The Casimir effect, a two-body interaction via vacuum fluctuations, is a fundamental property of quantum systems. In solid state physics it emerges as a long-range interaction between two impurity atoms via virtual phonons. In the classical limit for the impurity atoms in D dimensions the interaction is known to follow the universal power-law U (r) ∼ r −D . However, for finite masses of the impurity atoms on a lattice, it was predicted to be U (r) ∼ r −2D−1 at large distances. We examine how one power-law can change into another with increase of the impurity mass and in presence of an external potential. We provide the exact solution for the system in one-dimension. At large distances indeed U (r) ∼ r −3 for finite impurity masses, while for the infinite impurity masses or in an external potential it crosses over to U (r) ∼ r −1 . At short distances the Casimir interaction is not universal and depends on the impurity mass and the external potential.

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We develop a theory of the phonon mediated Casimir interaction between two point-like impurities, which is based on the single impurity scattering T -matrix approach. We show that the Casimir interaction at T = 0 falls off as a power law with the distance between the impurities. We find that the power in the weak and unitary phonon-impurity scattering limits differs, and we relate the power law to the low-energy properties of the single impurity scattering T -matrix. In addition, we consider the Casimir interaction at finite temperature and show that at finite temperatures the Casimir interaction becomes exponential at large distances.

The Sachdev-Ye-Kitaev (SYK) model describes interacting fermionic zero modes in zero spatial dimensions, e.g., quantum dot, with interactions strong enough to completely washout quasiparticle excitations in the infrared. In this paper we consider the complex-valued SYK model at initial temperature T and chemical potential μ coupled to a large reservoir by a quench at time t = 0. The reservoir is kept at zero temperature and charge neutrality. We find that the dynamics of the discharging process of the SYK quantum dot reveals a distinctive characteristic of the SYK non-Fermi liquid (nFl) state. In particular, we focus on the tunneling current induced by the quench. We show that the temperature dependent contribution to the current's half-life scales linearly in T at low temperatures for the SYK nFl state, while for the Fermi liquid it scales as T 2 .