Entanglement negativity in free-fermion systems: An overlap matrix approach

In this paper, we calculate the entanglement negativity in free-fermion systems by use of the overlap matrices. For a tripartite system, if the ground state can be factored into triples of modes, we show that the partially transposed reduced density matrix can be factorized and the entanglement negativity has a simple form. However, the factorability of the ground state in a tripartite system does not hold in general. In this situation, the partially transposed reduced density matrix can be expressed in terms of the Kronecker product of matrices. We explicitly compute the entanglement negativity for the Su-Schrieffer-Heeger model, the integer Quantum Hall state, and a homogeneous one-dimensional chain. We find that the entanglement negativity for the integer quantum Hall states shows an area law behavior. For the entanglement negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory. Our method provides a numerically feasible way to study the entanglement negativity in free-fermion systems.

Figure 1

The tripartite configuration of a dimerized Su-Schrieffer-Heeger model. The hopping amplitude of solid lines is $t_2$. One unit cell contains two orbitals that are pictured by $•$ and $\circ$. The system is divided into $A_1, A_2$ and $B$ parts by the red dashed lines. (a) Adjacent intervals for $A_1$ and $A_2$ regions. (b) Disjointed intervals for $A_1$ and $A_2$ regions.

Figure 2

The tripartite configuration of a integer quantum Hall state on a cylinder. The system is divided into $A_1, A_2$, and $B$. (a) Adjacent intervals for $A_1$ and $A_2$ regions. (b) Disjointed intervals for $A_1$ and $A_2$ regions.

Figure 3

The entanglement negativity $\mathcal{E}$ for two adjacent intervals ($A_1$ and $A_2$) in the integer quantum Hall state on a cylinder. The length of subsystem $A_2$ is $L_1$ and the compactified length on the cylinder is $L_y$. (a) $\mathcal{E}\left(L_y\right)$ for $L_1=0$ (blue), $L_1=1$ (green), and $L_1=10$ (red). (b) $\frac{\mathcal{E}\left(L_y\right)}{L_y}$ as a function of $L_1$. We set $L_x=100\pi$ that particles are restricted in the region $x⩽|L_x/2|$.

Figure 4

The entanglement negativity $\mathcal{E}$ for two disjointed intervals ($A_1$ and $A_2$) in the integer quantum Hall state on a cylinder. The length of subsystem $B$ and the compactified length on on the cylinder is $L_y$. (a) $\mathcal{E}\left(L_y\right)$ for $L_1=0$ (blue), $L_1=0.2$ (green), and $L_1=0.4$ (red). (b) $\frac{\mathcal{E}\left(L_y\right)}{L_y}$ as a function of $L_1$. We set $L_x=100\pi$ that particles are restricted in the region $x⩽|L_x/2|$.

Figure 5

Two tripartite configurations of 1D rings: (a) two adjacent intervals $A_1$ and $A_2$ with equal length $l$. (b) Two disjointed intervals $A_1$ and $A_2$ with equal length $l$ and the length between these two intervals is $d$.

Figure 6

(a) The entanglement negativity $\mathcal{E}(l/L)$ as a function of $l/L$ for two adjacent intervals in a homogeneous one-dimensional ring with $N$ particles. Two intervals are equal length $l$ and the total length of the ring is $L$. (b) $\mathcal{E}(l/L)-\mathcal{E}(1/4)$ as a function of $l/L$. Solid grey line is the scaling function from CFT calculation, $\mathcal{E}=\frac{1}{4}\ln \left[\tan \left(\pi \frac{l}{L}\right)\right]$.

Figure 7

The entanglement negativity $\mathcal{E}(d/l)$ as a function of $d/l$ for two disjointed intervals in a homogeneous one-dimensional ring with $N$ particles. Two disjointed intervals have equal length $l$ and the distance between two intervals is $d$.

Figure 8

(a) Su-Schrieffer-Heeger model on a one-dimensional ring. The hopping amplitude of dashed (solid) lines is $t_1\left(t_2\right)$. A unit cell contains two orbitals that are pictured by $•$ and $\circ$. The system is divided into $A$ and $B$ parts by the red dashed line. (b) The bipartite configuration of a ring with the lengths of $A$ and $B$ being equal. (c) The entanglement negativity $\mathcal{E}$ for an equally bipartite Su-Schrieffer-Heeger model with $t_1=t_2$ as a function of $\ln L$, where $L$ is the length of the ring.