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arXiv:2409.18898 · 2D Materials

Correlated states in super-moiré materials with a kernel polynomial quantics tensor cross interpolation algorithm

Adolfo O. Fumega, Marcel Niedermeier, José Luis Lado

Published Dec 11, 2024• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2053-1583/ad9d59

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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.
Moiré and super-moiré materials provide exceptional platforms to engineer exotic correlated quantum matter. The vast number of sites required to model moiré systems in real space remains a formidable challenge due to the immense computational resources required. Super-moiré materials push this requirement to the limit, where millions or even billions of sites need to be considered, a requirement beyond the capabilities of conventional methods for interacting systems. Here, we establish a methodology that allows solving correlated states in systems reaching a billion sites, which exploits tensor network representations of real-space Hamiltonians and self-consistent real-space mean-field equations. Our method combines a tensor network kernel polynomial method with quantics tensor cross-interpolation algorithm, enabling us to solve ultralarge models, including those whose single-particle Hamiltonian is too large to be stored explicitly. We demonstrate our methodology with super-moiré systems featuring spatially modulated hoppings, many-body interactions, and domain walls, showing that it allows access to self-consistent symmetry-broken states and spectral functions of real-space models reaching a billion sites. Our methodology provides a strategy to solve exceptionally large interacting problems, providing a widely applicable strategy to compute correlated super-moiré quantum matter.
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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 quantum twisting microscope experiments in twisted van der Waals heterostructures.
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.
Understanding the dynamics of open quantum many-body systems is a major problem in quantum matter. Specifically, efficiently solving the spectrum of the Liouvillian superoperator governing such dynamics remains a critical open challenge. Here, we put forward a method for solving the many-body Liouvillian spectrum and dynamics based on the non-Hermitian kernel polynomial method and tensor-network techniques. We demonstrate the faithfulness of our method by computing the dynamics of the dephasing quantum compass model with a gradient magnetic field and comparing it with exact results. In particular, we show that our method allows us to characterize the quantum Zeno crossover and the reduction of relaxation rate due to Stark localization in this model. We further demonstrate the ability of our method to go beyond exact results by exploring nearest-neighbor interaction effects on the Liouvillian dynamics, elucidating the interplay between Stark localization and many-body interactions. Our method provides an efficient solution to many-body Liouvillian spectrum and dynamics, establishing a methodology to explore large open quantum many-body systems. Published by the American Physical Society 2024