We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a C∗--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.
Benjamin Hinrichs, Jonas Lampart and Javier Valentín Martin
We study the ultraviolet problem for models of a finite-dimensional quantum mechanical system linearly coupled to a bosonic quantum field, such as the (many-)spin boson model or its rotating-wave approximation. If the state change of the system upon emission or absorption of a boson is either given by a normal matrix or by a 2-nilpotent one, which is the case for the previously named examples, we prove an optimal renormalization result. We complement it, by proving the norm resolvent convergence of appropriately regularized models to the renormalized one. Our method consists of a dressing transformation argument in the normal case and an appropriate interior boundary condition for the 2-nilpotent case.
We analyze the many-body Hamiltonian describing a mobile impurity immersed in a Bose-Einstein condensate (BEC). Using exact unitary transformations and rigorous error estimates, we show the validity of the Bogoliubov-Fröhlich Hamiltonian for the Bose polaron in the regime of moderately strong, repulsive interactions with a dilute BEC. Moreover, we calculate analytically the universal logarithmic correction to the ground state energy that arises from an impurity mediated phonon-phonon interaction.
In the Bogoliubov-Fröhlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.
We study a dilute system of N interacting bosons coupled to an impurity particle via a
pair potential in the Gross–Pitaevskii regime. We derive an expansion of the ground state energy up
to order one in the boson number, and show that the difference
of excited eigenvalues to the ground state is given by the eigenvalues
of the renormalized Bogoliubov–Fröhlich Hamiltonian in the limit of large boson numer.