Abstract:
We review the recent literature on lattice simulations for few- and many-body systems. We focus on methods and results that combine the framework of effective field theory with computational lattice methods. Lattice effective field theory is discussed for cold atoms as well as low-energy nucleons with and without pions. A number of different lattice formulations and computational algorithms are considered, and an effort is made to show common themes in studies of cold atoms and low-energy nuclear physics as well as common themes in work by different collaborations.

Abstract:
We prove a general theorem on spectral convexity with respect to particle number for 2N degenerate components of fermions. The number of spatial dimensions is arbitrary, and the system may be uniform or constrained by an external potential. We assume only that the interactions are governed by an SU(2N)-invariant two-body potential whose Fourier transform is negative definite. The convexity result implies that the ground state is in a 2N-particle clustering phase. We discuss implications for light nuclei as well as asymmetric nuclear matter in neutron stars.

Abstract:
The light quark-antiquark scattering Green's function is considered near a meson resonance peak. The Bethe-Salpeter equation is used to write formal expressions for the resonance width/mass ratio. Arguments are made concerning to what extent this ratio can be calculated perturbatively, and an upper bound is placed on the growth of this ratio as a function of radial excitation. Certain mesons and their radial excitations are considered, as well as the more general issue of classifying mesons in the quark model.

Abstract:
Using lattice effective field theory, we study the ground state binding energy of N distinct particles in two dimensions with equal mass interacting weakly via an attractive SU(N)-symmetric short range potential. We find that in the limit of zero range and large N, the ratio of binding energies B_{N}/B_{N-1} approaches the value 8.3(6).

Abstract:
We present recent results on lattice simulations using chiral effective field theory. In particular we discuss lattice simulations for dilute neutron matter at next-to-leading order and three-body forces in light nuclei at next-to-next-to-leading order.

Abstract:
We consider two-component fermions on the lattice in the unitarity limit. This is an idealized limit of attractive fermions where the range of the interaction is zero and the scattering length is infinite. Using Euclidean time projection, we compute the ground state energy using four computationally different but physically identical auxiliary-field methods. The best performance is obtained using a bounded continuous auxiliary field and a non-local updating algorithm called hybrid Monte Carlo. With this method we calculate results for 10 and 14 fermions at lattice volumes 4^3, 5^3, 6^3, 7^3, 8^3 and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is 0.292(12) times the ground state energy for non-interacting fermions. For 14 fermions the ratio is 0.329(5).

Abstract:
We present lattice results for spin-1/2 fermions at unitarity, where the effective range of the interaction is zero and the scattering length is infinite. We measure the spatial coherence of difermion pairs for a system of 6, 10, 14, 18, 22, 26 particles with equal numbers of up and down spins in a periodic cube. Using Euclidean time projection, we analyze ground state properties and transient behavior due to low-energy excitations. At asymptotically large values of t we see long-range order consistent with spontaneously broken U(1) fermion-number symmetry and a superfluid ground state. At intermediate times we see exponential decay in the t-dependent signal due to an unknown low-energy excitation. We probe this low-energy excitation further by calculating two-particle correlation functions. We find that the excitation has the properties of a chain of particles extending across the periodic lattice.

Abstract:
We review recent developments in non-perturbative field theory using modal field methods. We discuss Monte Carlo results as well as a new diagonalization technique known as the quasi-sparse eigenvector method.

Abstract:
We review a recently proposed approach to the problem of alternating signs for fermionic many body Monte Carlo simulations in finite temperature simulations. We derive an estimate for fermion wandering lengths and introduce the notion of permutation zones, special regions of the lattice where identical fermions may interchange and outside of which they may not. Using successively larger permutation zones, one can extrapolate to obtain thermodynamic observables in regimes where direct simulation is impossible.

Abstract:
We study the singular Landau surfaces of planar diagrams contributing to scattering of a massless quark and antiquark in 3+1 dimensions. In particular, we look at singularities which remain after integration with respect to the various angular degrees of freedom. We derive a general relation between these singularities and the singularities of quark- antiquark scattering in 1+1 dimensions. We then classify all Landau surfaces of the 1+1 dimensional system. Combining these results, we deduce that the singular surfaces of the angle- integrated 3+1 dimensional amplitude must satisfy at least one of three conditions, which we call the planar light-cone conditions. We discuss the extension of our results to non-perturbative processes by means of the non-perturbative operator product expansion. Our findings offer new insights into the connection between the 't Hooft model and large-N_c mesons in 3+1 dimensions and may prove useful in studies of confinement in relativistic meson systems.