Moving perturbation in a one-dimensional Fermi gas

We simulate a balanced attractively interacting two-component Fermi gas in a one-dimensional lattice perturbed with a moving potential well or barrier. Using the time-evolving block decimation (TEBD) method, we study different velocities of the perturbation and distinguish two velocity regimes based on clear differences in the time evolution of particle densities and the pair correlation function. We show that, in the slow regime, the densities deform as particles are either attracted by the potential well or repelled by the barrier, and a wave front of hole or particle excitations propagates at the maximum group velocity. Simultaneously, the initial pair correlations are broken and coherence over different sites is lost. In contrast, in the fast regime, the densities are not considerably deformed and the pair correlations are preserved.

Figure 1

Spin-up $\left(n_{i\uparrow }\right)$, spin-down $\left(n_{i\downarrow }\right)$, and doublon $\left(n_{i\uparrow \downarrow }\right)$ densities at times $t$ with $U=-4J$ (in practice, $n_{i\uparrow }$ and $n_{i\downarrow }$ overlap). On the first and third row, the density of spin-up particles is also shown for the equilibrium case with a static potential well or barrier $\left(n_{i\uparrow }^{\text{eq.}}\right)$. The first row shows a slow Gaussian well with $v=0.5J$ and the second row a fast one with $v=4J$. The third and fourth row show the same quantities in the case of a Lorentzian barrier. For the Gaussian, $V_0=-2J$ and for the Lorentzian, $V_0=2J$. A dashed black line indicates the perturbing potential multiplied by $0.1$.

Figure 2

Difference in the density of spin up particles with respect to the ground state as a function of position and time, $n_{i\uparrow }\left(t\right)-n_{i\uparrow }\left(0\right)$, for $U=-4J$. The perturbation is a Gaussian well with $V_0=-2J$ and $v=0.2J$ (left), $v=0.5J$ (middle), and $v=4J$ (right). The center of the perturbing potential is marked with a dashed black line.

Figure 3

Left: pair correlation $|C_{ij}|$ in the ground state in the middle part of the lattice for $U=-4J$. Right: the same quantity with $j$ fixed, $|C_{x,\frac{L}{2}}|$. A linear fit $f\left(x\right)=-\frac{1}{K_{\rho }}x+a$ gives the coefficients $K_{\rho }=1.22\pm 0.08$ and $a=-1.7\pm 0.1$ with errors given by the 95% confidence bounds. Data points after moving a potential well across the center of the lattice are also shown. For a well with $v=0.5J,|C_{x,\frac{L}{2}}|$ is shown at the time step when the perturbation has reached the site 80 and for $v=4J$ the site 72.

Figure 4

Phase ${\varphi }_{ij}$ at different time steps for $v=4J$ (a),(b) and $v=0.5J$ (c),(d) of the Gaussian potential well with $V_0=-2J$, when $U=-4J$.

Figure 5

Left: Phase difference ${\varphi }_{x,-x}$ for different values of $U$ and $v$. Right: The maximum of ${\varphi }_{ij}$ for different velocities of the perturbation given by the many-body simulations. The figure includes data for a Gaussian well with $V_0=-2J$ (right $y$ axis) and Lorentzian barriers with different heights from $V_0=10J$ to $V_0=0.5J$ (left $y$ axis). The black line is the result of the time integral for the Gaussian, $\Delta \varphi \left(v\right)=2V_0\sigma \frac{\sqrt{2\pi }}{v}$, and the blue line for the Lorentzian, $\Delta \varphi \left(v\right)=\frac{2\pi }{v}$. The time integral of the Lorentzian is independent of the height $V_0$ since its width is $\frac{1}{V_0}$.