The spin-boson model on compact manifolds: A functional analytic approach

The spin-boson model is a fundamental framework in quantum optics, condensed-matter physics, and the theory of open quantum systems. It describes a two-level quantum system interacting with a quantized bosonic field and captures essential processes such as radiative transitions, and phonon-assisted dynamics. Despite its simplicity, the model exhibits spectral properties, particularly in the non-perturbative regime. In this thesis, we formulate and analyze the spin-boson Hamiltonian on a compact Riemannian manifold. This geometric setting removes infrared divergences and allows the bosonic field to be expanded in terms of the discrete eigenmodes of the Laplace operator. Starting from non-relativistic QED under the dipole approximation, we derive the effective two-level Hamiltonian and couple it to the bosonic field defined on the manifold. The full Hamiltonian is constructed on the bosonic Fock space using creation and annihilation operators associated with the Laplacian eigenfunctions. A central tool in our analysis is a unitary transformation, which simplifies the interaction term and makes the Hamiltonian more accessible to spectral methods. Using this transformed operator, we obtain variational upper bounds and operator-theoretic lower bounds on the ground state energy. The main results establish three key properties of the spin-boson Hamiltonian in this setting: (i) its spectrum is purely discrete above the ground-state energy, (ii) the ground state is unique under natural assumptions on the parameters, and (iii) the ground-state degeneracy is at most two in general. Also, we show that, exactly as in the flat case, the ground state involves a finite number of bosons, which is a quite remarkable result.