Quantum-classical correspondence in spin-boson equilibrium states at arbitrary coupling

The equilibrium properties of nanoscale systems can deviate significantly from standard thermodynamics due to their coupling to an environment. For the generalised $\theta$-angled spin-boson model, we first derive a compact and general form of the classical equilibrium state including environmental corrections to all orders. Secondly, for the quantum spin-boson model we prove, by carefully taking a large spin limit, that Bohr's quantum-classical correspondence persists at all coupling strengths. This correspondence gives insight into the conditions for a coupled quantum spin to be well-approximated by a coupled classical spin-vector. Thirdly, we demonstrate that previously identified environment-induced 'coherences' in the equilibrium state of weakly coupled quantum spins, do not disappear in the classical case. Finally, we provide the first classification of the coupling parameter regimes for the spin-boson model, from weak to ultrastrong, both for the quantum case and the classical setting. Our results shed light on the interplay of quantum and mean force corrections in equilibrium states of the spin-boson model, and will help draw the quantum to classical boundary in a range of fields, such as magnetism and exciton dynamics.