Signatures of metal-insulator and topological phase transitions in the entanglement of one-dimensional disordered fermions

We study one-dimensional disordered fermions that either undergo metal-insulator transitions or topological phase transitions to become trivial Anderson insulators. We focus on using entanglement to elucidate how the spatial, momentum, and internal degrees of freedom of fermions are affected by the presence of disorder in such cases. We develop entanglement tools that reveal the existence of metallic states in the presence of disorder and further show clear signatures of the corresponding localization transition even in the presence of interactions. In systems where the internal degrees of freedom are coupled with the motion of the electrons, topological phases develop. We subject a topological insulator model to different types of disorder and discuss how the topological aspects of the system can be captured through entanglement, even at strong disorder.

Figure 1

Fourier components of the scattering potential (black, column 1), the spatial (blue, column 2) and momentum (red, column 3) entanglement spectrum, and the localization length (green, column 4) for three models: uncorrelated disorder (a)–(d); RDM (e)–(h); and RTM (i)–(l). For all of these plots, the disorder strength was fixed at ${\epsilon }_b=-0.5t$ and the lattice size is $N=1000$.

Figure 2

Disorder-averaged scaling of the spatial entanglement entropy for a sub-system region of size $\ell$ for the (a) RDM and the (b) RTM. The blue dots denote the numerical calculation when there are extended states in the system and the Fermi-level is tuned to the delocalized resonance state (with ${\epsilon }_b=-0.5$ for the RDM and ${\epsilon }_b=1.0$ for the RTM), the red line denotes the corresponding analytical expression and the green dots denote the result when the disorder strength is strong enough to localize all states. The number of sites for this calculation is $N=200,$ and the number of disorder realizations used is 100.

Figure 3

Momentum entanglement spectrum and entropy of the random dimer model: The momentum entanglement spectrum is shown in (a) as a function of the disorder strength and (b)–(e) as a function of the Fermi energy for different disorder strengths. The momentum entanglement entropy is shown in (f) as a function of the disorder strength and (g)–(j) as a function of the Fermi energy for different disorder strengths. The different disorder strengths are correlated with the locations of the horizontal lines in (a) and (f) and the colors of the lower plots correspond to the colored dashed lines in the top plots. For all of these plots, we used $N=1000$.

Figure 4

(a) Aubry-André potential in position space. (b) The corresponding Fourier components. (c) Momentum entanglement spectrum as a function of the Fermi energy. (d) Localization length as a function of energy. (c) and (d) are for $W=1$.

Figure 5

Entanglement of the Aubry-André model: (a) and (b) show the spatial and momentum entanglement spectrum as a function of the AAM potential height; (c) and (d) show the corresponding momentum and spatial entanglement entropy; (e) and (f) show the derivative of the momentum and spatial entanglement entropies with respect to $W$, both showing peaks at the phase transition when $W_c=2.$

Figure 6

The top two figures show (a) the logarithm of the configuration-basis inverse participation ratio $\eta$ and (b) the change in ground-state energy after twisting the boundary conditions $\left|\Delta E\right|$, both as a function of disorder strength. Each curve corresponds to a Hubbard interaction $U=0.0,0.5,1.0,2.0$, with the arrow denoting the sense in which the interaction increases. The two lower plots show the corresponding derivative of each curve so as to highlight the point where the transition occurs and how it shifts as the interaction is increased.

Figure 7

Many-body entanglement spectrum of the interacting Aubry-André model for: (top) a spatial entanglement cut and (bottom) a momentum entanglement cut. Each column corresponds to a Hubbard interaction strength $U=0.0,0.5,1.0,2.0$, respectively.

Figure 8

Entanglement entropies for the interacting Aubry-André model: (a) and (b) show the spatial and momentum entanglement entropy, respectively, as a function of disorder strength for $U=0.0,0.5,1.0,2.0$. The arrow shows the direction of increase of the interaction strength. (c) and (d) show the derivative of both types of entanglement entropy, which illustrate how the transition point shifts as a function of the interaction strength.

Figure 9

Entanglement of a chiral topological insulator in class AIII subjected to an Aubry-André potential. We show the spatial (left column), momentum (center column) and orbital (right column) entanglement plots. The top row corresponds to each type of entanglement in the $m\text{vs}W$ parameter space. The overlayed black curves denote the boundary of the region where the system is topological according to the nontrivial polarization of the system. The second, third and fourth rows show the entanglement spectra as a function of $W$, for three values of the mass parameter, which are shown in the top row by the colored lines.

Figure 10

Orbital entanglement spectrum of the lattice Dirac model in the clean limit as a function of the mass parameter. Each line corresponds to a fixed value of the momentum $k$ in the range $[0,2\pi ]$.

Figure 11

The top plot shows the spatial entanglement spectrum as a function of the mass parameter when the system is clean, for a lattice size $N=102$. The lower red plots show the hybrid momentum-orbital entanglement spectrum for various mass values shown on the horizontal axis of top figure.

Figure 12

(a) The large blue plot shows the spatial entanglement spectrum as a function of disorder when the mass parameter is set at $m=0$ for a lattice size $N=102$ and for the AA potential. The lower red plots show the hybrid momentum-orbital entanglement spectrum for various disorder strengths shown on the horizontal axis of top figure. (b) This set of subfigures correspond to the case when we use uncorrelated disorder with $N=150$ and $m=0$.