Simulation generates negative values of the specific heat.

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Simulation generates negative values for the specific heat, which is unphysical 
in my understanding.

I'd even say, I'd be very excited to see definition of the Specific Heat that 
is used in QUEST.

Original issue reported on code.google.com by iglovi...@gmail.com on 8 Nov 2013 at 3:16

Attachments:

• Lieb_t0_u-4_nx1.geom
• L10_mu1_1383880451.91.in
• Lieb_1383880451.91.out

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Original comment by iglovi...@gmail.com on 19 Dec 2013 at 11:54

• Added labels: Priority-Medium
• Removed labels: Priority-High

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Original comment by cxc639 on 25 Mar 2014 at 6:08

• Changed state: Started

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Original comment by iglovi...@gmail.com on 25 Mar 2014 at 8:42

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I uploaded the fix to the specific heat. As far as I can tell, everything was 
caused by the fact that
the energy did not include the U/2 contribution to the chemical potential. The 
fix was to
simply uncomment two lines of code!

As a result the code now prints an energy which contains
the "full" chemical potential (mu+U/2). I think this makes more sense. Another 
logical alternative 
would have been to exclude *both* mu and U/2. I prefer what is currently 
implemented since there
is a direct link between the printed energy and the specific heat independently 
of mu.
In any case, what we had now (energy with mu and no U/2) does not seem 
reasonable to me.

I have not tested the code at all. I am also attaching notes that should make 
easier to
see what's going on in the part of the code that computes the chemical 
potential. Can anyone
check that things are alright?

Cheers,

Simone

Original comment by simone.c...@gmail.com on 27 Mar 2014 at 2:53

Attachments:

• SpecificHeat.pdf
• SpecificHeat.tex

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Still gives negative values.

[1] Where is \sum_i \frac {U_i} {4} term in the derivation? (I will not buy 
argument that this is just "trivial" shift in energy. It should be included.)

[2] I believe we should not try to do optimizations when we should not do them, 
so I think we should have Hamiltonian in the particle-hole symmetric form and 
make chemical potential term part of the Hamiltonian. This will give us 16 
terms and not 4. It will be bunch of messy algebra, but such is life.

Original comment by iglovi...@gmail.com on 29 Mar 2014 at 12:09

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I am not following exactly how Cv is being calculated in QUEST, but I know
that in
the grand canonical ensemble, the change in \mu by changing the temperature
to
keep the density constant is important. See e.g., Eqs 6 and 7 of
http://arxiv.org/pdf/1204.1556.pdf

Best,

Ehsan

Original comment by ehsankha...@gmail.com on 29 Mar 2014 at 2:55

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Dear Vlad and Eshan,

Vlad : there is no need for that term for the specific heat. Set H' = H -C 
where C is a number independent of T. Evaluate <H'^2>-<H'>^2. You'll
easily find it is equal to <H^2>-<H>^2. I really do not see how that
term can make a difference provided we do things consistently and omit 
it in both <H^2> and <H>^2. I have been expanded the discussion a 
little bit in the new note I am attaching.

Eshan : What you're saying is the content of the last part of what I 
attached (Eq.15 in the attachment to comment #4 and Eq.17 in what I
attach to this one). What we are computing is the specific heat at
constant mu and we would have to add the extra pieces described in
your article or my note to get the specific heat at constant density.
However, **both specific heats should remain positive**
as they are both expressed as <A^2> - <A>^2. Do you agree? 

I'll look more into this in the afternoon and beef up the note with
pointers to where things happen in the code. I am a little frustrated
as Dave Cone went through quite a bit of test of this part of the
code a while ago and things seem to work alright (but then again
it was always the square lattice and I do not recall the tests he did).

Cheers,

Simone

Original comment by simone.c...@gmail.com on 29 Mar 2014 at 1:35

Attachments:

• SpecificHeat.pdf

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Why do we calculate specific heat per unit cell and not per site?

Original comment by iglovi...@gmail.com on 29 Mar 2014 at 7:01

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[1] You can get negative specific heat for the square lattice as well.

If you go to the "geom" folder and change value of U in the square.geom file 
you will get:

u = -2 => C = 0.16
u = -4 => C = -0.28

[2] Right now QUEST gives wrong values for the specific heat, but at it 
respects particle-hole symmetric form of the Hamiltonian.

C(u = 2) = C(u = -2) = 0.16
C(u = 4) = C(u = -4) = -0.28

[3] Am I right that Kinetic energy right now is defined as:
 KE = -\sum_{ij\sigma} t_{ij\sigma} (c^{\dagger}_{ij\sigma} c_{ij \sigma} + h.c) - \sum_i (mu_i + U_i / 2) n_i

?

Original comment by iglovi...@gmail.com on 29 Mar 2014 at 7:22

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Guys, attached is an extension of previous document where I point the relevant 
places in QUEST where things involved in the specific heat enters. I am hoping 
this is enough for you to get oriented and spot possible problems. By looking 
at the code I have not found any :(

Original comment by simone.c...@gmail.com on 29 Mar 2014 at 7:50

Attachments:

• SpecificHeat.pdf
• SpecificHeat.tex

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Vlad, 

concerning your question/comments:

-comment 8: why "per unit cell" and not "per site"? If you have a unit cell 
with inequivalent sites, I don't see
the "per site" definition making too much sense.

-comment 9 question[3]. Yes. See my last note for the definition of mu_i.

Simone

Original comment by simone.c...@gmail.com on 29 Mar 2014 at 7:55