Fermion confinement via quantum walks in (2+1)-dimensional and (3+1)-dimensional space-time

We analyze the properties of a two- and three-dimensional quantum walk that are inspired by the idea of a brane-world model put forward by Rubakov and Shaposhnikov [Phys. Lett. B 125, 136 (1983)]. In that model, particles are dynamically confined on the brane due to the interaction with a scalar field. We translated this model into an alternate quantum walk with a coin that depends on the external field, with a dependence which mimics a domain wall solution. As in the original model, fermions (in our case, the walker) become localized in one of the dimensions, not from the action of a random noise on the lattice (as in the case of Anderson localization) but from a regular dependence in space. On the other hand, the resulting quantum walk can move freely along the “ordinary” dimensions.

Figure 1

Probability distribution $\left|\right|\mathrm{\Psi }(t_j,x_p,y_q){\left|\right|}^2$ of the two-dimensional QW for a value $t=10$ of the time step and different values of $m$. The rest of the parameters are fixed to $\lambda =60,h=70$, with the lattice parameter $\epsilon =0.04$. The inset in the last subfigure also shows the projected density profile along each direction of the lattice (red dot-dashed line represents the $x$ direction and blue dashed line the $y$ direction). The initial condition is a Gaussian wave packet $\mathrm{\Psi }(0,x_p,y_q)=\sqrt{n(x_p,y_q)}\otimes {(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})}^{\top }$ centered at the point $(64,64)$, where the Gaussian distribution $n(x_p,y_q)$ has a width $\delta =0.1$.

Figure 2

Time evolution of the standard deviation divided by the time step, i.e., ${\sigma }_x\left(t\right)/t$ (in the inset) and ${\sigma }_y\left(t\right)/t$, calculated independently along the $x$ and $y$ directions, for a localized (red squares) and a free fermion (blue diamonds). The initial condition is a Gaussian wave packet $\mathrm{\Psi }(0,x_p,y_q)=\sqrt{n(x_p,y_q)}\otimes {(0,1)}^{\top }$ centered around $(128,128)$, and the parameters of the potential are $\lambda =60$ and $h=70$, with the lattice parameter $\epsilon =0.02$.

Figure 3

Density plot in 3D at time $j=12$ with Gaussian initial wave packet $\mathrm{\Psi }(0,x_p,y_q,z_r)=\sqrt{n(x_p,y_q,z_r)}\otimes {(1,i,1,i)}^{\top }$ centered around $(0,0)$ and for $m=0$.

Figure 4

Density plots in 3D at time $j=20$ with Gaussian initial wave packet $\mathrm{\Psi }(0,x_p,y_q,z_r)=\sqrt{n(x_p,y_q,z_r)}\otimes {(0,1,0,1)}^{\top }$ centered around $(0,0)$. The parameters of the potential are $\lambda =90,h=4$, and $m=11$. The two subfigures at the bottom display the $x\text{−}z$ side view (left) and the $x\text{−}y$ side view (right) of the 3D density plot.