# Determinant representations of spin-operator matrix elements in the XX spin chain and their applications. (arXiv:1709.00682v2 [cond-mat.stat-mech] UPDATED)

For the one-dimensional spin-1/2 XX model with either periodic or open

boundary conditions, it is shown by using a fermionic approach that the matrix

element of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between two

eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be

expressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whose

entries involve the coefficients of the canonical transformations diagonalizing

the model. In the special case of a homogeneous periodic XX chain, the matrix

element of $S^-_j$ reduces to a variant of the Cauchy determinant that can be

evaluated analytically to yield a factorized expression. The obtained compact

representations of these matrix elements are then applied to two physical

scenarios: (i) Nonlinear optical response of molecular aggregates, for which

the determinant representation of the transition dipole matrix elements between

eigenstates provides a convenient way to calculate the third-order nonlinear

responses for aggregates from small to large sizes compared with the optical

wavelength, and (ii) real-time dynamics of an interacting Dicke model

consisting 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 vanishing

intrabath coupling shows that the decay of the reduced bosonic occupation

number approaches a finite plateau value (in the long-time limit) that depends

on the ratio between the number of excitations and the total number of spins.

Our results can find useful applications in various "system-bath" systems, with

the system part inhomogeneously coupled to an interacting XX chain.