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

Abstract

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 $S_j^{-}$ ($S_j^{-}{S}_{j^{\prime }}^{+}$) 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 $N\le 16$ 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.

15 pages, 3 figures, to appear in Physical Review B

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Cited by in corpus (9)

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• Vibration-assisted exciton transfer in molecular aggregates strongly coupled to confined light fields
• Exact steady state of the open XX-spin chain: entanglement and transport properties
• Collective charging of an organic quantum battery
• Theory of exciton transport in molecular crystals strongly coupled to a cavity: A temperature-dependent variational approach
• Few-magnon excitations in a frustrated spin-$S$ ferromagnetic chain with single-ion anisotropy
• Relaxation of antiferromagnetic order and growth of Rényi entropy in a generalized Heisenberg star
• Dynamics of a periodic $XY$ chain coupled to a photon mode
• Ground-state and dynamical properties of a spin-$S$ Heisenberg star