2D Quantum Gravity from Quantum Entanglement

In quantum systems with many degrees of freedom the replica method is a useful tool to study the entanglement of arbitrary spatial regions. We apply it in a way that allows them to backreact. As a consequence, they become dynamical subsystems whose position, form, and extension are determined by their interaction with the whole system. We analyze, in particular, quantum spin chains described at criticality by a conformal field theory. Its coupling to the Gibbs’ ensemble of all possible subsystems is relevant and drives the system into a new fixed point which is argued to be that of the 2D quantum gravity coupled to this system. Numerical experiments on the critical Ising model show that the new critical exponents agree with those predicted by the formula of Knizhnik, Polyakov, and Zamolodchikov.

Figure 1

The critical ratio ${\lambda }^{-x}=C(\lambda L,8)/C(L,8)$ as a function of the fugacity $z$ for $L=96$, $\lambda =\frac{4}{3}$ and two replicas [see Eqs. (7, 8)]. The top and the bottom arrows indicate, respectively, the expected value ${\lambda }^{-4{\Delta }_{\sigma }^o}$ for the pure system ($z=0$) and ${\lambda }^{-4{\Delta }_{\sigma }-1}$ for the coupled system at $z=1$. ${\Delta }_{\sigma }^o$ and ${\Delta }_{\sigma }$ are related through the KPZ formula.

Figure 2

The two bottom lines are one-parameter fits of $[C(L,4)+C(L,8)]/2$ in the 1D setting at $z=1$ for $n=2$ and $n=3$. The top line is a similar fit in the 2D setting at $z=z_c$ for $n=2$. In this doubly logarithmic plot the data lie on straight lines. The slope of the top line is exactly $-4{\Delta }_{\sigma }$ while in the 1D settings is shifted by the integer $1-n$. In order to represent all data in a single plot, the 2D data are reduced by a factor of 10.

Figure 3

The link variable as a function of the size of the lattice. The solid curve is a fit to Eq. (9).