Entropy on a null surface for interacting quantum field theories and the Bousso bound

We study the vacuum-subtracted von Neumann entropy of a segment on a null plane. We argue that for interacting quantum field theories in more than two dimensions, this entropy has a simple expression in terms of the expectation value of the null components of the stress tensor on the null interval. More explicitly, $\mathrm{\Delta }S=2\pi \int d^{d-2}y{\int }_0^{1}d{x}^{+}g(x^{+})⟨T_{++}⟩$, where $g(x^{+})$ is a theory-dependent function. This function is constrained by general properties of quantum relative entropy. These constraints are enough to extend our recent free field proof of the quantum Bousso bound to the interacting case. This unusual expression for the entropy as the expectation value of an operator implies that the entropy is equal to the modular energy, $\mathrm{\Delta }S=⟨\mathrm{\Delta }K⟩$, where $K$ is the modular Hamiltonian. We explain how this equality is compatible with nonvanishing $\mathrm{\Delta }S$. Finally, we explicitly compute the function $g(x^{+})$ for theories that have a gravity dual.

Figure 1

The Rényi entropies for an interval $A$ involve the two point function of defect operators $D$ inserted at the end points of the interval. An operator in the $i$th CFT becomes an operator in the $(i+1)$th CFT when we go around the defect.

Figure 2

The functions $g(v)$ in the expression for the modular Hamiltonian of the null slab, for conformal field theories with a bulk dual. Here $d=2,3,4,8,\infty$ from bottom to top. Near the boundaries ($v\to 0$, $v\to 1$), we find $g\to 0$, $g^{\prime }\to \pm 1$, in agreement with the modular Hamiltonian of a Rindler wedge. We also note that the functions are concave (see Sec. 4). In particular, we see that $|g^{\prime }|\le 1$, in agreement with our general argument of Sec. 3.

Figure 3

Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a spacelike interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself.

Figure 4

The maximum value $E_{\max }(p)$ of $E$ for getting a surface that returns to the boundary (solid line). For comparison, we also plotted the line $E=p-1$ (dashed line). The extremal surface solutions of interest appear in the region $p>1$, $0<E<E_{\max }(p)$. Here, we have taken $d=3$.

Figure 5

Curves of constant $\mathrm{\Delta }{x}^{+}$ (black solid curves) and $\mathrm{\Delta }{x}^{-}$ (blue dashed curves), in the logarithmic parameter space defined by $(\log (p-1),-\log (E_{\max }(p)-E)/E_{\max }(p))$. The value $p=1$ maps to $-\infty$ and $p=\infty$ maps to $+\infty$ on the horizontal axis, while $E=0$ maps to 0 and $E=E_{\max }(p)$ maps to $+\infty$ on the vertical axis. The thick blue contour represents the null solutions with $\mathrm{\Delta }{x}^{-}=0$. Above this contour, the boundary interval is timelike. If $\mathrm{\Delta }{x}^{+}\gtrsim 15$ and we follow a contour of constant $\mathrm{\Delta }{x}^{+}$, we find two solutions with exact $\mathrm{\Delta }{x}^{-}=0$. For all contours of fixed $\mathrm{\Delta }{x}^{+}$, there exists an asymptotic null solution in the limit $p\to \infty$.

Figure 6

The vacuum-subtracted extremal surface area versus $\mathrm{\Delta }{x}^{-}$ for fixed $\mathrm{\Delta }{x}^{+}$ ($\mathrm{\Delta }{x}^{+}=20$ and $\mathrm{\Delta }{x}^{+}=10$ for $d=3$ is shown). This numerical simulation demonstrates that, for sufficiently large $\mathrm{\Delta }{x}^{+}$ (in $d=3$, the condition is $\mathrm{\Delta }{x}^{+}\gtrsim 15$), there exists a phase transition at finite $\mathrm{\Delta }{x}^{-}$ to a different, perturbative class of solutions. At smaller $\mathrm{\Delta }{x}^{+}$, there is no such phase transition.