  Understanding the collective behavior of a quantum many-body system, a system
composed of a large number of interacting microscopic degrees of freedom, is a
key aspect in many areas of contemporary physics. However, as a direct
consequence of the difficultly of the so-called many-body problem, many exotic
quantum phenomena involving extended systems, such as high temperature
superconductivity, remain not well understood on a theoretical level.
Entanglement renormalization is a recently proposed numerical method for the
simulation of many-body systems which draws together ideas from the
renormalization group and from the field of quantum information. By taking due
care of the quantum entanglement of a system, entanglement renormalization has
the potential to go beyond the limitations of previous numerical methods and to
provide new insight to quantum collective phenomena. This thesis comprises a
significant portion of the research development of ER following its initial
proposal. This includes exploratory studies with ER in simple systems of free
particles, the development of the optimisation algorithms associated to ER, and
the early applications of ER in the study of quantum critical phenomena and
frustrated spin systems.
