Entanglement in ground and excited states of gapped free-fermion systems and their relationship with Fermi surface and thermodynamic equilibrium properties

We study bipartite entanglement entropies in the ground and excited states of free-fermion models, where a staggered potential, ${\mu }_s$, induces a gap in the spectrum. Ground-state entanglement entropies satisfy the “area law”, and the “area-law” coefficient is found to diverge as a logarithm of the staggered potential, when the system has an extended Fermi surface at ${\mu }_s=0$. On the square lattice, we show that the coefficient of the logarithmic divergence depends on the Fermi surface geometry and its orientation with respect to the real-space interface between subsystems and is related to the Widom conjecture as enunciated by Gioev and Klich [Phys. Rev. Lett. 96, 100503 (2006)]. For point Fermi surfaces in two-dimension, the “area-law” coefficient stays finite as ${\mu }_s\to 0$. The von Neumann entanglement entropy associated with the excited states follows a “volume law” and allows us to calculate an entropy density function $s_V\left(e\right)$, which is substantially different from the thermodynamic entropy density function $s_T\left(e\right)$, when the lattice is bipartitioned into two equal subsystems but approaches the thermodynamic entropy density as the fraction of sites in the larger subsystem, that is integrated out, approaches unity.

Figure 1

von Neumann entanglement entropy in the ground state for different systems with linear size $L$ for (a) $d=1$, (b) $d=2$ uniform hopping model, (c) $d=2$ $\pi$-flux model, and (d) $d=3$ model. The area law is valid in all cases.

Figure 2

Coefficient of the “area-law” term associated with the von Neumann entropy in different dimensions, including the $\pi$-flux case in 2D.

Figure 3

Area-law coefficient for second Renyi entropy for spinless fermions in two dimensions. Series expansion results are compared with results from correlation matrix method.

Figure 4

Area law coefficient for second Renyi entropy for spinless fermions with $\pi$ flux in two dimensions. Series expansion results are compared with the correlation matrix method.

Figure 5

von Neumann entanglement entropy with different values of anisotropy $q=t_x/t_y$ on the square lattice.

Figure 6

Coefficient of the log-divergence in the area-law coefficient normalized to the 1D case and compared with the Widom conjecture.

Figure 7

Entanglement entropy for the excited states in the two-dimensional system scales with the volume of the system. Data are shown for $t/{\mu }_s=0.5$.

Figure 8

Entanglement entropy density of the excited states compared with the thermal entropy density. The results shown are for ${\mu }_s/t=0.5$.

Figure 9

Difference between thermal and von Neumann entanglement entropies at $e=0$ for $t/m{u}_s=0.5$ as a function of $N_A/N$. The dark circles represent the calculated data points. The line is a guide to the eye.