We present results of numerical simulations of the 2+1d Nambu - Jona-Lasinio model with a non-zero baryon chemical potential mu including the effects of a diquark source term. Diquark condensates, susceptibilities and masses are measured as functions of source strength j. The results suggest that diquark condensation does not take place in the high density phase mu>mu_c, but rather that the condensate scales non-analytically with j implying a line of critical points and long range phase coherence. Analogies are drawn with the low temperature phase of the 2d XY model. The spectrum of the spin-1/2 sector is also studied yielding the quasiparticle dispersion relation. There is no evidence for a non-zero gap; rather the results are characteristic of a normal Fermi liquid with Fermi velocity less than that of light. We conclude that the high density phase of the model describes a relativistic gapless thin film BCS superfluid.

We present results of a Monte Carlo simulation of a (2+1)-dimensional Nambu-Jona-Lasinio model. In the vacuum phase, the diquark condensate vanishes linearly as a function of diquark source j as expected, but simulations in a region with nonzero baryon density suggest a power-law scaling infinity j(alpha) and hence a critical system for all mu > mu(c). There is no signal for superfluidity. Comparisons are drawn with the pseudogap phase in cuprate superconductors. We also measure the dispersion relation E(k) for fermionic excitations, and find results consistent with a sharp Fermi surface. Any gap Delta is constrained to be much less than the constituent quark mass scale Sigma(0).

We study the global symmetries of SU(2) gauge theory with N flavors of staggered fermions in the presence of a chemical potential. We motivate the special interest of the case N = 1 (staggered) with fermions in the adjoint representation of the gauge group. We present results from numerical simulations with both hybrid Monte Carlo and the Two-Step Multi-Bosonic algorithm.

We identify the Goldstone modes appropriate to the low and high density phases of SU(2) lattice gauge theory with staggered fermions. We present hybrid Monte Carlo simulation results for susceptibilities on a 6^4 lattice at beta=1.5, m=0.05. We specify how to implement a diquark source term in a lattice simulation and present first measurements of the lattice diquark condensate as a function of the diquark source.

We discuss general aspects of the possibility of Bose condensation of diquark pairs in systems of dense matter. Lattice field theory simulations are presented for model four-fermion theories which are expected to manifest the phenomenon, and results from measurements of both diquark two-point and one-point functions presented. Whilst initial results are promising, there remain systematic effects needing to be understood.

We simulate the Gross-Neveu model in 2+1 dimensions at non-zero baryon density (chemical potential μ ≠ 0). It is possible to formulate this model with a real action and therefore to perform standard hybrid Monte Carlo simulations with μ ≠ 0 in the functional measure. We compare the physical observables from these simulations with simulations using the Glasgow method where the value of μ in the functional measure is fixed at a value μupd. We find that the observables are sensitive to the choice of μupd. We consider the implications of our findings for Glasgow method QCD simulations at μ ≠ 0. We demonstrate that the realisation of the Goldstone mechanism in the Gross-Neveu model is fundamentally different from that in QCD. We find that this difference explains why there is an unphysical transition in QCD simulations at μ ≠ 0 associated with the pion mass scale whereas the transition in the Gross-Neveu model occurs at a larger mass scale and is therefore consistent with theoretical predictions. We note classes of theories which are exceptions to the Vafa-Witten theorem which permits the possibility of formation of baryon number violating diquark condensates.

We review efforts to study, using the methods of lattice field theory, the phenomenon of diquark condensation via BCS pairing at a Fermi surface, which has been proposed as a mechanism for color superconductivity in dense quark matter. The particular models studied are the Gross-Neveu model and SU(2) lattice gauge theory; in both cases evidence for superfluidity at high density is presented. The behaviour expected for quarks in both fundamental and adjoint representations of SU(2) is contrasted.

The effectiveness of the Glasgow algorithm is explored via implementation in the 3d U(1) Gross-Neveu model and the realisation of the Goldstone mechanism in this model is compared and contrasted with its realisation in QCD.