Modulational instabilities in lattices with power-law hoppings and interactions

We study the occurrence of modulational instabilities in lattices with nonlocal power-law hoppings and interactions. Choosing as a case study the discrete nonlinear Schrödinger equation, we consider one-dimensional chains with power-law decaying interactions (with exponent $\alpha$) and hoppings (with exponent $\beta$): An extensive energy is obtained for $\alpha ,\beta >1$. We show that the effect of power-law interactions is that of shifting the onset of the modulational instabilities region for $\alpha >1$. At a critical value of the interaction strength, the modulational stable region shrinks to zero. Similar results are found for effectively short-range nonlocal hoppings ($\beta >2$): At variance, for longer-ranged hoppings ($1<\beta <2$) there is no longer any modulational stability. The hopping instability arises for $q=0$ perturbations, thus the system is most sensitive to the perturbations of the order of the system size. We also discuss the stability regions in the presence of the interplay between competing interactions - (e.g., attractive local and repulsive nonlocal interactions). We find that noncompeting nonlocal interactions give rise to a modulational instability emerging for a perturbing wave vector $q=\pi$ while competing nonlocal interactions may induce a modulational instability for a perturbing wave vector $0<q<\pi$. Since for $\alpha >1$ and $\beta >2$ the effects are similar to the effect produced on the stability phase diagram by finite range interactions and/or hoppings, we conclude that the modulational instability is ``genuinely'' long-ranged for $1<\beta <2$ nonlocal hoppings.

Figure 1

Stable (white) and unstable [gray, (red)] regions for the short-range DNLSE ($\overline{V}=0$ and only nearest-neighbor hoppings) for (a) $\overline{U}=2.5$ and (b) $\overline{U}=1.5$ in the $k$ (carrier wave momentum) $q$ (perturbing wave momentum) plane. The absolute value of the imaginary part of the frequencies $\Gamma (k;q)$ is depicted [from light to dark as $\Gamma (k;q)$ increases]. The continuous lines are equispaced isolines whose spacing is set to $0.2$. The thick (red) line indicates the position of $q_{\text{max}}$.

Figure 2

Stability in the $k$-$q$ plane (see caption of Fig. 1) for power-law interactions in presence of local nearest-neighbor hoppings with $\alpha =1.5$ and $\overline{U}=2.5$. We plot $\Gamma (k;q)$ for the four values of $\overline{V}=0.2$, $0.5$, 2, and 4 in panels $a$, $b$, $c$, and $d$, respectively.

Figure 3

The solid line represents $k_{\text{cr}}$ vs $\overline{V}$ from Eq. (27) (in the figure $\overline{U}=1$ and $\alpha =3$). The dots represent values obtained by direct simulation of the DNLSE, see the text for details (errors are smaller than the symbols). Notice that for $\overline{U}=1$ and $\alpha \to \infty$ one has $k_{\text{cr}}=\pi /2$ for $\overline{V}<0.5$ and $k_{\text{cr}}=0$ for $\overline{V}>1.5$. The dotted line is the analytical value of $k_{\text{cr}}$ vs $\overline{V}$ from Eq. (27) with $\alpha \to \infty$ and $\overline{U}=1$.

Figure 4

Time evolution (on the left) of the modulus of the order parameter $\phi$, defined in Eq. (28), for five different values (from left to right $k=77$, 76, 75, 74, 73, 72 in units of $2\pi /L$ with $L=512$) of carrier plane-wave wave vector for the DNLSE with parameters $\overline{U}=1$, $\overline{V}=1.25$, and $\alpha =3$. The lifetimes of the initial states are depicted in the right part of the figure. These times have been fitted with a function $\tau =\text{const}{(k-k_{\text{cr}})}^{-1/2}$ (dotted line) to obtain the a numerical estimates of $k_{\text{cr}}$ (vertical line).

Figure 5

Stability regions (see caption of Fig. 1) for the case of competing interactions with $\alpha =2$, $\overline{U}=-0.7$, and $\overline{V}=0.5$. The value of the most unstable $q$ which defines $k_{\text{cr}}$ is denoted by $q^{*}$.

Figure 6

Stability regions (see caption of Fig. 1) for the case of long-range hopping and local interaction $\overline{U}=1$ and $\overline{V}=0$. Panels (a), (b), (c), and (d) refer to the values of $\beta =4$, 3, 2, $1.5$.

Figure 7

Critical momentum $k_{\text{cr}}$ vs the power-law exponent $\beta$ of power-law hoppings with $\overline{V}=0$. Crosses are numerical data.

Figure 8

Panels (a) and (b) are contour plots of $\Gamma (k;q)$ (see caption of Fig. 1) for $\overline{U}=\overline{V}=1$ and $\alpha =2.5$, $\beta =\infty$ (long-range interaction) and $\alpha =\infty$, $\beta =2.5$ (long-range hopping). In panel (c) we depict the time evolution of the order parameter defined in Eq. (28) for the initial conditions indicated in the contour plots with points. As we can see the long-range hopping instability, upper (green) curve, takes a much longer time to set in than the long-range interaction instability, lower (blue) curve.

Figure 9

Contour plots of $\Gamma (k;q)$ (see caption of Fig. 1) for models including only on-site interactions ($U$) and next- and next-to-nearest-neighbor interactions ($V$ and $V_{j,j+2}=V_2$) and hoppings ($t=1$ and $t_{j,j+2}=t_2$), respectively. Panel (a) refers to values $U=-0.8$, $V=0.4$, $V_2=0.05$, and $t_2=0$ while panel (b) refers to values $U=1$, $V=V_2=0$, and $t_2=0.2$.

Figure 10

$k_{\text{cr}}$ vs $a/a_{dd}$ for a dipolar gas of ${}^{52}\mathrm{Cr}$ atoms. Solid lines refer to $\alpha =3$ (top solid line: $\rho =1$, bottom solid line: $\rho =100$), while dotted lines refer to $\alpha \to \infty$ (top dashed line: $\rho =1$, bottom dashed line: $\rho =100$). Parameters are ${\omega }_{\perp }=2\pi 100\text{Hz}$, $s=5,{\alpha }_{\text{orient}}=-1/2$, and $\lambda =0.7\mu \text{m}$. Inset: Corresponding value of $V/U$ vs $a/a_{dd}$ for $\rho =1$, solid line, and $\rho =100$, dotted line (the two lines practically coinciding with one another).

Figure 11

$k_{\text{cr}}$ vs $t_2/t_1$ for a model having nearest-neighbor and next-nearest-neighbor hoppings $t_1$ and $t_2$ ($V=0$).

Figure 12

Plot of ${\ell }_{\alpha }\left(0\right)$ [solid (red) line] and ${\ell }_{\alpha }\left(\pi \right)$ [dashed (green) line] as a function of $\alpha$.