Quantum Entanglement of Localized Excited States at Finite Temperature

Abstract

In this work we study the time evolutions of (Renyi) entanglement entropy of locally excited states in two dimensional conformal field theories (CFTs) at finite temperature. We consider excited states created by acting with local operators on thermal states and give both field theoretic and holographic calculations. In free field CFTs, we find that the growth of Renyi entanglement entropy at finite temperature is reduced compared to the zero temperature result by a small quantity proportional to the width of the localized excitations. On the other hand, in finite temperature CFTs with classical gravity duals, we find that the entanglement entropy approaches a characteristic value at late time. This behaviour does not occur at zero temperature. We also study the mutual information between the two CFTs in the thermofield double (TFD) formulation and give physical interpretations of our results.

Figures (12)

• Figure 1: Penrose diagram for an eternal AdS black hole.
• Figure 2: The reduced density matrix for our excited states is described by a path-integral on a cylinder with complex coordinates $x=\sigma+i\tau$. The two operators inserted at distance $x=-l$ from the cut $A$ are separated by $\Delta x=2i\epsilon$. $\text{Tr} \rho^n_A$ is computed as a partition function on the n-copies of these cylinders glued along $A$.
• Figure 3: The plot of time evolution of holographic entanglement entropy. It shows $S_A(t)-S_A(0)$ for the infinitely large subsystem $A$ (i.e. a semi half line) as a function of $t$. Blue: full HEE, Red: early time \ref{['earlyHEE']}, Yellow: late time \ref{['lateHEE']}. We set $R=4G_N=1,\epsilon=0.001,\beta=3$ and $M=0.1$.
• Figure 4: Reduced density matrix in the thermofield double state. CFT$_L$ corresponds to $x=i0$ and CFT$_R$ to $x=i\frac{\beta}{2}$
• Figure 5: Growth of the Renyi entanglement entropies for free scalar in TDF. Parameters $\epsilon/\beta=1/12$ and $b\neq 0$.