Entanglement hamiltonians in two-dimensional conformal field theory

TL;DR

This work classifies 2d CFT scenarios where the entanglement (modular) hamiltonian can be written as a local integral of the energy-momentum tensor, using an annulus mapping to extract the universal entanglement spectrum. It covers both time-independent setups (single and finite intervals, various geometries) and time-dependent quenches (global, local, and inhomogeneous), showing that the entanglement spectrum reduces to the boundary CFT data determined by the imposed boundary conditions, with universal leading growth tied to the annulus width W and subleading boundary entropies. The results connect the Rényi entropies to annulus partition functions and reveal how right- and left-moving sectors contribute differently in quenched dynamics, yielding explicit growth laws S_A(t) ~ (pi c/3β)t for global quenches and S_A(t) ~ (c/3) log t for local quenches, among others. These findings deepen the understanding of entanglement structure in 2d CFTs and provide a concrete framework for computing entanglement spectra via conformal mappings and boundary conformal data.

Abstract

We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy.

Figures (10)

• Figure 1: Euclidean space-time region for the path integral for reduced density matrix $\rho_A$ of the projected state $P_{\partial A}^\epsilon|\psi\rangle$. As usual the rows and columns of the density matrix are labelled by the values of the fields on the upper and lower edges of the slit along $A$ (shown in red.) The projection induces a boundary condition on the parts of the slit within $\epsilon$ of the boundary points between $A$ and $B$, shown in black. When the moments ${\rm Tr}\,\rho_A^n$ are computed, $n$ copies of this picture are sewn together cyclically along the red edges, but this leaves small black-edged holes around the boundary points. When $A$ is a single interval in an infinite system, the resulting manifold is topologically an annulus.
• Figure 2: Quasiparticle picture of the hamiltonian $K_A$ describing the entanglement between the semi-infinite intervals $A$ and $B$, after a global quench, as given in (\ref{['Kglobalapprox']}). Most of the entanglement is thermal, due to the R-moving particles (shown as solid lines) of pairs emitted from the interval $(-t,t)$, and reaching the subinterval $(0,2t)$ of $A$ at time $t$. The L-movers (shown as dashed lines) are correlated with these, and contribute similarly to $K_B$.
• Figure 3: Quasiparticle picture of the hamiltonian $K_A$ describing the entanglement between the semi-infinite intervals $A$ and $B$, after a local quench where two semi-infinite systems are joined together at $x=0$ at time $t=0$, as given in (\ref{['KA local large t']}). Most of the entanglement comes from R-moving particles of pairs emitted from near the junction, but also from R- and L-movers from near $x=-t$ and $x=t$ respectively.
• Figure 4: Euclidean space-times characterising the case of a single interval $A=(-R,R)$ on the infinite line, as discussed in Sec. \ref{['sec3a']}. In the left panel, the black dots denote the endpoints of the interval $A$. In the right panel the segments with $\textrm{Im}(w) =0$ and $\textrm{Im}(w) =2\pi$ are identified.
• Figure 5: Euclidean space-times describing the case of an interval $A=(-R,0)$ at the end of semi-infinite line, as discussed in Sec. \ref{['subsec:bdy']}. The thick black lines in the three panels are mapped into each other. In the right panel the segments on the lines $\textrm{Im}(w) =0$ and $\textrm{Im}(w) =2\pi$ are identified.