Quantized and maximum entanglement from sublattice symmetry

We observe that the many-body eigenstates of any quadratic, fermionic Hamiltonian with sublattice symmetry have quantized entanglement entropies between the sublattices: the entanglement comes in multiple singlets. Moreover, such systems always have a ground state that is maximally entangled between the two sublattices. In fact, we also show that under the same assumptions there always exists a (potentially distinct) basis of energy eigenstates that do not conserve the particle number in which each energy eigenstate is maximally entangled between the sublattices. No additional properties, such as translation invariance, are required. We also show that the quantization of ground-state entanglement may persist when interactions are introduced.

Figure 1

Exemplary geometries to which our results apply. The only condition is that the fermionic modes can be partitioned into two sets $A$ and $B$, here denoted by yellow and purple dots, such that the only nonzero tunneling amplitudes (dashed gray lines) are between the two sets.

Figure 2

Entanglement entropy of odd sites $S\left(B\right|E_s)$ [in units of $log\left(2\right)]$ in the energy eigenstate $|E_s\rangle$ vs energy $E_s$ for all $2^{12}$ eigenstates of a randomly chosen Hamiltonian $H_F$ on a chain of $N=12$ sites, with $A$ and $B$ corresponding to the even and odd lattice sites, respectively. The quantization of the entropy is clearly visible.

Figure 3

Entanglement entropy density of even sites in the thermodynamic limit in the ground state of the Hamiltonian $H_F^{XX}+\mu \widehat{N}$ as a function of the chemical potential $\mu$.

Figure 4

The sublattice entanglement entropy $S\left(B\right|E_{\mathrm{GS}})$ as a function of the interaction strength $g$ in the interacting model $H_{\mathrm{int}}$ defined in (8) for system size $N=12$.