Equilibration in long-range quantum spin systems from a BBGKY perspective
The time evolution of ℓ-spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of N 1/2 with the system size N, implying a separation of time scales in the large-system limit. For special parameter values and initial conditions, we can show analytically that closing the BBGKY hierarchy by neglecting ℓ-spin correlations never leads to equilibration, but gives rise to quasi-periodic time evolution with at most ℓ/2 independent frequencies. Moreover, for the same special parameter values and in the large- N limit, we solve the complete recursion relation (the full BBGKY hierarchy), observing a superexponential decay to equilibrium in rescaled time τ = tN -1/2. © 2012 IOP Publishing Ltd.