  Markov chain Monte Carlo algorithms are invaluable tools for exploring
stationary properties of physical systems, especially in situations where
direct sampling is unfeasible. Common implementations of Monte Carlo algorithms
employ reversible Markov chains. Reversible chains obey detailed balance and
thus ensure that the system will eventually relax to equilibrium. Detailed
balance is not necessary for convergence to equilibrium. We review
nonreversible Markov chains, which violate detailed balance, and yet still
relax to a given target stationary distribution. In particular cases,
nonreversible Markov chains are substantially better at sampling than the
conventional reversible Markov chains with up to a square root improvement in
the convergence time to the steady state. One kind of nonreversible Markov
chain is constructed from the reversible ones by enlarging the state space and
by modifying and adding extra transition rates to create non-reversible moves.
Because of the augmentation of the state space, such chains are often referred
to as lifted Markov Chains. We illustrate the use of lifted Markov chains for
efficient sampling for several examples. The examples include sampling on a
ring, sampling on a torus, the Ising model on a complete graph, and the
one-dimensional Ising model. We also provide a pseudocode implementation,
review related work, and discuss the applicability of such methods.
