Specific heat of the dissipative two-state system

The partition function of a two-state system coupled to a dissipative environment is studied. A functional-integral method is utilized to determine the specific heat for a spectral density J({ω}) of the environmental coupling which behaves as ωˢ at low frequencies. The Ohmic case (s=1), the sub-Ohmic case (0s1), and the super-Ohmic case (s>1) are treated. The range of damping parameters covers the region from weakly damped to heavily damped systems. For s>0, exact results are presented for all temperatures both in the weak- and strong-coupling limit and are compared with the corresponding results of the dilute-bounce-gas approximation. The qualitative behavior of the specific heat depends critically on the parameter s and on the coupling strength. For weak damping the specific heat shows a peak structure where both the ascent and descent characteristically depend on s. The peak structure vanishes if the coupling strength exceeds a critical value. For the special value {α}=1/2 of the coupling strength in the Ohmic case the exact analytical solution for all temperatures is presented. The case of a bias is also discussed.