  Following on from our previous work [Phys. Rev. Lett. 98, 166801 (2007)] we
examine the finite temperature magnetothermoelectric response in the vicinity
of a quantum critical point (QCP). We begin with general scaling considerations
relevant to an arbitrary QCP, either with or without Lorentz invariance, and in
arbitrary dimension. In view of the broad connections to high temperature
superconductivity, and cold atomic gases, we focus on the quantum critical
fluctuations of the relativistic Landau--Ginzburg theory. This paradigmatic
model arises in many contexts, and describes the (particle-hole symmetric)
superfluid--Mott insulator quantum phase transition in the Bose--Hubbard model.
The application of a magnetic field opens up a wide range of physical
observables, and we present a detailed overview of the charge and thermal
transport and thermodynamic response. We combine several different approaches
including the epsilon expansion and associated Quantum Boltzmann Equation
(QBE), entropy drift, and arguments based on Lorentz invariance. The results
differ markedly from the zero field case, and we include an extended discussion
of the finite thermal conductivity which emerges in the presence of a magnetic
field. We derive an integral equation that governs its response and explore the
crossover upon changing the magnetic field. This equation may be interpreted as
a projection equation in the low field limit, and clearly highlights the
important role of collision invariants (or zero modes) in the hydrodynamic
regime. Using an epsilon expansion around three-dimensions, our analytic and
numerical results interpolate between our previously published value and the
exact limit of two-dimensional relativistic magnetohydrodynamics.
