Anomalous transport and phonon renormalization in a chain with transverse and longitudinal vibrations

We study thermal transport in a chain of coupled atoms, which can vibrate in longitudinal as well as transverse directions. The particles interact through anharmonic potentials upto cubic order. The problem is treated quantum mechanically. We first calculate the phonon frequencies self-consistently taking into account the anharmonic interactions. We show that for all the modes, frequencies must have linear dispersion with wave-vector $q$ for small $q$ irrespective of their bare dispersions. We then calculate the phonon relaxation rates $\Gamma_i(q)$, where $i$ is the polarization index of the mode, in a self-consistent approximation based on second order perturbation diagrams. We find that the relaxation rate for the longitudinal phonon, $\Gamma_x(q) \propto q^{3/2}$, while that for the transverse phonon $\Gamma_y(q) \propto q^2$. The consequence of these results on the thermal conductivity $\kappa(N)$ of a chain of $N$ particles is that $\kappa(N) \propto N^{1/2}$

Continuous Functional Calculus for Quaternionic Bounded Normal Operators

In this article we give an approach to define continuous functional calculus for bounded quaternionic normal operators defined on a right quaternionic Hilbert space.Comment: Submitted to a journal. There was a gap in the previous version. We have corrected it and stated all the results for bounded cas

Quantum entanglement and Hawking temperature

The thermodynamic entropy of an isolated system is given by its von Neumann entropy. Over the last few years, there is an intense activity to understand thermodynamic entropy from the principles of quantum mechanics. More specifically, is there a relation between the (von Neumann) entropy of entanglement between a system and some (separate) environment is related to the thermodynamic entropy? It is difficult to obtain the relation for many body systems, hence, most of the work in the literature has focused on small number systems. In this work, we consider black-holes --- that are simple yet macroscopic systems --- and show that a direct connection could not be made between the entropy of entanglement and the Hawking temperature. In this work, within the adiabatic approximation, we explicitly show that the Hawking temperature is indeed given by the rate of change of the entropy of entanglement across a black hole's horizon with regard to the system energy. This is yet another numerical evidence to understand the key features of black hole thermodynamics from the viewpoint of quantum information theory.Comment: 10 pages, 5 figures (To appear in Eur. Phys. J. C