  Decoherence of a center spin or qubit in a spin bath is essentially
determined by the many-body bath evolution. We develop a cluster-correlation
expansion (CCE) theory for the spin bath dynamics relevant to the qubit
decoherence problem. A cluster correlation term is recursively defined as the
evolution of a group of bath spins divided by the cluster correlations of all
the subgroups. The so-defined correlation accounts for the authentic collective
excitations within a given group. The bath propagator is the product of all
possible cluster correlation terms. For a finite-time evolution as in the qubit
decoherence problem, a convergent result can be obtained by truncating the
expansion up to a certain cluster size. The two-spin cluster truncation of the
CCE corresponds to the pair-correlation approximation [PRB 74,195301(2006)]. In
terms of the standard linked cluster expansion, a cluster correlation term is
the infinite summation of all the connected diagrams with all and only the
spins in the group flip-flopped, and thus the expansion is exact whenever
converges. When the individual contribution of each higher-order correlation
term to the decoherence is small, as the usual case for relatively large baths
where the decoherence could complete well within the bath spin flip-flop time,
the CCE coincides with the cluster expansion [PRB 74,035322(2006)]. For small
baths, however, the qubit decoherence may not complete within the bath spin
flip-flop timescale and thus individual higher-order cluster correlations could
grow significant. In such cases, only the CCE converges to the exact coherent
dynamics of multi-spin clusters. We check the accuracy of the CCE in an exactly
solvable spin-chain model.
