  Theoretical description of anisotropic systems, such as layered
superconductors and coupled spin chains, is often a challenge due to the
different natures of interactions along different directions. As a model of
such a system, we present an analytical study of d-dimensional "nodes" arranged
as the vertices of a Bethe lattice, where each node has nonzero spatial
dimension and is described by an O(N) quantum rotor model, and there is hopping
between neighboring nodes. In the limit of large connectivity on the Bethe
lattice, the hopping can be treated by constructing a self-consistent effective
action for a single node. This procedure is akin to dynamical mean field
theory, but generalized so that spatial as well as quantum fluctuations are
taken into account on each node. The quantum phase transition is studied using
this effective action for both infinite and finite N. The importance of the
Perron-Frobenius uniform mode on the Bethe lattice is discussed, and its
elimination via an "infinite range hopping" term shifts the transition, leading
to nontrivial critical behavior. We calculate critical exponents and find that
the internode hopping reduces the upper and lower critical dimensions each by
one, indicating that--at least for the purposes of quantum criticality--the
large number of internode couplings is similar to adding a single extra
dimension to the theory describing a single node.
