https://doi.org/10.1007/s10773-020-04462-9 On Casimir Energy and Mutual Information in Non-relativistic Backgrounds M. Belyad 1 · M. Reza Tanhayi 1 Received: 22 January 2020 / Accepted: 27 March 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract We apply the AdS/CFT correspondence in considering the Casimir energy and some non- local entanglement measures in the non-relativistic backgrounds for general dynamical exponent z with and without hyperscaling violation exponent θ . In such background, we use holographic methods and compute the mutual information and tripartite information. We also consider the monogamy of holographic mutual information in the Schr¨ odinger-type geometry and by numerical analysis we show that this quantity is monogamous. Keywords Casimir energy · Mutual information · Holographic energy-momentum tensor 1 Introduction The Casimir effect describes non-trivial properties of the vacuum state in quantum field theory and it can be considered as one of the manifestations of macroscopic zero-point energy of the theory. From the experimental point of view this effect describes an attractive force between two electrically neutral parallel conductive planes placed in a vacuum, this force is due to the simple presence of the vacuum. Theoretically the Casimir effect can be described by the difference between zero-point fluctuation spectrum in the presence of these plates and in their absence. The key point of this effect is the boundary condition in a way that the zero-point energy of any relativistic field might be by boundary conditions. By the means of the quantized vacuum, the virtual particle-antiparticle pair can be created from vacuum and annihilated back to vacuum [1–10]. It is well known that in computing the vacuum energy of a given system the infini- ties appear which should be regularized and renormalized. There are many methods of regularization such as frequency cutoff regularization, Green’ s function method, zeta func- tion regularization, dimensional regularization and point-splitting method which have been studied in several papers. In the present study we follow Anti-de Sitter space/Conformal Field Theory (AdS/CFT) correspondence to consider this effect in the non-relativistic background.  M. Reza Tanhayi mtanhayi@ipm.ir 1 Department of Physics, Faculty of Basic Science, Islamic Azad University Central Tehran Branch (IAUCTB),Tehran, P.O. Box 14676-86831, Iran International Journal of Theoretical Physics (2020) 59:1905–1916 Published online: 20 2020 April