The SU(3) spin model with chemical potential corresponds to a simplified version of QCD with static quarks in the strong coupling regime. It has been studied previously as a testing ground for new methods aiming to overcome the sign problem of lattice QCD. In this work we show that the equation of state and the phase structure of the model can be fully determined to reasonable accuracy by a linked cluster expansion. In particular, we compute the free energy to 14-th order in the nearest neighbour coupling. The resulting predictions for the equation of state and the location of the critical end points agree with numerical determinations to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (1%) and   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (10%), respectively. While the accuracy for the critical couplings is still limited at the current series depth, the approach is equally applicable at zero and non-zero imaginary or real chemical potential, as well as to effective QCD Hamiltonians obtained by strong coupling and hopping expansions.

Germany

Jangho Kim

Anh Pham

Owe Philipsen

Jonas Scheunert

The SU(3) spin model with chemical potential by series expansion techniques

Lattice QCD with heavy quarks reduces to a three-dimensional effective theory of Polyakov loops, which is amenable to series expansion methods. We analyse the effective theory in the cold and dense regime for a general number of colours, N  c . In particular, we investigate the transition from a hadron gas to baryon condensation. For any finite lattice spacing, we find the transition to become stronger, i.e. ultimately first-order, as N  c  is made large. Moreover, in the baryon condensed regime, we find the pressure to scale as p ∼ N  c  through three orders in the hopping expansion. Such a phase differs from a hadron gas with p ∼   <math> <msubsup> <mi>N</mi> <mi>c</mi> <mn>0</mn> </msubsup> </math>  $$ {N}_c^0 $$ , or a quark gluon plasma, p ∼   <math> <msubsup> <mi>N</mi> <mi>c</mi> <mn>2</mn> </msubsup> </math>  $$ {N}_c^2 $$ , and was termed quarkyonic in the literature, since it shows both baryon-like and quark-like aspects. A lattice filling with baryon number shows a rapid and smooth transition from condensing baryons to a crystal of saturated quark matter, due to the Pauli principle, and is consistent with this picture. For continuum physics, the continuum limit needs to be taken before the large N  c  limit, which is not yet possible in practice. However, in the controlled range of lattice spacings and N  c -values, our results are stable when the limits are approached in this order. We discuss possible implications for physical QCD.