SciPost Submission Page
Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach
by Martín Zapata, Federico Finkel, Artemio González-López
Submission summary
| Authors (as registered SciPost users): | Artemio González-López |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2511.16473v1 (pdf) |
| Data repository: | https://doi.org/10.5281/zenodo.17652257 |
| Date submitted: | Nov. 21, 2025, 4 p.m. |
| Submitted by: | Artemio González-López |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- The paper is well structured, and the mathematical tools (orthogonal polynomials) are seamlessly integrated into the derivation of the physical results for the fermion density, making the presentation accessible to both mathematical physicists and condensed-matter theorists. 2- The analytical treatment is detailed and well supported. 3- The paper provides extensive comparisons of the analytical predictions with numerical results for several benchmark models (Rainbow, Krawtchouk, Cosine, and Rindler chains). The agreement shown in the figures is excellent.
Weaknesses
1- While the quantitative derivation of the depletion and saturation regimes is mathematically rigorous and well detailed, the physical implications of these phases could be more prominently discussed, possibly in connection with the specific models considered. 2- While the authors state that this is a "first step" toward an analytical treatment of entanglement entropy, the current paper focuses almost exclusively on the local density. Including a concrete example, such as an explicit entanglement-entropy calculation or a benchmark comparison for one of the models considered, would help clarify how the proposed discrete WKB approach can be leveraged for entanglement-related applications and would better contextualize its relevance for the entanglement community.
Report
This methodology allows to go beyond the results obtained with traditional approaches, which often rely on continuum limits or conformal field theory approximations. While those standard techniques are frequently restricted to low fillings or specific magnetic field regimes, the discrete WKB method can be applied for arbitrary values of such parameters. Moreover, the authors demonstrate that their method successfully captures interesting phenomena such as "depletion" (suppression of density) and "saturation" (areas where the density reaches its maximum), providing a rigorous theoretical basis for numerical observations in previous literature. This also allows to shed light on a related observed phenomena, namely the entanglement suppression in inhomogeneous XX spin chains.
This work is a high-quality, technically sound, and well-motivated paper. The study of inhomogeneous systems is currently a high-interest area in many-body physics, particularly concerning rainbow chains and other models where spatial curvature leads to non-trivial entanglement properties. The paper’s strength lies in its ability to bridge the gap between exact numerical results and heuristic field-theoretic predictions. By working directly with the discrete recurrence relations, the authors provide a more robust derivation that remains accurate even in regimes where the continuum approximation typically fails.
For these reasons, I think it meets the standards for publication in SciPost Physics
Requested changes
I ask the authors to respond to the weakness points raised above, either by incorporating minor revisions where appropriate or by clarifying why certain aspects are left for future work.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
2- The paper is well-written and clear.
3- The examples are numerous and compelling.
Weaknesses
2- Although the results naturally connect to the well-known Local Density Approximation (LDA), this connection is not explicitly discussed in the manuscript. Since Eq. (50) can essentially be recovered from an LDA argument, it would be helpful for readers if the authors briefly explained how their controlled WKB derivation clarifies the validity and limits of LDA.
Report
LDA is a method whereby the model’s parameters are assumed to vary slowly, such that in the continuum limit the local dispersion relation is written as the standard dispersion relation for the homogeneous chain with constant $J$ and $B$, but where the parameters are promoted to functions $J(x)$ and $B(x)$, i.e. $\epsilon_k(x) = 2J(x)\cos k + B(x) - \mu$ with a chemical potential $\mu$. The local Fermi momentum is thus $k_F(x) = \arccos\left(\frac{\mu - B(x)}{2J(x)}\right)$ and from there we essentially obtain the result of Eq. (50) from the paper. However, the LDA argument reproduces only the density profile and depletion/saturation regions, while the authors’ WKB approach goes further by also providing information on wavefunctions and spectral properties. Moreover, the LDA is often used without a clear controlled notion of what “slowly varying parameters” means. Here, the authors provide a clean controlled derivation of these results, obtained in particular by neglecting gradient corrections of order $a,J'(x)/J(x)$, thereby providing a quantitative understanding of regimes where the LDA works.
Moreover, Sec. 2.3 of Entanglement Hamiltonian and orthogonal polynomials, NPB 1020 (2025) 117185 (P.-A. Bernard et al.) contains a short discussion about the continuum limit of inhomogeneous chains within the LDA approximation. In particular, depletion and saturation zones are predicted for regions where the local Fermi velocity is not well defined, which coincide with the criteria for such regions derived by the authors here.
My point is that some of the authors’ results (density formula and criteria for depletion/saturation) are valuable and rigorous, and placing them explicitly in the context of existing approximations such as the LDA would further clarify their significance.
Requested changes
1- On the top of page 12, there are notations $\ell(\epsilon)$ on the right of the first two equations. I believe it should just be $\ell$.
2- I think that Remark 4.1 and Fig. 1 should be relegated to the relevant example section. As it is now, the full-page figure breaks the flow of the theory part of the paper. Also, I would reduce Fig. 1 to a few relevant curves supporting the authors’ claim, I do not think it needs to be that exhaustive.
3- At the bottom of page 16, there is “eq. (38)” written in plain, i.e., not a clickable \eqref appearing in red.
4- Provide a discussion on LDA, to put some of the results and methods used here in perspective, see report.
5- Add a reference to Entanglement Hamiltonian and orthogonal polynomials, NPB 1020 (2025) 117185 by P.-A. Bernard et al. for the continuum treatment of inhomogeneous chains, see discussion above.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)