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

Subir Sachdev

doi:10.1103/PhysRevB.55.142

paper

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

arXiv:cond-mat/9606083 · doi:10.1103/PhysRevB.55.142

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 $∣ T - T_{c} (t) ∣ ≪ T_{c} (t)$ , 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 $ε$ , 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 $ε$ expansion describes quantum relaxation at $ℏ ω \sim k_{B} T$ or larger, but fails for $ℏ ω \sim ε k_{B} T$ 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.

36 pages, 4 eps figures

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

Abstract

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