We study the phases of scalar field theories in thermal AdSd+1 spaces for d = 1, 2, 3. The analysis is done for theories with global O(N) symmetry for finite as well as large N. The symmetry-preserving and symmetry-breaking phases are identified as a function of the mass-squared of the scalar field and temperature. On the way we also describe a method for computing one-loop partition function for scalar field on thermal AdSd+1 for arbitrary d that reproduces results known in the literature. The derivation is based on the method of images and uses the eigenfunctions of the Laplacian on Euclidean AdS.

We study the phases of Yukawa theories at weak coupling and the Gross-Neveu models in AdS spaces at zero and finite temperature. Following the method used in [15], we first compute the one-loop partition functions, using the generalized eigenfunctions of the Dirac and Laplace operators on Euclidean AdS in the Poincaré coordinates. These functions satisfy desired periodicities under thermal identification. The method replicates results for partition functions known in the literature. We then study the phases of these field theories with fermions as regions in the corresponding parameter spaces at zero temperature. The phases and the corresponding phase boundaries are further identified as a function of the mass-squared of the scalar field and temperature for the Yukawa theories. While for the Gross-Neveu models, the changes in the phases as a function of the fermionic mass and the coupling constant at finite temperature are discussed. The Gross-Neveu-Yukawa model is studied for AdS4. We also note certain deviations from phases of these theories in flat space.