Energy relaxation in disordered charge and spin density waves

We investigate collective effects in the strong pinning model of disordered charge and spin density waves (CDWs and SDWs) in connection with heat relaxation experiments. We discuss the classical and quantum limits that contribute to two distinct contribution to the specific heat (a $C_v \sim T^{-2}$ contribution and a $C_v \sim T^{\alpha}$ contribution respectively), with two different types of disorder (strong pinning versus substitutional impurities). From the calculation of the two level system energy splitting distribution in the classical limit we find no slow relaxation in the commensurate case and a broad spectrum of relaxation times in the incommensurate case. In the commensurate case quantum effects restore a non vanishing energy relaxation, and generate stronger disorder effects in incommensurate systems. For substitutional disorder we obtain Friedel oscillations of bound states close to the Fermi energy. With negligible interchain couplings this explains the power-law specific heat $C_v \sim T^{\alpha}$ observed in experiments on CDWs and SDWs combined to the power-law susceptibility $\chi(T)\sim T^{-1+\alpha}$ observed in the CDW o-TaS$_3$.