Theory of finite-temperature crossovers near quantum critical points close to, or above, their upper-critical dimension

A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T = 0, t = 0, and in many cases there is a line of finite-temperature transitions at T = T-c(t), t < 0, with T-c(0) = 0. For the relativistic, n-component phi(4) continuum quantum field theory [which describes lattice quantum rotor (n greater than or equal to 2) and transverse field Ising (n = 1) models] the upper-critical dimension is d = 3, and for d < 3, epsilon = 3-d is the control parameter over the entire phase diagram. In the region \T - T-c(t)\ much less than T-c(t), we obtain an epsilon expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of root epsilon, module powers of ln epsilon, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (omega), the perturbative root epsilon expansion describes quantum relaxation at (h) over bar omega similar to k(B)T or larger, but fails for (h) over bar omega similar to root epsilon k(B)T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t = 0, for T > 0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.