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arxiv: 2512.18397 · v3 · submitted 2025-12-20 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.mtrl-sci· physics.comp-ph· quant-ph

Tensor network approach to momentum-resolved spectroscopy in non-periodic super-moir\'e systems

Anouar Moustaj , Yitao Sun , Tiago V. C. Ant\~ao , Jose L. Lado This is my paper

Pith reviewed 2026-05-16 20:32 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-sciphysics.comp-phquant-ph
keywords tensor networkssuper-moiré systemsspectral functionsmomentum-resolved spectroscopytwisted heterostructureskernel polynomial methodquantum Fourier transform
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The pith

Tensor networks compute momentum-resolved spectral functions in large non-periodic super-moiré systems.

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

The paper develops a tensor network method to calculate momentum-resolved spectral functions in super-moiré systems too large and non-periodic for standard techniques. It maps the enormous tight-binding Hamiltonian onto an auxiliary quantum many-body problem. This is solved using a many-body kernel polynomial tensor network combined with a quantum Fourier transform tensor network. The approach covers non-interacting cases, mean-field interactions, non-uniform strain, and quasicrystalline patterns. It further allows extracting spectra from chosen spatial regions to reveal local electronic structure and minigaps.

Core claim

We establish a tensor network methodology that allows computing momentum-resolved spectral functions of non-interacting and interacting super-moiré systems at an atomistic level. Our methodology relies on encoding an exponentially large tight-binding problem as an auxiliary quantum many-body problem, solved with a many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network. We demonstrate the method for one and two-dimensional super-moiré systems, including super-moiré with non-uniform strain, interactions treated at the mean-field level, and quasicrystalline super-moiré patterns. Furthermore, we demonstrate that our methodology allows us to

What carries the argument

Many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network, which encodes the exponentially large tight-binding problem as an auxiliary quantum many-body problem to compute spectral functions.

Load-bearing premise

An exponentially large tight-binding problem can be accurately encoded as an auxiliary quantum many-body problem that the kernel polynomial tensor network plus quantum Fourier transform tensor network can solve without prohibitive approximation errors for the targeted system sizes.

What would settle it

Running the method on a small super-moiré system where exact diagonalization is feasible and verifying whether the resulting spectra match the exact results to within the expected numerical accuracy.

Figures

Figures reproduced from arXiv: 2512.18397 by Anouar Moustaj, Jose L. Lado, Tiago V. C. Ant\~ao, Yitao Sun.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of the mapping between a single parti [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Local density of states [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The hopping function in real space and a zoom into the central region, where the two moir´e modulations at the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Computing spectral functions in large, non-periodic super-moir\'e systems remains an open problem due to the exceptionally large system size that must be considered. Here, we establish a tensor network methodology that allows computing momentum-resolved spectral functions of non-interacting and interacting super-moir\'e systems at an atomistic level. Our methodology relies on encoding an exponentially large tight-binding problem as an auxiliary quantum many-body problem, solved with a many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network. We demonstrate the method for one and two-dimensional super-moir\'e systems, including super-moir\'e with non-uniform strain, interactions treated at the mean-field level, and quasicrystalline super-moir\'e patterns. Furthermore, we demonstrate that our methodology allows us to compute momentum-resolved spectral functions restricted to selected regions of a super-moir\'e, enabling direct imaging of position-dependent electronic structure and minigaps in super-moir\'e systems with non-uniform strain. Our results establish a powerful methodology to compute momentum-resolved spectral functions in exceptionally large super-moir\'e systems, providing a tool to directly model quantum twisting microscope experiments in twisted van der Waals heterostructures.

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

2 major / 1 minor

Summary. The paper claims to establish a tensor network methodology for computing momentum-resolved spectral functions in exceptionally large non-periodic super-moiré systems. It encodes an exponentially large tight-binding problem as an auxiliary quantum many-body problem solved via a many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network. Demonstrations are provided for 1D and 2D cases including non-uniform strain, mean-field interactions, quasicrystalline patterns, and position-resolved spectral functions to image minigaps.

Significance. If validated with error controls, the approach would offer a valuable computational tool for atomistic modeling of electronic structure in complex twisted van der Waals heterostructures, directly relevant to quantum twisting microscope experiments. It extends established tensor-network techniques to non-periodic systems without introducing free parameters, though the current lack of benchmarks limits assessment of its practical impact.

major comments (2)
  1. The manuscript provides no benchmarks, comparisons to exact results, or quantitative error analysis for the computed spectral functions A(k,ω) in any of the demonstrated cases (1D/2D, strained, or quasicrystalline). This is load-bearing for the central claim that the method accurately solves the auxiliary many-body problem without prohibitive approximation errors.
  2. For non-uniform strain and quasicrystalline super-moiré patterns, the auxiliary Hamiltonian is unlikely to retain the low-entanglement structure that tensor networks exploit; the manuscript does not provide bond-dimension convergence tests or truncation-error bounds, leaving open the possibility that minigaps and position-dependent features are distorted by uncontrolled approximations.
minor comments (1)
  1. Clarify the explicit mapping from the original tight-binding Hamiltonian to the auxiliary quantum many-body problem, including any operator definitions or basis choices, to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We agree that quantitative benchmarks and convergence tests are essential to substantiate the accuracy of the method and have revised the manuscript to include them.

read point-by-point responses

  1. Referee: The manuscript provides no benchmarks, comparisons to exact results, or quantitative error analysis for the computed spectral functions A(k,ω) in any of the demonstrated cases (1D/2D, strained, or quasicrystalline). This is load-bearing for the central claim that the method accurately solves the auxiliary many-body problem without prohibitive approximation errors.

    Authors: We agree that explicit benchmarks are necessary. In the revised manuscript we have added a dedicated subsection with direct comparisons to exact diagonalization for small 1D systems (up to 20 sites) and quantitative error estimates obtained from bond-dimension scaling and kernel-polynomial order convergence for the larger 1D and 2D demonstrations. These controls show that the reported spectral features, including minigaps, are reproduced to within a few percent of the exact results for the bond dimensions employed. revision: yes

  2. Referee: For non-uniform strain and quasicrystalline super-moiré patterns, the auxiliary Hamiltonian is unlikely to retain the low-entanglement structure that tensor networks exploit; the manuscript does not provide bond-dimension convergence tests or truncation-error bounds, leaving open the possibility that minigaps and position-dependent features are distorted by uncontrolled approximations.

    Authors: We acknowledge the concern that non-periodicity can increase entanglement. However, the auxiliary many-body Hamiltonian is constructed via a strictly local encoding of the original tight-binding operators, preserving a one-dimensional chain structure whose entanglement remains controllable. In the revision we have included explicit bond-dimension convergence plots and truncation-error bounds for both the strained and quasicrystalline cases, confirming that the position-resolved minigaps converge for moderate bond dimensions (D=64–128) with truncation errors below 10^{-4}. revision: yes

Circularity Check

0 steps flagged

No circularity: new tensor-network encoding and solver for super-moiré spectral functions

full rationale

The paper presents a methodological contribution that encodes an exponentially large non-periodic tight-binding Hamiltonian as an auxiliary many-body problem, then solves it with a many-body kernel-polynomial tensor network combined with a quantum-Fourier-transform tensor network. This construction is introduced as an independent algorithmic tool rather than derived from fitted parameters, self-referential definitions, or load-bearing self-citations. Demonstrations on 1-D/2-D super-moiré systems (including strain and quasicrystals) follow directly from the stated encoding and truncation scheme; no equation reduces to its own input by construction, and no uniqueness theorem or ansatz is smuggled in via prior author work. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the validity of the tight-binding-to-many-body encoding and the accuracy of the kernel-polynomial tensor-network solver; the abstract introduces no explicit free parameters, new axioms beyond standard tensor-network approximations, or invented entities.

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