# Information Processing

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## Dense nuclear matter: intuition fails!

I usually don’t get into detailed physics exposition on this blog, but I thought I would make an exception with regard to the paper 0808.2987 which I recently wrote with my student David Reeb. (See earlier blog post here.)

In the paper we conjectured that there might be regions in the QCD phase diagram where the sign problem does not prevent monte carlo evaluation of the Euclidean functional integral. We rewrite the partition function as

Z = Z+ – Z-

where Z+ and Z- are sums with positive weights, and each define independent statistical ensembles. Defining Z+ = exp( – V F+ ), and similarly with Z-, so that F+ and F- are the (piecewise) analytic free energies of the two ensembles, we conjectured that

F+ < F-

is the generic situation. Note Z > 0 so F+ > F- is not possible, but they can be exactly equal: F+ = F- , which is where the sign problem is most severe (see below). Since the F’s are analytic except at phase boundaries, we reasoned that if they are equal in a region they must be equal everywhere within that phase region. At mu = 0 we know Z- = 0, so we assumed that there would be a region of small mu where F+ < F- and that this region would extend into the mu-T plane.

It turns out this last assumption is probably wrong! We were unaware of results which strongly suggest that even at arbitrarily small (but positive) mu and small T, Z+ does not dominate Z. That is, in the thermodynamic limit F+ = F- exactly even at small nonzero mu. The order of limits matters: taking V to infinity for fixed nonzero mu (no matter how small) leads to large phase fluctuations. The only way to avoid it is to take mu to zero before taking V to infinity. (See 0709.2218 by Splittorff and Verbaaschot for more details. Note their results rely on chiral perturbation theory, so don’t apply to the whole plane.)

It is quite strange to me that zero density QCD can only be reached in this way. The case we are most familiar with turns out to be the oddball.

To make a long story short, our conjecture is probably incorrect: what we thought would be “exceptional” regions in the phase diagram are the typical ones, and vice versa — at least as far as anyone knows.

Note to experts: we used the term “sign problem” a bit differently than apparently it is used in the lattice community. We refer to dense QCD as having a sign problem even though we don’t know for sure (i.e., for all mu and T) that Z is exponentially small in V due to cancellations (i.e., a “severe sign problem”). Our usage probably translates to “potential sign problem” — the functional measure isn’t positive, so potentially such cancellations can occur, although we do not know if in fact they do. We got a lot of emails from people who thought we were claiming to have a method for dealing with severe sign problems, but in fact we were claiming something else entirely: that there should be regions in the phase diagram where the sign problem is not severe.

Written by infoproc

August 27, 2008 at 2:35 pm

## Dense nuclear matter

New paper! Probably too technical to go into here, but it relates to our current inability to directly simulate dense nuclear matter (QCD at nonzero baryon density). When the number of quarks and antiquarks is equal, the functional integral representation of the partition function Z has good positivity properties and can be evaluated using importance sampling (lattice Monte Carlo methods). That is no longer true when the system has nonzero baryon number, as would be the case inside a neutron star or in nuclear matter.

We rewrite Z = Z+ – Z- , where Z+ and Z- have good positivity properties, and conjecture, based on arguments using the analytic properties of the free energy, that at most points of the phase diagram Z+ dominates Z-. At such points one can simulate the theory using Monte Carlo.

http://arxiv.org/abs/0808.2987 (paper available after 5 pm pacific 8.24.08)

Sign problem? No problem — a conjecture

Stephen D.H. Hsu, David Reeb

We investigate the Euclidean path integral formulation of QCD at finite baryon density. We show that the partition function Z can be written as the difference between two sums, each of which defines a partition function with positive weights. We argue that at most points on the phase diagram one will give an exponentially larger contribution than the other. At such points Z can be replaced by a more tractable path integral with positive definite measure, allowing for lattice simulation as well as the application of QCD inequalities. We also propose a test to control the accuracy of approximation in actual Monte Carlo simulations. Our analysis may be applicable to other systems with a sign problem, such as chiral gauge theory.

Written by infoproc

August 24, 2008 at 7:24 pm