Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice

A quantum critical (QC) fluid exhibits universal subleading corrections to the area law of its entanglement entropies. In two dimensions, when the partition involves a corner of angle $\theta$, the subleading term is logarithmic with coefficient $a_{\alpha }\left(\theta \right)$ for the $\alpha$-Rényi entropy. In the smooth limit $\theta \to \pi ,a_1\left(\theta \right)$ yields the central charge of the stress tensor when the QC point is described by a conformal field theory (CFT). For general Rényi indices and angles, $a_{\alpha }\left(\theta \right)$ is richer and few general results exist. We study $a_{\alpha }\left(\theta \right)$ focusing on two benchmark CFTs, the free Dirac fermion and boson. We perform numerical lattice calculations to obtain high precision results in $\theta ,\alpha$ regimes hitherto unexplored. We derive field theory estimates for $a_{\alpha }\left(\theta \right)$, including exact results, and demonstrate an excellent quantitative match with our numerical calculations. We also develop and test strong lower bounds, which apply to both free and interacting QC systems. Finally, we comment on the near collapse of $a_{\alpha }\left(\theta \right)$ for various theories, including interacting $O\left(N\right)$ models.

Figure 1

Examples of corners in the entangling boundary, superimposed on an underlying square lattice, with opening angles of $\theta =\pi /4$ (left) and $tan\theta =-2$ (right).

Figure 2

Lower bounds on $a_2\left(\theta \right)$ for the boson. $M=5$ lower bound ${\mathfrak{a}}_2^{\left(5\right)}\left(\theta \right)$, together with the corresponding series including terms up to ${(\theta -\pi )}^{10}$. The shaded region is not allowed. Also shown are the high-precision ansatz (22) and the BMW ansatz (21) for the entire corner function. The markers represent the lattice results, see Sec. 4.

Figure 3

Comparison of the field theory ansatze with the truncated series for $a_2^b\left(\theta \right)$ including terms up to order ${(\theta -\pi )}^{16}$. The series is expected to be a lower bound. Also shown is the small angle behavior ${\kappa }_2/\theta$.

Figure 4

${\sigma }_{\alpha }^{\left(p\right)}$ coefficients for the boson, see Table 3, normalized by their large-$p$ asymptotic value, $2{\kappa }_{\alpha }/{\pi }^{2p+3}$. The $p=0$ ones are not shown because they are substantially greater. The ${\kappa }_{\alpha }$ are given in Ref. [34].

Figure 5

Closed polygons in periodic boundary conditions in order to obtain the corner contribution from fitting the area law to the full entanglement entropy. Examples are shown for $tan\theta =-2$ (left) and for $tan\tilde{\theta }=2$ (right), which requires the subtraction the $\theta$ contribution.

Figure 6

Bipartitions $c_1,c_2,c_3$, and $c_4$ used to determine the corner entanglement for $\theta =\pi /2$ according to Eq. (34).

Figure 7

The entropy $S_1$ for free bosons as a function of $1/L$ for the case where region $A$ is a single site and the lattice has open boundary conditions (the same as the boundary conditions used in the NLCE). As expected, we see that the leading IR contribution to the entropy is $\mathcal{O}(1/L)$ when the boson is massless $\left(m=0\right)$. Similarly, the inset shows the single-site entropy for the massless fermion (on a lattice with antiperiodic boundary conditions) and illustrates that the leading IR contribution is $\mathcal{O}(1/L^3)$, which corresponds to a faster decay than the entropy for the free massless boson.

Figure 8

Angle dependence for the von Neumann $\left(\alpha =1\right)$ entropy. The lattice data agree very well with our new field theory based ansatz (22). For the field theory ansatz and series curves, see Sec. 3.

Figure 9

Corner entanglement for Rényi entropies for the free boson (left) and the Dirac fermion (right). The numerical data are compared to our new field theory ansatz (22), at $M=8$ (7) for the boson (fermion).

Figure 10

Collapsing the corner entanglement after dividing $a_{\alpha }\left(\theta \right)$ by the smooth-limit coefficient ${\sigma }_{\alpha }$ (left). The solid lines are obtained using field theory (22), while the markers are lattice data. We also show the $\pi /2$ data for the $O\left(N\right)$ Wilson-Fisher QCPs with Ising, XY and Heisenberg symmetry. The AdS/CFT result corresponds to a strongly-interacting CFT [35]. The solid lines show the new ansatz. Another way to collapse the data is to divide $a_{\alpha }\left(\theta \right)$ by the sharp limit coefficient ${\kappa }_{\alpha }$ (right), which achieves a better agreement for $\theta <\pi /2$.

Figure 11

Testing the boson-fermion duality at $\theta =\pi /2$. The data for fermions has been obtained by a fit to the area law using an exact diagonalization of the subsystem correlation matrix. For bosons, the NLCE was used. $a_{\infty }(\pi /2)$ was computed using the BMW ansatz [34] to be $0.0093\left(0.00715\right)$ for the fermion (boson) respectively. Both plots display the same data with different $\alpha$ prefactors, in order to resolve clearly the duality for both sides of $\alpha =1$.

Figure 12

Solid: corner function for holographic theories [35], normalized by the lower bound ${\mathfrak{a}}^{\left(5\right)}$. Dashed: the series including terms up to ${(\theta -\pi )}^{10}$, normalized by the lower bound.