We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in $d$ dimensions, the $n$th Rényi entropy $S_A^{(n)}$ behaves as $S_A^{(n)}=[f/(2\pi n{)}^{d-1}][\text{Area}(\partial A)/(d-2){\epsilon }^{d-2}](1+O(n))$ in the $n\to 0$ limit when the boundary of the entanglement domain $A$ is spherical with the UV cutoff $\epsilon$. The theory dependence is encapsulated in the cosmological constant $f$ in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain $A$. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive $n$. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.

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• Conformal field theory
• Effective field theory
• Entanglement in field theory
• Higher-dimensional field theories
• Symmetries