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arXiv:2602.21695

Combining matrix product states and mean-field theory to capture magnetic order in quasi-1D cuprates

Quentin Staelens, Daan Verraes, Daan Vrancken, Tom Braeckevelt, Jutho Haegeman, Véronique Van Speybroeck

Published Feb 25, 2026• 0 citations
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Abstract

We study quasi-one-dimensional strongly correlated materials using a multi-step approach based on density functional theory, downfolding techniques, and tensor-network simulations. The downfolding procedure yields effective multiband Hubbard models that capture the competition between electron hopping and local Coulomb interactions relevant to the system's low-energy properties. The resulting multiband Hubbard models are solved using matrix product states. Applied to Sr$_2$CuO$_3$, SrBaCuO$_3$, and Ba$_2$CuO$_3$, this purely one-dimensional treatment yields no long-range magnetic order, in contrast to the magnetic ordering observed experimentally. To account for this behavior, we extend the multi-step approach by incorporating interchain couplings through a self-consistent mean-field scheme. This combined approach stabilizes finite staggered magnetizations, providing a consistent description of magnetic order in agreement with experiment. For Sr$_2$CuO$_{3.5}$ and SrCuO$_2$, we also tested an approach proposed for ladder materials, however, we find that these materials are not well suited for this approach due to the small magnitude of the intraladder hopping parameters.

DOI
10.48550/arxiv.2602.21695

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High Resolution Image Download MS PowerPoint Slide We present a high-accuracy procedure for electronic structure calculations of strongly correlated materials. To address limitations in current electronic structure methods, we employ density functional theory in combination with the constrained random phase approximation to construct an effective multiband Hubbard model, which is subsequently solved using tensor networks. Our work focuses on one-dimensional and quasi-one-dimensional materials, for which we employ the machinery of matrix product states. We apply this framework to the conjugated polymers trans -polyacetylene and polythiophene, as well as the quasi-one-dimensional charge-transfer insulator Sr 2 CuO 3 . The predicted band gaps show quantitative agreement with state-of-the-art computational techniques and experimental measurements. Beyond band gaps, tensor networks provide access to a wide range of physically relevant properties, including spin magnetization and various excitation energies. Their flexibility supports the implementation of complex Hamiltonians with longer-range interactions, while the bond dimension enables systematic control over accuracy. Furthermore, the computational cost scales efficiently with system size, demonstrating the framework’s scalability.
Materials with strong electronic correlations often exhibit a superconducting phase in close competition with other insulating phases, which is outstandingly difficult to resolve, e.g., for a group of quasi-two-dimensional (Q2D) materials such as the cuprates, even for the simplified minimal model of these materials, the doped 2D Hubbard model. The present work shows how quasi-one-dimensional (Q1D) systems, 2D and three-dimensional (3D) arrays of weakly coupled 1D correlated electrons, are much more amenable to resolving such competition, treating both instabilities on equal footing. Using the recently established matrix product state plus mean field ( MPS + MF ) approach for fermions [Bollmark , ], we demonstrate that large systems can be reached readily in these systems, which opens up the thermodynamic regime via extrapolation. Focusing on basic model systems, 3D arrays of negative- U Hubbard chains with additional nearest-neighbor interaction V , we show that despite the MF component of the MPS + MF technique, we can reproduce the expected coexistence of the superconductivity and charge density wave at V = 0 for density n = 1 . We then show how we can tune away from coexistence by both tuning V and doping the system. This work thus paves the way to deploy two-channel MPS + MF theory on some highly demanding high- T c superconducting systems, such as 3D arrays of repulsive- U doped Hubbard ladders; we recently characterized the properties of such arrays in single-channel MPS + MF calculations [Bollmark , ].
Correlated electron states are at the root of many important phenomena including unconventional superconductivity (USC), where electron pairing arises from repulsive interactions.Computing the properties of correlated electrons, such as the critical temperature T c for the onset of USC, efficiently and reliably from the microscopic physics with quantitative methods remains a major challenge for almost all models and materials.In this theoretical work, we combine matrix product states (MPS) with static mean field (MF) to provide a solution to this challenge for quasi-one-dimensional (Q1D) systems: two-and threedimensional materials comprised of weakly coupled correlated 1D fermions.This MPS þ MF framework for the ground state and thermal equilibrium properties of Q1D fermions is developed and validated for attractive Hubbard systems first, and further enhanced via analytical field theory.We then deploy it to compute T c for superconductivity in 3D arrays of weakly coupled, doped, and repulsive Hubbard ladders.The MPS þ MF framework thus enables the quantitative study of USC and high-T c superconductivity-and potentially many more correlated phases-in fermionic Q1D systems based directly on their microscopic parameters, in ways inaccessible to previous methods.This approach further allows one to treat competing macroscopic orders, such as superconducting and insulating ones, on an equal footing.Benchmarks of the framework using auxiliary-field quantum Monte Carlo techniques show that the overestimation of, e.g., T c due to its mean-field component, is near constant in microscopic parameters.These features of the MPS þ MF approach to correlated fermions open up the possibility of designing deliberately optimized Q1D superconductors, from experiments in ultracold gases to synthesizing new materials.
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