Sluggish quantum mechanics of noninteracting fermions with spatially varying effective mass

TL;DR

This paper studies a class of one-dimensional quantum systems with position-dependent effective mass, analyzing eigenfunctions, propagators, and many-body fermion correlations, revealing novel kernels and density profiles relevant to engineered optical lattices.

Contribution

It introduces 'sluggish quantum mechanics' for systems with spatially varying effective mass and derives exact solutions for eigenfunctions, propagators, and fermion correlations, including a new kernel near the origin.

Findings

Eigenfunctions and propagators are obtained exactly for the system.

The many-body fermion ground state forms a determinantal point process.

The kernel near the origin is a novel sum of two Bessel kernels, not Bessel or Airy.

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.