Sluggish quantum mechanics of noninteracting fermions with spatially varying effective mass

Abstract

We analyze a class of one-dimensional quantum systems characterized by a position-dependent kinetic term arising as the continuum limit of an inhomogeneous tight-binding model with spatially varying hopping amplitudes. In this limit, the Schrodinger equation takes the so-called BenDaniel-Duke form with an effective mass, scaling as $m_{eff}(x) = m_{eff}|x|^{\alpha}$ with $\alpha > 0$, leading to a framework we term sluggish quantum mechanics, where particle motion is progressively suppressed at larger distances. Both without any external potential and with $V_{ext}(x)=\frac{1}{2}m_{eff}\omega^2 |x|^{\alpha+2}$, we obtain the eigenfunctions and the quantum propagators exactly. We then investigate the problem of $N$ noninteracting spinless fermions in the trap, determining the many-body ground-state wavefunction and the joint probability density function of the positions of the $N$ fermions. We show that the many-body quantum probability density in the ground state forms a determinantal point process whose correlation kernel can be computed for any $N$, giving access to the average density as well as higher order correlation functions for any finite $N$. Moreover, we analyze the scaling form of this kernel in the large $N$ limit in the bulk, near the edge, and close to the origin. Our results show that the scaled average density profile for large $N$ has a finite support symmetric with respect to the origin, but has a non-monotonic shape with a vanishing minimum at the origin for any $\alpha>0$. One of the key findings of our work is that the scaled kernel near the origin $x=0$ for $\alpha>0$ is neither the Bessel nor the Airy kernel (that are standard for trapped fermions), but is new, and is given by a sum of two Bessel kernels with different indices. Our results thus provide a framework relevant to engineered optical lattices with position-dependent tunneling.