Shell-Filling Effect in the Entanglement Entropies of Spinful Fermions

We consider the von Neumann and Rényi entropies of the one-dimensional quarter-filled Hubbard model. We observe that for periodic boundary conditions the entropies exhibit an unexpected dependence on system size: for $L=4\mathrm{mod}8$ the results are in agreement with expectations based on conformal field theory, while for $L=0\mathrm{mod}8$ additional contributions arise. We explain this observation in terms of a shell-filling effect and develop a conformal field theory approach to calculate the extra term in the entropies. Similar shell-filling effects in entanglement entropies are expected to be present in higher dimensions and for other multicomponent systems.

Figure 1

DMRG data for $S_1-2/3\ln L$ as a function of $x=\ell /L$ for $U=t$ and $L=24$, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64. The lower and upper branches corresponds to lattice lengths $L=4\mathrm{mod}8$ and $L=0\mathrm{mod}8$, respectively.

Figure 2

$\delta S_1\equiv S_1-\frac{2}{3}\ln [\frac{L}{\pi }\sin (\frac{\pi \ell }{L})]-c_1^{\prime }$ as a function of $x=\ell /L$ for $U=0.3t$ and $L=24$, 32, 40, 48, 56, 64. The constant $c_1^{\prime }=1.205$ has been adjusted by hand. The solid curve is $-F_1^{\prime }(x)$.