Bipartite entanglement entropy of the excited states of free fermions and harmonic oscillators

We study general entanglement properties of the excited states of the one-dimensional translational invariant free fermions and coupled harmonic oscillators. In particular, using the integrals of motion, we prove that these Hamiltonians independent of the gap (mass) have infinite excited states that can be described by conformal field theories with integer or half-integer central charges. In the case of free fermions, we also show that because of the huge degeneracy in the spectrum, even a gapless Hamiltonian can have excited states with an area law. Finally, we study the universal average entanglement entropy over eigenstates of the Hamiltonian of the free fermions introduced recently in Vidmar et al. [Phys. Rev. Lett. 119, 020601 (2017)]. In particular, using a duality relation, we map the average bipartite entanglement entropy over all the exponential numbers of eigenstates of a generic free fermion Hamiltonian to the problem of the calculation of the average multibipartite entanglement entropy over a single eigenstate of the $\mathit{XX}$ chain. This relation can be useful for experimental measurement of the universal average entanglement entropy. Part of our conclusions can be extended to the quantum spin chains that are associated with the free fermions via Jordan-Wigner transformation.

Figure 1

Top: Bipartite von Neumann entropy with respect to the logarithm of the size of the subsystem for the ground state of $I_m^{-}\left(XX\right)$ with $m=1,2,3$. Bottom: Bipartite von Neumann entropy with respect to the size of the subsystem for the ground state of $I_m^{-}\left(XX\right)$ with $m=\frac{L}{8},\frac{L}{4},\frac{L}{2}$, and $L=200$.

Figure 2

Bipartite von Neumann entropy with respect to the logarithm of the size of the subsystem for the ground state of $I_m^{+}$ with $m=1,2,3$ produced out of the critical Ising parameters.

Figure 3

Various averagings of the entanglement entropy of the eigenstates of free fermions ($\mathit{XX}$ chain) in a periodic chain with $L=28$ sites. Results are plotted as a function of the linear subsystem size $l$. In the inset, $s=\frac{S_{vN}(L/2)}{\frac{L}{2}ln2}$ is plotted as a function of the system size for $L\le 30$. As one can see, VHBR (blue square) and WA (yellow diamond) averagings converge to the number 0.5378 reported in Ref. [38]. However, the MAX (red triangle) and MIN (green triangle) do not converge to the same value.

Figure 4

Entanglement entropy of CHO with respect to the subsystem length ($\ell$) for $L=160$. We take $r\in \{2,4,6,\cdots ,L-1\}$ (blue circles), $r\in \{2,6,10,\cdots ,L-1\}$ (green squares), and $r\in \{2,10,18,\cdots ,L-1\}$ (red triangles). With this set of parameters, the volume-law behavior is apparent.

Figure 5

In this figure, the ratio of independent energies to total number of energies is plotted versus the system size. $\frac{N_{\text{ind}}}{N_T}$ decreases exponentially with size of the system. Inset: Plot of $\mathrm{\Delta }{E}_{\text{min}}$ versus the size of the system. This quantity gives us a measure to set a precision for finding the unequal energy levels of the spectrum. Note that for $L$ even, we have degeneracy even in the level of the energy modes, which is the main reason for stronger decay of $\frac{N_{\text{ind}}}{N_T}$ with respect to the $L$ odd case.