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arXiv:2512.18397 · Physical Review Research

Momentum-resolved spectral functions of super-moiré systems using tensor networks

Anouar Moustaj, Yitao Sun, Tiago V. C. Antao, Jose L. Lado

cond-mat.str-elcond-mat.mes-hallcond-mat.mtrl-sciphysics.comp-phquant-phSubmitted Dec 20, 2025• Announced Dec 23, 2025• 0 citations
v1 · Dec 20, 2025v2 · Dec 23, 2025v3 · May 11, 2026
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Abstract

Computing spectral functions in large, non-periodic super-moiré 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-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 compute momentum-resolved spectral functions restricted to selected regions of a super-moiré, enabling direct imaging of position-dependent electronic structure and minigaps in super-moiré systems with non-uniform strain. Our results establish a powerful methodology to compute momentum-resolved spectral functions in exceptionally large super-moiré systems, providing a tool to directly model scanning twisting microscope tunneling experiments in twisted van der Waals heterostructures.

Journal reference
Phys. Rev. Research 8, 023282 (2026)
DOI
10.1103/9btt-y8sh
Comments
10 pages, 3 figures, submitted to Physical Review Review Research. Comments are welcome

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Computing spectral functions in large, non-periodic super-moiré 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-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 compute momentum-resolved spectral functions restricted to selected regions of a super-moiré, enabling direct imaging of position-dependent electronic structure and minigaps in super-moiré systems with non-uniform strain. Our results establish a powerful methodology to compute momentum-resolved spectral functions in exceptionally large super-moiré systems, providing a tool to directly model scanning twisting microscope tunneling experiments in twisted van der Waals heterostructures.
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Understanding quantum materials at meso and even macroscopic scales requires tight-binding calculations on system sizes where explicit matrix representations become prohibitively costly. This represents a major bottleneck to rationalize phenomena in moiré and super-moiré heterostructures and quasicrystals. Here, we present a unified tensor-network methodology to solve tight-binding problems at exceptionally large scales, by mapping a system of N=2LN = 2^LN=2L sites onto a many-body problem of LLL pseudospin sites, which is subsequently solved with tensor network algorithms. For Hamiltonians with compressible real-space structure, the tensor network bond dimension remains modest, typically of order a few tens, independent of NNN. Tensor network representations of arbitrary hopping functions including long-range, spatially modulated, and twisted-layer couplings are built with quantics tensor cross interpolation, and all physical observables are evaluated entirely with tensor network algebra without explicit matrix storage or diagonalization. We demonstrate applications to spectral functions, momentum-space spectra via the tensor-network quantum Fourier transform, real-space topological invariants, real-time dynamics, correlation induced symmetry breaking with self-consistent mean-field calculations, non-Hermitian phenomena, and excitonic many-body physics. Our methodology enables routinely solving systems with billions of sites, by leveraging the tensor network compressibility of real-space structures, and establishing a flexible framework to study quantum matter at ultra-large length scales. The methodology is implemented in the open-source Julia package TensorBinding.
Non-Hermitian systems host unconventional topological phenomena, while real-space spectral calculations in three dimensions remain computationally challenging. Here, we develop a tensor-network framework that combines a quantics-based matrix-product-operator representation of large real-space tight-binding Hamiltonians with a non-Hermitian kernel polynomial method. The Hermitized spectral problem is retained within the compact tensor-network representation, enabling real-space spectral function calculations for systems exceeding one billion lattice sites. Using this approach, we investigate a three-dimensional non-Hermitian higher-order topological insulator and resolve its corner-mode spectral response on a billion-site lattice, demonstrating the scalability and computational reach of the method. By tuning the gain--loss strength, we identify distinct in-gap corner-mode features in the weak- and strong-gain--loss regimes. Our results establish a scalable tensor-network framework for real-space spectral calculations in exceptionally large non-Hermitian lattice systems.
Computing topological invariants in two-dimensional quasicrystals and supermoiré matter is a remarkable open challenge due to the absence of translational symmetry and the colossal number of sites inherent to these systems. Here, we establish a method to compute local topological invariants of exceptionally large systems using tensor networks, enabling the computation of invariants for Hamiltonians with hundreds of millions of sites, several orders of magnitude above the capabilities of conventional methodologies. Our approach leverages a tensor network representation of the density matrix using a Chebyshev tensor network algorithm, enabling large-scale calculations of topological markers in quasicrystalline and moiré systems. We demonstrate our methodology with two-dimensional quasicrystals featuring C_{8} and C_{10} rotational symmetries and mosaics of Chern phases. Our Letter establishes a powerful method to compute topological phases in exceptionally large-scale topological systems, providing the required tool to rationalize generic super-moiré and quasicrystalline topological matter.
Abstract Super-moiré materials represent a novel playground to engineer states of matter beyond the possibilities of conventional moiré materials. However, from the computational point of view, understanding correlated matter in these systems requires solving models with several millions of atoms, a formidable task for state-of-the-art methods. Conventional wavefunction methods for correlated matter scale with a cubic power with the number of sites, a major challenge for super-moiré materials. Here, we introduce a methodology capable of solving correlated states in super-moiré materials by combining a kernel polynomial method with a quantics tensor cross interpolation matrix product state algorithm. This strategy leverages a mapping of the super-moiré structure to a many-body Hilbert space, that is efficiently sampled with tensor cross interpolation with matrix product states, where individual evaluations are performed with a Chebyshev kernel polynomial algorithm. We demonstrate this approach with interacting super-moiré systems with up to several millions of atoms, showing its ability to capture correlated states in moiré-of-moiré systems and domain walls between different moiré systems. Our manuscript puts forward a widely applicable methodology to study correlated matter in ultra-long length scales, enabling rationalizing correlated super-moiré phenomena.