Motivated by an analogy with the spin anisotropies in the quantum XY chain and its reformulation in terms of spin-less Majorana fermions, its bosonic analogue, the spinanisotropic quantum spherical model, is introduced. The exact solution of the model permits to analyse the influence of the spin-anisotropy on the phase diagram and the universality of the critical behaviour in a new way, since the interactions of the quantum spins and their conjugate momenta create new effects. At zero temperature, a quantum critical line is found, which is in the same universality class as the thermal phase transition in the classical spherical model in d + 1 dimensions. The location of this quantum critical line shows a re-entrant quantum phase transition for dimensions 1 < d 2.065.

We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature T and of the quantum parameter g, which measures the strength of quantum fluctuations. While the von Neumann and the Rényi entropies exhibit the volume-law for any g and T , the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of g, T , the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any g, it exhibits the so-called sudden death, i.e., it is exactly zero for large enough T . The vanishing of the negativity defines a "death line", which we characterise analytically at small g. Importantly, for any finite T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit T → 0. Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.

The coherent quantum dynamics of a single bosonic spin variable, subject to a constraint derived from the quantum spherical model of a ferromagnet, and coupled to an external heat bath, is studied through the Lindblad equation for the reduced density matrix. Closed systems of equations of motion for several quantum observables are derived and solved exactly. The relationship to the single-mode Dicke model from quantum optics is discussed. The analysis of the interplay of the quantum fluctuation and the dissipation and their influence on the relaxation of the time-dependent magnetisation leads to the distinction of qualitatively different regimes of weak and strong quantum couplings. Considering the model's behaviour in an external field as a simple mean-field approximation of the dynamics of a quantum spherical ferromagnet, the magnetic phase diagram appears to be re-entrant and presents a quantum analogue of well-established classical examples of fluctuation-induced order.

The purely relaxational non-equilibrium dynamics of the quantum spherical model as described through a Lindblad equation is analysed. It is shown that the phenomenological requirements of reproducing the exact quantum equilibrium state as stationary solution and the associated classical Langevin equation in the classical limit g → 0 fix the form of the Lindblad dissipators, up to an overall time-scale. In the semi-classical limit, the models' behaviour becomes effectively the one of the classical analogue, with a dynamical exponent z = 2 indicating diffusive transport, and an effective temperature T eff , renormalised by the quantum coupling g. A different behaviour is found for a quantum quench, at zero temperature, deep into the ordered phase g ≪ g c (d), for d > 1 dimensions. Only for d = 2 dimensions, a simple scaling behaviour holds true, with a dynamical exponent z = 1 indicating ballistic transport, while for dimensions d = 2, logarithmic corrections to scaling arise. The spin-spin correlator, the growing length scale and the time-dependent susceptibility show the existence of several logarithmically different length scales.

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation-dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a markovian semi-classical approach, due to Bedeaux and Mazur, from a non-markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation-dissipation theorem is seen to be incompatible with a markovian dynamics. Possible applications to the quantum spherical model are discussed.