_________________________________________________________________________

R. Scalettar 
Thursday, September 15, 2011 7:27
paper

Dear Nils, Elena, Thereza, and Andreas,

I have read the paper again, as well as the recent comments and
suggestions.  In particular, Nils' main suggestion was

> we (including you) invest a few more hours
> trying to make the manuscript more interesting and relevant by a
> stronger focus on the most important messages.

I continue to find Figure 4 most intriguing-  there have been a lot of
plots of D(T) and Tneel(U) with which we lead off in Figures 1,2.  In
addition the first page, while very well written, is fairly familiar. 

I would like to think a bit more then about presentation.  Do you think
there is any way we can begin with Figure 4, or, even better, go into
more detail about its consequences?

As an example of "more detail" concerning the physics of Figure 4, we
have focussed on S and D in Thereza's runs (and to a lesser extent my
own).  But we have quite a bit of information on kinetic energy, near
neighbor (and long ranged) spin and density correlations etc.

Thereza, how difficult would it be to plot some of these quantities
versus entropy?  To be specific:  <SpinSpin>NearNeighbor,
<SpinSpin>MaximumSep, KE, <DenupDenup>NearNeighbor, and
<DenupDendn>NearNeighbor.  Will they all show monotonic development with
S so that the double occupancy is unique in its minimum?  It might be
that we win with either answer.  If other quantities are "boring" then
it brings the cool behavior of D into sharper focus.  If these other
quantities are "interesting", then...

So, again Thereza, is this a ton of work?  (I hope you have not
previously sent such figures!  I am at home now and cannot easily
run through my files of your figures.  I will do so tomorrow morning.)

Richard

_________________________________________________________________________

R. Scalettar 
Thursday, September 15, 2011 19:44
Re: Paper: scheme removed, finite-size study, new d=2 U/t=12.25 data, pseudogap

> In my view, our dimensional comparison profited a lot from new DQMC data
> for the square lattice obtained by Daniel Rost: now, the dimensional
> convergence is much clearer as a function of temperature (also due to
> extremely careful entropy estimates). I am not sure whether this
> contribution justifies his inclusion in the author list, but I would
> suggest that we are generous in this case.

I defer to your judgement on this.  Generousity is generally a good thing!

> We also looked at spectral data at weak coupling (U/t=3D4) for the
> square lattice and found a pseudogap below a characteristic temperature which
> might scale to the spin cross-over temperature (i.e., to the DMFT Neel
> temperature) in the thermodynamic limit. Do you think that this is
> possible?  We saw that there has been a controversy regarding the
> existence of a pseudogap for a long time, possibly starting with White's
> PRB 46, 5678 (1992) claim that gaps in DQMC data are finite-size effects.
> Any comments are welcome.

I am embarassed to say that I really don't know whether this was ever
resolved.  The issue, as I recall, is that one doesn't expect a
Mott gap until U exceeds roughly the bandwidth, so the gap one
observes at smaller U on a square lattice must be from the long
range AF correlations, ie a Slater gap.  But long range AF order
doesn't occur except at T=0.  So Steve argued the gap observed at
finite T and weak U was just because the correlation length xi had
exceeded the lattice size-  a finite size effect as Nils says.
It's really tricky to try to unravel something like that because
xi grows so rapidly- as exp(A/T).  Anyway, are we thinking of putting
something about this in the current paper?  It would be a shift in
focus...


_________________________________________________________________________

R. Scalettar 
Thursday, September 15, 2011 21:00
4x4 lattice is special

Nils-  see the attached pdf which explains why 4x4 is special.  In
particular, correlation functions for sites separated by (2,0) are
identical to those separated by (1,1).  This is not true for larger
square lattices.

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 0:26
language for paper

Dear Nils etal.,

In our current draft we discuss the challenges to looking for
antiferromagnetism, and especially long range order, given the spatial
inhomogeneities in optical lattices.  I propose below some language
which expands on that discussion.

I also am happy with the current versions of the manuscript, so
we can move forward with them if that's best.  Obviously we want
to do *something* fairly soon (my fault for all the delaying).

Richard

===================================================================

A key challenge to examining many body phenomena in optical lattices
is the inhomogeniety created by the confining potential.
  (WE ALREADY SAY SOMETHING LIKE THE ABOVE SENTENCE.  I INCLUDE IT
   HERE FOR COMPLETENESS.)
In two dimensions, in a situation where the fermionic density exceeds
half-filling in the trap center, the Mott insulating regime where strong
magnetic correlations will be present is an annulus around the cloud
center.  It is not clear whether such a geometry will be characterized
by one or two dimensional physics, or exhibit some complex dimensional
cross-over.  Indeed as has been discussed in [*], the behavior appears
to be quite subtle.  At short distances spin correlations follow a 2D
behavior.  At long distance, although significantly suppressed,
they do not approach 1D values as might be expected.  The best fit
appears to be to 1D behavior at a temperature quite a bit lower than the
nominal temperature of the 2D lattice.  In short, the spin correlations
are not well described by any single dimension or tempertature.

What these results emphasize is that in searching for signatures of
magnetic order in optical lattices, an analysis tool which does not rely
on specific dimension-dependent assumptions (like the appearance of long
range order), or even on a specific value of $T$ is required.  One might
imagine that the natural tool is the short range (near-neighbor) spin
correlation function itself.  The key observation of this paper is that
the double occupancy $D$, which is also easier to access experimentally,
already plays this role.  Specifically, we show that the dependence of
$D$ on the entropy density $s$ exhibits a universal (i.e. dimension
independent) minimum which separates the development of spin order at
small $s$ from the large $s$ regime dominated by charge physics.  Thus
we propose it as a powerful tool for searching for magnetism which can
avoid the subtle issues associated with inhomogeneity.

QUESTION:  IF WE INSERT SOME VARIANT OF THIS LANGUAGE, DO WE THEN WANT
TO MAKE THE PRESENT FIGURE 4 INTO FIGURE 1?  THAT MIGHT MAKE THE
NOVELTY OF OUR WORK STAND OUT BETTER.  WE WOULD THEN SHOW THE PRESENT
FIGURES 1,2,3 AS FIGURES 2,3,4 AND INTRODUCE APPROPRIATE LANGUAGE
TO ALERT THE READER ABOUT OUR STRATEGY: KEY CONCLUSION FIRST WITH
DETAILS TO FOLLOW.

MORE:  I AM STILL INTERESTED IN WHETHER KE(s),... CAN SHOW ANYTHING
OF INTEREST.  (SEE MY EMAIL FROM A DAY OR TWO AGO.)
THEREZA- DO YOU THINK THERE IS ANY HOPE THEY MIGHT
BE USEFUL AND, IF SO, HOW LONG TO MAKE SUCH PLOTS?  WE ARE OBVIOUSLY
UNDER SOME TIME PRESSURE TO RESPOND TO THE REFEREE REPORTS SINCE WE GOT
THEM BACK IN JULY.

[*] ``Magnetism and pairing of two-dimensional
trapped fermions", Simone Chiesa, Richard T. Scalettar, Christopher N.
Varney and Marcos Rigol,
Phys. Rev. Lett. {\bf 106}, 035301 (2011).

_________________________________________________________________________

Th. Paiva
Friday, September 16, 2011 0:44
Re: paper

Hi everybody,

Please find attached a pdf file with  some of the plts Richard suggested.
The ones I have not done yet are the longer distance correlations. I
suspect they
will have more noise, but I will check anyway.  Anyway, 
nearest-neighbor correlations will
not change much with system size, whereas longer nieghbors will change.

I do like the plots quite a lot! The kinetic energy shows a kink  at
s~ln(2). I wasn´t expecting something so clear!  Spin-spin correlations
also show a different behavior for s< ln(2) (sharp increase with s) and
s> ln(2) (almost zero). Would it go to zero for s > ln(2) within DMFT,
which has a sharper behavior for D? Can you calculate that?

Charge-charge correlations are similar to the spin spin correlations:
change fast with s for s<ln(2)
an approach 0.25 for s> ln(2).

All the data is for 6x6x6 and dt=0.05. I could not find 2d datat for
U=12.25. Did I do those runs?!


Em 9/15/2011 2:27 AM, Richard T. Scalettar escreveu:
> Dear Nils, Elena, Thereza, and Andreas,
>
> I have read the paper again, as well as the recent comments and
> suggestions. ...

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 1:03
Re: paper

Thank you Thereza.  You are amazing.  The KE is especially cool.
Does anyone understand it?!

Is it possible to replot
the right-hand panel of page 4 (derivative of nn spin correlation)
without the lowest entropy point and doing something to smooth/average
the data in the vicinity of the peak at s=log 2?  The negative slope at
lowest s certainly could just be noise...  These changes would make the
figure more dramatic (allow us to blow up the y-axis scale because no
need to run it to negative values...)

The big issue then is what to do with this data.  It becomes a much
more major rewrite of the paper, but we could include some of it.
For example,

[]  Current Figure 4 --->   Figure 1
[]  New Figure 2 showing some of the results Thereza just got
[]  Retain current Figures 2,3,4 as 3,4,5 or if not enough space
     eliminate one of them?

Again, I am not sure such a big reworking is optimal for us.
It's okay if others don't like it...


On Thu, 15 Sep 2011, Thereza C de L Paiva wrote:

>
> Hi everybody,
>
> Please find attached a pdf file with  some of the plts Richard suggested....

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 1:04
Re: paper

Incidentally, plotting the derivatives was a very nice touch.

On Thu, 15 Sep 2011, Thereza C de L Paiva wrote:

>
> Hi everybody,
>
> Please find attached a pdf file with  some of the plts Richard suggested....

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 1:06
all dimensions?!

Hmmm.  If we show these plots, do we want to show them just for d=3
or for other d as well?

On Thu, 15 Sep 2011, Thereza C de L Paiva wrote:

>
> Hi everybody,
>
> Please find attached a pdf file with  some of the plts Richard suggested....

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 4:06
paper

Dear Nils, Thereza, Elena, and Andreas,

The text I sent about D(s), if we want to use it, will need modification
in view of Thereza's new figures, at least to indicate that some
other quantities besides D might also be useful, and certainly show
a signal.  Perhaps the text is still fairly accurate in focusing on
D since the experimentalists like D.  (Do they measure KE?!)

Regarding dimensionality, I suspect that if D(s) has a signal at the
same s for all dimensions, probably all the quantities likewise
have signals at the same s.  What do you think?

This makes me wonder about general stat mech theory...  If we computed
quantities for the Ising model for example, and showed them as a
function of entropy, would data for different dimensions all line up?

Of course one can get some degree of line up by using T/z instead of T.
Would S be any better than that?

Richard

_________________________________________________________________________

Th. Paiva
Friday, September 16, 2011 8:04
Re: paper
Attached: Richardssuggestion.pdf‎ 

I am sending an updated version with U=12 (not 12.25) 10x10 data.


On Thu, 15 Sep 2011, Richard T. Scalettar wrote:

>
> Dear Nils, Thereza, Elena, and Andreas,
>
> The text I sent about D(s), if we want to use it, will need modification...

_________________________________________________________________________

N. Bluemer
Friday, September 16, 2011 16:05
back online, first feedback and suggestions / Re: paper

Dear Thereza and Richard,

due to travel and attending a retreat, I missed your fruitful exchange and 
will need a day or so to catch up. Just some initial comments:

Richard, thanks for the nice explanation about 4x4. We had already seen 
that the spectra at (pi/2, pi/2) and (pi,0) were identical within error 
bars which should be a consequence of this symmetry.

The computation of the kinetic energy was also on the task list I 
collected in Augsburg. It was clear that there had to be a significant 
signal since E=U*D+KE is a monotonic and rather smooth function of T 
(and, of course, also monotonic versus s) so that a negative slope of 
D(T) has to be offset by an extra positive slope in KE(T). However, it 
looks nicer than expected. I think we should definitely show some of 
this data since it is really physical: the gain in kinetic energy by 
exchange drives the AF correlations and increase in double occupancy.

I think the clear signals in the other observables are mostly due to 
the small slope in s(T) at s<~0.7 which implies that smooth functions 
f(T) will have an increased slope there when plotted as f(s). Drawing 
too many of such functions might hurt us. In the end, the reason why 
s=0.7 is special is, of course, that the half-filled Hubbard model can 
reach s<log(2) only by spin coherence (in any dimension)

The charge-charge data would seem more useful (as contrast to spin 
correlations) if plotted as a sum, i.e., < n_up (n_up+n_down)> in 
order to remove the spin-spin impact. Do you also have longer-range 
spin correlation data? Of course such data must show a stronger 
dependence on dimensionality, at least asymptotically.

In contrast to KE, D is a measurable quantity. Unfortunately, it is 
difficult to get the error bars below 0.01 (what we need). In fact, 
Antoine Georges (a possible referee) kept telling me in Augsburg 
that he thinks that k-resolved methods (addressing longer-range 
correlations) will be more useful.

Mark Jarrell suggested that the temporal correlation function 
<D(t) D(t')> would be interesting and accessible since it is 
essentially <KE(t) KE(t')>.

I think that this goes to far for the present project. We might, 
instead, compare spectra A(k,omega) or contrast m_stag, D, and 
<s_i s_j> for d=3.

Concerning the text: Richard, I think your proposal goes into the 
right direction. Of course, we will happily provide you with all 
sources (i.e. figures and tex-file) if you wanted to draft a 
reordered version. We will think more about it during the weekend.

Bests, Nils


On 16.09.2011, at 08:04, Thereza Cristina de Lacerda Paiva wrote:

>
> I am sending an updated version with U=12 (not 12.25) 10x10 data...

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 19:08
Re: paper

Cool!  Again, I would like to understand the fundamental theory of this
better.  For example, why so little dimensionality effect?  Especially 2D
vs 3D where one has a finite T transition and the other does not, yet they
look so incredibly similar.  I guess this question was there all along
when we were looking just at D(s) but somehow seeing the behavior
over and over in many quantities hit me more over the head.

Richard Scalettar
Professor, Physics Department
University of California, Davis 95616
phone 530-554-1605
fax   530-752-4717
email scalettar@physics.ucdavis.edu
http://leopard.physics.ucdavis.edu/rts/

On Fri, 16 Sep 2011, Thereza Cristina de Lacerda Paiva wrote:

>
> I am sending an updated version with U=12 (not 12.25) 10x10 data.

_________________________________________________________________________

Th. Paiva
Friday, September 16, 2011 19:18
Re: paper
Attached: dDds.eps

Please find attached a plot of dD/ds. It should have been the first
derivative to be done! It is amazing how the curves for 2d and 3D almost
collapse. These are data for fixed dtau, and 3d data does not have
a lot of statistics, as you could see from the fluctiations on
spin correlations.

Don't you think it is worth doing that with all data in Fig. 4?

On Fri, 16 Sep 2011, scalettar@physics.ucdavis.edu wrote:

>
> Cool!  Again, I would like to understand the fundamental theory of this
> better.  ...

_________________________________________________________________________

R. Scalettar 
Friday, September 16, 2011 20:05
Re: back online, first feedback and suggestions / Re: paper

> I think the clear signals in the other observables are mostly due to the
> small slope in s(T) at s<~0.7 which implies that smooth functions f(T)
> will have an increased slope there when plotted as f(s).

Good point!  Anything divided by zero will show a peak!  You make it
sound less interesting though.  Whose side are you on?

It is beginning to sound to me (especially hearing all the suggestions
from Mark etc) that we might need to start thinking about a second paper
to follow up on all this.  That would also raise the issue of whether
to show all of the dtau/L analysis in the supplement or in a different
manuscript.  Maybe we need it in the supplement because the current
referees asked about it.

_________________________________________________________________________

R. Scalettar 
Saturday, September 17, 2011 3:05
raw data

I just sent a plot of the boundary condition averaged data.  I thought
maybe you would also like to see the data from the individual boundary
conditions.  It's sort of amusing.  Note the large differences in 4^3
(more than 10%) for different boundary conditions, and much smaller
differences (about 1 %) on 8^3.  The table gives L followed by the
values for the double occupance with error bar for each of four boundary
conditions.  +++ means pbc in all three directions -++ means pbc in two
directions, apbc in the third, etc.

Remember here that U=4, beta=4, and rho=1.

  L        D+++         D---          D--+          D-++
 32   0.1507 0.0002 0.1716 0.0002  0.1513 0.0003 0.1617 0.0001
 40   0.1527 0.0003 0.1740 0.0002  0.1533 0.0003 0.1640 0.0002
 48   0.1535 0.0002 0.1747 0.0000  0.1546 0.0002 0.1648 0.0002
 64   0.1543 0.0002 0.1757 0.0001  0.1556 0.0002 0.1659 0.0002
 80   0.1551 0.0002 0.1766 0.0001  0.1566 0.0002 0.1663 0.0002

 32   0.1628 0.0001 0.1569 0.0002  0.1630 0.0002 0.1607 0.0001
 40   0.1648 0.0001 0.1591 0.0002  0.1651 0.0002 0.1625 0.0002
 48   0.1660 0.0001 0.1603 0.0001  0.1661 0.0002 0.1635 0.0002
 64   0.1668 0.0001 0.1613 0.0001  0.1670 0.0002 0.1648 0.0001
 80   0.1672 0.0001 0.1619 0.0001  0.1674 0.0001 0.1653 0.0001

 32   0.1613 0.0001 0.1629 0.0001  0.1618 0.0001 0.1624 0.0001
 40   0.1633 0.0001 0.1647 0.0001  0.1636 0.0001 0.1644 0.0001
 48   0.1643 0.0001 0.1658 0.0001  0.1647 0.0001 0.1655 0.0001
 64   0.1651 0.0001 0.1669 0.0001  0.1658 0.0001 0.1663 0.0001
 80   0.1658 0.0001 0.1673 0.0001  0.1663 0.0001 0.1669 0.0001

 
