论文标题
分级性和$ \ cal {pt} $ - 在平方晶格中的对称性
Fractionality and $\cal{PT}$- symmetry in a square lattice
论文作者
论文摘要
我们在平方晶格上研究了2D离散schrödinger方程的光谱稳定性，同时存在分数拉普拉斯和$ \ cal {pt} $对称性。为此，我们以封闭形式计算平面波频谱，这是增益参数和分数指数的函数。对频谱的检查表明，增益/损失参数的增加有利于复杂特征值的早期出现，因此，是破裂的$ {\ cal pt} $对称性的开始。另一方面，随着分数指数从统一降低，在临界值下，差距会打开上下频段，并且频谱变得真实。指数的进一步减小增加了间隙的宽度，并且系统保持在$ \ cal {pt} $ - 对称阶段，以至于分数指数的消失值。 Examination of the density of states and the participation ratio reinforce these observations and lead one to conclude that, unlike the standard, non-fractional case where the binary lattice is always in the broken $\cal{PT}$ phase, for the fractional case it is possible to have a symmetric $\cal{P}{\cal T}$ phase in the presence of a finite gain/loss parameter and a small enough fractional指数。
We study the spectral stability of a 2D discrete Schrödinger equation on a square lattice, in the simultaneous presence of a fractional Laplacian and $\cal{PT}$ symmetry. For that purpose, we compute the plane-wave spectrum in closed form, as a function of the gain/loss parameter and the fractional exponent. Examination of the spectrum reveals that an increase of the gain/loss parameter favors the early appearance of complex eigenvalues, thus is, the onset of a broken ${\cal PT}$ symmetry. On the other hand, as the fractional exponent decreases from unity, at a critical value a gap opens up separating the upper and lower bands, and the spectrum becomes real. Further decrease of the exponent increases the width of the gap and the system remains in the $\cal{PT}$-symmetric phase down to a vanishing value of the fractional exponent. Examination of the density of states and the participation ratio reinforce these observations and lead one to conclude that, unlike the standard, non-fractional case where the binary lattice is always in the broken $\cal{PT}$ phase, for the fractional case it is possible to have a symmetric $\cal{P}{\cal T}$ phase in the presence of a finite gain/loss parameter and a small enough fractional exponent.
