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

Abstract

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 $ϕ^4$ continuum quantum field theory (which describes lattice quantum rotor ($n \geq 2$) and transverse field Ising ($n=1$) models) the upper critical dimension is $d=3$, and for $d<3$, $ε=3-d$ is the control parameter over the entire phase diagram. In the region $|T - T_c (t)| \ll T_c (t)$, we obtain an $ε$ expansion for coupling constants which then are input as arguments of known {\em classical, tricritical,} crossover functions. In the high $T$ region of the continuum theory, an expansion in integer powers of $\sqrtε$, 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 $\sqrtε$ expansion describes quantum relaxation at $\hbar ω\sim k_B T$ or larger, but fails for $\hbar ω\sim \sqrtε k_B T$ 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.