and
add some terms which are very small and
then we can use perturbation theory this
is what we do in quantum field theory or
calculations and related methods we
can also use their usual way function.
For example mean field theory does that or
dynamical mean Field Theory D.F.T.
density functional or
variation among the color of these or
some methods based on
variational wave functions.
Of we can also truncate Hilbert
space identify low energy and
throughout high energy modes and
keep just the low energy part this is what
we do in India more do tensor products
state peps Miral and related that.
We can also transform our
interacting problem into space and
on some over for free bows ons
which can be solved very easily but
we have to some over
many many of such songs.
And this some can be up can be
evaluated using quantum Monte Carlo for
example using Metropolis algorithm.
But all of these methods except for
the first one.
Of some limitations we cannot apply
them in every place the first method can
be applied only to three systems but
the other methods like D.M. or G.
is considered as an exact method but
only when the system is quasi
one dimensional meaning that the system
can be extended a long way direction but
the width of the system along
what has to be a small like for
sites five sites things like that and
quantum Monte-Carlo can be applied
only to systems that don't have sign.
I will talk about sine problem later but
only to specific problems we can
apply it at zero temperature for
other problems we have we cannot
go to larger systems or we cannot.
Decrease temperature below some thirty
short so all of these problems have
limitations and it's very difficult
to solve quantum problems.
And even if we can solve the systems for
larger values of and like and around
four hundred five hundred we still need to
take the thermodynamic limit we need to do
find its size scaling for example to
see what happens in them and then.
In this talk I want to tell you about an
alternative method that we have to develop
recently and
we call it local temperature on sites and
you will see what we have given me but
the idea is quite simple if we
have a system let's call that M..
The dimension of the Hilbert space scales
exponentially with the system size.
And if we know the density
matrix associated with this big
system then we can find
average of all operators or
correlation functions just by using growth
in principle we can do that in practice
we can't if if V.M.
is big Well what is row M.
formally we can rule right through M.
as the minus beta zero eight you
waited by some constant and H.
M.
is the Hamiltonian of the system which is
I consider to be local so
we have just local interactions and
local happenings in
the system as you see so
this is the total density matrix
that describes the whole system but
now let's consider a sub system inside
this big system let's call that.
A I assume a is a small enough
such that dimension of A D.
A which is much much smaller than
the mission of the whole system
is below the limit that I
talked about in the past slight
like to the thirty six something
like that that we can still handle.
OK if we know the reduced in-city matrix I
will define all these quantities later but
if we have produced a city matrix
associated with a whose dimension is
the A we can evaluate the average and
expectation value of all operators
that are defined inside a.
Just by knowing RHO A We don't need to
know root total just a would be no.
And we can always write to the minus K.
you can define K.
to be minus the log of a.
And in general you can expand K.
in terms of electron operators it can
have what drastic terms quartic terms and
higher order terms.
But I will show in this talk that for
systems for
Hamiltonians that are local and
have translation invariance.
And when they are dept
we have this on sites K.
can be written as some
local inverse temperature
some some functions some smooth function
times the hopping the hopping amplitude
that we had in the Hamiltonian plus
some function some another function.
Times the interaction that we
have in the Hamiltonian so
it's related to the Hamiltonian density
somehow and if we know what the form of K.
S.
We can easily evaluate these quantities
because dimension of row is small.
Why is this interesting because
Row A is embedded inside a much
bigger system so what we're doing here
is a thermodynamic limit calculations.
And the form of data that.
Showed previously is like this
near the boundaries of a it
of vanishes linearly and for
away from the boundary much very.
Inside the bulk it becomes more or
less equal to one over
the temperature of the entire system.
Which.
The Hamiltonian has this for.
The.
Reduced in-city matrix and
the distance has to be measured.
As you see this is then tangling surface
or the boundary of region a this is X.
a call zero X.
a call one X.
acquittals one point five X.
equals two X.
two two point five three and so and for
this onsite beta at this
point has to be evaluated for
exit polls one point five here to be
evaluated three at the middle point
OK what are the applications of
this reduced in situ Matrix or
local temperature on first of all by
solving a finite subsystem subsystem.
A is finite we get information about.
An infinite system basically we get
local information about an infinite
system we cannot talk about
operators who is this the nce or
of bigger than the size of A but if they
are both inside a we can talk about.
And this paradigm can be applied to any
of the previously mentioned approach it
can be applied to quantum Monte Carlo and
I will show that it will fix the sign
problem it doesn't fully solve
the problem but it makes the behavior
much better exponentially.
And it also enhances the more India Margy
the best Bond dimensions that
people can consider the moment is like
ten thousand five thousand ten thousand
that's state of the art calculations but
we can increase that
up to one million if he used
this idea I will show you this.
OK And this is the outline of the rest
of the talk I will talk about reduced
in-city matrix I will define
what you do since the matrix is.
I will discuss how we can find
reduced density matrix and
present an algorithm for that then I
will talk about local temperature.
And.
I will.
Tell you about applications of
local temperature and sites and
show many examples.
And of finally I will give some numerical
demonstration of local temperature on so
that's why we that's a good approximation
OK what is a reduced in
situ matrix first of all.
If you have a quantum system the quantum
system can be in two different
situations in the first case it
can be a pure state meaning that
we can describe the system
just by a vector.
Like ground state size zero and
the average of all operators from quantum
mechanics we know that we can
find them in this way but we can.
Write this quantity as
trace of time zero as well.
The road is this projector operator but
there is no need to do this
when we have just a victor.
Working with vectors is usually easier but
we can formally write it
in the really as well.
Now let's consider we have another state
which is a linear superposition
of many states like C.
zero size zero plus the ones I want and
so of.
The density matrix in this case can still
be written as five one wife I want and
such states are still pure although
we have some over many states but
it's pure because we have some the victors
and again the average of operators can be
written as trace of over zero one
which is formally written in this way.
Now let's consider the second situation
let's consider you have an ant song like
like a thermal and some and
the system can be in state or
in density matrix size zero size
zero by chance one square or
side one side one weights chancy one
square and so on this is a different
situation now I sum over
different density matrices not
worked and this is different from
five one five one the very important.
And the average of operators in this
case is different from this case here we
have of their going to terms as well but
here we just have their going out so
these are called mixed states and
associated with mixed state we
can distinguish mixed states and
clear States by using
the so-called entropy for
normal entropy the entropy
associated with pure state is
always zero because if
the diagonal is density matrix
all elements on that they're going to or
zero except for one of them which is one.
And one times log one is zero but for
mixed states the they're going
to elements are not all zero or
one some of them are non-zero and
less than one so
the entanglement entropy in
this case would be nonzero or
entry so we're.
Reduced in city matrix Now let's
imagine we want to find the expectation
value of some operator that
is defined inside a only so
all can be represented in this way
it's zero which acts on a non-trivial
out say a doesn't do
anything is just identity.
In this case average of zero
is obviously trace of a B.
the entire system times times
the density matrix associated with
the entire system but
because over has the simple form we can
directly take trace over beside
because doesn't act on the B.
side and write this as trace of a a I
was this part this if we call this
part through a then average of all can
be written as trace of a zero a row a.
Row A In this case place for
the row of density matrix.
So we call this reduced in-city matrix and
if we know reduced in city matrix
then we can find the average of
any operator that lives inside a.
That's.
The good thing about density matrix and
the dimension of rho is
much smaller than really.
OK if Roe A.B. is pure the question is.
A pure two for example if you start
with ground state and we take trace
over the density matrix associated grounds
if we get a pure state after we take trace
the answer is no we can start
with a state which is pure so
the entropy zero and then we take trace
we obtain a and for Row A The entropy
is non-zero so we generate entropy by
tracing out on the use of freedom.
Let me give you one example
let's consider two Spence and
the interaction between these two spins
is his number interaction as wonder
as to its ground state is
the single state as you see.
And.
For this state we have the vector
we can find the density matrix
reduced in situ makes his how can we do
that it's very simple we just need to
do this you need three transform ation
this is subsystem a this is subsystem B.
We just need to change if we can't
on substate be too broad a store and
stop state B.
This is a unitary transformation so
no information is lost by why is this good
because this thing is a matrix is a two
by two matrix and the being a four by one
victim and then reduced density
matrix is simply this Times itself.
So reduced density matrix in this case
would be its bar times as bar which is
a matrix now and at the end we get
this as the reduced in-city matrix but
this reduced density matrix has two
eigenvalues one half and one half so
the entropy or entanglement entropy in
this case is locked to we started with
a pure state with a single state whose
entropy total entropy was Europe but when
we trace over the right part we get locked
to entropy so we have generated untruth.
Now let me talk about applications
of the reduced city matrix.
The obvious.
Application is that reduced in
a city matrix has all information
about subsystem.
Any operator that is defined so.
We can find its expectation value or
correlation function of different
operators if we know route.
And it's the mention is much smaller than
the dimension of the entire system and
that's another advantage and
by that I mean we have D.N.A.
and we have D B We assume a is much less
than that entire system so da time D.B.
which is the size of the total
system is much bigger than a square.
And because of that working with
subsystem a directly is much easier.
And finally reduced density matrix can be
used to truncate the Hilbert space and
operators in subsystem say why should
we use reduced density matrix not
the Hamiltonian the reason is I just
mentioned the density reduced and
city matrix has all information about that
subsystem because it has all information
so we have to truncate based on that
we need to they agonize the density
matrix we have the total wave function
from that total wave function we can
do this reshaping and by reshaping we
can find the reduced density matrix
then we can dabble is reduced density
matrix it has some values and
our values are between zero and one and
let's imagine we have sorted out in values
and we have them first hold Lambda Chi
plus one is less than thirty whole and
everything lambda want to learn the.
Above that we keep all these lambda us and
we throw out the rest of them
and we can put all these are going victors
next to each other and instead of operator
a subsystem a we can work with ot all
day where all till does truncated and
its dimension is even smaller
than dimension of A or.
So dimension of the Row A which
was D A has been truncated
to cry now so
these are Cry dimensional operators.
And we can show that the correlation
functions of these truncated
operators is the same as correlation
a function of the on truncated operators
plus corrections of order epsilon So
if epsilon is small enough like
ten to the minus five we can
make sure that we have kept.
Important information about
reduced estimations So
we need reduced in-city matrix
to truncate Hilbert's and
another application of the reduced
in-city Matrix is that.
In the standard approach we usually
consider periodic boundary condition over
the system and then we pretty Other
consider periodic on the condition
the maximum distance along
extraction is not L X it is X.
over to because these two points are
identified they are close to each other we
have to wrap it around so
that's the maximum distance but
if we use local temperature on sites or
reduced in city matrix these two
sides are not identified any more so
the maximum distance that we can use for
like fitting extracting scaling
dimension and global is X.
Not Alex half so we get in for
each the mission it's twice in practice.
OK The big question now is how shall
we compute the reduced incident Matrix
how can we find it there
are many ways to find
reduced insta matrix the obvious
Where is exactly I don't zation
which means you have to
solve the big problem first.
That's the stupid way we first find the.
Wave function associated with a very
big system then we take trace
over degrees of freedom which are outside
and at the end of the day we get through.
But what's the point of working with a if
you have already solved the problem.
Another method that I will.
Describe in this thought is a variation
of principle there is another method to
fine reduce the city matrix and that of.
That using D.M. orgy in the a more G..
We use another idea to find
the reduced in-city matrix.
How do we do that is the following
let's consider this big system.
And let's imagine we are interested
in the reduced instead to make shift
associated with these sites.
Obviously we have to solve this
big system in order to find row
associated with this part exactly but
what we do India Margie's that in
the infinite steps of the a more G.
is to approximate the reduced in-city
matrix of this big system with the reduced
in-city matrix of this smaller system so
we take Row A and
we put of the same number of sites on the
right we solve this part completely and
then we take trace OK this is much better
than solving this infinite system but
it's still difficult because we have
to work with operators which are D.
A squared dimensional the A for
the left for
the right is a still difficult and
it is an approximation
it's not correct because it's not
natural system that we're interested.
OK for free for Myans we can
op then reduce the city matrix
exactly and very easily how
can we do that first of all.
Note that the total density
matrix of the minus beta eight H.
is the hopping matrix T M and c and c M.
And Green's function is
the inverse of each of the beta T.
plus identity that's the green's
function that we can solve and
because this is a non interacting
problem these matrices by and
is the number of sites in the system
we can do that very quickly.
And of Similarly you can write wrote
A as it is a minus a overs E.A..
And because it's a non
interacting problem K.
has only quartic terms quadratic
terms it doesn't have quarter
but what is ha we have a physical
constraint here what is the physical
constraint I just told you how to
compute Green's function for the system.
We can also use reduced in-city matrix
to find the green's function and
these two should match with each other
these two answers should be consistent
with each other so the GA which is up pain
from this method has to be equal the G.
that we have obtained from this method but
what is the G.
of ten from this method
its identity divided by H.
plus identity right because
this is the beta T.
I have each and if you equate these two.
Because this D.
you can solve.
That equation and H.
has this for the simple so we can find
reduced in-city matrix entanglement
Hamiltonian or what some people call
modular Hamiltonian in the story.
How both reduced in a city matrix of
interacting firms that's the hard problem
because they are known and
took very hard to find.
As you remember from
a statistical physics or
from quantum field theory we
can map thermo of problems
into path integral where the time
direction imaginary time direction
represent represent temperature and
we have this for
so the reduced in-city matrix is some
over all degrees of freedom there is no
degree of freedom left it's just a number
of function is just a number and
the density matrix is a density so
it's a matrix and
it has two degrees of freedom it's like a
round tube tensors So it has incoming and
outgoing degrees of freedom it can be
formally written as the same path integral
but with two constraints the constraint
is that side one side at T.
call Zero has to be equal to this one and.
To be side with those two constraints
we can opt in total density
matrix as well how about reduce the city
matrix for reduced in Semitics we have
to do something in between we sum
over all degrees of freedom which
are inside region be
outside a sum over them but
we keep in coming and
outgoing degrees of freedom for
region a only for region and
this is another formal
representation of reduced in-city matrix
but let me show them in a simple cartoon.
This is the partition function by the way
this is an infinite cylinder or a.
Taurus so don't consider these
boundaries it's like they don't exist so
there is no boundary left here and
all this close or compact many.
Fall is just a number because
there is no degree of freedom left
this is partition function.
If we make a caught.
Along called zero in this
system we get two edges and
those two edges each represent one degree
of freedom so this gives us a matrix
because we have two degrees of freedom
this is the total density matrix
and this is the reduced in
city matrix the density matrix
associated with region A because
we have nude some part B.
of the system side B.
but we have kept side a open so
we sum over all degrees of freedom
except for these two edges.
So on to the first I didn't do anything
but now let's imagine temperature is zero.
If temperature is zero we cannot
distinguish between this system and
this system remember the circle
friends of this system is beta and
beta goes to infinity when temperature
is you want to work temperature.
So when this length is twenty we can't
distinguish between these two manifold so
basically this is equal to that but
what is this path integral
this path integral can be imagined as an
evolution operator which evolves this red
edge into Green EDGE But what is
the evolution operator that does this for
us it's like to pyro Taisha and if we
know the generator of rotation operator.
That gives us the reduced in-city matrix
because that's the solution operator
the the angle is to part and
the the Says the rotation operator but
what is the rotation operator
rotation operator is nothing but X.
times P.T. minus times P.
X..
And if we evaluate this as T.
call zero or
tell called zero we don't have the second
term we just have
the first with what is P.T.
it's Hamiltonian operator and
we have to put firmly the last year or
speed of light years well
this is exact when we have.
Lawrence symmetry if we have
conformal symmetry the mapping
that I just mentioned is
can be always done even for
find it before we can always map this man
to fall into this and then write this.
To the minus two Pike K.
operator.
Book So for conformal systems and
for some more and
symmetric or Lorentz invariance systems
this computation this calculation is
exact This was done like forty years ago.
Is called regular Hamiltonian but
in a different context I mean at that time
they were not interested in untangle they
were interested in other things like
what is the temperature that you feel when
your distance from a black hole is X.
or from the horizon basically X..
So that is called only to temperature or
group problem.
To.
OK So the bottom line is that the reduced
in-city matrix can be retained in this way
as I promised before it's like each
other minus Hamiltonian density times
some smooth function very smooth
function decays linearly vanishes
linearly when we get close to the boundary
the way from the boundary groups and
if we have find a temperature
it will saturate and
the distances they mention should
be measured in this way but
what is that Miltonian density that I
just mentioned if you have hopping model
in one day for example this is
the Hamiltonian for each term for
each fixed I We call this
operator Hamiltonian density so
that's the density of Hamilton and if you
have Hobert Albert interaction in two D.
We have four different local
terms we have hopping along X.
hopping along why we have onsite
interaction and we have chemical potential
so we have four different local
terms in the Hamiltonian.
And the form of beta at
finite temperature instead of
growing nearly up to infinity near one and
the other
which eventually becomes something
like that it will saturate at some.
Value and that value is one over to zero
the temperature of the initial system so
this is the profile that
we expect when we have.
Find a temperature in the system OK.
Now let me tell you about
the algorithm to find the.
Local temperature onset I told you that
there exists some function like beta
which vanishes linearly near
the boundary and becomes a constant.
Well inside the ball but I didn't tell you
what the exact values of that function is
in order to find that we can look at
this as a variation of problem we can
consider beta to be some smooth function
with a small privateers like it's we can
consider its slope to be variational and
we can put other variables if you wish.
And what we do is we can find
the average of operators
with respect to this variation August for
this initial guess.
And.
We know that because Row A is
a reduced in-city matrix associated
with a translation O.E.M.
variant system the system has to exhibit
translation invariance which means
that if I take operator at X.
and find its expectation value it
has to be equal to the average
of the same operator but
at different point so this should be X.
independent that means
translation invariance but
if you have not chosen beta properly
you cannot satisfy this constraint so
you play with beta such that
this construct is satisfied and
when you satisfy that constraint you have
reached the solution you can also add
a cost term to your Hamiltonian and that
energy what is the energy of the system
because we have translation in various
energy is nothing but the SOM over of
different local terms at one
side OK at some specific X..
Which of X.
you want because we.
I have made sure that we have translation
in Vegas already you can always
also add another constraint which is
the entropy or entanglement at U.B.S.
of fitted with the region and
the total cost front cost function that
you have is something like this this is
something similar to the free energy is
like we have minus like temperature times.
And trippy free energy
subject to this constraint so
we try to minimize the cost function and
the solutions of this cost function or
these bake up for us.
OK.
For example if you have the Hubbard
model we can consider the.
Energy per side to be average
of these two operators.
And the four operators that
we can consider to inforce
translation symmetry of can be
chosen to be these four operators
we can make sure that these
exhibit translation you know.
You can do better than that but.
That's good enough OK so
what is that we cations let me.
Go into this part.
This is.
A system which has fifty sites in it and
we have
consider periodic boundary condition
with the periodic Bulmer condition of
the red that you see that you have more or
less uniform of value for
this quantity this is one dimensional
three for me and change by the way.
And if you consider open
boundary condition for
fifty sites you see many
oscillations in the system so
it's not translation of the invariant
when you have boundaries.
And the Blue Dogs blue line which
can be barely distinguished from Red
is the local temperature onslaughts for
fifty sites what I have done here is that
I have considered some profile for
beta and played with it slope and.
Change the slope on pillar of make sure
that I have translation invariance and
as you see it agrees with this but
there is still some difference and
you may wonder why I don't exactly get
that remember local temperature on South
is related to the thermodynamic limit this
red line is associated for
fifty sites fifty sites and
infinite the big chains or
not exactly the same in the next layer.
I have zoomed in over.
Those two things this is
periodic boundary condition for
fifty site this is local temperature and
starts.
And this is two hundred sites
two hundred sites is closer to
thermodynamic limit than this and
as you said sea local temperature and
such is already much bigger than
periodic boundary condition for
that size it's much closer to two
hundred sites and I just consider.
Local form of the reduced in city Matrix
just nearest neighbor hopping but
with different values of beat
you can do better by considering
next nearest neighbor hopping or
certain thing is.
A pain this value exactly like playing but
this is already good you see
the difference is in the fourth that.
OK as another application I
have considered eight sites
of the one dimensional Heisenberg chain so
I have an infinite
Heisenberg chain but I focus on
just eight sites on a subsystem and
for this subsystem I start with this
guess for the reduced in-city matrix
I just consider quadratic terms as I
doubt is to do to suit the symmetry and
I have not considered higher order
terms just with this simple answer and
I played with it such that I
get the optimum point where
this is the correlation this is
the optimum primitives that opting for
j I.J. as you see its previous local
only nearest neighbor is big and
all other or very small complex.
And as you see the S I that
S J Has translation invariance
as one that is two equals as two that
is called is three that is four and
four other values you can see that and
you can use this value to octane
the energy per site and energy per site in
this case that you up to is this number
which magically calls more or less within
the Arab or to this number and this number
is the exact answer in the thermodynamic
limit that's the best dance out solution.
So you can opting thermodynamic limit just
by exactly a condition of eight sites
which can be done in a fraction of
a second OK if you already know this but
the you can wait like ten
seconds to obtain this.
Result.
OK with just me those data
the data that I showed you
we can find the entanglement entropy.
As a function of subsystem size we have
eight sides so we can find in time
at entropy associated with one side
two sites up to eight sites and
this is what we get this slope of
this is this number divided by three.
And this number is called central charge
theory predicts it to be exactly one
what we are pain is one point zero two and
again remember I didn't
consider the higher order terms
just consider that simple to but
if you consider higher order terms and
if you consider instead of sides like
twelve Sides which is still doable
even sixteen sites you get one.
There easily.
How about the scaling dimension of
operators This is S I that is J But
this is log log plot.
Because it's an anti for magnet you
have all solutions like this but
if you just consider one three and
problem and
connect the slope is point nine seven and.
Theory predicts it to be one
the scaling dimension so
you get the scaling dimension
very easily just a solving.
In other ministers like D.M. or G..
If you want to solve the exact
scaling dimension you have to go to
many sites like three hundred sites and
wait like many hours to
obtain this result OK
as a second example I want to talk about
quantum Monte Carlo and show that this.
Method somehow.
Reduces the severe ity
of the sine problem.
First of all what is the of
idea behind quantum Monte Carlo
first of all if you have Harvard model for
example we have an interacting system but
there is a very nice trick called Harbor
district tunnel vision transformation
which transforms our interacting
problem which is exponentially hard
to solve in two and
on some over many many many
non interacting problems which each of
them can be solved very quickly but
the problem is we have many of them the
number of terms that we have here is two
to the end times number of slices that you
have considered the time direction and
this is already much bigger than
the Hilbert space the size of the Hilbert
space was just to do the and now we have
to work with such such a big number but
there is a very clever idea
to evaluate this song.
And its Monte Carlo and what Monte Carlo
say is that OK these weights
these Peiffer by if they are all positive.
Then we can use Monte Carlo but what
is the idea behind Monte Carlo itself.
Monte Carlo classical or just Monte Carlo
method is nothing but taking into groves.
When we want to take in take rolls we
usually use Riemann song but remains some
is not very clever if your search
function decays at long distances for
example for the girls and wave function we
have to spend so much time at its tails
which are already very small and
on important you can just ignore them so
in Monte Carlo using
Metro Police algorithm we first.
Identify the important
part of the function and
we just spent time near the peaks or
near important parts and
by doing that we evaluate
the sum of very quickly but
there is a condition here the function
has to be positive definite and
it has to decay to zero quickly enough and
it works we just need a few
thousand samples but.
What if you are still function is not
that nice What is your function is
like sine function which fluctuates
strongly its sign fluctuates for
such functions that fluctuate inside
the convergence is very slow and
you have to consider many many
many samples like millions or
billions of samples and
this is called sine problem because
your function is oscillate for
the Harvard model only at half feeling and
when you have nearest neighbor
hopping this they are all positive for
other values of chemical potential
you can show numerically or
theoretically that these votes
are not positive definite some
of them are positive some are negative so
we have so much problem.
And because of that quantum
Monte Carlo cannot be applied to this
problem way from have only at half
wing and way from happening at large.
Temperatures.
Temperatures we still can
with low temperature.
OK.
In Quanah Monte-Carlo people have shown
that the average sign or average sign of
this quote fission goes to zero as each
of the minus one over temperature so
if you try to ten temperature to
zero Beta becomes infinity and
average time become zero times the volume
of the system if you try to study
a larger system again you have
sign problem times some constant.
And how many samples do you need
to make sure that these small
it's scales as one over the average sign
so the number of samples that you need
to consider is this big debate as you know
this is an exponentially large number so
this is an N.P. hard problem and
there is no general solution to that.
And here I don't want to present
a generic solution what they want to
say is that when we work with
local temperature onslaughts.
Instead of applying quantum Monte Carlo
to the total density matrix
we can similarly apply a quantum on
to called the reduced and matrix and
if we do that then the average sign just
like the other problem behaves in this way
but instead of having baiters
zero we have beta average
where beta average is
the average of the beta profile.
And you can show that beta average is
larger than the minimum of beta zero and
pi times and X.
over six.
If your system along X.
direction I assume along with other.
Long X.
direction.
If your system sizes and X.
subsystem size is an X.
It cannot be bigger than this even
if you assume that the temperature
of the entire system is exactly zero.
But it's such a way to sample which
means the average sine goes to zero
exponentially with this subsystem
volume not with the entire system so
you can still take the thermodynamic
limit by keeping your subsystem finite
and the average sign saturate
as a function of because you so
you increase your betas you know beter
average at some point will plateau
doesn't change and at zero temperature
the profile of beta is like this it's
values PI over four times and X.
and this is one over to
zero finite temperature and
this is that average sign
that we have obtained for
Quanta Monte Carlo calculation
with local temperature on sots and
the periodic Panu condition this is for
six by six sites
at chemical potential calls to minus
two point sixteen which gives us this
feeling Fraction It's like one eight The
Open which is the interesting doping in
of high temperature superconductors that's
the reason we have considered this and
we have considered you to be six and
as you see the.
Value of beta beyond six gives us
moralists zero average sign so
forth so we cannot study the system
beyond five equals five or
six if we use product manager condition
but if we use local temperature and
such as you see platters at some point
you still have some problem you know
every sign is not exactly one but it's
a finite number and it remains more or
less constant This is where the six
by six of us system this is for
eight by eight the number of samples
was like just ten thousand so.
You don't it's not as this moves at the
other one because the system sizes bigger
a number of samples is like and
thousand it has to be bigger to get the.
Plots But anyways this is for
local temperature and
starts again as you see it
tends to plateau at some point.
And for the other one it
becomes zero very quickly.
So it seems that.
Some problem is not a big issue if you use
local temperature which is a very good
news because though cover model
describes I think most people believe
that it captures the essential physics
of high temperature superconductors and
there is no general consensus
on the theory of high T.C.
Kuper it's because we have not been
able to solve it by any method we
have applied different methods but we
cannot rely on those methods for example
quantum Monte Carlo the temperature that
we have considered so far is like four or
five Was that mean the hopping
amplitude in Q prates
is around one forty million electron
world which means fifteen hundred killed.
And when we say Beta is five it means
the temperature of the system is fifteen
hundred divided by five it's like three
hundred and the transition temperature
of superconductivity the optimum value
is like one hundred round one hundred so
three hundred is already much above the
transition amplitude of superconductor so
we don't get any information about
the ground state or about the face because
it's the temperature is too high we
have to consider beta to be at least
like twenty or thirty to access
the ground state properties so.
Other methods have been applied like him
already but the West is quits narrow
it's not in good so
OK this is another application for D.M.
or G I just I mentioned
previously that indium or G.
the idea to find reduced in city
matrix and then use it for truncation
is to double the system we have subsystem
a we put something similar to it and
we solve the entire thing and from
that mean for the reduced A to make X.
for the top two.
This is of local temperature
answers applied to D.M. or G.
as you see the central
charge that we get here
is this number which is very close
to the theoretical prediction.
And.
And why is that so
because India Margie they said we have to.
Solve a much bigger system in terms of
the Hilbert space dimension to find
the reduced insta matrix but if we already
know what the reduced insta matrix is
we can skip that part and
that's the expensive part India Margy not
more than one thousand nine percent of the
fraction of the time is spent in lunches
there going on zation and we can skip
that that's a very good news and
if we compare the computation
time of the year more G.
With respect to the local temperature and
starts you see that we get so much gain
and it gets even better when we go to
a larger values of the bond dimension
five hundred almost five hundred one
thousand three thousand four thousand and
as you see the ratio becomes bigger and
which means the density of the.
Bunn dimension can be considered as big as
these numbers the exact number depends on
how many symmetries you
have in the problem.
So one million is possible that's
that's very good because the current.
Typical values of Bonderman is around five
to ten ten thousand we can increase that
a factor of one hundred which is very
good and the larger the value of.
The better accuracy that you
get about the ground state.
And now let me show you some
very few cases about the.
About this method I said that the.
Beta has this for which is.
The X.
but the slope depends on
primitives of the problem for
example if you have some stagger chemical
potential in one that gives you a massive
theory and this slope starts from
the theoretical prediction but
it becomes smaller and
smaller if when you turn on mass
when mass is like twenty times bigger
than your hopping it's quite small but
local temperature on such this is still
applicable even for such a massive point.
And this is another formula that
I mentioned I said that beta
is to PIO over V X V X when the system
is math massless where V.X.
is the Fermi velocity if we change
chemical potential in this one dimensional
not interacting chain we
can change the velocity and
as you see this is the slope
versus want to be X.
and they match quite nicely.
OK this is the local this is
better that we have opting for
a two dimensional system which
has surfaced doesn't have of.
Lorentz symmetry for example.
And as you see the beta for
different components for hopping along X.
Y.
and for the on site mass they fall more or
less the same care but there are some
differences and here I want to
convince you that the time it Hamiltonian
or reduce the matrix is local and
in order to show that let's consider
whether we have long range hopping along
the boundary or not and whether we have
longer hopping normal to the boundary or
not here this is the long range
hopping along the boundary.
In this direction a long way direction and
as you see it's pretty
local only these three numbers
are north and it decays very fast.
And along X.
direction it's even more local
only the onsite term and
the hopping term are non-zero and becomes
zero very quickly you have already you can
have also consider a honeycomb model which
has emergent Lorentz symmetry because we
have to be there are cons again
as you see we have very local.
Sorts the boundary can be arm chair or
the exact this is for the zigzag edge and.
This is a local temperature this is heat
heat for beta not temperature as you
see for different boundaries we
can find the local temperature.
And.
Another way to very far these results
is to use quantum Monte Carlo there is
a an algorithm proposed by
Tauren Grover in two thousand and
thirteen which gives us reduced in-city
matrix exactly no one had tried this
algorithm before but we tried it and
this is the result that we get for
hopping and for on site of a harbor term.
Of four different values of you
different temperatures different system
sizes you see and they all seem to.
Follow the local temperature answer.
And in order to see the locality
of hopping along the edge
this is the hopping between this site and
nearby sites you see only to nearest
neighbor and to this site we have.
Like point one hopping to all other terms
it's practically zero so it's a local.
Density matrix and as you see.
For the middle point we have
hopping to these points.
Nearby sites.
This is for another case another
you another temperature.
And the last thing that
I want to mention is
comparing results of local temperature
onslaughts and periodic boundary condition
an open boundary condition for
interacting systems this is.
For.
Periodic boundary condition with
Monte-Carlo yellow line is for
open boundary condition as you
see we have so much for air it.
And blue is for
local temperature on sites so
they match mourners nicely especially if
you go it's likely away from the boundary.
It's hard to distinguish them so if it
is consistent with the standard methods.
And this is for the on site hopping term
average of an eye opener down I
apologies this is not shown here for
this value of you and for
three different boundary.
Conditions.
This is for you it calls for
local temperature on sots and
preoccupying new condition for
a much bigger system.
If we consider more samples then
the convert just becomes better for
both but they are pretty close to each
other this is for the hopping term.
Along X.
This is hopping along why and
this is for the on site.
Interaction.
And as a conclusion local time
pressure on sorts is a novel method
that can up pain thermodynamic limits but.
Studies define IT system that's
the whole point here it resolves
sine problem for
system sizes that we are concerned about
like twelve by twelve systems or
sixteen by sixteen sides and
the very interesting question that
we are currently working on is.
Can be opting phase
diagram of the high Q C.Q.
plates it seems the answer is yes
we can using this method but.
We will publish these NATO hopefully very
soon and you will see the the answer.
And this method can enhance
the more general lated methods
we can consider gigantic
one dimensions if we.
Use local temperature and
it can also be applied to other fields
like lattice Q CD which also
suffers from quantum Monte Carlo.
It seems that it can work in this case
as well especially that in this case we
were in symmetry as an exact match.
Or for massless for Myans we have
conformal symmetry and all one can
also go beyond the call temperature
on sites I said that the local or
the modular home of Tony or
entanglement Hamiltonian that gives or
reduced into the matrix is local you can
add terms which are not quite local and
don't exist in your Hamiltonian like
next nearest neighbor hopping or
third neighbor hoping the cost
of adding such terms is near.
So you have solved a problem
which is exponentially
complicated in a linear
fashion you can add like.
As many helpings as you want or
as many interactions as you
want the cost is just a year.
And or you can opt in to more or
less accurate results and
thank you very much for your attention I.
Write.
Thank you
very much that's a very excellent question
actually we have tried that what we
have obtained these that or let's add some
onsite disorder to the helping model for
example and see whether the onsite
term follows local temperature and
such or not as an on some it does so
if you consider many many terms and you
take average over them it still follows
the local temperature on such in
the sense I mean that the beta
is a smooth function it goes to
zero linearly near the boundaries.
And all for each configuration for.
The local Hamiltonians is still local so
the local temperature so what you can.
Write is that the reduced insta Matrix
can still be represented as a local.
Operator it with a minus key or
keys local but the problem is you cannot
use the translation invariance that
they mentioned earlier to find
those coefficients if you consider those
coefficients as variational parameters.
If you solve your system and then look
at the reduced estimate is local but
the problem is self-consistent
very difficult to find but
as an estate testicle problem yet
they if you take an all sound over them
they fall local temperature on its own.
All people have been working on
the theoretical part of not I mean
these methods but on a reduced and city
matrix finding reduced in-city matrix and
of a modular Hamiltonian it
seems to be very difficult
a problem if you if you add
some part of Asians or strong
perturbations small part of Asians can
still be and the strong part of Asians.
So.
As a new new miracle tool I think
it can tell us something about
black holes as well because of what
we do in black hole physics or
what people doing black hole physically is
that they try to trace out black hole and
everything inside horizon and
they just look outside so for
them sub system is huge it's outside
black hole it's very big and subsystem B.
is inside black hole that's the black box
and we want to get some information by
looking outside the system of as
a numerical talk my guess is yes but
as a theoretical method I mean people
have been working on this very.
Smart people are working on
this in that community and
it seems that theoretically
it's very difficult problem but
the interesting message of this work
is that local temperature on sites or
generalized real or Hamiltonian
is correct even if we break trans
break Lorentz symmetry even if we break
conformal symmetry that's a that's
the very interesting message of this work
that they didn't know before they couldn't
prove that they couldn't prove that but
now we have a numerical proof at least for
some classes class of models
that we have considered.
That's for the dope Hubbard model at the
moment unfortunately I'm not allude to.
To show those results but
yes we have applied our.
Technique to finite Talking to find
a chemical potential of the Hubbard
model which seems to capture
the essential physics of the C.Q.
Brits that's the interesting part
that from the beginning we were
aiming to all focus on that system.