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For the one-dimensional spin-1/2 XX model with either periodic or openboundary conditions, it is shown by using a fermionic approach that the matrixelement of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between twoeigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can beexpressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whoseentries involve the coefficients of the canonical transformations diagonalizingthe model. In the special case of a homogeneous periodic XX chain, the matrixelement of $S^-_j$ reduces to a variant of the Cauchy determinant that can beevaluated analytically to yield a factorized expression. The obtained compactrepresentations of these matrix elements are then applied to two physicalscenarios: (i) Nonlinear optical response of molecular aggregates, for whichthe determinant representation of the transition dipole matrix elements betweeneigenstates provides a convenient way to calculate the third-order nonlinearresponses for aggregates from small to large sizes compared with the opticalwavelength, and (ii) real-time dynamics of an interacting Dicke modelconsisting of a single bosonic mode coupled to a one-dimensional XX spin bath.In this setup, full quantum calculation up to $N\leq 16$ spins for vanishingintrabath coupling shows that the decay of the reduced bosonic occupationnumber approaches a finite plateau value (in the long-time limit) that dependson the ratio between the number of excitations and the total number of spins.Our results can find useful applications in various "system-bath" systems, withthe system part inhomogeneously coupled to an interacting XX chain.

Determinant representations of spin-operator matrix elements in the XX spin chain and their applications