Entanglement negativity in two-dimensional free lattice models

We study the scaling properties of the ground-state entanglement between finite subsystems of infinite two-dimensional free lattice models, as measured by the logarithmic negativity. For adjacent regions with a common boundary, we observe that the negativity follows a strict area law for a lattice of harmonic oscillators, whereas for fermionic hopping models the numerical results indicate a multiplicative logarithmic correction. In this latter case we conjecture a formula for the prefactor of the area-law violating term, which is entirely determined by the geometries of the Fermi surface and the boundary between the subsystems. The conjecture is tested against numerical results and a good agreement is found.

Figure 1

Different tripartitions of a 1D chain.

Figure 2

Different choices for subsystems $A_1$ and $A_2$ with their common boundary $\partial A_{12}$ shown in red. The blue rectangle corresponds to the boundary $\partial A$ of subsystem $A=A_1\cup A_2$.

Figure 3

Logarithmic negativity per surface area for the harmonic lattice, between two halves of a $\ell \times \ell$ square with vertical (left) and horizontal (right) partitioning (note the different vertical scales). The couplings $k_y$ are varied while the other parameters are fixed as $k_x=1$ and ${\mathrm{\Omega }}_0={10}^{-3}$.

Figure 4

Logarithmic ratios per linear size $-{\mathcal{E}}_n/\ell$ for the 2D hopping model with vertical (left) and horizontal (right) partitioning and various couplings $t_y$. The data is plotted on a logarithmic horizontal scale. The dashed lines have slopes given by $2/9{\sigma }_a$ and $2/9{\sigma }_b$ for $n_o=3$ (up) and by $1/4{\sigma }_a$ and $1/4{\sigma }_b$ for $n_e=4$ (down), respectively, see Eq. (39).

Figure 5

Logarithmic ratios per linear size $-{\mathcal{E}}_n/\ell$ for the 2D hopping model with corner (left) and embedded (right) partitioning and various couplings $t_y$. The data is plotted on a logarithmic horizontal scale. The dashed lines have slopes given by $2/9{\sigma }_c$ and $2/9{\sigma }_d$ for $n_o=3$ (up) and by $1/4{\sigma }_c$ and $1/4{\sigma }_d$ for $n_e=4$ (down), respectively, see Eq. (40).

Figure 6

The Fermi surface of the infinite 2D hopping model for various anisotropies $t_y$ and $t_x=1$.