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=${\mathrm{T}}_{\mathrm{c}}$(t), t${\mathrm{T}}_{\mathrm{c}}$(0)=0. For the relativistic, n-component ${\mathrm{\phi }}^4$ continuum quantum field theory [which describes lattice quantum rotor (n⩾2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d${\mathrm{T}}_{\mathrm{c}}$(t)|≪${\mathrm{T}}_{\mathrm{c}}$(t), we obtain an ε 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 $\sqrt{\epsilon }$, modulo powers of ln ε, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (ω), the perturbative $\sqrt{\epsilon }$ expansion describes quantum relaxation at ℏω∼${\mathrm{k}}_{\mathrm{B}}$T or larger, but fails for ℏω∼$\sqrt{\epsilon }$ ${\mathrm{k}}_{\mathrm{B}}$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.