Exact solution of the Heisenberg equation of motion for the surface spin in a semi-infinite S=1/2 XY chain at infinite temperatures

Dynamics of the transverse Ising model with next-nearest-neighbor interactions

We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.

Why irreversibility is not a sufficient condition for ergodicity

Khinchin's theorem of ergodicity is examined by means of linear response theory. The resulting ergodic condition shows that, contrary to the theorem, irreversibility is not a sufficient condition for ergodicity. By the recurrence relations method, we prove that irreversibility is broader in scope than ergodicity, showing why it can only be a necessary condition for ergodicity.

Slow algebraic relaxation in quartic potentials and related results

We present a detailed report [see S. Sen et al., Phys. Rev. Lett. 77, 4855 (1996)] of our numerical and analytical studies on the relaxation of a classical particle in the potentials V(x)= +/-x(2)/2+x(4)/4. Both of the approaches confirm that at all temperatures, the relaxation functions (e.g., velocity relaxation function and position relaxation function) decay asymptotically in time t as sin(omega(0)t)/t. Numerically calculated power spectra of the relaxation functions show a gradual transition with increasing temperature from a single sharp peak located at the harmonic frequency omega(0) to a broad continuous band. The 1/t relaxation is also found when V(x) is a polynomial in powers of x(2) with a nonvanishing coefficient accompanying the x(4) term in V(x). Numerical calculations show that in the cases in which the leading term in V(x) behaves as x(2n) with integer n, the asymptotic relaxation exhibits 1/t(phi) decay where phi =1/(n-1). We briefly discuss the analytical approaches to relaxation studies in these strongly anharmonic systems using direct solution of the equation of motion and using the continued fraction formalism approach for relaxation studies. We show that the study of the dynamics of strongly anharmonic oscillators poses unique difficulties when studied via the continued fraction or any other time-series construction based approaches. We close with comments on the physical processes in which the insights presented in this work may be applicable.