Since present Monte Carlo algorithms for lattice QCD may become trapped in a fixed topological charge sector, it is important to understand the effect of calculating at fixed topology. In this work, we show that although the restriction to a fixed to
pological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite size effects due to contributions from all $theta$-vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology.
Numerical simulations of strongly correlated electron systems suffer from the notorious fermion sign problem which has prevented progress in understanding if systems like the Hubbard model display high-temperature superconductivity. Here we show how
the fermion sign problem can be solved completely with meron-cluster methods in a large class of models of strongly correlated electron systems, some of which are in the extended Hubbard model family and show s-wave superconductivity. In these models we also find that on-site repulsion can even coexist with a weak chemical potential without introducing sign problems. We argue that since these models can be simulated efficiently using cluster algorithms they are ideal for studying many of the interesting phenomena in strongly correlated electron systems.
The phase diagrams of cuprate superconductors and of QCD at non-zero baryon chemical potential are qualitatively similar. The Neel phase of the cuprates corresponds to the chirally broken phase of QCD, and the high-temperature superconducting phase c
orresponds to the color superconducting phase. In the SO(5) theory for the cuprates the $SO(3)_s$ spin rotational symmetry and the $U(1)_{em}$ gauge symmetry of electromagnetism are dynamically unified. This suggests that the $SU(2)_L otimes SU(2)_R otimes U(1)_B$ chiral symmetry of QCD and the $SU(3)_c$ color gauge symmetry may get unified to SO(10). Dynamical enhancement of symmetry from $SO(2)_s otimes Z(2)$ to $SO(3)_s$ is known to occur in anisotropic antiferromagnets. In these systems the staggered magnetization flops from an easy 3-axis into the 12-plane at a critical value of the external magnetic field. Similarly, the phase transitions in the SO(5) and SO(10) models are flop transitions of a ``superspin. Despite this fact, a renormalization group flow analysis in $4-epsilon$ dimensions indicates that a point with full SO(5) or SO(10) symmetry exists neither in the cuprates nor in QCD.
Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin s
ystems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.
