Modular Hamiltonian for holographic excited states

In this work we study the Tomita-Takesaki construction for a family of excited states that, in a strongly coupled CFT—at large $N$—correspond to coherent states in an asymptotically AdS spacetime geometry. We compute the modular flow and modular Hamiltonian associated to these excited states in the Rindler wedge and for a ball shaped entangling surface. Using holography, one can compute the bulk modular flow and construct the Tomita-Takesaki theory for these cases. We also discuss generalizations of the entanglement regions in the bulk and how to evaluate the modular Hamiltonian in a large N approximation. Finally, we extend the holographic Banks, Douglas, Horowitz and Matinec (BDHM) formula to compute the modular evolution of operators in the corresponding CFT algebra, and propose this as a more general prescription.

Figure 1

A depiction of the computation of $T{r}_{{\overline{\mathrm{\Sigma }}}_A}|{\mathrm{\Psi }}_{\lambda }⟩⟨{\mathrm{\Psi }}_{\lambda }|$ in (2.6) is shown. The figure in the left represents the gluing of the states, (2.1) and (2.2), by tracing over ${\overline{\mathrm{\Sigma }}}_A$, the complement of ${\mathrm{\Sigma }}_A\subset \mathrm{\Sigma }$, shown in dark grey. The CFT external source $\lambda$, as well as conditions ${\varphi }^{\pm }$, provide boundary conditions for the path integral in the bulk. The blue line corresponds to the extremal surface ${\gamma }_A$. On the right, an example where a $\zeta$ Killing vector that runs as an angle from ${\mathrm{\Sigma }}_A^{-}$ to ${\mathrm{\Sigma }}_A^{+}$ pivoting around the blue point ${\gamma }_A$ is shown. All but radial and euclidean time coordinates have been suppressed.

Figure 2

(a) Closed symmetric Schwinger-Keldysh (sSK) path in the complex $t$-plane. The horizontal lines represent real time evolution. The vertical lines give imaginary time evolution, and the intervals $I_{\pm }$ have identical lengths equal to $\beta /2$. The insertion of sources in the vertical lines generate excitations over the (vacuum) thermal state. (b) The radial Rindler coordinate is shown so that one can see that the L and R pieces are connected. Sources ${\lambda }_{\pm }$ are turned on in the corresponding Euclidean regions $E_{\pm }$. Notice that the red point in (a) is now a red line, representing a spacelike $\mathrm{\Sigma }$ surface. The black arrow represents the path ordering of the operators given by $\mathcal{P}$, according to the parameter of the curve ($\theta$); it should not be confused with the time direction; for instance on the right side, the time parameter grows against the sense of the arrow (recall that $T_{-}<T_{+}$).

Figure 3

Upon building the modular Hamiltonian via an Euclidean path integral, the picture shows its evolution on the L/R wedges given by the operator ${\mathrm{\Delta }}^{is}$.

Figure 4

The figure shows a subregion $A$ of a system defined on the white plane and its causal diamond, as well as its entanglement wedge inside its holographic dual. Assuming a bulk timelike killing vector $\zeta$, whose flows are shown in red, one can define the spacelike surfaces ${\mathrm{\Sigma }}_A$. The blue lines are the flows that live on the boundary.

Figure 5

(a) A piece of Euclidean evolution cut at regions ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$ understood depicted as the matrix element $⟨n|{\rho }_{\lambda }|m⟩$ of a density matrix ${\rho }_{\lambda }$. (b) The same geometry can be instead understood as the coefficient $⟪n\tilde{m}|{\mathrm{\Psi }}_{\lambda }⟫$ of a ket $|{\mathrm{\Psi }}_{\lambda }⟫$ defined in the TFD Hilbert space $\mathcal{H}\otimes \tilde{H}$.

Figure 6

(a) Under the CHM map $\mathcal{U}$, one can also build a geometric interpretation of the modular operator and evolution for a ball in a CFT. Notice that the Euclidean evolution resembles a circle near $R$ but departs from this behavior at greater distances. (b) A closed contour in complex modular evolution can also be geometrically interpreted.