Many-body entanglement in a topological chiral ladder

We find that the topological phase transition in a chiral ladder is characterized by dramatic signatures in many body entanglement entropy between the legs, close to half filling. The value of entanglement entropy for various fillings close to half filling is identical, at the critical point, but splays out on either side, thus showing a sharp signature at the transition point. A second signature is provided by the change in entanglement entropy when a particle is added (or subtracted) from half filling which turns out to be exactly $-log2$ in the trivial phase but zero in the topological phase. A microscopic understanding of tendencies to form singlets along the rungs in the trivial phase, and along the diagonals in the topological phase, is afforded by a study of concurrence. At the topological phase transition the magnitude of the derivative of the average concurrence of all the rungs shows a sharp peak. Also, at the critical point, the average concurrence is the same for various fillings close to half filling but splays out on either side, just like entanglement entropy. Entanglement entropy also offers a striking signature at the vortex-to-Meissner phase transition inherent in this system.

Figure 1

The ladder system consists of two legs $\mathbf{a}$ (red) and $\mathbf{b}$ (blue) with uniform magnetic flux $\varphi$ per plaquette. $t$ and $t^{\prime }$ are the hopping strength along the leg and the rungs, respectively. $t_d$ is the diagonal hopping amplitude in each plaquette.

Figure 2

Lower energy band (green color) and higher energy band (red color) for ladder system for set of parameters $\xi =0.5,{\xi }_d=0$ with (a) $\phi =0.2\pi$, (b) $\phi =0.3\pi$, (c) $\phi =0.4\pi$, (d) $\phi =0.5\pi$.

Figure 3

Variation of Chiral current ($\mathbf{J}$) for one particle ground state with parameters (a) $\xi$ and $\phi$, (b) $\xi$ and ${\xi }_d$, i.e., diagonal hopping for a constant $\phi =\frac{\pi }{2}$.

Figure 4

(a) Energy spectra for ladder system with varying ${\xi }_d$. Squared amplitudes of the coefficients of wave function (${{\psi }_i}^2$) for edge sates are show in (b) and (c), and for a typical bulk state in (d), with ${\xi }_d$ fixed at 1.5. The other parameters $\xi =1$, number of rungs $L=50, \phi =\frac{\pi }{2}$ are common to all figures. The indices first run through 1 to $L$ among the ‘a’ sites, and then again 1 to $L$, through ‘b’ sites, hence the edge states show signals at one edge and close to the center of the figure, which is also an edge of the ladder.

Figure 5

Visualizing the SSH model as being made up of two-site unit cells.

Figure 6

The change in winding number with respect to change in ${\xi }_d$: (a) Trivial phase ${\xi }_d<\xi$ with winding number zero, (b) the critical point ${\xi }_d=\xi$ with undefined winding number, and (c) the topological phase ${\xi }_d\ge \xi$ with winding number one.

Figure 7

Ladder system with horizontal division(light blue lines) of two subsystems. The above leg is subsystem $\mathbf{a}$ and lower leg corresponds to subsystem $\mathbf{b}$.

Figure 8

Entropy of subsystem a ($S_a$) of one particle ground state as a function of (a) $\xi$ and $\phi$ and (b) $\xi$ and ${\xi }_d$, i.e., diagonal hopping and $\phi =\frac{\pi }{2}$.

Figure 9

The subsystem entanglement entropy ($S_a$) for the horizontal cut (Fig. 1) as a function of ${\xi }_d$, close to half filling. Number of rungs $\text{L}=1000, \phi =\frac{\pi }{2}$, and $\xi =1$ with open boundary conditions (OBC) imposed. $S_a$ takes on the same value for various fillings close to half filling at the critical point. $S_a$ attains a minimum right after the topological phase transition $({\xi }_d=1)$. The top inset shows only the data for half filling in an extended region.

Figure 10

$\mathrm{\Delta }S$ with varying ${\xi }_d$ with $\xi =1.0, L=1000$ for different flux values under open boundary conditions (OBC).

Figure 11

Ladder system with vertical division (light blue lines) of two subsystems. The left ladder is subsystem $\mathbf{a}$ and right ladder corresponds to subsystem $\mathbf{b}$.

Figure 12

Subsystem entanglement entropy ($S_a$) with ${\xi }_d$ for $L=2000, \xi =1, \phi =\frac{\pi }{2}$ with open boundary conditions.

Figure 13

Subsystem entanglement entropy ($S_a$) with ${\xi }_d$ for $L=1000, \xi =1, \phi =\frac{\pi }{2}$ with periodic boundary conditions (PBC).

Figure 14

(a) Subsystem entanglement entropy of ladder system ($\mathrm{\Delta }{S}_a$) with ${\xi }_d$ OBC. (b) Subsystem entanglement entropy ($\mathrm{\Delta }{S}_a$) with ${\xi }_d$ PBC.

Figure 15

Average concurrence of diagonal (violet), rungs (green), and edge states (blue) of a system with varying ${\xi }_d$ for half filling with system parameters as $\xi =1.0, L=1000$, and $\phi =\frac{\pi }{2}$ under open boundary conditions.

Figure 16

Average concurrence of rungs with ${\xi }_d$ near half filling. Also, the derivative of concurrence (green) featuring the trivial-to-topological transition. Inset shows a similar feature as Fig. 3 for average rungs concurrence for $L=1000, \phi =\frac{\pi }{2}, \xi =1$ and open boundary conditions.

Figure 17

Average concurrence of rungs as a function of filling, for various diagonal hopping. The parameters are $\xi =1.0, L=200$, and $\phi =\frac{\pi }{2}$ and open boundary conditions are imposed.