Renormalization group flow of entanglement entropy to thermal entropy

Utilizing the holographic technique, we investigate how the entanglement entropy evolves along the renormalization group flow. After introducing a new generalized temperature which satisfies the thermodynamicslike law even in the IR regime, we find that the renormalized entropy and the generalized temperature in the IR limit approach the thermal entropy and thermodynamic temperature of a real thermal system. This result implies that the microscopic quantum entanglement entropy in the IR region leads to the thermodynamic relation up to small quantum corrections caused by the quantum entanglement near the entangling surface. Intriguingly, this IR feature of the entanglement entropy universally happens regardless of the detail of the dual field theory and the shape of the entangling surface. We check this IR universality with a most general geometry called the hyperscaling violation geometry which is dual to a relativistic nonconformal field theory.

Figure 1

(a) The inverse of the entanglement temperature depending on the subsystem size and (b) its RG flow.

Figure 2

The renormalized entropy and its derivative depending on the subsystem size. Figure 1 shows that the renormalized entropy monotonically decreases along the RG flow. Since its derivative approaches zero in Fig 1, we see that the renormalized entropy converges into a certain value ($\approx -435$) as $l\to \infty$.

Figure 3

The value of the test function $S_T$. In the IR limit, it approaches to $S_T\approx -359$, which is almost equal to the value of $-(d-2)a_{\mathrm{UV}}$.