Entanglement entropy of low-lying excitation in localized interacting system: Signature of Fock space delocalization

The properties of the entanglement entropy (EE) of low-lying excitations in one-dimensional disordered interacting systems are studied. The ground state EE shows a clear signature of localization, while low-lying excitations show a significantly longer localization length. The dependence of the localization as a function of interaction strength and sample length is studied using the density matrix renormalization group (DMRG). This behavior corresponds to the presence of the predicted many-particle critical energy in the vicinity of the ground state energy. The implications of these results on experiments are discussed.

Figure 1

Upper panel: The EE of a metallic system ($W=0.7$ and $U=-1$) as function of $L_A$, for different system sizes $L$. The symbols correspond to the DMRG results, where $◯,L=100$; $\square ,L=200$; $\diamond ,L=300$; $△,L=500$; and $▽,L=700$. The colors correspond to the excitation number (continuous black: ground state, dashed red: first excitation, long dashed green: second excitation, dot-dashed blue: third excitation). The thick dashed line corresponds to Eq. (3). Lower panel: The EE scaled as $X_A=L_A/L$. The thick line corresponds to Eq. (5) with $n_0=1.82$. Inset: The EE for $L_A=0.2L$ as a function of $L$. The symbols correspond to the numerical EE, while the lines correspond to a fit to $c^{\left(i\right)}+(1/6)ln\left(L\right)$.

Figure 2

The EE of a weakly disordered system (upper panel, ${\xi }_{MP}\sim 100$) and strongly disordered systems (lower panel, ${\xi }_{MP}\sim 50$) as a function of $L_A$, for different $L$. The symbols represent DMRG results as in Fig. 1. The different excitations are indicated by color and line shape. Upper panel: The full symbols correspond to $U=0.6$ and $W=0.5$ ($\xi \sim 400,{\xi }_{MP}\sim 100$), while empty symbols to $U=1.8$ and $W=0.15$ ($\xi \sim 4500,{\xi }_{MP}\sim 100$). Lower panel: Full symbols $U=0.6$ and $W=0.7$ ($\xi \sim 200,{\xi }_{MP}\sim 50$), empty symbols $U=1.8$ and $W=0.25$ ($\xi \sim 1600,{\xi }_{MP}\sim 50$).

Figure 3

The third ESEE for the weakly disordered system (${\xi }_{MP}\sim 100$) depicted in the upper panel of Fig. 2 as a function of $X_A$. Again, the symbols correspond to the DMRG results as in Fig. 1 ($U=1.8$ open symbols, $U=0.6$ filled symbols). For comparison the GSEE of $L=700$ is depicted by black left triangle symbols. The dashed green curve corresponds to Eq. (5) with $n_0=2.1$ and $c^{\left(i\right)}=0.53$, while the red curve corresponds to Eq. (3) with $c_g=1.22$.