Kavli Affiliate: Hirosi Ooguri
| First 5 Authors: Yuya Kusuki, Hirosi Ooguri, Sridip Pal, ,
| Summary:
We use the thermal effective theory to prove that, for the vacuum state in
any conformal field theory in $d$ dimensions, the $n$-th R’enyi entropy
$S_A^{(n)}$ behaves as $S_A^{(n)} = frac{f}{(2pi n)^{d-1}} frac{ {rm
Area}(partial A)}{(d-2)epsilon^{d-2}}left(1+O(n)right)$ in the $n
rightarrow 0$ 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
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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