Lifshitz formulas for finite-density Casimir effect

Fujii, Daisuke; Nakayama, Katsumasa*; Suzuki, Kei   

The Lifshitz formula has long served as a foundational framework for analyzing the Casimir effect under finite-temperature conditions. In this work, we extend its applicability to situations involving finite chemical potential, thereby incorporating quantum field contributions beyond thermal effects. We illustrate the broad utility of the resulting generalized formula by exploring characteristic features of the Casimir effect across a variety of setups, including different boundary conditions, temperature regimes, spatial dimensionalities, and chemical potential imbalances. This extended framework is particularly relevant to systems such as dense quark matter and Dirac/Weyl semimetals, where the chemical potential functions as a tunable parameter that influences the Casimir force.