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REVIEW 2 major objections 6 minor 38 references

Probing disorder-induced Fisher information matrix and Cram\'{e}r-Rao bound by STM

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that the normalized local density of states measured by STM is a probability density, and that its Fisher information matrices impose Cramér-Rao lower bounds on the local energy variance and the spatial variance of…

desk verdict Correct, clean application of standard Fisher-information math to a genuinely new object—normalized STM LDOS—but the universal-applicability claims outrun the demonstrated single-particle, energy-independent-tip regime. read the letter →

arxiv 2505.22965 v1 pith:BRMIHGSX submitted 2025-05-29 cond-mat.str-el cond-mat.dis-nn

Lucas A. Oliveira , Wei Chen This is my paper
classification cond-mat.str-elcond-mat.dis-nn
keywords FisherinformationmatrixCramér-RaoboundlocaldensityofstatesscanningtunnelingmicroscopydisordergeometryCherninsulatortight-bindingmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper treats the normalized local density of states $\rho(x,E)$ measured by STM as a conditional probability density: the probability that an electron at position $x$ has energy $E$ within the probe window. From this single object it builds two Fisher information matrices—one in real space and one in energy space—and shows that the corresponding Cramér-Rao bounds constrain the variance of local energy and the variance of electron position. Because the bounds are derived for any normalized probability density, the authors argue they hold for any host material and any type of disorder, and they demonstrate the quantities on disordered tight-binding models of a metal and a Chern insulator. The payoff is a new, model-free way to interpret STM conductance maps as information geometry of the disordered solid.

What carries the argument

The load-bearing object is the normalized local density of states $\rho(x,E)=\tilde{\rho}(x,E)/\int_{E_c}^{0}dE\,\tilde{\rho}(x,E)$, a conditional probability density whose variation defines the two Fisher informations: $I_{\mu\nu}(x,E_c)=\int dE\,\partial_\mu\rho\,\partial_\nu\rho/\rho$ in real space and $I_E(E)=\sum_x (\partial_E\rho)^2/\rho$ in energy space. The bounds are carried by the positive semidefiniteness of the correlation matrix built from $(f-\langle f\rangle,\partial_\mu\ln\rho)$, and by the Cauchy-Schwarz inequality in the energy-space case, which converts the overlap of normalized LDOS at nearby positions or energies into an exact variance bound. Numerically, spatial derivatives are evaluated by central differences along lattice vectors, and the energy derivatives of the delta functions entering $\rho$ are computed with a Lorentzian broadening.

What would settle it

Modify the same tight-binding models by replacing the constant tunneling prefactor in Eq. (4) with an energy-dependent $|M(E)|^2\rho_{\mathrm{tip}}(0,E)$ and recompute the ratio in Eq. (10); the resulting object is no longer the electron probability density, and the Fisher information and Cramér-Rao bounds computed from the normalized conductance would be bounds on a different quantity, which a direct comparison with the true quantum variances would expose.

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Extended reading notes

Core claim

The central claim is that the STM differential conductance, after summing over sublattices and normalizing by its energy integral, is exactly the electron probability density $\rho(x,E)$ of an electron at position $x$ with energy $E$, provided the tunneling matrix element and tip density of states are energy-independent. Treating $E$ as the random variable and $x$ as the parameter yields a real-space Fisher information matrix $I_{\mu\nu}(x,E_c)$, whose Cramér-Rao bound reads $\mathrm{Var}[f] \ge \partial_\mu\langle f\rangle I^{\mu\nu}\partial_\nu\langle f\rangle$; choosing $f(E)=E$ bounds the local energy variance at each site. Treating $x$ as the random variable and $E$ as the parameter yields an energy-space Fisher information $I_E(E)$, whose bound $\mathrm{Var}[x_\mu] \ge (\partial_E\langle x_\mu\rangle)^2/I_E$ bounds the spatial variance at fixed energy. The paper verifies numerically in 2D metals and in Chern insulators in both topological phases that these quantities are positive, that disorder induces nonzero Fisher information and a curved information geometry, and that the bounds are respected at every site and every energy.

Load-bearing premise

The load-bearing premise is that the STM differential conductance is proportional to the single-particle local density of states with an energy-independent tunneling matrix element and tip density of states, so the normalized conductance equals the electron's position-energy probability density.

Editorial extensions

If this is right

  • Any STM dataset that resolves $dI/dV(x,E)$ over a grid can be processed through the ratio in Eq. (10) to produce maps of the real-space Fisher information and its volume form, making impurity-induced curvature of space a directly measurable field.
  • Wherever the Fisher information matrix is invertible, the local energy variance is bounded below by the gradient of the local energy via $\mathrm{Var}[E] \ge \partial_\mu\langle E\rangle I^{\mu\nu}\partial_\nu\langle E\rangle$, a bound that matters only where disorder makes the Fisher information nonzero.
  • At fixed energy, the spatial variance of electrons is bounded below by $(\partial_E\langle x_\mu\rangle)^2/I_E$, which is nonzero precisely in disordered systems, so disorder strictly raises the lower bound on position fluctuations compared with a homogeneous system.
  • Because the derivation uses only that $\rho$ is a normalized probability density, the two Cramér-Rao bounds apply equally to metals, insulators, semiconductors, and topological phases, as illustrated by the Chern insulator results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is to use the real-space Fisher information texture as a model-free fingerprint of the impurity configuration: since $I_{\mu\nu}$ is largest where $\rho$ changes sharply, STM maps could in principle be inverted to locate or classify disorder without assuming a specific Hamiltonian.
  • The same probability-density construction could be carried over to other scanning spectroscopies that produce normalized maps over a coordinate and a bias parameter, whenever a legitimate probability interpretation exists; this is an extrapolation beyond the paper's solid-state examples.
  • The larger energy variances and Cramér-Rao values seen in the topologically nontrivial Chern phase could in principle serve as a disorder-sensitive indicator of topology, but the paper does not claim such a diagnostic; a systematic scan over disorder strength would be needed to test it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes a new information-geometric analysis of STM-measured local density of states (LDOS). It normalizes the LDOS to form a position-conditional energy probability density and an energy-conditional position probability mass function, then defines a real-space Fisher information matrix (FIM) and an energy-space FIM. From these it derives Cramér-Rao bounds that lower-bound the local energy variance (Eq. (20)) and the spatial position variance (Eq. (37)). The formalism is demonstrated numerically on disordered 2D tight-binding models of a metal and of a Chern insulator in both topological phases.

Significance. The mathematical core of the paper is sound and self-contained: the derivations of Eqs. (15)-(20) and (33)-(36) are correct, no parameters are fitted, and the numerical positivity checks are consistent with the inequalities. The distribution-free character of the Cramér-Rao bounds is a genuine strength, and the idea of using the FIM as a disorder-induced metric on real space and energy space is novel for STM analysis. The main weakness is the bridge between measured tunneling conductance and an electron probability density: the extraction protocol in Eq. (10) requires energy-independent tunneling prefactors, and the Born-probability interpretation in Eq. (5) is a single-particle statement. Because the paper claims applicability to any material and any disorder, this bridge is load-bearing and needs to be either justified or explicitly qualified.

major comments (2)
  1. [II A, Eqs. (4) and (10)] The experimental extraction protocol divides the conductance by its energy integral. This cancels the prefactor in Eq. (4) only under the stated assumption that |M|^2 and the tip DOS are energy-independent. For real STM tips with energy-dependent DOS or matrix elements, the extracted object is rho_exp(x,E) = g(E) rho(x,E) / integral g(E') rho(x,E') dE', which is a different probability density; the Cramér-Rao bound computed from it need not bound the true electron energy or position variance. Please either justify the energy-independence assumption for a concrete experimental setup, or restrict the claims and discuss the resulting systematic error.
  2. [I, Eq. (5), and Conclusions] The Born-probability interpretation is introduced via single-particle eigenstates |a_n(x)|^2 in Eq. (5). For interacting or correlated materials, STM measures the single-particle spectral function, a many-body object, which is not a joint position-energy probability density for an electron. The statements in the Introduction and Conclusions that the formalism applies to 'any kind of material' and 'any types of long or short range defects' therefore overreach beyond the non-interacting tight-binding demonstration in Sections II C and III C. Please narrow the universality claims or provide an explicit argument for how the formalism extends to interacting systems.
minor comments (6)
  1. [Abstract] The abstract contains a typo, 'bonuds' should be 'bounds'.
  2. [II A, Eq. (11)] For a discrete spectrum, rho(x,E) is a sum of delta functions, so the integral in Eq. (11) is formal unless a broadening is specified. The Lorentzian regularization is introduced only for the energy-space FIM in Eq. (32); please state explicitly how the real-space FIM is regularized in the numerics or define it directly through the fidelity expansion in Eq. (12) with broadened LDOS.
  3. [Fig. 2 caption] The quantity described as 'the CRB' is actually the difference between the energy variance and the Cramér-Rao bound, Var(E) - d<I>^T I^{-1} d<I>. Consider renaming it 'CRB gap' or 'excess variance' to avoid confusion.
  4. [III A, Eq. (31)] For consistency with Eq. (28), the PMF should be expressed in terms of |a_n(x)|^2 as defined in Eq. (7), including the sums over sublattice and orbital/spin indices, rather than |a_{n gamma}(x)|^2.
  5. [II B, Eq. (24)] The symbol f is used both for the arbitrary energy function in Eq. (14) and for the Fermi function in Eq. (24); please rename one of them to avoid ambiguity.
  6. [II B, text below Eq. (20)] The phrase 'under some unitary transformation U that changes the coordinates' is misleading because a unitary transformation in Hilbert space does not change spatial coordinates. This should be rephrased as a coordinate reparametrization or a rotation of the lattice axes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FIM/CRB derivation is self-contained, and the two self-citations are motivational rather than load-bearing.

full rationale

The paper's central derivation is self-contained and not circular. The normalized LDOS is defined in Eqs. (8)-(9) from the eigenstate expansion, the real-space and energy-space Fisher information matrices are defined in Eqs. (11) and (29), and the Cramér-Rao bounds in Eqs. (20) and (36)-(37) are derived explicitly from positive-semidefiniteness of the Gram matrix (Eqs. 16-19) and from the Cauchy-Schwarz inequality (Eq. 35), following standard statistics. No parameter is fitted to data in order to make the inequalities hold; the bounds follow from the PDF normalization for any positive normalized density, and the numerical checks simply verify a theorem rather than test a fitted prediction. The two self-citations, Refs. 27 and 38, are used only for contextual motivation (a real-space quantum metric analogy and prior studies of Chern markers under disorder) and are not load-bearing for the FIM-CRB result. The remaining assumptions, such as energy-independent tunneling matrix element and tip DOS in Eq. (4) and the single-particle Born interpretation of LDOS in Eq. (5), are physical modeling assumptions about what STM measures, not circular reductions of the derived inequality to its own inputs. Thus the paper warrants a low circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central inequalities are derived from standard probability theory and the definition of normalized LDOS. The only hand-chosen inputs are model parameters for the numerical demonstrations, and the main domain assumptions are the STM proportionality, the non-interacting Born interpretation, and zero-temperature filling. No new physical entities are introduced.

free parameters (4)
  • Impurity potential range U_imp = random in (0,1), applied to 10% of sites
    Hand-chosen disorder model used in the numerical demonstrations of Eqs. (25) and (26); not fitted to data and not needed for the central inequality.
  • Chemical potential mu = 0.1
    Hand-chosen model parameter setting the filling in the tight-binding examples; affects only the numerical illustrations.
  • Mass term M = -2 and 2
    Hand-chosen values that place the Chern insulator in the topological and trivial phases for comparison; not part of the central derivation.
  • Lorentzian broadening eta = not stated
    Used in Eq. (32) to regularize delta functions in the energy-space FIM; the value is not specified, which affects exact numerical reproduction.
assumptions (5)
  • standard math Cramér-Rao inequality for any probability distribution and any function of the random variable
    Used as the core theorem in Eqs. (20) and (36); the derivation is reproduced in Sections II.B and III.B following van den Bos.
  • domain assumption Normalized LDOS rho(x,E) is a conditional probability density
    Defined in Eq. (8) via normalization by n(x,Ec); non-negativity and integration to 1 hold by construction, but the physical interpretation as electron probability relies on single-particle wavefunctions.
  • domain assumption STM conductance is proportional to LDOS with an energy-independent prefactor
    Stated around Eq. (4): the tunneling matrix element |M|^2 and tip DOS are independent of energy; this is needed for Eq. (10) to extract rho(x,E) from raw conductance.
  • domain assumption Non-interacting single-particle tight-binding description of electrons
    The models in Eqs. (25) and (26) and the use of eigenstate amplitudes as probability amplitudes assume no electron-electron interactions.
  • domain assumption Zero-temperature filling of states with Ec < E < 0 in the STM window
    The sums over n are restricted to filled states in the measured energy window; this idealization underpins the normalized LDOS in the numerical examples.

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Pith. "Pith review of Probing disorder-induced Fisher information matrix and Cram\'{e}r-Rao bound by STM." pith.science (2026). https://pith.science/paper/BRMIHGSX

@misc{pith2026250522965,
  author       = {Pith},
  title        = {Pith review of: Probing disorder-induced Fisher information matrix and Cram\'er-Rao bound by STM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRMIHGSX}},
  note         = {Machine review of arXiv:2505.22965}
}
read the original abstract

The electronic local density of states of solids, if normalized correctly, represents the probability density that the electron at a specific position has a particular energy. Because this probability density can vary in space in disordered systems, we propose that one can either treat the energy as a random variable and position as an external parameter to construct a real space Fisher information matrix, or treat the position as a random variable and energy as an external parameter to construct an energy space Fisher information, both quantify the variation of local density of states caused by the disorder. The corresponding Cram\'{e}r-Rao bounds in these two scenarios set a limit on the energy variance and the position variance of electrons, respectively, pointing to new interpretations of STM measurements. Our formalism thus bring the notion of information geometry into STM measurements, as demonstrated explicitly by lattice models of metals and topological insulators.

Figures

Figures reproduced from arXiv: 2505.22965 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of the notation in this paper using hexag [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Numerical results for the real space information geometry in disordered 2D metal (top), 2D Chern insulator in the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy space information geometry for disordered [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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