Finite-Size Effects on Symmetry Restoration in a Rotating Scalar System with Spatially Dependent Thermal Self-Energy
We investigate $\mathbb{Z}_2$ symmetry restoration in a real $λϕ^4$ theory confined to a finite cylindrical region undergoing rigid rotation. The finite transverse extent is imposed through Dirichlet boundary conditions, resulting in a discrete Fourier-Bessel spectrum and an explicit radial dependence of the fluctuation propagator. Using the background-field method, we derive the one-loop effective potential while retaining the spatial structure induced by the finite geometry. At finite temperature, rotation modifies the thermal mode energies through angular-momentum-dependent shifts, and the coincident-point propagator generates a position-dependent thermal self-energy that is incorporated through ring resummation. We use the resulting spatially resolved effective potential to determine the symmetry-restoration temperature $T_c(Ω,R)$ and examine its dependence on the angular velocity and transverse size. The results show that the finite transverse size of the system modifies the symmetry-restoration temperature, leading to a nontrivial dependence of $T_c(Ω,R)$ on $R$. An additional scaling behavior emerges when the normalized critical temperature is expressed in terms of the boundary velocity $ΩR$: systems with different transverse sizes exhibit the same relative modification of $T_c$ when their boundary velocities are equal. This scaling persists after the inclusion of the position-dependent thermal self-energy through ring resummation, indicating that the $ΩR$ dependence of the rotational response is maintained in the interacting finite system.