Lessons from the 3d U(1) Gross-Neveu Model

In QCD the fermion determinant is complex for chemical potential, μ non-zero, therefore generating an ensemble at μ 6= 0 is impracticable. The fermion number density, J0, measures the excess of quarks relative to anti-quarks and is expected to start to rise from zero at some value, μo. The onset, μo corresponds to the point where the phase of nuclear matter is more energetically favourable than the vacuum state (for which J0 = 0). Chiral symmetry restoration occurs at μc and we expect [2] that μ0 ∼ μc ' mp/3, where mp is the proton mass. Quenched simulations, on the other hand, predict μc ' mπ/2 suggesting that in the limit where the bare quark mass, mq → 0 chiral symmetry is restored for any μ 6= 0. This result is unphysical because the pion should not couple to the baryon chemical potential. First attempts to simulate full QCD using the Glasgow method [3] also produced perplexing results [4]. On an 8 lattice with bare quark mass mq = 0.01 at β = 5.1 we found μo ' 0.1 which differs considerably from the strong coupling analysis [5] prediction μc ' 0.65 for β = 5.0. Furthermore the scaling of μo with mq was consistent with a Goldstone boson controlling the onset. This result has motivated an assessment of the effectiveness of the Glasgow method via its implementation in a simpler model. The method involves expansion of the Grand Canonical Partition Function (GCPF) as a polynomial in the fugacity variable (e ). Consider the conventional expression for the GCPF in lattice QCD