arxiv: 2602.12417 · v2 · submitted 2026-02-12 · ❄️ cond-mat.str-el · quant-ph

Information lattice approach to the metal-insulator transition

William Skoglund , Elton Giacomelli , Yiqi Yang , Jens H. Bardarson , Erik van Loon This is my paper

Pith reviewed 2026-05-16 01:51 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph

keywords informationlatticescaletemperaturecorrelationmetal-insulatorquantumtransition

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The pith

The information lattice distinguishes metals from insulators via power-law versus exponential decay of information per scale in 1D tight-binding models.

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

In quantum systems, phase transitions are often tracked with correlation functions that require choosing specific observables to measure. The information lattice instead tracks how much information is stored at different length scales in a model-independent way. The authors apply this tool to a basic one-dimensional chain where electrons hop between sites. In the metallic regime at low temperature, information per scale falls off slowly as a power law and shows oscillations similar to Friedel oscillations in electron density. In the insulating regime or at high temperature, the same quantity falls off rapidly in an exponential manner. The contrast supplies a signature of the metal-insulator transition that does not depend on picking particular measurements.

Core claim

We find that the information per scale follows a power law in metals at low temperature and that Friedel-like oscillations are visible in the information lattice. At high temperature or in insulators at low temperature, the information per scale decays exponentially.

Load-bearing premise

The information lattice construction accurately captures scale-dependent information content in an observable-independent manner, and the observed power-law versus exponential distinction is sufficient to identify the metal-insulator transition in the studied models.

Figures

Figures reproduced from arXiv: 2602.12417 by Elton Giacomelli, Erik van Loon, Jens H. Bardarson, William Skoglund, Yiqi Yang.

**Figure 2.** Figure 2: FIG. 2. Illustration of three subsets [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: shows the information per scale i ℓ for two positions of the chemical potential, with the corresponding electronic spectrum shown on the right. The information per scale i ℓ decays in all cases with ℓ, but the decay changes both quantitatively and qualitatively. At high temperatures, the metal and insulator are very similar because the actual occupation numbers of the states are similar, and i ℓ decays e… view at source ↗

**Figure 5.** Figure 5: FIG. 5. Dependence of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Inverse decay length [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: shows how the wavelength λ extracted from these oscillations in i ℓ n depends on µ. The dashed line shows the expectation for ideal Friedel oscillations, λ = kF /π, where the analytical dependence of kF on µ is given in Appendix A for the infinite chain at zero temperature. Note that in the one-dimensional chain considered here, the Fermi surface is entirely described by the scalar kF . We find that λ −1… view at source ↗

**Figure 8.** Figure 8: shows a comparison of i ℓ n for an odd and even length SSH chain, with the chemical potential close to the middle of the bulk gap, µ = 0.05. Whereas i ℓ n is similar in the bulk for both chains, there are clear differences close to the boundary where i ℓ n increases. The zero-energy-mode in the odd chain is spatially localized close to the edge (at large n), and this is reflected in the substantial increa… view at source ↗

**Figure 9.** Figure 9: FIG. 9. Role of finite-size effects, in the SSH Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Decay in the metal and insulator, for the SSH model, [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. Finite size effects, staggered potential [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Decay length versus temperature in the SSH model [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗

read the original abstract

Correlation functions and correlation lengths are frequently used to describe phase transitions in quantum systems, but they require an explicit choice of observables. The recently introduced information lattice instead provides an observable-independent way to identify where and at which scale information is contained in quantum lattice models. Here, we use it to study the difference between the metallic and insulating regime of one-dimensional noninteracting tight-binding chains. We find that the information per scale follows a power law in metals at low temperature and that Friedel-like oscillations are visible in the information lattice. At high temperature or in insulators at low temperature, the information per scale decays exponentially. Thus, the information lattice is a useful tool for analyzing the metal-insulator transition.

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.

Desk Editor's Note private letter to a colleague

The information lattice largely reproduces the expected Green's function decay in these 1D chains rather than offering a distinctly new observable-independent diagnostic.

read the letter

The paper applies the information lattice to the metal-insulator transition in 1D noninteracting tight-binding chains. It reports that information per scale follows a power law with Friedel oscillations in metals at low temperature, while it decays exponentially in insulators at low T or in metals at high T. This application is new. The authors take a recently introduced tool and use it to map out the transition through scaling behaviors that had not been documented for this method before. The paper does well in keeping the presentation straightforward and in highlighting the observable-independent nature of the approach in principle. The reported distinction between power-law and exponential regimes is clear from the abstract and presumably backed by the calculations in the full text. The main limitation is that these models are noninteracting, so the density matrix encodes the Green's function decay directly. The information lattice essentially reproduces the known 1/r versus exponential behavior of correlations. It does not demonstrate a genuinely new way to identify the transition that avoids observable choice in a meaningful way beyond what is already standard. The numerics appear sound based on the claims, with no signs of circular reasoning. Details on system sizes would help confirm robustness, but the overall pattern holds up. This paper is for condensed matter theorists interested in information-theoretic methods for quantum phases. A reader exploring new tools for phase transitions could get value from seeing the concrete scaling results. It is less critical for those focused on 1D chains specifically. I recommend sending it for peer review. The work is honest and the method deserves further testing in other contexts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the validity of the recently introduced information lattice as an observable-independent descriptor; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)

domain assumption The information lattice provides an observable-independent measure of information content at different scales in quantum lattice models.
This is the core premise invoked by the abstract to justify the method.

pith-pipeline@v0.9.0 · 5423 in / 1164 out tokens · 162472 ms · 2026-05-16T01:51:38.354835+00:00 · methodology

discussion (0)

Reference graph

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(C2) are used, which are all equal to 1 2 and there- fore contribute zero to the information (half-filling), so I ℓ=0 n = 0

More precisely, I=I ℓ=2 n = 2 + 2xlog2(x) + 2(1−x) log 2(1−x).(C3) ForI ℓ=0 n , the 1×1 subblocks on the diagonal of Eq. (C2) are used, which are all equal to 1 2 and there- fore contribute zero to the information (half-filling), so I ℓ=0 n = 0. ForI ℓ=1 n , the upper and lower 2×2 block of Eq. (C2) are used, which are identical by symmetry. This block 1 ...

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