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Equilibration in long-range quantum spin systems from a BBGKY perspective

The time evolution of $\ell$-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 $\ell$-spin correlations does never lead to equilibration, but gives rise to quasi-periodic time evolution with at most $\ell/2$ 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 $τ=tN^{-1/2}$.