QCD is constructed as a lattice gauge theory in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. The resulting quantum link model for QCD is formulated with a fifth Euclidean dimension, whose extent resembles the inverse gauge coupling of the resulting four-dimensional theory after dimensional reduction. The inclusion of quarks is natural in Shamir’s variant of Kaplan’s fermion method, which does not require fine-tuning to approach the chiral limit. A rishon representation in terms of fermionic constituents of the gluons is derived and the quantum link Hamiltonian for QCD with a $U\left(N\right)\mathrm{}$ gauge symmetry is expressed in terms of glueball, meson and constituent quark operators. The new formulation of QCD is promising both from an analytic and from a computational point of view.

We present a general strategy to solve the notorious fermion sign problem using cluster algorithms. The method applies to various systems in the Hubbard model family as well as to relativistic fermions. Here it is illustrated for nonrelativistic lattice fermions. A configuration of fermion world lines is decomposed into clusters that contribute independently to the fermion permutation sign. A cluster whose flip changes the sign is referred to as a meron. Configurations containing meron clusters contribute $0$ to the path integral, while all other configurations contribute $1$. The cluster representation describes the partition function as a gas of clusters in the zero-meron sector.

We construct an efficient Monte Carlo algorithm that overcomes the severe signal-to-noise ratio problems and helps us to accurately compute the conformal dimensions of large-$Q$ fields at the Wilson-Fisher fixed point in the $O(2)$ universality class. Using it, we verify a recent proposal that conformal dimensions of strongly coupled conformal field theories with a global $U(1)$ charge can be obtained via a series expansion in the inverse charge $1/Q$. We find that the conformal dimensions of the lowest operator with a fixed charge $Q$ are almost entirely determined by the first few terms in the series.

We study quantum critical behavior in three dimensional lattice Gross-Neveu models containing two four-component massless Dirac fermions. We focus on two models with $SU(2)$ flavor symmetry and either a $Z_2$ or a $U(1)$ chiral symmetry. Both models could not be studied earlier due to sign problems. We use the fermion bag approach which is free of sign problems and compute critical exponents at the phase transitions. We estimate $\nu =0.83(1)$, $\eta =0.62(1)$, ${\eta }_{\psi }=0.38(1)$ in the $Z_2$ and $\nu =0.849(8)$, $\eta =0.633(8)$, ${\eta }_{\psi }=0.373(3)$ in the $U(1)$ model.