Stripe and superconducting order competing in the Hubbard model on a square lattice studied by a combined variational Monte Carlo and tensor network method

Andrew S. Darmawan, Yusuke Nomura, Youhei Yamaji, Masatoshi Imada

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11 Citations (Scopus)


The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the d-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration (δ) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10, for the on-site interaction U and the transfer t, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t. We have identified distinct phases including a uniform d-wave superconducting phase for 0.17δ0.22 and a stripe charge/spin ordered phase for δ0.17 with the stripe period depending on δ, together with presumable spatially coexisting antiferromagnetic and stripe order for δ0.07 and coexisting stripe and d-wave superconductivity for 0.07δ0.17. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around δ=0.17 in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.

Original languageEnglish
Article number205132
JournalPhysical Review B
Issue number20
Publication statusPublished - 2018 Nov 16
Externally publishedYes


ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

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