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arxiv: 2604.11173 · v1 · submitted 2026-04-13 · ❄️ cond-mat.stat-mech · quant-ph

Sluggish quantum mechanics of noninteracting fermions with spatially varying effective mass

Giuseppe Del Vecchio Del Vecchio , Manas Kulkarni , Satya N. Majumdar , Sanjib Sabhapandit This is my paper

Pith reviewed 2026-05-10 15:32 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords noninteracting fermionsposition-dependent massdeterminantal point processBessel kernelcorrelation kerneloptical latticesinhomogeneous quantum systemsground-state density
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The pith

Noninteracting fermions with position-dependent effective mass have a correlation kernel near the origin that is a sum of two Bessel kernels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies one-dimensional quantum particles whose effective mass grows as a power of distance from the origin, arising from the continuum limit of a tight-binding chain with varying hoppings. For N noninteracting fermions placed in a matching trap potential, the ground-state positions form a determinantal point process whose kernel is obtained in closed form for any finite N. In the large-N scaling limit the kernel near the origin takes a previously unknown form given by the sum of two Bessel kernels of different orders, producing an average density that is symmetric and of finite support yet vanishes at the center. The exact solutions also yield the single-particle eigenfunctions and the quantum propagator both in free space and in the trap.

Core claim

For alpha greater than zero the scaled kernel of the determinantal process near x equals zero is neither the standard Bessel nor Airy kernel but equals the sum of two Bessel kernels with distinct indices; this implies that the large-N average density has a non-monotonic profile with a vanishing minimum at the origin while remaining of finite width.

What carries the argument

the scaled correlation kernel near the origin expressed as the sum of two Bessel kernels with different indices, which determines all position statistics of the ground-state fermions

If this is right

  • The average density for large particle number is non-monotonic, symmetric about the origin, and exactly zero at the center.
  • All higher-order correlation functions of the fermion positions follow from the determinantal kernel for any finite N.
  • Exact eigenfunctions and the time-dependent propagator are available for the single-particle problem both with and without the external trap.
  • The construction directly models optical lattices in which tunneling strength varies with site position.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same power-law mass variation may produce analogous new kernels in higher-dimensional or interacting settings if the continuum limit remains controllable.
  • Direct comparison between the tight-binding lattice model and the continuum predictions for moderate N would test the accuracy of the BenDaniel-Duke approximation.
  • The vanishing central density offers a clear signature that could be sought in cold-atom experiments with engineered position-dependent tunneling.

Load-bearing premise

The continuum limit of the inhomogeneous tight-binding model with power-law hopping produces the BenDaniel-Duke equation with m_eff proportional to |x|^alpha, and the chosen trap potential permits exact closed-form solutions.

What would settle it

Numerical or experimental measurement of the ground-state density for large N in an optical lattice whose hopping amplitudes realize m_eff(x) proportional to |x|^alpha, checking whether the density reaches a minimum of zero at the center while staying finite and symmetric elsewhere.

Figures

Figures reproduced from arXiv: 2604.11173 by Giuseppe Del Vecchio Del Vecchio, Manas Kulkarni, Sanjib Sabhapandit, Satya N. Majumdar.

Figure 2
Figure 2. Figure 2: The eigenvalue spectrum λk with k = 0,1,2,... is divided into even and odd sectors: λ2n = λ even n and λ2n+1 = λ odd n with n = 0,1,2,..., which are interlaced. The even and odd eigenvalues are given by Eqs. (54) and (60) respectively. B. Remarks on the corresponding classical stochastic problem It is useful to compare the quantum probabilities |ψn(x)| 2 , given in Eq. (63) and illustrated in [PITH_FULL_I… view at source ↗
Figure 3
Figure 3. Figure 3: Density profile ρN(x) [see Eq. (124)], obtained by adding Eqs. (127) and (128) for different numbers of fermions (N) with µ = 1 and α = 1. we obtain ρ ± N (x) as ρ + N (x) = 1 N  2µ α +2 −γ µ e −z Γ(⌈ N 2 ⌉+1) Γ(⌈ N 2 ⌉ −γ) ×  L (−γ+1) ⌈ N 2 ⌉−1 (z)L (−γ) ⌈ N 2 ⌉−1 (z)−L (−γ+1) ⌈ N 2 ⌉−2 (z)L (−γ) ⌈ N 2 ⌉ (z)  , (127) and ρ − N (x) = 1 N  2µ α +2 −γ µ z 2γ e −z Γ(⌊ N 2 ⌋+1) Γ(⌊ N 2 ⌋+γ) ×  L (γ+1) ⌊… view at source ↗
Figure 4
Figure 4. Figure 4: The solid lines plot the scaled density profile obtained [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript analyzes one-dimensional noninteracting fermions with position-dependent effective mass m_eff(x) = m_eff |x|^α (α > 0) arising as the continuum limit of an inhomogeneous tight-binding model. Exact eigenfunctions and propagators are derived for the BenDaniel-Duke Schrödinger equation in both the free case and with the trap V_ext(x) = (1/2) m_eff ω² |x|^{α+2}. For N fermions in the trap, the ground-state many-body wavefunction yields a determinantal point process whose correlation kernel is obtained exactly via the Christoffel-Darboux identity; large-N scaling limits of the kernel are analyzed in the bulk, at the edge, and near the origin, where the scaled kernel is shown to be a sum of two Bessel kernels with different indices rather than the standard Airy or Bessel kernels.

Significance. If the derivations hold, this provides a rare exactly solvable inhomogeneous quantum system with closed-form eigenfunctions, finite-N kernels, and explicit scaling limits, including a novel determinantal kernel at the origin. The exact solvability and parameter-free asymptotic analysis constitute a clear strength, with direct relevance to engineered optical lattices featuring position-dependent tunneling amplitudes.

minor comments (3)
  1. [§2] §2: The continuum expansion from the discrete tight-binding Hamiltonian to the BenDaniel-Duke operator is sketched but would benefit from an explicit display of the second-order terms in the hopping expansion to confirm the precise form of m_eff(x).
  2. [§4.2] §4.2: The statement that the origin kernel is 'new' is supported by the asymptotic analysis, but a short comparison table or explicit functional form contrasting it with the standard Bessel kernel (e.g., via the indices ν) would aid readability.
  3. [Figure 3] Figure 3 caption: The scaling variable for the origin kernel is not defined in the caption; it should be stated explicitly (e.g., ξ = N^β x) to match the text in §5.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary, and positive assessment of our work. We are pleased that the exact solvability, closed-form kernels, and the novel sum-of-Bessel structure at the origin are recognized as strengths with relevance to engineered optical lattices. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from continuum limit and exact solutions

full rationale

The paper begins with the standard continuum expansion of the inhomogeneous tight-binding Hamiltonian to obtain the BenDaniel-Duke operator with m_eff(x) = m_eff |x|^α. The external trap is deliberately chosen as V_ext(x) = (1/2) m_eff ω² |x|^{α+2} so that a change of variables reduces the eigenvalue problem exactly to the Bessel equation, yielding closed-form eigenfunctions |x|^β J_ν(λ |x|^γ) and Y_ν. The N-fermion ground-state kernel is then constructed via the Christoffel-Darboux identity applied to these explicit functions. Large-N scaling limits (bulk, edge, origin) follow from standard asymptotic expansions of the Bessel functions in microscopic variables. No step reduces by construction to a fitted parameter, self-citation chain, or redefinition of the target result; the central claim (new origin kernel as sum of two Bessel kernels) is obtained directly from the asymptotics without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of non-relativistic quantum mechanics and the continuum limit of a tight-binding model; no free parameters are fitted to data, no new particles or forces are postulated, and the trap potential is chosen by hand to enable solvability rather than derived from first principles.

axioms (2)
  • domain assumption The continuum limit of an inhomogeneous tight-binding model with spatially varying hopping amplitudes yields the BenDaniel-Duke form of the Schrödinger equation with m_eff(x) = m_eff |x|^α.
    Explicitly stated as the starting point of the analysis.
  • ad hoc to paper The external potential is chosen as V_ext(x) = 1/2 m_eff ω² |x|^{α+2} to permit exact solvability.
    The specific power is selected to match the mass variation and allow closed-form eigenfunctions.

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Reference graph

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