<p>We study a stratified braneworld model in which a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((d+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional asymptotically AdS bulk contains <i>N</i> parallel AdS<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(_d\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>d</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> end-of-the-world branes at fixed radii. Using the gravitational replica trick and the quantum extremal surface (QES) prescription, we compute the holographic entanglement entropy of regions on the branes. We find multiple <i>island</i> configurations—one per brane layer—and discrete transitions between them: as the region grows, new islands successively appear on deeper branes when their would-be entropy would exceed the horizon area of that layer, producing a multistep (<i>staircase</i>) Page curve. In the limit of many backreacting layers, the system becomes equivalent to a single thick brane with smoothly varying warp factor <i>A</i>(<i>z</i>) and energy density <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\rho (z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We apply these ideas explicitly to stratified black hole and wormhole geometries, finding that the competition between connected extremal surfaces and island surfaces reproduces the Page curve and restores unitarity.</p>

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Entanglement Entropy, Islands, and Multistep Page Curves in a Stratified AdS Braneworld

  • Alberto Miró Morán

摘要

We study a stratified braneworld model in which a \((d+1)\) ( d + 1 ) -dimensional asymptotically AdS bulk contains N parallel AdS \(_d\) d end-of-the-world branes at fixed radii. Using the gravitational replica trick and the quantum extremal surface (QES) prescription, we compute the holographic entanglement entropy of regions on the branes. We find multiple island configurations—one per brane layer—and discrete transitions between them: as the region grows, new islands successively appear on deeper branes when their would-be entropy would exceed the horizon area of that layer, producing a multistep (staircase) Page curve. In the limit of many backreacting layers, the system becomes equivalent to a single thick brane with smoothly varying warp factor A(z) and energy density \(\rho (z)\) ρ ( z ) . We apply these ideas explicitly to stratified black hole and wormhole geometries, finding that the competition between connected extremal surfaces and island surfaces reproduces the Page curve and restores unitarity.