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Physical review. B./Physical review. B

Quantum Monte Carlo study of a bilayer U(2)×U(2) -symmetric Hubbard model

Yosef Caplan, Dror Orgad

Published Oct 18, 2023• 2 citations
Publisher

Abstract

We carry out a sign-problem-free quantum Monte Carlo calculation of a bilayer model with a repulsive intralayer Hubbard interaction and a ferromagnetic interlayer interaction. The latter breaks the global $SU(2)$ spin rotational symmetry but preserves a $U(2)\ifmmode\times\else\texttimes\fi{}U(2)$ invariance under mixing of same-spin electrons between layers. We show that despite the difference in symmetry, the bilayer model exhibits the same qualitative features found in the single-layer Hubbard model. These include stripe phases, whose nature is sensitive to the presence of next-nearest-neighbor hopping, a maximum in the Knight shift that moves to lower temperatures with increasing hole doping, and lack of evidence for intralayer $d$-wave superconductivity. Instead, we find a superconducting phase, coexisting with stripes, whose critical temperature traces a dome as a function of doping and is due to interlayer spin-polarized pairing that is induced by the ferromagnetic interaction.

DOI
10.1103/physrevb.108.165131

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Quantum Monte Carlo study of a bilayer U(2)×U(2)U(2)\times U(2)U(2)×U(2) symmetric Hubbard model
We carry out a sign-problem-free quantum Monte Carlo calculation of a bilayer model with a repulsive intra-layer Hubbard interaction and a ferromagnetic inter-layer interaction. The latter breaks the global SU(2)SU(2)SU(2) spin rotational symmetry but preserves a U(2)×U(2)U(2)\times U(2)U(2)×U(2) invariance under mixing of same-spin electrons between layers. We show that despite the difference in symmetry, the bilayer model exhibits the same qualitative features found in the single-layer Hubbard model. These include stripe phases, whose nature is sensitive to the presence of next-nearest-neighbor hopping, a maximum in the Knight shift that moves to lower temperatures with increasing hole doping, and lack of evidence for intra-layer d-wave superconductivity. Instead, we find a superconducting phase, coexisting with stripes, whose critical temperature traces a dome as a function of doping and is due to inter-layer spin-polarized pairing that is induced by the ferromagnetic interaction.
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The single band, two-dimensional Hubbard Hamiltonian has been extensively studied as a model for high temperature superconductivity. While quantum Monte Carlo simulations within the dynamic cluster approximation are now providing considerable evidence for a ddd-wave superconducting state at low temperature, such a transition remains well out of reach of finite lattice simulations because of the ``sign problem.'' We show here that a bilayer Hubbard model, in which one layer is electron doped and one layer is hole doped, can be studied to lower temperatures and exhibits an interesting signal of ddd-wave pairing. The results of our simulations bear resemblance to a recent report on the magnetic and superconducting properties of Ba2Ca3Cu4O8F2{\mathrm{Ba}}_{2}{\mathrm{Ca}}_{3}{\mathrm{Cu}}_{4}{\mathrm{O}}_{8}{\mathrm{F}}_{2}Ba2​Ca3​Cu4​O8​F2​ which contains both electron and hole doped CuO2\mathrm{Cu}{\mathrm{O}}_{2}CuO2​ planes. We also explore the phase diagram of bilayer models in which each sheet is at half-filling.
Competing orders represent a central challenge in understanding strongly correlated systems. In this work, we employ projector quantum Monte Carlo simulations to study a sign-problem-free bilayer extended Hubbard model. In this model, a charge stripe phase, characterized by a peak at (0±2πδ,0) is induced by highly anisotropic interlayer antiferromagnetic spin-exchange coupling Jz, and strongly suppressed by the decreasing spin-exchange anisotropy (i.e., introducing the spin-flip term J⊥); in contrast, the introduction of J⊥ favors the emergence of interlayer pairing superconductivity. We further demonstrate that the anisotropy of the interlayer spin-exchange directly governs the competition between these two phases, while the on-site interaction U plays a complex role in tuning both the charge stripe and superconductivity. Furthermore, against the background of charge stripes we discover the spin stripes, characterized by twice the period of charge stripes, indicating the profound internal connection between them. Our work identifies the key factors driving charge stripe formation, highlights the sensitivity of both the charge stripe and superconducting phases to interaction parameters, and thereby provides valuable insights into competing orders in strongly correlated systems.
The bilayer Hubbard model with an intralayer hopping ttt and an interlayer hopping ${t}_{\ensuremath{\perp}}$ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half filling, if the splitting between the bonding and antibonding bands ${t}_{\ensuremath{\perp}}/t$ is small, the gap has B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave) symmetry, but when the splitting becomes larger, A1g{A}_{1g}A1g​ (${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave) pairing is favored. We also find that in the ${s}^{\ifmmode\pm\else\textpm\fi{}}$ pairing region, the pairing is driven by interlayer spin fluctuations and that Tc{T}_{c}Tc​ is enhanced.
The search for superconductivity with unconventional pairing symmetry has been a central focus in the study of strongly correlated electron systems. In this work, we report a numerically exact study of the spin-triplet pairing in a two-dimensional Hubbard model with repulsive interactions, employing the determinant quantum Monte Carlo method. The model includes next-nearest-neighbor and third-nearest-neighbor hopping terms and maintains spin balance. By tuning the fermion filling close to a type-II Van Hove singularity (VHS) in the model, we numerically investigate the ordering tendencies of several possible pairing channels with different symmetries. Our numerical results provide clear evidence of the spin-triplet ppp-wave pairing instability approaching low temperatures, as revealed by the vertex contribution to the pairing susceptibility. This signature becomes increasingly pronounced as the interaction strength increases in the weak to intermediate regime. We further find that, near the type-II VHS, the dominant spin-spin correlations in the system are ferromagnetic, suggesting its close relation to the spin-triplet pairing instability. Our findings offer a reliable approach to realize spin-triplet ppp-wave superfluidity in the repulsive Hubbard model from an unbiased numerical perspective.